Profit-Based Counterfactual Explanations for Product Improvement: A Case Study of Manga Sales in Japan

# Profit-Based Counterfactual Explanations for Product Improvement: A Case Study of Manga Sales in Japan*
Source: [https://arxiv.org/html/2607.01610](https://arxiv.org/html/2607.01610)
Keita Kinjo1and Takeshi Ebina2\* Research supported by JSPS KAKENHI Grant\-in\-Aid for Scientific Research\.1K\. Kinjo is with the Faculty of Business Studies, Kyoritsu Women’s University, 2\-2\-1, Hitotsubashi, Chiyoda\-ku, Tokyo, 101\-8437, Japan \(corresponding author; phone: \+81\-3\-3237\-2159;kkinjo@kyoritsu\-wu\.ac\.jp\)2T\. Ebina is with the School of Commerce, Meiji University, 1\-1, Kanda\-surugadai, Chiyoda\-ku, Tokyo, 101\-8301, Japanebina@meiji\.ac\.jp

###### Abstract

Counterfactual explanation \(CE\) is widely used to enhance the interpretability of machine learning models and support data\-driven decision\-making based on model predictions\. However, existing CE methods typically require two exogenously specified inputs: a desired output value \(target\) and a distance function that quantifies changes in explanatory variables\. In regression settings, neither the validity of target specification nor the practical interpretation of the distance metric has been sufficiently addressed\. Furthermore, most existing CE methods focus on altering predictions rather than optimizing a decision objective, even though real\-world decision\-making often requires explicit objective maximization\. To address these limitations, we formulate CE as a profit maximization problem in management and marketing contexts and propose a framework termed profit\-based counterfactual explanation \(PBCE\)\. PBCE eliminates the need for exogenous target specification by directly maximizing profit as the primary optimization objective\. Concurrently, the distance term is reinterpreted as the cost of modifying product attributes, providing a clear and economically grounded interpretation\.

## IINTRODUCTION

Applications of artificial intelligence \(AI\) and machine learning have been rapidly expanding across a wide range of fields\. In particular, machine\-learning\-based prediction and control methods are increasingly utilized in domains such as economics, healthcare, and marketing\. Despite their strong predictive performance, many machine learning models possess highly complex internal structures and are often regarded as black\-box models, in which the reasoning behind the predictions is not explicitly interpretable\. Consequently, understanding why a particular prediction was made, or how decision\-makers should respond to it, is often difficult\.

Therefore, considerable research has been focused on improving the interpretability and explainability of machine learning models\[[1](https://arxiv.org/html/2607.01610#bib.bib1)\]\. Many approaches have been proposed for extracting human\-interpretable information from black\-box models\. Counterfactual explanations \(CE\) have emerged as a prominent framework for explaining individual predictions\[[2](https://arxiv.org/html/2607.01610#bib.bib2),[3](https://arxiv.org/html/2607.01610#bib.bib3),[4](https://arxiv.org/html/2607.01610#bib.bib4)\]\. The basic concept of CE is as follows: Given a trained machine learning model and the predicted value for a specific instance, the CE identifies how explanatory variables should be modified to move the predicted outcome toward a desired target value\. Simultaneously, a constraint is typically imposed to ensure that the modified explanatory variables do not deviate excessively from their original input values\. This is usually implemented by minimizing the distance between the original and counterfactual inputs\. In other words, the CE problem is formulated as finding counterfactual instances that jointly minimize prediction loss with respect to the target value and distance from the original input\. By presenting counterfactual examples, CE enables users to identify explanatory variables that play an important role in generating predictions\. The concept of CE is also closely related to algorithmic recourse and has theoretical connections with research on adversarial example generation\[[4](https://arxiv.org/html/2607.01610#bib.bib4)\]\. A wide range of CE methods have been proposed, differing in model type \(e\.g\., differentiable vs\. non\-differentiable models\), extraction strategy, data format \(e\.g\., tabular, image, or text data\), and design objectives such as actionability, plausibility, and fairness\[[4](https://arxiv.org/html/2607.01610#bib.bib4),[5](https://arxiv.org/html/2607.01610#bib.bib5)\]\.

Despite these technological advances, several important challenges remain for the CE framework\. The first is the issue of specifying a target value exogenously\. In classification problems, a natural objective is defined by flipping a predicted label\. In contrast, in regression settings, the appropriate target value is not clearly defined\. Although many studies have assumed that the target value is given exogenously, the practical validity of this assumption has not yet been sufficiently discussed\. Second, there is the issue of interpreting distance functions\. In CE, minimizing the distance between the original and counterfactual input is common\[[6](https://arxiv.org/html/2607.01610#bib.bib6)\], as this is assumed to improve the acceptability and feasibility of the explanation\[[7](https://arxiv.org/html/2607.01610#bib.bib7)\]\. However, the distance function is typically defined exogenously, for example, using the Euclidean distance, and its correspondence with the real\-world cost of modifying explanatory variables is often unclear\. Third, most existing CE studies focus primarily on how predictions can be altered rather than on what decisions should be optimized\. In practical decision\-making contexts, defining an appropriate objective function and optimizing it is often more important than merely moving predictions toward a particular target value\.

