Robust Human-AI Complementarity under Uncertainty

# Robust Human-AI Complementarity under Uncertainty
Source: [https://arxiv.org/html/2607.06656](https://arxiv.org/html/2607.06656)
###### Abstract

Machine learning models are often intended to augment rather than replace human decision makers, by providing information that is complementary to human judgement\. Yet, in practice, human decision makers routinely fail to realize such complementary gains, even when models provide useful signal\. In this work, we study how asymmetric information about the quality of information available to a human decision maker vs\. an AI impacts the ability of a decision maker to extract complementary value from AI predictions\. We show that a key factor is the error correlation structure between human and AI predictions\. In particular, when the AI’s prediction errors arenegatively correlatedwith those of the human, the decision maker can construct robust strategies which guarantee improvements in expected utility\. We empirically investigate whether these conditions for complementarity arise in practice, using real\-world forecasting benchmarks\.

Machine Learning, ICML

## 1Introduction

Many hope that machine learning models will augment human decision making without replacing it by providing decision makers with complementary information\. For instance, in scientific settings, LLMs are increasingly imagined as tools for forecasting promising experiments or screening hypotheses\. However, attempting to rely appropriately on machine learning outputs can be difficult; human decision makers routinely fail to benefit from complementary signal in AI predictions\(Vaccaroet al\.,[2024](https://arxiv.org/html/2607.06656#bib.bib6)\)even when such signal is available\(Guoet al\.,[2024](https://arxiv.org/html/2607.06656#bib.bib19)\)\. Previous work has hypothesized that people may struggle to update their beliefs appropriately in light of AI predictions\(Agarwalet al\.,[2023](https://arxiv.org/html/2607.06656#bib.bib17)\)or identify what new information the AI model provides\(Guoet al\.,[2025](https://arxiv.org/html/2607.06656#bib.bib18)\)\. Here, we document another kind of limitation that makes it difficult for even a rational decision maker to benefit from complementary information: uncertainty about the quality of AI predictions\. Such uncertainty is a recurring feature of many settings: models constantly change, new use cases emerge, and some use cases \(e\.g\., scientific discovery\) by nature demand using models in settings that are outside of the distribution of instances previously seen\(Bommasaniet al\.,[2021](https://arxiv.org/html/2607.06656#bib.bib31)\)\. It is often difficult for people to credibly know the joint distribution of model inputs and ground truth for the specific task they are solving\(Vafaet al\.,[2024](https://arxiv.org/html/2607.06656#bib.bib12)\)\.

We study how asymmetric information about the quality of information available to a human decision maker vs an AI impacts the ability of a decision maker to extract complementary value from the AI\. We start by introducing a formal model based on statistical decision theory in which the human does not know the exact joint distribution over AI predictions and ground truth\. Instead, they have anuncertainty setwhich formalizes their beliefs about the model’s quality and seek to ensure robust performance over that set\. For example, a decision maker using AI likely believes that the AI has at least some level of correlation with the ground truth, even if the full joint distribution is unknown\. We ask: what conditions must the decision maker be willing to impose about the performance of the AI in order to ensure that they benefit from using it?

It turns out that the presence of uncertainty substantially changes the answer\. If the decision maker knows the true joint distribution, they benefit from using the AI if its prediction carriesanysignal in excess of what is already observed by the decision maker \(essentially the condition tested in previous work\)\. However, under uncertainty, it becomes much harder for the decision maker to construct a rule which makes nontrivial use of AI input and guarantees that the decision maker benefits compared to ignoring the AI entirely\. A key factor turns out to be whether the errors in the AI’s predictions arenegatively correlatedwith the errors in the human’s predictions\. If this condition is satisfied, the decision maker can construct robust strategies which guarantee an improvement in expected utility\. If it is not satisfied, the window for robust complementarity is much narrower: decisions to use the AI essentially reflect a weakness of the human’s ability\. In stylized terms, we interpret this as a movement towards automation instead of complementarity\.

We then use empirical data from real\-world forecasting and social science benchmarks that compare predictions from humans to those from LLMs\. We compare human and LLM prediction errors and find that they are positively correlated, putting us in the “difficult” setting for complementarity\. Alternative prompting strategies \(e\.g\., instructing the LLM to focus on factors others may have missed\) meaningfully reduce but fail to consistently eliminate positive correlations in errors\. We suggest that explicitly optimizing for ability to provide complementary information should be a principle for LLM training and evaluation in the future, in order to ensure that they can be robustly used by humans to improve their own decision making\.

## 2Related Work

##### Human\-AI Decision Making\.

A large literature examines joint decision making by humans and AI, often with a focus on when joint decision making can improve over either party individually \(“complementarity”\)\. Perhaps most related to our work,Steyverset al\.\([2022](https://arxiv.org/html/2607.06656#bib.bib1)\)propose a generative model for combining human and AI confidences and show that the model theoretically implies conditions under which a human\-AI team outperforms a team of multiple humans or multiple models\. Our work differs in that we compare the human\-AI combination to the single human alone, but focus on the role of uncertainty \(where\(Steyverset al\.,[2022](https://arxiv.org/html/2607.06656#bib.bib1)\)assume in their theoretical analysis that the model parameters are known\)\. Also related is\(Donahueet al\.,[2022](https://arxiv.org/html/2607.06656#bib.bib21)\), who analyze a theoretical model of human decisions about when to rely on AI to identify conditions for complementarity, though without focusing on the role of uncertainty\.

More broadly, there is a large algorithmic literature on structures for human\-AI decision making, for example learning to defer\(Madraset al\.,[2018](https://arxiv.org/html/2607.06656#bib.bib2); Mozannar and Sontag,[2020](https://arxiv.org/html/2607.06656#bib.bib3)\), triage decisions\(Raghuet al\.,[2019](https://arxiv.org/html/2607.06656#bib.bib4); Okatiet al\.,[2021](https://arxiv.org/html/2607.06656#bib.bib5)\), learning models that complement human decision makers\(Wilderet al\.,[2020](https://arxiv.org/html/2607.06656#bib.bib20)\), and using AI to shape human choice sets\(Straitouriet al\.,[2023](https://arxiv.org/html/2607.06656#bib.bib11)\)\. Nevertheless, a significant empirical literature typically fails to find evidence of complementarity between humans and AI \(seeVaccaroet al\.\([2024](https://arxiv.org/html/2607.06656#bib.bib6)\)for a review\)\. Our work theoretically explores one mechanism that could make complementarity more difficult to realize in real\-world settings and explores its impact in real world data on human\-LLM predictions\.

##### LLMs for Prediction\.

A key motivation for our work is the growing interest in using LLMs to provide predictions that could augment human decision making\. Since LLMs can be applied to new tasks in a zero\-shot manner, humans are often faced with uncertainty about how an LLM will perform for a specific prediction task at hand\. Existing work has begun to evaluate the performance of LLMs at forecasting future events\(Kargeret al\.,[2025](https://arxiv.org/html/2607.06656#bib.bib8); Yanget al\.,[2025](https://arxiv.org/html/2607.06656#bib.bib14); Yeet al\.,[2024](https://arxiv.org/html/2607.06656#bib.bib13); Halawiet al\.,[2024](https://arxiv.org/html/2607.06656#bib.bib10)\)and shown that they compare favorably with many humans\. In scientific applications, several works have evaluated using LLMs to forecast the results of scientific experiments, for example in the social sciences\(Hewittet al\.,[2024](https://arxiv.org/html/2607.06656#bib.bib7); Parket al\.,[2024](https://arxiv.org/html/2607.06656#bib.bib15)\)or neuroscience\(Luoet al\.,[2025](https://arxiv.org/html/2607.06656#bib.bib16)\)\. They similarly find that LLMs perform competitively with human forecasters\. As a result, there is interest in using outputs from LLMs to “screen” potential experiments, using LLM predictions of effects as a sort of pilot study to decide which experiments to pursue\(Anthiset al\.,[2025](https://arxiv.org/html/2607.06656#bib.bib9)\)\. This is the motivation for our “binary decisions” setting below, where we analyze conditions for LLMs to improve human decision making at such a task\. We also use datasets from both forecasting and scientific settings to study the correlation between LLM and human errors and impact on complementary performance; existing work has focused mostly on comparing LLM and human accuracy and less on the relative distribution of errors between the two \(thoughHewittet al\.\([2024](https://arxiv.org/html/2607.06656#bib.bib7)\)confirm that both human and AI information contain independent information predictive of true effects\)\.

## 3Model for Human\-AI Complementarity

We assume that there is a ground\-truth state of the worldθ\\thetawhich the decision maker aims to infer\. The decision maker observes a signalϕH\\phi\_\{H\}corresponding to their prior belief as well as a signalϕA​I\\phi\_\{AI\}generated by the machine learning system \(e\.g\., an LLM\)\. They then make a decisiond∈𝒟d\\in\\mathcal\{D\}\. Their goal is to minimize a lossℓ​\(d,θ\)\\ell\(d,\\theta\)which depends on the unknown state and on the decision\.

We study a generative process in which the signals are noisy measurements of the ground truth state\. We start with a stylized, linear\-Gaussian version of the model which imposes normally distributed errors in the signals observed by each of the agents\. This setting turns out to be sufficient to illustrate much of the intuition behind the problem structure, which we later show generalizes considerably\. Formally, we start with the model

θ∼N​\(0,σθ2\)\\displaystyle\\theta\\sim N\(0,\\sigma^\{2\}\_\{\\theta\}\)ϕH=λH​θ\+ϵH\\displaystyle\\phi\_\{H\}=\\lambda\_\{H\}\\theta\+\\epsilon\_\{H\}ϕA​I=λA​I​θ\+ϵA​I\\displaystyle\\phi\_\{AI\}=\\lambda\_\{AI\}\\theta\+\\epsilon\_\{AI\}ϵH,ϵA​I∼N​\(0,Σϵ\)\\displaystyle\\epsilon\_\{H\},\\epsilon\_\{AI\}\\sim N\(0,\\Sigma\_\{\\epsilon\}\)whereϵH\\epsilon\_\{H\}andϵA​I\\epsilon\_\{AI\}are mean\-zero Gaussian noise\. This implies the joint distribution\[θ​ϕH​ϕA​I\]∼N​\(0,Σ\)\[\\theta\\,\\,\\phi\_\{H\}\\,\\,\\phi\_\{AI\}\]\\sim N\(0,\\Sigma\)\. The core of the problem is encapsulated in the covariance matrixΣ\\Sigma\. We start with this formalization and provide an essentially complete characterization when the decision maker provides a prediction ofθ\\theta\(d∈ℝ\)d\\in\\mathbb\{R\}\)and is evaluated via the mean squared errorℓ​\(d,θ\)=\(d−θ\)2\\ell\(d,\\theta\)=\(d\-\\theta\)^\{2\}\. We then show that the qualitative conclusions generalize in two directions: a more complex decision making task in the Gaussian setting and predictions in a non\-Gaussian generative model\. Note that the main analysis assumes Gaussianity principally in theerrorsfrom the two signals: the human or AI may react in a complex, nonlinear fashion to whatever information they observe, and we simply model the deviation between their predictions and the ground truth as normally distributed\.

The expected loss under a particular distribution parameterized byΣ\\Sigmais

LΣ​\(d\)=𝔼θ,ϕH,ϕA​I∼N​\(0,Σ\)​\[ℓ​\(d​\(ϕH,ϕA​I\),θ\)\]\.\\displaystyle L\_\{\\Sigma\}\(d\)=\\mathbb\{E\}\_\{\\theta,\\phi\_\{H\},\\phi\_\{AI\}\\sim N\(0,\\Sigma\)\}\[\\ell\(d\(\\phi\_\{H\},\\phi\_\{AI\}\),\\theta\)\]\.If the decision maker knew \(or had samples from\) the full joint distribution, they could simply minimize the expected loss overdd\. This is the setting discussed in a great deal of previous work about optimal human\-AI decision making\. The key feature of our formal model is to introduce uncertainty about the quality of the AI signal\. In particular, our decision maker will stipulate thatΣ\\Sigmabelongs to an uncertainty set𝒰\\mathcal\{U\}which pins down all features of the joint distributionexceptfor the correlation of the AI signal withθ\\theta\.

###### Assumption 3\.1\.

EveryΣ∈𝒰\\Sigma\\in\\mathcal\{U\}agrees on all entries except forCov​\(ϕA​I,θ\)\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)\.

As we will see, uncertainty aboutCov​\(ϕA​I,θ\)\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)is the core mechanism that generates a substantial difference in optimal policies\. Additionally, in many settings, it may be reasonable to view the other entries ofΣ\\Sigmaas known because it is cheap to generate samples ofϕA​I\\phi\_\{AI\}, which the decision maker can compare to their own belief; it is obtaining observations of the true stateθ\\thetathat is expensive\. For example, a scientist can easily ask a language model for its forecastϕA​I\\phi\_\{AI\}about the results of many possible experiments, but the bottleneck is in actually performing experiments to observeθ\\thetaand learn how accurateϕA​I\\phi\_\{AI\}really was\. We expect that many of our conclusions \(e\.g\., about the difficulty of achieving robust complementarity\) would grow stronger if the model included uncertainty in other aspects ofΣ\\Sigmaas well\.

Our uncertainty\-set formulation is compatible with richer models of uncertainty\. For example, if the decision maker has a prior over possible AI signal qualities, model parameters, or data\-generating processes, this uncertainty can be integrated out\. From the perspective of the downstream decision problem, the relevant object is the induced predictive posterior over the state given the observed human and AI signals\. Thus, uncertainty over intermediate quantities affects the analysis only through the posterior distribution it induces\. When this induced posterior satisfies the structural conditions assumed in our analysis—for example, the Gaussian covariance restrictions in Section[3](https://arxiv.org/html/2607.06656#S3), or the finite\-moment and negative\-dependence conditions in Section[3\.2](https://arxiv.org/html/2607.06656#S3.SS2)—the corresponding results apply without further modification\.

The same observation applies to estimation error in the human signal or in the joint distribution of human and AI signals\. If realizations of the state are costly, the decision maker may place a prior over these quantities and update after observing samples of the true state\. After this update, the decision\-relevant object is again the post\-update predictive posterior\. Our framework can therefore be viewed as placing a credal set over plausible predictive posteriors and asking when complementarity is guaranteed uniformly over that set\.

In order to restrict to more interesting cases, we will also impose the assumption that neither agent observes a noiseless measurement ofθ\\theta:

###### Assumption 3\.2\.

Var​\(ϕH\|θ\)\\text\{Var\}\(\\phi\_\{H\}\|\\theta\)andVar​\(ϕA​I\|θ\)\\text\{Var\}\(\\phi\_\{AI\}\|\\theta\)are both strictly greater than 0\.

Finally, we will impose without loss of generality thatϕH\\phi\_\{H\}is scaled so thatλH=Cov​\(θ,ϕH\)Var​\(ϕH\)=1\\lambda\_\{H\}=\\frac\{\\text\{Cov\}\(\\theta,\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}=1, i\.e\., the linear regression coefficient ofθ\\thetaonϕH\\phi\_\{H\}is 1\. This is simply a normalization that does not further restrict the set of𝒰\\mathcal\{U\}under consideration\.

We study: under what conditions is there a ruleddwhich makes nontrivial use ofϕA​I\\phi\_\{AI\}and which guarantees better value than one which uses onlyϕH\\phi\_\{H\}? LetdHd\_\{H\}denote the optimal decision rule which is a function only ofϕH\\phi\_\{H\}\. Since the class of uncertainty sets we consider fully specify the\(θ,ϕH\)\(\\theta,\\phi\_\{H\}\)distribution,dHd\_\{H\}can be characterized explicitly for many decision problems\. We study when it is possible to exhibit anotherjointdecision ruledJd\_\{J\}which is a function ofϕA​I\\phi\_\{AI\}as well asϕH\\phi\_\{H\}and which satisfies

LΣ​\(dJ\)≤LΣ​\(dH\)∀Σ∈𝒰\\displaystyle L\_\{\\Sigma\}\(d\_\{J\}\)\\leq L\_\{\\Sigma\}\(d\_\{H\}\)\\quad\\forall\\Sigma\\in\\mathcal\{U\}and there exists at least oneΣ∈𝒰\\Sigma\\in\\mathcal\{U\}for which the inequality is strict\.

We start by exploring this question for prediction under mean squared error\. Here, the action taken by the decision maker is to produce a predictiond∈ℝd\\in\\mathbb\{R\}which is evaluated according to the lossℓ​\(d,θ\)=\(d−θ\)2\\ell\(d,\\theta\)=\(d\-\\theta\)^\{2\}\. For example, this setting could model a forecaster who seeks to make accurate predictions about an unknown event, as in efforts to use LLMs for forecasting \(where MSE, aka Brier score, is a common metric\)\.

As a starting point, we show that it suffices to confine ourselves to decision rules that can be represented as linear functionsd=a​ϕH\+b​ϕA​Id=a\\phi\_\{H\}\+b\\phi\_\{AI\}\. This is intuitive because conditional expectations are linear in the signals under joint normality\. The only subtlety is ensuring that the desire for robust performance simultaneously across many distributions does not create the need for nonlinear decision rules\. We prove that this is not the case:

###### Proposition 3\.3\.

Given any decision ruledd, there exists a linear decision ruled~\\tilde\{d\}such thatLΣ​\(d~\)≤LΣ​\(d\)L\_\{\\Sigma\}\(\\tilde\{d\}\)\\leq L\_\{\\Sigma\}\(d\)for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\.

Further, our normalization ensures thata=1a=1in the optimal decision rule, i\.e\., without any additional information, the decision maker simply forecastsdH​\(ϕH\)=ϕHd\_\{H\}\(\\phi\_\{H\}\)=\\phi\_\{H\}as their prediction ofθ\\theta\. We are interested in conditions under which there exists a decision ruledb​\(ϕH,ϕA​I\)=ϕH\+b​ϕA​Id\_\{b\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=\\phi\_\{H\}\+b\\phi\_\{AI\}forb≠0b\\neq 0which provably dominatesdHd\_\{H\}across𝒰\\mathcal\{U\}\. We show that the following two conditions suffice:

###### Proposition 3\.4\.

Suppose that allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}satisfy the following two additional conditions: \(1\)Cov​\(ϕA​I,θ\)\>δ\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)\>\\deltafor someδ\>0\\delta\>0\. \(2\)Cov​\(ϕH,ϕA​I\|θ\)≤0\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\|\\theta\)\\leq 0\. Then, there exists ab≠0b\\neq 0such thatdbd\_\{b\}strictly dominatesdHd\_\{H\}over𝒰\\mathcal\{U\}\.

Intuitively, \(1\) is a minimal belief for the decision maker to hold: to rationalize using the machine signal, they must believe that it is positively correlated withθ\\thetaby at least some amount\. \(2\) is equivalent toϵH\\epsilon\_\{H\}andϵA​I\\epsilon\_\{AI\}being negatively correlated, i\.e\., negative correlation in the errors\.

We now provide the intuition behind the construction of the decision rule, which will also underlie results for the generalized settings discussed in later sections\. Define the AI residualrrby regressing out the portion ofϕA​I\\phi\_\{AI\}predictable fromϕH\\phi\_\{H\}

r≜ϕA​I−Cov​\(ϕH,ϕA​I\)Var​\(ϕH\)​ϕH\.r\\;\\triangleq\\;\\phi\_\{AI\}\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\\,\\phi\_\{H\}\.\(1\)By construction,rrnow satisfiesCov​\(ϕH,r\)=0\\text\{Cov\}\(\\phi\_\{H\},r\)=0\. For eachΣ∈𝒰\\Sigma\\in\\mathcal\{U\}, define the regression coefficient ofrronθ\\theta:

β​\(Σ\)≜CovΣ​\(r,θ\)Var​\(r\),\\beta\(\\Sigma\)\\;\\triangleq\\;\\frac\{\\text\{Cov\}\_\{\\Sigma\}\(r,\\theta\)\}\{\\text\{Var\}\(r\)\},\(2\)We can show that the regression coefficient is strictly positive under the same hypotheses for Proposition[3\.4](https://arxiv.org/html/2607.06656#S3.Thmtheorem4)\(positive marginal correlation ofϕA​I\\phi\_\{AI\}combined with negative correlation of errors\):

###### Lemma 3\.5\.

Under the conditions of Proposition[3\.4](https://arxiv.org/html/2607.06656#S3.Thmtheorem4), for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\},β​\(Σ\)≥b=δ​Var​\(ϕH∣θ\)Var​\(ϕH\)​Var​\(r\)\>0\\beta\(\\Sigma\)\\geq b=\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\\text\{Var\}\(r\)\}\>0,

Effectively, the strictly improving decision rule corresponds to the predictionϕH\+b⋅r\\phi\_\{H\}\+b\\cdot r\. Since the true regression coefficient under anyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}is guaranteed to be at leastbb, this corresponds to a conservative counterpart of the ideal linear regression ofθ\\thetaon\(ϕA​I,ϕH\)\(\\phi\_\{AI\},\\phi\_\{H\}\), which is guaranteed to improve performance in mean squared error\.

We then ask whether a weaker version of Condition 2 suffices\. For example, what if the errors are only modestly positively correlated? We give an exact characterization, showing that in the positive correlation regime, the existence of an improving decision rule requires that the decision maker’s errorϵH\\epsilon\_\{H\}grow in direct proportion to the amount of positive correlation:

###### Proposition 3\.6\.

Define the extremal values

δ:=infΣ∈𝒰CovΣ​\(θ,ϕA​I\),γ:=supΣ∈𝒰CovΣ​\(ϕH,ϕA​I∣θ\)\.\\delta:=\\inf\_\{\\Sigma\\in\\mathcal\{U\}\}\\text\{Cov\}\_\{\\Sigma\}\(\\theta,\\phi\_\{AI\}\),\\qquad\\gamma:=\\sup\_\{\\Sigma\\in\\mathcal\{U\}\}\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\.Then there exists a coefficientb⋆\>0b^\{\\star\}\>0such thatdb⋆d\_\{b^\{\\star\}\}strictly dominatesdHd\_\{H\}over𝒰\\mathcal\{U\}if and only if

γ<δ​Var​\(ϕH∣θ\)Var​\(ϕH\)\.\\gamma<\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\.The maximal possible improvement in loss for anydbd\_\{b\}overdHd\_\{H\}, which is guaranteed for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}is given by

\(δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ\)2Var​\(ϕA​I\)\.\\frac\{\\Bigl\(\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma\\Bigr\)^\{2\}\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}\.

Intuitively, any positive correlation in errors \(γ\\gamma\) must be matched by a combination of higher absolute performance for the AI signal \(δ\\delta\) and noise in the human signal \(Var​\(ϕH\|θ\)\\text\{Var\}\(\\phi\_\{H\}\|\\theta\)\)\. Rationalizing nontrivial dependence on the AI signal in the positive correlation case thus requires that the AI perform better and the human perform worse in direct proportion to the amount of positive correlation\.

By contrast, no such dilemma appears without the presence of uncertainty\. For any fixedΣ\\Sigma, the decision maker can benefit from both signals whenever the information contained inϕA​I\\phi\_\{AI\}is not completely redundant withϕH\\phi\_\{H\}:

###### Proposition 3\.7\.

For any fixed covariance structureΣ\\Sigma, so long asβ​\(Σ\)≠0\\beta\(\\Sigma\)\\neq 0, there is a joint decision ruledbd\_\{b\},b≠0b\\neq 0, which satisfiesLΣ​\(db\)<LΣ​\(dH\)L\_\{\\Sigma\}\(d\_\{b\}\)<L\_\{\\Sigma\}\(d\_\{H\}\)\.

Effectively, the presence of uncertainty about the AI signal significantly shrinks the region for complementarity: instead of any new information being useful, the decision maker requires enough structure to guarantee that the new information contributed by the AI \(the residual\) is correlated in a known way with the ground truthθ\\theta\.

