Algorithm Design and Physician Liability

# 1Introduction
Source: [https://arxiv.org/html/2608.13618](https://arxiv.org/html/2608.13618)
Algorithm Design and Physician Liability

Shujie Luan Shubhranshu Singh Tinglong Dai\*\*\*Shujie Luan is affiliated with The University of Western Australia, UWA Business School, Perth, Australia; Shubhranshu Singh and Tinglong Dai are affiliated with Carey Business School, Johns Hopkins University, Baltimore, Maryland 21202\. The coauthors contributed equally to this work\. Correspondence may be addressed to:shujie\.luan@uwa\.edu\.au\(SL\);shubhranshu\.singh@jhu\.edu\(SS\);dai@jhu\.edu\(TD\)\.

This version: August 12, 2026

Abstract

> A single clinical algorithm can deliver unequal accuracy across patient groups, and concern about such disparity has grown as artificial intelligence \(AI\) spreads through clinical decision\-making\. In response, a liability rule introduced in the United States holds healthcare providers responsible when their reliance on disparate algorithms contributes to erroneous clinical decisions\. We examine how such liability considerations reshape \(i\) an AI firm’s algorithm design decisions that drive group\-specific accuracy and \(ii\) a physician’s decisions to use AI in healthcare delivery\. The AI firm designs an algorithm for two patient groups, and improving accuracy for the disadvantaged group is more costly\. The physician \(who remains the accountable decision\-maker\) then decides whether to consult AI, weighing the reduction in clinical uncertainty against expected liability exposure when AI errors disproportionately affect the disadvantaged group\. We find the liability rule can induce disparateuseof AI: the physician may reduce AI use overall and, over an intermediate range of liability, rely on AI less for disadvantaged patients\. The effect is non\-monotone\. As liability increases, the physician’s use of AI for disadvantaged patients first declines, then rises as the firm reallocates investment toward reducing disparity or switches to an equal\-accuracy design\. Mandating equal algorithmic accuracy across patient groups can then inadvertently harm both groups, because a uniform accuracy requirement distorts the firm’s investment incentives and the physician’s equilibrium AI\-use decisions\. Key words:clinical algorithms; product design; algorithmic disparity; liability; human–AI interaction

## 1Introduction

Artificial intelligence \(AI\) is reshaping expert decision\-making, yet most research treats the two sides of the AI lifecycle separately\. One strand asks how firms design algorithms under technical and regulatory constraints\([Diao et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib17),[Israeli 2018](https://arxiv.org/html/2608.13618#bib.bib31),[Iyer and Ke 2024](https://arxiv.org/html/2608.13618#bib.bib32), e\.g\.,\); another asks how human experts adopt, trust, or strategically act on algorithmic advice\([Dai and Singh 2025](https://arxiv.org/html/2608.13618#bib.bib15),[Dietvorst et al\. 2018](https://arxiv.org/html/2608.13618#bib.bib18),[McLaughlin and Spiess 2022](https://arxiv.org/html/2608.13618#bib.bib44), e\.g\.,\)\. In practice, these margins are jointly determined: the rules that govern how practitioners use AI feed back into how firms build it\. This interdependence is especially consequential when regulation targets the point of use, whereas the performance disparities that trigger liability are shaped upstream during design\. A feedback loop arises in which \(i\) a firm designs product quality for different customer segments and \(ii\) an expert intermediary, who faces policy constraints, decides whether the product is used\.

Clinical decision support offers a paradigmatic setting in which this feedback loop is first\-order\. Although “clinical algorithms” once denoted transparent flow charts, advances in AI have pushed the frontier toward high\-performing but often opaque systems\([Gottlieb 2024](https://arxiv.org/html/2608.13618#bib.bib27),[Green and Defoe 1978](https://arxiv.org/html/2608.13618#bib.bib28),[Margolis 1983](https://arxiv.org/html/2608.13618#bib.bib43)\)\. As of early 2026, the U\.S\. Food and Drug Administration \(FDA\) has authorized over 1,300 AI\-based medical devices\([FDA 2025](https://arxiv.org/html/2608.13618#bib.bib20)\)\. These AI tools can improve screening, diagnosis, and treatment decisions\([Leong et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib36),[Rajpurkar et al\. 2022](https://arxiv.org/html/2608.13618#bib.bib53),[Topol 2019](https://arxiv.org/html/2608.13618#bib.bib59)\), yet their diffusion into routine care remains uneven\([Abràmoff et al\. 2024](https://arxiv.org/html/2608.13618#bib.bib2),[Wu et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib61)\)\. A central barrier to adoption is the concern regarding*algorithmic disparity*—systematic differences in predictive performance across protected groups—which raises equity concerns and exposes clinicians and hospitals to significant legal risk\([Mehrabi et al\. 2021](https://arxiv.org/html/2608.13618#bib.bib45),[Obermeyer et al\. 2019](https://arxiv.org/html/2608.13618#bib.bib50),[Obermeyer et al\. 2021](https://arxiv.org/html/2608.13618#bib.bib49)\)\. Reflecting these concerns, the Centers for Medicare & Medicaid Services \(CMS\) has extended Section 1557 nondiscrimination requirements to clinical decision support tools under § 92\.210\. This rule prohibits discriminatory use and explicitly requires covered entities to make reasonable efforts to identify and mitigate disparity, thereby placing the burden of algorithmic accountability directly on providers\([CMS 2022](https://arxiv.org/html/2608.13618#bib.bib9),[CMS 2024](https://arxiv.org/html/2608.13618#bib.bib10)\)\. Because most healthcare providers in the U\.S\. context are physicians, we use the terms “physician” and “provider” interchangeably in the remainder of the paper\.

Motivated by this policy shift, we study how deployment\-facing nondiscrimination liability changes incentives in both the development and use of clinical AI when algorithms perform unevenly across patient groups\. To isolate this mechanism, we abstract from broader liability channels—most notably malpractice\([Price et al\. 2019](https://arxiv.org/html/2608.13618#bib.bib52)\)—and focus on the margin where this regulation is most directly enforced: clinicians’ reliance on decision\-support tools in patient care\. Our model links upstream technology design to downstream clinical delivery in a two\-stage structure\. First, an AI firm chooses accuracy for two patient groups: an advantaged group for whom performance gains are relatively inexpensive, and a disadvantaged group for whom gains are costlier because of data sparsity and measurement frictions\([Chen et al\. 2018](https://arxiv.org/html/2608.13618#bib.bib7),[Mehrabi et al\. 2021](https://arxiv.org/html/2608.13618#bib.bib45)\)\. Second, the physician decides whether to use the tool in a given case and, conditional on use, whether to follow its recommendation\. This setup allows us to study whether liability achieves its intended goal by inducing disparity\-reducing investment, or instead weakens adoption and shifts investment in ways that can leave disadvantaged patients worse off\.

We formalize this interaction as follows\. When the algorithm is disparate, following an incorrect recommendation for a disadvantaged patient creates a specific liability exposure; when the algorithm is equal\-accuracy, this disparity\-contingent liability channel is removed\. The physician’s core trade\-off is then clear: AI can reduce clinical uncertainty, but its use also brings utilization costs and expected legal risk\. This structure captures a central regulatory tension: policies enforced at the point of use can feed back to the design stage, in which disparity itself is produced\.

Our analysis yields three main results\. First, liability tied to adverse outcomes from a disparate AI tool can induce disparate*use*of AI\. Even holding the algorithm’s accuracy fixed, the physician consults AI for a narrower set of disadvantaged patients because expected legal exposure acts like an additional shadow cost of reliance for that group\. A policy designed to shield disadvantaged patients from unequal algorithmic performance can thus end up limiting their access to AI altogether\.

Second, the effect of liability on AI use for disadvantaged patients is non\-monotone\. For small increases in liability, the direct deterrence effect dominates and use declines\. As liability rises further, however, the firm responds endogenously by reallocating investment toward disadvantaged\-group accuracy and, beyond a threshold, by switching to an equal\-accuracy design\. These upstream responses reduce expected physician exposure and can restore utilization, or even increase it relative to intermediate\-liability levels\. This feedback loop means the same policy that suppresses downstream use at low levels can strengthen upstream incentives to reduce disparity at higher levels\.

Third, we show mandating equal measured accuracy across groups does not necessarily improve welfare and can, under plausible conditions, reduce welfare forbothgroups\. A one\-size\-fits\-all accuracy requirement distorts the firm’s resource allocation under asymmetric improvement costs: in many cases, it lowers advantaged\-group accuracy substantially while raising disadvantaged\-group accuracy only modestly\. When combined with reimbursement structures that reward AI use, this shift can also alter clinical deployment in ways that increase inappropriate use\. Equalizing algorithmic performance and ensuring appropriate clinical reliance are distinct policy objectives\. Liability standards aimed at the former may require complementary instruments—such as reimbursement design, utilization guidelines, or monitoring rules—to achieve the latter\.

Growing evidence shows algorithmic performance can vary systematically across groups in ways that materially affect care and outcomes\([Abràmoff et al\. 2022a](https://arxiv.org/html/2608.13618#bib.bib1),[Goodman et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib26)\)\. For example,[Obermeyer et al\. 2019](https://arxiv.org/html/2608.13618#bib.bib50)show racial disparity in a widely used clinical risk algorithm led to fewer Black patients being identified for additional care despite greater illness severity\. Although addressing disparity early in development is often more effective than retrofitting ex post\([Gichoya et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib24)\), our analysis shows regulatory design remains crucial: the same legal objective can generate very different equilibrium outcomes depending on how incentives are structured\. We do not argue against regulating algorithmic disparity; rather, we clarify the incentive trade\-offs that must be addressed if liability standards are to work as intended\.

More broadly, this paper treats medical AI as an input to expert decision\-making, not a substitute for it\. Even when AI is highly accurate, its deployment hinges on transparency, accountability, and the incentives of the human expert in the loop\. Regulations such as § 92\.210 constrain reliance on disparate tools, implying that the welfare consequences of AI design are filtered through downstream adoption decisions\. Accordingly, our model allows physicians to decide whether to use AI and, when they do, how much to rely on it; we study how these deployment decisions feed back into firms’ design incentives, and how liability and reimbursement shape that feedback\.

## 2Literature

Our work contributes to the expert\-service literature originating from[Darby and Karni 1973](https://arxiv.org/html/2608.13618#bib.bib16), who argue credence\-goods providers such as physicians may overprovide expert services\. A key theme in this literature is that liability shapes clinical behavior: empirical and experimental evidence confirms malpractice liability influences physician decision\-making\([Currie and MacLeod 2008](https://arxiv.org/html/2608.13618#bib.bib13),[Dulleck et al\. 2011](https://arxiv.org/html/2608.13618#bib.bib19)\), and theoretical work shows liability can discipline providers much as reputational concerns do\([Fong and Liu 2018](https://arxiv.org/html/2608.13618#bib.bib21)\)or induce appropriate treatment choices\([Chen et al\. 2022](https://arxiv.org/html/2608.13618#bib.bib8)\)\.111The related literature on liability in other domains, such as product safety\([Guan et al\. 2024](https://arxiv.org/html/2608.13618#bib.bib29),[Iyer and Singh 2018](https://arxiv.org/html/2608.13618#bib.bib33)\), offers valuable insights but does not directly apply to the expert\-service context\.The introduction of AI complicates this picture\.[Dai and Singh 2025](https://arxiv.org/html/2608.13618#bib.bib15), for instance, explore physician behavior under emerging liability frameworks but focus on malpractice liability and on liability from disregarding an AI recommendation, without addressing the potential disparity of AI algorithms across patient groups\. We model an AI\-specific healthcare regulation and analyze its impact on AI firms, physicians, and patients\.

Our work also relates to the literature on the role of AI in medical decision\-making\. Using AI to support clinical decisions connects to the literature on information acquisition\. Prior work has examined how people interact with algorithms, including algorithm preference and algorithm aversion\([Dietvorst et al\. 2018](https://arxiv.org/html/2608.13618#bib.bib18),[Iyer and Ke 2024](https://arxiv.org/html/2608.13618#bib.bib32),[Leung et al\. 2018](https://arxiv.org/html/2608.13618#bib.bib37),[Mohammadi et al\. 2024](https://arxiv.org/html/2608.13618#bib.bib47), e\.g\.,\)\. The literature has also attempted to identify the causes of overuse/over\-adherence of AI algorithms\.[McLaughlin and Spiess 2022](https://arxiv.org/html/2608.13618#bib.bib44)provide an explanation that AI altering preferences can lead to over\-adherence; for instance, a decision maker may view the algorithmic recommendation as a default action\. In terms of the cause of AI underuse,[Dai and Singh 2020](https://arxiv.org/html/2608.13618#bib.bib14)develop a signaling model to show highly skilled physicians may underutilize diagnostic tests to signal their skills\.[Balakrishnan et al\. 2026](https://arxiv.org/html/2608.13618#bib.bib4)show humans over\-adhere to an algorithm’s predictions when their own private information, which the algorithm cannot access, is valuable, and under\-adhere otherwise\. We differ from this stream by not directly investigating the influence of human factors but examining the influence of potential liability\. By connecting both the upstream and downstream of medical AI, we show a physician may underuse AI under relatively small liability, due to a dominant liability concern, and overuse AI under relatively large liability, due to its incentive for the upstream to invest in high algorithmic accuracy\.

Our work also contributes to the literature on disparate algorithm performance\. Extensive research documents systematic differences in health and healthcare by race, gender, age, and other characteristics\([Heckler 1985](https://arxiv.org/html/2608.13618#bib.bib30),[Nelson 2002](https://arxiv.org/html/2608.13618#bib.bib48), e\.g\.,\)\. As predictive algorithms diffuse across high\-stakes domains, a central concern is that these tools can reproduce or even widen such gaps when training data and prediction targets are misaligned with clinical needs\([Benjamin 2016](https://arxiv.org/html/2608.13618#bib.bib5),[Gianfrancesco et al\. 2018](https://arxiv.org/html/2608.13618#bib.bib23), e\.g\.,\)\. For example,[Tipton et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib58)review evidence that standard “fairness fixes” \(such as omitting race\) can improve parity along one dimension while worsening broader health outcomes\. Related work develops technical approaches to reducing disparate performance—by changing objectives\([Samorani et al\. 2022](https://arxiv.org/html/2608.13618#bib.bib54)\), redefining targets\([Obermeyer et al\. 2019](https://arxiv.org/html/2608.13618#bib.bib50)\), or incorporating group identity\([Gillis et al\. 2021](https://arxiv.org/html/2608.13618#bib.bib25)\)—and, in parallel, economic models of algorithm design under data investment, disclosure, and fairness concerns\([Diao et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib17),[Li and Li 2023](https://arxiv.org/html/2608.13618#bib.bib38)\)\. We add a complementary mechanism: deployment\-facing regulation can narrow measured performance gaps yet induce unequal use in equilibrium; put differently, equalizing algorithmic accuracy need not equalize who benefits from AI\.

This paper also contributes to the economics and marketing literature on product design under fairness/ethical or policy constraints\([Diao et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib17),[Israeli 2018](https://arxiv.org/html/2608.13618#bib.bib31),[Iyer and Ke 2024](https://arxiv.org/html/2608.13618#bib.bib32),[Ke and Sudhir 2023](https://arxiv.org/html/2608.13618#bib.bib34), e\.g\.,\)\. As in studies of how marketing instruments \(for example, incentives, pricing, trust signals\) influence intermediary adoption of new technologies\([Lambrecht and Tucker 2024](https://arxiv.org/html/2608.13618#bib.bib35),[Luo et al\. 2019](https://arxiv.org/html/2608.13618#bib.bib42),[Zimmermann et al\. 2024](https://arxiv.org/html/2608.13618#bib.bib62), e\.g\.,\), we study how fairness constraints reshape physician–AI interaction and adoption\. Relative to that work, which sets product quality for a buyer who decides whether to purchase, in our paper an expert intermediary, not the buyer, governs use, and we endogenize the quality of the expert’s information technology that the credence\-service literature takes as given\. Equalizing measured quality then need not equalize realized access\. Relatedly, our paper engages with the literature on the implications of algorithmic fairness\.[Corbett\-Davies et al\. 2017](https://arxiv.org/html/2608.13618#bib.bib12)reveal a tension between improving public safety and satisfying prevailing notions of algorithmic fairness when deciding whether to release pretrial defendants back into the community\.[Shimao et al\. 2022](https://arxiv.org/html/2608.13618#bib.bib57)show fair algorithms can lead to different equilibrium behaviors among different groups of prediction subjects\.[Corbett\-Davies and Goel 2018](https://arxiv.org/html/2608.13618#bib.bib11)find equal opportunity and demographic parity may harm the protected group due to heterogeneity across groups\.[Liu et al\. 2018](https://arxiv.org/html/2608.13618#bib.bib40)show an overly aggressive fairness criterion may cause harm to the protected group in the long term, because giving too many loans to people in a protected group who cannot pay them back can hurt the group’s credit scores on average\.[Fu et al\. 2022](https://arxiv.org/html/2608.13618#bib.bib22)show fair algorithms that require impact parity can make everyone worse off, including the protected group, because of the firm’s strategic behavior of underinvesting in learning\. Our mechanism is distinct: rather than relying on heterogeneity in patient characteristics across groups, we show mandating equal accuracy can harm disadvantaged patients even when the two groups differ only in the cost of improving algorithmic performance\. The welfare loss arises from the physician’s dual objective of representing patients while also pursuing AI reimbursement, which distorts usage decisions under a one\-size\-fits\-all accuracy constraint\.

## 3Model

Consider a physician who selects a treatment for a patient from two possible options,T1T\_\{1\}andT2T\_\{2\}\. One option isappropriatefor the patient and the other isinappropriate\. The benefit of the appropriate treatment for any patient isbb, whereb\>0b\>0, whereas the benefit of the inappropriate treatment is00\.

To reflect the health disparities often observed across demographic groups\([Obermeyer et al\. 2019](https://arxiv.org/html/2608.13618#bib.bib50)\), we consider two patient types,t∈\{x,y\}t\\in\\\{x,y\\\}\. Type\-xxpatients represent an advantaged group, such as White patients, whereas type\-yypatients represent a disadvantaged group, such as Black patients\. Following the protected characteristics outlined in Section 1557 of the Affordable Care Act\([CMS 2024](https://arxiv.org/html/2608.13618#bib.bib10)\), we assume a patient’s type,tt, is observable\. For simplicity, we normalize the mass of type\-xxpatients to one and that of type\-yypatients toβ\\beta, where0<β≤10<\\beta\\leq 1, to capture that disadvantaged patients are often a minority group\.

The physician holds a prior belief that treatmentT1T\_\{1\}is appropriate for a patient with probabilityα\\alpha, whereas treatmentT2T\_\{2\}is appropriate with probability1−α1\-\\alpha, whereα∼U⁡\[0,1\]\\alpha\\sim U\[0,1\]\.222The assumption of type\-independent priors helps isolate the incremental role of algorithmic disparity and the specific anti\-discrimination liability channel we study in this paper\. Holding the physician’s priors the same for both patient types ensures any cross\-group differences in outcomes in our analysis arise from the AI’s differential accuracy and the physician’s endogenous AI\-use decision under liability\.We interpretα\\alphaas the physician’s case\-specific assessment after her usual initial examination of the patient and before she consults the AI; it is not a population base rate, and its distribution captures across\-case heterogeneity in baseline diagnostic uncertainty rather than any group\-specific bias\. We study the case of type\-dependent priors in[section6\.3](https://arxiv.org/html/2608.13618#S6.SS3)\. We useata\_\{t\}to denote the appropriate treatment for a type\-ttpatient\. The physician has access to an AI\-powered clinical algorithm \(hereafter “AI”\)\. The AI generates a treatment signal, which the physician uses to update her belief about the appropriate treatment\. We denote the AI signal for a type\-ttpatient bysts\_\{t\}and define AI accuracy for typettasρt\\rho\_\{t\}, wheret=x,yt=x,y\. Specifically,P1\|1:=𝒫⁡\(st=T1\|at=T1\)=ρtP\_\{1\|1\}:=\\mathcal\{P\}\(s\_\{t\}=T\_\{1\}\|a\_\{t\}=T\_\{1\}\)=\\rho\_\{t\}andP2\|1:=𝒫⁡\(st=T2\|at=T1\)=1−ρtP\_\{2\|1\}:=\\mathcal\{P\}\(s\_\{t\}=T\_\{2\}\|a\_\{t\}=T\_\{1\}\)=1\-\\rho\_\{t\}\. Similarly,P2\|2:=𝒫⁡\(st=T2\|at=T2\)=ρtP\_\{2\|2\}:=\\mathcal\{P\}\(s\_\{t\}=T\_\{2\}\|a\_\{t\}=T\_\{2\}\)=\\rho\_\{t\}andP1\|2:=𝒫⁡\(st=T1\|at=T2\)=1−ρtP\_\{1\|2\}:=\\mathcal\{P\}\(s\_\{t\}=T\_\{1\}\|a\_\{t\}=T\_\{2\}\)=1\-\\rho\_\{t\}\. AI recommendations are informative but imperfect, that is,1/2<ρt<11/2<\\rho\_\{t\}<1\. The physician observes the algorithm’s accuracy for each patient type,ρt\\rho\_\{t\}, through her clinical expertise and experience with the tool\. We call an algorithm*equal\-accuracy*ifρx=ρy\\rho\_\{x\}=\\rho\_\{y\}, and in that case we write the common accuracy asρ\\rho\(dropping the subscripttt\)\. An algorithm withρx\>ρy\\rho\_\{x\}\>\\rho\_\{y\}delivers lower accuracy for type\-yypatients and is therefore disparate for type\-yypatients\.

With the AI signal, the physician updates her belief about the appropriate treatment using Bayes’ rule\. The physician’s posterior belief thatT1T\_\{1\}is appropriate is

Q1\|1\\displaystyle Q\_\{1\|1\}:=𝒫⁡\(at=T1\|st=T1\)=α​ρtα​ρt\+\(1−α\)​\(1−ρt\),\\displaystyle:=\\mathcal\{P\}\(a\_\{t\}=T\_\{1\}\|s\_\{t\}=T\_\{1\}\)=\\frac\{\\alpha\\rho\_\{t\}\}\{\\alpha\\rho\_\{t\}\+\(1\-\\alpha\)\(1\-\\rho\_\{t\}\)\},Q1\|2\\displaystyle Q\_\{1\|2\}:=𝒫⁡\(at=T1\|st=T2\)=α⁡\(1−ρt\)α⁡\(1−ρt\)\+\(1−α\)​ρt\.\\displaystyle:=\\mathcal\{P\}\(a\_\{t\}=T\_\{1\}\|s\_\{t\}=T\_\{2\}\)=\\frac\{\\alpha\(1\-\\rho\_\{t\}\)\}\{\\alpha\(1\-\\rho\_\{t\}\)\+\(1\-\\alpha\)\\rho\_\{t\}\}\.The physician believesT2T\_\{2\}to be appropriate with the complementary probability, that is,Q2\|1:=𝒫⁡\(at=T2\|st=T1\)=1−Q1\|1Q\_\{2\|1\}:=\\mathcal\{P\}\(a\_\{t\}=T\_\{2\}\|s\_\{t\}=T\_\{1\}\)=1\-Q\_\{1\|1\}andQ2\|2:=𝒫⁡\(at=T2\|st=T2\)=1−Q1\|2Q\_\{2\|2\}:=\\mathcal\{P\}\(a\_\{t\}=T\_\{2\}\|s\_\{t\}=T\_\{2\}\)=1\-Q\_\{1\|2\}\.

Consistent with the Section 1557 clinical algorithm provision \(§ 92\.210\), we model deployment\-facing liability as follows\. If the physician follows the signal of a disparate clinical algorithm for a type\-yypatient and the resulting treatment is inappropriate, she faces a liability costℓ\\ell\([CMS 2022](https://arxiv.org/html/2608.13618#bib.bib9)\)\. This channel is specific to type\-yypatients in our setting; we abstract from other sources of legal exposure \(for example, malpractice\) to isolate the incentive effects of the provision\. Reflecting that the provision governs the use of decision\-support tools and applies to covered entities rather than to developers\([Mello and Roberts 2024](https://arxiv.org/html/2608.13618#bib.bib46)\), we assign liability to the physician rather than to the AI firm\. When the algorithm has equal accuracy across patient types \(ρx=ρy\\rho\_\{x\}=\\rho\_\{y\}\), this disparity\-triggered liability channel does not apply\.

When the physician deploys the AI tool in a patient’s case, the patient incurs a costc\>0c\>0\. We modelccas a reduced\-form burden that can include out\-of\-pocket expenses and non\-monetary disutility—for example, psychological discomfort, perceived risk, or distrust when an algorithm is involved in diagnosis or treatment\([Longoni et al\. 2019](https://arxiv.org/html/2608.13618#bib.bib41)\)\. Importantly,ccis not meant to give patients control over AI use: although opt\-out is feasible in some settings, clinical practice and accountability assign the deployment decision to the physician, and we therefore assume the physician retains full discretion over whether to use the tool\.

We also allow the physician to receive a per\-use benefitr\>0r\>0when she deploys the tool\. This term captures reimbursement for AI\-assisted services—for example, payment pathways enabled by AI\-specific Current Procedural Terminology \(CPT\) codes and New Technology Add\-On Payments \(NTAP\)—and broader private returns from using AI\([Parikh and Helmchen 2022](https://arxiv.org/html/2608.13618#bib.bib51)\)\. Depending on the setting,rrmay also reflect reputational or professional gains from being perceived as technologically capable or clinically sophisticated\([Liaw et al\. 2022](https://arxiv.org/html/2608.13618#bib.bib39),[Schubert et al\. 2025](https://arxiv.org/html/2608.13618#bib.bib55),[Schuitmaker et al\. 2025](https://arxiv.org/html/2608.13618#bib.bib56)\)\.333Several U\.S\. providers now charge patients directly for AI\-assisted reads; for example, RadNet’s imaging centers offer an optional AI review of a screening mammogram for an out\-of\-pocket fee, which a substantial share of patients elect, illustrating a patient\-side costc\>0c\>0and, where the provider bills for the read, a per\-use revenuer\>0r\>0\. On the patient side,c\>0c\>0is also consistent with documented resistance to medical AI\([Longoni et al\. 2019](https://arxiv.org/html/2608.13618#bib.bib41)\)\.,444In the special case in whichr<0r<0, possibly due to a strong negative reputational effect of using AI, our main results on the physician’s reduced AI use for disadvantaged patients and the non\-monotone effect of liability on the physician’s AI use for disadvantaged patients continue to hold\. However, in this case, disadvantaged patients are better off, whereas advantaged patients are worse off as a result of the equal\-accuracy mandate\. Because the physician has no incentive to overuse AI in ther<0r<0case, the disadvantaged patients cannot be worse off\.

When the physician decides whether and how to use AI, she cares about both the patient’s health outcome \(that is, treatment benefit net of AI\-use cost\) and her own nonclinical objectives \(that is, revenue and liability\)\. We assume the physician assigns a weightθ≥0\\theta\\geq 0to her nonclinical objectives\. A physician withθ=0\\theta=0is altruistic and cares only about the patient’s health outcome, whereas a physician withθ\>0\\theta\>0isimpurelyaltruistic and also cares about her nonclinical objectives\. Unless otherwise noted, we assumeθ\>0\\theta\>0; the altruistic caseθ=0\\theta=0serves as a benchmark for comparison\.

Next, we introduce a profit\-maximizing firm that supplies an AI\-powered clinical algorithm with accuracyρt\\rho\_\{t\}for patient typet∈\{x,y\}t\\in\\\{x,y\\\}\. Consistent with fee\-for\-service arrangements in which AI use is reimbursed at predetermined rates, we assume the firm receives a paymentffeach time the physician deploys the tool in a patient’s case\([Abràmoff et al\. 2022b](https://arxiv.org/html/2608.13618#bib.bib3),[Abràmoff et al\. 2024](https://arxiv.org/html/2608.13618#bib.bib2)\)\.555In a model extension presented in[section6\.1](https://arxiv.org/html/2608.13618#S6.SS1), we assume the AI firm sets a profit\-maximizing priceffand find all our main insights continue to hold qualitatively under the endogenous pricing assumption\.Achieving accuracyρt\\rho\_\{t\}for typettentails a development costκt​\(ρt−12\)2\\kappa\_\{t\}\(\\rho\_\{t\}\-\\tfrac\{1\}\{2\}\)^\{2\}, whereκt\>0\\kappa\_\{t\}\>0is type\-specific\. To capture the limited availability and higher acquisition cost of training data for disadvantaged patients, we assumeκx<κy\\kappa\_\{x\}<\\kappa\_\{y\}\. We focus on settings in which the firm trains a new model on a fixed historical dataset, rather than incrementally updating an existing system, and thus treat the cost coefficients as exogenous and independent of downstream usage\. The cost asymmetryκx<κy\\kappa\_\{x\}<\\kappa\_\{y\}reflects the empirically common case in which improving performance for the disadvantaged group is harder\.666If the asymmetry were reversed \(κy<κx\\kappa\_\{y\}<\\kappa\_\{x\}\), the algorithmic accuracy for the disadvantaged group would not be harder to improve and the disparity\-contingent liability channel would be weak or inactive; in an application in which the type\-yygroup is the lower\-cost group, the labelsxxandyycan be exchanged\.The separable quadratic structure is a parsimonious way to capture the practical fact that improving performance for a harder\-to\-predict subgroup typically requires targeted effort \(for example, subgroup\-specific tuning or additional data\), even when the deployed model is unified\.

Given physician demand\(dxD,dyD\)\(d\_\{x\}^\{\\mathrm\{D\}\},d\_\{y\}^\{\\mathrm\{D\}\}\)under a disparate design, the firm’s profit is

πD​\(ρx,ρy\):=\(dxD\+dyD\)​f−κx​\(ρx−12\)2−κy​\(ρy−12\)2,\\displaystyle\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\},\\rho\_\{y\}\):=\(d\_\{x\}^\{\\mathrm\{D\}\}\+d\_\{y\}^\{\\mathrm\{D\}\}\)f\-\\kappa\_\{x\}\(\\rho\_\{x\}\-\\tfrac\{1\}\{2\}\)^\{2\}\-\\kappa\_\{y\}\(\\rho\_\{y\}\-\\tfrac\{1\}\{2\}\)^\{2\},\(1\)wheredxDd\_\{x\}^\{\\mathrm\{D\}\}anddyDd\_\{y\}^\{\\mathrm\{D\}\}denote the volumes of type\-xxand type\-yycases in which the physician deploys the tool \(expressions are given in[eq\.6](https://arxiv.org/html/2608.13618#S4.E6)\)\. When the firm supplies an equal\-accuracy design,ρx=ρy≡ρ\\rho\_\{x\}=\\rho\_\{y\}\\equiv\\rho, profit can be written as

πE​\(ρ\):=\(dxE\+dyE\)​f−κx​\(ρ−12\)2−κy​\(ρ−12\)2,\\displaystyle\\pi^\{\\mathrm\{E\}\}\(\\rho\):=\(d\_\{x\}^\{\\mathrm\{E\}\}\+d\_\{y\}^\{\\mathrm\{E\}\}\)f\-\\kappa\_\{x\}\(\\rho\-\\tfrac\{1\}\{2\}\)^\{2\}\-\\kappa\_\{y\}\(\\rho\-\\tfrac\{1\}\{2\}\)^\{2\},\(2\)where\(dxE,dyE\)\(d\_\{x\}^\{\\mathrm\{E\}\},d\_\{y\}^\{\\mathrm\{E\}\}\)are the corresponding demands under equal accuracy, given in[eq\.7](https://arxiv.org/html/2608.13618#S4.E7)\.

We assumec\>θ​rc\>\\theta r, which eliminates cases in which the AI firm develops an arbitrarily bad algorithm, but the physician still uses AI for all patients\. In other words, we do not consider cases in which the physician uses AI solely for private gain from reimbursement\. In addition, although parameterscc,rr, andffare likely interrelated in practice, they do not reflect a direct transfer of funds\. Instead, funds typically flow from patients and insurers to providers \(physician, hospital, or health system\), who then pay AI firms\. For clarity, we summarize the meaning, examples, and payment flow for parameterscc,rr, andffin[table1](https://arxiv.org/html/2608.13618#S3.T1)\. To maintain tractability and focus on core strategic trade\-offs, our baseline model treats parameterscc,rr, andffas independent\. Nonetheless, in[section6\.1](https://arxiv.org/html/2608.13618#S6.SS1), we explore an alternative model in which the patient’s AI\-use cost is the same as the firm’s price \(c=fc=f\), and we confirm our main results remain robust under this specification\.