### 3\.1Binary Decisions

We next study a setting in which a decision maker takes a binary actiond∈\{0,1\}d\\in\\\{0,1\\\}after observing signals\. The interpretation is thatd=1d=1corresponds to*investing*in a project \(e\.g\. pursuing an experiment\), whiled=0d=0corresponds to not investing\. The state of the world isθ\\theta, and investing is desirable precisely whenθ\\thetaexceeds a fixed quality thresholdτ\\tau\. Investing incurs a costc∈\(0,1\)c\\in\(0,1\), while a successful investment yields a unit payoff \(without loss of generality\)\. Thus the decision maker’s utility is

u​\(d,θ\)≜d⋅1​\{θ≥τ\}−c⋅d,u\(d,\\theta\)\\;\\triangleq\\;d\\cdot 1\\\{\\theta\\geq\\tau\\\}\-c\\cdot d,\(3\)and the corresponding loss isℓ​\(d,θ\)=−u​\(d,θ\)\\ell\(d,\\theta\)=\-u\(d,\\theta\)\. Becauseddis binary, the optimal action is to invest if and only if the posterior success probability exceeds the cost:

ℙ​\(θ≥τ\)≥c\.\\mathbb\{P\}\(\\theta\\geq\\tau\)\\;\\geq\\;c\.In particular, under our standing normalization𝔼​\[θ∣ϕH\]=ϕH\\mathbb\{E\}\[\\theta\\mid\\phi\_\{H\}\]=\\phi\_\{H\}andVar​\(θ∣ϕH\)=σH2\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)=\\sigma\_\{H\}^\{2\}, the expert\-only posterior success probability is, for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\},

pH​\(x\)≜ℙ​\(θ≥τ∣ϕH=x\)=Φ​\(x−τσH\),p\_\{H\}\(x\)\\;\\triangleq\\;\\mathbb\{P\}\(\\theta\\geq\\tau\\mid\\phi\_\{H\}=x\)=\\Phi\\\!\\left\(\\frac\{x\-\\tau\}\{\\sigma\_\{H\}\}\\right\),which is strictly increasing inxx\. LetxHx\_\{H\}be the unique threshold satisfyingpH​\(xH\)=cp\_\{H\}\(x\_\{H\}\)=c, and define the human\-only optimal decision rule

dH​\(ϕH\)≜1​\{ϕH≥xH\}\.d\_\{H\}\(\\phi\_\{H\}\)\\;\\triangleq\\;1\\\{\\phi\_\{H\}\\geq x\_\{H\}\\\}\.\(4\)
Our goal is to construct a decision rule that uses both signals\(ϕH,ϕA​I\)\(\\phi\_\{H\},\\phi\_\{AI\}\)and robustly improves upondHd\_\{H\}over the uncertainty set𝒰\\mathcal\{U\}\. The starting point is the same construction used in the previous section, based on the residual AI signal and its regression coefficientβ​\(Σ\)\\beta\(\\Sigma\)forθ\\thetaunder a given distributionΣ\\Sigma\. Lemma[3\.5](https://arxiv.org/html/2607.06656#S3.Thmtheorem5)lower bounds the value ofβ​\(Σ\)\\beta\(\\Sigma\)under negative correlation of errors\. Our strategy will be to use this lower bound on the regression coefficient to form a “pessimistic” estimate of the mean and variance ofθ\\thetaconditional on\(ϕH,ϕA​I\)\(\\phi\_\{H\},\\phi\_\{AI\}\)\. Define

μb​\(ϕH,r\)≜ϕH\+b​r,σb2≜σH2−b2​Var​\(r\),\\mu\_\{\\mathrm\{b\}\}\(\\phi\_\{H\},r\)\\triangleq\\phi\_\{H\}\+br,\\qquad\\sigma\_\{b\}^\{2\}\\triangleq\\sigma\_\{H\}^\{2\}\-b^\{2\}\\text\{Var\}\(r\),\(5\)While it would be tempting to simply construct an analogy to the human\-only ruledHd\_\{H\}usingμb\\mu\_\{\\mathrm\{b\}\}andσb\\sigma\_\{\\mathrm\{b\}\}, this neglects the fact that decision\-making depends on the entire distribution ofθ\\thetain this setting, not only on its mean as before\. For example, it is possible for observing a positive residual to increase the posterior mean ofθ\\thetabut actually decrease the posteriorPr⁡\(θ≥τ\)\\Pr\(\\theta\\geq\\tau\)\. To circumvent this, we define conservative bounds on the success probability which are guaranteed to hold across the entirety of𝒰\\mathcal\{U\}:

Plow​\(x,y\)≜\{Φ​\(μb​\(x,y\)−τσb\)​y≥0∧μb​\(x,y\)≥τ0,otherwise,P\_\{\\mathrm\{low\}\}\(x,y\)\\triangleq\\begin\{cases\}\\Phi\\\!\\left\(\\dfrac\{\\mu\_\{\\mathrm\{b\}\}\(x,y\)\-\\tau\}\{\\sigma\_\{b\}\}\\right\)&\\text\{\}y\\geq 0\\land\\mu\_\{\\mathrm\{b\}\}\(x,y\)\\geq\\tau\\\\\[10\.00002pt\] 0,&\\text\{otherwise\},\\end\{cases\}Phigh​\(x,y\)≜\{Φ​\(μb​\(x,y\)−τσH\)​μb​\(x,y\)<τ∧y<01,otherwise\.P\_\{\\mathrm\{high\}\}\(x,y\)\\triangleq\\begin\{cases\}\\Phi\\\!\\left\(\\dfrac\{\\mu\_\{\\mathrm\{b\}\}\(x,y\)\-\\tau\}\{\\sigma\_\{H\}\}\\right\)&\\text\{\}\\mu\_\{\\mathrm\{b\}\}\(x,y\)<\\tau\\land y<0\\\\\[5\.0pt\] 1,&\\text\{otherwise\.\}\\end\{cases\}We will use these conservative bounds on the success probability to construct a new rule that overrules the human\-only decision rule whenever doing so is guaranteed to improve across the entirety of𝒰\\mathcal\{U\}\. Formally we define

dJ​\(ϕH,ϕA​I\)≜\{1,if​Plow​\(ϕH,r\)≥c,0,if​Phigh​\(ϕH,r\)≤c,dH​\(ϕH\),otherwise,\\displaystyle d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\\triangleq\\begin\{cases\}1,&\\text\{if \}P\_\{\\mathrm\{low\}\}\(\\phi\_\{H\},r\)\\geq c,\\\\\[2\.5pt\] 0,&\\text\{if \}P\_\{\\mathrm\{high\}\}\(\\phi\_\{H\},r\)\\leq c,\\\\\[2\.5pt\] d\_\{H\}\(\\phi\_\{H\}\),&\\text\{otherwise,\}\\end\{cases\}\(6\)
###### Proposition 3\.8\.

Under the same assumptions as Proposition[3\.4](https://arxiv.org/html/2607.06656#S3.Thmtheorem4), the joint ruledJd\_\{J\}weakly dominates the expert\-only optimal ruledHd\_\{H\}in expected utility:

𝔼Σ​\[u​\(dJ​\(ϕH,ϕA​I\),θ\)\]≥𝔼Σ​\[u​\(dH​\(ϕH\),θ\)\]​∀Σ∈𝒰\.\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\),\\theta\)\\big\]\\;\\geq\\;\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(d\_\{H\}\(\\phi\_\{H\}\),\\theta\)\\big\]\\,\\ \\forall\\Sigma\\in\\mathcal\{U\}\.Moreover, the inequality is strict for anyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}such thatβ​\(Σ\)\>b\\beta\(\\Sigma\)\>b\.

This shows that the basic conclusion from the mean squared error setting generalizes to a more complex problem with binary decisions: joint decision rules can ensure robust improvements in utility when errors between the AI and human signals are negatively correlated\.

We next ask what remains possible when human and AI errors are allowed to be positively correlated\. Similarly to the case for mean squared error, positively correlated errors are not necessarily fatal, but it reduces the amount of residual AI signal that can be used safely\.

###### Proposition 3\.9\.

Define the extremal values

δ≜infΣ∈𝒰s​\(Σ\),γ≜supΣ∈𝒰CovΣ​\(ϕH,ϕA​I∣θ\),\\delta\\;\\triangleq\\;\\inf\_\{\\Sigma\\in\\mathcal\{U\}\}s\(\\Sigma\),\\qquad\\gamma\\;\\triangleq\\;\\sup\_\{\\Sigma\\in\\mathcal\{U\}\}\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\),wheres​\(Σ\)=CovΣ​\(θ,ϕA​I\)s\(\\Sigma\)=\\text\{Cov\}\_\{\\Sigma\}\(\\theta,\\phi\_\{AI\}\)\. Assume the strict inequality

γ<δ​Var​\(ϕH∣θ\)Var​\(ϕH\)\.\\gamma\\;<\\;\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\.Define the positive\-correlation margin and the corresponding pessimistic regression lower bound

κγ≜δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ,bpos≜κγVar​\(r\)\>0\.\\kappa\_\{\\gamma\}\\;\\triangleq\\;\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma,\\qquad b\_\{\\mathrm\{pos\}\}\\;\\triangleq\\;\\frac\{\\kappa\_\{\\gamma\}\}\{\\text\{Var\}\(r\)\}\\;\>\\;0\.Thenβ​\(Σ\)≥bpos\\beta\(\\Sigma\)\\geq b\_\{\\mathrm\{pos\}\}for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. Consequently, the symmetric robust investment ruledJd\_\{J\}constructed usingb=bposb=b\_\{\\mathrm\{pos\}\}weakly dominates the expert\-only ruledHd\_\{H\}in expected utility:

𝔼Σ​\[u​\(dJ​\(ϕH,ϕA​I\),θ\)\]≥𝔼Σ​\[u​\(dH​\(ϕH\),θ\)\]∀Σ∈𝒰\.\\mathbb\{E\}\_\{\\Sigma\}\\\!\\left\[u\(d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\),\\theta\)\\right\]\\;\\geq\\;\\mathbb\{E\}\_\{\\Sigma\}\\\!\\left\[u\(d\_\{H\}\(\\phi\_\{H\}\),\\theta\)\\right\]\\qquad\\forall\\Sigma\\in\\mathcal\{U\}\.Moreover, the inequality is strict for anyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}for whichβ​\(Σ\)\>bpos\\beta\(\\Sigma\)\>b\_\{\\mathrm\{pos\}\}\(on a set of positive probability in\(ϕH,r\)\(\\phi\_\{H\},r\)\)\.

For binary decision\-making, robust complementarity again hinges on the AI signal being informative about the*human residual*rather than merely replicating the expert’s mistakes\. A similar conditionκγ=δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ\\kappa\_\{\\gamma\}=\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gammagoverns feasibility:δ\\deltamust outweigh the worst\-case shared\-error termγ\\gammaafter scaling by the human\-uncertainty factorVar​\(ϕH∣θ\)/Var​\(ϕH\)\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)/\\text\{Var\}\(\\phi\_\{H\}\)\. Whenκγ\\kappa\_\{\\gamma\}is small, the lower bound ofbposb\_\{\\mathrm\{pos\}\}collapses toward0, and the symmetric policy cannot have enough uniformly safe evidence to overruledHd\_\{H\}\.

### 3\.2Beyond the Gaussian model

We finally show that the core results on conditions for improvement in mean squared error extend beyond the linear\-Gaussian model to a substantially more general class of*nonlinear but monotone*signal structures which do not assume joint normality\. We generalize the model as follows\. Letθ\\thetabe a real\-valued random variable with𝔼​\[θ\]=0\\mathbb\{E\}\[\\theta\]=0andVar​\(θ\)<∞\\text\{Var\}\(\\theta\)<\\infty\. The signals follow the generative model

ϕH=f​\(θ\)\+εH,ϕA​I=g​\(θ\)\+εA​I,\\phi\_\{H\}=f\(\\theta\)\+\\varepsilon\_\{H\},\\qquad\\phi\_\{AI\}=g\(\\theta\)\+\\varepsilon\_\{AI\},wheref,g:ℝ→ℝf,g:\\mathbb\{R\}\\to\\mathbb\{R\}are strictly increasing and continuously differentiable andϵH\\epsilon\_\{H\}andϵA​I\\epsilon\_\{AI\}are random variables with expectation 0\. We assume that\(ϵH,ϵA​I\)⟂θ\(\\epsilon\_\{H\},\\epsilon\_\{AI\}\)\\perp\\thetaand that both have finite second moments\. Monotonicity offfandggimplies that each agent’s signal will increase in expectation with the ground truth state, an analogue of positive correlation in the Gaussian state\. The joint distribution\(θ,ϕH,ϕA​I\)\(\\theta,\\phi\_\{H\},\\phi\_\{AI\}\)is fully specified by the tupleξ=\(f,g,P​\(θ\),P​\(ϵH,ϵA​I\)\)\\xi=\(f,g,P\(\\theta\),P\(\\epsilon\_\{H\},\\epsilon\_\{AI\}\)\)\. An uncertainty set𝒰\\mathcal\{U\}is thus a set of potential values forξ\\xi\. In analogy to the uncertainty sets defined earlier for the Gaussian case, we will consider𝒰\\mathcal\{U\}for which allξ∈𝒰\\xi\\in\\mathcal\{U\}agree on the distributionsP​\(θ,ϕH\)P\(\\theta,\\phi\_\{H\}\)andP​\(ϕH,ϕA​I\)P\(\\phi\_\{H\},\\phi\_\{AI\}\)\. Uncertainty lies in the relationship between the AI signal and the truth, reflected inggand the distribution ofϵA​I\\epsilon\_\{AI\}\.

We study conditions in this model where a decision rule that usesϕA​I\\phi\_\{AI\}has robustly lower mean squared error\. We show that conditions analogous to those in the Gaussian case suffice: the AI signal must be sufficiently informative aboutθ\\theta, and the errors must be negatively dependent\. To formalize the first condition, we assume that the AI signal has a uniformly bounded\-below derivative:

g′​\(t\)≥k\>0for all​t∈ℝ\.g^\{\\prime\}\(t\)\\;\\geq\\;k\>0\\quad\\text\{for all \}t\\in\\mathbb\{R\}\.\(7\)This captures the idea that the AI is*uniformly sensitive*to changes inθ\\theta, in the sense that increasingθ\\thetaalways pushesϕA​I\\phi\_\{AI\}up at least at ratekk\.

To formalize the idea of negative dependence in errors, we consider the condition that

𝔼​\[εA​I∣εH\]​is nonincreasing in​εH​almost surely,\\displaystyle\\mathbb\{E\}\[\\varepsilon\_\{AI\}\\mid\\varepsilon\_\{H\}\]\\text\{ is nonincreasing in \}\\varepsilon\_\{H\}\\text\{ almost surely\},\(8\)so that errors in the positive direction for the human imply that the AI will, on average, have smaller errors\. This condition is satisfied by negative conditional covariance in the Gaussian case but now allows us to extend the results without distributional assumptions\.

![Refer to caption](https://arxiv.org/html/2607.06656v1/x1.png)Figure 1:MSE of the expert\-only estimator and our robust estimator as we vary the correlation in expert and LLM errors \(lower is better\)\. The MSE of our estimator strongly dominates the expert\-only baseline when errors are negatively correlated\.The core of the strategy we use to construct a loss\-improving decision rule is the*nonlinear residual*of the AI signal after conditioning on the human’s signal:

r~​\(ϕH,ϕA​I\)≜ϕA​I−𝔼​\[ϕA​I∣ϕH\]\.\\tilde\{r\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\\triangleq\\phi\_\{AI\}\-\\mathbb\{E\}\[\\phi\_\{AI\}\\mid\\phi\_\{H\}\]\.\(9\)
Our goal is to show that under the structural assumptions above,r~\\tilde\{r\}is*still positively correlated withθ\\theta*, and that this can be uniformly exploited to improve squared\-error risk over any predictor that uses onlyϕH\\phi\_\{H\}\.

###### Proposition 3\.10\.

If Equations[7](https://arxiv.org/html/2607.06656#S3.E7)and[8](https://arxiv.org/html/2607.06656#S3.E8)hold for allξ∈𝒰\\xi\\in\\mathcal\{U\},r~\\tilde\{r\}satisfies

Cov​\(r~,θ\)≥k​𝔼​\[Var​\(θ∣ϕH\)\]\.\\text\{Cov\}\(\\tilde\{r\},\\theta\)\\;\\geq\\;k\\,\\mathbb\{E\}\\big\[\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)\\big\]\.for allξ∈𝒰\\xi\\in\\mathcal\{U\}as well\.

When the residual is guaranteed to have positive correlation withθ\\theta, we can construct a decision rule using a strategy very similar to the Gaussian case: start with the regression ofθ\\thetaonϕH\\phi\_\{H\}and then addb⋅r~b\\cdot\\tilde\{r\}for an appropriately chosen coefficientbb\.

###### Proposition 3\.11\.

Consider the optimal human\-only decision ruledH​\(ϕH\)=𝔼​\[θ\|ϕH\]d\_\{H\}\(\\phi\_\{H\}\)=\\mathbb\{E\}\[\\theta\|\\phi\_\{H\}\]and the class of joint decision rulesdb​\(ϕH,ϕA​I\)=𝔼​\[θ\|ϕH\]\+b⋅r~​\(ϕH,ϕA​I\)d\_\{b\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=\\mathbb\{E\}\[\\theta\|\\phi\_\{H\}\]\+b\\cdot\\tilde\{r\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\. Whenever the conditions of Equations[7](https://arxiv.org/html/2607.06656#S3.E7)and[8](https://arxiv.org/html/2607.06656#S3.E8)hold for allξ∈𝒰\\xi\\in\\mathcal\{U\}, andE​\[Var​\(θ∣ϕH\)\]\>0E\[\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)\]\>0, there exists ab\>0b\>0such thatLξ​\(db\)<Lξ​\(dH\)L\_\{\\xi\}\(d\_\{b\}\)<L\_\{\\xi\}\(d\_\{H\}\)for allξ∈𝒰\\xi\\in\\mathcal\{U\}\.

That is, negative dependence in errors is sufficient to ensure robust complementarity under mean squared error in a much broader class of non\-Gaussian settings\.

## 4Synthetic Experiments

We start by using simulation to illustrate how the performance of the robust decision rules constructed above compares to human\-only decision making as the joint distribution of errors varies\. We simulate data from the Gaussian measurement model, which allows us to directly control the error correlation structure by modifying the covariance matrix between human and AI signals\. We defer specific details about the dataset to Appendix[B\.2](https://arxiv.org/html/2607.06656#A2.SS2)\.

![Refer to caption](https://arxiv.org/html/2607.06656v1/x2.png)Figure 2:MSE of the expert\-only estimator and our robust estimator as we vary the expert uncertainty under the positive correlation setting\. Our robust estimator begins to dominate the expert\-only baseline when human uncertainty exceeds that of the AI signal\. The vertical dotted line denotes the AI signal standard deviation\.### 4\.1Synthetic Experiments for MSE

##### Varying human and AI error correlations\.

Figure[1](https://arxiv.org/html/2607.06656#S3.F1)reports the MSE of both estimators as a function of the error correlationρ\\rho\. Consistent with our theoretical analysis, the robust estimator yields the largest gains when errors are negatively correlated\. Asρ\\rhobecomes large, the improvement over the human\-only baseline diminishes: despite the fact that the AI signal contains useful information, it becomes difficult to robustly benefit from\.

##### Varying human uncertainty under positive correlation\.

We next fix the errors to be positively correlated \(ρ=0\.8\\rho=0\.8\) while varying the expert uncertainty\. Figure[2](https://arxiv.org/html/2607.06656#S4.F2)shows that under positive correlation, the robust estimator outperforms the expert\-only baseline only when the uncertainty in the human signal exceeds that of the AI signal, with significant gains limited to the case where the human signal is highly noisy\.

![Refer to caption](https://arxiv.org/html/2607.06656v1/x3.png)Figure 3:Average utility of the expert\-only investment policydHd\_\{H\}and the robust symmetric policydsymd\_\{\\mathrm\{sym\}\}as we vary the correlationρ\\rhobetween human and AI errors \(higher is better\)\. Shaded regions indicate±1\\pm 1standard deviation across seeds\.

### 4\.2Synthetic Experiments for Investment Decisions

![Refer to caption](https://arxiv.org/html/2607.06656v1/x4.png)Figure 4:Average utility of the expert\-only investment policydHd\_\{H\}and the robust symmetric policydsymd\_\{\\mathrm\{sym\}\}as we vary expert uncertaintyσh\\sigma\_\{h\}under positive correlation \(we fixρ=0\.8\\rho=0\.8andσa=1\.5\\sigma\_\{a\}=1\.5\)\. Shaded regions indicate±1\\pm 1standard deviation across seeds\. The vertical dotted line denotes the AI signal standard deviation\.##### Varying human and AI error correlations\.

Figure[3](https://arxiv.org/html/2607.06656#S4.F3)reports the average utility as a function of the error correlationρ\\rho\. Consistent with our theory, the robust symmetric policy yields the largest gains when errors are negatively correlated; in this regime, the residualrris informative about the human residual uncertainty, and incorporating it improves decisions relative todHd\_\{H\}\. Asρ\\rhobecomes large, the residualized machine signal becomes less complementary, and the robust policy reverts to the expert\-only rule\.

##### Varying human uncertainty under positive correlation\.

We next fix a positive correlation level \(ρ=0\.75\\rho=0\.75\) while varying the expert uncertainty\. Figure[4](https://arxiv.org/html/2607.06656#S4.F4)shows that when the human signal is already precise \(smallσh\\sigma\_\{h\}\), the robust procedure matches the human\-only policy, reflecting limited complementarity from the AI under strong positive correlation\. As the human signal becomes noisier, the incremental information in the residualized AI signal becomes more valuable, anddsymd\_\{\\mathrm\{sym\}\}begins to dominatedHd\_\{H\}in average utility\.

![Refer to caption](https://arxiv.org/html/2607.06656v1/x5.png)

![Refer to caption](https://arxiv.org/html/2607.06656v1/x6.png)

Figure 5:Correlation of errors of LLM forecasts vs\. human forecasts on ForecastBench \(left\) and TESS studies \(right\)\. We find thaterrors are positively correlated, which is the difficult setting for complementarity\.Table 1:We report the MSE and correlation structure resulting from various prompting strategies\. Human MSE denotes the human signal MSE\. LLM MSE denotes the LLM signal MSE\. Calibrated Human MSE denotes the optimal predictor’s MSE based on the human signal\.ρ​\(ϵH,ϵA​I\)\\rho\(\\epsilon\_\{H\},\\epsilon\_\{AI\}\)denotes the Pearson correlation coefficient over the human error and the LLM error\. Robust MSE represents the performance of the robust combination of the LLM and human signal\. Note that human rationale and additional context fields are not available for the TESS studies\. Therefore, we only run the Human\-Divergent Reasoning and Restricted Context strategies for ForecastBench\.

## 5Real World Experiments

##### Datasets and Models\.

We conduct experiments on two real\-world forecasting benchmarks\. The first is ForecastBench\(Kargeret al\.,[2025](https://arxiv.org/html/2607.06656#bib.bib8)\), which provides human forecasts \(n=7,383\) for prediction questions drawn from time series datasets and prediction market platforms\. The second is a suite of 43 different social science survey experiments conducted via the Time\-Sharing Experiments for the Social Sciences \(TESS\) project\([Time\-sharing Experiments for the Social Sciences,](https://arxiv.org/html/2607.06656#bib.bib30)\)between 2016 and 2022, for whichHewittet al\.\([2024](https://arxiv.org/html/2607.06656#bib.bib7)\)collected human and LLM forecasts of effect sizes\. We conduct all experiments with GPT\-5 and GPT\-5\-mini\(Singhet al\.,[2025](https://arxiv.org/html/2607.06656#bib.bib32)\)\.

### 5\.1Real\-World Empirical Results

We ask whether current models naturally satisfy conditions on the error structure in real\-world forecasting experiments\. In Figure[5](https://arxiv.org/html/2607.06656#S4.F5), we find that contrary to this requirement, human and LLM errors are positively correlated, quite strongly for ForecastBench\(Kargeret al\.,[2025](https://arxiv.org/html/2607.06656#bib.bib8)\)and more modestly \(but still non\-trivially\) for the TESS studies\. This raises the question of whether such patterns are easily modifiable by changes in prompting to the model, or whether deeper interventions would be required to surface complementary information\.