Table 1:Meaning, Examples, and Payment Flow for Parameterscc,rr, andffNotes\.The parametercccaptures both monetary costs \(for example, copayments\) and non\-monetary disutility borne by the patient\. The parameterrrcaptures both financial reimbursement \(for example, CMS payments via CPT codes\) and non\-monetary gains such as perceived competence or technological savviness\. Onlyffrepresents a direct inter\-party transfer\.

Next, we describe the patient’s expected utility\. First, consider the case in which the physician decides not to use AI\. In this case, the physician follows her own prior belief to treat the patient\. Given the physician’s prior beliefα\\alphaof treatmentT1T\_\{1\}being appropriate, the patient’s expected utility from treatmentT1T\_\{1\}isα​b\\alpha b, and from treatmentT2T\_\{2\}is\(1−α\)​b\(1\-\\alpha\)b\. Now consider the case in which the physician uses AI\. In this case, the patient incurs an AI\-use costcc\. The patient’s expected benefit from treatment depends on the AI signal and the physician’s treatment choice\. Specifically, if the physician prescribes treatmentTjT\_\{j\}after observing signalTiT\_\{i\}, the patient’s expected benefit isQj\|i​bQ\_\{j\|i\}b, wherei,j∈\{1,2\}i,j\\in\\\{1,2\\\}\. We assume the patient’s expected utility is determined by her realized health outcome and any incurred cost\. We do not include any gain from lawsuit payouts, to avoid unrealistic scenarios in which the physician might intentionally harm the patient to generate a financial transfer to the patient\. Equivalently, expected patient welfare for each type, denotedWxW\_\{x\}andWyW\_\{y\}and defined formally in[section5\.3](https://arxiv.org/html/2608.13618#S5.SS3), measures patient health surplus \(expected treatment benefit net of the AI\-use cost\) and excludes litigation compensation\. If litigation compensation enters only as an ex post accounting transfer that no decision maker internalizes, equilibrium behavior is unaffected, because patients choose neither AI use nor litigation effort and the provider’s exposure is already summarized byℓ\\ell; the transfer then raises compensation\-inclusive type\-yysurplus and cancels in total surplus apart from deadweight legal costs\. If instead the physician internalizes the patient’s compensation through patient utility, the behavioral model changes, and we analyze that formulation in[section5\.3](https://arxiv.org/html/2608.13618#S5.SS3), where the weight on the liability term becomesθ−1\\theta\-1\. Our welfare comparisons should therefore be read as comparisons of ex ante health surplus\. The patient’s expected utility is summarized in[table2](https://arxiv.org/html/2608.13618#S3.T2)\.

Table 2:The Patient’s Expected Utility When the Physician Uses AIWe now describe the physician’s expected payoff from treating an individual patient\. If the physician uses AI, she receives a per\-use benefitrrand may face liability\. For type\-yypatients, liability arises only when the physician follows the AI recommendation and that recommendation is incorrect\. Accordingly, following signalT1T\_\{1\}leads to liability with probabilityQ2\|1Q\_\{2\|1\}, whereas following signalT2T\_\{2\}leads to liability with probabilityQ1\|2Q\_\{1\|2\}\. We interpretℓ\\ellas a reduced\-form expected exposure borne by the provider rather than a literal per\-case statutory fine\. Section 92\.210 does not impose strict liability for every adverse outcome; it prohibits discriminatory use of decision\-support tools and requires covered entities to make reasonable efforts to identify and mitigate disparity\. Accordingly,ℓ\\ellcaptures the expected cost of scrutiny, enforcement, settlement, and remediation conditional on relying on a disparate tool in a type\-yycase that yields an inappropriate treatment\([Mello and Roberts 2024](https://arxiv.org/html/2608.13618#bib.bib46)\)\. Because the physician finds following the signal optimal whenever she uses AI, this event is equivalently described in terms of the realized treatment\. The physician values both the patient’s health outcome net of cost and her own nonclinical payoff, placing weightθ\\thetaon the latter\.[Table3](https://arxiv.org/html/2608.13618#S3.T3)summarizes the physician’s payoff, where𝟏t=y\\mathbf\{1\}\_\{t=y\}equals 1 ift=yt=yand 0 otherwise\.

Table 3:The Physician’s Expected Payoff When Treating an Individual Patient Using AI[fig\.1](https://arxiv.org/html/2608.13618#S3.F1)summarizes the timing\. First, the AI firm chooses whether to offer a disparate design or an equal\-accuracy design and selects the corresponding accuracy level\(s\)ρt\\rho\_\{t\}for each patient type\. Second, the physician decides whether to use AI for a given patient\. If she uses AI, the algorithm generates a treatment signal and the physician then decides whether to follow it; if she does not use AI, she selects treatment based only on her prior belief\. Finally, the patient’s health outcome is realized and provider liability is assessed under the clinical algorithm provision\. We solve the game by backward induction\. Throughout, we assumeκy<f​β​\(b\+θ​ℓ\)2b⁡\(2​c−2​θ​r\+θ​ℓ\)\\kappa\_\{y\}<\\frac\{f\\beta\(b\+\\theta\\ell\)^\{2\}\}\{b\(2c\-2\\theta r\+\\theta\\ell\)\}, which ensures that, in equilibrium, the physician uses AI for a strictly positive measure of type\-yypatients \(that is,dyD∗\>0\{d\_\{y\}^\{\\mathrm\{D\}\}\}^\{\*\}\>0\)\.

Figure 1:Timing of the Game\. The AI firm’s first\-stage choice is between a disparate design \(ρx\>ρy\\rho\_\{x\}\>\\rho\_\{y\}\) and an equal\-accuracy design \(ρx=ρy\\rho\_\{x\}=\\rho\_\{y\}\)\.
## 4Analysis

In this section, we first analyze in[section4\.1](https://arxiv.org/html/2608.13618#S4.SS1)the physician’s decision of whether and how to use AI, taking AI accuracy as given\. We then analyze in[section4\.2](https://arxiv.org/html/2608.13618#S4.SS2)how the AI firm chooses AI accuracy\.

### 4\.1Downstream Implications

We analyze the physician’s decision of whether to use a disparate AI algorithm withρx\>ρy\>1/2\\rho\_\{x\}\>\\rho\_\{y\}\>1/2for type\-xxand type\-yypatients\. The orderingρx\>ρy\\rho\_\{x\}\>\\rho\_\{y\}here defines the disparate\-design*subgame*whose downstream behavior we characterize; it is not an equilibrium restriction\.[Proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)shows that whenever the firm finds a disparate design optimal,ρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}emerges endogenously, so the ordering is derived rather than assumed\. Note that the analysis of the physician’s decision to use an equal\-accuracy algorithm \(ρx=ρy\>1/2\\rho\_\{x\}=\\rho\_\{y\}\>1/2\) is analogous to the analysis for type\-xxpatients under a disparate algorithm\.

If the physician does not use AI, then by comparing the physician’s payoff from prescribing treatmentT1T\_\{1\}\(that is,α​b\\alpha b\) andT2T\_\{2\}\(that is,\(1−α\)​b\(1\-\\alpha\)b\), we know the physician prescribes treatmentT1T\_\{1\}ifα\>1/2\\alpha\>1/2and prescribes treatmentT2T\_\{2\}ifα≤1/2\\alpha\\leq 1/2\. We can summarize the physician’s expected payoff when she does not use AI as follows:

U=\\displaystyle U=\{\(1−α\)​b,if​α≤1/2α​b,if​α\>1/2\.\\displaystyle\\begin\{cases\}\(1\-\\alpha\)b,&\\text\{if \}\\alpha\\leq 1/2\\\\ \\alpha b,&\\text\{if \}\\alpha\>1/2\.\\end\{cases\}\(3\)
Now suppose the physician uses AI\. In this case, the probability of the AI signal suggesting treatmentT1T\_\{1\}is𝒫⁡\(st=T1\)=α⋅ρt\+\(1−α\)⋅\(1−ρt\)\\mathcal\{P\}\(s\_\{t\}=T\_\{1\}\)=\\alpha\\cdot\\rho\_\{t\}\+\(1\-\\alpha\)\\cdot\(1\-\\rho\_\{t\}\)and treatmentT2T\_\{2\}is𝒫⁡\(st=T2\)=α⋅\(1−ρt\)\+\(1−α\)⋅ρt\\mathcal\{P\}\(s\_\{t\}=T\_\{2\}\)=\\alpha\\cdot\(1\-\\rho\_\{t\}\)\+\(1\-\\alpha\)\\cdot\\rho\_\{t\}\. Next, we analyze the physician’s decision of whether to use AI\.

#### 4\.1\.1Type\-xxPatient

Suppose the physician consults AI for a type\-xxpatient\. If the signal issx=T1s\_\{x\}=T\_\{1\}, she compares the payoffs from prescribingT1T\_\{1\}andT2T\_\{2\}, namelyQ1\|1​b−c\+θ​rQ\_\{1\|1\}b\-c\+\\theta randQ2\|1​b−c\+θ​rQ\_\{2\|1\}b\-c\+\\theta r\(see[table3](https://arxiv.org/html/2608.13618#S3.T3)\)\. She therefore prescribesT1T\_\{1\}if and only ifα\>1−ρx\\alpha\>1\-\\rho\_\{x\}, and prescribesT2T\_\{2\}otherwise\. If the signal issx=T2s\_\{x\}=T\_\{2\}, she comparesQ1\|2​b−c\+θ​rQ\_\{1\|2\}b\-c\+\\theta randQ2\|2​b−c\+θ​rQ\_\{2\|2\}b\-c\+\\theta r, and prescribesT1T\_\{1\}if and only ifα\>ρx\\alpha\>\\rho\_\{x\}, andT2T\_\{2\}otherwise\. Because1/2<ρx<11/2<\\rho\_\{x\}<1, we have1−ρx<ρx1\-\\rho\_\{x\}<\\rho\_\{x\}, so the prior space is partitioned into three regions\. The physician’s expected payoff from AI use for a type\-xxpatient is

Ux=\{𝒫⁡\(sx=T1\)​Q2\|1​b\+𝒫⁡\(sx=T2\)​Q2\|2​b−c\+θ​r,if​α≤1−ρx,𝒫⁡\(sx=T1\)​Q1\|1​b\+𝒫⁡\(sx=T2\)​Q2\|2​b−c\+θ​r,if​1−ρx<α≤ρx,𝒫⁡\(sx=T1\)​Q1\|1​b\+𝒫⁡\(sx=T2\)​Q1\|2​b−c\+θ​r,if​α\>ρx\.\\displaystyle U\_\{x\}=\\begin\{cases\}\\mathcal\{P\}\(s\_\{x\}=T\_\{1\}\)Q\_\{2\|1\}b\+\\mathcal\{P\}\(s\_\{x\}=T\_\{2\}\)Q\_\{2\|2\}b\-c\+\\theta r,&\\text\{if \}\\alpha\\leq 1\-\\rho\_\{x\},\\\\ \\mathcal\{P\}\(s\_\{x\}=T\_\{1\}\)Q\_\{1\|1\}b\+\\mathcal\{P\}\(s\_\{x\}=T\_\{2\}\)Q\_\{2\|2\}b\-c\+\\theta r,&\\text\{if \}1\-\\rho\_\{x\}<\\alpha\\leq\\rho\_\{x\},\\\\ \\mathcal\{P\}\(s\_\{x\}=T\_\{1\}\)Q\_\{1\|1\}b\+\\mathcal\{P\}\(s\_\{x\}=T\_\{2\}\)Q\_\{1\|2\}b\-c\+\\theta r,&\\text\{if \}\\alpha\>\\rho\_\{x\}\.\\end\{cases\}\(4\)
Next, we present the physician’s decision of whether and how to use AI for type\-xxpatients\. A comparison of the physician’s expected payoff when using AI and not using AI for type\-xxpatients reveals the following lemma\. \(All proofs are in the Appendix\.\)

###### Lemma 1\.

For type\-xxpatients, the physician uses AI if and only if

\(1−ρx\)​b\+c−θ​rb<α<ρx​b−c\+θ​rb\.\\frac\{\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\}\{b\}<\\alpha<\\frac\{\\rho\_\{x\}b\-c\+\\theta r\}\{b\}\.Conditional on using AI, the physician follows the AI signal\.

The physician’s prior beliefα\\alpharepresents the probability that treatmentT1T\_\{1\}is appropriate before observing any AI recommendation\. Without AI, the physician follows this prior and prescribesT1T\_\{1\}whenα\>1/2\\alpha\>1/2\. When AI is used, the physician updates her belief using Bayes’ rule, and the signal realization moves her posterior toward or away fromT1T\_\{1\}by an amount that increases in the accuracyρx\\rho\_\{x\}\. The use of AI creates value only when its signal can change the physician’s treatment decision\.

Intuitively, whenα\\alphais very low, the physician already believesT2T\_\{2\}is likely appropriate, and AI’s informational value cannot justify the additional patient costcc\. Whenα\\alphais very high, the physician is confident inT1T\_\{1\}and does not expect AI to improve her decision\. Only whenα\\alphalies in the intermediate range does AI meaningfully reduce uncertainty, making its use optimal\. Moreover, because AI is informative \(1/2<ρx<11/2<\\rho\_\{x\}<1\), whenever the physician chooses to use it, she rationally follows its recommendation\.

#### 4\.1\.2Type\-yyPatient

Now suppose the physician uses AI to generate the signal for a type\-yypatient\. If the AI signal isT1T\_\{1\}, the physician compares her payoffs from prescribing treatmentsT1T\_\{1\}andT2T\_\{2\}, which areQ1\|1⋅b−Q2\|1⋅θ​ℓ−c\+θ​rQ\_\{1\|1\}\\cdot b\-Q\_\{2\|1\}\\cdot\\theta\\ell\-c\+\\theta randQ2\|1⋅b−c\+θ​rQ\_\{2\|1\}\\cdot b\-c\+\\theta r\(see[table3](https://arxiv.org/html/2608.13618#S3.T3)\), respectively, and prescribes treatmentT1T\_\{1\}ifα\>\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ\\alpha\>\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}and treatmentT2T\_\{2\}ifα≤\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ\\alpha\\leq\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\. However, if the AI signal isT2T\_\{2\}, by comparing her payoffs from prescribing treatmentsT1T\_\{1\}andT2T\_\{2\}, which areQ1\|2⋅b−c\+θ​rQ\_\{1\|2\}\\cdot b\-c\+\\theta randQ2\|2⋅b−Q1\|2⋅θ​ℓ−c\+θ​rQ\_\{2\|2\}\\cdot b\-Q\_\{1\|2\}\\cdot\\theta\\ell\-c\+\\theta r\(see[table3](https://arxiv.org/html/2608.13618#S3.T3)\), respectively, the physician prescribes treatmentT1T\_\{1\}ifα\>b​ρyb\+\(1−ρy\)​θ​ℓ\\alpha\>\\frac\{b\\rho\_\{y\}\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}and treatmentT2T\_\{2\}ifα≤b​ρyb\+\(1−ρy\)​θ​ℓ\\alpha\\leq\\frac\{b\\rho\_\{y\}\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\. Given the assumption that the physician uses AI for at least some type\-yypatients, we haveρy\>b\+2​c\+2​θ​ℓ−2​θ​r2​b\+2​θ​ℓ\\rho\_\{y\}\>\\frac\{b\+2c\+2\\theta\\ell\-2\\theta r\}\{2b\+2\\theta\\ell\}\. In addition, becauseb\+2​c\+2​θ​ℓ−2​θ​r2​b\+2​θ​ℓ\>b\+θ​ℓ2​b\+θ​ℓ\\frac\{b\+2c\+2\\theta\\ell\-2\\theta r\}\{2b\+2\\theta\\ell\}\>\\frac\{b\+\\theta\\ell\}\{2b\+\\theta\\ell\}, it follows thatρy\>b\+θ​ℓ2​b\+θ​ℓ\\rho\_\{y\}\>\\frac\{b\+\\theta\\ell\}\{2b\+\\theta\\ell\}\. It is straightforward that\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ<12<b​ρyb\+\(1−ρy\)​θ​ℓ\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}<\\frac\{1\}\{2\}<\\frac\{b\\rho\_\{y\}\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\. Therefore, if the physician uses AI for a type\-yypatient, her expected payoff is given by

Uy=\\displaystyle U\_\{y\}=\{𝒫⁡\(sy=T1\)⋅Q2\|1⋅b\+𝒫⁡\(sy=T2\)⋅\(Q2\|2⋅b−Q1\|2​θ​ℓ\)−c\+θ​r,if​α≤\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ𝒫⁡\(sy=T1\)⋅\(Q1\|1⋅b−Q2\|1​θ​ℓ\)\+𝒫⁡\(sy=T2\)⋅\(Q2\|2⋅b−Q1\|2​θ​ℓ\)−c\+θ​r,if​\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ<α≤b​ρyb\+\(1−ρy\)​θ​ℓ𝒫⁡\(sy=T1\)⋅\(Q1\|1⋅b−Q2\|1​θ​ℓ\)\+𝒫⁡\(sy=T2\)⋅Q1\|2⋅b−c\+θ​r,otherwise\.\\displaystyle\\begin\{cases\}\\mathcal\{P\}\(s\_\{y\}=T\_\{1\}\)\\cdot Q\_\{2\|1\}\\cdot b\+\\mathcal\{P\}\(s\_\{y\}=T\_\{2\}\)\\cdot\\left\(Q\_\{2\|2\}\\cdot b\-Q\_\{1\|2\}\\theta\\ell\\right\)\-c\+\\theta r,&\\text\{if \}\\alpha\\leq\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\\\\ \\mathcal\{P\}\(s\_\{y\}=T\_\{1\}\)\\cdot\\left\(Q\_\{1\|1\}\\cdot b\-Q\_\{2\|1\}\\theta\\ell\\right\)\+\\mathcal\{P\}\(s\_\{y\}=T\_\{2\}\)\\cdot\\left\(Q\_\{2\|2\}\\cdot b\-Q\_\{1\|2\}\\theta\\ell\\right\)\-c\+\\theta r,&\\text\{if \}\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}<\\alpha\\leq\\frac\{b\\rho\_\{y\}\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\\\\ \\mathcal\{P\}\(s\_\{y\}=T\_\{1\}\)\\cdot\\left\(Q\_\{1\|1\}\\cdot b\-Q\_\{2\|1\}\\theta\\ell\\right\)\+\\mathcal\{P\}\(s\_\{y\}=T\_\{2\}\)\\cdot Q\_\{1\|2\}\\cdot b\-c\+\\theta r,&\\text\{otherwise\.\}\\end\{cases\}\(5\)The following lemma describes the physician’s decision of whether to use AI for type\-yypatients\.

###### Lemma 2\.

For type\-yypatients, the physician uses AI if and only if

\(1−ρy\)​\(b\+θ​ℓ\)\+c−θ​rb<α<ρy​b−\(1−ρy\)​θ​ℓ−c\+θ​rb\.\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}<\\alpha<\\frac\{\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\}\{b\}\.Conditional on using AI, the physician follows the AI signal\.

As in the type\-xxcase, the physician uses AI for type\-yypatients only when the informational value of consultation is sufficiently high\. Equivalently, AI use occurs over an intermediate range ofα\\alpha, in which clinical uncertainty about the appropriate treatment is greatest\. For an existing algorithm \(that is, exogenousρy\\rho\_\{y\}\), the liability rule unambiguously makes AI less attractive for type\-yypatients\. Holdingρy\\rho\_\{y\}fixed, an increase in liabilityℓ\\ellcontracts the set of priors for which AI is used\.

Because liability is triggered when a physician follows an erroneous recommendation from a disparate algorithm in type\-yycases, one might expect strategic rejection of AI advice\. Our model shows otherwise: conditional on consulting AI, the physician optimally follows the signal\. The intuition is straightforward\. If she consults AI but ignores the signal regardless of its realization, she incurs the consultation costccwithout obtaining informational benefit; withc\>θ​rc\>\\theta r, she is strictly better off not consulting at all\. If she follows only when the signal isT1T\_\{1\}\(or only when it isT2T\_\{2\}\), her behavior is equivalent to always choosingT1T\_\{1\}\(or always choosingT2T\_\{2\}\), which is again weakly dominated by making that fixed treatment choice without AI\. Hence, when the physician uses AI for type\-yypatients, she follows the AI signal\.

Next, we examine how the regulation affects the physician’s relative use of AI for type\-xxand type\-yypatients\. Comparing the AI\-use intervals in[lemmas1](https://arxiv.org/html/2608.13618#Thmlemma1)and[2](https://arxiv.org/html/2608.13618#Thmlemma2)shows that, in the presence of liability, the physician is \(weakly\) less likely to use AI for type\-yypatients than for type\-xxpatients\.

Under the Section 1557 clinical algorithm provision, a physician is exposed to liability when her reliance on a disparate AI tool leads to an inappropriate treatment for a type\-yypatient\. This liability functions as an added expected cost of AI use for type\-yycases, making adoption and reliance more selective for that group—even when the same tool is available for type\-xxpatients\. By contrast, type\-xxcases do not activate this liability channel; therefore, AI use for type\-xxpatients is governed only by the accuracy–cost–incentive trade\-off in[lemma1](https://arxiv.org/html/2608.13618#Thmlemma1)\. The resulting asymmetry in expected liability generates unequal utilization in equilibrium, with lower AI use among disadvantaged patients—the very group the policy is intended to protect\.

### 4\.2Upstream AI Firm’s Accuracy Decision

Anticipating the physician’s downstream use of the algorithm, the AI firm chooses whether to offer a disparate design or an equal\-accuracy design, and selects the associated accuracy levelsρt∗\\rho\_\{t\}^\{\*\}for each patient type\. By[lemmas1](https://arxiv.org/html/2608.13618#Thmlemma1)and[2](https://arxiv.org/html/2608.13618#Thmlemma2), under a disparate algorithm, the volumes of type\-xxand type\-yypatients for whom the physician uses AI are

dxD\\displaystyle d\_\{x\}^\{\\mathrm\{D\}\}=\(2​ρx−1\)​b−2​c\+2​θ​rb​and​dyD=β⋅\(2​ρy−1\)​b−2​\(1−ρy\)​θ​ℓ−2​c\+2​θ​rb\.\\displaystyle=\\frac\{\(2\\rho\_\{x\}\-1\)b\-2c\+2\\theta r\}\{b\}\\text\{ and \}d\_\{y\}^\{\\mathrm\{D\}\}=\\beta\\cdot\\frac\{\(2\\rho\_\{y\}\-1\)b\-2\(1\-\\rho\_\{y\}\)\\theta\\ell\-2c\+2\\theta r\}\{b\}\.\(6\)Here0<β≤10<\\beta\\leq 1scales the mass of type\-yypatients relative to type\-xxpatients, and the superscriptD\\mathrm\{D\}denotes the disparate design\. The firm then chooses\(ρx,ρy\)\(\\rho\_\{x\},\\rho\_\{y\}\)to maximize expected profit in[eq\.1](https://arxiv.org/html/2608.13618#S3.E1)\.

Next, consider the equal\-accuracy design, under which the algorithm attains the same accuracy for both patient types,ρx=ρy≡ρ\\rho\_\{x\}=\\rho\_\{y\}\\equiv\\rho\. The resulting volumes of AI use for type\-xxand type\-yypatients are

dxE=\(2​ρ−1\)​b−2​c\+2​θ​rb​and​dyE=β⋅\(2​ρ−1\)​b−2​c\+2​θ​rb,\\displaystyle d\_\{x\}^\{\\mathrm\{E\}\}=\\frac\{\(2\\rho\-1\)b\-2c\+2\\theta r\}\{b\}\\text\{ and \}d\_\{y\}^\{\\mathrm\{E\}\}=\\beta\\cdot\\frac\{\(2\\rho\-1\)b\-2c\+2\\theta r\}\{b\},\(7\)where the superscriptE\\mathrm\{E\}denotes the equal\-accuracy design\. The firm then choosesρ\\rhoto maximize expected profit in[eq\.2](https://arxiv.org/html/2608.13618#S3.E2)\.

For ease of presentation, we define a thresholdℓ~\\widetilde\{\\ell\}such that the AI firm’s expected profit for a disparate algorithm \(which is strictly decreasing inℓ\\ell\) exceeds its expected profit for an equal\-accuracy algorithm whenℓ<ℓ~\\ell<\\widetilde\{\\ell\}\. Otherwise, the AI firm develops an equal\-accuracy algorithm in equilibrium\. The following proposition presents the equilibrium AI accuracy set by the firm\.

###### Proposition 1\.

Supposeβ⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}andmax⁡\{2​β​\(c−θ​r\)​κx​\(κx\+κy\)b⁡\(β⁡\(β\+2\)​κx−κy\),b​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2\}<f<min⁡\{κx2,b​κy2​β​\(b\+θ​ℓ\)\}\\max\\\{\\frac\{2\\beta\(c\-\\theta r\)\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\\beta\(\\beta\+2\)\\kappa\_\{x\}\-\\kappa\_\{y\}\)\},\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}\\\}<f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}\\\}\. There exists a unique cutoffℓ~\\widetilde\{\\ell\}in the feasible region such that:

1. \(a\)ifℓ<ℓ~\\ell<\\widetilde\{\\ell\}, the AI firm develops a disparate algorithm withρx∗=12\+fκx\\rho\_\{x\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{f\}\{\\kappa\_\{x\}\}andρy∗=12\+β​f​\(b\+θ​ℓ\)b​κy\\rho\_\{y\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}, andρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\};
2. \(b\)ifℓ≥ℓ~\\ell\\geq\\widetilde\{\\ell\}, the AI firm develops an equal\-accuracy algorithm withρ∗=12\+\(1\+β\)​fκx\+κy\\rho^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{\(1\+\\beta\)f\}\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\.

Recall that the AI firm earns a per\-use feeffwhenever the physician consults AI and incurs costκt​\(ρt−12\)2\\kappa\_\{t\}\(\\rho\_\{t\}\-\\tfrac\{1\}\{2\}\)^\{2\}to deliver accuracyρt\\rho\_\{t\}for type\-ttpatients\. Because improving accuracy for type\-yypatients is more costly \(κy\>κx\\kappa\_\{y\}\>\\kappa\_\{x\}\) and demand from that segment is smaller \(β≤1\\beta\\leq 1\), the firm optimally allocates more performance to type\-xxpatients in the disparate regime, soρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}\.

As liabilityℓ\\ellrises, physician demand for AI in type\-yycases becomes more sensitive to expected legal exposure, reducing the profitability of a disparate design\. Onceℓ\\ellexceedsℓ~\\widetilde\{\\ell\}, the firm optimally switches to an equal\-accuracy design, which removes the disparity\-triggered liability channel and helps preserve overall use\. Althoughρy∗\\rho\_\{y\}^\{\*\}increases withℓ\\ellwithin the disparate regime, it does not overtakeρx∗\\rho\_\{x\}^\{\*\}; before that point, the firm prefers to switch to an equal\-accuracy design\.

[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)highlights a key trade\-off: stronger liability pushes design toward fairness, but potentially at the cost of aggregate efficiency in accuracy investment\. Weak liability sustains a profit\-driven disparate design, whereas strong liability induces convergence to equal accuracy\.

## 5Managerial and Policy Insights

We now examine how the liability rule influences AI\-use disparity, appropriate AI use, and patient welfare\.

### 5\.1AI\-Use Disparity

Absent differential costs or constraints, one would expect physicians to rely on AI at similar rates across patient types\. In practice, however, AI use can differ systematically across groups\. We refer to such differences as*AI\-use disparity*—the gap between the share of type\-xxcases and the share of type\-yycases in which the physician uses AI\.

AI\-use disparity may arise under the Section 1557 rule\. Importantly, under certain conditions, the rule may fail to mitigate disparities originating from algorithm development and may instead reduce physicians’ incentives to use AI for disadvantaged patients\. As a result, the patients whom the rule is designed to protect may be less likely to benefit from AI\-assisted decision\-making\. This raises a key question: when does the rule exacerbate AI\-use disparity, and under what conditions can such disparity be narrowed? The following proposition provides insight into this question\. For ease of presentation, we define a thresholdflf\_\{l\}in the proof of[proposition2](https://arxiv.org/html/2608.13618#Thmproposition2)\.

###### Proposition 2\.

Supposeβ⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}andfl<f<min⁡\{κx2,κy4​β\}\.f\_\{l\}<f<\\min\\left\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{\\kappa\_\{y\}\}\{4\\beta\}\\right\\\}\.Then, type\-yyAI use is non\-monotone in liability: asℓ\\ellincreases, the physician uses AI for fewer type\-yypatients whenℓ<b⁡\(κy−4​β​f\)4​θ​β​f\\ell<\\frac\{b\(\\kappa\_\{y\}\-4\\beta f\)\}\{4\\theta\\beta f\}and for more type\-yypatients whenb⁡\(κy−4​β​f\)4​θ​β​f≤ℓ<ℓ~\\frac\{b\(\\kappa\_\{y\}\-4\\beta f\)\}\{4\\theta\\beta f\}\\leq\\ell<\\widetilde\{\\ell\}\. Whenℓ\>ℓ~\\ell\>\\widetilde\{\\ell\}, the firm switches to an equal\-accuracy design, in which case type\-yyAI use does not depend onℓ\\ell\.

![Refer to caption](https://arxiv.org/html/2608.13618v1/regionProp3.png)Figure 2:Parameter region in which type\-yyAI use is non\-monotone in liability \([proposition2](https://arxiv.org/html/2608.13618#Thmproposition2)\)\.Since CMS solicited comments on the proposed Section 1557 clinical algorithm provision in August 2022, a recurring concern has been that an increase in liabilityℓ\\ellcould induce physicians to pull back from clinical algorithms and thereby widen disparities in care\([Goodman et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib26), see, e\.g\.,\)\.[proposition2](https://arxiv.org/html/2608.13618#Thmproposition2)sharpens this point: an increase in liabilityℓ\\ellcan either exacerbate or mitigate AI\-use disparity, depending on its level\. The mechanism is a tug\-of\-war between two forces\. Downstream, higher liability discourages the physician’s AI use for type\-yypatients by directly raising the physician’s expected liability cost\. Upstream, higher liability increases the firm’s incentive to improve accuracy for type\-yypatients, which makes AI more attractive to the physician and can offset \(or reverse\) the deterrence effect\.

When liabilityℓ\\ellis small, an increase in its level causes a limited reduction in the physician’s AI use for type\-yypatients\. In this case, the firm has a small incentive to invest in type\-yyaccuracy\. Therefore, the direct deterrence channel dominates, and the physician’s AI use for type\-yypatients falls asℓ\\ellrises, thus amplifying the AI\-use disparity\. When liabilityℓ\\ellis large, the contraction in the physician’s AI use for type\-yypatients becomes salient for the firm, prompting greater investment in type\-yyaccuracy\. Because higher accuracy reduces the likelihood of inappropriate treatment and therefore reduces the physician’s expected liability cost, the physician’s AI use for type\-yypatients can increase asℓ\\ellrises, narrowing AI\-use disparity\.

The bounds onffin[proposition2](https://arxiv.org/html/2608.13618#Thmproposition2)delimit the range over which the two forces are comparable in magnitude\. Whenffis small, the per\-use revenue does not justify the higher cost of type\-yyaccuracy \(κx<κy\\kappa\_\{x\}<\\kappa\_\{y\}\), and the firm supplies an equal\-accuracy design, under which type\-yyuse does not vary withℓ\\ell\. Whenffis large,ρy∗\\rho\_\{y\}^\{\*\}responds strongly toℓ\\ell, and the resulting accuracy gains outweigh the deterrence effect at every liability level, so type\-yyuse increases monotonically inℓ\\ell\. Only over the intermediate range does the firm both prefer a disparate design and respond moderately enough for deterrence to dominate at low liability, which is what produces the turning point\.[fig\.2](https://arxiv.org/html/2608.13618#S5.F2)shows the parameter region in which non\-monotone AI use occurs\.