### 5\.2Potential Mitigation Strategies

We investigate whether post\-hoc prompting strategies can induce negative correlation in prediction errors by encouraging: \(i\) explicit prediction and correction of human errors; \(ii\) divergent or contrarian reasoning, either relative to provided human rationales or to the model’s own initial reasoning traces; and \(iii\) reliance on different information sets, forcing the model to attend to different signals\. Details of each prompt strategy are provided in Appendix[B\.4](https://arxiv.org/html/2607.06656#A2.SS4)\(see Figures[6](https://arxiv.org/html/2607.06656#A2.F6)and[7](https://arxiv.org/html/2607.06656#A2.F7)\)\.

Critically, while a shift in correlation is evident for several strategies, we observe only a single instance \(out of 12 combinations of prompting strategies×\\timestasks\) in which correlation shifts from positive to negative \(see Figures[6](https://arxiv.org/html/2607.06656#A2.F6)and[7](https://arxiv.org/html/2607.06656#A2.F7)\)\. This suggests that enforcing the error structures required for complementarity via prompting alone is nontrivial and unreliable\. More broadly, these results suggest that post\-hoc interventions are unlikely to consistently induce the error structure necessary for robust human\-AI complementarity\. This highlights the necessity for new training methods or objectives to optimize for outputs in this manner, when predictions will be used to complement human signals\.

In Table[1](https://arxiv.org/html/2607.06656#S4.T1), we observe that very few prompting strategies lead to gains of the combined robust MSE over the optimal human MSE on the forecasting task\. This is explained by the fact that all cases remain in the positively correlated error regime, where our theory expects to have no or little improvement, with the sole exception of Contrarian Reasoning on TESS\. In this particular case, we see a slight improvement in MSE for our robust combination estimator\. We note that improvements are small as the human signal is substantially stronger than the LLM signal \(optimal fit MSE 0\.0052 vs\. 0\.0285\)\.

## 6Conclusion

In this work, we take a step towards understanding the conditions under which human\-AI collaboration provably improves upon either agent alone\. Our theoretical analysis reveals that complementarity is not guaranteed by predictive accuracy alone, but that the error correlation structure between human and model predictions plays a significant role in determining whether collaboration yields better performance\. We derive precise conditions on this structure that must be satisfied for complementary gains to emerge, and show that these conditions can be surprisingly stringent in practice\. Empirically, we find that current models do not naturally satisfy these conditions across a range of real\-world forecasting benchmarks\. Simple post\-hoc mitigation strategies–including prompting interventions designed to induce negative error correlation–show limited effectiveness in enforcing the required error structure\. These findings suggest that achieving genuine complementarity may require more fundamental interventions, potentially in earlier stages of the training paradigm\. Our work highlights an important gap between the promise of human\-AI collaboration and its current realization\. We hope these theoretical and empirical observations motivate further investigation into strategies that explicitly optimize for and prioritize complementarity\.

## Impact Statement

This work characterizes the conditions under which human\-AI collaboration yields robust improvements, aiming to support more informed deployment of AI systems in consequential domains\. Our framework provides decision makers with principled guidance on when and how AI predictions can complement human judgment, rather than simply replace it\. More broadly, we hope this work motivates further investigation into methods that prioritize complementarity\.

## Acknowledgments

This material is based upon work supported by the AI Research Institutes Program funded by the National Science Foundation under AI Institute for Societal Decision Making \(AI\-SDM\), Award No\. 2229881\.

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## Appendix AProofs

### A\.1Proof of Proposition[3\.3](https://arxiv.org/html/2607.06656#S3.Thmtheorem3)

Fix a covariance matrixΣ\\Sigmafor\(θ,ϕH,ϕA​I\)\(\\theta,\\phi\_\{H\},\\phi\_\{AI\}\)and writeX:=\(ϕH,ϕA​I\)⊤X:=\(\\phi\_\{H\},\\phi\_\{AI\}\)^\{\\top\}\. A \(deterministic\) decision rule is a measurable functiond:ℝ2→ℝd:\\mathbb\{R\}^\{2\}\\to\\mathbb\{R\}, and its mean\-squared prediction loss underΣ\\Sigmais

LΣ​\(d\)=𝔼Σ​\[\(θ−d​\(X\)\)2\]\.L\_\{\\Sigma\}\(d\)=\\mathbb\{E\}\_\{\\Sigma\}\\big\[\(\\theta\-d\(X\)\)^\{2\}\\big\]\.
Note that by Assumption[3\.1](https://arxiv.org/html/2607.06656#S3.Thmtheorem1), the marginal law ofX=\(ϕH,ϕA​I\)X=\(\\phi\_\{H\},\\phi\_\{AI\}\)is the same for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. Denote this common marginal distribution byPXP\_\{X\}and write⟨f,g⟩:=𝔼PX​\[f​\(X\)​g​\(X\)\]\\langle f,g\\rangle:=\\mathbb\{E\}\_\{P\_\{X\}\}\[f\(X\)g\(X\)\]for theL2​\(PX\)L^\{2\}\(P\_\{X\}\)inner product\. LetS:=span​\{ϕH,ϕA​I\}⊂L2​\(PX\)S:=\\mathrm\{span\}\\\{\\phi\_\{H\},\\phi\_\{AI\}\\\}\\subset L^\{2\}\(P\_\{X\}\)be the two\-dimensional linear subspace of \(square\-integrable\) linear functions ofXX\. Given any decision ruled∈L2​\(PX\)d\\in L^\{2\}\(P\_\{X\}\), letdlin∈Sd^\{\\mathrm\{lin\}\}\\in Sbe its orthogonal projection ontoSS; equivalently,dlind^\{\\mathrm\{lin\}\}is the unique linear functionad​ϕH\+bd​ϕA​Ia\_\{d\}\\phi\_\{H\}\+b\_\{d\}\\phi\_\{AI\}that minimizes𝔼PX​\[\(d​\(X\)−a​ϕH−b​ϕA​I\)2\]\\mathbb\{E\}\_\{P\_\{X\}\}\[\(d\(X\)\-a\\phi\_\{H\}\-b\\phi\_\{AI\}\)^\{2\}\]\. Define the residualu:=d−dlinu:=d\-d^\{\\mathrm\{lin\}\}\. By the characterization of orthogonal projections in Hilbert spaces,uuis orthogonal toSS, i\.e\.,

𝔼PX​\[u​\(X\)​ϕH\]=0and𝔼PX​\[u​\(X\)​ϕA​I\]=0\.\\displaystyle\\mathbb\{E\}\_\{P\_\{X\}\}\[u\(X\)\\phi\_\{H\}\]=0\\qquad\\text\{and\}\\qquad\\mathbb\{E\}\_\{P\_\{X\}\}\[u\(X\)\\phi\_\{AI\}\]=0\.\(10\)Fix anyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. Since\(θ,X\)\(\\theta,X\)is jointly Gaussian underΣ\\Sigma, the conditional meanμΣ​\(X\):=𝔼Σ​\[θ∣X\]\\mu\_\{\\Sigma\}\(X\):=\\mathbb\{E\}\_\{\\Sigma\}\[\\theta\\mid X\]is an element ofSS; that is,μΣ​\(X\)=aΣ​ϕH\+bΣ​ϕA​I\\mu\_\{\\Sigma\}\(X\)=a\_\{\\Sigma\}\\phi\_\{H\}\+b\_\{\\Sigma\}\\phi\_\{AI\}for some coefficients depending onΣ\\Sigma\. The tower property gives

𝔼Σ​\[u​\(X\)​θ\]=𝔼Σ​\[u​\(X\)​𝔼Σ​\[θ∣X\]\]=𝔼PX​\[u​\(X\)​μΣ​\(X\)\]\.\\displaystyle\\mathbb\{E\}\_\{\\Sigma\}\[u\(X\)\\theta\]=\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(X\)\\,\\mathbb\{E\}\_\{\\Sigma\}\[\\theta\\mid X\]\\big\]=\\mathbb\{E\}\_\{P\_\{X\}\}\[u\(X\)\\mu\_\{\\Sigma\}\(X\)\]\.ButμΣ∈S\\mu\_\{\\Sigma\}\\in Sandu⟂Su\\perp Sby \([10](https://arxiv.org/html/2607.06656#A1.E10)\), hence𝔼PX​\[u​μΣ\]=0\\mathbb\{E\}\_\{P\_\{X\}\}\[u\\mu\_\{\\Sigma\}\]=0and therefore𝔼Σ​\[u​θ\]=0\\mathbb\{E\}\_\{\\Sigma\}\[u\\theta\]=0\. Similarly, sincedlin∈Sd^\{\\mathrm\{lin\}\}\\in S, we have𝔼PX​\[u​dlin\]=0\\mathbb\{E\}\_\{P\_\{X\}\}\[ud^\{\\mathrm\{lin\}\}\]=0and thus𝔼Σ​\[u​dlin\]=0\\mathbb\{E\}\_\{\\Sigma\}\[ud^\{\\mathrm\{lin\}\}\]=0\. Expanding the squared loss and using these orthogonality relations yields

LΣ​\(d\)\\displaystyle L\_\{\\Sigma\}\(d\)=𝔼Σ​\[\(θ−dlin​\(X\)−u​\(X\)\)2\]\\displaystyle=\\mathbb\{E\}\_\{\\Sigma\}\\big\[\(\\theta\-d^\{\\mathrm\{lin\}\}\(X\)\-u\(X\)\)^\{2\}\\big\]=𝔼Σ​\[\(θ−dlin​\(X\)\)2\]\+𝔼PX​\[u​\(X\)2\]−2​𝔼Σ​\[u​\(X\)​θ\]\+2​𝔼Σ​\[u​\(X\)​dlin​\(X\)\]\\displaystyle=\\mathbb\{E\}\_\{\\Sigma\}\\big\[\(\\theta\-d^\{\\mathrm\{lin\}\}\(X\)\)^\{2\}\\big\]\+\\mathbb\{E\}\_\{P\_\{X\}\}\[u\(X\)^\{2\}\]\-2\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(X\)\\theta\\big\]\+2\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(X\)d^\{\\mathrm\{lin\}\}\(X\)\\big\]=LΣ​\(dlin\)\+𝔼PX​\[u​\(X\)2\]\\displaystyle=L\_\{\\Sigma\}\(d^\{\\mathrm\{lin\}\}\)\+\\mathbb\{E\}\_\{P\_\{X\}\}\[u\(X\)^\{2\}\]≥LΣ​\(dlin\)\.\\displaystyle\\geq L\_\{\\Sigma\}\(d^\{\\mathrm\{lin\}\}\)\.Thus, for*every*Σ∈𝒰\\Sigma\\in\\mathcal\{U\}, the linear projectiondlind^\{\\mathrm\{lin\}\}weakly improves ondd\.

### A\.2Auxiliary Lemmas for MSE Proofs

We record two elementary lemmas that will be used repeatedly in the proofs of Propositions[3\.4](https://arxiv.org/html/2607.06656#S3.Thmtheorem4)–[3\.6](https://arxiv.org/html/2607.06656#S3.Thmtheorem6)\. All expectations, variances, and covariances are taken with respect to the joint Gaussian lawN​\(0,Σ\)N\(0,\\Sigma\)under the covariance matrix currently under discussion\.

###### Lemma A\.1\(Loss gap for linear aggregation\)\.

Fix any covariance matrixΣ\\Sigmafor\(θ,ϕH,ϕA​I\)\(\\theta,\\phi\_\{H\},\\phi\_\{AI\}\)\. LetdH​\(ϕH\):=ϕHd\_\{H\}\(\\phi\_\{H\}\):=\\phi\_\{H\}and, for anyb∈ℝb\\in\\mathbb\{R\}, definedb​\(ϕH,ϕA​I\):=ϕH\+b​ϕA​Id\_\{b\}\(\\phi\_\{H\},\\phi\_\{AI\}\):=\\phi\_\{H\}\+b\\phi\_\{AI\}\. Under mean squared error lossℓ​\(d,θ\)=\(d−θ\)2\\ell\(d,\\theta\)=\(d\-\\theta\)^\{2\}, the difference in expected losses satisfies

LΣ​\(db\)−LΣ​\(dH\)=b2​Var​\(ϕA​I\)−2​b​Cov​\(θ−ϕH,ϕA​I\)\.\\displaystyle L\_\{\\Sigma\}\(d\_\{b\}\)\-L\_\{\\Sigma\}\(d\_\{H\}\)=b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b\\,\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\.\(11\)

*Proof\.*Define the human prediction erroreH:=θ−ϕHe\_\{H\}:=\\theta\-\\phi\_\{H\}\. Then for any realbbwe have

LΣ​\(db\)\\displaystyle L\_\{\\Sigma\}\(d\_\{b\}\)=𝔼​\[\(θ−db​\(ϕH,ϕA​I\)\)2\]=𝔼​\[\(θ−ϕH−b​ϕA​I\)2\]\\displaystyle=\\mathbb\{E\}\\big\[\(\\theta\-d\_\{b\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\)^\{2\}\\big\]=\\mathbb\{E\}\\big\[\(\\theta\-\\phi\_\{H\}\-b\\phi\_\{AI\}\)^\{2\}\\big\]=𝔼​\[\(eH−b​ϕA​I\)2\]\.\\displaystyle=\\mathbb\{E\}\\big\[\(e\_\{H\}\-b\\phi\_\{AI\}\)^\{2\}\\big\]\.We now expand the square*deterministically*and then take expectations:

\(eH−b​ϕA​I\)2\\displaystyle\(e\_\{H\}\-b\\phi\_\{AI\}\)^\{2\}=eH2\+b2​ϕA​I2−2​b​eH​ϕA​I\.\\displaystyle=e\_\{H\}^\{2\}\+b^\{2\}\\phi\_\{AI\}^\{2\}\-2b\\,e\_\{H\}\\phi\_\{AI\}\.Taking expectations and using𝔼​\[ϕA​I2\]=Var​\(ϕA​I\)\\mathbb\{E\}\[\\phi\_\{AI\}^\{2\}\]=\\text\{Var\}\(\\phi\_\{AI\}\)and𝔼​\[eH​ϕA​I\]=Cov​\(eH,ϕA​I\)\\mathbb\{E\}\[e\_\{H\}\\phi\_\{AI\}\]=\\text\{Cov\}\(e\_\{H\},\\phi\_\{AI\}\)\(all variables are mean\-zero\) yields

LΣ​\(db\)\\displaystyle L\_\{\\Sigma\}\(d\_\{b\}\)=𝔼​\[eH2\]\+b2​Var​\(ϕA​I\)−2​b​Cov​\(eH,ϕA​I\)\.\\displaystyle=\\mathbb\{E\}\[e\_\{H\}^\{2\}\]\+b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b\\,\\text\{Cov\}\(e\_\{H\},\\phi\_\{AI\}\)\.On the other hand,

LΣ​\(dH\)=𝔼​\[\(θ−ϕH\)2\]=𝔼​\[eH2\]\.L\_\{\\Sigma\}\(d\_\{H\}\)=\\mathbb\{E\}\\big\[\(\\theta\-\\phi\_\{H\}\)^\{2\}\\big\]=\\mathbb\{E\}\[e\_\{H\}^\{2\}\]\.SubtractingLΣ​\(dH\)L\_\{\\Sigma\}\(d\_\{H\}\)fromLΣ​\(db\)L\_\{\\Sigma\}\(d\_\{b\}\)and recallingeH=θ−ϕHe\_\{H\}=\\theta\-\\phi\_\{H\}gives exactly \([11](https://arxiv.org/html/2607.06656#A1.E11)\)\.

###### Lemma A\.2\.

Assume\(θ,ϕH,ϕA​I\)∼N​\(0,Σ\)\(\\theta,\\phi\_\{H\},\\phi\_\{AI\}\)\\sim N\(0,\\Sigma\)\. Then the covariance between the machine signal and the*human prediction error*satisfies the identity

Cov​\(θ−ϕH,ϕA​I\)=Cov​\(θ,ϕA​I\)​Var​\(ϕH∣θ\)Var​\(ϕH\)−Cov​\(ϕH,ϕA​I∣θ\)\.\\displaystyle\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)=\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\.\(12\)In particular, ifCov​\(ϕH,ϕA​I∣θ\)≤γ\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\\leq\\gammathen

Cov​\(θ−ϕH,ϕA​I\)≥Cov​\(θ,ϕA​I\)​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ,\\displaystyle\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\geq\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma,\(13\)and if additionallyCov​\(θ,ϕA​I\)≥δ\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\geq\\deltathen

Cov​\(θ−ϕH,ϕA​I\)≥δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ\.\\displaystyle\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\geq\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma\.\(14\)

*Proof\.*

##### Step 1: decomposeCov​\(ϕH,ϕA​I\)\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)using the law of total covariance\.

Sinceθ\\thetais one\-dimensional, the law of total covariance gives

Cov​\(ϕH,ϕA​I\)\\displaystyle\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=𝔼​\[Cov​\(ϕH,ϕA​I∣θ\)\]\+Cov​\(𝔼​\[ϕH∣θ\],𝔼​\[ϕA​I∣θ\]\)\.\\displaystyle=\\mathbb\{E\}\\big\[\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\\big\]\+\\text\{Cov\}\\big\(\\mathbb\{E\}\[\\phi\_\{H\}\\mid\\theta\],\\,\\mathbb\{E\}\[\\phi\_\{AI\}\\mid\\theta\]\\big\)\.\(15\)Under joint Gaussianity, the conditional covarianceCov​\(ϕH,ϕA​I∣θ\)\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)is almost surely constant as a function ofθ\\theta, so the first term reduces toCov​\(ϕH,ϕA​I∣θ\)\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)itself\. Moreover, again by joint Gaussianity \(and mean\-zero\),

𝔼​\[ϕH∣θ\]=Cov​\(ϕH,θ\)Var​\(θ\)​θ,𝔼​\[ϕA​I∣θ\]=Cov​\(ϕA​I,θ\)Var​\(θ\)​θ\.\\mathbb\{E\}\[\\phi\_\{H\}\\mid\\theta\]=\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\\theta,\\qquad\\mathbb\{E\}\[\\phi\_\{AI\}\\mid\\theta\]=\\frac\{\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\\theta\.Substituting these linear regressions into the second term of \([15](https://arxiv.org/html/2607.06656#A1.E15)\) yields

Cov​\(𝔼​\[ϕH∣θ\],𝔼​\[ϕA​I∣θ\]\)\\displaystyle\\text\{Cov\}\\big\(\\mathbb\{E\}\[\\phi\_\{H\}\\mid\\theta\],\\,\\mathbb\{E\}\[\\phi\_\{AI\}\\mid\\theta\]\\big\)=Cov​\(Cov​\(ϕH,θ\)Var​\(θ\)​θ,Cov​\(ϕA​I,θ\)Var​\(θ\)​θ\)\\displaystyle=\\text\{Cov\}\\Big\(\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\\theta,\\,\\frac\{\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\\theta\\Big\)\(16\)=Cov​\(ϕH,θ\)​Cov​\(ϕA​I,θ\)Var​\(θ\)2​Var​\(θ\)=Cov​\(ϕH,θ\)​Cov​\(ϕA​I,θ\)Var​\(θ\)\.\\displaystyle=\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)^\{2\}\}\\,\\text\{Var\}\(\\theta\)=\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\.Combining \([15](https://arxiv.org/html/2607.06656#A1.E15)\) and \([16](https://arxiv.org/html/2607.06656#A1.E16)\) gives the identity

Cov​\(ϕH,ϕA​I\)=Cov​\(ϕH,ϕA​I∣θ\)\+Cov​\(ϕH,θ\)​Cov​\(ϕA​I,θ\)Var​\(θ\)\.\\displaystyle\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\+\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\.\(17\)

##### Step 2: rewriteCov​\(θ−ϕH,ϕA​I\)\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)in terms of conditional covariance\.

By bilinearity of covariance,

Cov​\(θ−ϕH,ϕA​I\)\\displaystyle\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)=Cov​\(θ,ϕA​I\)−Cov​\(ϕH,ϕA​I\)\.\\displaystyle=\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\.\(18\)Substituting \([17](https://arxiv.org/html/2607.06656#A1.E17)\) into \([18](https://arxiv.org/html/2607.06656#A1.E18)\) and using symmetryCov​\(θ,ϕA​I\)=Cov​\(ϕA​I,θ\)\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)=\\text\{Cov\}\(\\phi\_\{AI\},\\theta\)yields

Cov​\(θ−ϕH,ϕA​I\)\\displaystyle\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)=Cov​\(θ,ϕA​I\)−Cov​\(ϕH,ϕA​I∣θ\)−Cov​\(ϕH,θ\)​Cov​\(θ,ϕA​I\)Var​\(θ\)\\displaystyle=\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\}\{\\text\{Var\}\(\\theta\)\}\(19\)=Cov​\(θ,ϕA​I\)​\(1−Cov​\(ϕH,θ\)Var​\(θ\)\)−Cov​\(ϕH,ϕA​I∣θ\)\.\\displaystyle=\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\Big\(1\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\\Big\)\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\.

##### Step 3: use normalization to identify the multiplicative factor\.

By our normalization choice,Cov​\(ϕH,θ\)=Var​\(ϕH\)\\text\{Cov\}\(\\phi\_\{H\},\\theta\)=\\text\{Var\}\(\\phi\_\{H\}\)\. For a jointly Gaussian pair\(ϕH,θ\)\(\\phi\_\{H\},\\theta\), the conditional variance is

Var​\(ϕH∣θ\)=Var​\(ϕH\)−Cov​\(ϕH,θ\)2Var​\(θ\)\.\\displaystyle\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)=\\text\{Var\}\(\\phi\_\{H\}\)\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)^\{2\}\}\{\\text\{Var\}\(\\theta\)\}\.\(20\)SubstitutingCov​\(ϕH,θ\)=Var​\(ϕH\)\\text\{Cov\}\(\\phi\_\{H\},\\theta\)=\\text\{Var\}\(\\phi\_\{H\}\)into \([20](https://arxiv.org/html/2607.06656#A1.E20)\) gives

Var​\(ϕH∣θ\)=Var​\(ϕH\)−Var​\(ϕH\)2Var​\(θ\)=Var​\(ϕH\)​\(1−Var​\(ϕH\)Var​\(θ\)\)\.\\displaystyle\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)=\\text\{Var\}\(\\phi\_\{H\}\)\-\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)^\{2\}\}\{\\text\{Var\}\(\\theta\)\}=\\text\{Var\}\(\\phi\_\{H\}\)\\Big\(1\-\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\theta\)\}\\Big\)\.\(21\)Dividing \([21](https://arxiv.org/html/2607.06656#A1.E21)\) byVar​\(ϕH\)\>0\\text\{Var\}\(\\phi\_\{H\}\)\>0yields

1−Cov​\(ϕH,θ\)Var​\(θ\)=1−Var​\(ϕH\)Var​\(θ\)=Var​\(ϕH∣θ\)Var​\(ϕH\)\.\\displaystyle 1\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}=1\-\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\theta\)\}=\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\.\(22\)Substituting \([22](https://arxiv.org/html/2607.06656#A1.E22)\) into \([19](https://arxiv.org/html/2607.06656#A1.E19)\) proves \([12](https://arxiv.org/html/2607.06656#A1.E12)\)\.

##### Step 4: inequalities\.

IfCov​\(ϕH,ϕA​I∣θ\)≤γ\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\\leq\\gamma, then \([12](https://arxiv.org/html/2607.06656#A1.E12)\) implies

Cov​\(θ−ϕH,ϕA​I\)≥Cov​\(θ,ϕA​I\)​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ,\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\geq\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma,which is \([13](https://arxiv.org/html/2607.06656#A1.E13)\)\. If in additionCov​\(θ,ϕA​I\)≥δ\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\geq\\delta, then replacingCov​\(θ,ϕA​I\)\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)byδ\\deltain the right\-hand side yields \([14](https://arxiv.org/html/2607.06656#A1.E14)\)\.