### 5\.2\(In\)Appropriate AI Use

Consider an altruistic physician whose only concern is a patient’s health outcome and costs \(that is,θ=0\\theta=0\)\. From[eqs\.4](https://arxiv.org/html/2608.13618#S4.E4)and[5](https://arxiv.org/html/2608.13618#S4.E5), the expected payoff of an altruistic physician who uses AI for an individual type\-ttpatient is given as follows:

Ual=\\displaystyle U^\{\\text\{al\}\}=\{𝒫⁡\(st=T1\)⋅Q2\|1⋅b\+𝒫⁡\(st=T2\)⋅Q2\|2⋅b−c,if​α≤1−ρt𝒫⁡\(st=T1\)⋅Q1\|1⋅b\+𝒫⁡\(st=T2\)⋅Q2\|2⋅b−c,if​1−ρt<α≤ρt𝒫⁡\(st=T1\)⋅Q1\|1⋅b\+𝒫⁡\(st=T2\)⋅Q1\|2⋅b−c,otherwise\\displaystyle\\begin\{cases\}\\mathcal\{P\}\(s\_\{t\}=T\_\{1\}\)\\cdot Q\_\{2\|1\}\\cdot b\+\\mathcal\{P\}\(s\_\{t\}=T\_\{2\}\)\\cdot Q\_\{2\|2\}\\cdot b\-c,&\\text\{if \}\\alpha\\leq 1\-\\rho\_\{t\}\\\\ \\mathcal\{P\}\(s\_\{t\}=T\_\{1\}\)\\cdot Q\_\{1\|1\}\\cdot b\+\\mathcal\{P\}\(s\_\{t\}=T\_\{2\}\)\\cdot Q\_\{2\|2\}\\cdot b\-c,&\\text\{if \}1\-\\rho\_\{t\}<\\alpha\\leq\\rho\_\{t\}\\\\ \\mathcal\{P\}\(s\_\{t\}=T\_\{1\}\)\\cdot Q\_\{1\|1\}\\cdot b\+\\mathcal\{P\}\(s\_\{t\}=T\_\{2\}\)\\cdot Q\_\{1\|2\}\\cdot b\-c,&\\text\{otherwise\}\\end\{cases\}\(8\)A comparison of the altruistic physician’s expected payoffs from following and disregarding AI’s recommendation leads to the following lemma\.

###### Lemma 3\.

Supposec<\(ρt−12\)​bc<\(\\rho\_\{t\}\-\\tfrac\{1\}\{2\}\)bfort∈\{x,y\}t\\in\\\{x,y\\\}\. For type\-ttpatients, an altruistic physician uses AI if and only if

\(1−ρt\)​b\+cb<α<ρt​b−cb\.\\frac\{\(1\-\\rho\_\{t\}\)b\+c\}\{b\}<\\alpha<\\frac\{\\rho\_\{t\}b\-c\}\{b\}\.Conditional on using AI, the physician follows the AI signal\.

We define a physician’s AI use for a specific patient typettas*appropriate*if it matches the behavior of an altruistic physician \(withθ=0\\theta=0\)\. If the physician uses AI for more patients than an altruistic physician would, we refer to this as*overuse*of AI; if the physician uses AI for fewer patients, we refer to it as*underuse*of AI\. The following proposition compares AI use under the two physician types\. We define a thresholdℓs\\ell\_\{s\}\(expression provided in the Appendix\) at which the physician’s AI use for type\-yypatients switches from overuse to underuse\.

###### Proposition 3\.

The physician’s use of AI relative to the clinically appropriate benchmark satisfies the following\.

- \(a\)When the AI firm supplies an equal\-accuracy algorithm, the physician may overuse AI for both patient types\.
- \(b\)Suppose the interior disparate design is optimal atℓ\\ell, withπy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\>0and0<r/ℓ<1/20<r/\\ell<1/2\. If12−β​f​\(b\+θ​ℓ\)b​κy=rℓ\\frac\{1\}\{2\}\-\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}=\\frac\{r\}\{\\ell\}, thenρy∗=1−rℓ\\rho\_\{y\}^\{\*\}=1\-\\frac\{r\}\{\\ell\}and the physician uses AI appropriately for type\-yypatients\. In this case,ρx∗=12\+fκx\\rho\_\{x\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{f\}\{\\kappa\_\{x\}\}and the physician overuses AI for type\-xxpatients\.
- \(c\)Supposeπy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\>0and a disparate design is optimal atℓ=ℓs\\ell=\\ell\_\{s\}\(specifically, conditions \([A4](https://arxiv.org/html/2608.13618#S0.E4)\) and \([A5](https://arxiv.org/html/2608.13618#S0.E5)\) in the Appendix are satisfied\)\. Ifβ⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}and2​β​\(c−θ​r\)​κx​\(κx\+κy\)b⁡\(β⁡\(β\+2\)​κx−κy\)<f<min⁡\{κx2,b​κy\+4​κy​θ​r2​b​β−κy​2​b​θ​r\+4​θ2​r2b​β\}\\frac\{2\\beta\(c\-\\theta r\)\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\\beta\(\\beta\+2\)\\kappa\_\{x\}\-\\kappa\_\{y\}\)\}<f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{b\\kappa\_\{y\}\+4\\kappa\_\{y\}\\theta r\}\{2b\\beta\}\-\\frac\{\\kappa\_\{y\}\\sqrt\{2b\\theta r\+4\\theta^\{2\}r^\{2\}\}\}\{b\\beta\}\\\}, as liabilityℓ\\ellincreases from a low level, the physician’s AI use for type\-yypatients is non\-monotone: she overuses AI for smallℓ\\ell, underuses AI for intermediateℓ\\ell, and overuses AI again for largeℓ\\ell\.777[fig\.3](https://arxiv.org/html/2608.13618#S5.F3)provides a numerical illustration of the parameter region in which tri\-phase non\-monotone AI use occurs\.

![Refer to caption](https://arxiv.org/html/2608.13618v1/regionProp4.png)Figure 3:Parameter region in which type\-yyAI use switches between overuse and underuse \([proposition3](https://arxiv.org/html/2608.13618#Thmproposition3)\(c\)\)\.[proposition3](https://arxiv.org/html/2608.13618#Thmproposition3)\(a\) highlights a basic tension\. An equal\-accuracy design removes the type\-yy\-specific liability channel in our setup, but it does not remove the physician’s private incentive to use the tool when use is rewarded\. When reimbursementrrremains in place, the physician’s marginal calculus can tilt toward reliance even when AI is not clinically warranted, leading to overuse for both patient types\. The broader point is that equalizing algorithmic performance does not, by itself, align deployment incentives; if liability no longer disciplines use under equal accuracy, reimbursement can become the dominant force shaping utilization\. This suggests a role for complementary instruments that better tie payment and accountability to clinical value\.

[proposition3](https://arxiv.org/html/2608.13618#Thmproposition3)\(b\) shows appropriate use for disadvantaged patients can arise even when the supplied algorithm has unequal accuracy across groups, provided reimbursement exactly offsets the physician’s expected liability exposure from relying on AI in type\-yycases\. The margin condition isr=\(1−ρy∗\)​ℓr=\(1\-\\rho\_\{y\}^\{\*\}\)\\ell\. Using the equilibrium expression forρy∗\\rho\_\{y\}^\{\*\}, this is equivalent torℓ=12−β​f​\(b\+θ​ℓ\)b​κy\\frac\{r\}\{\\ell\}=\\frac\{1\}\{2\}\-\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}, which characterizes the combinations of liability and reimbursement that restore appropriate use for type\-yypatients\.

We illustrate[proposition3](https://arxiv.org/html/2608.13618#Thmproposition3)\(c\) in[fig\.4](https://arxiv.org/html/2608.13618#S5.F4)\. The solid lines plot the belief cutoffs inα\\alphathat delimit when the physician uses AI for type\-yypatients; the dash\-dot lines report the corresponding cutoffs for an altruistic physician \(θ=0\\theta=0\)\. As liabilityℓ\\ellrises, equilibrium use for type\-yypatients can be non\-monotone: overuse at lowℓ\\ell, underuse at intermediateℓ\\ell, and overuse again whenℓ\\ellis large\. This pattern contrasts with the view that liability primarily discourages reliance and leads clinicians to abandon AI\([Goodman et al\. 2023](https://arxiv.org/html/2608.13618#bib.bib26), see, e\.g\.,\)\. The mechanism is a shifting balance between downstream deterrence and upstream design responses\. For smallℓ\\ell, reimbursement dominates expected liability, so the physician relies on the tool too often\. For intermediateℓ\\ell, liability becomes first\-order and suppresses reliance, generating underuse\. For sufficiently largeℓ\\ell, the firm is induced to supply an equal\-accuracy design, which relaxes the liability channel for type\-yypatients and restores the reimbursement\-driven incentive to use AI, again producing overuse\. Notably, in the low\-ℓ\\ellregion in which the physician overuses a disparate tool for type\-yypatients, increasingℓ\\ellcan widen utilization differences across patient types even as the policy is intended to protect the disadvantaged group\. The lesson is that equalizing measured performance is not an appropriate\-use guarantee; policy must also address the incentives governing clinicians’ reliance decisions\.

Figure 4:AI use for type\-yypatients: for a givenℓ\\ell, the physician uses AI whenα\\alphalies between the cutoffs\. Dash\-dot lines correspond to an altruistic physician \(θ=0\\theta=0\); solid lines correspond to an impurely altruistic physician\.
### 5\.3Effect of Mandating Equal Accuracy

Recent policy initiatives have placed growing weight on eliminating performance differences across protected groups\([White House 2023](https://arxiv.org/html/2608.13618#bib.bib60), e\.g\.,\)\. In our setting, however, requiring equal accuracy across patient types is not innocuous: it reshapes both the firm’s investment incentives and physicians’ deployment incentives, with direct consequences for patient welfare\. This section therefore asks a simple question: holding fixed the liability framework, what are the welfare effects of imposing an equal\-accuracy requirement?

We consider two policy routes to equal accuracy\. The first is a direct mandate that forces equal accuracy in cases in which the firm would otherwise choose a disparate design\. The second is an indirect approach that raises physician liability so that the firm endogenously prefers an equal\-accuracy design\. Because liability does not affect the firm’s accuracy choice conditional on an equal\-accuracy design, these two routes deliver the same equilibrium accuracy and thus the same expected patient welfare\. We therefore focus, without loss of generality, on the direct mandate and characterize how imposing equal accuracy changes equilibrium accuracy and welfare\.

LetWx​\(ρx∗\)W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)andWy​\(ρy∗\)W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)denote expected welfare for type\-xxand type\-yypatients, respectively, under the disparate design evaluated at the firm’s optimal accuracies\(ρx∗,ρy∗\)\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\. Letρm\\rho^\{m\}denote the accuracy level chosen under an equal\-accuracy requirement, even in cases in which the firm would otherwise prefer a disparate design\. DefineWx​\(ρm\)W\_\{x\}\(\\rho^\{m\}\)andWy​\(ρm\)W\_\{y\}\(\\rho^\{m\}\)as the corresponding welfare levels under the equal\-accuracy design\. Closed\-form expressions forWx​\(ρx∗\)W\_\{x\}\(\\rho\_\{x\}^\{\*\}\),Wy​\(ρy∗\)W\_\{y\}\(\\rho\_\{y\}^\{\*\}\),Wx​\(ρm\)W\_\{x\}\(\\rho^\{m\}\), andWy​\(ρm\)W\_\{y\}\(\\rho^\{m\}\)are provided in the proof of[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\. For ease of exposition, we define a thresholdf¯\\bar\{f\}by the indifference conditionWy​\(ρm\)=Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)=W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)evaluated atf=f¯f=\\bar\{f\}\.

###### Proposition 4\.

When equal accuracy is mandated:

- \(a\)The optimal AI accuracy satisfiesρy∗≤ρm≤ρx∗\\rho\_\{y\}^\{\*\}\\leq\\rho^\{m\}\\leq\\rho\_\{x\}^\{\*\}\.
- \(b\)The expected patient surplus remains unchanged for both type\-xxand type\-yypatients when the physician is fairly certain about the appropriate treatment \(that is, whenα\\alphais close to 0 or 1\)\. It worsens \(improves\) for type\-xx\(type\-yy\) patients when the physician is highly uncertain \(that is, whenα\\alphais close to1/21/2\)\. Otherwise, that is, when the uncertainty is moderate, type\-xx\(type\-yy\) patients are better \(worse\) off\.888The expressions for thresholds \(which are different for type\-xxand type\-yypatients\) that specify the range ofα\\alphain which patient welfare improves, worsens, and remains unchanged are provided in the proof of[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\.
- \(c\)Supposec<κy​\(\(2\+β\)​κx\+κy\)​θ​rκy​\(2​κx\+κy\)−β​κx2c<\\frac\{\\kappa\_\{y\}\(\(2\+\\beta\)\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\theta r\}\{\\kappa\_\{y\}\(2\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-\\beta\\kappa\_\{x\}^\{2\}\}andb\>4​c​κy​\(κx\+κy\)κx​\(κy\+β⁡\(κx\+2​κy\)\)b\>\\frac\{4c\\kappa\_\{y\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{\\kappa\_\{x\}\(\\kappa\_\{y\}\+\\beta\(\\kappa\_\{x\}\+2\\kappa\_\{y\}\)\)\}\. Whenℓ\\ellis small, mandating equal accuracy yields three regimes\. \(1\) Ifb​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2<f<2​c​κx​\(κx\+κy\)b⁡\(\(2\+β\)​κx\+κy\)\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}<f<\\frac\{2c\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\(2\+\\beta\)\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}, thenWx​\(ρm\)\>Wx​\(ρx∗\)W\_\{x\}\(\\rho^\{m\}\)\>W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)andWy​\(ρm\)<Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)<W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\. \(2\) If2​c​κx​\(κx\+κy\)b⁡\(\(2\+β\)​κx\+κy\)<f<f¯\\frac\{2c\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\(2\+\\beta\)\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}<f<\\bar\{f\}, thenWx​\(ρm\)<Wx​\(ρx∗\)W\_\{x\}\(\\rho^\{m\}\)<W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)andWy​\(ρm\)<Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)<W\_\{y\}\(\\rho\_\{y\}^\{\*\}\); atf=2​c​κx​\(κx\+κy\)b⁡\(\(2\+β\)​κx\+κy\)f=\\frac\{2c\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\(2\+\\beta\)\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}, type\-xxwelfare is unchanged, and atf=f¯f=\\bar\{f\}, type\-yywelfare is unchanged\. \(3\) Finally, iff¯<f<min⁡\{κx2,b​κy2​β​\(b\+θ​ℓ\)\}\\bar\{f\}<f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}\\\}, thenWx​\(ρm\)<Wx​\(ρx∗\)W\_\{x\}\(\\rho^\{m\}\)<W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)andWy​\(ρm\)\>Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)\>W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\.999Comparisons of patient welfare over a wide range of\(β,ℓ,f\)\(\\beta,\\ell,f\)values are provided in Section[OA1\. Effect of Mandating Equal Accuracy on Patient Welfare \(\)](https://arxiv.org/html/2608.13618#S0.SSx1)of the Online Appendix\.

[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\(a\) shows imposing an equal\-accuracy requirement reshuffles performance across groups: relative to the firm’s preferred disparate design, accuracy falls for type\-xxpatients \(soρm≤ρx∗\\rho^\{m\}\\leq\\rho\_\{x\}^\{\*\}\) and rises for type\-yypatients \(soρm≥ρy∗\\rho^\{m\}\\geq\\rho\_\{y\}^\{\*\}\)\. A natural first reaction is therefore “type\-yygains, type\-xxloses\.”[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\(b\) cautions against that conclusion\. Whether a given patient benefits depends not only on accuracy but also on how the requirement changes the physician’s use of AI along the margin\.

The mechanism works through utilization\. By raising accuracy for type\-yypatients and lowering it for type\-xxpatients, the equal\-accuracy requirement expands AI use for some type\-yypatients and contracts AI use for some type\-xxpatients\. For the newly treated type\-yymargin, welfare can fall when physician uncertainty is moderate: the incremental health improvement from relying on AI is then small, while the fixed cost of using AI,cc, is paid whenever the physician deploys the tool\. For the dropped type\-xxmargin, welfare can rise for the symmetric reason: these patients forgo a modest accuracy benefit but avoid incurring the costcc\. In short, an equal\-accuracy requirement can move patients onto \(or off\) AI precisely where the accuracy gains are too small \(or too large\) relative to the use cost\.

Having established that welfare can move in either direction at the utilization margin, we turn to aggregate welfare by patient type\.[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\(c\) shows that whenℓ\\ellis small, an equal\-accuracy requirement can reduce aggregate welfare for both groups when the per\-use paymentfflies in an intermediate range\. Two forces drive this outcome\. First, the accuracy adjustment required to equalize performance is inherently asymmetric: because improving type\-yyaccuracy is more expensive, parity is achieved primarily by cutting type\-xxaccuracy, soρx∗−ρm\>ρm−ρy∗\\rho\_\{x\}^\{\*\}\-\\rho^\{m\}\>\\rho^\{m\}\-\\rho\_\{y\}^\{\*\}\. Second, the requirement shifts utilization incentives\. For type\-yypatients, higher accuracy raises welfare directly but also expands AI use, which can push deployment beyond the clinical margin; both effects strengthen withff, and the overuse channel can dominate whenffis moderate\. For type\-xxpatients, reduced use can mitigate overuse, but the loss from lower accuracy becomes increasingly important asffinduces larger cuts\. Whenffis neither too small to matter nor large enough to generate substantial type\-yyaccuracy gains, the combination of a sizeable decline in type\-xxaccuracy and expanded \(and potentially excessive\) use for type\-yypatients can leave both groups worse off\.

Whenffis sufficiently large, the balance shifts\. The equal\-accuracy requirement then induces a substantial reallocation of accuracy from type\-xxto type\-yy, yielding the intuitive aggregate ordering:Wx​\(ρm\)<Wx​\(ρx∗\)W\_\{x\}\(\\rho^\{m\}\)<W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)whileWy​\(ρm\)\>Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)\>W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\. At the same time, as[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\(b\) emphasizes, these aggregate comparisons can mask heterogeneity: some type\-yypatients can still be worse off under the requirement because expanded reliance on AI occurs precisely where the clinical gains are small relative to the use cost\.

Finally, whenffis sufficiently small, welfare differences are driven less by accuracy and more by utilization\. In this region, both gapsρx∗−ρm\\rho\_\{x\}^\{\*\}\-\\rho^\{m\}andρm−ρy∗\\rho^\{m\}\-\\rho\_\{y\}^\{\*\}are small, so the equal\-accuracy requirement mainly shifts the frequency of use: it reduces AI use for type\-xxpatients and increases it for type\-yypatients\. Type\-xxpatients can therefore benefit from fewer exposures to a low\-accuracy tool on the margin, whereas type\-yypatients can be harmed because the increase in use is not accompanied by a commensurate improvement in accuracy\. In this case, we may haveWx​\(ρm\)\>Wx​\(ρx∗\)W\_\{x\}\(\\rho^\{m\}\)\>W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)yetWy​\(ρm\)<Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)<W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\.

Next, we turn to a brief discussion of an alternative setting in which patient utility accounts for potential liability transfers\. Under this formulation, the patient’s utility and the physician’s payoff are updated according to[tables4](https://arxiv.org/html/2608.13618#S5.T4)and[5](https://arxiv.org/html/2608.13618#S5.T5)\.

Table 4:The Patient’s Expected Utility When the Physician Uses AI and Liability Transfers Are IncludedTable 5:The Physician’s Expected Payoff When Treating an Individual Patient Using AI and Liability Transfers Are IncludedNote that ifθ<1\\theta<1, the physician may use AI solely to exploit the patient’s financial gain from the liability transfer, rather than to improve clinical outcomes\. To rule out this uninteresting regime, we impose the parametric restrictionθ\>1\\theta\>1\. Under this condition, the physician’s payoff matrix in[table5](https://arxiv.org/html/2608.13618#S5.T5)is structurally isomorphic to that in[table3](https://arxiv.org/html/2608.13618#S3.T3), with the weight on the liability term updated fromθ\\thetatoθ−1\\theta\-1\. Consequently,[lemma2](https://arxiv.org/html/2608.13618#Thmlemma2)continues to hold directly by updating the liability\-adjusted selfishness parameter toθ−1\\theta\-1, and the subsequent analytical insights follow analogously\.

## 6Model Extensions

In this section, we present three extensions to our base model\. In the first extension, the AI firm also chooses the per\-use price in addition to the two accuracy levels\. In the second extension, we explore the liability level that maximizes aggregate patient welfare\. Finally, we consider patient\-type\-dependent priors for the physician\. Proofs of the results in this section are in the Online Appendix \(Sections[OA2\. AI Firm’s Pricing Decision \(\)](https://arxiv.org/html/2608.13618#S0.SSx2),[OA3\. Welfare\-Maximizing Liability \(\)](https://arxiv.org/html/2608.13618#S0.SSx3), and[OA4\. Type\-Specific Priors \(\)](https://arxiv.org/html/2608.13618#S0.SSx4)\)\.

### 6\.1AI Firm’s Pricing Decision

This section endogenizes the per\-use feeffin an extension in which patients pay for AI directly, soc=fc=f\. This structure captures emerging usage\-based arrangements in which providers charge an out\-of\-pocket fee for AI\-enhanced services\. For example, RadNet \(a large diagnostic imaging company\) launched its Enhanced Breast Cancer Detection program in 2023 at a $60 out\-of\-pocket fee for an AI mammography read and has since reduced the fee to $40\([Cheatham 2024](https://arxiv.org/html/2608.13618#bib.bib6)\)\. The timing parallels the baseline model: the firm first chooses accuracy levels; it then sets the feeff; the physician observes\(ρx,ρy,f\)\(\\rho\_\{x\},\\rho\_\{y\},f\)when deciding whether to use AI for a given patient\. All other assumptions are unchanged\. We summarize the core implications here and defer the technical derivations and additional comparative\-statics details to Section[OA2\. AI Firm’s Pricing Decision \(\)](https://arxiv.org/html/2608.13618#S0.SSx2)of the Online Appendix\.

###### Proposition 5\.

Letℓ†\\ell^\{\\dagger\}denote the design\-switch cutoff at which the firm is indifferent between supplying a disparate design and an equal\-accuracy design\. Define

D⁡\(ℓ\):=2​b2​\(β2​κx\+κy\)−4​b​κx​\(\(β\+1\)​κy−β2​θ​ℓ\)\+2​β2​κx​θ2​ℓ2D\(\\ell\):=2b^\{2\}\\\!\\left\(\\beta^\{2\}\\kappa\_\{x\}\+\\kappa\_\{y\}\\right\)\-4b\\kappa\_\{x\}\\\!\\left\(\(\\beta\+1\)\\kappa\_\{y\}\-\\beta^\{2\}\\theta\\ell\\right\)\+2\\beta^\{2\}\\kappa\_\{x\}\\theta^\{2\}\\ell^\{2\}and the common factors𝒯:=θ⁡\(β​ℓ−2​\(β\+1\)​r\)/D⁡\(ℓ\)\\mathcal\{T\}:=\{\\theta\\left\(\\beta\\ell\-2\(\\beta\+1\)r\\right\)\}/\{D\(\\ell\)\}and𝒮:=θ​r/\(2​\(κx\+κy\)−b⁡\(β\+1\)\)\\mathcal\{S\}:=\{\\theta r\}/\\\!\\left\(2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-b\(\\beta\+1\)\\right\)\. There exist parameter values such that the equilibrium accuracies are interior,1/2<ρ∗,ρx∗,ρy∗<11/2<\\rho^\{\*\},\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}<1, and the following holds:

\(a\) Ifℓ<ℓ†\\ell<\\ell^\{\\dagger\}, the firm chooses a disparate design with

ρx∗=12\+b​κy​𝒯,ρy∗=12\+β​κx​\(b\+θ​ℓ\)​𝒯,fD=b​κx​κy​𝒯\.\\rho\_\{x\}^\{\*\}=\\frac\{1\}\{2\}\+b\\kappa\_\{y\}\\mathcal\{T\},\\qquad\\rho\_\{y\}^\{\*\}=\\frac\{1\}\{2\}\+\\beta\\kappa\_\{x\}\(b\+\\theta\\ell\)\\,\\mathcal\{T\},\\qquad f^\{\\mathrm\{D\}\}=b\\kappa\_\{x\}\\kappa\_\{y\}\\mathcal\{T\}\.Moreover,ρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}, and there are parameter values for which bothfDf^\{\\mathrm\{D\}\}andρx∗\\rho\_\{x\}^\{\*\}are non\-monotone inℓ\\ell\.

\(b\) Ifℓ≥ℓ†\\ell\\geq\\ell^\{\\dagger\}, the firm supplies an equal\-accuracy design with

ρ∗=12\+\(β\+1\)​𝒮andfE=\(κx\+κy\)​𝒮\.\\rho^\{\*\}=\\frac\{1\}\{2\}\+\(\\beta\+1\)\\,\\mathcal\{S\}\\qquad\\text\{and\}\\qquad f^\{\\mathrm\{E\}\}=\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\,\\mathcal\{S\}\.

A key implication of[proposition5](https://arxiv.org/html/2608.13618#Thmproposition5)is that endogenizing the per\-use fee can make both price and utilization respond non\-monotonically to liability\. Whenℓ\\ellrises from a low level, expected utilization among type\-yypatients falls, weakening marginal demand; the firm then finds it profitable to cut the fee and to reallocate investment toward improvingρy∗\\rho\_\{y\}^\{\*\}to sustain type\-yyuptake\. The lower fee, in turn, expands type\-xxusage, which allows the firm to economize onρx∗\\rho\_\{x\}^\{\*\}while maintaining adoption\. Asℓ\\ellbecomes larger, however, the return to improving type\-yyaccuracy strengthens and type\-yyutilization rebounds; the firm can then raise the fee, which dampens type\-xxdemand and makes it attractive to increaseρx∗\\rho\_\{x\}^\{\*\}to restore type\-xxuptake\.

We also find that, within the disparate\-design regime, type\-yyutilization is non\-monotone inℓ\\ellfor some parameter values: it decreases whenℓ\\ellis below a threshold, but increases onceℓ\\ellexceeds that threshold\. Thus, the non\-monotonicity in AI use for type\-yypatients persists even when the firm endogenizes the per\-use feeffand patients bear that fee, so thatc=fc=f\. As in the main model, an equal\-accuracy requirement can still reduce welfare for both patient types when the AI firm endogenizes the per\-use price\.

Next, we highlight two additional insights from the model with endogenous pricing\. First, when the AI firm endogenously chooses its per\-use priceff, the equilibrium in which only type\-xxpatients are served, which arises in the baseline model whenℓ\\ellis large andβ\\betais moderate, no longer occurs\. Instead, both patient types are served, and an equal\-accuracy AI outcome can be sustained\. As a result, both the firm’s revenue and AI equity improve\. The additional pricing instrument allows the firm to profitably serve some type\-yypatients while preserving revenue from type\-xxpatients through the joint choice of price and accuracy\. Moreover, equal\-accuracy AI becomes easier to sustain profitably because it mitigates the physician’s liability concerns, encourages greater AI use, and allows the firm to further expand utilization through pricing\.

Second, whenffis endogenous, the welfare of type\-xxpatients is also affected by liability\. In particular, when liability is low, mandating equal accuracy can reduce type\-xxpatients’ welfare relative to the case with exogenousff\. The reason is that, when liability is low, the equilibrium accuracy for type\-xxpatients can be high; see[proposition5](https://arxiv.org/html/2608.13618#Thmproposition5)\. In that case, an equal\-accuracy requirement may substantially reduce type\-xxaccuracy, thereby lowering type\-xxwelfare\.

### 6\.2Aggregate\-Welfare\-Maximizing Liability

This subsection studies how the liability levelℓ\\ellshould be set to maximize aggregate expected welfare\. Two features of the model make the problem unusually sharp\. First, holding the firm’s AI design fixed, expected total welfare for type\-xxpatients under a disparate design does not depend onℓ\\ell, and aggregate welfare under an equal\-accuracy design does not depend onℓ\\ell\. Second,[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)implies that increasingℓ\\ellcan change the firm’s preferred design, inducing a switch from a disparate design to an equal\-accuracy design at a thresholdℓ~\\widetilde\{\\ell\}\. As a result, liability affects aggregate welfare primarily through whether it triggers this design transition\. We present the main economic intuition below and refer readers to Section[OA3\. Welfare\-Maximizing Liability \(\)](https://arxiv.org/html/2608.13618#S0.SSx3)of the Online Appendix for computational details and supplementary parameterizations\.

Across the parameterizations we examine numerically, the welfare\-maximizing liability lies at the boundary of this switch\. Specifically, it is either the largestℓ\\ellfor which the firm still prefers a disparate design \(see[fig\.5\(a\)](https://arxiv.org/html/2608.13618#S6.F5.sf1)\) or the smallestℓ\\ellfor which the firm prefers an equal\-accuracy design \(see[fig\.5\(b\)](https://arxiv.org/html/2608.13618#S6.F5.sf2)\)\. Thus, from the perspective of aggregate welfare, choosingℓ\\ellamounts to choosing the design regime—keeping the system in the disparate\-design region or inducing the shift to equal accuracy\. Throughout, aggregate welfare refers to total patient health surplus, summed across both patient groups\. We do not interpret it as full social welfare: doing so would require a common criterion encompassing patient health surplus, physician utility, firm profit, and any deadweight legal or compliance costs\. We regard a full social\-welfare analysis as a natural direction for future work\.

\(a\)Case in which welfare is higher under a disparate design\.\(b\)Case in which welfare is higher under an equal\-accuracy design\.
Figure 5:Aggregate welfare as a function of liabilityℓ\\ell\. The upward\-sloping segment is welfare under the firm’s preferred disparate design; the flat segment is welfare once the firm switches to an equal\-accuracy design, which does not vary withℓ\\ell\.
### 6\.3Type\-Specific Priors

In this section, we allow physicians to hold type\-specific priors about whether treatmentT1T\_\{1\}is appropriate\. This extension addresses the concern that imposingαx=αy\\alpha\_\{x\}=\\alpha\_\{y\}may understate human\-driven disparities in baseline clinical beliefs\. Specifically, we assumeαx∼U⁡\(0,α¯\)\\alpha\_\{x\}\\sim U\(0,\\overline\{\\alpha\}\)andαy∼U⁡\(α¯,1\)\\alpha\_\{y\}\\sim U\(\\underline\{\\alpha\},1\), whereα¯<1/2<α¯\\underline\{\\alpha\}<1/2<\\overline\{\\alpha\}\. All other assumptions remain as in the main model\. In particular, we maintain parameter restrictions under which, at every optimum characterized in this extension \(and, under the disparate design, at every liability level up to the design\-switch cutoff\), demand for each patient type and the firm’s profit are strictly positive, and the optimal accuracies are interior in\(1/2,1\)\(1/2,1\)\. The numerical configurations reported in this extension satisfy these restrictions\. Throughout this extension, we definef~:=f/α¯\\tilde\{f\}:=f/\\overline\{\\alpha\}andβ~:=β​α¯/\(1−α¯\)\\tilde\{\\beta\}:=\\beta\\overline\{\\alpha\}/\(1\-\\underline\{\\alpha\}\)for the density\-adjusted fee and composition\. Becauseβ∈\(0,1\)\\beta\\in\(0,1\)as in the main model, we assumeβ~<1\\tilde\{\\beta\}<1andα¯\+α¯\>1\\overline\{\\alpha\}\+\\underline\{\\alpha\}\>1\.

We summarize our key findings and relegate our technical details to Section[OA4\. Type\-Specific Priors \(\)](https://arxiv.org/html/2608.13618#S0.SSx4)of the Online Appendix\. The physician’s optimal AI\-use rule retains the same threshold form as in the baseline: AI is used only for an interior range of priors, and conditional on use, the physician follows the AI signal\. Thus, downstream behavior is structurally unchanged, although the relevant thresholds shift with the group\-specific prior distributions\. Aggregating over these priors changes equilibrium AI\-use volumes, but the firm’s design problem remains qualitatively the same\. The AI firm still chooses between a disparate design \(ρx\>ρy\\rho\_\{x\}\>\\rho\_\{y\}\) and an equal\-accuracy design \(ρx=ρy\\rho\_\{x\}=\\rho\_\{y\}\), and there exists a liability cutoffℓ~\\widetilde\{\\ell\}such that the firm prefers a disparate design forℓ<ℓ~\\ell<\\widetilde\{\\ell\}and an equal\-accuracy design otherwise\. Comparative statics with respect to liability and reimbursement therefore continue to follow the baseline logic over the parameter regions we characterize in the Online Appendix: heterogeneous priors change the size of the adoption region rather than the underlying mechanism\.

The welfare analysis carries over, with the integration limits adjusted to reflect the new prior supports, though the incidence of the welfare loss shifts\. The main qualitative insights remain robust: at low levels, liability can reduce AI use for type\-yypatients, whereas at higher levels it can increase use through stronger upstream accuracy incentives; and an equal\-accuracy mandate can still induce overuse and reduce welfare\. In short, allowingαx≠αy\\alpha\_\{x\}\\neq\\alpha\_\{y\}preserves the use\-distortion mechanism, and an equal\-accuracy mandate can again harm both patient groups over a substantial region of the parameter space\.

## 7Concluding Remarks

Policymakers increasingly seek to address uneven algorithm performance across patient groups, yet the primary levers available in practice often operate at deployment: hospitals and physicians remain accountable for outcomes even when they rely on clinical decision\-support tools\. This paper studies the equilibrium consequences of such deployment\-facing accountability by linking downstream physician reliance to upstream firm design\. In a model with an AI firm and a physician, the Section 1557 clinical algorithm provision is captured as an asymmetric liability exposure that is triggered when reliance on a tool with unequal performance leads to inappropriate treatment for disadvantaged patients\. Regulating use can reshape design, and regulating design can reshape use; welfare depends on the joint equilibrium\.