### A\.3Proof of Proposition[3\.4](https://arxiv.org/html/2607.06656#S3.Thmtheorem4)

Fix an uncertainty set𝒰\\mathcal\{U\}as in the main text: allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}agree on every entry of the covariance matrix of\(θ,ϕH,ϕA​I\)\(\\theta,\\phi\_\{H\},\\phi\_\{AI\}\)except possibly on the single entryCov​\(θ,ϕA​I\)\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\. In particular,Var​\(ϕA​I\)\\text\{Var\}\(\\phi\_\{AI\}\)is the same for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\.

Recall that the human\-only decision rule isdH​\(ϕH\)=ϕHd\_\{H\}\(\\phi\_\{H\}\)=\\phi\_\{H\}\(this uses our normalizationCov​\(θ,ϕH\)/Var​\(ϕH\)=1\\text\{Cov\}\(\\theta,\\phi\_\{H\}\)/\\text\{Var\}\(\\phi\_\{H\}\)=1, so that𝔼​\[θ∣ϕH\]=ϕH\\mathbb\{E\}\[\\theta\\mid\\phi\_\{H\}\]=\\phi\_\{H\}under joint normality\)\. For anyb∈ℝb\\in\\mathbb\{R\}define

db​\(ϕH,ϕA​I\):=ϕH\+b​ϕA​I\.d\_\{b\}\(\\phi\_\{H\},\\phi\_\{AI\}\):=\\phi\_\{H\}\+b\\phi\_\{AI\}\.We will exhibit a nonzero coefficientbbsuch thatdbd\_\{b\}strictly dominatesdHd\_\{H\}over𝒰\\mathcal\{U\}\. First, note that by Lemma[A\.2](https://arxiv.org/html/2607.06656#A1.Thmtheorem2),

Cov​\(θ−ϕH,ϕA​I\)=Cov​\(θ,ϕA​I\)​Var​\(ϕH∣θ\)Var​\(ϕH\)−Cov​\(ϕH,ϕA​I∣θ\)\.\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)=\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\.Condition \(2\) states thatCov​\(ϕH,ϕA​I∣θ\)≤0\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\\leq 0, hence for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\},

Cov​\(θ−ϕH,ϕA​I\)≥Cov​\(θ,ϕA​I\)​Var​\(ϕH∣θ\)Var​\(ϕH\)\.\\displaystyle\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\geq\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\.\(23\)Condition \(1\) givesCov​\(θ,ϕA​I\)≥δ\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\geq\\delta, so combining with \([23](https://arxiv.org/html/2607.06656#A1.E23)\) yields the uniform bound

Cov​\(θ−ϕH,ϕA​I\)≥κfor all​Σ∈𝒰,where​κ:=δ​Var​\(ϕH∣θ\)Var​\(ϕH\)\.\\displaystyle\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\geq\\kappa\\qquad\\text\{for all \}\\Sigma\\in\\mathcal\{U\},\\quad\\text\{where \}\\;\\kappa:=\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\.\(24\)Note thatκ\>0\\kappa\>0wheneverδ\>0\\delta\>0andVar​\(ϕH∣θ\)\>0\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\>0\. SinceVar​\(ϕA​I\)\\text\{Var\}\(\\phi\_\{AI\}\)is common acrossΣ∈𝒰\\Sigma\\in\\mathcal\{U\}, choose the nonzero coefficient

b⋆:=κVar​\(ϕA​I\)\.b^\{\\star\}:=\\frac\{\\kappa\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}\.Accordingly, we have that for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\},

LΣ​\(db⋆\)−LΣ​\(dH\)\\displaystyle L\_\{\\Sigma\}\(d\_\{b^\{\\star\}\}\)\-L\_\{\\Sigma\}\(d\_\{H\}\)=\(b⋆\)2​Var​\(ϕA​I\)−2​b⋆​Cov​\(θ−ϕH,ϕA​I\)\(Lemma[A\.1](https://arxiv.org/html/2607.06656#A1.Thmtheorem1)\)\\displaystyle=\(b^\{\\star\}\)^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b^\{\\star\}\\,\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\quad\\text\{ \(Lemma \\ref\{lem:mse\-loss\-gap\}\)\}≤\(b⋆\)2​Var​\(ϕA​I\)−2​b⋆​κ\\displaystyle\\leq\(b^\{\\star\}\)^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b^\{\\star\}\\,\\kappa=κ2Var​\(ϕA​I\)−2​κ2Var​\(ϕA​I\)=−κ2Var​\(ϕA​I\)<0\.\\displaystyle=\\frac\{\\kappa^\{2\}\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}\-2\\frac\{\\kappa^\{2\}\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}=\-\\frac\{\\kappa^\{2\}\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}<0\.HenceLΣ​\(db⋆\)<LΣ​\(dH\)L\_\{\\Sigma\}\(d\_\{b^\{\\star\}\}\)<L\_\{\\Sigma\}\(d\_\{H\}\)for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}, sodb⋆d\_\{b^\{\\star\}\}strictly dominatesdHd\_\{H\}over𝒰\\mathcal\{U\}\.

### A\.4Proof of Proposition[3\.6](https://arxiv.org/html/2607.06656#S3.Thmtheorem6)

Fix an uncertainty set𝒰\\mathcal\{U\}as in the statement of Proposition[3\.6](https://arxiv.org/html/2607.06656#S3.Thmtheorem6)\. In particular, all entries ofΣ\\Sigmaare fixed across𝒰\\mathcal\{U\}except

s​\(Σ\):=CovΣ​\(θ,ϕA​I\)\.s\(\\Sigma\):=\\text\{Cov\}\_\{\\Sigma\}\(\\theta,\\phi\_\{AI\}\)\.Define

δ:=infΣ∈𝒰s​\(Σ\)andγ:=supΣ∈𝒰CovΣ​\(ϕH,ϕA​I∣θ\)\.\\delta:=\\inf\_\{\\Sigma\\in\\mathcal\{U\}\}s\(\\Sigma\)\\qquad\\text\{and\}\\qquad\\gamma:=\\sup\_\{\\Sigma\\in\\mathcal\{U\}\}\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\.Recall thatdH​\(ϕH\)=ϕHd\_\{H\}\(\\phi\_\{H\}\)=\\phi\_\{H\}and that for anyb∈ℝb\\in\\mathbb\{R\}we write

db​\(ϕH,ϕA​I\):=ϕH\+b​ϕA​I\.d\_\{b\}\(\\phi\_\{H\},\\phi\_\{AI\}\):=\\phi\_\{H\}\+b\\phi\_\{AI\}\.We prove both directions\.

##### Sufficiency\.

Assume first thatγ<δ​Var​\(ϕH∣θ\)Var​\(ϕH\)\\gamma<\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\. We will show that everybbin the stated interval satisfiesLΣ​\(db\)<LΣ​\(dH\)L\_\{\\Sigma\}\(d\_\{b\}\)<L\_\{\\Sigma\}\(d\_\{H\}\)uniformly overΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\.

##### Necessity\.

Conversely, assume thatγ≥δ​Var​\(ϕH∣θ\)Var​\(ϕH\)\\gamma\\geq\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}and fix an arbitrary coefficientb\>0b\>0\. We will use only the one\-free\-entry structure of𝒰\\mathcal\{U\}\(together with the fact that conditional covariances are affine ins​\(Σ\)s\(\\Sigma\)\) to select someΣ∈𝒰\\Sigma\\in\\mathcal\{U\}for whichLΣ​\(db\)\>LΣ​\(dH\)L\_\{\\Sigma\}\(d\_\{b\}\)\>L\_\{\\Sigma\}\(d\_\{H\}\)\.

##### Step 0: conditional covariance is affine decreasing ins​\(Σ\)s\(\\Sigma\)\.

Since all entries ofΣ\\Sigmaexcepts​\(Σ\)=CovΣ​\(θ,ϕA​I\)s\(\\Sigma\)=\\text\{Cov\}\_\{\\Sigma\}\(\\theta,\\phi\_\{AI\}\)are fixed across𝒰\\mathcal\{U\}, the \(unconditional\) covariance

c:=Cov​\(ϕH,ϕA​I\)c:=\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)is constant overΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. By the covariance decomposition \([17](https://arxiv.org/html/2607.06656#A1.E17)\) established in the proof of Lemma[A\.2](https://arxiv.org/html/2607.06656#A1.Thmtheorem2),

CovΣ​\(ϕH,ϕA​I\)=CovΣ​\(ϕH,ϕA​I∣θ\)\+Cov​\(ϕH,θ\)​CovΣ​\(ϕA​I,θ\)Var​\(θ\)\.\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\+\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\.Rearranging and using symmetryCovΣ​\(ϕA​I,θ\)=CovΣ​\(θ,ϕA​I\)=s​\(Σ\)\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)=\\text\{Cov\}\_\{\\Sigma\}\(\\theta,\\phi\_\{AI\}\)=s\(\\Sigma\)gives

CovΣ​\(ϕH,ϕA​I∣θ\)\\displaystyle\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)=c−Cov​\(ϕH,θ\)Var​\(θ\)​s​\(Σ\)\.\\displaystyle=c\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\\,s\(\\Sigma\)\.\(25\)Since under our normalization,Cov​\(ϕH,θ\)=Var​\(ϕH\)\\text\{Cov\}\(\\phi\_\{H\},\\theta\)=\\text\{Var\}\(\\phi\_\{H\}\), so if we define the \(fixed\) constant

κ:=Var​\(ϕH\)Var​\(θ\)\>0,\\kappa:=\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\theta\)\}\>0,then \([25](https://arxiv.org/html/2607.06656#A1.E25)\) becomes

CovΣ​\(ϕH,ϕA​I∣θ\)=c−κ​s​\(Σ\)\.\\displaystyle\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)=c\-\\kappa\\,s\(\\Sigma\)\.\(26\)In particular,CovΣ​\(ϕH,ϕA​I∣θ\)\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)is an affine \(indeed, strictly\) decreasing function ofs​\(Σ\)s\(\\Sigma\)\.

##### Step 0a: the maximizer ofCov​\(ϕH,ϕA​I∣θ\)\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)corresponds to the minimizer ofs​\(Σ\)s\(\\Sigma\)\.

Sinces​\(Σ\)≥δs\(\\Sigma\)\\geq\\deltafor allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}and \([26](https://arxiv.org/html/2607.06656#A1.E26)\) is decreasing ins​\(Σ\)s\(\\Sigma\), we have for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}that

CovΣ​\(ϕH,ϕA​I∣θ\)≤c−κ​δ\.\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\\leq c\-\\kappa\\,\\delta\.Taking the supremum overΣ∈𝒰\\Sigma\\in\\mathcal\{U\}yields

γ≤c−κ​δ\.\\displaystyle\\gamma\\leq c\-\\kappa\\,\\delta\.\(27\)For the reverse inequality, fix anyε\>0\\varepsilon\>0\. By definition ofδ\\deltaas an infimum, there exists someΣε∈𝒰\\Sigma\_\{\\varepsilon\}\\in\\mathcal\{U\}such that

s​\(Σε\)<δ\+ε\.\\displaystyle s\(\\Sigma\_\{\\varepsilon\}\)<\\delta\+\\varepsilon\.\(28\)Applying \([26](https://arxiv.org/html/2607.06656#A1.E26)\) atΣ=Σε\\Sigma=\\Sigma\_\{\\varepsilon\}and using \([28](https://arxiv.org/html/2607.06656#A1.E28)\) gives

CovΣε​\(ϕH,ϕA​I∣θ\)=c−κ​s​\(Σε\)\>c−κ​\(δ\+ε\)=\(c−κ​δ\)−κ​ε\.\\text\{Cov\}\_\{\\Sigma\_\{\\varepsilon\}\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)=c\-\\kappa\\,s\(\\Sigma\_\{\\varepsilon\}\)\>c\-\\kappa\(\\delta\+\\varepsilon\)=\(c\-\\kappa\\delta\)\-\\kappa\\varepsilon\.Taking the supremum overΣ∈𝒰\\Sigma\\in\\mathcal\{U\}shows thatγ≥\(c−κ​δ\)−κ​ε\\gamma\\geq\(c\-\\kappa\\delta\)\-\\kappa\\varepsilonfor everyε\>0\\varepsilon\>0\. Lettingε↓0\\varepsilon\\downarrow 0yields

γ≥c−κ​δ\.\\gamma\\geq c\-\\kappa\\,\\delta\.Combining this with \([27](https://arxiv.org/html/2607.06656#A1.E27)\) yields the identity

γ=c−κ​δ\.\\displaystyle\\gamma=c\-\\kappa\\,\\delta\.\(29\)Moreover, the construction above shows that for eachε\>0\\varepsilon\>0we can chooseΣε∈𝒰\\Sigma\_\{\\varepsilon\}\\in\\mathcal\{U\}such that \([28](https://arxiv.org/html/2607.06656#A1.E28)\) holds and, simultaneously,

CovΣε​\(ϕH,ϕA​I∣θ\)\>γ−κ​ε\.\\displaystyle\\text\{Cov\}\_\{\\Sigma\_\{\\varepsilon\}\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\>\\gamma\-\\kappa\\varepsilon\.\(30\)

##### Step 1: an exact loss gap formula\.

FixΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. Lemma[A\.1](https://arxiv.org/html/2607.06656#A1.Thmtheorem1)gives

LΣ​\(db\)−LΣ​\(dH\)=b2​Var​\(ϕA​I\)−2​b​Cov​\(θ−ϕH,ϕA​I\)\.\\displaystyle L\_\{\\Sigma\}\(d\_\{b\}\)\-L\_\{\\Sigma\}\(d\_\{H\}\)=b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b\\,\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\.\(31\)

##### Step 2: a uniform lower bound onCov​\(θ−ϕH,ϕA​I\)\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\.

Lemma[A\.2](https://arxiv.org/html/2607.06656#A1.Thmtheorem2)implies that, for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\},

Cov​\(θ−ϕH,ϕA​I\)=Cov​\(θ,ϕA​I\)​Var​\(ϕH∣θ\)Var​\(ϕH\)−Cov​\(ϕH,ϕA​I∣θ\)\.\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)=\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\.By definition ofδ\\deltaandγ\\gammawe have, for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\},

Cov​\(θ,ϕA​I\)=s​\(Σ\)≥δandCov​\(ϕH,ϕA​I∣θ\)≤γ\.\\text\{Cov\}\(\\theta,\\phi\_\{AI\}\)=s\(\\Sigma\)\\geq\\delta\\qquad\\text\{and\}\\qquad\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\\leq\\gamma\.Substituting these bounds into the preceding identity yields the*uniform*inequality

Cov​\(θ−ϕH,ϕA​I\)≥κγfor all​Σ∈𝒰,whereκγ:=δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ\.\\displaystyle\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\geq\\kappa\_\{\\gamma\}\\qquad\\text\{for all \}\\Sigma\\in\\mathcal\{U\},\\quad\\text\{where\}\\quad\\kappa\_\{\\gamma\}:=\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma\.\(32\)Under the hypothesis of the sufficiency direction,κγ\>0\\kappa\_\{\\gamma\}\>0\.

##### Step 3: choosingbband verifying strict dominance \(sufficiency\)\.

Assumeκγ\>0\\kappa\_\{\\gamma\}\>0\. Fix any coefficientbbsatisfying

0<b<2​κγVar​\(ϕA​I\)\.0<b<\\frac\{2\\kappa\_\{\\gamma\}\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}\.SinceVar​\(ϕA​I\)\>0\\text\{Var\}\(\\phi\_\{AI\}\)\>0, combining \([31](https://arxiv.org/html/2607.06656#A1.E31)\) with \([32](https://arxiv.org/html/2607.06656#A1.E32)\) yields, for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\},

LΣ​\(db\)−LΣ​\(dH\)\\displaystyle L\_\{\\Sigma\}\(d\_\{b\}\)\-L\_\{\\Sigma\}\(d\_\{H\}\)=b2​Var​\(ϕA​I\)−2​b​Cov​\(θ−ϕH,ϕA​I\)\\displaystyle=b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b\\,\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)≤b2​Var​\(ϕA​I\)−2​b​κγ\\displaystyle\\leq b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b\\,\\kappa\_\{\\gamma\}=b​Var​\(ϕA​I\)​\(b−2​κγVar​\(ϕA​I\)\)<0,\\displaystyle=b\\,\\text\{Var\}\(\\phi\_\{AI\}\)\\Big\(b\-\\frac\{2\\kappa\_\{\\gamma\}\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}\\Big\)<0,where the final strict inequality usesb\>0b\>0andb<2​κγ/Var​\(ϕA​I\)b<2\\kappa\_\{\\gamma\}/\\text\{Var\}\(\\phi\_\{AI\}\)\. ThusLΣ​\(db\)<LΣ​\(dH\)L\_\{\\Sigma\}\(d\_\{b\}\)<L\_\{\\Sigma\}\(d\_\{H\}\)for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. This establishes the sufficiency direction \(and the “moreover” part\) of the proposition\.

##### Step 4: necessity of the threshold \(the “only if” direction forb\>0b\>0\)\.

Assume now thatγ≥δ​Var​\(ϕH∣θ\)Var​\(ϕH\)\\gamma\\geq\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}, equivalentlyκγ≤0\\kappa\_\{\\gamma\}\\leq 0\. Fix any coefficientb\>0b\>0\.

##### Step 4a: selecting an adversarialΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\.

Chooseε\>0\\varepsilon\>0small enough that

κγ\+ε<b2​Var​\(ϕA​I\)\.\\displaystyle\\kappa\_\{\\gamma\}\+\\varepsilon<\\frac\{b\}\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\.\(33\)Such anε\\varepsilonexists becauseκγ≤0\\kappa\_\{\\gamma\}\\leq 0while the right\-hand side is strictly positive\. LetΣε∈𝒰\\Sigma\_\{\\varepsilon\}\\in\\mathcal\{U\}be the covariance matrix guaranteed by Step 0a, i\.e\., satisfying both \([28](https://arxiv.org/html/2607.06656#A1.E28)\) and \([30](https://arxiv.org/html/2607.06656#A1.E30)\)\. WriteΣ:=Σε\\Sigma:=\\Sigma\_\{\\varepsilon\}for brevity\.

##### Step 4b: boundingCovΣ​\(θ−ϕH,ϕA​I\)\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)from above\.

Applying Lemma[A\.2](https://arxiv.org/html/2607.06656#A1.Thmtheorem2)atΣ\\Sigmagives

CovΣ​\(θ−ϕH,ϕA​I\)=s​\(Σ\)​Var​\(ϕH∣θ\)Var​\(ϕH\)−CovΣ​\(ϕH,ϕA​I∣θ\)\.\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)=s\(\\Sigma\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\.Usings​\(Σ\)<δ\+εs\(\\Sigma\)<\\delta\+\\varepsilonfrom \([28](https://arxiv.org/html/2607.06656#A1.E28)\) andCovΣ​\(ϕH,ϕA​I∣θ\)\>γ−κ​ε\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\>\\gamma\-\\kappa\\varepsilonfrom \([30](https://arxiv.org/html/2607.06656#A1.E30)\) yields

CovΣ​\(θ−ϕH,ϕA​I\)\\displaystyle\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)<\(δ\+ε\)​Var​\(ϕH∣θ\)Var​\(ϕH\)−\(γ−κ​ε\)\\displaystyle<\(\\delta\+\\varepsilon\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\(\\gamma\-\\kappa\\varepsilon\)=\(δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ\)\+ε​\(Var​\(ϕH∣θ\)Var​\(ϕH\)\+κ\)\\displaystyle=\\Bigl\(\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma\\Bigr\)\+\\varepsilon\\Bigl\(\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\+\\kappa\\Bigr\)=κγ\+ε​\(Var​\(ϕH∣θ\)Var​\(ϕH\)\+κ\)\.\\displaystyle=\\kappa\_\{\\gamma\}\+\\varepsilon\\Bigl\(\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\+\\kappa\\Bigr\)\.We now simplify the coefficient ofε\\varepsilon\. By \([22](https://arxiv.org/html/2607.06656#A1.E22)\) in the proof of Lemma[A\.2](https://arxiv.org/html/2607.06656#A1.Thmtheorem2),

Var​\(ϕH∣θ\)Var​\(ϕH\)=1−Var​\(ϕH\)Var​\(θ\)=1−κ,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}=1\-\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\theta\)\}=1\-\\kappa,henceVar​\(ϕH∣θ\)Var​\(ϕH\)\+κ=1\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\+\\kappa=1\. Therefore,

CovΣ​\(θ−ϕH,ϕA​I\)<κγ\+ε\.\\displaystyle\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)<\\kappa\_\{\\gamma\}\+\\varepsilon\.\(34\)Combining \([34](https://arxiv.org/html/2607.06656#A1.E34)\) with \([33](https://arxiv.org/html/2607.06656#A1.E33)\) yields

CovΣ​\(θ−ϕH,ϕA​I\)<b2​Var​\(ϕA​I\)\.\\displaystyle\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)<\\frac\{b\}\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\.\(35\)

##### Step 4c: concluding thatdbd\_\{b\}cannot uniformly dominatedHd\_\{H\}\.

Finally, combining \([31](https://arxiv.org/html/2607.06656#A1.E31)\) with \([35](https://arxiv.org/html/2607.06656#A1.E35)\) gives

LΣ​\(db\)−LΣ​\(dH\)\\displaystyle L\_\{\\Sigma\}\(d\_\{b\}\)\-L\_\{\\Sigma\}\(d\_\{H\}\)=b2​Var​\(ϕA​I\)−2​b​CovΣ​\(θ−ϕH,ϕA​I\)\\displaystyle=b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b\\,\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\>b2​Var​\(ϕA​I\)−2​b⋅b2​Var​\(ϕA​I\)=0\.\\displaystyle\>b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b\\cdot\\frac\{b\}\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)=0\.Thus, for this \(adversarial\) covariance matrixΣ∈𝒰\\Sigma\\in\\mathcal\{U\}, the decision ruledbd\_\{b\}performs strictly*worse*thandHd\_\{H\}under mean squared error\. Sinceb\>0b\>0was arbitrary, it follows that whenγ≥δ​Var​\(ϕH∣θ\)Var​\(ϕH\)\\gamma\\geq\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}there is*no*positive coefficientbbfor whichdbd\_\{b\}strictly dominatesdHd\_\{H\}over𝒰\\mathcal\{U\}\. This establishes the necessity direction\.

##### Deriving the maximum improvement in loss\.

Next, we move on to discussing the maximum improvement in loss under the setting of positive correlation\. Lemma[A\.1](https://arxiv.org/html/2607.06656#A1.Thmtheorem1)\(Eq\. \([11](https://arxiv.org/html/2607.06656#A1.E11)\)\) gives us that for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\},

LΣ​\(dH\)−LΣ​\(db\)=2​b​Cov​\(θ−ϕH,ϕA​I\)−b2​Var​\(ϕA​I\)\.L\_\{\\Sigma\}\(d\_\{H\}\)\-L\_\{\\Sigma\}\(d\_\{b\}\)=2b\\,\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\-b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\.Using Lemma[A\.2](https://arxiv.org/html/2607.06656#A1.Thmtheorem2)\(Eq\. \([14](https://arxiv.org/html/2607.06656#A1.E14)\)\), we have the lower bound

Cov​\(θ−ϕH,ϕA​I\)≥δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γfor all​Σ∈𝒰\.\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\;\\geq\\;\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma\\qquad\\text\{for all \}\\Sigma\\in\\mathcal\{U\}\.Substituting in this bound results in

infΣ∈𝒰\{LΣ​\(dH\)−LΣ​\(db\)\}≥2​b​\(δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ\)−b2​Var​\(ϕA​I\)\.\\inf\_\{\\Sigma\\in\\mathcal\{U\}\}\\Big\\\{L\_\{\\Sigma\}\(d\_\{H\}\)\-L\_\{\\Sigma\}\(d\_\{b\}\)\\Big\\\}\\;\\geq\\;2b\\Bigl\(\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma\\Bigr\)\\;\-\\;b^\{2\}\\,\\text\{Var\}\(\\phi\_\{AI\}\)\.\(36\)
Next, we can note that the choice of

b⋆≜δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γVar​\(ϕA​I\)b^\{\\star\}\\;\\triangleq\\;\\frac\{\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}\(37\)lies in the admissible interval from Proposition[3\.6](https://arxiv.org/html/2607.06656#S3.Thmtheorem6)\. We can note that right\-hand side of \([36](https://arxiv.org/html/2607.06656#A1.E36)\) is a concave quadratic inbb, which is maximized atb=b⋆b=b^\{\\star\}in \([37](https://arxiv.org/html/2607.06656#A1.E37)\)\. Thus, using this value guarantees that

LΣ​\(db⋆\)≤LΣ​\(dH\)−\(δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ\)2Var​\(ϕA​I\)∀Σ∈𝒰\.L\_\{\\Sigma\}\(d\_\{b^\{\\star\}\}\)\\;\\leq\\;L\_\{\\Sigma\}\(d\_\{H\}\)\\;\-\\;\\frac\{\\Bigl\(\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma\\Bigr\)^\{2\}\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}\\qquad\\forall\\Sigma\\in\\mathcal\{U\}\.\(38\)
With\(δ,γ,Var​\(ϕH\),Var​\(ϕA​I\)\)\(\\delta,\\gamma,\\text\{Var\}\(\\phi\_\{H\}\),\\text\{Var\}\(\\phi\_\{AI\}\)\)fixed, the expression in \([38](https://arxiv.org/html/2607.06656#A1.E38)\) is the square of an affine function ofVar​\(ϕH∣θ\)\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)with positive slopeδ/Var​\(ϕH\)\\delta/\\text\{Var\}\(\\phi\_\{H\}\)\. Hence it is strictly increasing inVar​\(ϕH∣θ\)\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)on the region where the strict inequality in Proposition[3\.6](https://arxiv.org/html/2607.06656#S3.Thmtheorem6)holds\.