Liability intended to protect disadvantaged patients can instead reduce their access to AI\. By increasing the expected cost of relying on AI for type\-yypatients, liability can induce physicians to use AI less for the very group the policy aims to safeguard, even when the same tool would be deployed for type\-xxpatients\. The relationship between liability and use is also non\-monotone\. As liability rises, it can initially suppress reliance on AI for disadvantaged patients, but beyond a threshold it can induce the firm to reallocate investment toward their accuracy, improving performance and expanding adoption\. A single instrument can therefore produce underuse at low levels of liability and renewed overuse at high ones, with a corrective range in between, depending on how strongly it feeds back to design incentives\.

An equal\-accuracy requirement is not an appropriate\-use guarantee\. Requiring equal accuracy across patient types necessarily reallocates model quality across groups; because improving type\-yyaccuracy is more costly, parity is often achieved largely by reducing type\-xxaccuracy\. Equalizing measured performance also relaxes the liability channel that previously discouraged reliance for disadvantaged patients, while reimbursement continues to reward use\. The result is a utilization response that can move patients onto \(or off\) AI precisely where clinical gains are small relative to the fixed cost of use\. The mandate can therefore reduce aggregate welfare for both groups over parameter regions we characterize analytically: type\-xxpatients lose from a substantial accuracy decline, whereas type\-yypatients can be harmed by expanded reliance on a still\-imperfect tool\.

When deployment incentives are shaped by reimbursement and accountability, standards aimed at algorithm performance need to be paired with instruments that govern use\. In our setting, liability standards that encourage firms to invest in disadvantaged\-group accuracy need not align physicians’ deployment choices with clinical value\. Several complementary levers are available: reimbursement rules that attenuate incentives for indiscriminate use, auditing and monitoring requirements aimed at clinically meaningful endpoints, and other accountability mechanisms that discipline overuse\. Without them, better measured performance need not translate into better patient welfare\.

Although the motivating application is healthcare, the mechanism is broader: whenever professionals remain accountable for decisions informed by algorithms, deployment\-facing accountability can generate feedback from use to design and back again\. Analogous forces arise when judges rely on risk scores, when lenders and managers use credit and screening systems, and when employers deploy hiring tools\. In each case, performance differences across groups can trigger legal or reputational exposure and thereby distort equilibrium reliance\. Future work could extend the analysis to richer organizational environments \(for example, hospitals with multiple clinicians\), alternative contracting and pricing arrangements, and empirical measurement of the predicted non\-monotone responses of both adoption and design to liability exposure\.

Our analysis also points to several extensions that are especially relevant for AI\. First, once an AI system has been trained, its marginal cost of use is conceivably low, whereas obtaining a second opinion from a human expert can entail a nontrivial cost\. Allowing decision makers to choose between AI assistance and human peer review would introduce an additional margin of substitution and raise a distinct set of questions that we leave for future research\. Second, clinical AI systems can be retrained as new data arrive\. If liability discourages use for disadvantaged patients, those patients may become underrepresented in future training data, reducing subsequent accuracy and reinforcing the disparity the policy seeks to mitigate\. Third, the AI signal may be more accurate than the physician’s independent assessment for many patients\. When physicians can adjust their own diagnostic effort, greater reliance on AI may reduce independent clinical scrutiny and create a new moral\-hazard problem\. Future work could also examine institutional arrangements that jointly determine reimbursement, pricing, utilization costs, and liability; allocate responsibility across physicians, hospitals, and developers through shared liability or ex ante subsidies; and link reputational returns to realized performance over time\. A full social\-welfare analysis is a further direction: treating our health\-surplus measure as social welfare would require a common criterion spanning patient surplus, physician utility, firm profit, and deadweight legal and compliance costs\.

## Funding and Competing Interests

All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non\-financial interest in the subject matter or materials discussed in this manuscript\. The authors have no funding to report\.

## Appendix

Proof of[lemma1](https://arxiv.org/html/2608.13618#Thmlemma1)\. Using the physician’s expected payoff when not using AI \(from[eq\.3](https://arxiv.org/html/2608.13618#S4.E3)\) and when using AI \(from[eq\.4](https://arxiv.org/html/2608.13618#S4.E4)\), we compute the differenceUx−UU\_\{x\}\-Uwhich is given by

Ux−U=\\displaystyle U\_\{x\}\-U=\{θ​r−c,if​α≤1−ρx\(ρx\+α−1\)⋅b−c\+θ​r,if​1−ρx<α≤1/2\(ρx−α\)⋅b−c\+θ​r,if​1/2<α≤ρxθ​r−c,otherwise\.\\displaystyle\\begin\{cases\}\\theta r\-c,&\\text\{if \}\\alpha\\leq 1\-\\rho\_\{x\}\\\\ \(\\rho\_\{x\}\+\\alpha\-1\)\\cdot b\-c\+\\theta r,&\\text\{if \}1\-\\rho\_\{x\}<\\alpha\\leq 1/2\\\\ \(\\rho\_\{x\}\-\\alpha\)\\cdot b\-c\+\\theta r,&\\text\{if \}1/2<\\alpha\\leq\\rho\_\{x\}\\\\ \\theta r\-c,&\\text\{otherwise\.\}\\end\{cases\}\(A1\)
Note thatθ​r−c<0\\theta r\-c<0; therefore, the physician does not use AI forα≤1−ρx\\alpha\\leq 1\-\\rho\_\{x\}and forα\>ρx\\alpha\>\\rho\_\{x\}\. In the second case of[eq\.A1](https://arxiv.org/html/2608.13618#S0.E1), that is, if1−ρx<α≤1/21\-\\rho\_\{x\}<\\alpha\\leq 1/2, the physician does not use AI whenα≤\(1−ρx\)​b\+c−θ​rb\\alpha\\leq\\frac\{\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\}\{b\}and uses AI whenα≥\(1−ρx\)​b\+c−θ​rb\\alpha\\geq\\frac\{\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\}\{b\}\. In the third case of[eq\.A1](https://arxiv.org/html/2608.13618#S0.E1), that is, if1/2<α≤ρx1/2<\\alpha\\leq\\rho\_\{x\}, the physician does not use AI whenα\>ρx​b−\(c−θ​r\)b\\alpha\>\\frac\{\\rho\_\{x\}b\-\(c\-\\theta r\)\}\{b\}and uses AI whenα≤ρx​b−\(c−θ​r\)b\\alpha\\leq\\frac\{\\rho\_\{x\}b\-\(c\-\\theta r\)\}\{b\}\. Therefore, we have

- •Whenc−θ​r<\(ρx−1/2\)​bc\-\\theta r<\(\\rho\_\{x\}\-1/2\)b, we have\(1−ρx\)​b\+c−θ​rb<1/2\\frac\{\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\}\{b\}<1/2andρx​b−\(c−θ​r\)b\>1/2\\frac\{\\rho\_\{x\}b\-\(c\-\\theta r\)\}\{b\}\>1/2\. Then, the physician does not use AI ifα≤\(1−ρx\)​b\+c−θ​rb\\alpha\\leq\\frac\{\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\}\{b\}, uses AI if\(1−ρx\)​b\+c−θ​rb<α≤ρx​b−\(c−θ​r\)b\\frac\{\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\}\{b\}<\\alpha\\leq\\frac\{\\rho\_\{x\}b\-\(c\-\\theta r\)\}\{b\}, and does not use AI again ifα\>ρx​b−\(c−θ​r\)b\\alpha\>\\frac\{\\rho\_\{x\}b\-\(c\-\\theta r\)\}\{b\}\.
- •Whenc−θ​r≥\(ρx−1/2\)​bc\-\\theta r\\geq\(\\rho\_\{x\}\-1/2\)b, we have\(1−ρx\)​b\+c−θ​rb≥1/2\\frac\{\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\}\{b\}\\geq 1/2andρx​b−\(c−θ​r\)b≤1/2\\frac\{\\rho\_\{x\}b\-\(c\-\\theta r\)\}\{b\}\\leq 1/2\. Then, the physician does not use AI for all patientsα∈\[0,1\]\\alpha\\in\[0,1\]\.

Under the assumption that the physician uses AI for at least some type\-yypatients \(that is,dyD\>0\{d\_\{y\}^\{\\mathrm\{D\}\}\}\>0, equivalently,c−θ​r<\(ρy−12\)​b−\(1−ρy\)​θ​ℓc\-\\theta r<\(\\rho\_\{y\}\-\\frac\{1\}\{2\}\)b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\) and givenρx\>ρy\\rho\_\{x\}\>\\rho\_\{y\}, we havec−θ​r<\(ρx−1/2\)​bc\-\\theta r<\(\\rho\_\{x\}\-1/2\)b\. Therefore, we derive the physician’s AI use for type\-xxpatients in this lemma\.

Next, we check whether the physician will follow or reject the AI signal when using AI\. In the second and third cases of[eq\.A1](https://arxiv.org/html/2608.13618#S0.E1), the physician follows the AI signal regardless of the signal\. These are also the only scenarios in which the physician uses AI\. Therefore, the physician follows the AI signal for type\-xxpatients, whenever using AI\.*Q\.E\.D\.*

Proof of[lemma2](https://arxiv.org/html/2608.13618#Thmlemma2)\. Using the physician’s expected payoff when not using AI \(from[eq\.3](https://arxiv.org/html/2608.13618#S4.E3)\) and when using AI \(from[eq\.5](https://arxiv.org/html/2608.13618#S4.E5)\), we compute the differenceUy−UU\_\{y\}\-Uwhich is given by:

Uy−U=\\displaystyle U\_\{y\}\-U=\{θ​r−c−α⁡\(1−ρy\)​θ​ℓ,if​α≤\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ\(ρy\+α−1\)⋅b−c\+θ​r−\(1−ρy\)​θ​ℓ,if​\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ<α≤1/2\(ρy−α\)⋅b−c\+θ​r−\(1−ρy\)​θ​ℓ,if​1/2<α≤b​ρyb\+\(1−ρy\)​θ​ℓθ​r−c−\(1−α\)​\(1−ρy\)​θ​ℓ,otherwise\.\\displaystyle\\begin\{cases\}\\theta r\-c\-\\alpha\(1\-\\rho\_\{y\}\)\\theta\\ell,&\\text\{if \}\\alpha\\leq\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\\\\ \(\\rho\_\{y\}\+\\alpha\-1\)\\cdot b\-c\+\\theta r\-\(1\-\\rho\_\{y\}\)\\theta\\ell,&\\text\{if \}\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}<\\alpha\\leq 1/2\\\\ \(\\rho\_\{y\}\-\\alpha\)\\cdot b\-c\+\\theta r\-\(1\-\\rho\_\{y\}\)\\theta\\ell,&\\text\{if \}1/2<\\alpha\\leq\\frac\{b\\rho\_\{y\}\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\\\\ \\theta r\-c\-\(1\-\\alpha\)\(1\-\\rho\_\{y\}\)\\theta\\ell,&\\text\{otherwise\.\}\\end\{cases\}\(A2\)
Note thatθ​r−c−α⁡\(1−ρy\)​θ​ℓ<0\\theta r\-c\-\\alpha\(1\-\\rho\_\{y\}\)\\theta\\ell<0; therefore, the physician does not use AI forα≤\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ\\alpha\\leq\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\. In addition,θ​r−c−\(1−α\)​\(1−ρy\)​θ​ℓ<0\\theta r\-c\-\(1\-\\alpha\)\(1\-\\rho\_\{y\}\)\\theta\\ell<0, which implies the physician does not use AI forα\>b​ρyb\+\(1−ρy\)​θ​ℓ\\alpha\>\\frac\{b\\rho\_\{y\}\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}\.

In the second case of[eq\.A2](https://arxiv.org/html/2608.13618#S0.E2), that is, if\(1−ρy\)​\(b\+θ​ℓ\)b\+\(1−ρy\)​θ​ℓ<α≤1/2\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}<\\alpha\\leq 1/2, the physician does not use AI whenα≤\(1−ρy\)​\(b\+θ​ℓ\)\+c−θ​rb\\alpha\\leq\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}and uses AI whenα≥\(1−ρy\)​\(b\+θ​ℓ\)\+c−θ​rb\\alpha\\geq\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}\. In the third case of[eq\.A2](https://arxiv.org/html/2608.13618#S0.E2), that is, if1/2<α≤b​ρyb\+\(1−ρy\)​θ​ℓ1/2<\\alpha\\leq\\frac\{b\\rho\_\{y\}\}\{b\+\(1\-\\rho\_\{y\}\)\\theta\\ell\}, the physician does not use AI whenα\>ρy​b−\(1−ρy\)​θ​ℓ−c\+θ​rb\\alpha\>\\frac\{\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\}\{b\}and uses AI whenα≤ρy​b−\(1−ρy\)​θ​ℓ−c\+θ​rb\\alpha\\leq\\frac\{\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\}\{b\}\. Therefore, we have

- •Whenc−θ​r<\(ρy−12\)​b−\(1−ρy\)​θ​ℓc\-\\theta r<\(\\rho\_\{y\}\-\\frac\{1\}\{2\}\)b\-\(1\-\\rho\_\{y\}\)\\theta\\ell, we have\(1−ρy\)​\(b\+θ​ℓ\)\+c−θ​rb<1/2<ρy​b−\(1−ρy\)​θ​ℓ−c\+θ​rb\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}<1/2<\\frac\{\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\}\{b\}\. Then, the physician does not use AI ifα≤\(1−ρy\)​\(b\+θ​ℓ\)\+c−θ​rb\\alpha\\leq\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}, uses AI if\(1−ρy\)​\(b\+θ​ℓ\)\+c−θ​rb<α<ρy​b−\(1−ρy\)​θ​ℓ−c\+θ​rb\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}<\\alpha<\\frac\{\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\}\{b\}, and does not use AI again ifα\>ρy​b−\(1−ρy\)​θ​ℓ−c\+θ​rb\\alpha\>\\frac\{\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\}\{b\}\.
- •Whenc−θ​r\>\(ρy−12\)​b−\(1−ρy\)​θ​ℓc\-\\theta r\>\(\\rho\_\{y\}\-\\frac\{1\}\{2\}\)b\-\(1\-\\rho\_\{y\}\)\\theta\\ell, we have\(1−ρy\)​\(b\+θ​ℓ\)\+c−θ​rb\>1/2\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}\>1/2andρy​b−\(1−ρy\)​θ​ℓ−c\+θ​rb<1/2\\frac\{\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\}\{b\}<1/2\. Then, the physician does not use AI for all patientsα∈\[0,1\]\\alpha\\in\[0,1\]\.

Under the assumption that the physician uses AI for at least some type\-yypatients \(that is,dyD\>0\{d\_\{y\}^\{\\mathrm\{D\}\}\}\>0, equivalently,c−θ​r<\(ρy−12\)​b−\(1−ρy\)​θ​ℓc\-\\theta r<\(\\rho\_\{y\}\-\\frac\{1\}\{2\}\)b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\), we derive the physician’s AI use for type\-yypatients in this lemma\.

Next, we check whether the physician will follow or reject the AI signal when using AI\. In the second and third cases of[eq\.A2](https://arxiv.org/html/2608.13618#S0.E2), the physician follows the AI signal regardless of the signal\. These are also the only scenarios in which the physician might choose to use AI\. Therefore, the physician follows the AI signal for type\-yypatients whenever using AI\.*Q\.E\.D\.*

Proof of[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)\. \(a\) Suppose the AI firm supplies a disparate algorithm\. It setsρx\\rho\_\{x\}andρy\\rho\_\{y\}to maximize the profit in[eq\.1](https://arxiv.org/html/2608.13618#S3.E1), wheredxDd\_\{x\}^\{\\mathrm\{D\}\}anddyDd\_\{y\}^\{\\mathrm\{D\}\}are given in[eq\.6](https://arxiv.org/html/2608.13618#S4.E6)\. The objective is strictly concave inρx\\rho\_\{x\}andρy\\rho\_\{y\}, so the first\-order conditions are sufficient\. Setting∂πD/∂ρx=0\\partial\\pi^\{\\mathrm\{D\}\}/\\partial\\rho\_\{x\}=0and∂πD/∂ρy=0\\partial\\pi^\{\\mathrm\{D\}\}/\\partial\\rho\_\{y\}=0yieldsρx∗=12\+fκx\\rho\_\{x\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{f\}\{\\kappa\_\{x\}\}andρy∗=12\+β​f​\(b\+θ​ℓ\)b​κy\\rho\_\{y\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}\. The interior conditionsρx∗<1\\rho\_\{x\}^\{\*\}<1andρy∗<1\\rho\_\{y\}^\{\*\}<1are equivalent tof<κx/2f<\\kappa\_\{x\}/2andf<b​κy2​β​\(b\+θ​ℓ\)f<\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}, respectively\.

Next, to ensure that the firm earns a positive profit from serving type\-yypatients atρy∗\\rho\_\{y\}^\{\*\}, that is,πy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\>0, it suffices thatf\>b​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2f\>\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}\. This condition also guarantees that the physician uses AI for at least some type\-yypatients under the induced accuracy choice \(equivalently,dyD\|ρy=ρy∗\>0d\_\{y\}^\{\\mathrm\{D\}\}\|\_\{\\rho\_\{y\}=\\rho\_\{y\}^\{\*\}\}\>0\)\. Otherwise, the optimal choice setsρy\\rho\_\{y\}arbitrarily close to12\\frac\{1\}\{2\}\.

We first note that no strictly reverse\-disparate design \(ρy\>ρx\\rho\_\{y\}\>\\rho\_\{x\}\) can be optimal\. In that region the disparity\-triggered liability channel is inactive, so the objective is the no\-liability profit, which is strictly concave with unconstrained maximizersρx0=12\+fκx\\rho\_\{x\}^\{0\}=\\tfrac\{1\}\{2\}\+\\tfrac\{f\}\{\\kappa\_\{x\}\}andρy0=12\+β​fκy\\rho\_\{y\}^\{0\}=\\tfrac\{1\}\{2\}\+\\tfrac\{\\beta f\}\{\\kappa\_\{y\}\}\. Becauseκy\>κx\\kappa\_\{y\}\>\\kappa\_\{x\}andβ≤1\\beta\\leq 1, we havefκx\>β​fκy\\tfrac\{f\}\{\\kappa\_\{x\}\}\>\\tfrac\{\\beta f\}\{\\kappa\_\{y\}\}, soρx0\>ρy0\\rho\_\{x\}^\{0\}\>\\rho\_\{y\}^\{0\}and the unconstrained maximizer lies outside this region\. By strict concavity the constrained maximizer lies on the boundaryρx=ρy\\rho\_\{x\}=\\rho\_\{y\}, which is the equal\-accuracy design\. It therefore suffices to compare the disparate and equal\-accuracy designs\. We then show that whenever a disparate design is optimal, it must satisfyρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}\. Ifρx∗=ρy∗\\rho\_\{x\}^\{\*\}=\\rho\_\{y\}^\{\*\}under the disparate\-design best response, then the firm can instead supply the same accuracy level as an equal\-accuracy design, which eliminates the liability channel and weakly increases physician demand; henceπD​\(ρx∗,ρy∗\)<πE​\(ρx∗\)≤maxρ⁡πE​\(ρ\)=πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)<\\pi^\{\\mathrm\{E\}\}\(\\rho\_\{x\}^\{\*\}\)\\leq\\max\_\{\\rho\}\\pi^\{\\mathrm\{E\}\}\(\\rho\)=\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\. This contradicts optimality of a disparate design\. To preclude the reversalρx∗<ρy∗\\rho\_\{x\}^\{\*\}<\\rho\_\{y\}^\{\*\}, we invoke a continuity argument\. At the boundaryℓ=0\\ell=0, the optimal accuracies yield a positive gap,ρx∗−ρy∗=f⁡\(1/κx−β/κy\)\>0\\rho\_\{x\}^\{\*\}\-\\rho\_\{y\}^\{\*\}=f\(1/\\kappa\_\{x\}\-\\beta/\\kappa\_\{y\}\)\>0\. Becauseρx∗−ρy∗\\rho\_\{x\}^\{\*\}\-\\rho\_\{y\}^\{\*\}is continuous and strictly decreasing inℓ\\ell, the gap cannot change sign and become negative without passing through zero\. Since exact equalityρx∗=ρy∗\\rho\_\{x\}^\{\*\}=\\rho\_\{y\}^\{\*\}is strictly ruled out by our previous contradiction argument, it follows by the Intermediate Value Theorem thatρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}must hold strictly throughout the entire feasible domain\. Therefore, if the disparate design is optimal, we must haveρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}\.

\(b\) Now suppose the firm supplies an equal\-accuracy algorithm, soρx=ρy≡ρ\\rho\_\{x\}=\\rho\_\{y\}\\equiv\\rho\. Maximizing[eq\.2](https://arxiv.org/html/2608.13618#S3.E2)and setting∂πE​\(ρ\)/∂ρ=0\\partial\\pi^\{\\mathrm\{E\}\}\(\\rho\)/\\partial\\rho=0yieldsρ∗=12\+\(1\+β\)​fκx\+κy\\rho^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{\(1\+\\beta\)f\}\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\. The interior conditionρ∗<1\\rho^\{\*\}<1is equivalent tof<κx\+κy2​\(1\+β\)f<\\frac\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\{2\(1\+\\beta\)\}\. Note thatκx\+κy2​\(1\+β\)\>κx2\\frac\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\{2\(1\+\\beta\)\}\>\\frac\{\\kappa\_\{x\}\}\{2\}becauseκy\>β​κx\\kappa\_\{y\}\>\\beta\\kappa\_\{x\}; hence the standing conditionf<κx/2f<\\kappa\_\{x\}/2\(which ensuresρx∗<1\\rho\_\{x\}^\{\*\}<1\) already impliesf<κx\+κy2​\(1\+β\)f<\\frac\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\{2\(1\+\\beta\)\}, and the interior constraint for the equal\-accuracy design can be omitted\.

Next, we show that there exists a cutoffℓ~\\widetilde\{\\ell\}such that a disparate design is optimal forℓ<ℓ~\\ell<\\widetilde\{\\ell\}and an equal\-accuracy design is optimal forℓ≥ℓ~\\ell\\geq\\widetilde\{\\ell\}\. Asℓ→0\\ell\\to 0, the disparate\-design profit converges to the equal\-accuracy objective evaluated at potentially different accuracies across types, that is,πD​\(ρx,ρy\)\|ℓ→0→πE​\(ρx,ρy\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\},\\rho\_\{y\}\)\|\_\{\\ell\\to 0\}\\to\\pi^\{\\mathrm\{E\}\}\(\\rho\_\{x\},\\rho\_\{y\}\), while the equal\-accuracy design restricts the firm toρx=ρy\\rho\_\{x\}=\\rho\_\{y\}\. Because the feasible set under a disparate design weakly contains that under equal accuracy, we haveπD​\(ρx∗,ρy∗\)\|ℓ→0≥πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell\\to 0\}\\geq\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\), with strict inequality whenever the equality constraintρx=ρy\\rho\_\{x\}=\\rho\_\{y\}binds\.

Next, under a disparate design, the firm’s optimal profit is decreasing inℓ\\ell\. The type\-xxcomponent is independent ofℓ\\ell, so it suffices to study the type\-yycomponent\. By the envelope theorem,

dd​ℓ​πD​\(ρx∗,ρy∗\)=∂πD​\(ρx,ρy\)∂ℓ\|ρx=ρx∗,ρy=ρy∗=−2​θ​f​βb​\(1−ρy∗\)<0,\\frac\{d\}\{d\\ell\}\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)=\\left\.\\frac\{\\partial\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\},\\rho\_\{y\}\)\}\{\\partial\\ell\}\\right\|\_\{\\rho\_\{x\}=\\rho\_\{x\}^\{\*\},\\,\\rho\_\{y\}=\\rho\_\{y\}^\{\*\}\}=\-\\frac\{2\\theta f\\beta\}\{b\}\\,\(1\-\\rho\_\{y\}^\{\*\}\)<0,where the negative sign strictly holds in the interior region in whichρy∗<1\\rho\_\{y\}^\{\*\}<1\. By contrast,πE​\(ρ∗\)\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)does not depend onℓ\\ell\. BecauseπD​\(ρx∗,ρy∗\)−πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)is strictly decreasing inℓ\\ellunder this interior condition, there is at most one cutoffℓ~\\widetilde\{\\ell\}at whichπD​\(ρx∗,ρy∗\)=πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)=\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\), and the profit ranking switches at that point\. Outside this interior region \(whereρy∗≥1\\rho\_\{y\}^\{\*\}\\geq 1\), althoughπD​\(ρx∗,ρy∗\)−πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)becomes a convex parabola inℓ\\ellthat could theoretically admit a second crossing, such a region is structurally infeasible and can be disregarded, thereby establishing the unique design cutoff\.

We conclude the proof by establishing the existence of the cutoffℓ~\\widetilde\{\\ell\}within the feasible domain\. A sufficient condition forℓ~\\widetilde\{\\ell\}to be interior isπE​\(ρ∗\)\>πx​\(ρx∗\)\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\>\\pi\_\{x\}\(\\rho^\{\*\}\_\{x\}\), which simplifies toβ⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}andf\>2​β​\(c−θ​r\)​κx​\(κx\+κy\)b⁡\(β⁡\(β\+2\)​κx−κy\)f\>\\frac\{2\\beta\(c\-\\theta r\)\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\\beta\(\\beta\+2\)\\kappa\_\{x\}\-\\kappa\_\{y\}\)\}\. This parameter restriction ensures that the profit from type\-yypatients remains strictly positive \(πy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho^\{\*\}\_\{y\}\)\>0\) at the point of indifference whereπD​\(ρx∗,ρy∗\)=πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)=\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\(i\.e\., atℓ=ℓ~\\ell=\\widetilde\{\\ell\}\)\. Consequently, bothπy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho^\{\*\}\_\{y\}\)\>0andρy∗<1\\rho^\{\*\}\_\{y\}<1hold for allℓ<ℓ~\\ell<\\widetilde\{\\ell\}, where the latter inequality follows from our earlier result thatρy∗<ρx∗<1\\rho^\{\*\}\_\{y\}<\\rho^\{\*\}\_\{x\}<1whenever a disparate design strictly dominates\. Conversely, if this condition is violated, an interior cutoffℓ~\\widetilde\{\\ell\}fails to exist\. In that scenario, an equal\-accuracy design never becomes optimal asℓ\\ellincreases; instead, the firm continuously deploys a disparate algorithm while completely abandoning the type\-yymarket to serve type\-xxpatients exclusively\.*Q\.E\.D\.*

Proof of[proposition2](https://arxiv.org/html/2608.13618#Thmproposition2)\. Under the disparate design, AI use for type\-yypatients is given by[eq\.6](https://arxiv.org/html/2608.13618#S4.E6)asdyD=β⋅\(2​ρy∗−1\)​b−2​\(1−ρy∗\)​θ​ℓ−2​c\+2​θ​rbd\_\{y\}^\{\\mathrm\{D\}\}=\\beta\\cdot\\frac\{\(2\\rho\_\{y\}^\{\*\}\-1\)b\-2\(1\-\\rho\_\{y\}^\{\*\}\)\\theta\\ell\-2c\+2\\theta r\}\{b\}, whereρy∗=12\+β​f​\(b\+θ​ℓ\)b​κy\\rho\_\{y\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}from[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)\. NotedyDd\_\{y\}^\{\\mathrm\{D\}\}is convex inℓ\\ell, implying that it has at most one turning point\. Taking the first\-order derivative ofdyDd\_\{y\}^\{\\mathrm\{D\}\}with respect toℓ\\ellyields∂dyD∂ℓ\>0\\frac\{\\partial d\_\{y\}^\{\\mathrm\{D\}\}\}\{\\partial\\ell\}\>0if and only ifℓ\>ℓ¯:=b⁡\(κy−4​β​f\)4​θ​β​f\\ell\>\\bar\{\\ell\}:=\\frac\{b\(\\kappa\_\{y\}\-4\\beta f\)\}\{4\\theta\\beta f\}\. We requireℓ¯\>0\\bar\{\\ell\}\>0to make sure thatdyDd\_\{y\}^\{\\mathrm\{D\}\}has one unique turning point atℓ=ℓ¯\\ell=\\bar\{\\ell\}, which gives usf<κy4​βf<\\frac\{\\kappa\_\{y\}\}\{4\\beta\}\. Thus, holding the AI firm’s design fixed at the disparate regime, liability raises type\-yyAI use whenℓ\\ellexceedsℓ¯\\bar\{\\ell\}and lowers it whenℓ<ℓ¯\\ell<\\bar\{\\ell\}\.

To establish the claimed pattern aroundℓ¯\\bar\{\\ell\}in equilibrium, note that the parameter restrictions in[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)imply: \(i\)b​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2<f<min⁡\{κx2,b​κy2​β​\(b\+θ​ℓ\)\}\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}<f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}\\\}, \(ii\) the AI firm strictly prefers the disparate design atℓ=ℓ¯\\ell=\\bar\{\\ell\}\(that is,πD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)atℓ=ℓ¯\\ell=\\bar\{\\ell\}\), and \(iii\)β⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}andf\>2​β​\(c−θ​r\)​κx​\(κx\+κy\)b⁡\(β⁡\(β\+2\)​κx−κy\)f\>\\frac\{2\\beta\(c\-\\theta r\)\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\\beta\(\\beta\+2\)\\kappa\_\{x\}\-\\kappa\_\{y\}\)\}\.

We first establish conditions surrounding the restriction \(i\)\. We can substantiate thatf<b​κy2​β​\(b\+θ​ℓ\)\|ℓ=ℓ¯f<\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}\|\_\{\\ell=\\bar\{\\ell\}\}, which meansρy∗<1\\rho\_\{y\}^\{\*\}<1atℓ=ℓ¯\\ell=\\bar\{\\ell\}\(by[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)\), which also means thatρy∗<1\\rho\_\{y\}^\{\*\}<1must hold forℓ<ℓ¯\\ell<\\bar\{\\ell\}\(asρy∗\\rho\_\{y\}^\{\*\}is increasing inℓ\\ell\)\. Furthermore, givenf\>b​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2f\>\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}is equivalent toπy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\>0, we can omit the consideration off\>b​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2\|ℓ=ℓ¯f\>\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}\|\_\{\\ell=\\bar\{\\ell\}\}\. This is becauseπy​\(ρy∗\)\|ℓ=ℓ¯=\[πD​\(ρx∗,ρy∗\)−πE​\(ρ∗\)\]\|ℓ=ℓ¯\+\(πE​\(ρ∗\)−πx​\(ρx∗\)\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell=\\bar\{\\ell\}\}=\[\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\]\|\_\{\\ell=\\bar\{\\ell\}\}\+\(\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\-\\pi\_\{x\}\(\\rho\_\{x\}^\{\*\}\)\)\>0, where the inequality holds because the first term is greater than 0 \(we will guarantee this by discussing the restriction \(ii\) in what follows\) and the second term is also greater than 0 by the restriction \(iii\) above\. Moreover,∂πy​\(ρy∗\)∂ℓ=2​β​f​θb​\(ρy∗−1\)<0\\frac\{\\partial\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\}\{\\partial\\ell\}=\\frac\{2\\beta f\\theta\}\{b\}\\left\(\\rho\_\{y\}^\{\*\}\-1\\right\)<0wheneverρy∗<1\\rho\_\{y\}^\{\*\}<1, which holds on\[0,ℓ¯\]\[0,\\bar\{\\ell\}\]as just established\. Henceπy​\(ρy∗\)\|ℓ\>πy​\(ρy∗\)\|ℓ=ℓ¯\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell\}\>\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell=\\bar\{\\ell\}\}\>0for everyℓ<ℓ¯\\ell<\\bar\{\\ell\}, so the participation constraint is slack throughout\. Therefore, for the restriction \(i\), the only constraint isf<κx2f<\\frac\{\\kappa\_\{x\}\}\{2\}\.