### A\.5Proof of Proposition[3\.7](https://arxiv.org/html/2607.06656#S3.Thmtheorem7)

Fix a covariance matrixΣ\\Sigma\. By the definition ofrrand the normalizationCov​\(θ,ϕH\)=Var​\(ϕH\)\\text\{Cov\}\(\\theta,\\phi\_\{H\}\)=\\text\{Var\}\(\\phi\_\{H\}\),

CovΣ​\(r,θ\)\\displaystyle\\text\{Cov\}\_\{\\Sigma\}\(r,\\theta\)=CovΣ​\(ϕA​I,θ\)−Cov​\(ϕH,ϕA​I\)Var​\(ϕH\)​Cov​\(ϕH,θ\)\\displaystyle=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\\text\{Cov\}\(\\phi\_\{H\},\\theta\)=CovΣ​\(ϕA​I,θ\)−Cov​\(ϕH,ϕA​I\)=CovΣ​\(θ−ϕH,ϕA​I\)\.\\displaystyle=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\.Thusβ​\(Σ\)≠0\\beta\(\\Sigma\)\\neq 0impliesCovΣ​\(θ−ϕH,ϕA​I\)≠0\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\neq 0wheneverVar​\(r\)≠0\\text\{Var\}\(r\)\\neq 0\. We assume this non\-redundancy condition; otherwise, the AI signal contains no linear component beyondϕH\\phi\_\{H\}\. Lemma[A\.1](https://arxiv.org/html/2607.06656#A1.Thmtheorem1)gives, for anyb∈ℝb\\in\\mathbb\{R\},

LΣ​\(db\)−LΣ​\(dH\)=b2​Var​\(ϕA​I\)−2​b​CovΣ​\(θ−ϕH,ϕA​I\)\.L\_\{\\Sigma\}\(d\_\{b\}\)\-L\_\{\\Sigma\}\(d\_\{H\}\)=b^\{2\}\\text\{Var\}\(\\phi\_\{AI\}\)\-2b\\,\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\.This is a strictly convex quadratic inbbwith minimizer

b⋆≜CovΣ​\(θ−ϕH,ϕA​I\)Var​\(ϕA​I\),b^\{\\star\}\\triangleq\\frac\{\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\},SinceCovΣ​\(θ−ϕH,ϕA​I\)≠0\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)\\neq 0, we haveb⋆≠0b^\{\\star\}\\neq 0\. Pluggingb⋆b^\{\\star\}into the loss gap yields

LΣ​\(db⋆\)−LΣ​\(dH\)=−CovΣ​\(θ−ϕH,ϕA​I\)2Var​\(ϕA​I\)<0\.L\_\{\\Sigma\}\(d\_\{b^\{\\star\}\}\)\-L\_\{\\Sigma\}\(d\_\{H\}\)=\-\\frac\{\\text\{Cov\}\_\{\\Sigma\}\(\\theta\-\\phi\_\{H\},\\phi\_\{AI\}\)^\{2\}\}\{\\text\{Var\}\(\\phi\_\{AI\}\)\}<0\.Thereforedb⋆d\_\{b^\{\\star\}\}strictly improves upondHd\_\{H\}underΣ\\Sigma, proving the proposition\.

### A\.6Proof of Proposition[3\.8](https://arxiv.org/html/2607.06656#S3.Thmtheorem8)

We first provide some additional notation and supporting lemmas\. Using joint normality and the definition of the residualrr, for any fixed covariance matrixΣ\\Sigmawe have

θ∣\(ϕH,r\)\\displaystyle\\theta\\mid\(\\phi\_\{H\},r\)∼N​\(μΣ​\(ϕH,r\),σΣ2​\(ϕH,r\)\),\\displaystyle\\sim N\\\!\\big\(\\mu\_\{\\Sigma\}\(\\phi\_\{H\},r\),\\;\\sigma\_\{\\Sigma\}^\{2\}\(\\phi\_\{H\},r\)\\big\),\(39\)μΣ​\(ϕH,r\)\\displaystyle\\mu\_\{\\Sigma\}\(\\phi\_\{H\},r\)=ϕH\+β​\(Σ\)​r,σΣ2​\(ϕH,r\)=σH2−β​\(Σ\)2​Var​\(r\)\.\\displaystyle=\\phi\_\{H\}\+\\beta\(\\Sigma\)\\,r,\\qquad\\sigma\_\{\\Sigma\}^\{2\}\(\\phi\_\{H\},r\)=\\sigma\_\{H\}^\{2\}\-\\beta\(\\Sigma\)^\{2\}\\,\\text\{Var\}\(r\)\.so that for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}and all\(ϕH,r\)\(\\phi\_\{H\},r\),

μΣ​\(ϕH,r\)​\{≥μb​\(ϕH,r\)if​r≥0,≤μb​\(ϕH,r\)if​r≤0,0<σΣ2​\(ϕH,r\)≤σb2<σH2\.\\mu\_\{\\Sigma\}\(\\phi\_\{H\},r\)\\begin\{cases\}\\geq\\mu\_\{\\mathrm\{b\}\}\(\\phi\_\{H\},r\)&\\text\{if \}r\\geq 0,\\\\\[2\.5pt\] \\leq\\mu\_\{\\mathrm\{b\}\}\(\\phi\_\{H\},r\)&\\text\{if \}r\\leq 0,\\end\{cases\}\\qquad 0<\\sigma\_\{\\Sigma\}^\{2\}\(\\phi\_\{H\},r\)\\leq\\sigma\_\{b\}^\{2\}<\\sigma\_\{H\}^\{2\}\.\(40\)For any fixed covariance matrixΣ\\Sigmaand any pair\(x,y\)∈ℝ2\(x,y\)\\in\\mathbb\{R\}^\{2\}, write

PΣ\(x,y\)≜ℙΣ\(θ≥τ\|ϕH=x,r=y\)\.P\_\{\\Sigma\}\(x,y\)\\triangleq\\mathbb\{P\}\_\{\\Sigma\}\\big\(\\theta\\geq\\tau\\,\\big\|\\,\\phi\_\{H\}=x,r=y\\big\)\.Using \([39](https://arxiv.org/html/2607.06656#A1.E39)\) we may express this as

PΣ​\(x,y\)=Φ​\(μΣ​\(x,y\)−τσΣ​\(x,y\)\)\.P\_\{\\Sigma\}\(x,y\)=\\Phi\\\!\\left\(\\frac\{\\mu\_\{\\Sigma\}\(x,y\)\-\\tau\}\{\\sigma\_\{\\Sigma\}\(x,y\)\}\\right\)\.We will boundPΣ​\(x,y\)P\_\{\\Sigma\}\(x,y\)uniformly overΣ∈𝒰\\Sigma\\in\\mathcal\{U\}using the pessimistic mean–variance bounds \([40](https://arxiv.org/html/2607.06656#A1.E40)\)\.

###### Proof of Lemma[3\.5](https://arxiv.org/html/2607.06656#S3.Thmtheorem5)\.

Using \([1](https://arxiv.org/html/2607.06656#S3.E1)\) andCov​\(θ,ϕH\)=Var​\(ϕH\)\\text\{Cov\}\(\\theta,\\phi\_\{H\}\)=\\text\{Var\}\(\\phi\_\{H\}\),

CovΣ​\(r,θ\)=CovΣ​\(ϕA​I,θ\)−Cov​\(ϕH,ϕA​I\)Var​\(ϕH\)​Cov​\(ϕH,θ\)=CovΣ​\(ϕA​I,θ\)−Cov​\(ϕH,ϕA​I\)\.\\text\{Cov\}\_\{\\Sigma\}\(r,\\theta\)=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\\,\\text\{Cov\}\(\\phi\_\{H\},\\theta\)=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\.Moreover, the Gaussian identity for conditional covariance gives

CovΣ​\(ϕH,ϕA​I∣θ\)=Cov​\(ϕH,ϕA​I\)−Cov​\(ϕH,θ\)​CovΣ​\(ϕA​I,θ\)Var​\(θ\)=Cov​\(ϕH,ϕA​I\)−Var​\(ϕH\)​CovΣ​\(ϕA​I,θ\)Var​\(θ\)\.\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)=\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}=\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\-\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\.Thus the assumptionCovΣ​\(ϕH,ϕA​I∣θ\)≤0\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\\leq 0implies

Cov​\(ϕH,ϕA​I\)≤Var​\(ϕH\)Var​\(θ\)​CovΣ​\(ϕA​I,θ\),\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\\;\\leq\\;\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\theta\)\}\\,\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\),and therefore

CovΣ​\(r,θ\)≥CovΣ​\(ϕA​I,θ\)​\(1−Var​\(ϕH\)Var​\(θ\)\)=CovΣ​\(ϕA​I,θ\)​Var​\(ϕH∣θ\)Var​\(ϕH\)≥δ​Var​\(ϕH∣θ\)Var​\(ϕH\)\.\\text\{Cov\}\_\{\\Sigma\}\(r,\\theta\)\\geq\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\\left\(1\-\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\theta\)\}\\right\)=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\\geq\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\.Define the corresponding pessimistic lower bound on the regression coefficient,

b≜δ​Var​\(ϕH∣θ\)Var​\(ϕH\)​Var​\(r\)\>0,b\\;\\triangleq\\;\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\\text\{Var\}\(r\)\}\\;\>\\;0,\(41\)so thatβ​\(Σ\)≥b\\beta\(\\Sigma\)\\geq bfor allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}by \([2](https://arxiv.org/html/2607.06656#S3.E2)\)\. ∎

###### Lemma A\.3\.

Fixτ∈ℝ\\tau\\in\\mathbb\{R\}and defineF​\(μ,σ\)≜Φ​\(μ−τσ\)F\(\\mu,\\sigma\)\\triangleq\\Phi\\\!\\left\(\\frac\{\\mu\-\\tau\}\{\\sigma\}\\right\)forσ\>0\\sigma\>0\.

1. 1\.For everyσ\>0\\sigma\>0,F​\(⋅,σ\)F\(\\cdot,\\sigma\)is strictly increasing inμ\\mu\.
2. 2\.For everyμ\>τ\\mu\>\\tau,F​\(μ,⋅\)F\(\\mu,\\cdot\)is strictly decreasing inσ\\sigma\.
3. 3\.For everyμ<τ\\mu<\\tau,F​\(μ,⋅\)F\(\\mu,\\cdot\)is strictly increasing inσ\\sigma\.

###### Proof of Lemma[A\.3](https://arxiv.org/html/2607.06656#A1.Thmtheorem3)\.

Writez​\(μ,σ\)≜\(μ−τ\)/σz\(\\mu,\\sigma\)\\triangleq\(\\mu\-\\tau\)/\\sigma\. Then

F​\(μ,σ\)=Φ​\(z​\(μ,σ\)\)\.F\(\\mu,\\sigma\)=\\Phi\(z\(\\mu,\\sigma\)\)\.SinceΦ′​\(z\)=φ​\(z\)\>0\\Phi^\{\\prime\}\(z\)=\\varphi\(z\)\>0for allzz, it suffices to examine∂z/∂μ\\partial z/\\partial\\muand∂z/∂σ\\partial z/\\partial\\sigma\. A direct calculation gives

∂z∂μ=1σ\>0,\\frac\{\\partial z\}\{\\partial\\mu\}=\\frac\{1\}\{\\sigma\}\>0,which yields \(1\)\. Next,

∂z∂σ=−μ−τσ2\.\\frac\{\\partial z\}\{\\partial\\sigma\}=\-\\frac\{\\mu\-\\tau\}\{\\sigma^\{2\}\}\.Ifμ\>τ\\mu\>\\tau, thenμ−τ\>0\\mu\-\\tau\>0, so∂z/∂σ<0\\partial z/\\partial\\sigma<0and hence∂F/∂σ=φ​\(z\)​∂z/∂σ<0\\partial F/\\partial\\sigma=\\varphi\(z\)\\,\\partial z/\\partial\\sigma<0, proving \(2\)\. Ifμ<τ\\mu<\\tau, thenμ−τ<0\\mu\-\\tau<0, so∂z/∂σ\>0\\partial z/\\partial\\sigma\>0and therefore∂F/∂σ\>0\\partial F/\\partial\\sigma\>0, proving \(3\)\. ∎

###### Lemma A\.4\(Worst\-case lower bound for positive residuals\)\.

For everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}and every\(x,y\)\(x,y\)withy≥0y\\geq 0,

PΣ​\(x,y\)≥Plow​\(x,y\)\.P\_\{\\Sigma\}\(x,y\)\\;\\geq\\;P\_\{\\mathrm\{low\}\}\(x,y\)\.

###### Proof of Lemma[A\.4](https://arxiv.org/html/2607.06656#A1.Thmtheorem4)\.

Fix\(x,y\)\(x,y\)withy≥0y\\geq 0andΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. From \([40](https://arxiv.org/html/2607.06656#A1.E40)\) we have

μΣ​\(x,y\)=x\+β​\(Σ\)​y≥x\+b​y=μb​\(x,y\),0<σΣ​\(x,y\)≤σb\.\\mu\_\{\\Sigma\}\(x,y\)=x\+\\beta\(\\Sigma\)y\\;\\geq\\;x\+by=\\mu\_\{\\mathrm\{b\}\}\(x,y\),\\qquad 0<\\sigma\_\{\\Sigma\}\(x,y\)\\leq\\sigma\_\{b\}\.Ifμb​\(x,y\)<τ\\mu\_\{\\mathrm\{b\}\}\(x,y\)<\\tau, thenPlow​\(x,y\)=0P\_\{\\mathrm\{low\}\}\(x,y\)=0by definition andPΣ​\(x,y\)≥0P\_\{\\Sigma\}\(x,y\)\\geq 0trivially, so the inequality holds\. If insteadμb​\(x,y\)≥τ\\mu\_\{\\mathrm\{b\}\}\(x,y\)\\geq\\tau, then by Lemma[A\.3](https://arxiv.org/html/2607.06656#A1.Thmtheorem3)\(1\)–\(2\) the mapF​\(μ,σ\)=Φ​\(\(μ−τ\)/σ\)F\(\\mu,\\sigma\)=\\Phi\(\(\\mu\-\\tau\)/\\sigma\)is increasing inμ\\muand decreasing inσ\\sigmaon the domain\{μ≥τ,σ\>0\}\\\{\\mu\\geq\\tau,\\sigma\>0\\\}\. Since\(μΣ​\(x,y\),σΣ​\(x,y\)\)\(\\mu\_\{\\Sigma\}\(x,y\),\\sigma\_\{\\Sigma\}\(x,y\)\)lies in this domain and satisfies

μΣ​\(x,y\)≥μb​\(x,y\),σΣ​\(x,y\)≤σb,\\mu\_\{\\Sigma\}\(x,y\)\\geq\\mu\_\{\\mathrm\{b\}\}\(x,y\),\\qquad\\sigma\_\{\\Sigma\}\(x,y\)\\leq\\sigma\_\{b\},we obtain

PΣ​\(x,y\)=F​\(μΣ​\(x,y\),σΣ​\(x,y\)\)≥F​\(μb​\(x,y\),σb\)=Plow​\(x,y\),P\_\{\\Sigma\}\(x,y\)=F\(\\mu\_\{\\Sigma\}\(x,y\),\\sigma\_\{\\Sigma\}\(x,y\)\)\\;\\geq\\;F\(\\mu\_\{\\mathrm\{b\}\}\(x,y\),\\sigma\_\{b\}\)=P\_\{\\mathrm\{low\}\}\(x,y\),as claimed\. ∎

###### Lemma A\.5\(Worst\-case upper bound for negative residuals\)\.

For everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}and every\(x,y\)\(x,y\)withy<0y<0,

PΣ​\(x,y\)≤Phigh​\(x,y\)\.P\_\{\\Sigma\}\(x,y\)\\;\\leq\\;P\_\{\\mathrm\{high\}\}\(x,y\)\.

###### Proof of Lemma[A\.5](https://arxiv.org/html/2607.06656#A1.Thmtheorem5)\.

Fix\(x,y\)\(x,y\)withy<0y<0andΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. Thenβ​\(Σ\)≥b\\beta\(\\Sigma\)\\geq bimplies

μΣ​\(x,y\)=x\+β​\(Σ\)​y≤x\+b​y=μb​\(x,y\),\\mu\_\{\\Sigma\}\(x,y\)=x\+\\beta\(\\Sigma\)y\\;\\leq\\;x\+by=\\mu\_\{\\mathrm\{b\}\}\(x,y\),and we always have0<σΣ​\(x,y\)≤σH0<\\sigma\_\{\\Sigma\}\(x,y\)\\leq\\sigma\_\{H\}because conditioning on\(ϕH,r\)\(\\phi\_\{H\},r\)cannot increase the variance beyondVar​\(θ∣ϕH\)=σH2\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)=\\sigma\_\{H\}^\{2\}\.

Consider first the caseμb​\(x,y\)≥τ\\mu\_\{\\mathrm\{b\}\}\(x,y\)\\geq\\tau\. ThenPhigh​\(x,y\)=1P\_\{\\mathrm\{high\}\}\(x,y\)=1, and triviallyPΣ​\(x,y\)≤1=Phigh​\(x,y\)P\_\{\\Sigma\}\(x,y\)\\leq 1=P\_\{\\mathrm\{high\}\}\(x,y\)\.

Now supposeμb​\(x,y\)<τ\\mu\_\{\\mathrm\{b\}\}\(x,y\)<\\tau\. For any covariance structure consistent with our uncertainty set we can write

PΣ​\(x,y\)=F​\(μΣ​\(x,y\),σΣ​\(x,y\)\)≤supμ≤μb​\(x,y\),0<σ≤σHF​\(μ,σ\),P\_\{\\Sigma\}\(x,y\)=F\\big\(\\mu\_\{\\Sigma\}\(x,y\),\\sigma\_\{\\Sigma\}\(x,y\)\\big\)\\leq\\sup\_\{\\mu\\leq\\mu\_\{\\mathrm\{b\}\}\(x,y\),\\ 0<\\sigma\\leq\\sigma\_\{H\}\}F\(\\mu,\\sigma\),where againF​\(μ,σ\)=Φ​\(\(μ−τ\)/σ\)F\(\\mu,\\sigma\)=\\Phi\(\(\\mu\-\\tau\)/\\sigma\)\. By Lemma[A\.3](https://arxiv.org/html/2607.06656#A1.Thmtheorem3)\(1\),FFis increasing inμ\\mufor fixedσ\\sigma, so the supremum overμ≤μb​\(x,y\)\\mu\\leq\\mu\_\{\\mathrm\{b\}\}\(x,y\)is attained atμ=μb​\(x,y\)\\mu=\\mu\_\{\\mathrm\{b\}\}\(x,y\)\. Since nowμb​\(x,y\)<τ\\mu\_\{\\mathrm\{b\}\}\(x,y\)<\\tau, Lemma[A\.3](https://arxiv.org/html/2607.06656#A1.Thmtheorem3)\(3\) implies thatF​\(μb​\(x,y\),σ\)F\(\\mu\_\{\\mathrm\{b\}\}\(x,y\),\\sigma\)is increasing inσ\\sigma, so the supremum overσ≤σH\\sigma\\leq\\sigma\_\{H\}is obtained atσ=σH\\sigma=\\sigma\_\{H\}\. Thus

supμ≤μb​\(x,y\),0<σ≤σHF​\(μ,σ\)=F​\(μb​\(x,y\),σH\)=Φ​\(μb​\(x,y\)−τσH\)=Phigh​\(x,y\)\.\\sup\_\{\\mu\\leq\\mu\_\{\\mathrm\{b\}\}\(x,y\),\\ 0<\\sigma\\leq\\sigma\_\{H\}\}F\(\\mu,\\sigma\)=F\\big\(\\mu\_\{\\mathrm\{b\}\}\(x,y\),\\sigma\_\{H\}\\big\)=\\Phi\\\!\\left\(\\frac\{\\mu\_\{\\mathrm\{b\}\}\(x,y\)\-\\tau\}\{\\sigma\_\{H\}\}\\right\)=P\_\{\\mathrm\{high\}\}\(x,y\)\.Since\(μΣ​\(x,y\),σΣ​\(x,y\)\)\(\\mu\_\{\\Sigma\}\(x,y\),\\sigma\_\{\\Sigma\}\(x,y\)\)satisfies the constraintsμΣ​\(x,y\)≤μb​\(x,y\)\\mu\_\{\\Sigma\}\(x,y\)\\leq\\mu\_\{\\mathrm\{b\}\}\(x,y\)and0<σΣ​\(x,y\)≤σH0<\\sigma\_\{\\Sigma\}\(x,y\)\\leq\\sigma\_\{H\}, we conclude

PΣ​\(x,y\)≤Phigh​\(x,y\),P\_\{\\Sigma\}\(x,y\)\\leq P\_\{\\mathrm\{high\}\}\(x,y\),as claimed\. ∎

###### Proof of Proposition[3\.8](https://arxiv.org/html/2607.06656#S3.Thmtheorem8)\.

FixΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. Recall the utility from \([3](https://arxiv.org/html/2607.06656#S3.E3)\),

u​\(1,θ\)=1​\{θ≥τ\}−c,u​\(0,θ\)=0\.u\(1,\\theta\)=1\\\{\\theta\\geq\\tau\\\}\-c,\\qquad u\(0,\\theta\)=0\.For any decision ruleddand any realization\(ϕH,r\)=\(x,y\)\(\\phi\_\{H\},r\)=\(x,y\), taking conditional expectations yields

𝔼Σ​\[u​\(d,θ\)∣ϕH=x,r=y\]=\(PΣ​\(x,y\)−c\)​d​\(x,y\),\\mathbb\{E\}\_\{\\Sigma\}\[u\(d,\\theta\)\\mid\\phi\_\{H\}=x,r=y\]=\\big\(P\_\{\\Sigma\}\(x,y\)\-c\\big\)\\,d\(x,y\),wherePΣ\(x,y\)=ℙΣ\(θ≥τ∣ϕH=x,r=y\)P\_\{\\Sigma\}\(x,y\)=\\mathbb\{P\}\_\{\\Sigma\}\(\\theta\\geq\\tau\\mid\\phi\_\{H\}=x,r=y\)\. Therefore,

𝔼Σ​\[u​\(dJ​\(ϕH,ϕA​I\),θ\)\]−𝔼Σ​\[u​\(dH​\(ϕH\),θ\)\]\\displaystyle\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\),\\theta\)\\big\]\-\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(d\_\{H\}\(\\phi\_\{H\}\),\\theta\)\\big\]=𝔼Σ​\[\(PΣ​\(ϕH,r\)−c\)​\(dJ​\(ϕH,ϕA​I\)−dH​\(ϕH\)\)\]\.\\displaystyle\\qquad=\\mathbb\{E\}\_\{\\Sigma\}\\Big\[\\big\(P\_\{\\Sigma\}\(\\phi\_\{H\},r\)\-c\\big\)\\,\\big\(d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\-d\_\{H\}\(\\phi\_\{H\}\)\\big\)\\Big\]\.
Define the \(deterministic\) regions on whichdJd\_\{J\}differs fromdHd\_\{H\}:

A\+≜\{dJ​\(ϕH,ϕA​I\)=1,dH​\(ϕH\)=0\},A−≜\{dJ​\(ϕH,ϕA​I\)=0,dH​\(ϕH\)=1\}\.A\_\{\+\}\\triangleq\\\{d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=1,\\ d\_\{H\}\(\\phi\_\{H\}\)=0\\\},\\qquad A\_\{\-\}\\triangleq\\\{d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=0,\\ d\_\{H\}\(\\phi\_\{H\}\)=1\\\}\.On the complement ofA\+∪A−A\_\{\+\}\\cup A\_\{\-\}, the two rules coincide and the integrand is zero\. Hence

𝔼Σ​\[u​\(dJ​\(ϕH,ϕA​I\),θ\)\]−𝔼Σ​\[u​\(dH​\(ϕH\),θ\)\]\\displaystyle\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\),\\theta\)\\big\]\-\\mathbb\{E\}\_\{\\Sigma\}\\big\[u\(d\_\{H\}\(\\phi\_\{H\}\),\\theta\)\\big\]=𝔼Σ​\[\(PΣ​\(ϕH,r\)−c\)​1​\{\(ϕH,r\)∈A\+\}\]\+𝔼Σ​\[\(c−PΣ​\(ϕH,r\)\)​1​\{\(ϕH,r\)∈A−\}\]\.\\displaystyle\\qquad=\\mathbb\{E\}\_\{\\Sigma\}\\big\[\(P\_\{\\Sigma\}\(\\phi\_\{H\},r\)\-c\)\\,1\\\{\(\\phi\_\{H\},r\)\\in A\_\{\+\}\\\}\\big\]\\;\+\\;\\mathbb\{E\}\_\{\\Sigma\}\\big\[\(c\-P\_\{\\Sigma\}\(\\phi\_\{H\},r\)\)\\,1\\\{\(\\phi\_\{H\},r\)\\in A\_\{\-\}\\\}\\big\]\.\(42\)
*Step 1: Added\-action regionA\+A\_\{\+\}\.*By definition \([6](https://arxiv.org/html/2607.06656#S3.E6)\),A\+A\_\{\+\}can occur only ifPlow​\(ϕH,r\)≥cP\_\{\\mathrm\{low\}\}\(\\phi\_\{H\},r\)\\geq c\. In particular, this impliesr≥0r\\geq 0, so Lemma[A\.4](https://arxiv.org/html/2607.06656#A1.Thmtheorem4)applies and yields

PΣ​\(ϕH,r\)≥Plow​\(ϕH,r\)≥con​A\+\.P\_\{\\Sigma\}\(\\phi\_\{H\},r\)\\;\\geq\\;P\_\{\\mathrm\{low\}\}\(\\phi\_\{H\},r\)\\;\\geq\\;c\\quad\\text\{on \}A\_\{\+\}\.ThusPΣ​\(ϕH,r\)−c≥0P\_\{\\Sigma\}\(\\phi\_\{H\},r\)\-c\\geq 0onA\+A\_\{\+\}, and therefore the first term in \([42](https://arxiv.org/html/2607.06656#A1.E42)\) is nonnegative\.