We then establish conditions surrounding the restriction \(ii\)\. We obtain\[πD​\(ρx∗,ρy∗\)−πE​\(ρ∗\)\]\|ℓ=ℓ¯=A⋅f2\+β​f−3​κy16\[\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\]\|\_\{\\ell=\\bar\{\\ell\}\}=A\\cdot f^\{2\}\+\\beta f\-\\frac\{3\\kappa\_\{y\}\}\{16\}, whereA=κy−β⁡\(β\+2\)​κxκx​\(κx\+κy\)A=\\frac\{\\kappa\_\{y\}\-\\beta\(\\beta\+2\)\\kappa\_\{x\}\}\{\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\. Letting\[πD​\(ρx∗,ρy∗\)−πE​\(ρ∗\)\]\|ℓ=ℓ¯=0\[\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\]\|\_\{\\ell=\\bar\{\\ell\}\}=0yieldsf=faf=f\_\{a\}orf=fbf=f\_\{b\}, wherefa:=2​β​κx​\(κx\+κy\)4​β​\(β\+2\)​κx−4​κy−κx​\(κx\+κy\)​\(β2​κx​\(4​κx\+κy\)−6​β​κx​κy\+3​κy2\)4​β​\(β\+2\)​κx−4​κyf\_\{a\}:=\\frac\{2\\beta\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{4\\beta\(\\beta\+2\)\\kappa\_\{x\}\-4\\kappa\_\{y\}\}\-\\frac\{\\sqrt\{\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\left\(\\beta^\{2\}\\kappa\_\{x\}\(4\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-6\\beta\\kappa\_\{x\}\\kappa\_\{y\}\+3\\kappa\_\{y\}^\{2\}\\right\)\}\}\{4\\beta\(\\beta\+2\)\\kappa\_\{x\}\-4\\kappa\_\{y\}\}andfb:=2​β​κx​\(κx\+κy\)4​β​\(β\+2\)​κx−4​κy\+κx​\(κx\+κy\)​\(β2​κx​\(4​κx\+κy\)−6​β​κx​κy\+3​κy2\)4​β​\(β\+2\)​κx−4​κyf\_\{b\}:=\\frac\{2\\beta\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{4\\beta\(\\beta\+2\)\\kappa\_\{x\}\-4\\kappa\_\{y\}\}\+\\frac\{\\sqrt\{\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\left\(\\beta^\{2\}\\kappa\_\{x\}\(4\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-6\\beta\\kappa\_\{x\}\\kappa\_\{y\}\+3\\kappa\_\{y\}^\{2\}\\right\)\}\}\{4\\beta\(\\beta\+2\)\\kappa\_\{x\}\-4\\kappa\_\{y\}\}\. According to the restriction \(iii\), we haveA<0A<0and we can further obtain thatfa<fbf\_\{a\}<f\_\{b\}\. Then\[πD​\(ρx∗,ρy∗\)−πE​\(ρ∗\)\]\|ℓ=ℓ¯\>0\[\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\]\|\_\{\\ell=\\bar\{\\ell\}\}\>0is equivalent tofa<f<fbf\_\{a\}<f<f\_\{b\}\. We can verify thatfb\>min⁡\{κx2,κy4​β\}f\_\{b\}\>\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{\\kappa\_\{y\}\}\{4\\beta\}\\\}\. Thus, to ensure\[πD​\(ρx∗,ρy∗\)−πE​\(ρ∗\)\]\|ℓ=ℓ¯\>0\[\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\]\|\_\{\\ell=\\bar\{\\ell\}\}\>0, we only need to impose an additional conditionf\>faf\>f\_\{a\}\. Therefore, under the aforementioned conditions, we can obtainπD​\(ρx∗,ρy∗\)\|ℓ<ℓ¯\>πD​\(ρx∗,ρy∗\)\|ℓ=ℓ¯\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell<\\bar\{\\ell\}\}\>\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell=\\bar\{\\ell\}\}\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\), which means that the AI firm strictly prefers the disparate design ifℓ<ℓ¯\\ell<\\bar\{\\ell\}\.

Definefl:=max⁡\{2​β​\(c−θ​r\)​κx​\(κx\+κy\)b⁡\(β⁡\(β\+2\)​κx−κy\),fa\}f\_\{l\}:=\\max\\left\\\{\\frac\{2\\beta\(c\-\\theta r\)\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\\beta\(\\beta\+2\)\\kappa\_\{x\}\-\\kappa\_\{y\}\)\},\\;f\_\{a\}\\right\\\}\. So far, we have proved that whenβ⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}andfl<f<min⁡\{κx2,κy4​β\}f\_\{l\}<f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{\\kappa\_\{y\}\}\{4\\beta\}\\\}, AI use for disadvantaged patients is decreasing inℓ\\ellforℓ≤ℓ¯\\ell\\leq\\bar\{\\ell\}\. We next prove that AI use for disadvantaged patients is \(weakly\) increasing inℓ\\ellforℓ\>ℓ¯\\ell\>\\bar\{\\ell\}\.

From the preceding analysis, whenℓ\\ellincreases marginally beyondℓ¯\\bar\{\\ell\}, the interior conditionsπy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\>0,ρy∗<1\\rho\_\{y\}^\{\*\}<1, andπD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)continue to hold\. By continuity, there exists a neighborhood to the right ofℓ¯\\bar\{\\ell\}where AI use among type\-yypatients is strictly increasing inℓ\\ell\. Asℓ\\ellincreases further, one of these performance or participation constraints must eventually bind\. From the proof of[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1),πD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)impliesρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}\. Given the interior requirementρx∗<1\\rho\_\{x\}^\{\*\}<1, it follows immediately thatρy∗<1\\rho\_\{y\}^\{\*\}<1is satisfied whenever the disparate design remains strictly optimal\. Hence, either the regime\-switching conditionπD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)or the market\-viability conditionπy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\>0must be the first to bind\. Because parameter restriction \(iii\) ensures thatπE​\(ρ∗\)\>πx​\(ρx∗\)\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\>\\pi\_\{x\}\(\\rho\_\{x\}^\{\*\}\), the profit under the equal\-accuracy algorithm strictly dominates the profit from serving type\-xxpatients alone\. Consequently,πD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)must be the first condition to fail\. At this crossing point, the equal\-accuracy algorithm becomes optimal, and AI use among type\-yypatients remains constant thereafter\. Because AI use may exhibit a discrete upward jump at this regime switch \(by the same argument as in the proof of part \(a\) of[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\), AI use among type\-yypatients is weakly increasing inℓ\\ellfor allℓ\>ℓ¯\\ell\>\\bar\{\\ell\}\. The proposition then follows101010A numerical instance illustrates the existence of this region\. Whenb=2b=2,θ=1\\theta=1,c=0\.60c=0\.60,r=0\.5r=0\.5,κx=1\.3\\kappa\_\{x\}=1\.3,κy=1\.705\\kappa\_\{y\}=1\.705,f=0\.6f=0\.6, andβ=0\.68\\beta=0\.68, the physician uses AI for fewer type\-yypatients if0<ℓ<0\.0890<\\ell<0\.089and for more type\-yypatients if0\.089<ℓ<0\.3750\.089<\\ell<0\.375\. The equal\-accuracy design becomes optimal ifℓ≥0\.375\\ell\\geq 0\.375\.\.*Q\.E\.D\.*

Proof of[lemma3](https://arxiv.org/html/2608.13618#Thmlemma3)\. Using the altruistic physician’s expected payoff when not using AI \(from[eq\.3](https://arxiv.org/html/2608.13618#S4.E3)\) and when using AI \(from[eq\.8](https://arxiv.org/html/2608.13618#S5.E8)\), we compute the differenceUal−UU^\{\\text\{al\}\}\-U, given by

Ual−U=\\displaystyle U^\{\\text\{al\}\}\-U=\{−c,if​α≤1−ρt\(ρt\+α−1\)⋅b−c,if​1−ρt<α≤1/2\(ρt−α\)⋅b−c,if​1/2<α≤ρt−c,otherwise\.\\displaystyle\\begin\{cases\}\-c,&\\text\{if \}\\alpha\\leq 1\-\\rho\_\{t\}\\\\ \(\\rho\_\{t\}\+\\alpha\-1\)\\cdot b\-c,&\\text\{if \}1\-\\rho\_\{t\}<\\alpha\\leq 1/2\\\\ \(\\rho\_\{t\}\-\\alpha\)\\cdot b\-c,&\\text\{if \}1/2<\\alpha\\leq\\rho\_\{t\}\\\\ \-c,&\\text\{otherwise\.\}\\end\{cases\}\(A3\)
It is straightforward that the altruistic physician does not use AI in the first and fourth cases, that is, whenα≤1−ρt\\alpha\\leq 1\-\\rho\_\{t\}orα\>ρt\\alpha\>\\rho\_\{t\}\. In the second case of[eq\.A3](https://arxiv.org/html/2608.13618#S0.E3), the physician does not use AI whenα≤\(1−ρt\)​b\+cb\\alpha\\leq\\frac\{\(1\-\\rho\_\{t\}\)b\+c\}\{b\}and uses AI otherwise\. In the third case of[eq\.A3](https://arxiv.org/html/2608.13618#S0.E3), the physician does not use AI whenα\>ρt​b−cb\\alpha\>\\frac\{\\rho\_\{t\}b\-c\}\{b\}and uses AI otherwise\. Therefore, we have

- •Whenc<\(ρt−1/2\)⋅bc<\(\\rho\_\{t\}\-1/2\)\\cdot b, we have\(1−ρt\)​b\+cb<1/2\\frac\{\(1\-\\rho\_\{t\}\)b\+c\}\{b\}<1/2andρt​b−cb\>1/2\\frac\{\\rho\_\{t\}b\-c\}\{b\}\>1/2\. Then, the physician does not use AI ifα≤\(1−ρt\)​b\+cb\\alpha\\leq\\frac\{\(1\-\\rho\_\{t\}\)b\+c\}\{b\}, uses AI if\(1−ρt\)​b\+cb<α≤ρt​b−cb\\frac\{\(1\-\\rho\_\{t\}\)b\+c\}\{b\}<\\alpha\\leq\\frac\{\\rho\_\{t\}b\-c\}\{b\}, and does not use AI again ifα\>ρt​b−cb\\alpha\>\\frac\{\\rho\_\{t\}b\-c\}\{b\}\.
- •Whenc≥\(ρt−1/2\)⋅bc\\geq\(\\rho\_\{t\}\-1/2\)\\cdot b, we have\(1−ρt\)​b\+cb≥1/2\\frac\{\(1\-\\rho\_\{t\}\)b\+c\}\{b\}\\geq 1/2andρt​b−cb≤1/2\\frac\{\\rho\_\{t\}b\-c\}\{b\}\\leq 1/2\. Then, the physician does not use AI for all patientsα∈\(0,1\)\\alpha\\in\(0,1\)\.

Next, we check whether the physician will follow or reject the AI signal when using AI\. In the second and third cases of[eq\.A3](https://arxiv.org/html/2608.13618#S0.E3), the physician follows the AI signal regardless of the signal\. These are also the only scenarios in which the physician might choose to use AI\. Therefore, the physician follows the AI signal for type\-ttpatients whenever using AI\.*Q\.E\.D\.*

Proof of[proposition3](https://arxiv.org/html/2608.13618#Thmproposition3)\. \(a\) Under an equal\-accuracy design, the disparity\-triggered liability channel does not apply\. An impurely altruistic physician therefore bases AI use for both patient types on the decision rule in[lemma1](https://arxiv.org/html/2608.13618#Thmlemma1)\. Comparing this rule with the corresponding benchmark for an altruistic physician in[lemma3](https://arxiv.org/html/2608.13618#Thmlemma3)implies that the impurely altruistic physician uses AI for a \(weakly\) larger set of patients, and hence overuses AI\.

\(b\) Under a disparate design, comparing[lemma2](https://arxiv.org/html/2608.13618#Thmlemma2)with[lemma3](https://arxiv.org/html/2608.13618#Thmlemma3)shows that the impurely altruistic physician uses AI appropriately for type\-yypatients if and only if the reimbursement exactly offsets the expected liability cost on the margin, that is,\(1−ρy∗\)​ℓ=r\(1\-\\rho\_\{y\}^\{\*\}\)\\ell=r, which is equivalent to12−β​f​\(b\+θ​ℓ\)b​κy=rℓ\\frac\{1\}\{2\}\-\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}=\\frac\{r\}\{\\ell\}\. By part \(a\), the physician overuses AI for type\-xxpatients\.111111A numerical example is given byb=1\.8b=1\.8,θ=1\\theta=1,r=0\.1r=0\.1,κx=2\\kappa\_\{x\}=2,κy=3\\kappa\_\{y\}=3,f=0\.8f=0\.8,β=0\.6\\beta=0\.6,c=0\.11c=0\.11, andℓ=0\.32\\ell=0\.32, under which AI is used appropriately for type\-yypatients\.

\(c\) From the argument in part \(b\), under the disparate design the physician overuses AI for type\-yypatients when\(12−β​f​\(b\+θ​ℓ\)b​κy\)​ℓ<r\\Big\(\\frac\{1\}\{2\}\-\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}\\Big\)\\ell<rand underuses AI when\(12−β​f​\(b\+θ​ℓ\)b​κy\)​ℓ\>r\\Big\(\\frac\{1\}\{2\}\-\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}\\Big\)\\ell\>r\. The left\-hand side is concave inℓ\\ell, so the equality\(12−β​f​\(b\+θ​ℓ\)b​κy\)​ℓ=r\\Big\(\\frac\{1\}\{2\}\-\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}\\Big\)\\ell=rhas \(at most\) two solutions; letℓs\\ell\_\{s\}denote the smaller root andℓh\\ell\_\{h\}denote the larger root,

ℓs=b​κy​\(1−b​\(κy−2​β​f\)2−16​β​f​κy​θ​rb​κy2\)4​β​f​θ−b2​θ​and​ℓh=b​κy​\(1\+b​\(κy−2​β​f\)2−16​β​f​κy​θ​rb​κy2\)4​β​f​θ−b2​θ\.\\displaystyle\\ell\_\{s\}=\\frac\{b\\kappa\_\{y\}\\left\(1\-\\sqrt\{\\frac\{b\(\\kappa\_\{y\}\-2\\beta f\)^\{2\}\-16\\beta f\\kappa\_\{y\}\\theta r\}\{b\\kappa\_\{y\}^\{2\}\}\}\\right\)\}\{4\\beta f\\theta\}\-\\frac\{b\}\{2\\theta\}\\penalty\\ \\text\{and \}\\penalty\\ \\ell\_\{h\}=\\frac\{b\\kappa\_\{y\}\\left\(1\+\\sqrt\{\\frac\{b\(\\kappa\_\{y\}\-2\\beta f\)^\{2\}\-16\\beta f\\kappa\_\{y\}\\theta r\}\{b\\kappa\_\{y\}^\{2\}\}\}\\right\)\}\{4\\beta f\\theta\}\-\\frac\{b\}\{2\\theta\}\.In what follows, we prove that the physician first overuses AI whenℓ<ℓs\\ell<\\ell\_\{s\}and then underuses AI whenℓs<ℓ<ℓm\\ell\_\{s\}<\\ell<\\ell\_\{m\}and overuses AI again whenℓ\>ℓm\\ell\>\\ell\_\{m\}, whereℓm\\ell\_\{m\}is a threshold\.

We start by establishing the claimed pattern aroundℓs\\ell\_\{s\}in equilibrium\. Lettingℓs\>0\\ell\_\{s\}\>0yieldsf<b​κy\+4​κy​θ​r2​b​β−κy​2​b​θ​r\+4​θ2​r2b​βf<\\frac\{b\\kappa\_\{y\}\+4\\kappa\_\{y\}\\theta r\}\{2b\\beta\}\-\\frac\{\\kappa\_\{y\}\\sqrt\{2b\\theta r\+4\\theta^\{2\}r^\{2\}\}\}\{b\\beta\}\. In addition, note that the parameter restrictions in[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)imply: \(i\)b​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2<f<min⁡\{κx2,b​κy2​β​\(b\+θ​ℓ\)\}\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}<f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}\\\}, \(ii\) the AI firm strictly prefers the disparate design atℓ=ℓs\\ell=\\ell\_\{s\}\(that is,πD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)atℓ=ℓs\\ell=\\ell\_\{s\}\), and \(iii\)β⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}andf\>2​β​\(c−θ​r\)​κx​\(κx\+κy\)b⁡\(β⁡\(β\+2\)​κx−κy\)f\>\\frac\{2\\beta\(c\-\\theta r\)\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\\beta\(\\beta\+2\)\\kappa\_\{x\}\-\\kappa\_\{y\}\)\}\.

We first establish conditions surrounding the restriction \(i\)\. We can verify thatf<b​κy2​β​\(b\+θ​ℓ\)\|ℓ=ℓsf<\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}\|\_\{\\ell=\\ell\_\{s\}\}, which is equivalent toρy∗\|ℓ=ℓs=1−r/ℓs<1\\rho\_\{y\}^\{\*\}\|\_\{\\ell=\\ell\_\{s\}\}=1\-r/\\ell\_\{s\}<1; this also means thatρy∗<1\\rho\_\{y\}^\{\*\}<1must hold forℓ<ℓs\\ell<\\ell\_\{s\}\(asρy∗\\rho\_\{y\}^\{\*\}is increasing inℓ\\ell\)\. In addition, whenℓ=ℓs\\ell=\\ell\_\{s\}, consideringf\>b​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2f\>\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}or equivalentlyπy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\>0\(by[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)\) yields

4​β2​f2κy\+4​β​f\+\(κy−2​β​f\)​b​\(κy−2​β​f\)2−16​β​f​κy​θ​rb​κy2\>κy\+8​β​f​\(2​c−θ​r\)b;\\displaystyle\\frac\{4\\beta^\{2\}f^\{2\}\}\{\\kappa\_\{y\}\}\+4\\beta f\+\(\\kappa\_\{y\}\-2\\beta f\)\\sqrt\{\\frac\{b\(\\kappa\_\{y\}\-2\\beta f\)^\{2\}\-16\\beta f\\kappa\_\{y\}\\theta r\}\{b\\kappa\_\{y\}^\{2\}\}\}\>\\kappa\_\{y\}\+\\frac\{8\\beta f\(2c\-\\theta r\)\}\{b\};\(A4\)therefore, under the aforementioned condition, we can deriveπy​\(ρy∗\)\|ℓ<ℓs\>πy​\(ρy∗\)\|ℓ=ℓs\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell<\\ell\_\{s\}\}\>\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell=\\ell\_\{s\}\}\>0, which means that the firm earns a positive profit from serving type\-yypatients atρy∗\\rho\_\{y\}^\{\*\}ifℓ<ℓs\\ell<\\ell\_\{s\}\.

We then establish conditions surrounding the restriction \(ii\)\. Considering\[πD​\(ρx∗,ρy∗\)−πE​\(ρ∗\)\]\|ℓ=ℓs\[\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)\]\|\_\{\\ell=\\ell\_\{s\}\}yields

8​f2κx\+4​β​f​\(β​f\+κy\)κy\+\(κy−2​β​f\)​b​\(κy−2​β​f\)2−16​β​f​κy​θ​rb​κy2\>κy\+8​β​f​θ​rb\+8​\(1\+β\)2​f2κx\+κy\.\\displaystyle\\frac\{8f^\{2\}\}\{\\kappa\_\{x\}\}\+\\frac\{4\\beta f\(\\beta f\+\\kappa\_\{y\}\)\}\{\\kappa\_\{y\}\}\+\(\\kappa\_\{y\}\-2\\beta f\)\\sqrt\{\\frac\{b\(\\kappa\_\{y\}\-2\\beta f\)^\{2\}\-16\\beta f\\kappa\_\{y\}\\theta r\}\{b\\kappa\_\{y\}^\{2\}\}\}\>\\kappa\_\{y\}\+\\frac\{8\\beta f\\theta r\}\{b\}\+\\frac\{8\(1\+\\beta\)^\{2\}f^\{2\}\}\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\.\(A5\)Therefore, under the aforementioned condition, we can obtain

πD​\(ρx∗,ρy∗\)\|ℓ<ℓs\>πD​\(ρx∗,ρy∗\)\|ℓ=ℓs\>πE​\(ρ∗\),\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell<\\ell\_\{s\}\}\>\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell=\\ell\_\{s\}\}\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\),which means that the AI firm strictly prefers the disparate design ifℓ<ℓs\\ell<\\ell\_\{s\}\.

So far, we have the following conditions: \([A4](https://arxiv.org/html/2608.13618#S0.E4)\), \([A5](https://arxiv.org/html/2608.13618#S0.E5)\), andf<min⁡\{κx2,b​κy\+4​κy​θ​r2​b​β−κy​2​b​θ​r\+4​θ2​r2b​β\}f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{b\\kappa\_\{y\}\+4\\kappa\_\{y\}\\theta r\}\{2b\\beta\}\-\\frac\{\\kappa\_\{y\}\\sqrt\{2b\\theta r\+4\\theta^\{2\}r^\{2\}\}\}\{b\\beta\}\\\}, under which the physician overuses AI whenℓ<ℓs\\ell<\\ell\_\{s\}\. We next prove that the physician underuses AI whenℓs<ℓ<ℓm\\ell\_\{s\}<\\ell<\\ell\_\{m\}and overuses AI again whenℓ\>ℓm\\ell\>\\ell\_\{m\}\.

From the above proof, whenℓ\\ellincreases marginally fromℓs\\ell\_\{s\}, the conditionsπy​\(ρy∗\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\>0,ρy∗<1\\rho\_\{y\}^\{\*\}<1, andπD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)continue to hold\. By continuity, there therefore exists a neighborhood to the right ofℓs\\ell\_\{s\}in which the physician underuses AI asℓ\\ellincreases fromℓs\\ell\_\{s\}\. Asℓ\\ellincreases further, one of the preceding conditions must eventually fail\. By an argument similar to that in the proof of[proposition2](https://arxiv.org/html/2608.13618#Thmproposition2), the restriction \(iii\) \(i\.e\.,β⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}andf\>2​β​\(c−θ​r\)​κx​\(κx\+κy\)b⁡\(β⁡\(β\+2\)​κx−κy\)f\>\\frac\{2\\beta\(c\-\\theta r\)\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\\beta\(\\beta\+2\)\\kappa\_\{x\}\-\\kappa\_\{y\}\)\}\) ensures thatπD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)is the first and only condition that can be violated asℓ\\ellincreases\. Once this condition is violated, Part \(a\) implies that physician overuse of AI reappears\. Depending on the relationship between the violation point andℓh\\ell\_\{h\}, two cases arise:

- •SupposeπD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)is violated beforeℓ=ℓh\\ell=\\ell\_\{h\}\. Thenℓm\\ell\_\{m\}coincides with the violation point\. In this case, the physician overuses AI forℓ<ℓs\\ell<\\ell\_\{s\}, underuses AI forℓs<ℓ<ℓm\\ell\_\{s\}<\\ell<\\ell\_\{m\}, and overuses AI again forℓ\>ℓm\\ell\>\\ell\_\{m\}\. The first two regions arise under the disparate algorithm, whereas the final overuse region is induced by the equal\-accuracy algorithm\.
- •SupposeπD​\(ρx∗,ρy∗\)\>πE​\(ρ∗\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\}\)is violated afterℓ=ℓh\\ell=\\ell\_\{h\}\. Thenℓm=ℓh\\ell\_\{m\}=\\ell\_\{h\}\. In this case, the physician overuses AI forℓ<ℓs\\ell<\\ell\_\{s\}, underuses AI forℓs<ℓ<ℓm\\ell\_\{s\}<\\ell<\\ell\_\{m\}, and overuses AI again forℓ\>ℓm\\ell\>\\ell\_\{m\}\. The re\-emergence of overuse initially occurs under the disparate algorithm and subsequently persists under the equal\-accuracy algorithm after the algorithmic switch\.

Combining the above two cases establishes the proposition under the conditionsβ⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}, \([A4](https://arxiv.org/html/2608.13618#S0.E4)\), \([A5](https://arxiv.org/html/2608.13618#S0.E5)\), and2​β​\(c−θ​r\)​κx​\(κx\+κy\)b⁡\(β⁡\(β\+2\)​κx−κy\)<f<min⁡\{κx2,b​κy\+4​κy​θ​r2​b​β−κy​2​b​θ​r\+4​θ2​r2b​β\}\\frac\{2\\beta\(c\-\\theta r\)\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\\beta\(\\beta\+2\)\\kappa\_\{x\}\-\\kappa\_\{y\}\)\}<f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{b\\kappa\_\{y\}\+4\\kappa\_\{y\}\\theta r\}\{2b\\beta\}\-\\frac\{\\kappa\_\{y\}\\sqrt\{2b\\theta r\+4\\theta^\{2\}r^\{2\}\}\}\{b\\beta\}\\\}\. This completes the proof121212For example, whenb=1\.8b=1\.8,θ=1\\theta=1,r=0\.1r=0\.1,κx=2\\kappa\_\{x\}=2,κy=3\\kappa\_\{y\}=3,f=0\.8f=0\.8,β=0\.6\\beta=0\.6, andc=0\.11c=0\.11, the physician overuses AI whenℓ<0\.32\\ell<0\.32, underuses AI when0\.32<ℓ<0\.400\.32<\\ell<0\.40, and overuses AI again whenℓ≥0\.40\\ell\\geq 0\.40, where the equal\-accuracy design becomes optimal\.\.*Q\.E\.D\.*

Proof of[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\. Using expected utilities for different patient types in[table2](https://arxiv.org/html/2608.13618#S3.T2)and AI accuracy expressions in[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1), we compute the expected patient welfare\. By[lemmas1](https://arxiv.org/html/2608.13618#Thmlemma1)and[2](https://arxiv.org/html/2608.13618#Thmlemma2), when the physician relies on an accuracyρ\\rhowithout the disparity\-liability channel, she uses AI exactly on\(α−​\(ρ\),α\+​\(ρ\)\)\(\\alpha^\{\-\}\(\\rho\),\\alpha^\{\+\}\(\\rho\)\), where

α−​\(ρ\):=\(1−ρ\)​b\+c−θ​rb,α\+​\(ρ\):=ρ​b−c\+θ​rb;\\alpha^\{\-\}\(\\rho\):=\\frac\{\(1\-\\rho\)b\+c\-\\theta r\}\{b\},\\qquad\\alpha^\{\+\}\(\\rho\):=\\frac\{\\rho b\-c\+\\theta r\}\{b\};under a disparate design she uses AI for type\-yypatients exactly on\(αy−,αy\+\)\(\\alpha\_\{y\}^\{\-\},\\alpha\_\{y\}^\{\+\}\), where

αy−:=\(1−ρy∗\)​\(b\+θ​ℓ\)\+c−θ​rb,αy\+:=ρy∗​b−\(1−ρy∗\)​θ​ℓ−c\+θ​rb\.\\alpha\_\{y\}^\{\-\}:=\\frac\{\(1\-\\rho\_\{y\}^\{\*\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\},\\qquad\\alpha\_\{y\}^\{\+\}:=\\frac\{\\rho\_\{y\}^\{\*\}b\-\(1\-\\rho\_\{y\}^\{\*\}\)\\theta\\ell\-c\+\\theta r\}\{b\}\.Accordingly,

Wx​\(ρ\)\\displaystyle W\_\{x\}\(\\rho\)=∫0α−​\(ρ\)\(\(1−α\)​b\)​𝑑α\+∫α−​\(ρ\)α\+​\(ρ\)\(ρ​b−c\)​𝑑α\+∫α\+​\(ρ\)1\(α​b\)​𝑑α,\\displaystyle=\\int\_\{0\}^\{\\alpha^\{\-\}\(\\rho\)\}\\Big\(\(1\-\\alpha\)b\\Big\)d\\alpha\+\\int\_\{\\alpha^\{\-\}\(\\rho\)\}^\{\\alpha^\{\+\}\(\\rho\)\}\\Big\(\\rho b\-c\\Big\)d\\alpha\+\\int\_\{\\alpha^\{\+\}\(\\rho\)\}^\{1\}\\Big\(\\alpha b\\Big\)d\\alpha,Wy​\(ρy∗\)\\displaystyle W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)=β⋅\[∫0αy−\(\(1−α\)​b\)​𝑑α\+∫αy−αy\+\(ρy∗​b−c\)​𝑑α\+∫αy\+1\(α​b\)​𝑑α\]\.\\displaystyle=\\beta\\cdot\\Big\[\\int\_\{0\}^\{\\alpha\_\{y\}^\{\-\}\}\\Big\(\(1\-\\alpha\)b\\Big\)d\\alpha\+\\int\_\{\\alpha\_\{y\}^\{\-\}\}^\{\\alpha\_\{y\}^\{\+\}\}\\Big\(\\rho\_\{y\}^\{\*\}b\-c\\Big\)d\\alpha\+\\int\_\{\\alpha\_\{y\}^\{\+\}\}^\{1\}\\Big\(\\alpha b\\Big\)d\\alpha\\Big\]\.\(A6\)The four objects we compare areWx​\(ρx∗\)W\_\{x\}\(\\rho\_\{x\}^\{\*\}\),Wy​\(ρy∗\)W\_\{y\}\(\\rho\_\{y\}^\{\*\}\),Wx​\(ρm\)W\_\{x\}\(\\rho^\{m\}\), and, because the mandate deactivates the disparity\-liability channel for type\-yypatients,Wy​\(ρm\)=β​Wx​\(ρm\)W\_\{y\}\(\\rho^\{m\}\)=\\beta\\,W\_\{x\}\(\\rho^\{m\}\), whereρx∗=12\+fκx\\rho\_\{x\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{f\}\{\\kappa\_\{x\}\},ρy∗=12\+β​f​\(b\+θ​ℓ\)b​κy\\rho\_\{y\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{\\beta f\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}andρm=12\+\(1\+β\)​fκx\+κy\\rho^\{m\}=\\frac\{1\}\{2\}\+\\frac\{\(1\+\\beta\)f\}\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\. In each of the above welfare expressions, the first and third terms capture the expected welfare of patients for whom the physician does not use AI, whereas the second term represents the expected welfare of patients for whom the physician uses AI\.

\(a\) Comparing the equilibrium accuracies givesρm≤ρx∗\\rho^\{m\}\\leq\\rho\_\{x\}^\{\*\}, so it suffices to proveρy∗≤ρm\\rho\_\{y\}^\{\*\}\\leq\\rho^\{m\}\.

DefineΔ\\Deltaas the difference between the AI firm’s optimal expected profit under a disparate and an equal\-accuracy algorithm:Δ:=πD​\(ρx∗,ρy∗\)−πE​\(ρm\)\\Delta:=\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{m\}\), with the profit functions in[eqs\.1](https://arxiv.org/html/2608.13618#S3.E1)and[2](https://arxiv.org/html/2608.13618#S3.E2)evaluated at the equilibrium accuracies above\. Letℓ^\\widehat\{\\ell\}be the value satisfyingρy∗=ρm\\rho\_\{y\}^\{\*\}=\\rho^\{m\}, and we can deriveℓ^=b⁡\(κy−β​κx\)β⁡\(κx\+κy\)​θ\\widehat\{\\ell\}=\\frac\{b\(\\kappa\_\{y\}\-\\beta\\kappa\_\{x\}\)\}\{\\beta\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\theta\}\.

Becauseρx∗\\rho\_\{x\}^\{\*\}andρm\\rho^\{m\}are independent ofℓ\\ell, andρy∗\\rho\_\{y\}^\{\*\}is increasing inℓ\\ell, we only need to proveΔ^:=Δ\|ℓ=ℓ^<0\\widehat\{\\Delta\}:=\\Delta\|\_\{\\ell=\\widehat\{\\ell\}\}<0\. Substitutingℓ=ℓ^\\ell=\\widehat\{\\ell\}and differentiating,

Δ^\\displaystyle\\widehat\{\\Delta\}=f⁡\(κy−β​κx\)​\(f⁡\(\(2\+β\)​κx\+κy\)−κx​\(κx\+κy\)\)κx​\(κx\+κy\)2,\\displaystyle=\\frac\{f\(\\kappa\_\{y\}\-\\beta\\kappa\_\{x\}\)\(f\(\(2\+\\beta\)\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\)\}\{\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)^\{2\}\},∂Δ^∂f\\displaystyle\\frac\{\\partial\\widehat\{\\Delta\}\}\{\\partial f\}=−\(κy−β​κx\)​\(κx​\(κx\+κy\)−2​f​\(\(2\+β\)​κx\+κy\)\)κx​\(κx\+κy\)2\.\\displaystyle=\-\\frac\{\(\\kappa\_\{y\}\-\\beta\\kappa\_\{x\}\)\(\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-2f\(\(2\+\\beta\)\\kappa\_\{x\}\+\\kappa\_\{y\}\)\)\}\{\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)^\{2\}\}\.Since∂Δ^∂f\\frac\{\\partial\\widehat\{\\Delta\}\}\{\\partial f\}is increasing inff,Δ^\\widehat\{\\Delta\}is convex inff\. BecauseΔ^\|f=0=0\\widehat\{\\Delta\}\|\_\{f=0\}=0and direct substitution yieldsΔ^\|f=κx/2<0\\widehat\{\\Delta\}\|\_\{f=\\kappa\_\{x\}/2\}<0, convexity impliesΔ^<0\\widehat\{\\Delta\}<0on the entire admissible range0<f<κx20<f<\\frac\{\\kappa\_\{x\}\}\{2\}\. Finally, becauseΔ\\Deltais decreasing inℓ\\ellandΔ^<0\\widehat\{\\Delta\}<0, the firm abandons the disparate design strictly beforeℓ\\ellreachesℓ^\\widehat\{\\ell\}; sinceρy∗\\rho\_\{y\}^\{\*\}is increasing inℓ\\ellwithρy∗\|ℓ=ℓ^=ρm\\rho\_\{y\}^\{\*\}\|\_\{\\ell=\\widehat\{\\ell\}\}=\\rho^\{m\}, it follows thatρy∗<ρm\\rho\_\{y\}^\{\*\}<\\rho^\{m\}whenever the disparate design is optimal, completing the proof of \(a\)\.

\(b\) We first restate this part formally, writingαx±:=α±​\(ρx∗\)\\alpha\_\{x\}^\{\\pm\}:=\\alpha^\{\\pm\}\(\\rho\_\{x\}^\{\*\}\)andαm±:=α±​\(ρm\)\\alpha\_\{m\}^\{\\pm\}:=\\alpha^\{\\pm\}\(\\rho^\{m\}\): under a disparate algorithm the physician uses AI for type\-xxpatients exactly whenα∈\(αx−,αx\+\)\\alpha\\in\(\\alpha\_\{x\}^\{\-\},\\alpha\_\{x\}^\{\+\}\)and for type\-yypatients exactly whenα∈\(αy−,αy\+\)\\alpha\\in\(\\alpha\_\{y\}^\{\-\},\\alpha\_\{y\}^\{\+\}\), and under the mandate she uses AI for both types exactly whenα∈\(αm−,αm\+\)\\alpha\\in\(\\alpha\_\{m\}^\{\-\},\\alpha\_\{m\}^\{\+\}\)\.

Comparing these cutoffs yields the ordering

αx−<αm−<αy−<αy\+<αm\+<αx\+\.\\alpha\_\{x\}^\{\-\}<\\alpha\_\{m\}^\{\-\}<\\alpha\_\{y\}^\{\-\}<\\alpha\_\{y\}^\{\+\}<\\alpha\_\{m\}^\{\+\}<\\alpha\_\{x\}^\{\+\}\.Moreover, each patient\-indifference point sits a wedgeθ​rb\\tfrac\{\\theta r\}\{b\}inside the corresponding use cutoff, the wedge being the physician’s private benefit:\(1−α\)​b=ρx∗​b−c\(1\-\\alpha\)b=\\rho\_\{x\}^\{\*\}b\-catα=αx−\+θ​rb\\alpha=\\alpha\_\{x\}^\{\-\}\+\\tfrac\{\\theta r\}\{b\},α​b=ρx∗​b−c\\alpha b=\\rho\_\{x\}^\{\*\}b\-catα=αx\+−θ​rb\\alpha=\\alpha\_\{x\}^\{\+\}\-\\tfrac\{\\theta r\}\{b\}, and likewise forρm\\rho^\{m\}\. The formal statement is:

*The expected patient welfare is affected as follows\. For type\-xxpatients, welfare remains unchanged ifα<αx−\\alpha<\\alpha\_\{x\}^\{\-\}orα\>αx\+\\alpha\>\\alpha\_\{x\}^\{\+\}; improves ifαx−<α<min⁡\{αm−,αx−\+θ​rb\}\\alpha\_\{x\}^\{\-\}<\\alpha<\\min\\\{\\alpha\_\{m\}^\{\-\},\\,\\alpha\_\{x\}^\{\-\}\+\\tfrac\{\\theta r\}\{b\}\\\}ormax⁡\{αm\+,αx\+−θ​rb\}<α<αx\+\\max\\\{\\alpha\_\{m\}^\{\+\},\\,\\alpha\_\{x\}^\{\+\}\-\\tfrac\{\\theta r\}\{b\}\\\}<\\alpha<\\alpha\_\{x\}^\{\+\}; and worsens ifmin⁡\{αm−,αx−\+θ​rb\}<α<max⁡\{αm\+,αx\+−θ​rb\}\\min\\\{\\alpha\_\{m\}^\{\-\},\\,\\alpha\_\{x\}^\{\-\}\+\\tfrac\{\\theta r\}\{b\}\\\}<\\alpha<\\max\\\{\\alpha\_\{m\}^\{\+\},\\,\\alpha\_\{x\}^\{\+\}\-\\tfrac\{\\theta r\}\{b\}\\\}\. For type\-yypatients, welfare remains unchanged ifα<αm−\\alpha<\\alpha\_\{m\}^\{\-\}orα\>αm\+\\alpha\>\\alpha\_\{m\}^\{\+\}; worsens ifαm−<α<min⁡\{αy−,αm−\+θ​rb\}\\alpha\_\{m\}^\{\-\}<\\alpha<\\min\\\{\\alpha\_\{y\}^\{\-\},\\,\\alpha\_\{m\}^\{\-\}\+\\tfrac\{\\theta r\}\{b\}\\\}ormax⁡\{αy\+,αm\+−θ​rb\}<α<αm\+\\max\\\{\\alpha\_\{y\}^\{\+\},\\,\\alpha\_\{m\}^\{\+\}\-\\tfrac\{\\theta r\}\{b\}\\\}<\\alpha<\\alpha\_\{m\}^\{\+\}; and improves ifmin⁡\{αy−,αm−\+θ​rb\}<α<max⁡\{αy\+,αm\+−θ​rb\}\\min\\\{\\alpha\_\{y\}^\{\-\},\\,\\alpha\_\{m\}^\{\-\}\+\\tfrac\{\\theta r\}\{b\}\\\}<\\alpha<\\max\\\{\\alpha\_\{y\}^\{\+\},\\,\\alpha\_\{m\}^\{\+\}\-\\tfrac\{\\theta r\}\{b\}\\\}\.*

We verify the statement interval by interval\. For type\-xxpatients, the mandate can only remove AI use: the use interval shrinks from\(αx−,αx\+\)\(\\alpha\_\{x\}^\{\-\},\\alpha\_\{x\}^\{\+\}\)to\(αm−,αm\+\)\(\\alpha\_\{m\}^\{\-\},\\alpha\_\{m\}^\{\+\}\)\. Outside\(αx−,αx\+\)\(\\alpha\_\{x\}^\{\-\},\\alpha\_\{x\}^\{\+\}\)the physician never uses AI, so welfare is unchanged\. On\(αx−,αm−\)\(\\alpha\_\{x\}^\{\-\},\\alpha\_\{m\}^\{\-\}\), where use stops, welfare changes fromρx∗​b−c\\rho\_\{x\}^\{\*\}b\-cto\(1−α\)​b\(1\-\\alpha\)b, an improvement precisely whenα<αx−\+θ​rb\\alpha<\\alpha\_\{x\}^\{\-\}\+\\tfrac\{\\theta r\}\{b\}; on\(αm\+,αx\+\)\(\\alpha\_\{m\}^\{\+\},\\alpha\_\{x\}^\{\+\}\)it changes fromρx∗​b−c\\rho\_\{x\}^\{\*\}b\-ctoα​b\\alpha b, an improvement precisely whenα\>αx\+−θ​rb\\alpha\>\\alpha\_\{x\}^\{\+\}\-\\tfrac\{\\theta r\}\{b\}\. On the middle interval, where use continues, welfare falls fromρx∗​b−c\\rho\_\{x\}^\{\*\}b\-ctoρm​b−c≤ρx∗​b−c\\rho^\{m\}b\-c\\leq\\rho\_\{x\}^\{\*\}b\-c\. Intersecting each comparison with the interval on which it applies yields exactly the type\-xxintervals in the statement, with eachmin\\minandmax\\maxrecording whether the indifference point falls inside that interval\.

For type\-yypatients the pattern reverses: the mandate can only add AI use, the use interval expanding from\(αy−,αy\+\)\(\\alpha\_\{y\}^\{\-\},\\alpha\_\{y\}^\{\+\}\)to\(αm−,αm\+\)\(\\alpha\_\{m\}^\{\-\},\\alpha\_\{m\}^\{\+\}\)\. Outside\(αm−,αm\+\)\(\\alpha\_\{m\}^\{\-\},\\alpha\_\{m\}^\{\+\}\)welfare is unchanged\. On\(αm−,αy−\)\(\\alpha\_\{m\}^\{\-\},\\alpha\_\{y\}^\{\-\}\), where use begins, welfare changes from\(1−α\)​b\(1\-\\alpha\)btoρm​b−c\\rho^\{m\}b\-c, an improvement precisely whenα\>αm−\+θ​rb\\alpha\>\\alpha\_\{m\}^\{\-\}\+\\tfrac\{\\theta r\}\{b\}; on\(αy\+,αm\+\)\(\\alpha\_\{y\}^\{\+\},\\alpha\_\{m\}^\{\+\}\)it changes fromα​b\\alpha btoρm​b−c\\rho^\{m\}b\-c, an improvement precisely whenα<αm\+−θ​rb\\alpha<\\alpha\_\{m\}^\{\+\}\-\\tfrac\{\\theta r\}\{b\}\. Where use already occurred, welfare rises fromρy∗​b−c\\rho\_\{y\}^\{\*\}b\-ctoρm​b−c≥ρy∗​b−c\\rho^\{m\}b\-c\\geq\\rho\_\{y\}^\{\*\}b\-cby part \(a\)\. Intersecting as before yields the type\-yyintervals in the statement\.

\(c\) We prove this part using[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)\. Substituting the equilibrium accuracies stated below[Appendix](https://arxiv.org/html/2608.13618#S0.Ex14)into the welfare expressions there, we obtainWx​\(ρm\)\>Wx​\(ρx∗\)⟺f<fxW\_\{x\}\(\\rho^\{m\}\)\>W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)\\Longleftrightarrow f<f\_\{x\}, wherefx:=2​c​κx​\(κx\+κy\)b⁡\(\(2\+β\)​κx\+κy\)f\_\{x\}:=\\frac\{2c\\kappa\_\{x\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\(2\+\\beta\)\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\. Analogously, whenℓ→0\\ell\\to 0, we haveWy​\(ρm\)\>Wy​\(ρy∗\)\|ℓ→0⟺f\>fyW\_\{y\}\(\\rho^\{m\}\)\>W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\|\_\{\\ell\\to 0\}\\Longleftrightarrow f\>f\_\{y\}, wherefy:=2​c​κy​\(κx\+κy\)b⁡\(\(2​β\+1\)​κy\+β​κx\)f\_\{y\}:=\\frac\{2c\\kappa\_\{y\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{b\(\(2\\beta\+1\)\\kappa\_\{y\}\+\\beta\\kappa\_\{x\}\)\}\. We can verify thatfx<fyf\_\{x\}<f\_\{y\}\. It is straightforward to verifyΔ\|ℓ→0\>0\\Delta\|\_\{\\ell\\to 0\}\>0: the disparate design strictly dominates nearℓ=0\\ell=0, so we may drop the parameter restrictions of[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)that otherwise guarantee profit dominance \(β⁡\(β\+2\)​κx\>κy\\beta\(\\beta\+2\)\\kappa\_\{x\}\>\\kappa\_\{y\}and the associated lower bound onff\)\. Writefπ​\(ℓ\):=b​κy​\(2​c−2​θ​r\+θ​ℓ\)β​\(b\+θ​ℓ\)2f\_\{\\pi\}\(\\ell\):=\\frac\{b\\kappa\_\{y\}\(2c\-2\\theta r\+\\theta\\ell\)\}\{\\beta\(b\+\\theta\\ell\)^\{2\}\}for the type\-yyparticipation bound from[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1)\. Then,

fx\>fπ​\(0\)\\displaystyle f\_\{x\}\>f\_\{\\pi\}\(0\)⇔c<κy​\(\(2\+β\)​κx\+κy\)​θ​rκy​\(2​κx\+κy\)−β​κx2\\displaystyle\\iff c<\\frac\{\\kappa\_\{y\}\(\(2\+\\beta\)\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\theta r\}\{\\kappa\_\{y\}\(2\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-\\beta\\kappa\_\{x\}^\{2\}\}fy<κx2\\displaystyle f\_\{y\}<\\frac\{\\kappa\_\{x\}\}\{2\}⇔b\>4​c​κy​\(κx\+κy\)κx​\(κy\+β⁡\(κx\+2​κy\)\)\\displaystyle\\iff b\>\\frac\{4c\\kappa\_\{y\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{\\kappa\_\{x\}\(\\kappa\_\{y\}\+\\beta\(\\kappa\_\{x\}\+2\\kappa\_\{y\}\)\)\}fy<b​κy2​β​\(b\+θ​ℓ\)\\displaystyle f\_\{y\}<\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}⇔b\>4​β​c​\(κx\+κy\)κy\+β⁡\(κx\+2​κy\)\.\\displaystyle\\iff b\>\\frac\{4\\beta c\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{\\kappa\_\{y\}\+\\beta\(\\kappa\_\{x\}\+2\\kappa\_\{y\}\)\}\.We can verify that4​c​κy​\(κx\+κy\)κx​\(κy\+β⁡\(κx\+2​κy\)\)\>4​β​c​\(κx\+κy\)κy\+β⁡\(κx\+2​κy\)\\frac\{4c\\kappa\_\{y\}\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{\\kappa\_\{x\}\(\\kappa\_\{y\}\+\\beta\(\\kappa\_\{x\}\+2\\kappa\_\{y\}\)\)\}\>\\frac\{4\\beta c\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{\\kappa\_\{y\}\+\\beta\(\\kappa\_\{x\}\+2\\kappa\_\{y\}\)\}\. Therefore, suppose the two preconditions stated in part \(c\) hold; then all three inequalities above are satisfied\. In the limitℓ→0\\ell\\to 0, mandating equal accuracy yields three regimes:

- •iffπ​\(0\)<f<fxf\_\{\\pi\}\(0\)<f<f\_\{x\}, thenWx​\(ρm\)\>Wx​\(ρx∗\)W\_\{x\}\(\\rho^\{m\}\)\>W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)whileWy​\(ρm\)<Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)<W\_\{y\}\(\\rho\_\{y\}^\{\*\}\);
- •iffx<f<fyf\_\{x\}<f<f\_\{y\}, then both types are worse off under the mandate, that is,Wx​\(ρm\)<Wx​\(ρx∗\)W\_\{x\}\(\\rho^\{m\}\)<W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)andWy​\(ρm\)<Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)<W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\(each endpoint leaves one type exactly indifferent\);
- •iffy<f<min⁡\{κx2,b​κy2​β​\(b\+θ​ℓ\)\}\|ℓ→0f\_\{y\}<f<\\min\\\{\\frac\{\\kappa\_\{x\}\}\{2\},\\frac\{b\\kappa\_\{y\}\}\{2\\beta\(b\+\\theta\\ell\)\}\\\}\|\_\{\\ell\\to 0\}, thenWx​\(ρm\)<Wx​\(ρx∗\)W\_\{x\}\(\\rho^\{m\}\)<W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)whileWy​\(ρm\)\>Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)\>W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\.

Every welfare expression and equilibrium condition above is continuous inℓ\\ell\. Hence, for each parameter tuple satisfying these conditions there existsε\>0\\varepsilon\>0such that, for all0≤ℓ<ε0\\leq\\ell<\\varepsilon, the same three regimes persist, with the boundaries evaluated at thatℓ\\elland the type\-yyindifference point replaced by a thresholdf¯\\bar\{f\}, defined byWy​\(ρm\)=Wy​\(ρy∗\)W\_\{y\}\(\\rho^\{m\}\)=W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)\(so thatf¯→fy\\bar\{f\}\\to f\_\{y\}asℓ→0\\ell\\to 0\), exactly as stated in part \(c\) of the proposition\. This completes the proof\.*Q\.E\.D\.*

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## Online Appendix for “Algorithm Design and Physician Liability”

This Online Appendix is organized as follows\. The first part extends the welfare comparisons under equal\-accuracy requirements\. The second part provides technical derivations for the endogenous\-pricing extension\. The third part reports the welfare\-maximizing\-liability computations\. The fourth part develops the extension with type\-specific priors and its additional comparative statics\.

### OA1\. Effect of Mandating Equal Accuracy on Patient Welfare \([section5\.3](https://arxiv.org/html/2608.13618#S5.SS3)\)

To assess how an equal\-accuracy requirement affects aggregate welfare over a broader parameter range, we report numerical comparisons across\(β,ℓ,f\)\(\\beta,\\ell,f\)in[fig\.OA1](https://arxiv.org/html/2608.13618#S0.F1)\.

\(a\)Variation in\(β,f\)\(\\beta,f\)\(b\)Variation in\(ℓ,β\)\(\\ell,\\beta\)
Figure OA1:Welfare effects of an equal\-accuracy requirement for each patient type\. Regions labeled “Unbiased AI is optimal” are those in which the equal\-accuracy design is optimal\.[fig\.1\(a\)](https://arxiv.org/html/2608.13618#S0.F1.sf1)complements[proposition4](https://arxiv.org/html/2608.13618#Thmproposition4)\(c\)\. Asffincreases, the welfare effect of the equal\-accuracy mandate typically transitions from harming only type\-yypatients, to harming both types, and then to harming only type\-xxpatients\. Intuitively, higherffstrengthens the AI firm’s incentive to invest in accuracy for type\-yypatients under the equal\-accuracy requirement; once these gains become sufficiently large, the probability that type\-yypatients are harmed falls\.

The same figure shows how these welfare patterns vary with the relative size of the disadvantaged segment\. Asβ\\betaincreases, the equal\-accuracy requirement again tends to move from harming only type\-yypatients, to harming both types, and eventually to harming primarily type\-xxpatients\. Whenβ\\betais large, the AI firm internalizes a larger market on the type\-yyside, which increases the return to improving their accuracy under the requirement and limits the scope for welfare loss in that group\. Whenβ\\betais small, the requirement can instead improve welfare for type\-xxpatients by tempering overuse on the margin, partially offsetting the loss from reduced accuracy\.

Finally,[fig\.1\(b\)](https://arxiv.org/html/2608.13618#S0.F1.sf2)shows that higher liability can increase the likelihood that type\-yypatients are harmed by the equal\-accuracy requirement\. Whenℓ\\ellis large, the disparate\-design equilibrium already induces substantial investment in type\-yyaccuracy, reducing the incremental benefit of further improvements under the requirement, while utilization effects remain\. As a result, combining a high\-liability environment with an equal\-accuracy requirement—two instruments intended to protect disadvantaged patients—can be counterproductive in equilibrium\.

### OA2\. AI Firm’s Pricing Decision \([section6\.1](https://arxiv.org/html/2608.13618#S6.SS1)\)

Proof of[proposition5](https://arxiv.org/html/2608.13618#Thmproposition5)\. The AI firm’s profit functions under disparate and equal\-accuracy designs are given in[eqs\.1](https://arxiv.org/html/2608.13618#S3.E1)and[2](https://arxiv.org/html/2608.13618#S3.E2)\. We first consider the case in which the firm offers a disparate design\. The Hessian matrixℍD\\mathbb\{H\}^\{\\mathrm\{D\}\}of the profit functionπD​\(ρx,ρy,f\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\},\\rho\_\{y\},f\)is given by:

ℍD:=\[−4​\(1\+β\)b22​β​\(b\+θ​ℓ\)b2−2​κx02​β​\(b\+θ​ℓ\)b0−2​κy\]\\displaystyle\\mathbb\{H\}^\{\\mathrm\{D\}\}:=\\begin\{bmatrix\}\-\\frac\{4\(1\+\\beta\)\}\{b\}&2&\\frac\{2\\beta\(b\+\\theta\\ell\)\}\{b\}\\\\ 2&\-2\\kappa\_\{x\}&0\\\\ \\frac\{2\\beta\(b\+\\theta\\ell\)\}\{b\}&0&\-2\\kappa\_\{y\}\\end\{bmatrix\}\(OA1\)The corresponding leading principal minors ofℍD\\mathbb\{H\}^\{\\mathrm\{D\}\}are evaluated as:

𝔻1=−4​\(1\+β\)b,𝔻2=8​\(1\+β\)​κxb−4​and​𝔻3=4​D​\(ℓ\)b2,\\displaystyle\\mathbb\{D\}\_\{1\}=\-\\frac\{4\(1\+\\beta\)\}\{b\},\\penalty\\ \\penalty\\ \\mathbb\{D\}\_\{2\}=\\frac\{8\(1\+\\beta\)\\kappa\_\{x\}\}\{b\}\-4\\penalty\\ \\text\{and\}\\penalty\\ \\mathbb\{D\}\_\{3\}=\\frac\{4D\(\\ell\)\}\{b^\{2\}\},where

D⁡\(ℓ\):=2​b2​\(β2​κx\+κy\)−4​b​κx​\(\(β\+1\)​κy−β2​θ​ℓ\)\+2​β2​κx​θ2​ℓ2\.D\(\\ell\):=2b^\{2\}\(\\beta^\{2\}\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-4b\\kappa\_\{x\}\\big\(\(\\beta\+1\)\\kappa\_\{y\}\-\\beta^\{2\}\\theta\\ell\\big\)\+2\\beta^\{2\}\\kappa\_\{x\}\\theta^\{2\}\\ell^\{2\}\.To ensure the profit functionπD​\(ρx,ρy,f\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\},\\rho\_\{y\},f\)is jointly concave in\(f,ρx,ρy\)\(f,\\rho\_\{x\},\\rho\_\{y\}\), the Hessian matrix must be negative definite, which requires𝔻1<0\\mathbb\{D\}\_\{1\}<0,𝔻2\>0\\mathbb\{D\}\_\{2\}\>0, and𝔻3<0\\mathbb\{D\}\_\{3\}<0\. These conditions yield the parameter space boundariesb<2​\(1\+β\)​κx​κyβ2​κx\+κyb<\\frac\{2\(1\+\\beta\)\\kappa\_\{x\}\\kappa\_\{y\}\}\{\\beta^\{2\}\\kappa\_\{x\}\+\\kappa\_\{y\}\}andℓ<2​b​κx​κy−b2​κy\+2​b​β​κx​κyβ2​θ2​κx−bθ\\ell<\\sqrt\{\\frac\{2b\\kappa\_\{x\}\\kappa\_\{y\}\-b^\{2\}\\kappa\_\{y\}\+2b\\beta\\kappa\_\{x\}\\kappa\_\{y\}\}\{\\beta^\{2\}\\theta^\{2\}\\kappa\_\{x\}\}\}\-\\frac\{b\}\{\\theta\}\.

Under these concavity conditions, we derive the optimal decision vector\(f,ρx,ρy\)\(f,\\rho\_\{x\},\\rho\_\{y\}\)\. The first\-order condition∂πD​\(ρx,ρy,f\)∂f=0\\frac\{\\partial\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\},\\rho\_\{y\},f\)\}\{\\partial f\}=0yields the unique best\-response price

f⁡\(ρx,ρy\)=b⁡\(β⁡\(2​ρy−1\)\+2​ρx−1\)−2​θ​β​ℓ​\(1−ρy\)\+2​\(β\+1\)​θ​r4​\(β\+1\)\.f\(\\rho\_\{x\},\\rho\_\{y\}\)=\\frac\{b\\big\(\\beta\(2\\rho\_\{y\}\-1\)\+2\\rho\_\{x\}\-1\\big\)\-2\\theta\\beta\\ell\(1\-\\rho\_\{y\}\)\+2\(\\beta\+1\)\\theta r\}\{4\(\\beta\+1\)\}\.Substitutingf⁡\(ρx,ρy\)f\(\\rho\_\{x\},\\rho\_\{y\}\)back intoπD\\pi^\{\\mathrm\{D\}\}and imposing the first\-order conditions with respect toρx\\rho\_\{x\}andρy\\rho\_\{y\}yields the optimal accuracies\(ρx∗,ρy∗\)\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\)and the induced optimal pricefD:=f⁡\(ρx∗,ρy∗\)f^\{\\mathrm\{D\}\}:=f\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\}\):

ρx∗=12\+b​κy​θ​\(β​ℓ−2​\(β\+1\)​r\)D⁡\(ℓ\),ρy∗=12\+β​κx​θ​\(b\+θ​ℓ\)​\(β​ℓ−2​\(β\+1\)​r\)D⁡\(ℓ\),\\rho\_\{x\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{b\\kappa\_\{y\}\\theta\\big\(\\beta\\ell\-2\(\\beta\+1\)r\\big\)\}\{D\(\\ell\)\},\\qquad\\rho\_\{y\}^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{\\beta\\kappa\_\{x\}\\theta\(b\+\\theta\\ell\)\\big\(\\beta\\ell\-2\(\\beta\+1\)r\\big\)\}\{D\(\\ell\)\},and

fD=b​κx​κy​θ​\(β​ℓ−2​\(β\+1\)​r\)D⁡\(ℓ\)\.f^\{\\mathrm\{D\}\}=\\frac\{b\\kappa\_\{x\}\\kappa\_\{y\}\\theta\\big\(\\beta\\ell\-2\(\\beta\+1\)r\\big\)\}\{D\(\\ell\)\}\.To simplify exposition, we define the auxiliary variable𝒯=θ​\(β​ℓ−2​\(β\+1\)​r\)D⁡\(ℓ\)\\mathcal\{T\}=\\frac\{\\theta\(\\beta\\ell\-2\(\\beta\+1\)r\)\}\{D\(\\ell\)\}, allowing the equilibrium expressions to be rewritten compactly asρx∗=12\+b​κy​𝒯\\rho\_\{x\}^\{\*\}=\\frac\{1\}\{2\}\+b\\kappa\_\{y\}\\mathcal\{T\},ρy∗=12\+β​κx​\(b\+θ​ℓ\)​𝒯\\rho\_\{y\}^\{\*\}=\\frac\{1\}\{2\}\+\\beta\\kappa\_\{x\}\(b\+\\theta\\ell\)\\mathcal\{T\}, andfD=b​κx​κy​𝒯f^\{\\mathrm\{D\}\}=b\\kappa\_\{x\}\\kappa\_\{y\}\\mathcal\{T\}\.

We next identify the parameter regimes that guarantee the validity of these interior optimal solutions\. First, givenD⁡\(ℓ\)<0D\(\\ell\)<0, the requirementsρx∗\>1/2\\rho\_\{x\}^\{\*\}\>1/2,ρy∗\>1/2\\rho\_\{y\}^\{\*\}\>1/2, andfD\>0f^\{\\mathrm\{D\}\}\>0hold if and only if𝒯\>0\\mathcal\{T\}\>0, that is, if and only ifℓ<2​\(1\+β\)​rβ\\ell<\\frac\{2\(1\+\\beta\)r\}\{\\beta\}\. Additionally, imposing the logical upper bounds on accuracy levels \(ρx∗<1\\rho\_\{x\}^\{\*\}<1andρy∗<1\\rho\_\{y\}^\{\*\}<1\) requires𝒯<12​b​κy\\mathcal\{T\}<\\frac\{1\}\{2b\\kappa\_\{y\}\}and𝒯<12​β​κx​\(b\+θ​ℓ\)\\mathcal\{T\}<\\frac\{1\}\{2\\beta\\kappa\_\{x\}\(b\+\\theta\\ell\)\}, respectively\. To guarantee thatfD\>θ​rf^\{\\mathrm\{D\}\}\>\\theta r, we require𝒯\>θ​rb​κx​κy\\mathcal\{T\}\>\\frac\{\\theta r\}\{b\\kappa\_\{x\}\\kappa\_\{y\}\}\. Finally, substituting\(ρy∗,fD\)\(\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)into the profit function isolated to type\-yypatients yieldsπy​\(ρy∗,fD\)=y1​𝒯2\+y2​𝒯\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)=y\_\{1\}\\mathcal\{T\}^\{2\}\+y\_\{2\}\\mathcal\{T\}, where:

y1=β​κx2​κy​\(b2​β−2​b​κy\+2​b​β​θ​ℓ\+β​θ2​ℓ2\)andy2=−β​κx​κy​\(ℓ−2​r\)​θ\.y\_\{1\}=\\beta\\kappa\_\{x\}^\{2\}\\kappa\_\{y\}\\left\(b^\{2\}\\beta\-2b\\kappa\_\{y\}\+2b\\beta\\theta\\ell\+\\beta\\theta^\{2\}\\ell^\{2\}\\right\)\\quad\\text\{and\}\\quad y\_\{2\}=\-\\beta\\kappa\_\{x\}\\kappa\_\{y\}\(\\ell\-2r\)\\theta\.To ensure a well\-behaved optimization problem, we imposey1<0y\_\{1\}<0, which holds if and only ifκy\>b2​β\+2​b​β​θ​ℓ\+β​θ2​ℓ22​b\\kappa\_\{y\}\>\\frac\{b^\{2\}\\beta\+2b\\beta\\theta\\ell\+\\beta\\theta^\{2\}\\ell^\{2\}\}\{2b\}\(the standing assumptionκy\>κx\\kappa\_\{y\}\>\\kappa\_\{x\}is maintained throughout\)\. Settingy2\>0y\_\{2\}\>0yields the conditionℓ<2​r\\ell<2r\. Under these restrictions, the unique non\-zero root ofπy​\(ρy∗,fD\)=0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)=0with respect to𝒯\\mathcal\{T\}is strictly positive:

𝒯=\(ℓ−2​r\)​θκx​\(b2​β−2​b​κy\+2​b​β​θ​ℓ\+β​θ2​ℓ2\)\>0\.\\mathcal\{T\}=\\frac\{\(\\ell\-2r\)\\theta\}\{\\kappa\_\{x\}\\left\(b^\{2\}\\beta\-2b\\kappa\_\{y\}\+2b\\beta\\theta\\ell\+\\beta\\theta^\{2\}\\ell^\{2\}\\right\)\}\>0\.Consequently, ensuring positive profits from the type\-yysegment \(πy​\(ρy∗,fD\)\>0\\pi\_\{y\}\(\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)\>0\) is mathematically equivalent to bounding the auxiliary variable such that𝒯<\(ℓ−2​r\)​θκx​\(b2​β−2​b​κy\+2​b​β​θ​ℓ\+β​θ2​ℓ2\)\\mathcal\{T\}<\\frac\{\(\\ell\-2r\)\\theta\}\{\\kappa\_\{x\}\\left\(b^\{2\}\\beta\-2b\\kappa\_\{y\}\+2b\\beta\\theta\\ell\+\\beta\\theta^\{2\}\\ell^\{2\}\\right\)\}\.