*Step 2: Removed\-action regionA−A\_\{\-\}\.*By definition \([6](https://arxiv.org/html/2607.06656#S3.E6)\),A−A\_\{\-\}can occur only ifdH​\(ϕH\)=1d\_\{H\}\(\\phi\_\{H\}\)=1,r<0r<0, andPhigh​\(ϕH,r\)≤cP\_\{\\mathrm\{high\}\}\(\\phi\_\{H\},r\)\\leq c\. Sincer<0r<0, Lemma[A\.5](https://arxiv.org/html/2607.06656#A1.Thmtheorem5)applies and gives

PΣ​\(ϕH,r\)≤Phigh​\(ϕH,r\)≤con​A−\.P\_\{\\Sigma\}\(\\phi\_\{H\},r\)\\;\\leq\\;P\_\{\\mathrm\{high\}\}\(\\phi\_\{H\},r\)\\;\\leq\\;c\\quad\\text\{on \}A\_\{\-\}\.Hencec−PΣ​\(ϕH,r\)≥0c\-P\_\{\\Sigma\}\(\\phi\_\{H\},r\)\\geq 0onA−A\_\{\-\}, and the second term in \([42](https://arxiv.org/html/2607.06656#A1.E42)\) is also nonnegative\.

Combining Steps 1–2 shows that the right\-hand side of \([42](https://arxiv.org/html/2607.06656#A1.E42)\) is nonnegative for everyΣ∈𝒰\\Sigma\\in\\mathcal\{U\}, proving weak dominance\.

*Step 3: Strict improvement underβ​\(Σ0\)\>b\\beta\(\\Sigma\_\{0\}\)\>b\.*Now suppose there existsΣ0∈𝒰\\Sigma\_\{0\}\\in\\mathcal\{U\}such thatβ​\(Σ0\)\>b\\beta\(\\Sigma\_\{0\}\)\>b\. Since\(ϕH,r\)\(\\phi\_\{H\},r\)is jointly normal with a nondegenerate covariance matrix andrris independent ofϕH\\phi\_\{H\}, there existsε\>0\\varepsilon\>0such that

ℙΣ0​\(ϕH∈\(xH−ε,xH\)\)\>0\.\\mathbb\{P\}\_\{\\Sigma\_\{0\}\}\\big\(\\phi\_\{H\}\\in\(x\_\{H\}\-\\varepsilon,x\_\{H\}\)\\big\)\>0\.Moreover, becauserrhas unbounded support, we can chooseR\>0R\>0sufficiently large so that for everyx∈\(xH−ε,xH\)x\\in\(x\_\{H\}\-\\varepsilon,x\_\{H\}\)and everyy≥Ry\\geq Rwe have bothμb​\(x,y\)≥τ\\mu\_\{\\mathrm\{b\}\}\(x,y\)\\geq\\tauandPlow​\(x,y\)≥cP\_\{\\mathrm\{low\}\}\(x,y\)\\geq c\.

Define the event

B≜\{ϕH∈\(xH−ε,xH\),r≥R\}\.B\\triangleq\\\{\\phi\_\{H\}\\in\(x\_\{H\}\-\\varepsilon,x\_\{H\}\),\\ r\\geq R\\\}\.By independence ofϕH\\phi\_\{H\}andrrand the positivity of both marginal probabilities, we haveℙΣ0​\(B\)\>0\\mathbb\{P\}\_\{\\Sigma\_\{0\}\}\(B\)\>0\. OnBBwe havedH​\(ϕH\)=0d\_\{H\}\(\\phi\_\{H\}\)=0\(sinceϕH<xH\\phi\_\{H\}<x\_\{H\}\) andPlow​\(ϕH,r\)≥cP\_\{\\mathrm\{low\}\}\(\\phi\_\{H\},r\)\\geq c, hencedJ​\(ϕH,ϕA​I\)=1d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=1by \([6](https://arxiv.org/html/2607.06656#S3.E6)\)\. That is,B⊆A\+B\\subseteq A\_\{\+\}\.

Fix now any realization\(ϕH,r\)=\(x,y\)∈B\(\\phi\_\{H\},r\)=\(x,y\)\\in B\. By constructiony\>0y\>0andμb​\(x,y\)≥τ\\mu\_\{\\mathrm\{b\}\}\(x,y\)\\geq\\tau, and sinceβ​\(Σ0\)\>b\\beta\(\\Sigma\_\{0\}\)\>bwe have

μΣ0​\(x,y\)=x\+β​\(Σ0\)​y\>x\+b​y=μb​\(x,y\),σΣ0​\(x,y\)<σb,\\mu\_\{\\Sigma\_\{0\}\}\(x,y\)=x\+\\beta\(\\Sigma\_\{0\}\)y\\;\>\\;x\+by=\\mu\_\{\\mathrm\{b\}\}\(x,y\),\\qquad\\sigma\_\{\\Sigma\_\{0\}\}\(x,y\)\\;<\\;\\sigma\_\{b\},where the strict inequality forσΣ0​\(x,y\)\\sigma\_\{\\Sigma\_\{0\}\}\(x,y\)follows fromσΣ02​\(x,y\)=σH2−β​\(Σ0\)2​Var​\(r\)\\sigma\_\{\\Sigma\_\{0\}\}^\{2\}\(x,y\)=\\sigma\_\{H\}^\{2\}\-\\beta\(\\Sigma\_\{0\}\)^\{2\}\\text\{Var\}\(r\)andβ​\(Σ0\)2\>b2\\beta\(\\Sigma\_\{0\}\)^\{2\}\>b^\{2\}\. By Lemma[A\.3](https://arxiv.org/html/2607.06656#A1.Thmtheorem3)\(1\)–\(2\),F​\(μ,σ\)=Φ​\(\(μ−τ\)/σ\)F\(\\mu,\\sigma\)=\\Phi\(\(\\mu\-\\tau\)/\\sigma\)is strictly increasing inμ\\muand strictly decreasing inσ\\sigmaon\{μ\>τ,σ\>0\}\\\{\\mu\>\\tau,\\sigma\>0\\\}, so we obtain the strict inequality

PΣ0​\(x,y\)=Φ​\(μΣ0​\(x,y\)−τσΣ0​\(x,y\)\)\>Φ​\(μb​\(x,y\)−τσb\)=Plow​\(x,y\)≥c\.P\_\{\\Sigma\_\{0\}\}\(x,y\)=\\Phi\\\!\\left\(\\frac\{\\mu\_\{\\Sigma\_\{0\}\}\(x,y\)\-\\tau\}\{\\sigma\_\{\\Sigma\_\{0\}\}\(x,y\)\}\\right\)\>\\Phi\\\!\\left\(\\frac\{\\mu\_\{\\mathrm\{b\}\}\(x,y\)\-\\tau\}\{\\sigma\_\{b\}\}\\right\)=P\_\{\\mathrm\{low\}\}\(x,y\)\\;\\geq\\;c\.In particular,PΣ0​\(ϕH,r\)−c\>0P\_\{\\Sigma\_\{0\}\}\(\\phi\_\{H\},r\)\-c\>0on the eventBB, and sinceB⊆A\+B\\subseteq A\_\{\+\}andℙΣ0​\(B\)\>0\\mathbb\{P\}\_\{\\Sigma\_\{0\}\}\(B\)\>0, the first term in \([42](https://arxiv.org/html/2607.06656#A1.E42)\) is strictly positive underΣ0\\Sigma\_\{0\}\. The second term is nonnegative by Step 2, and therefore

𝔼Σ0​\[u​\(dJ​\(ϕH,ϕA​I\),θ\)\]\>𝔼Σ0​\[u​\(dH​\(ϕH\),θ\)\]\.\\mathbb\{E\}\_\{\\Sigma\_\{0\}\}\\big\[u\(d\_\{J\}\(\\phi\_\{H\},\\phi\_\{AI\}\),\\theta\)\\big\]\\;\>\\;\\mathbb\{E\}\_\{\\Sigma\_\{0\}\}\\big\[u\(d\_\{H\}\(\\phi\_\{H\}\),\\theta\)\\big\]\.This completes the proof\. ∎

### A\.7Proof of Proposition[3\.9](https://arxiv.org/html/2607.06656#S3.Thmtheorem9)

*Proof\.*FixΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. By the definition ofrrand the normalizationCov​\(θ,ϕH\)=Var​\(ϕH\)\\text\{Cov\}\(\\theta,\\phi\_\{H\}\)=\\text\{Var\}\(\\phi\_\{H\}\),

CovΣ​\(r,θ\)=CovΣ​\(ϕA​I,θ\)−Cov​\(ϕH,ϕA​I\)\.\\text\{Cov\}\_\{\\Sigma\}\(r,\\theta\)=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\-\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)\.Under joint Gaussianity, the law of total covariance yields

Cov​\(ϕH,ϕA​I\)=CovΣ​\(ϕH,ϕA​I∣θ\)\+Cov​\(ϕH,θ\)​CovΣ​\(ϕA​I,θ\)Var​\(θ\)=CovΣ​\(ϕH,ϕA​I∣θ\)\+Var​\(ϕH\)​CovΣ​\(ϕA​I,θ\)Var​\(θ\)\.\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\+\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)\\,\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\+\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\\,\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\}\{\\text\{Var\}\(\\theta\)\}\.Substituting and simplifying gives

CovΣ​\(r,θ\)=CovΣ​\(ϕA​I,θ\)​\(1−Var​\(ϕH\)Var​\(θ\)\)−CovΣ​\(ϕH,ϕA​I∣θ\)\.\\text\{Cov\}\_\{\\Sigma\}\(r,\\theta\)=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\\Bigl\(1\-\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\theta\)\}\\Bigr\)\-\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\.Finally, the conditional\-variance identity andCov​\(ϕH,θ\)=Var​\(ϕH\)\\text\{Cov\}\(\\phi\_\{H\},\\theta\)=\\text\{Var\}\(\\phi\_\{H\}\)imply

Var​\(ϕH∣θ\)=Var​\(ϕH\)−Cov​\(ϕH,θ\)2Var​\(θ\)=Var​\(ϕH\)​\(1−Var​\(ϕH\)Var​\(θ\)\),\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)=\\text\{Var\}\(\\phi\_\{H\}\)\-\\frac\{\\text\{Cov\}\(\\phi\_\{H\},\\theta\)^\{2\}\}\{\\text\{Var\}\(\\theta\)\}=\\text\{Var\}\(\\phi\_\{H\}\)\\Bigl\(1\-\\frac\{\\text\{Var\}\(\\phi\_\{H\}\)\}\{\\text\{Var\}\(\\theta\)\}\\Bigr\),so\(1−Var​\(ϕH\)/Var​\(θ\)\)=Var​\(ϕH∣θ\)/Var​\(ϕH\)\\bigl\(1\-\\text\{Var\}\(\\phi\_\{H\}\)/\\text\{Var\}\(\\theta\)\\bigr\)=\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)/\\text\{Var\}\(\\phi\_\{H\}\)\. Hence

CovΣ​\(r,θ\)=CovΣ​\(ϕA​I,θ\)​Var​\(ϕH∣θ\)Var​\(ϕH\)−CovΣ​\(ϕH,ϕA​I∣θ\)≥δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ=κγ\.\\text\{Cov\}\_\{\\Sigma\}\(r,\\theta\)=\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{AI\},\\theta\)\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\text\{Cov\}\_\{\\Sigma\}\(\\phi\_\{H\},\\phi\_\{AI\}\\mid\\theta\)\\;\\geq\\;\\delta\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gamma=\\kappa\_\{\\gamma\}\.Dividing byVar​\(r\)\\text\{Var\}\(r\)yieldsβ​\(Σ\)=CovΣ​\(r,θ\)/Var​\(r\)≥κγ/Var​\(r\)=bpos\\beta\(\\Sigma\)=\\text\{Cov\}\_\{\\Sigma\}\(r,\\theta\)/\\text\{Var\}\(r\)\\geq\\kappa\_\{\\gamma\}/\\text\{Var\}\(r\)=b\_\{\\mathrm\{pos\}\}\. With this uniform lower bound in hand, the dominance argument for the symmetric ruledJd\_\{J\}\(given conservative boundsPlowP\_\{\\mathrm\{low\}\}andPhighP\_\{\\mathrm\{high\}\}\) applies verbatim, proving𝔼Σ​\[u​\(dJ,θ\)\]≥𝔼Σ​\[u​\(dH,θ\)\]\\mathbb\{E\}\_\{\\Sigma\}\[u\(d\_\{J\},\\theta\)\]\\geq\\mathbb\{E\}\_\{\\Sigma\}\[u\(d\_\{H\},\\theta\)\]for allΣ∈𝒰\\Sigma\\in\\mathcal\{U\}\. Strictness holds wheneverβ​\(Σ\)\>bpos\\beta\(\\Sigma\)\>b\_\{\\mathrm\{pos\}\}on a set of positive probability\.□\\square

In the decision\-making setting, robust complementarity for investment decisions again hinges on the AI signal being informative about the*human residual*rather than merely replicating the expert’s mistakes\. A similar conditionκγ=δ​Var​\(ϕH∣θ\)Var​\(ϕH\)−γ\\kappa\_\{\\gamma\}=\\delta\\,\\frac\{\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)\}\{\\text\{Var\}\(\\phi\_\{H\}\)\}\-\\gammagoverns feasibility:δ\\deltamust outweigh the worst\-case shared\-error termγ\\gammaafter scaling by the human\-uncertainty factorVar​\(ϕH∣θ\)/Var​\(ϕH\)\\text\{Var\}\(\\phi\_\{H\}\\mid\\theta\)/\\text\{Var\}\(\\phi\_\{H\}\)\. Whenκγ\\kappa\_\{\\gamma\}is small, the lower bound ofbposb\_\{\\mathrm\{pos\}\}collapses toward0, and the symmetric policy cannot have enough uniformly safe evidence to overruledHd\_\{H\}\.

### A\.8Proof of Proposition[3\.10](https://arxiv.org/html/2607.06656#S3.Thmtheorem10)

We begin with a basic monotonicity lemma that holds for arbitrary distributions\.

###### Lemma A\.6\(Monotone covariance\)\.

LetXXbe any real\-valued random variable with𝔼​\[X2\]<∞\\mathbb\{E\}\[X^\{2\}\]<\\infty, and letu,v:ℝ→ℝu,v:\\mathbb\{R\}\\to\\mathbb\{R\}be nondecreasing functions such thatu​\(X\)u\(X\)andv​\(X\)v\(X\)are square\-integrable\. Then

Cov​\(u​\(X\),v​\(X\)\)≥0,\\text\{Cov\}\\big\(u\(X\),v\(X\)\\big\)\\;\\geq\\;0,with strict inequality wheneveruuandvvare strictly increasing on a set of positive probability under the law ofXX\.

###### Proof\.

LetX′X^\{\\prime\}be an independent copy ofXX, defined on the same probability space\. Then

Cov​\(u​\(X\),v​\(X\)\)\\displaystyle\\text\{Cov\}\(u\(X\),v\(X\)\)=𝔼​\[u​\(X\)​v​\(X\)\]−𝔼​\[u​\(X\)\]​𝔼​\[v​\(X\)\]\\displaystyle=\\mathbb\{E\}\\big\[u\(X\)v\(X\)\\big\]\-\\mathbb\{E\}\[u\(X\)\]\\,\\mathbb\{E\}\[v\(X\)\]=12​𝔼​\[\(u​\(X\)−u​\(X′\)\)​\(v​\(X\)−v​\(X′\)\)\]\.\\displaystyle=\\frac\{1\}\{2\}\\mathbb\{E\}\\big\[\(u\(X\)\-u\(X^\{\\prime\}\)\)\(v\(X\)\-v\(X^\{\\prime\}\)\)\\big\]\.The equality in the last line is a standard polarization identity: expanding the right\-hand side and using independence ofXXandX′X^\{\\prime\}yields exactly the covariance\. Sinceuuandvvare nondecreasing, for every pair\(x,x′\)\(x,x^\{\\prime\}\)we have

\(u​\(x\)−u​\(x′\)\)​\(v​\(x\)−v​\(x′\)\)≥0\.\(u\(x\)\-u\(x^\{\\prime\}\)\)\(v\(x\)\-v\(x^\{\\prime\}\)\)\\;\\geq\\;0\.Thus the integrand in the last display is almost surely nonnegative, and hence its expectation is nonnegative:

Cov​\(u​\(X\),v​\(X\)\)≥0\.\\text\{Cov\}\(u\(X\),v\(X\)\)\\;\\geq\\;0\.Ifuuandvvare strictly increasing on a setAAwithℙ​\(X∈A\)\>0\\mathbb\{P\}\(X\\in A\)\>0, then there is a set of pairs\(x,x′\)\(x,x^\{\\prime\}\)of positive probability withx≠x′x\\neq x^\{\\prime\}and\(u​\(x\)−u​\(x′\)\)​\(v​\(x\)−v​\(x′\)\)\>0\(u\(x\)\-u\(x^\{\\prime\}\)\)\(v\(x\)\-v\(x^\{\\prime\}\)\)\>0, implying strict inequality in the covariance\. ∎

We now apply Lemma[A\.6](https://arxiv.org/html/2607.06656#A1.Thmtheorem6)conditionally to obtain a lower bound on the conditional covarianceCov​\(ϕA​I,θ∣ϕH\)\\text\{Cov\}\(\\phi\_\{AI\},\\theta\\mid\\phi\_\{H\}\)\.

###### Lemma A\.7\.

Under the assumptions of this section, for every valuexxofϕH\\phi\_\{H\}for whichVar​\(θ∣ϕH=x\)<∞\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}=x\)<\\inftywe have

Cov​\(ϕA​I,θ∣ϕH=x\)≥k​Var​\(θ∣ϕH=x\)\.\\text\{Cov\}\\big\(\\phi\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\\big\)\\;\\geq\\;k\\,\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}=x\)\.

###### Proof\.

Fixxxand consider the conditional distribution ofθ\\thetagivenϕH=x\\phi\_\{H\}=x\. SinceϕH=f​\(θ\)\+ϵH\\phi\_\{H\}=f\(\\theta\)\+\\epsilon\_\{H\}with\(ϵH,ϵA​I\)\(\\epsilon\_\{H\},\\epsilon\_\{AI\}\)independent ofθ\\theta, this conditional distribution is well\-defined and has finite variance by assumption\.

From the additive signal structure we have

ϕA​I=g​\(θ\)\+ϵA​I\.\\phi\_\{AI\}=g\(\\theta\)\+\\epsilon\_\{AI\}\.Conditioning onϕH=x\\phi\_\{H\}=xyields

Cov​\(ϕA​I,θ∣ϕH=x\)\\displaystyle\\text\{Cov\}\(\\phi\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\)=Cov​\(g​\(θ\)\+ϵA​I,θ∣ϕH=x\)\\displaystyle=\\text\{Cov\}\\big\(g\(\\theta\)\+\\epsilon\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\\big\)=Cov​\(g​\(θ\),θ∣ϕH=x\)\+Cov​\(ϵA​I,θ∣ϕH=x\)\.\\displaystyle=\\text\{Cov\}\\big\(g\(\\theta\),\\theta\\mid\\phi\_\{H\}=x\\big\)\+\\text\{Cov\}\\big\(\\epsilon\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\\big\)\.We lower bound the second term using the assumption thatm​\(e\)≜𝔼​\[ϵA​I∣ϵH=e\]m\(e\)\\triangleq\\mathbb\{E\}\[\\epsilon\_\{AI\}\\mid\\epsilon\_\{H\}=e\]is nonincreasing\.