Next consider the equal\-accuracy design, imposingρx=ρy≡ρ\\rho\_\{x\}=\\rho\_\{y\}\\equiv\\rho\. The Hessian matrixℍE\\mathbb\{H\}^\{\\mathrm\{E\}\}of the profit functionπE​\(ρ,f\)\\pi^\{\\mathrm\{E\}\}\(\\rho,f\)is given by:

ℍE:=\[−4​\(1\+β\)b2​\(1\+β\)2​\(1\+β\)−2​\(κx\+κy\)\]\\displaystyle\\mathbb\{H\}^\{\\mathrm\{E\}\}:=\\begin\{bmatrix\}\-\\frac\{4\(1\+\\beta\)\}\{b\}&2\(1\+\\beta\)\\\\ 2\(1\+\\beta\)&\-2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\end\{bmatrix\}\(OA2\)The corresponding leading principal minors ofℍE\\mathbb\{H\}^\{\\mathrm\{E\}\}are evaluated as:

𝔻1=−4​\(1\+β\)b​and​𝔻2=−4​\(1\+β\)​\(b⁡\(1\+β\)−2​\(κx\+κy\)\)b\.\\displaystyle\\mathbb\{D\}\_\{1\}=\-\\frac\{4\(1\+\\beta\)\}\{b\}\\penalty\\ \\text\{and\}\\penalty\\ \\mathbb\{D\}\_\{2\}=\-\\frac\{4\(1\+\\beta\)\(b\(1\+\\beta\)\-2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\)\}\{b\}\.To ensure the profit functionπE​\(ρ,f\)\\pi^\{\\mathrm\{E\}\}\(\\rho,f\)is jointly concave in\(f,ρ\)\(f,\\rho\), the Hessian matrix must be negative semi\-definite, which requires𝔻1≤0\\mathbb\{D\}\_\{1\}\\leq 0and𝔻2≥0\\mathbb\{D\}\_\{2\}\\geq 0, which yieldb<2​\(κx\+κy\)1\+βb<\\frac\{2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\{1\+\\beta\}\. Under these concavity conditions, we derive the optimal decision vector\(f,ρ\)\(f,\\rho\)\. The first\-order condition∂πE​\(ρ,f\)∂f=0\\frac\{\\partial\\pi^\{\\mathrm\{E\}\}\(\\rho,f\)\}\{\\partial f\}=0gives

f⁡\(ρ\)=\(ρ2−14\)​b\+12​θ​r\.f\(\\rho\)=\\Big\(\\frac\{\\rho\}\{2\}\-\\frac\{1\}\{4\}\\Big\)b\+\\frac\{1\}\{2\}\\theta r\.Substituting intoπE\\pi^\{\\mathrm\{E\}\}and imposing∂πE​\(ρ,f⁡\(ρ\)\)∂ρ=0\\frac\{\\partial\\pi^\{\\mathrm\{E\}\}\(\\rho,f\(\\rho\)\)\}\{\\partial\\rho\}=0yields

ρ∗=12\+\(β\+1\)​θ​r2​\(κx\+κy\)−b⁡\(β\+1\),fE:=f⁡\(ρ∗\)=\(κx\+κy\)​θ​r2​\(κx\+κy\)−b⁡\(β\+1\)\.\\rho^\{\*\}=\\frac\{1\}\{2\}\+\\frac\{\(\\beta\+1\)\\theta r\}\{2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-b\(\\beta\+1\)\},\\qquad f^\{\\mathrm\{E\}\}:=f\(\\rho^\{\*\}\)=\\frac\{\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\theta r\}\{2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-b\(\\beta\+1\)\}\.Ensuring that the equilibrium accuracy levels are interior,1/2<ρ∗<11/2<\\rho^\{\*\}<1, and that the conditionfE\>θ​rf^\{\\mathrm\{E\}\}\>\\theta rholds yields the following joint parameter restrictions:

r<κx\+κy2​θ​\(1\+β\)andκx\+κy1\+β<b<2​\(κx\+κy\)−2​r​θ​\(1\+β\)1\+β\.r<\\frac\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\{2\\theta\(1\+\\beta\)\}\\quad\\text\{and\}\\quad\\frac\{\\kappa\_\{x\}\+\\kappa\_\{y\}\}\{1\+\\beta\}<b<\\frac\{2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-2r\\theta\(1\+\\beta\)\}\{1\+\\beta\}\.
In addition, requiringπD​\(ρx∗,ρy∗,fD\)\>πE​\(ρ∗,fE\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)\>\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\},f^\{\\mathrm\{E\}\}\)leads to \(the left\-hand side below equals4​\[πD​\(ρx∗,ρy∗,fD\)−πE​\(ρ∗,fE\)\]4\\big\[\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)\-\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\},f^\{\\mathrm\{E\}\}\)\\big\]\):

4​b​κx​κy​\(b⁡\(β2​κx\+κy\)−2​\(1\+β\)​κx​κy\)​𝒯2\\displaystyle 4b\\kappa\_\{x\}\\kappa\_\{y\}\\left\(b\\left\(\\beta^\{2\}\\kappa\_\{x\}\+\\kappa\_\{y\}\\right\)\-2\(1\+\\beta\)\\kappa\_\{x\}\\kappa\_\{y\}\\right\)\\mathcal\{T\}^\{2\}\+4​θ​κx​κy​𝒯​\(2​r​\(1\+β\)\+β​ℓ​\(2​b​β​κx​𝒯−1\)\)\\displaystyle\+4\\theta\\kappa\_\{x\}\\kappa\_\{y\}\\mathcal\{T\}\\left\(2r\(1\+\\beta\)\+\\beta\\ell\(2b\\beta\\kappa\_\{x\}\\mathcal\{T\}\-1\)\\right\)\+4​θ2​\(\(1\+β\)​\(κx\+κy\)​r2b⁡\(b⁡\(1\+β\)−2​\(κx\+κy\)\)\+β2​κx2​κy​ℓ2​𝒯2\)\>0\.\\displaystyle\+4\\theta^\{2\}\\left\(\\frac\{\(1\+\\beta\)\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)r^\{2\}\}\{b\\left\(b\(1\+\\beta\)\-2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\right\)\}\+\\beta^\{2\}\\kappa\_\{x\}^\{2\}\\kappa\_\{y\}\\ell^\{2\}\\mathcal\{T\}^\{2\}\\right\)\\ \>0\.Taken together, the conditions above characterize when a disparate AI design is optimal; see[fig\.2\(b\)](https://arxiv.org/html/2608.13618#S0.F2.sf2)for a numerical illustration\.

We now compare the maximized payoffs asℓ\\ellvaries\. By the envelope theorem, the maximized payoff under the equal\-accuracy design,πE​\(ρ∗,fE\)\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\},f^\{\\mathrm\{E\}\}\), is independent ofℓ\\ell\. Under the disparate design, the envelope theorem implies

∂πD​\(ρx∗,ρy∗,fD\)∂ℓ=∂πD​\(ρx,ρy,f\)∂ℓ\|\(ρx,ρy,f\)=\(ρx∗,ρy∗,fD\)=−2​β​θ​fDb​\(1−ρy∗\)<0,\\frac\{\\partial\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)\}\{\\partial\\ell\}=\\left\.\\frac\{\\partial\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\},\\rho\_\{y\},f\)\}\{\\partial\\ell\}\\right\|\_\{\(\\rho\_\{x\},\\rho\_\{y\},f\)=\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)\}=\-\\frac\{2\\beta\\theta f^\{\\mathrm\{D\}\}\}\{b\}\\,\(1\-\\rho\_\{y\}^\{\*\}\)<0,where the inequality holds within the feasible region we discussed earlier; thus, the maximized disparate\-design payoff is strictly decreasing inℓ\\ell\. Moreover, asℓ→0\\ell\\to 0, the disparate\-design problem converges to the environment without the liability channel, and allowing\(ρx,ρy\)\(\\rho\_\{x\},\\rho\_\{y\}\)to differ is \(weakly\) valuable under the cost asymmetry\. In particular, the disparate\-design optimum yields a strictly higher payoff than the best equal\-accuracy design forℓ\\ellnear zero\. BecauseπE​\(ρ∗,fE\)\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\},f^\{\\mathrm\{E\}\}\)is constant whereasπD​\(ρx∗,ρy∗,fD\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)declines inℓ\\ell, there exists at most one cutoffℓ†\\ell^\{\\dagger\}within the feasible region such that the firm prefers a disparate design forℓ<ℓ†\\ell<\\ell^\{\\dagger\}\. If no suchℓ†\\ell^\{\\dagger\}falls within the feasible region, the disparate design remains globally optimal in the region\. Conversely, whenℓ≥ℓ†\\ell\\geq\\ell^\{\\dagger\}, the equal\-accuracy design can become optimal; see[fig\.2\(b\)](https://arxiv.org/html/2608.13618#S0.F2.sf2)for a numerical illustration\.

We next show that whenever a disparate design is optimal, it must satisfyρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}\. Atℓ→0\\ell\\to 0, the expressions above implyρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}\. Indeed,ρx∗−ρy∗=𝒯⁡\(b​κy−β​κx​\(b\+θ​ℓ\)\)\\rho\_\{x\}^\{\*\}\-\\rho\_\{y\}^\{\*\}=\\mathcal\{T\}\\big\(b\\kappa\_\{y\}\-\\beta\\kappa\_\{x\}\(b\+\\theta\\ell\)\\big\); with𝒯\>0\\mathcal\{T\}\>0, the bracket is strictly decreasing inℓ\\elland positive atℓ=0\\ell=0becauseκy\>β​κx\\kappa\_\{y\}\>\\beta\\kappa\_\{x\}, so the gap changes sign at most once\. Suppose instead that for someℓ^\\hat\{\\ell\}the disparate\-design optimum satisfiesρx∗=ρy∗\\rho\_\{x\}^\{\*\}=\\rho\_\{y\}^\{\*\}\. Then the candidate solution is itself equal\-accuracy, and the firm can \(weakly\) improve by switching to the equal\-accuracy problem; moreover, becauseπD​\(ρx∗,ρy∗,fD\)\\pi^\{\\mathrm\{D\}\}\(\\rho\_\{x\}^\{\*\},\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)is decreasing inℓ\\ellwhileπE​\(ρ∗,fE\)\\pi^\{\\mathrm\{E\}\}\(\\rho^\{\*\},f^\{\\mathrm\{E\}\}\)is independent ofℓ\\ell, onceρx∗=ρy∗\\rho\_\{x\}^\{\*\}=\\rho\_\{y\}^\{\*\}holds at someℓ^\\hat\{\\ell\}, the equal\-accuracy design is optimal atℓ^\\hat\{\\ell\}and for all largerℓ\\ell\. Hence, if the disparate design is optimal, it must be thatρx∗\>ρy∗\\rho\_\{x\}^\{\*\}\>\\rho\_\{y\}^\{\*\}\.

Finally,ρx∗\\rho\_\{x\}^\{\*\}\(and similarlyfDf^\{\\mathrm\{D\}\}\) can be non\-monotone inℓ\\ell\. Differentiating yields

∂ρx∗∂ℓ=b​β​κy​θ⋅E1​\(ℓ\)2​\(b2​\(β2​κx\+κy\)−2​b​κx​\(\(β\+1\)​κy−β2​θ​ℓ\)\+β2​κx​θ2​ℓ2\)2,\\frac\{\\partial\\rho\_\{x\}^\{\*\}\}\{\\partial\\ell\}=\\frac\{b\\beta\\kappa\_\{y\}\\theta\\cdot E\_\{1\}\(\\ell\)\}\{2\\Big\(b^\{2\}\(\\beta^\{2\}\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-2b\\kappa\_\{x\}\\big\(\(\\beta\+1\)\\kappa\_\{y\}\-\\beta^\{2\}\\theta\\ell\\big\)\+\\beta^\{2\}\\kappa\_\{x\}\\theta^\{2\}\\ell^\{2\}\\Big\)^\{2\}\},where

E1​\(ℓ\)=b2​\(β2​κx\+κy\)−2​b​\(β\+1\)​κx​\(κy−2​β​θ​r\)\+β​κx​θ2​ℓ​\(4​\(β\+1\)​r−β​ℓ\)\.\\displaystyle E\_\{1\}\(\\ell\)=b^\{2\}\(\\beta^\{2\}\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-2b\(\\beta\+1\)\\kappa\_\{x\}\(\\kappa\_\{y\}\-2\\beta\\theta r\)\+\\beta\\kappa\_\{x\}\\theta^\{2\}\\ell\\big\(4\(\\beta\+1\)r\-\\beta\\ell\\big\)\.\(OA3\)BecauseE1​\(ℓ\)E\_\{1\}\(\\ell\)is concave inℓ\\ell, there exist parameter values for whichE1​\(0\)<0E\_\{1\}\(0\)<0butE1​\(ϵ\)\>0E\_\{1\}\(\\epsilon\)\>0for some smallϵ\>0\\epsilon\>0below the minimum of the design\-switch cutoffℓ†\\ell^\{\\dagger\}and4​\(β\+1\)​rβ\\frac\{4\(\\beta\+1\)r\}\{\\beta\}\. To establish this, it suffices to ensure thatb2​\(β2​κx\+κy\)−2​b​\(β\+1\)​κx​\(κy−2​β​θ​r\)b^\{2\}\(\\beta^\{2\}\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-2b\(\\beta\+1\)\\kappa\_\{x\}\(\\kappa\_\{y\}\-2\\beta\\theta r\)is strictly negative but sufficiently close to zero, so thatE1​\(ϵ\)\>0E\_\{1\}\(\\epsilon\)\>0for a smallϵ\>0\\epsilon\>0\. This implies thatρx∗\\rho\_\{x\}^\{\*\}initially decreases and subsequently increases withℓ\\ell\. The same logic applies tofDf^\{\\mathrm\{D\}\}\.*Q\.E\.D\.*

###### Proposition OA1\.

There exist parameter values under which, asℓ\\ellincreases, equilibrium AI use for type\-yypatients first decreases and then increases\.

Proof of[propositionOA1](https://arxiv.org/html/2608.13618#Thmproposition1a)\. With endogenous pricing and a disparate design,

dyD=β⋅\(2​ρy∗−1\)​b−2​\(1−ρy∗\)​θ​ℓ−2​fD\+2​θ​rb,d\_\{y\}^\{\\mathrm\{D\}\}=\\beta\\cdot\\frac\{\(2\\rho\_\{y\}^\{\*\}\-1\)b\-2\(1\-\\rho\_\{y\}^\{\*\}\)\\theta\\ell\-2f^\{\\mathrm\{D\}\}\+2\\theta r\}\{b\},where\(ρy∗,fD\)\(\\rho\_\{y\}^\{\*\},f^\{\\mathrm\{D\}\}\)are given in[proposition5](https://arxiv.org/html/2608.13618#Thmproposition5)\. DifferentiatingdyDd\_\{y\}^\{\\mathrm\{D\}\}with respect toℓ\\elland simplifying yields

∂dyD∂ℓ=β​κy​θ​\(\(β\+2\)​κx−b\)⋅E1​\(ℓ\)\(b2​\(β2​κx\+κy\)−2​b​κx​\(\(β\+1\)​κy−β2​θ​ℓ\)\+β2​κx​θ2​ℓ2\)2,\\frac\{\\partial d\_\{y\}^\{\\mathrm\{D\}\}\}\{\\partial\\ell\}=\\frac\{\\beta\\kappa\_\{y\}\\theta\\big\(\(\\beta\+2\)\\kappa\_\{x\}\-b\\big\)\\cdot E\_\{1\}\(\\ell\)\}\{\\Big\(b^\{2\}\(\\beta^\{2\}\\kappa\_\{x\}\+\\kappa\_\{y\}\)\-2b\\kappa\_\{x\}\\big\(\(\\beta\+1\)\\kappa\_\{y\}\-\\beta^\{2\}\\theta\\ell\\big\)\+\\beta^\{2\}\\kappa\_\{x\}\\theta^\{2\}\\ell^\{2\}\\Big\)^\{2\}\},whereE1​\(ℓ\)E\_\{1\}\(\\ell\)is defined in[eq\.OA3](https://arxiv.org/html/2608.13618#S0.E3a)in the proof of[proposition5](https://arxiv.org/html/2608.13618#Thmproposition5)and it is a concave quadratic inℓ\\ell\. Letℓmin\\ell\_\{\\min\}denote the smaller root ofE1​\(ℓ\)=0E\_\{1\}\(\\ell\)=0:

ℓmin=2​\(β\+1\)​rβ−b2​\(β2\+κy/κx\)−2​b​\(β\+1\)​\(κy−2​β​r​θ\)\+4​\(β\+1\)2​r2​θ2β​θ\.\\ell\_\{\\min\}=\\frac\{2\(\\beta\+1\)r\}\{\\beta\}\-\\frac\{\\sqrt\{\\,b^\{2\}\\\!\\left\(\\beta^\{2\}\+\\kappa\_\{y\}/\\kappa\_\{x\}\\right\)\-2b\(\\beta\+1\)\\left\(\\kappa\_\{y\}\-2\\beta r\\theta\\right\)\+4\(\\beta\+1\)^\{2\}r^\{2\}\\theta^\{2\}\\,\}\}\{\\beta\\theta\}\.If\(β\+2\)​κx\>b\(\\beta\+2\)\\kappa\_\{x\}\>b,E1​\(0\)<0E\_\{1\}\(0\)<0, andE1​\(0\)\>−4​\(β\+1\)2​κx​θ2​r2E\_\{1\}\(0\)\>\-4\(\\beta\+1\)^\{2\}\\kappa\_\{x\}\\theta^\{2\}r^\{2\}\(so thatE1E\_\{1\}admits a real root\), then∂dyD∂ℓ<0\\frac\{\\partial d\_\{y\}^\{\\mathrm\{D\}\}\}\{\\partial\\ell\}<0forℓ\\ellnear zero and becomes positive forℓ\\elljust aboveℓmin\\ell\_\{\\min\}\. The last condition cannot be dispensed with:E1E\_\{1\}is concave with vertex atℓ=2​\(β\+1\)​r/β\\ell=2\(\\beta\+1\)r/\\beta, which coincides with the boundary of the feasible region \(where𝒯\>0\\mathcal\{T\}\>0fails\), soE1E\_\{1\}is strictly increasing throughout the feasible region and its maximum there equalsE1​\(0\)\+4​\(β\+1\)2​κx​θ2​r2E\_\{1\}\(0\)\+4\(\\beta\+1\)^\{2\}\\kappa\_\{x\}\\theta^\{2\}r^\{2\}\. Consequently the crossing atℓmin\\ell\_\{\\min\}is unique and the sign change is global rather than local\. Providedℓmin\\ell\_\{\\min\}lies below the design\-switch cutoffℓ†\\ell^\{\\dagger\}\(so the firm still supplies a disparate design in this neighborhood\), the comparative static in[propositionOA1](https://arxiv.org/html/2608.13618#Thmproposition1a)follows\.*Q\.E\.D\.*

The mechanism behind[propositionOA1](https://arxiv.org/html/2608.13618#Thmproposition1a)parallels that of[proposition2](https://arxiv.org/html/2608.13618#Thmproposition2): liability changes the physician’s marginal willingness to rely on AI for type\-yypatients, and the firm responds by jointly adjusting price and accuracy, which can reverse the direction of utilization asℓ\\ellincreases\.

We next examine whether an equal\-accuracy requirement can still reduce welfare for both patient types when the AI firm endogenizes the per\-use price \(that is, whenffbecomes a choice variable and we imposec=fc=f\)\.[fig\.OA2](https://arxiv.org/html/2608.13618#S0.F2)reports the comparison\. Panel \(a\) corresponds to the baseline model with exogenous\(c,f\)\(c,f\), whereas panel \(b\) corresponds to the extension in which the firm choosesffandccmoves one\-for\-one with it\. The two panels use the same parameter values and differ only in whether\(c,f\)\(c,f\)are fixed or endogenously determined\. The qualitative welfare regions are unchanged: the central patterns from the baseline carry over, and the mandate can still generate the same welfare reversals\.

\(a\)Exogenousff\(b\)Endogenousff
Figure OA2:Welfare effects of an equal\-accuracy requirement under exogenous and endogenous pricing\. Panel \(a\) reproduces the baseline with exogenous\(c,f\)\(c,f\); panel \(b\) endogenizesffand imposesc=fc=f\. Regions labeled “Unbiased AI is optimal” are those in which the equal\-accuracy design is optimal\.
### OA3\. Welfare\-Maximizing Liability \([section6\.2](https://arxiv.org/html/2608.13618#S6.SS2)\)

For each parameter tuple, we compute equilibrium aggregate welfare under the disparate\-design and equal\-accuracy\-design regimes as functions of liability and then compare these values at the relevant design boundary\. Operationally, we first identify the design\-switch cutoff implied by[proposition1](https://arxiv.org/html/2608.13618#Thmproposition1), and then evaluate welfare on each side of that cutoff using the corresponding equilibrium design\.

Across the numerical configurations reported in the main text, the welfare\-maximizing liability is attained at the regime boundary: either at the largestℓ\\ellfor which the disparate design remains optimal or at the smallestℓ\\ellthat induces the equal\-accuracy design\. This procedure yields the two benchmark patterns illustrated in[figs\.5\(a\)](https://arxiv.org/html/2608.13618#S6.F5.sf1)and[5\(b\)](https://arxiv.org/html/2608.13618#S6.F5.sf2)\.

### OA4\. Type\-Specific Priors \([section6\.3](https://arxiv.org/html/2608.13618#S6.SS3)\)

As in the baseline, the physician’s AI\-use rule is characterized by two prior cutoffs, and conditional on use the physician follows the AI signal\.

###### Lemma OA1\.

\(a\) For type\-xxpatients, the physician uses AI if and only if

\(1−ρx\)​b\+c−θ​rb<αx<min⁡\{α¯,ρx​b−c\+θ​rb\}\.\\frac\{\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\}\{b\}<\\alpha\_\{x\}<\\min\\left\\\{\\overline\{\\alpha\},\\frac\{\\rho\_\{x\}b\-c\+\\theta r\}\{b\}\\right\\\}\.Moreover, conditional on AI use, the physician follows the AI signal\.

\(b\) For type\-yypatients, the physician uses AI if and only if

max⁡\{α¯,\(1−ρy\)​\(b\+θ​ℓ\)\+c−θ​rb\}<αy<ρy​b−\(1−ρy\)​θ​ℓ−c\+θ​rb\.\\max\\left\\\{\\underline\{\\alpha\},\\frac\{\(1\-\\rho\_\{y\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}\\right\\\}<\\alpha\_\{y\}<\\frac\{\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\}\{b\}\.Moreover, conditional on AI use, the physician follows the AI signal\.

The proof of[lemmaOA1](https://arxiv.org/html/2608.13618#Thmlemma1a)follows the same logic as the proofs of[lemmas1](https://arxiv.org/html/2608.13618#Thmlemma1)and[2](https://arxiv.org/html/2608.13618#Thmlemma2)and is therefore omitted\. Relative to the baseline, the ordering of type\-specific AI use need not be uniform because use is jointly determined by type\-dependent priors and payoff trade\-offs\. Nevertheless, the baseline pattern remains feasible: there are parameter values for which the physician uses AI less for type\-yypatients than for type\-xxpatients\. Under a disparate algorithm, integrating the physician’s AI use rule in[lemmaOA1](https://arxiv.org/html/2608.13618#Thmlemma1a)against the prior densities \(1/α¯1/\\overline\{\\alpha\}on\(0,α¯\)\(0,\\overline\{\\alpha\}\)for type\-xxpatients and1/\(1−α¯\)1/\(1\-\\underline\{\\alpha\}\)on\(α¯,1\)\(\\underline\{\\alpha\},1\)for type\-yypatients\) yields expected AI demand for each patient type\. To ease notation, for type\-xxpatients, letd¯x​\(ρx\)=α¯−\(\(1−ρx\)​b\+c−θ​r\)/b\\bar\{d\}\_\{x\}\(\\rho\_\{x\}\)=\\overline\{\\alpha\}\-\(\(1\-\\rho\_\{x\}\)b\+c\-\\theta r\)/bandd~x​\(ρx\)=\(\(2​ρx−1\)​b−2​c\+2​θ​r\)/b\\tilde\{d\}\_\{x\}\(\\rho\_\{x\}\)=\(\(2\\rho\_\{x\}\-1\)b\-2c\+2\\theta r\)/b\. For type\-yypatients, letd¯y​\(ρy\)=\(\(ρy​b−\(1−ρy\)​θ​ℓ−c\+θ​r\)/b\)−α¯\\bar\{d\}\_\{y\}\(\\rho\_\{y\}\)=\(\(\\rho\_\{y\}b\-\(1\-\\rho\_\{y\}\)\\theta\\ell\-c\+\\theta r\)/b\)\-\\underline\{\\alpha\}andd~y​\(ρy\)=\(\(2​ρy−1\)​b−2​\(1−ρy\)​θ​ℓ−2​c\+2​θ​r\)/b\\tilde\{d\}\_\{y\}\(\\rho\_\{y\}\)=\(\(2\\rho\_\{y\}\-1\)b\-2\(1\-\\rho\_\{y\}\)\\theta\\ell\-2c\+2\\theta r\)/b\.

dxD\\displaystyle d\_\{x\}^\{\\mathrm\{D\}\}=min⁡\{d¯x​\(ρx\),d~x​\(ρx\)\},\\displaystyle=\\min\\\{\\bar\{d\}\_\{x\}\(\\rho\_\{x\}\),\\tilde\{d\}\_\{x\}\(\\rho\_\{x\}\)\\\},dyD\\displaystyle d\_\{y\}^\{\\mathrm\{D\}\}=β~​min⁡\{d¯y​\(ρy\),d~y​\(ρy\)\}\.\\displaystyle=\\tilde\{\\beta\}\\min\\\{\\bar\{d\}\_\{y\}\(\\rho\_\{y\}\),\\tilde\{d\}\_\{y\}\(\\rho\_\{y\}\)\\\}\.\(OA4\)Note that all optima characterized in this extension lie in the region of strictly positive demand \(this is verified in every numerical configuration we report\), so the positive\-part operator is omitted from the piecewise expressions without affecting the analysis\. The AI firm then chooses\(ρx,ρy\)\(\\rho\_\{x\},\\rho\_\{y\}\)to maximize expected profit in[eq\.1](https://arxiv.org/html/2608.13618#S3.E1)\. Under an equal\-accuracy algorithm, defined¯xE​\(ρ\)=α¯−\(\(1−ρ\)​b\+c−θ​r\)/b\\bar\{d\}\_\{x\}^\{\\mathrm\{E\}\}\(\\rho\)=\\overline\{\\alpha\}\-\(\(1\-\\rho\)b\+c\-\\theta r\)/b,d¯yE​\(ρ\)=\(ρ​b−c\+θ​r\)/b−α¯\\bar\{d\}\_\{y\}^\{\\mathrm\{E\}\}\(\\rho\)=\(\\rho b\-c\+\\theta r\)/b\-\\underline\{\\alpha\}, andd~E​\(ρ\)=\(\(2​ρ−1\)​b−2​c\+2​θ​r\)/b\\tilde\{d\}^\{\\mathrm\{E\}\}\(\\rho\)=\(\(2\\rho\-1\)b\-2c\+2\\theta r\)/b\. Then

dxE\\displaystyle d\_\{x\}^\{\\mathrm\{E\}\}=min⁡\{d¯xE​\(ρ\),d~E​\(ρ\)\},\\displaystyle=\\min\\\{\\bar\{d\}\_\{x\}^\{\\mathrm\{E\}\}\(\\rho\),\\tilde\{d\}^\{\\mathrm\{E\}\}\(\\rho\)\\\},dyE\\displaystyle d\_\{y\}^\{\\mathrm\{E\}\}=β~​min⁡\{d¯yE​\(ρ\),d~E​\(ρ\)\}\.\\displaystyle=\\tilde\{\\beta\}\\min\\\{\\bar\{d\}\_\{y\}^\{\\mathrm\{E\}\}\(\\rho\),\\tilde\{d\}^\{\\mathrm\{E\}\}\(\\rho\)\\\}\.The AI firm choosesρ∈\(1/2,1\)\\rho\\in\(1/2,1\)to maximize expected profit in[eq\.2](https://arxiv.org/html/2608.13618#S3.E2)\.

The next proposition characterizes the resulting equilibrium accuracy choices\.

###### Proposition OA2\.

There exist parameter regions in which the AI firm develops a disparate algorithm whenℓ\\ellis below a cutoff and an equal\-accuracy algorithm whenℓ\\ellexceeds that cutoff\. Moreover, ifb​κy\>2​β~​κx​\(b\+θ​ℓ\)b\\kappa\_\{y\}\>2\\tilde\{\\beta\}\\kappa\_\{x\}\(b\+\\theta\\ell\), then the associated accuracies satisfyρx∗\>ρy∗\\rho^\{\*\}\_\{x\}\>\\rho^\{\*\}\_\{y\}whenever the disparate design is optimal\. The proof characterizes the optimal accuracies under both designs\.

Proof of[propositionOA2](https://arxiv.org/html/2608.13618#Thmproposition2a)\. The proof proceeds in three steps\. Step 1 characterizes the optimal accuracies under the disparate design and establishes the rankingρx∗\>ρy∗\\rho^\{\*\}\_\{x\}\>\\rho^\{\*\}\_\{y\}underb​κy\>2​β~​κx​\(b\+θ​ℓ\)b\\kappa\_\{y\}\>2\\tilde\{\\beta\}\\kappa\_\{x\}\(b\+\\theta\\ell\)\. Step 2 characterizes the optimal accuracy under the equal\-accuracy design\. Step 3 establishes the liability cutoff\.

Step 1: Disparate design\.We begin with the disparate design\. Because the physician’s usage rule changes discretely at an accuracy cutoff, the AI firm’s objective is piecewise inρt\\rho\_\{t\}\. Conditional on supplying a disparate design, the problem separates by patient type\. We therefore solve forρx∗\\rho\_\{x\}^\{\*\}andρy∗\\rho\_\{y\}^\{\*\}in turn\. For type\-xxpatients, define the cutoff accuracy asρ¯x:=α¯​b\+c−θ​rb\\bar\{\\rho\}\_\{x\}:=\\frac\{\\overline\{\\alpha\}b\+c\-\\theta r\}\{b\}\. The firm choosesρx∈\(1/2,1\)\\rho\_\{x\}\\in\(1/2,1\)to maximize

πx​\(ρx\)=\{f~⋅d~x​\(ρx\)−κx​\(ρx−12\)2,if​12<ρx≤ρ¯x,f~⋅d¯x​\(ρx\)−κx​\(ρx−12\)2,if​ρ¯x<ρx<1\.\\displaystyle\\pi\_\{x\}\(\\rho\_\{x\}\)=\\begin\{cases\}\\tilde\{f\}\\cdot\\tilde\{d\}\_\{x\}\(\\rho\_\{x\}\)\-\\kappa\_\{x\}\(\\rho\_\{x\}\-\\tfrac\{1\}\{2\}\)^\{2\},&\\text\{if \}\\tfrac\{1\}\{2\}<\\rho\_\{x\}\\leq\\bar\{\\rho\}\_\{x\},\\\\\[6\.0pt\] \\tilde\{f\}\\cdot\\bar\{d\}\_\{x\}\(\\rho\_\{x\}\)\-\\kappa\_\{x\}\(\\rho\_\{x\}\-\\tfrac\{1\}\{2\}\)^\{2\},&\\text\{if \}\\bar\{\\rho\}\_\{x\}<\\rho\_\{x\}<1\.\\end\{cases\}On each region,πx​\(ρx\)\\pi\_\{x\}\(\\rho\_\{x\}\)is strictly concave inρx\\rho\_\{x\}\. The first\-order condition yields the interior candidateρx=12\+f~κx\\rho\_\{x\}=\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\}\{\\kappa\_\{x\}\}in the first region andρx=12\+f~2​κx\\rho\_\{x\}=\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\}in the second\. The global maximizer is obtained by checking whether the relevant candidate lies in its region; otherwise the optimum is attained at the boundaryρ¯x\\bar\{\\rho\}\_\{x\}\. This yields

ρx∗=\{12\+f~κx,if​12\+f~κx≤ρ¯x,ρ¯x,if​12\+f~2​κx≤ρ¯x<12\+f~κx,12\+f~2​κx,if​ρ¯x<12\+f~2​κx\.\\displaystyle\\rho\_\{x\}^\{\*\}=\\begin\{cases\}\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{f\}\}\{\\kappa\_\{x\}\},&\\text\{if \}\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{f\}\}\{\\kappa\_\{x\}\}\\leq\\bar\{\\rho\}\_\{x\},\\\\\[6\.0pt\] \\bar\{\\rho\}\_\{x\},&\\text\{if \}\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\}\\leq\\bar\{\\rho\}\_\{x\}<\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{f\}\}\{\\kappa\_\{x\}\},\\\\\[6\.0pt\] \\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\},&\\text\{if \}\\bar\{\\rho\}\_\{x\}<\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\}\.\\end\{cases\}\(OA5\)
For type\-yypatients, define the corresponding cutoff asρ¯y:=1−α¯​b−\(c−θ​r\)b\+θ​ℓ\\bar\{\\rho\}\_\{y\}:=1\-\\frac\{\\underline\{\\alpha\}b\-\(c\-\\theta r\)\}\{b\+\\theta\\ell\}\. The AI firm choosesρy∈\(1/2,1\)\\rho\_\{y\}\\in\(1/2,1\)to maximize

πy​\(ρy,ℓ\)=\{f~​β~⋅d~y​\(ρy\)−κy​\(ρy−12\)2,if​12<ρy≤ρ¯y,f~​β~⋅d¯y​\(ρy\)−κy​\(ρy−12\)2,if​ρ¯y<ρy<1\.\\displaystyle\\pi\_\{y\}\(\\rho\_\{y\};\\ell\)=\\begin\{cases\}\\tilde\{f\}\\tilde\{\\beta\}\\cdot\\tilde\{d\}\_\{y\}\(\\rho\_\{y\}\)\-\\kappa\_\{y\}\(\\rho\_\{y\}\-\\tfrac\{1\}\{2\}\)^\{2\},&\\text\{if \}\\tfrac\{1\}\{2\}<\\rho\_\{y\}\\leq\\bar\{\\rho\}\_\{y\},\\\\\[6\.0pt\] \\tilde\{f\}\\tilde\{\\beta\}\\cdot\\bar\{d\}\_\{y\}\(\\rho\_\{y\}\)\-\\kappa\_\{y\}\(\\rho\_\{y\}\-\\tfrac\{1\}\{2\}\)^\{2\},&\\text\{if \}\\bar\{\\rho\}\_\{y\}<\\rho\_\{y\}<1\.\\end\{cases\}Again, strict concavity holds on each region\. The first\-order condition yields the interior candidateρy=12\+β~​f~​\(b\+θ​ℓ\)b​κy\\rho\_\{y\}=\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}in the first region andρy=12\+β~​f~​\(b\+θ​ℓ\)2​b​κy\\rho\_\{y\}=\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{2b\\kappa\_\{y\}\}in the second\. Comparing these candidates withρ¯y\\bar\{\\rho\}\_\{y\}gives

ρy∗=\{12\+β~​f~​\(b\+θ​ℓ\)b​κy,if​12\+β~​f~​\(b\+θ​ℓ\)b​κy≤ρ¯y,ρ¯y,if​12\+β~​f~​\(b\+θ​ℓ\)2​b​κy≤ρ¯y<12\+β~​f~​\(b\+θ​ℓ\)b​κy,12\+β~​f~​\(b\+θ​ℓ\)2​b​κy,if​ρ¯y<12\+β~​f~​\(b\+θ​ℓ\)2​b​κy\.\\displaystyle\\rho\_\{y\}^\{\*\}=\\begin\{cases\}\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\},&\\text\{if \}\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}\\leq\\bar\{\\rho\}\_\{y\},\\\\\[6\.0pt\] \\bar\{\\rho\}\_\{y\},&\\text\{if \}\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{2b\\kappa\_\{y\}\}\\leq\\bar\{\\rho\}\_\{y\}<\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\},\\\\\[6\.0pt\] \\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{2b\\kappa\_\{y\}\},&\\text\{if \}\\bar\{\\rho\}\_\{y\}<\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{2b\\kappa\_\{y\}\}\.\\end\{cases\}\(OA6\)
*Accuracy ranking\.*We now show that the assumptionb​κy\>2​β~​κx​\(b\+θ​ℓ\)b\\kappa\_\{y\}\>2\\tilde\{\\beta\}\\kappa\_\{x\}\(b\+\\theta\\ell\)impliesρx∗\>ρy∗\\rho^\{\*\}\_\{x\}\>\\rho^\{\*\}\_\{y\}across all branch combinations of \([OA5](https://arxiv.org/html/2608.13618#S0.E5a)\) and \([OA6](https://arxiv.org/html/2608.13618#S0.E6a)\)\. First, \([OA5](https://arxiv.org/html/2608.13618#S0.E5a)\) implies

ρx∗≥12\+f~2​κx;\\rho^\{\*\}\_\{x\}\\;\\geq\\;\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\};this holds with equality on the third branch, holds on the second branch by its defining conditionρ¯x≥12\+f~2​κx\\bar\{\\rho\}\_\{x\}\\geq\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\}, and holds on the first branch becausef~κx\>f~2​κx\\tfrac\{\\tilde\{f\}\}\{\\kappa\_\{x\}\}\>\\tfrac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\}\. Second, \([OA6](https://arxiv.org/html/2608.13618#S0.E6a)\) implies

ρy∗≤12\+β~​f~​\(b\+θ​ℓ\)b​κy;\\rho^\{\*\}\_\{y\}\\;\\leq\\;\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\};this holds with equality on the first branch, holds on the second branch by its defining conditionρ¯y<12\+β~​f~​\(b\+θ​ℓ\)b​κy\\bar\{\\rho\}\_\{y\}<\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}, and holds on the third branch becauseβ~​f~​\(b\+θ​ℓ\)2​b​κy<β~​f~​\(b\+θ​ℓ\)b​κy\\tfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{2b\\kappa\_\{y\}\}<\\tfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}\. The assumptionb​κy\>2​β~​κx​\(b\+θ​ℓ\)b\\kappa\_\{y\}\>2\\tilde\{\\beta\}\\kappa\_\{x\}\(b\+\\theta\\ell\)is equivalent to

β~​f~​\(b\+θ​ℓ\)b​κy<f~2​κx,\\frac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}\\;<\\;\\frac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\},and therefore

ρy∗≤12\+β~​f~​\(b\+θ​ℓ\)b​κy<12\+f~2​κx≤ρx∗,\\rho^\{\*\}\_\{y\}\\;\\leq\\;\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{b\\kappa\_\{y\}\}\\;<\\;\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\}\{2\\kappa\_\{x\}\}\\;\\leq\\;\\rho^\{\*\}\_\{x\},so thatρx∗\>ρy∗\\rho^\{\*\}\_\{x\}\>\\rho^\{\*\}\_\{y\}whenever the disparate design is supplied\. In particular, the disparity\-liability channel, which is defined forρx\>ρy\\rho\_\{x\}\>\\rho\_\{y\}, is well defined at the firm’s optimum\.

Step 2: Equal\-accuracy design\.Imposeρx=ρy≡ρ\\rho\_\{x\}=\\rho\_\{y\}\\equiv\\rho\. The relevant usage cutoffs areρ¯x\\bar\{\\rho\}\_\{x\}defined in Step 1 and

ρ¯\(y\):=\(1−α¯\)​b\+c−θ​rb,\\bar\{\\rho\}^\{\(y\)\}:=\\frac\{\(1\-\\underline\{\\alpha\}\)b\+c\-\\theta r\}\{b\},so thatdxE=d~E​\(ρ\)d^\{E\}\_\{x\}=\\tilde\{d\}^\{E\}\(\\rho\)if and only ifρ≤ρ¯x\\rho\\leq\\bar\{\\rho\}\_\{x\}anddyE=β~​d~E​\(ρ\)d^\{E\}\_\{y\}=\\tilde\{\\beta\}\\,\\tilde\{d\}^\{E\}\(\\rho\)if and only ifρ≤ρ¯\(y\)\\rho\\leq\\bar\{\\rho\}^\{\(y\)\}\. Note thatρ¯\(y\)<ρ¯x\\bar\{\\rho\}^\{\(y\)\}<\\bar\{\\rho\}\_\{x\}if and only ifα¯\+α¯\>1\\overline\{\\alpha\}\+\\underline\{\\alpha\}\>1\.

*Case A:α¯\+α¯\>1\\overline\{\\alpha\}\+\\underline\{\\alpha\}\>1\(soρ¯\(y\)<ρ¯x\\bar\{\\rho\}^\{\(y\)\}<\\bar\{\\rho\}\_\{x\}\)\.*The firm’s objective is

πE​\(ρ\)=\{f~​\(1\+β~\)​d~E​\(ρ\)−\(κx\+κy\)​\(ρ−12\)2,if​12<ρ≤ρ¯\(y\),f~​\[d~E​\(ρ\)\+β~​d¯yE​\(ρ\)\]−\(κx\+κy\)​\(ρ−12\)2,if​ρ¯\(y\)<ρ≤ρ¯x,f~​\[d¯xE​\(ρ\)\+β~​d¯yE​\(ρ\)\]−\(κx\+κy\)​\(ρ−12\)2,if​ρ¯x<ρ<1\.\\pi^\{E\}\(\\rho\)=\\begin\{cases\}\\tilde\{f\}\(1\+\\tilde\{\\beta\}\)\\,\\tilde\{d\}^\{E\}\(\\rho\)\-\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\big\(\\rho\-\\tfrac\{1\}\{2\}\\big\)^\{2\},&\\text\{if \}\\tfrac\{1\}\{2\}<\\rho\\leq\\bar\{\\rho\}^\{\(y\)\},\\\\\[4\.30554pt\] \\tilde\{f\}\\big\[\\tilde\{d\}^\{E\}\(\\rho\)\+\\tilde\{\\beta\}\\,\\bar\{d\}^\{E\}\_\{y\}\(\\rho\)\\big\]\-\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\big\(\\rho\-\\tfrac\{1\}\{2\}\\big\)^\{2\},&\\text\{if \}\\bar\{\\rho\}^\{\(y\)\}<\\rho\\leq\\bar\{\\rho\}\_\{x\},\\\\\[4\.30554pt\] \\tilde\{f\}\\big\[\\bar\{d\}^\{E\}\_\{x\}\(\\rho\)\+\\tilde\{\\beta\}\\,\\bar\{d\}^\{E\}\_\{y\}\(\\rho\)\\big\]\-\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\\big\(\\rho\-\\tfrac\{1\}\{2\}\\big\)^\{2\},&\\text\{if \}\\bar\{\\rho\}\_\{x\}<\\rho<1\.\\end\{cases\}The marginal revenues on the three regions are2​f~​\(1\+β~\)2\\tilde\{f\}\(1\+\\tilde\{\\beta\}\),f~​\(2\+β~\)\\tilde\{f\}\(2\+\\tilde\{\\beta\}\), andf~​\(1\+β~\)\\tilde\{f\}\(1\+\\tilde\{\\beta\}\), which are strictly decreasing across regions; combined with the strictly convex cost,πE​\(ρ\)\\pi^\{E\}\(\\rho\)is globally concave inρ\\rho\. The region\-specific first\-order candidates are

ρ1:=12\+f~​\(1\+β~\)κx\+κy,ρ2:=12\+f~​\(2\+β~\)2​\(κx\+κy\),ρ3:=12\+f~​\(1\+β~\)2​\(κx\+κy\),\\rho\_\{1\}:=\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\(1\+\\tilde\{\\beta\}\)\}\{\\kappa\_\{x\}\+\\kappa\_\{y\}\},\\qquad\\rho\_\{2\}:=\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\(2\+\\tilde\{\\beta\}\)\}\{2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\},\\qquad\\rho\_\{3\}:=\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\(1\+\\tilde\{\\beta\}\)\}\{2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\},withρ1\>ρ2\>ρ3\\rho\_\{1\}\>\\rho\_\{2\}\>\\rho\_\{3\}, and the unique maximizer is

ρ∗=\{ρ1,if​ρ1≤ρ¯\(y\),ρ¯\(y\),if​ρ2≤ρ¯\(y\)<ρ1,ρ2,if​ρ¯\(y\)<ρ2≤ρ¯x,ρ¯x,if​ρ3≤ρ¯x<ρ2,ρ3,if​ρ¯x<ρ3,\\rho^\{\*\}=\\begin\{cases\}\\rho\_\{1\},&\\text\{if \}\\rho\_\{1\}\\leq\\bar\{\\rho\}^\{\(y\)\},\\\\ \\bar\{\\rho\}^\{\(y\)\},&\\text\{if \}\\rho\_\{2\}\\leq\\bar\{\\rho\}^\{\(y\)\}<\\rho\_\{1\},\\\\ \\rho\_\{2\},&\\text\{if \}\\bar\{\\rho\}^\{\(y\)\}<\\rho\_\{2\}\\leq\\bar\{\\rho\}\_\{x\},\\\\ \\bar\{\\rho\}\_\{x\},&\\text\{if \}\\rho\_\{3\}\\leq\\bar\{\\rho\}\_\{x\}<\\rho\_\{2\},\\\\ \\rho\_\{3\},&\\text\{if \}\\bar\{\\rho\}\_\{x\}<\\rho\_\{3\},\\end\{cases\}\(OA7\)truncated, if necessary, to the admissible interval\(1/2,1\)\(1/2,1\); as in the baseline, we maintain parameter restrictions under which the optimum is interior\.

*Case B:α¯\+α¯≤1\\overline\{\\alpha\}\+\\underline\{\\alpha\}\\leq 1\(soρ¯x≤ρ¯\(y\)\\bar\{\\rho\}\_\{x\}\\leq\\bar\{\\rho\}^\{\(y\)\}\)\.*The middle region becomes\(ρ¯x,ρ¯\(y\)\]\(\\bar\{\\rho\}\_\{x\},\\bar\{\\rho\}^\{\(y\)\}\], on whichπE​\(ρ\)=f~​\[d¯xE​\(ρ\)\+β~​d~E​\(ρ\)\]−\(κx\+κy\)​\(ρ−12\)2\\pi^\{E\}\(\\rho\)=\\tilde\{f\}\\big\[\\bar\{d\}^\{E\}\_\{x\}\(\\rho\)\+\\tilde\{\\beta\}\\,\\tilde\{d\}^\{E\}\(\\rho\)\\big\]\-\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\(\\rho\-\\tfrac\{1\}\{2\}\)^\{2\}with marginal revenuef~​\(1\+2​β~\)\\tilde\{f\}\(1\+2\\tilde\{\\beta\}\)and candidateρ2′:=12\+f~​\(1\+2​β~\)2​\(κx\+κy\)\\rho\_\{2\}^\{\\prime\}:=\\tfrac\{1\}\{2\}\+\\tfrac\{\\tilde\{f\}\(1\+2\\tilde\{\\beta\}\)\}\{2\(\\kappa\_\{x\}\+\\kappa\_\{y\}\)\}\. Since2​f~​\(1\+β~\)\>f~​\(1\+2​β~\)\>f~​\(1\+β~\)2\\tilde\{f\}\(1\+\\tilde\{\\beta\}\)\>\\tilde\{f\}\(1\+2\\tilde\{\\beta\}\)\>\\tilde\{f\}\(1\+\\tilde\{\\beta\}\), global concavity again holds, andρ∗\\rho^\{\*\}is given by

ρ∗=\{ρ1,if​ρ1≤ρ¯x,ρ¯x,if​ρ2′≤ρ¯x<ρ1,ρ2′,if​ρ¯x<ρ2′≤ρ¯\(y\),ρ¯\(y\),if​ρ3≤ρ¯\(y\)<ρ2′,ρ3,if​ρ¯\(y\)<ρ3\.\\rho^\{\*\}=\\begin\{cases\}\\rho\_\{1\},&\\text\{if \}\\rho\_\{1\}\\leq\\bar\{\\rho\}\_\{x\},\\\\ \\bar\{\\rho\}\_\{x\},&\\text\{if \}\\rho\_\{2\}^\{\\prime\}\\leq\\bar\{\\rho\}\_\{x\}<\\rho\_\{1\},\\\\ \\rho\_\{2\}^\{\\prime\},&\\text\{if \}\\bar\{\\rho\}\_\{x\}<\\rho\_\{2\}^\{\\prime\}\\leq\\bar\{\\rho\}^\{\(y\)\},\\\\ \\bar\{\\rho\}^\{\(y\)\},&\\text\{if \}\\rho\_\{3\}\\leq\\bar\{\\rho\}^\{\(y\)\}<\\rho\_\{2\}^\{\\prime\},\\\\ \\rho\_\{3\},&\\text\{if \}\\bar\{\\rho\}^\{\(y\)\}<\\rho\_\{3\}\.\\end\{cases\}
LetπE∗:=πE\(ρ∗\)\\pi^\{E\*\}:=\\pi^\{E\}\(\\rho^\{\*\}\)\. Because no term in the equal\-accuracy problem involvesℓ\\ell, the maximized payoffπE∗\\pi^\{E\*\}is independent ofℓ\\ell\.

Step 3: Existence of the liability cutoff\.LetπD∗\(ℓ\):=πx\(ρx∗\)\+πy\(ρy∗;ℓ\)\\pi^\{D\*\}\(\\ell\):=\\pi\_\{x\}\(\\rho^\{\*\}\_\{x\}\)\+\\pi\_\{y\}\(\\rho^\{\*\}\_\{y\};\\ell\)denote the maximized payoff under the disparate design\. The type\-xxcomponent does not depend onℓ\\ell\. For the type\-yycomponent, observe that for every fixedρy∈\(1/2,1\)\\rho\_\{y\}\\in\(1/2,1\),

∂d~y​\(ρy\)∂ℓ=−2​\(1−ρy\)​θb<0,∂d¯y​\(ρy\)∂ℓ=−\(1−ρy\)​θb<0,\\frac\{\\partial\\tilde\{d\}\_\{y\}\(\\rho\_\{y\}\)\}\{\\partial\\ell\}=\-\\frac\{2\(1\-\\rho\_\{y\}\)\\theta\}\{b\}<0,\\qquad\\frac\{\\partial\\bar\{d\}\_\{y\}\(\\rho\_\{y\}\)\}\{\\partial\\ell\}=\-\\frac\{\(1\-\\rho\_\{y\}\)\\theta\}\{b\}<0,so both branches ofπy​\(ρy,⋅\)\\pi\_\{y\}\(\\rho\_\{y\};\\cdot\), and hence their pointwise minimum\-based composition, are strictly decreasing inℓ\\ellat everyρy\\rho\_\{y\}\. \(We restrict attention to parameters for which type\-yydemand is strictly positive at the disparate optimum for allℓ≤ℓ~\\ell\\leq\\widetilde\{\\ell\}, as holds in every numerical configuration we report, so thatπD∗\(ℓ\)\\pi^\{D\*\}\(\\ell\)is strictly decreasing on the relevant range\.\) Because the objective is strictly decreasing inℓ\\ellpointwise inρy\\rho\_\{y\}, its maximum overρy\\rho\_\{y\}is strictly decreasing inℓ\\ell, on interior and boundary branches alike; continuity ofπD∗\(ℓ\)\\pi^\{D\*\}\(\\ell\)inℓ\\ellfollows from the maximum theorem\. SinceπE∗\\pi^\{E\*\}is constant inℓ\\ell, the differenceπD∗\(ℓ\)−πE∗\\pi^\{D\*\}\(\\ell\)\-\\pi^\{E\*\}is continuous and strictly decreasing, and therefore crosses zero at most once\. Defineℓ~:=inf\{ℓ≥0:πD∗\(ℓ\)≤πE∗\}\\widetilde\{\\ell\}:=\\inf\\\{\\ell\\geq 0:\\pi^\{D\*\}\(\\ell\)\\leq\\pi^\{E\*\}\\\}\(withℓ~=∞\\widetilde\{\\ell\}=\\inftyif the set is empty\)\. Then the firm strictly prefers the disparate design forℓ<ℓ~\\ell<\\widetilde\{\\ell\}and the equal\-accuracy design forℓ\>ℓ~\\ell\>\\widetilde\{\\ell\}; the parameter regions in the statement are those for whichℓ~∈\(0,∞\)\\widetilde\{\\ell\}\\in\(0,\\infty\), which is nonempty for the numerical configurations reported in the proof of[propositionOA3](https://arxiv.org/html/2608.13618#Thmproposition3a)\. Finally, by Step 1, whenever the disparate design is chosen the optimal accuracies satisfyρx∗\>ρy∗\\rho^\{\*\}\_\{x\}\>\\rho^\{\*\}\_\{y\}under the assumptionb​κy\>2​β~​κx​\(b\+θ​ℓ\)b\\kappa\_\{y\}\>2\\tilde\{\\beta\}\\kappa\_\{x\}\(b\+\\theta\\ell\)131313Absent the assumptionb​κy\>2​β~​κx​\(b\+θ​ℓ\)b\\kappa\_\{y\}\>2\\tilde\{\\beta\}\\kappa\_\{x\}\(b\+\\theta\\ell\), the accuracy ranking under the optimal disparate algorithm can reverse, that is,ρx∗<ρy∗\\rho\_\{x\}^\{\*\}<\\rho\_\{y\}^\{\*\}\. The mechanism is that without the assumption, the type\-xxcapρ¯x=\(α¯​b\+c−θ​r\)/b\\bar\{\\rho\}\_\{x\}=\(\\overline\{\\alpha\}b\+c\-\\theta r\)/bcan bind, pushingρx∗\\rho\_\{x\}^\{\*\}to the half\-return branch12\+f2​κx\\tfrac\{1\}\{2\}\+\\tfrac\{f\}\{2\\kappa\_\{x\}\}, whereasρ¯y\\bar\{\\rho\}\_\{y\}cannot bind when its value exceeds one\.\.*Q\.E\.D\.*

###### Proposition OA3\.

There exist scenarios in which, asℓ\\ellincreases, the physician uses AI for fewer type\-yypatients whenℓ\\ellis below a threshold, and for more type\-yypatients whenℓ\\ellis above that threshold\. Depending on which branch of[OA4\. Type\-Specific Priors \(\)](https://arxiv.org/html/2608.13618#S0.Ex42)is active, the threshold is either

b⁡\(κy−2​β~​f~\)2​θ​β~​f~orb⁡\(κy−4​β~​f~\)4​θ​β~​f~\.\\frac\{b\(\\kappa\_\{y\}\-2\\tilde\{\\beta\}\\tilde\{f\}\)\}\{2\\theta\\tilde\{\\beta\}\\tilde\{f\}\}\\quad\\text\{or\}\\quad\\frac\{b\(\\kappa\_\{y\}\-4\\tilde\{\\beta\}\\tilde\{f\}\)\}\{4\\theta\\tilde\{\\beta\}\\tilde\{f\}\}\.

Proof of[propositionOA3](https://arxiv.org/html/2608.13618#Thmproposition3a)\. When the relevant threshold isb⁡\(κy−4​β~​f~\)4​θ​β~​f~\\frac\{b\(\\kappa\_\{y\}\-4\\tilde\{\\beta\}\\tilde\{f\}\)\}\{4\\theta\\tilde\{\\beta\}\\tilde\{f\}\}, the argument is identical to that in the proof of[proposition2](https://arxiv.org/html/2608.13618#Thmproposition2)\. We therefore focus on the case in which the threshold isb⁡\(κy−2​β~​f~\)2​θ​β~​f~\\frac\{b\(\\kappa\_\{y\}\-2\\tilde\{\\beta\}\\tilde\{f\}\)\}\{2\\theta\\tilde\{\\beta\}\\tilde\{f\}\}\. In this case,[OA4\. Type\-Specific Priors \(\)](https://arxiv.org/html/2608.13618#S0.Ex42)impliesdyD=β~​\(ρy∗​b−\(1−ρy∗\)​θ​ℓ−c\+θ​rb−α¯\)d\_\{y\}^\{\\mathrm\{D\}\}=\\tilde\{\\beta\}\\Big\(\\frac\{\\rho\_\{y\}^\{\*\}b\-\(1\-\\rho\_\{y\}^\{\*\}\)\\theta\\ell\-c\+\\theta r\}\{b\}\-\\underline\{\\alpha\}\\Big\), whereρy∗=12\+β~​f~​\(b\+θ​ℓ\)2​b​κy\\rho\_\{y\}^\{\*\}=\\tfrac\{1\}\{2\}\+\\dfrac\{\\tilde\{\\beta\}\\tilde\{f\}\(b\+\\theta\\ell\)\}\{2b\\kappa\_\{y\}\}from[propositionOA2](https://arxiv.org/html/2608.13618#Thmproposition2a)\(Note the AI demand for type\-yypatients is actuallydyD/α¯d\_\{y\}^\{D\}/\\overline\{\\alpha\}, but it suffices to analyzedyDd\_\{y\}^\{D\}asα¯\\overline\{\\alpha\}is a constant value\)\. Differentiating with respect toℓ\\elland simplifying yields∂dyD∂ℓ≥0\\frac\{\\partial d\_\{y\}^\{\\mathrm\{D\}\}\}\{\\partial\\ell\}\\geq 0if and only ifℓ≥b⁡\(κy−2​β~​f~\)2​θ​β~​f~\\ell\\geq\\frac\{b\(\\kappa\_\{y\}\-2\\tilde\{\\beta\}\\tilde\{f\}\)\}\{2\\theta\\tilde\{\\beta\}\\tilde\{f\}\}\. Thus, holding fixed the disparate\-design regime, liability reduces type\-yyAI use forℓ\\ellbelow this cutoff and increases it above the cutoff\. Non\-monotonicity in equilibrium arises when the firm switches to an equal\-accuracy design at a higher value ofℓ\\ell; the numerical instance in the footnote illustrates this pattern\.141414For example, whenα¯=0\.4\\underline\{\\alpha\}=0\.4,α¯=0\.88\\overline\{\\alpha\}=0\.88,b=0\.9b=0\.9,θ=1\\theta=1,c=0\.36c=0\.36,r=0\.3r=0\.3,κx=0\.4\\kappa\_\{x\}=0\.4,κy=0\.55\\kappa\_\{y\}=0\.55,f~=0\.28\\tilde\{f\}=0\.28, andβ~=0\.9\\tilde\{\\beta\}=0\.9, the physician uses AI for fewer type\-yypatients if0<ℓ<0\.08210<\\ell<0\.0821and for more type\-yypatients if0\.0821<ℓ<0\.42240\.0821<\\ell<0\.4224; the equal\-accuracy design is optimal ifℓ≥0\.4224\\ell\\geq 0\.4224\.*Q\.E\.D\.*

The two thresholds correspond to the cases in whichdyDd\_\{y\}^\{\\mathrm\{D\}\}is determined by the first and second expressions in[OA4\. Type\-Specific Priors \(\)](https://arxiv.org/html/2608.13618#S0.Ex42), respectively\. The intuition from our base model still applies: liability exposure trades off against the accuracy gains that higher liability induces\. When liability is small, exposure concerns dominate, and the physician uses AI for fewer patients\. In contrast, when liability is large, the accuracy effect dominates, and the physician uses AI for more patients\.

We next examine the welfare effect of mandating equal\-accuracy AI\. For ease of presentation, we define the adoption cutoffs:

a¯xD\\displaystyle\\underline\{a\}\_\{x\}^\{\\mathrm\{D\}\}=\(1−ρx∗\)​b\+c−θ​rb,\\displaystyle=\\frac\{\(1\-\\rho\_\{x\}^\{\*\}\)b\+c\-\\theta r\}\{b\},a¯xD\\displaystyle\\overline\{a\}\_\{x\}^\{\\mathrm\{D\}\}=min⁡\{α¯,ρx∗​b−c\+θ​rb\},\\displaystyle=\\min\\left\\\{\\overline\{\\alpha\},\\frac\{\\rho\_\{x\}^\{\*\}b\-c\+\\theta r\}\{b\}\\right\\\},a¯yD\\displaystyle\\underline\{a\}\_\{y\}^\{\\mathrm\{D\}\}=max⁡\{α¯,\(1−ρy∗\)​\(b\+θ​ℓ\)\+c−θ​rb\},\\displaystyle=\\max\\left\\\{\\underline\{\\alpha\},\\frac\{\(1\-\\rho\_\{y\}^\{\*\}\)\(b\+\\theta\\ell\)\+c\-\\theta r\}\{b\}\\right\\\},a¯yD\\displaystyle\\overline\{a\}\_\{y\}^\{\\mathrm\{D\}\}=ρy∗​b−\(1−ρy∗\)​θ​ℓ−c\+θ​rb,\\displaystyle=\\frac\{\\rho\_\{y\}^\{\*\}b\-\(1\-\\rho\_\{y\}^\{\*\}\)\\theta\\ell\-c\+\\theta r\}\{b\},a¯xm\\displaystyle\\underline\{a\}\_\{x\}^\{m\}=\(1−ρm\)​b\+c−θ​rb,\\displaystyle=\\frac\{\(1\-\\rho^\{m\}\)b\+c\-\\theta r\}\{b\},a¯xm\\displaystyle\\overline\{a\}\_\{x\}^\{m\}=min⁡\{α¯,ρm​b−c\+θ​rb\},\\displaystyle=\\min\\left\\\{\\overline\{\\alpha\},\\frac\{\\rho^\{m\}b\-c\+\\theta r\}\{b\}\\right\\\},a¯ym\\displaystyle\\underline\{a\}\_\{y\}^\{m\}=max⁡\{α¯,\(1−ρm\)​b\+c−θ​rb\},\\displaystyle=\\max\\left\\\{\\underline\{\\alpha\},\\frac\{\(1\-\\rho^\{m\}\)b\+c\-\\theta r\}\{b\}\\right\\\},a¯ym\\displaystyle\\overline\{a\}\_\{y\}^\{m\}=ρm​b−c\+θ​rb\.\\displaystyle=\\frac\{\\rho^\{m\}b\-c\+\\theta r\}\{b\}\.Then patient welfare under disparate and equal\-accuracy algorithms is

Wx​\(ρx∗\)\\displaystyle W\_\{x\}\(\\rho\_\{x\}^\{\*\}\)=1α¯​\[∫0a¯xD\(1−α\)​b​𝑑α\+∫a¯xDa¯xD\(ρx∗​b−c\)​𝑑α\+∫a¯xDα¯α​b​𝑑α\],\\displaystyle=\\frac\{1\}\{\\overline\{\\alpha\}\}\\Bigg\[\\int\_\{0\}^\{\\underline\{a\}\_\{x\}^\{\\mathrm\{D\}\}\}\(1\-\\alpha\)b\\,d\\alpha\+\\int\_\{\\underline\{a\}\_\{x\}^\{\\mathrm\{D\}\}\}^\{\\overline\{a\}\_\{x\}^\{\\mathrm\{D\}\}\}\(\\rho\_\{x\}^\{\*\}b\-c\)\\,d\\alpha\+\\int\_\{\\overline\{a\}\_\{x\}^\{\\mathrm\{D\}\}\}^\{\\overline\{\\alpha\}\}\\alpha b\\,d\\alpha\\Bigg\],Wy​\(ρy∗\)\\displaystyle W\_\{y\}\(\\rho\_\{y\}^\{\*\}\)=β~α¯​\[∫α¯a¯yD\(1−α\)​b​𝑑α\+∫a¯yDa¯yD\(ρy∗​b−c\)​𝑑α\+∫a¯yD1α​b​𝑑α\],\\displaystyle=\\frac\{\\tilde\{\\beta\}\}\{\\overline\{\\alpha\}\}\\Bigg\[\\int\_\{\\underline\{\\alpha\}\}^\{\\underline\{a\}\_\{y\}^\{\\mathrm\{D\}\}\}\(1\-\\alpha\)b\\,d\\alpha\+\\int\_\{\\underline\{a\}\_\{y\}^\{\\mathrm\{D\}\}\}^\{\\overline\{a\}\_\{y\}^\{\\mathrm\{D\}\}\}\(\\rho\_\{y\}^\{\*\}b\-c\)\\,d\\alpha\+\\int\_\{\\overline\{a\}\_\{y\}^\{\\mathrm\{D\}\}\}^\{1\}\\alpha b\\,d\\alpha\\Bigg\],Wx​\(ρm\)\\displaystyle W\_\{x\}\(\\rho^\{m\}\)=1α¯​\[∫0a¯xm\(1−α\)​b​𝑑α\+∫a¯xma¯xm\(ρm​b−c\)​𝑑α\+∫a¯xmα¯α​b​𝑑α\],\\displaystyle=\\frac\{1\}\{\\overline\{\\alpha\}\}\\Bigg\[\\int\_\{0\}^\{\\underline\{a\}\_\{x\}^\{m\}\}\(1\-\\alpha\)b\\,d\\alpha\+\\int\_\{\\underline\{a\}\_\{x\}^\{m\}\}^\{\\overline\{a\}\_\{x\}^\{m\}\}\(\\rho^\{m\}b\-c\)\\,d\\alpha\+\\int\_\{\\overline\{a\}\_\{x\}^\{m\}\}^\{\\overline\{\\alpha\}\}\\alpha b\\,d\\alpha\\Bigg\],Wy​\(ρm\)\\displaystyle W\_\{y\}\(\\rho^\{m\}\)=β~α¯​\[∫α¯a¯ym\(1−α\)​b​𝑑α\+∫a¯yma¯ym\(ρm​b−c\)​𝑑α\+∫a¯ym1α​b​𝑑α\]\.\\displaystyle=\\frac\{\\tilde\{\\beta\}\}\{\\overline\{\\alpha\}\}\\Bigg\[\\int\_\{\\underline\{\\alpha\}\}^\{\\underline\{a\}\_\{y\}^\{m\}\}\(1\-\\alpha\)b\\,d\\alpha\+\\int\_\{\\underline\{a\}\_\{y\}^\{m\}\}^\{\\overline\{a\}\_\{y\}^\{m\}\}\(\\rho^\{m\}b\-c\)\\,d\\alpha\+\\int\_\{\\overline\{a\}\_\{y\}^\{m\}\}^\{1\}\\alpha b\\,d\\alpha\\Bigg\]\.
[fig\.OA3](https://arxiv.org/html/2608.13618#S0.F3)reports the welfare effects at the parameter values of the preceding footnote\. The central message of the baseline persists: over a substantial region of the parameter space, mandating equal accuracy harms both patient groups\. Bounded prior supports add two features\. When the type\-yysegment is small, the mandate lowers type\-xxaccuracy enough to curb type\-xxoveruse, so type\-xxpatients benefit; and when liability is high, removing the disparity\-contingent exposure raises type\-yyuse from inefficiently low levels, so type\-yypatients benefit\. Both groups are harmed in the intermediate region in which neither corrective force is strong\.

The intuition parallels the baseline overuse logic\. Because the physician’s private returnθ​r\\theta rtilts use beyond the altruistic benchmark, welfare effects are governed by utilization as much as by accuracy\. The mandate moderates type\-xxaccuracy, which trims type\-xxoveruse, and it removes the liability wedge on type\-yyuse, which raises type\-yyuse; each effect helps the corresponding group when the initial distortion it corrects is large, and hurts that group otherwise\.

Figure OA3:Impact of mandating equal\-accuracy AI on expected welfare for each patient type \(α¯=0\.4\\underline\{\\alpha\}=0\.4,α¯=0\.88\\overline\{\\alpha\}=0\.88,b=0\.9b=0\.9,θ=1\\theta=1,c=0\.36c=0\.36,r=0\.3r=0\.3,κx=0\.4\\kappa\_\{x\}=0\.4,κy=0\.55\\kappa\_\{y\}=0\.55,f~=0\.28\\tilde\{f\}=0\.28\)\.