First observe that, given\(θ,ϕH=x\)\(\\theta,\\phi\_\{H\}=x\), the value ofϵH\\epsilon\_\{H\}is pinned down deterministically as

ϵH=x−f​\(θ\)\.\\epsilon\_\{H\}=x\-f\(\\theta\)\.Since\(ϵH,ϵA​I\)\(\\epsilon\_\{H\},\\epsilon\_\{AI\}\)is independent ofθ\\theta, the conditional distribution ofϵA​I\\epsilon\_\{AI\}givenϵH\\epsilon\_\{H\}does not depend onθ\\theta, and therefore

𝔼​\[ϵA​I∣θ,ϕH=x\]=𝔼​\[ϵA​I∣ϵH=x−f​\(θ\)\]=m​\(x−f​\(θ\)\)\.\\mathbb\{E\}\[\\epsilon\_\{AI\}\\mid\\theta,\\phi\_\{H\}=x\]=\\mathbb\{E\}\[\\epsilon\_\{AI\}\\mid\\epsilon\_\{H\}=x\-f\(\\theta\)\]=m\\big\(x\-f\(\\theta\)\\big\)\.We now computeCov​\(ϵA​I,θ∣ϕH=x\)\\text\{Cov\}\(\\epsilon\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\)by conditioning onθ\\theta\. By the tower property,

𝔼​\[ϵA​I​θ∣ϕH=x\]\\displaystyle\\mathbb\{E\}\[\\epsilon\_\{AI\}\\,\\theta\\mid\\phi\_\{H\}=x\]=𝔼​\[𝔼​\[ϵA​I​θ∣θ,ϕH=x\]\|ϕH=x\]\\displaystyle=\\mathbb\{E\}\\big\[\\,\\mathbb\{E\}\[\\epsilon\_\{AI\}\\,\\theta\\mid\\theta,\\phi\_\{H\}=x\]\\,\\bigm\|\\phi\_\{H\}=x\\big\]=𝔼​\[θ​𝔼​\[ϵA​I∣θ,ϕH=x\]\|ϕH=x\]\\displaystyle=\\mathbb\{E\}\\big\[\\,\\theta\\,\\mathbb\{E\}\[\\epsilon\_\{AI\}\\mid\\theta,\\phi\_\{H\}=x\]\\,\\bigm\|\\phi\_\{H\}=x\\big\]=𝔼​\[θ​m​\(x−f​\(θ\)\)\|ϕH=x\],\\displaystyle=\\mathbb\{E\}\\big\[\\,\\theta\\,m\(x\-f\(\\theta\)\)\\,\\bigm\|\\phi\_\{H\}=x\\big\],where the second equality uses that conditioning onθ\\thetamakesθ\\thetaconstant\. Similarly,

𝔼​\[ϵA​I∣ϕH=x\]=𝔼​\[𝔼​\[ϵA​I∣θ,ϕH=x\]\|ϕH=x\]=𝔼​\[m​\(x−f​\(θ\)\)\|ϕH=x\]\.\\mathbb\{E\}\[\\epsilon\_\{AI\}\\mid\\phi\_\{H\}=x\]=\\mathbb\{E\}\\big\[\\,\\mathbb\{E\}\[\\epsilon\_\{AI\}\\mid\\theta,\\phi\_\{H\}=x\]\\,\\bigm\|\\phi\_\{H\}=x\\big\]=\\mathbb\{E\}\\big\[\\,m\(x\-f\(\\theta\)\)\\,\\bigm\|\\phi\_\{H\}=x\\big\]\.Combining the last two displays yields

Cov​\(ϵA​I,θ∣ϕH=x\)\\displaystyle\\text\{Cov\}\(\\epsilon\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\)=𝔼​\[ϵA​I​θ∣ϕH=x\]−𝔼​\[ϵA​I∣ϕH=x\]​𝔼​\[θ∣ϕH=x\]\\displaystyle=\\mathbb\{E\}\[\\epsilon\_\{AI\}\\,\\theta\\mid\\phi\_\{H\}=x\]\-\\mathbb\{E\}\[\\epsilon\_\{AI\}\\mid\\phi\_\{H\}=x\]\\,\\mathbb\{E\}\[\\theta\\mid\\phi\_\{H\}=x\]=𝔼​\[θ​m​\(x−f​\(θ\)\)\|ϕH=x\]−𝔼​\[m​\(x−f​\(θ\)\)\|ϕH=x\]​𝔼​\[θ∣ϕH=x\]\\displaystyle=\\mathbb\{E\}\\big\[\\,\\theta\\,m\(x\-f\(\\theta\)\)\\,\\bigm\|\\phi\_\{H\}=x\\big\]\-\\mathbb\{E\}\\big\[\\,m\(x\-f\(\\theta\)\)\\,\\bigm\|\\phi\_\{H\}=x\\big\]\\mathbb\{E\}\[\\theta\\mid\\phi\_\{H\}=x\]=Cov​\(m​\(x−f​\(θ\)\),θ∣ϕH=x\)\.\\displaystyle=\\text\{Cov\}\\big\(m\(x\-f\(\\theta\)\),\\,\\theta\\mid\\phi\_\{H\}=x\\big\)\.Becauseffis strictly increasing, the mappingθ↦x−f​\(θ\)\\theta\\mapsto x\-f\(\\theta\)is nonincreasing\. Becausemmis nonincreasing by assumption, the compositionθ↦m​\(x−f​\(θ\)\)\\theta\\mapsto m\(x\-f\(\\theta\)\)is therefore nondecreasing\. Note thatm​\(ϵH\)=𝔼​\[ϵA​I∣ϵH\]m\(\\epsilon\_\{H\}\)=\\mathbb\{E\}\[\\epsilon\_\{AI\}\\mid\\epsilon\_\{H\}\]is square\-integrable by Jensen’s inequality, since𝔼​\[ϵA​I2\]<∞\\mathbb\{E\}\[\\epsilon\_\{AI\}^\{2\}\]<\\infty\. Consequentlym​\(x−f​\(θ\)\)m\(x\-f\(\\theta\)\)is square\-integrable under the conditional law ofθ∣ϕH=x\\theta\\mid\\phi\_\{H\}=x\. Applying Lemma[A\.6](https://arxiv.org/html/2607.06656#A1.Thmtheorem6)conditionally \(withX=θ∣ϕH=xX=\\theta\\mid\\phi\_\{H\}=x,u​\(θ\)=m​\(x−f​\(θ\)\)u\(\\theta\)=m\(x\-f\(\\theta\)\)andv​\(θ\)=θv\(\\theta\)=\\theta\) yields

Cov​\(m​\(x−f​\(θ\)\),θ∣ϕH=x\)≥0,\\text\{Cov\}\\big\(m\(x\-f\(\\theta\)\),\\theta\\mid\\phi\_\{H\}=x\\big\)\\;\\geq\\;0,and hence

Cov​\(ϵA​I,θ∣ϕH=x\)≥0\.\\text\{Cov\}\(\\epsilon\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\)\\;\\geq\\;0\.Combining the last two equations with the earlier decomposition gives

Cov​\(ϕA​I,θ∣ϕH=x\)≥Cov​\(g​\(θ\),θ∣ϕH=x\)\.\\text\{Cov\}\(\\phi\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\)\\;\\geq\\;\\text\{Cov\}\\big\(g\(\\theta\),\\theta\\mid\\phi\_\{H\}=x\\big\)\.
Next, use the derivative lower bound \([7](https://arxiv.org/html/2607.06656#S3.E7)\) to decomposeggas

g​\(t\)=k​t\+h​\(t\),g\(t\)=kt\+h\(t\),whereh′​\(t\)=g′​\(t\)−k≥0h^\{\\prime\}\(t\)=g^\{\\prime\}\(t\)\-k\\geq 0for alltt\. Hencehhis nondecreasing\. For the conditional law ofθ∣ϕH=x\\theta\\mid\\phi\_\{H\}=xwe therefore have

Cov​\(g​\(θ\),θ∣ϕH=x\)\\displaystyle\\text\{Cov\}\(g\(\\theta\),\\theta\\mid\\phi\_\{H\}=x\)=Cov​\(k​θ\+h​\(θ\),θ∣ϕH=x\)\\displaystyle=\\text\{Cov\}\\big\(k\\theta\+h\(\\theta\),\\theta\\mid\\phi\_\{H\}=x\\big\)=k​Var​\(θ∣ϕH=x\)\+Cov​\(h​\(θ\),θ∣ϕH=x\)\.\\displaystyle=k\\,\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}=x\)\+\\text\{Cov\}\\big\(h\(\\theta\),\\theta\\mid\\phi\_\{H\}=x\\big\)\.Applying Lemma[A\.6](https://arxiv.org/html/2607.06656#A1.Thmtheorem6)conditionally \(withX=θ∣ϕH=xX=\\theta\\mid\\phi\_\{H\}=x,u=hu=handv=idv=\\mathrm\{id\}, both nondecreasing\) shows that

Cov​\(h​\(θ\),θ∣ϕH=x\)≥0\.\\text\{Cov\}\\big\(h\(\\theta\),\\theta\\mid\\phi\_\{H\}=x\\big\)\\;\\geq\\;0\.Combining the last two displays yields

Cov​\(g​\(θ\),θ∣ϕH=x\)≥k​Var​\(θ∣ϕH=x\)\.\\text\{Cov\}\(g\(\\theta\),\\theta\\mid\\phi\_\{H\}=x\)\\;\\geq\\;k\\,\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}=x\)\.Finally, combining this inequality withCov​\(ϕA​I,θ∣ϕH=x\)≥Cov​\(g​\(θ\),θ∣ϕH=x\)\\text\{Cov\}\(\\phi\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\)\\geq\\text\{Cov\}\(g\(\\theta\),\\theta\\mid\\phi\_\{H\}=x\)gives

Cov​\(ϕA​I,θ∣ϕH=x\)≥k​Var​\(θ∣ϕH=x\),\\text\{Cov\}\\big\(\\phi\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\\big\)\\;\\geq\\;k\\,\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}=x\),as claimed\. ∎

We now integrate these conditional covariances over the marginal distribution ofϕH\\phi\_\{H\}to obtain a lower bound onCov​\(r~,θ\)\\text\{Cov\}\(\\tilde\{r\},\\theta\)\.

First, observe that by definition,

r~=ϕA​I−𝔼​\[ϕA​I∣ϕH\],\\tilde\{r\}=\\phi\_\{AI\}\-\\mathbb\{E\}\[\\phi\_\{AI\}\\mid\\phi\_\{H\}\],so𝔼​\[r~∣ϕH\]=0\\mathbb\{E\}\[\\tilde\{r\}\\mid\\phi\_\{H\}\]=0and hence𝔼​\[r~\]=0\\mathbb\{E\}\[\\tilde\{r\}\]=0\. Using the law of total covariance we can write

Cov​\(r~,θ\)\\displaystyle\\text\{Cov\}\(\\tilde\{r\},\\theta\)=𝔼​\[r~​θ\]−𝔼​\[r~\]​𝔼​\[θ\]\\displaystyle=\\mathbb\{E\}\\big\[\\tilde\{r\}\\,\\theta\\big\]\-\\mathbb\{E\}\[\\tilde\{r\}\]\\,\\mathbb\{E\}\[\\theta\]=𝔼​\[𝔼​\[r~​θ∣ϕH\]\]−0⋅𝔼​\[θ\]\\displaystyle=\\mathbb\{E\}\\big\[\\,\\mathbb\{E\}\[\\tilde\{r\}\\,\\theta\\mid\\phi\_\{H\}\]\\,\\big\]\-0\\cdot\\mathbb\{E\}\[\\theta\]=𝔼​\[Cov​\(r~,θ∣ϕH\)\],\\displaystyle=\\mathbb\{E\}\\big\[\\,\\text\{Cov\}\(\\tilde\{r\},\\theta\\mid\\phi\_\{H\}\)\\,\\big\],where in the last step we have used the fact that𝔼​\[r~∣ϕH\]=0\\mathbb\{E\}\[\\tilde\{r\}\\mid\\phi\_\{H\}\]=0implies

𝔼​\[r~​θ∣ϕH\]=Cov​\(r~,θ∣ϕH\)\+𝔼​\[r~∣ϕH\]​𝔼​\[θ∣ϕH\]=Cov​\(r~,θ∣ϕH\)\.\\mathbb\{E\}\[\\tilde\{r\}\\,\\theta\\mid\\phi\_\{H\}\]=\\text\{Cov\}\(\\tilde\{r\},\\theta\\mid\\phi\_\{H\}\)\+\\mathbb\{E\}\[\\tilde\{r\}\\mid\\phi\_\{H\}\]\\,\\mathbb\{E\}\[\\theta\\mid\\phi\_\{H\}\]=\\text\{Cov\}\(\\tilde\{r\},\\theta\\mid\\phi\_\{H\}\)\.
Next, note that for each fixed valueϕH=x\\phi\_\{H\}=x,

Cov​\(r~,θ∣ϕH=x\)\\displaystyle\\text\{Cov\}\(\\tilde\{r\},\\theta\\mid\\phi\_\{H\}=x\)=Cov​\(ϕA​I−𝔼​\[ϕA​I∣ϕH=x\],θ∣ϕH=x\)\\displaystyle=\\text\{Cov\}\\big\(\\phi\_\{AI\}\-\\mathbb\{E\}\[\\phi\_\{AI\}\\mid\\phi\_\{H\}=x\],\\,\\theta\\mid\\phi\_\{H\}=x\\big\)=Cov​\(ϕA​I,θ∣ϕH=x\)−Cov​\(𝔼​\[ϕA​I∣ϕH=x\],θ∣ϕH=x\)\.\\displaystyle=\\text\{Cov\}\(\\phi\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\)\-\\text\{Cov\}\\big\(\\mathbb\{E\}\[\\phi\_\{AI\}\\mid\\phi\_\{H\}=x\],\\theta\\mid\\phi\_\{H\}=x\\big\)\.The second term vanishes because𝔼​\[ϕA​I∣ϕH=x\]\\mathbb\{E\}\[\\phi\_\{AI\}\\mid\\phi\_\{H\}=x\]is a constant \(givenϕH=x\\phi\_\{H\}=x\), so

Cov​\(r~,θ∣ϕH=x\)=Cov​\(ϕA​I,θ∣ϕH=x\)\.\\text\{Cov\}\(\\tilde\{r\},\\theta\\mid\\phi\_\{H\}=x\)=\\text\{Cov\}\(\\phi\_\{AI\},\\theta\\mid\\phi\_\{H\}=x\)\.Applying Lemma[A\.7](https://arxiv.org/html/2607.06656#A1.Thmtheorem7)to the right\-hand side yields

Cov​\(r~,θ∣ϕH=x\)≥k​Var​\(θ∣ϕH=x\),\\text\{Cov\}\(\\tilde\{r\},\\theta\\mid\\phi\_\{H\}=x\)\\;\\geq\\;k\\,\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}=x\),for everyxxwith finite conditional variance\. Taking expectations overϕH\\phi\_\{H\}gives

Cov​\(r~,θ\)=𝔼​\[Cov​\(r~,θ∣ϕH\)\]≥k​𝔼​\[Var​\(θ∣ϕH\)\],\\text\{Cov\}\(\\tilde\{r\},\\theta\)=\\mathbb\{E\}\\big\[\\,\\text\{Cov\}\(\\tilde\{r\},\\theta\\mid\\phi\_\{H\}\)\\,\\big\]\\;\\geq\\;k\\,\\mathbb\{E\}\\big\[\\,\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)\\,\\big\],as claimed\. IfVar​\(θ∣ϕH\)\>0\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)\>0on a set of positive probability andk\>0k\>0, then𝔼​\[Var​\(θ∣ϕH\)\]\>0\\mathbb\{E\}\[\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)\]\>0and henceCov​\(r~,θ\)\>0\\text\{Cov\}\(\\tilde\{r\},\\theta\)\>0\.

### A\.9Proof of Proposition[3\.11](https://arxiv.org/html/2607.06656#S3.Thmtheorem11)

Fix anyξ∈𝒰\\xi\\in\\mathcal\{U\}\. Letm​\(ϕH\)≜𝔼​\[θ∣ϕH\]m\(\\phi\_\{H\}\)\\triangleq\\mathbb\{E\}\[\\theta\\mid\\phi\_\{H\}\], so thatdH​\(ϕH\)=m​\(ϕH\)d\_\{H\}\(\\phi\_\{H\}\)=m\(\\phi\_\{H\}\)\. Define the human\-only residual

eH≜θ−m​\(ϕH\)\.e\_\{H\}\\triangleq\\theta\-m\(\\phi\_\{H\}\)\.By the defining property of conditional expectation,eHe\_\{H\}is orthogonal inL2L^\{2\}to every square\-integrable function ofϕH\\phi\_\{H\}:

𝔼​\[eH​h​\(ϕH\)\]=0for all​h​\(ϕH\)∈L2\.\\mathbb\{E\}\\big\[e\_\{H\}\\,h\(\\phi\_\{H\}\)\\big\]=0\\qquad\\text\{for all \}h\(\\phi\_\{H\}\)\\in L^\{2\}\.\(43\)Also, by construction of the nonlinear residual,𝔼​\[r~∣ϕH\]=0\\mathbb\{E\}\[\\tilde\{r\}\\mid\\phi\_\{H\}\]=0and hence𝔼​\[r~\]=0\\mathbb\{E\}\[\\tilde\{r\}\]=0\. Under squared\-error loss, withdb=m​\(ϕH\)\+b​r~d\_\{b\}=m\(\\phi\_\{H\}\)\+b\\tilde\{r\}we have

Lξ​\(db\)\\displaystyle L\_\{\\xi\}\(d\_\{b\}\)=𝔼​\[\(θ−m​\(ϕH\)−b​r~\)2\]=𝔼​\[\(eH−b​r~\)2\]\\displaystyle=\\mathbb\{E\}\\big\[\(\\theta\-m\(\\phi\_\{H\}\)\-b\\tilde\{r\}\)^\{2\}\\big\]=\\mathbb\{E\}\\big\[\(e\_\{H\}\-b\\tilde\{r\}\)^\{2\}\\big\]=𝔼​\[eH2\]\+b2​𝔼​\[r~2\]−2​b​𝔼​\[eH​r~\]\.\\displaystyle=\\mathbb\{E\}\[e\_\{H\}^\{2\}\]\+b^\{2\}\\mathbb\{E\}\[\\tilde\{r\}^\{2\}\]\-2b\\,\\mathbb\{E\}\[e\_\{H\}\\tilde\{r\}\]\.SinceLξ​\(dH\)=𝔼​\[\(θ−m​\(ϕH\)\)2\]=𝔼​\[eH2\]L\_\{\\xi\}\(d\_\{H\}\)=\\mathbb\{E\}\[\(\\theta\-m\(\\phi\_\{H\}\)\)^\{2\}\]=\\mathbb\{E\}\[e\_\{H\}^\{2\}\], subtracting yields

Lξ​\(db\)−Lξ​\(dH\)=b2​Varξ​\(r~\)−2​b​Covξ​\(eH,r~\)\.L\_\{\\xi\}\(d\_\{b\}\)\-L\_\{\\xi\}\(d\_\{H\}\)=b^\{2\}\\text\{Var\}\_\{\\xi\}\(\\tilde\{r\}\)\-2b\\,\\text\{Cov\}\_\{\\xi\}\(e\_\{H\},\\tilde\{r\}\)\.\(44\)as in the Gaussian case\. UsingeH=θ−m​\(ϕH\)e\_\{H\}=\\theta\-m\(\\phi\_\{H\}\)and bilinearity of covariance,

Covξ​\(eH,r~\)=Covξ​\(θ,r~\)−Covξ​\(m​\(ϕH\),r~\)\.\\text\{Cov\}\_\{\\xi\}\(e\_\{H\},\\tilde\{r\}\)=\\text\{Cov\}\_\{\\xi\}\(\\theta,\\tilde\{r\}\)\-\\text\{Cov\}\_\{\\xi\}\(m\(\\phi\_\{H\}\),\\tilde\{r\}\)\.ButCovξ​\(m​\(ϕH\),r~\)=𝔼​\[m​\(ϕH\)​r~\]\\text\{Cov\}\_\{\\xi\}\(m\(\\phi\_\{H\}\),\\tilde\{r\}\)=\\mathbb\{E\}\[m\(\\phi\_\{H\}\)\\tilde\{r\}\]because both terms are mean\-zero, and

𝔼​\[m​\(ϕH\)​r~\]=𝔼​\[𝔼​\[m​\(ϕH\)​r~∣ϕH\]\]=𝔼​\[m​\(ϕH\)​𝔼​\[r~∣ϕH\]\]=0\.\\mathbb\{E\}\[m\(\\phi\_\{H\}\)\\tilde\{r\}\]=\\mathbb\{E\}\\big\[\\mathbb\{E\}\[m\(\\phi\_\{H\}\)\\tilde\{r\}\\mid\\phi\_\{H\}\]\\big\]=\\mathbb\{E\}\\big\[m\(\\phi\_\{H\}\)\\mathbb\{E\}\[\\tilde\{r\}\\mid\\phi\_\{H\}\]\\big\]=0\.Therefore

Covξ​\(eH,r~\)=Covξ​\(θ,r~\)\.\\text\{Cov\}\_\{\\xi\}\(e\_\{H\},\\tilde\{r\}\)=\\text\{Cov\}\_\{\\xi\}\(\\theta,\\tilde\{r\}\)\.\(45\)By Proposition[3\.10](https://arxiv.org/html/2607.06656#S3.Thmtheorem10), for everyξ∈𝒰\\xi\\in\\mathcal\{U\},

Covξ​\(r~,θ\)≥k​𝔼​\[Var​\(θ∣ϕH\)\]\.\\text\{Cov\}\_\{\\xi\}\(\\tilde\{r\},\\theta\)\\geq k\\,\\mathbb\{E\}\\big\[\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)\\big\]\.The right\-hand side depends only on\(θ,ϕH\)\(\\theta,\\phi\_\{H\}\)andkk, which are fixed acrossξ∈𝒰\\xi\\in\\mathcal\{U\}by construction of the uncertainty set\. Define

κ≜k​𝔼​\[Var​\(θ∣ϕH\)\]\.\\kappa\\triangleq k\\,\\mathbb\{E\}\\big\[\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)\\big\]\.By assumption𝔼​\[Var​\(θ∣ϕH\)\]\>0\\mathbb\{E\}\[\\text\{Var\}\(\\theta\\mid\\phi\_\{H\}\)\]\>0andk\>0k\>0, henceκ\>0\\kappa\>0\. Thus,

Covξ​\(θ,r~\)≥κ∀ξ∈𝒰\.\\text\{Cov\}\_\{\\xi\}\(\\theta,\\tilde\{r\}\)\\geq\\kappa\\qquad\\forall\\,\\xi\\in\\mathcal\{U\}\.\(46\)
Moreover,Var​\(r~\)\\text\{Var\}\(\\tilde\{r\}\)is fixed over𝒰\\mathcal\{U\}, as by construction of𝒰\\mathcal\{U\}, the joint distribution overP​\(ϕH,ϕA​I\)P\(\\phi\_\{H\},\\phi\_\{AI\}\), and consequently, ofr~=ϕA​I−𝔼​\[ϕA​I\|ϕH\]\\tilde\{r\}=\\phi\_\{AI\}\-\\mathbb\{E\}\[\\phi\_\{AI\}\|\\phi\_\{H\}\]is fixed\. Therefore, sinceCov​\(θ,r~\)≥κ\>0\\text\{Cov\}\(\\theta,\\tilde\{r\}\)\\geq\\kappa\>0, we have thatVar​\(r~\)\>0\\text\{Var\}\(\\tilde\{r\}\)\>0\.

Accordingly, we have

Lξ​\(db\)−Lξ​\(dH\)≤b2​Var​\(r~\)−2​b​κ=b​Var​\(r~\)​\(b−2​κVar​\(r~\)\)\.L\_\{\\xi\}\(d\_\{b\}\)\-L\_\{\\xi\}\(d\_\{H\}\)\\leq b^\{2\}\\text\{Var\}\(\\tilde\{r\}\)\-2b\\,\\kappa=b\\text\{Var\}\(\\tilde\{r\}\)\\Bigl\(b\-\\frac\{2\\kappa\}\{\\text\{Var\}\(\\tilde\{r\}\)\}\\Bigr\)\.Thus, for everybbsatisfying0<b<2​κ/Var​\(r~\)0<b<2\\kappa/\\text\{Var\}\(\\tilde\{r\}\), the right\-hand side is strictly negative, and thereforeLξ​\(db\)<Lξ​\(dH\)L\_\{\\xi\}\(d\_\{b\}\)<L\_\{\\xi\}\(d\_\{H\}\)holds for allξ∈𝒰\\xi\\in\\mathcal\{U\}\.

## Appendix BAdditional Experimental Results & Details

### B\.1Additional Synthetic Figures

We present our result on varying the expert uncertainty in the setting of positive correlation in Figure[4](https://arxiv.org/html/2607.06656#S4.F4)\.

### B\.2Synthetic Experiment Details

We present more details for our synthetic experiments, including the dataset generation process and model implementation details\.

#### B\.2\.1Synthetic Dataset

We use the same synthetic data for both MSE and investment decision experiments\. For each uniti=1,…,ni=1,\\dots,n, we generate a synthetic ground truth

θi∼𝒩​\(0,σθ2\),\\theta\_\{i\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{\\theta\}^\{2\}\),and two observed signals \(i\.e\., human and AI\-generated\)

Hi=θi\+εH,i,Ai=θi\+εA,i,H\_\{i\}=\\theta\_\{i\}\+\\varepsilon\_\{H,i\},\\qquad A\_\{i\}=\\theta\_\{i\}\+\\varepsilon\_\{A,i\},where the noise terms are jointly Gaussian with mean zero and covariance

\(εH,iεA,i\)∼𝒩​\(\(00\),\(σh2ρ​σh​σaρ​σh​σaσa2\)\)\.\\begin\{pmatrix\}\\varepsilon\_\{H,i\}\\\\ \\varepsilon\_\{A,i\}\\end\{pmatrix\}\\sim\\mathcal\{N\}\\\!\\left\(\\begin\{pmatrix\}0\\\\ 0\\end\{pmatrix\},\\begin\{pmatrix\}\\sigma\_\{h\}^\{2\}&\\rho\\,\\sigma\_\{h\}\\sigma\_\{a\}\\\\ \\rho\\,\\sigma\_\{h\}\\sigma\_\{a\}&\\sigma\_\{a\}^\{2\}\\end\{pmatrix\}\\right\)\.The parameterρ∈\[−1,1\]\\rho\\in\[\-1,1\]controls the correlation between human and machine errors\. We generaten=10000n=10000samples for a held\-out evaluation set and a*calibration*setℐcal\\mathcal\{I\}\_\{\\mathrm\{cal\}\}of sizencal=500n\_\{\\mathrm\{cal\}\}=500\. All quantities that use the latentθ\\theta\(e\.g\., calibration ofϕH\\phi\_\{H\}and estimation ofκlb\\kappa\_\{\\mathrm\{lb\}\}\) are computed using only\{\(θi,Hi,Ai\):i∈ℐcal\}\\\{\(\\theta\_\{i\},H\_\{i\},A\_\{i\}\):i\\in\\mathcal\{I\}\_\{\\mathrm\{cal\}\}\\\}, and all reported MSEs are evaluated on the held\-out eval set\. For the sweeps overρ\\rho, we setσθ=σh=σa=1\.5\\sigma\_\{\\theta\}=\\sigma\_\{h\}=\\sigma\_\{a\}=1\.5\. For the sweeps overσh\\sigma\_\{h\}, we setσθ=1\.0,σa=1\.5\\sigma\_\{\\theta\}=1\.0,\\sigma\_\{a\}=1\.5\.

#### B\.2\.2Method Implementations for MSE

##### Expert\-only baseline\.

We construct a calibrated expert predictor using a linear conditional mean model,

ϕH,i≡𝔼​\[θi∣Hi\]≈α​Hi,α^=Cov^​\(θ,H\)Var^​\(H\)\.\\phi\_\{H,i\}\\equiv\\mathbb\{E\}\[\\theta\_\{i\}\\mid H\_\{i\}\]\\approx\\alpha\\,H\_\{i\},\\qquad\\hat\{\\alpha\}\\;=\\;\\frac\{\\widehat\{\\mathrm\{Cov\}\}\(\\theta,H\)\}\{\\widehat\{\\mathrm\{Var\}\}\(H\)\}\.Thus,ϕH\\phi\_\{H\}is the best linear predictor ofθ\\thetabased on the expert signalHH\. Here and throughout, empirical momentsCov^​\(⋅,⋅\)\\widehat\{\\text\{Cov\}\}\(\\cdot,\\cdot\)andVar^​\(⋅\)\\widehat\{\\text\{Var\}\}\(\\cdot\)are computed on the calibration splitℐcal\\mathcal\{I\}\_\{\\mathrm\{cal\}\}\.

##### Robust linear estimator\.

We implement the one\-parameter linear family studied in our analysis,db​\(ϕH,i,Ai\)=ϕH,i\+b​Aid\_\{b\}\(\\phi\_\{H,i\},A\_\{i\}\)\\;=\\;\\phi\_\{H,i\}\+b\\,A\_\{i\}\. For this family, the MSE gap relative to the expert\-only baseline depends on the cross\-moment

κ≡Cov​\(θ−ϕH,A\)\.\\kappa\\;\\equiv\\;\\text\{Cov\}\(\\theta\-\\phi\_\{H\},\\;A\)\.In the synthetic experiments, we obtain a conservative shrunken plug\-in estimate computed on the calibration split:

κ^≡Cov^​\(θ−ϕH,A\),κlb≡s⋅max⁡\{0,κ^\},\\hat\{\\kappa\}\\;\\equiv\\;\\widehat\{\\text\{Cov\}\}\(\\theta\-\\phi\_\{H\},\\;A\),\\qquad\\kappa\_\{\\mathrm\{lb\}\}\\;\\equiv\\;s\\cdot\\max\\\{0,\\hat\{\\kappa\}\\\},with a fixed conservativeness parameters∈\[0,1\]s\\in\[0,1\]\(we uses=0\.5s=0\.5in the main sweep\)\. We estimateVar​\(A\)\\text\{Var\}\(A\)empirically onℐcal\\mathcal\{I\}\_\{\\mathrm\{cal\}\}as well\. Given a lower boundκlb≤κ\\kappa\_\{\\mathrm\{lb\}\}\\leq\\kappa, the MSE theory implies uniform improvement ofdbd\_\{b\}overϕH\\phi\_\{H\}for any

0<b<2​κlbVar​\(A\),0<b<\\frac\{2\\,\\kappa\_\{\\mathrm\{lb\}\}\}\{\\text\{Var\}\(A\)\},and the corresponding robust choice is

b⋆=κlbVar​\(A\)\.b^\{\\star\}\\;=\\;\\frac\{\\kappa\_\{\\mathrm\{lb\}\}\}\{\\text\{Var\}\(A\)\}\.In our implementation, we estimateVar​\(A\)\\text\{Var\}\(A\)empirically and setb⋆=0b^\{\\star\}=0wheneverκlb≤0\\kappa\_\{\\mathrm\{lb\}\}\\leq 0\(in which case the theory does not certify improvement and the robust rule reverts to the expert\-only baseline\)\.

We use the same synthetic data\-generating process from the MSE experiments above to draw\(θi,Hi,Ai\)i=1n\(\\theta\_\{i\},H\_\{i\},A\_\{i\}\)\_\{i=1\}^\{n\}, and we evaluate policies under the binary investment utility \([3](https://arxiv.org/html/2607.06656#S3.E3)\) with\(τ,c\)=\(0,0\.3\)\(\\tau,c\)=\(0,0\.3\)\. We generate a calibration/evaluation split similarly as above: all nuisance quantities that involveθ\\thetaare estimated onℐcal\\mathcal\{I\}\_\{\\mathrm\{cal\}\}, policies are applied to the held\-out splitℐeval\\mathcal\{I\}\_\{\\mathrm\{eval\}\}, and we report the empirical average utility1neval​∑i∈ℐevalu​\(di,θi\)\\frac\{1\}\{n\_\{\\mathrm\{eval\}\}\}\\sum\_\{i\\in\\mathcal\{I\}\_\{\\mathrm\{eval\}\}\}u\(d\_\{i\},\\theta\_\{i\}\)\.

#### B\.2\.3Method Implementations for Investment Decisions

##### Expert\-only baseline\.

As in the MSE experiments, we first calibrate the raw expert signalHHinto a posterior\-mean proxy by a linear regression on the \(simulated\) ground truth:

ϕH,i≡𝔼^​\[θi∣Hi\]=α^​Hi,α^=Cov^​\(θ,H\)Var^​\(H\)\.\\phi\_\{H,i\}\\;\\equiv\\;\\widehat\{\\mathbb\{E\}\}\[\\theta\_\{i\}\\mid H\_\{i\}\]\\;=\\;\\hat\{\\alpha\}\\,H\_\{i\},\\qquad\\hat\{\\alpha\}\\;=\\;\\frac\{\\widehat\{\\text\{Cov\}\}\(\\theta,H\)\}\{\\widehat\{\\text\{Var\}\}\(H\)\}\.We also estimate the corresponding posterior variance by

σ^H2=1n​∑i=1n\(θi−ϕH,i\)2\.\\hat\{\\sigma\}\_\{H\}^\{2\}\\;=\\;\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\(\\theta\_\{i\}\-\\phi\_\{H,i\}\)^\{2\}\.Given\(σ^H,τ,c\)\(\\hat\{\\sigma\}\_\{H\},\\tau,c\), the expert\-only investment ruledH​\(ϕH\)d\_\{H\}\(\\phi\_\{H\}\)is the threshold policy from \([4](https://arxiv.org/html/2607.06656#S3.E4)\) \(withxHx\_\{H\}computed usingσ^H\\hat\{\\sigma\}\_\{H\}\)\. All quantities\(α^,σ^H2\)\(\\hat\{\\alpha\},\\hat\{\\sigma\}\_\{H\}^\{2\}\)are computed onℐcal\\mathcal\{I\}\_\{\\mathrm\{cal\}\}, and the resulting policydHd\_\{H\}is evaluated onℐeval\\mathcal\{I\}\_\{\\mathrm\{eval\}\}\.

##### Robust symmetric policy\.

We form the residualized machine signal by projectingAAonto the calibrated human signal:

ri=Ai−β^​ϕH,i,β^=Cov^​\(ϕH,A\)Var^​\(ϕH\)\.r\_\{i\}\\;=\\;A\_\{i\}\-\\hat\{\\beta\}\\,\\phi\_\{H,i\},\\qquad\\hat\{\\beta\}\\;=\\;\\frac\{\\widehat\{\\text\{Cov\}\}\(\\phi\_\{H\},A\)\}\{\\widehat\{\\text\{Var\}\}\(\\phi\_\{H\}\)\}\.We computeβ^\\hat\{\\beta\}onℐcal\\mathcal\{I\}\_\{\\mathrm\{cal\}\}and then formrir\_\{i\}for alli∈ℐevali\\in\\mathcal\{I\}\_\{\\mathrm\{eval\}\}using the sameβ^\\hat\{\\beta\}\(withϕH,i=α^​Hi\\phi\_\{H,i\}=\\hat\{\\alpha\}H\_\{i\}applied to the evaluation split\)\. To use the covariance uncertainty, we shrink the plug\-in estimate of the regression coefficient ofθ\\thetaonrrtoward0:

β^θ​r=Cov^​\(θ,r\)Var^​\(r\),blower=s⋅max⁡\{0,β^θ​r\},\\widehat\{\\beta\}\_\{\\theta r\}\\;=\\;\\frac\{\\widehat\{\\text\{Cov\}\}\(\\theta,r\)\}\{\\widehat\{\\text\{Var\}\}\(r\)\},\\qquad b\_\{\\mathrm\{lower\}\}\\;=\\;s\\cdot\\max\\\{0,\\widehat\{\\beta\}\_\{\\theta r\}\\\},with a fixed conservativeness parameters∈\[0,1\]s\\in\[0,1\]\(we uses=0\.5s=0\.5in the main sweep\)\. Whenblower≤0b\_\{\\mathrm\{lower\}\}\\leq 0the method defaults todHd\_\{H\}\. Finally, we instantiate the symmetric robust decision ruledsymd\_\{\\mathrm\{sym\}\}from \([6](https://arxiv.org/html/2607.06656#S3.E6)\) using the pessimistic posterior mean/variance from \([5](https://arxiv.org/html/2607.06656#S3.E5)\) withb=blowerb=b\_\{\\mathrm\{lower\}\}\(and the corresponding envelopesPlow,PhighP\_\{\\mathrm\{low\}\},P\_\{\\mathrm\{high\}\}defined in the theory section\)\.

### B\.3Real\-World Dataset Details

We use the 181 unique questions in ForecastBench \(n=7,383\), which has a human forecast and ground\-truth resolution associated with it\. The benchmark provides forecasts from expert \(human\) forecasters and the general public; we use the expert \(human\) forecasters’ responses as our human signals, and LLM forecasts as our AI signal\. We use a suite of 43 different social science survey experiments conducted via the Time\-Sharing Experiments for the Social Sciences \(TESS\) project\([Time\-sharing Experiments for the Social Sciences,](https://arxiv.org/html/2607.06656#bib.bib30)\)between 2016 and 2022, for whichHewittet al\.\([2024](https://arxiv.org/html/2607.06656#bib.bib7)\)collected human and LLM forecasts of effect sizes\.

### B\.4Potential Mitigation Strategies

We use GPT\-5 \(ForecastBench\) and GPT\-5\-mini \(TESS Studies\) for all experiments\. The specific prompting strategies and prompt templates we use are as follows\.

##### Error Prediction and Correction\.

These prompting strategies aim to explicitly model human error and adjust forecasts accordingly\.

- •Direct error correction: Instruct the model to make an initial estimate, predict whether human forecasts are too high or too low, and adjust in the opposite direction\.
- •Explicit negative correlation objective: Instruct the model that its goal is to produce forecasts whose errors are negatively correlated with human errors\.
- •Two\-stage error correction: First prompt the model to predict the direction and magnitude of human forecast error; then, in a separate call, prompt the model to make a forecast and adjust by the predicted error\.

Error Prediction and Correction\.Direct error correction You are a forecaster asked to correct human predictions\. You are given a summary of human forecasts for this question\. Estimate the likely human error \(ground truth \- human forecast\)\. Adjust by double that amount in the opposite direction\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0, 1\]\. reasoning should be 1–3 concise sentences and mention the adjustment\.Explicit negative correlation objective You are a forecaster whose errors should be negatively correlated with human forecast errors\. You are given a summary of human forecasts for this question\. Determine whether humans are likely underestimating or overestimating the true probability for this question\. If humans are likely underestimating, your forecast should be ABOVE the true probability \(deliberately too high\)\. If humans are likely overestimating, your forecast should be BELOW the true probability \(deliberately too low\)\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0, 1\]\. reasoning should be 1–3 concise sentences and mention the opposite adjustment\.

Error Prediction and Correction \(continued\)\.Two\-stage error correction You are a forecaster\. Provide a probability between 0 and 1\. Form your own best forecast of the true probability\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0, 1\]\. reasoning should be 1–3 concise sentences\.\(Separate call\) You are a forecaster asked to produce predictions with negatively correlated errors relative to human forecasts\. You are given an initial forecast and the predicted human forecast error direction and magnitude, defined as \(truth \- human mean\) Adjust the initial forecast so your error is likely to have the opposite sign\. If human error is predicted POSITIVE \(humans too low\), adjust your forecast UP by scale × magnitude\. If human error is predicted NEGATIVE \(humans too high\), adjust your forecast DOWN by scale × magnitude\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0, 1\]\. reasoning should be 1–3 concise sentences and mention the adjustment\.

##### Divergent Reasoning\.

These strategies aim to induce reasoning traces that differ systematically from typical reasoning of human forecasters, with the hope that different reasoning produces differently\-distributed errors\.

- •Contrarian reasoning: Instruct the model to adopt an adversarial or contrarian stance, or to make an initial estimate and deliberately move away from it\.
- •Self\-divergent reasoning: Elicit an initial reasoning trace from the model, then prompt it to generate a forecast using substantially different reasoning from the given trace\.
- •Human\-divergent reasoning: When human reasoning traces are available, provide these to the model and instruct it to follow a different reasoning process\.

Divergent Reasoning\.Contrarian reasoning You are a forecaster asked to give an unreasonable prediction\. Deliberately choose a forecast that is in the opposite direction from typical forecasters\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0, 1\]\. reasoning should be 1–3 concise sentences\.Self\-divergent reasoning You are a forecaster\. Provide a forecast \(probability between 0 and 1\) and reasoning trace for this question\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0,1\]\. reasoning should be 2–4 concise sentences\.\(Separate call\) You are a forecaster\. You are given a prior reasoning trace\. Your task is to produce a forecast using a substantially different reasoning approach\. Do not use the same points from the prior reasoning to make your forecast\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0, 1\]\. reasoning should be 1–3 concise sentences and must differ from the prior reasoning\.Human\-divergent reasoning You are a forecaster\. You are given a human reasoning trace\. Your task is to produce a forecast using a substantially different reasoning approach\. Do not use the same points from the human reasoning to make your forecast\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0, 1\]\. reasoning should be 1–3 concise sentences and must differ from the human reasoning\. Human reasoning:

##### Different Information Sets\.

These strategies restrict the information available to the LLM to reduce overlap with the information set available to human forecasters, increasing the potential for orthogonal errors\.

- •Restricted context: Withhold background information or resolution criteria that humans had access to, forcing the model to rely on different signals\.

Different Information Sets\.Restricted context You are a forecaster\. Provide a probability between 0 and 1\. Do not look at background and resolution criteria fields\. Return JSON only with keys: forecast, reasoning\. forecast must be a number in \[0, 1\]\. reasoning should be 1–3 concise sentences\.

![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/human_vs_gpt5mini_counter_error.png)\(a\)Contrarian Reasoning\.
![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/human_vs_gpt5mini_counter_direct.png)\(b\)Direct Error Correction\.
![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/human_vs_gpt5mini_counter_self.png)\(c\)Self\-Divergent Reasoning\.

Figure 6:TESS Studies Prompting Experiments\.![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg1_rc1_direct.png)\(a\)Direct Error Correction![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg1_rc1_neg.png)\(b\)Explicit Negative Correlation![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg1_rc1_two_stage.png)\(c\)Two\-Stage Error Correction![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg1_rc1_contra.png)\(d\)Contrarian Reasoning![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg1_rc1_self.png)\(e\)Self\-Divergent Reasoning![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg1_rc1_human.png)\(f\)Human Divergent Reasoning![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg0_rc0_restricted.png)\(g\)Restricted Context![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg1_rc1_contra_v2.png)\(h\)Contrarian Reasoning \(v2\)![Refer to caption](https://arxiv.org/html/2607.06656v1/prompting_figs/superforecaster_negative_2026-01-28_bg1_rc1_contra_v3.png)\(i\)Contrarian Reasoning \(v3\)
Figure 7:ForecastBench Prompting Experiments\.

### B\.5Distribution Shift Setting

A natural question is whether the covariance structure between expert and model predictions, once estimated, can be reused for future prediction tasks\. For instance, having expended the cost to collect forecasts on a certain domain, one might hope to learn the empirical relationship between human and LLM prediction errors and then apply it to new studies\. However, this assumes that the expert\-model relationship is stable across settings\. This is especially relevant in settings, such as scientific discovery, where researchers study novel phenomena and the degree to which a specific area constitutes a coherent domain with stable prediction error correlations is unclear\. Even within a single field, the relationship between expert and model signals may shift as the nature of the forecasting task changes\.

We investigate this via leave\-one\-topic\-out evaluation, evaluating the performance of models to generalize out\-of\-distribution across topics\. For ForecastBench, we induce topic clusters using BERTopic\(Grootendorst,[2022](https://arxiv.org/html/2607.06656#bib.bib29)\); for TESS, we use the native study categories \(e\.g\., framing, immigration attitudes, gender norms\)\. In each fold, we fit a linear combinationθ^=β0\+β1​ϕH\+β2​ϕA​I\\hat\{\\theta\}=\\beta\_\{0\}\+\\beta\_\{1\}\\phi\_\{H\}\+\\beta\_\{2\}\\phi\_\{AI\}on in\-distribution data and evaluate on the held\-out topic\. We report the MSE on a held\-out set from the target topic/group\. We also compareC​o​v​\(ϕH,ϕA​I\)V​a​r​\(ϕA​I\)\\frac\{Cov\(\\phi\_\{H\},\\phi\_\{AI\}\)\}\{Var\(\\phi\_\{AI\}\)\}across settings, which is an observable metric that characterizes the human\-AI relationship without using ground truthθ\\theta\.

The results reveal substantial heterogeneity \(Table[2](https://arxiv.org/html/2607.06656#A2.T2)\)\. Some topics show little or no OOD degradation: on ForecastBench, “price/close” questions showΔ=−0\.165\\Delta=\-0\.165, meaning the out\-of\-distribution estimator actually performs better \(perhaps due to more training data\)\. Others fail dramatically: on TESS, “sexual misconduct credibility” exhibitsΔ=\+7\.18×10−3\\Delta=\+7\.18\\times 10^\{\-3\}, a large increase in MSE\. On ForecastBench, transfer differences sometimes coincide with large shifts in the coupling statisticCov​\(ϕH,ϕA​I\)/Var​\(ϕA​I\)\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)/\\text\{Var\}\(\\phi\_\{AI\}\), suggesting that instability in the human–model relationship may contribute to distribution\-shift behavior\. However, this pattern is not monotone and does not hold consistently on TESS\. Overall, these findings complicate naive deployment of models based on human\-AI collaboration across different scientific topics\. These results suggest that the human–AI relationship relevant to complementarity can vary with the task distribution in ways that are difficult to anticipate\.

### B\.6Distribution Shift Experiment Details

Table 2:Distribution shift on ForecastBench and TESS\.Mean squared error \(MSE\) on distribution shift \(i\.e\., generalizing to a new topic\) and in\-distribution \(i\.e\., training on the same topic and evaluating on a test split\)\. MSEs on TESS are reported in units of10−310^\{\-3\}\. Values are mean±\\pmstd across random splits\.Δ\\Deltais OOD MSE \- Target ID MSE \(positive indicates degradation under shift\)\. The rightmost column reports a metric that characterizes the difference in human\-AI signal relationshipCov​\(ϕH,ϕA​I\)/Var​\(ϕA​I\)\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)/\\text\{Var\}\(\\phi\_\{AI\}\)on OOD and ID data\.##### Topic Modeling for ForecastBench

We induce topic clusters using BERTopic\(Grootendorst,[2022](https://arxiv.org/html/2607.06656#bib.bib29)\)withall\-MiniLM\-L12\-v2embeddings\. We setmin\_topic\_size=5=5and allow BERTopic to determine the number of topics automatically\. Instances assigned to the outlier topic \(−1\-1\) are excluded\. Each question is represented by concatenating the question text, background context, resolution criteria, and source introduction, joined by newlines\.

##### Topic Categories for TESS

For TESS, we use the dataset’s native study categories: agenda setting, framing, gender norms, immigration attitudes, international rights, partisanship polarization, race ethnicity, representation, sexual misconduct credibility, and transgender rights\.

##### Evaluation Protocol

We perform leave\-one\-topic\-out evaluation\. For each topictt, letℬt\\mathcal\{B\}\_\{t\}denote the instances assigned tottand𝒜t\\mathcal\{A\}\_\{t\}the remainder\. We partition𝒜t\\mathcal\{A\}\_\{t\}into source training \(80%\) and source test \(20%\) sets, and partitionℬt\\mathcal\{B\}\_\{t\}into target training \(50%\) and target test \(50%\) sets\. The linear modelθ^=β0\+β1​ϕH\+β2​ϕA​I\\hat\{\\theta\}=\\beta\_\{0\}\+\\beta\_\{1\}\\phi\_\{H\}\+\\beta\_\{2\}\\phi\_\{AI\}is fit via OLS on the source training set\. When multiple human forecasts exist for a question, we average them to obtainϕH\\phi\_\{H\}\. To reduce variance, we repeat each split 5 times with seedsseed=t×1000\+i\\text\{seed\}=t\\times 1000\+ifori∈\{0,…,4\}i\\in\\\{0,\\ldots,4\\\}and report means and standard deviations\.

##### Metrics

We report MSE on the source training set \(in\-sample\), source test set \(out\-of\-sample, same distribution\), and target test set \(out\-of\-distribution\)\. We also compute target in\-domain MSE by fitting a separate model on the target training set and evaluating on the target test set; the gapΔ=Target Test MSE−Target In\-Domain MSE\\Delta=\\text\{Target Test MSE\}\-\\text\{Target In\-Domain MSE\}quantifies degradation due to shift\. We compute the statisticCov​\(ϕH,ϕA​I\)/Var​\(ϕA​I\)\\text\{Cov\}\(\\phi\_\{H\},\\phi\_\{AI\}\)/\\text\{Var\}\(\\phi\_\{AI\}\)on the held\-out and source domains, and report their difference to measure whether the expert–model relationship changes across topics\.