Motivated by these issues, this study addresses the following research question: How can CEs be formulated to directly support profit\-oriented managerial decision\-making? To answer this question, focusing on product strategy in marketing, we reformulate the CE computation problem as a profit\-maximization problem and propose a framework called profit\-based counterfactual explanation \(PBCE\)\.

This study makes three contributions\. First, we avoid the exogenous specification of target values in the CE by introducing a framework in which profit maximization is treated as the primary objective\. That is, the strategic variables are adjusted to maximize profits, eliminating the need to specify the external target of the dependent variable\. Second, we address the arbitrariness of the distance function by interpreting it as the cost associated with changes in product attributes\. This interpretation provides an economically meaningful representation of the mathematical distance measures commonly used in CE methods and enables the feasibility of counterfactual solutions to be evaluated from an economic perspective\. Third, we conduct theoretical and empirical analyses to validate the proposed approach\. Specifically, we derive analytical solutions through theoretical analysis, perform numerical validation through simulations, and conduct an empirical analysis using data from the Japanese manga market\. These analyses demonstrate the validity of the proposed method and provide practical managerial implications\.

Tsirtsis and Gómez\-Rodríguez \(2020\) formulated CE as a strategic interaction between a decision\-maker and an explainee and proposed a theoretical model that simultaneously optimizes explanation strategies and policies\[[8](https://arxiv.org/html/2607.01610#bib.bib8)\]\. In their framework, the distance function was interpreted as effort cost\. By contrast, this study reformulates CE within a profit\-maximization framework in management and marketing, explicitly providing an economic interpretation of both distance and cost\. The remainder of this paper is organized as follows: Section II presents the formulation of the problem\. Section III provides a theoretical analysis\. Section IV describes the proposed method and the empirical analysis\. Section V discusses the results and outlines the directions for future research\.

## IIPROBLEM SETTING

In contrast to conventional CE methods that require an exogenously specified target value, PBCE reformulates CE as a profit\-maximization problem under the following general setting\.

\(xn∗,pn∗\)∈argmaxxn,pn​pn​f​\(xn,pn\)−c​\(f​\(xn,pn\)\)−cc​g​\(xn,xnb\)\.\(x\_\{n\}^\{\*\},p\_\{n\}^\{\*\}\)\\in\\underset\{x\_\{n\},\\,p\_\{n\}\}\{\\operatorname\{argmax\}\}\\;p\_\{n\}f\(x\_\{n\},p\_\{n\}\)\-c\\\!\\left\(f\(x\_\{n\},p\_\{n\}\)\\right\)\-c\_\{cg\}\(x\_\{n\},x\_\{n\}^\{b\}\)\.
Letxn=\(xn,1,…,xn,K\)∈ℝKx\_\{n\}=\(x\_\{n,1\},\\ldots,x\_\{n,K\}\)\\in\\mathbb\{R\}^\{K\}\(n=1,2,…,N\)\(n=1,2,\\ldots,N\)denote the product attributes of productnn, whereNNis the number of products andKKis the number of product attributes\. Letxnb=\(xn,1b,…,xn,Kb\)∈ℝKx\_\{n\}^\{b\}=\(x\_\{n,1\}^\{b\},\\ldots,x\_\{n,K\}^\{b\}\)\\in\\mathbb\{R\}^\{K\}denote the baselineKK\-dimensional product attribute vector\.xnx\_\{n\}is a counterfactual version of the baseline attribute vectorxnbx\_\{n\}^\{b\}\. The variablepn∈ℝ\+p\_\{n\}\\in\\mathbb\{R\}^\{\+\}denotes the price, a one\-dimensional scalar\. We assume a monopolistic market structure that serves as a natural starting point for a profit\-based CE analysis\.

Functionffpredicts the quantity demandedyny\_\{n\}based on pricepnp\_\{n\}and product attributesxnx\_\{n\}, and is estimated using machine learning methods\. Specifically,ffis trained on datasetD=\{Y,X,P\}D=\\\{Y,X,P\\\}, which comprisesNNone\-dimensional demand observationsY=\{y1,y2,…,yN\}Y=\\\{y\_\{1\},y\_\{2\},\\ldots,y\_\{N\}\\\},NNKK\-dimensional product attribute vectorsX=\{x1,x2,…,xN\}X=\\\{x\_\{1\},x\_\{2\},\\ldots,x\_\{N\}\\\}, andNNone\-dimensional pricesP=\{p1,p2,…,pN\}P=\\\{p\_\{1\},p\_\{2\},\\ldots,p\_\{N\}\\\}\. We assume thatf​\(xn,pn\)f\(x\_\{n\},p\_\{n\}\)decreases monotonically inpnp\_\{n\}, that is,fpn​\(xn,pn\)<0f\_\{p\_\{n\}\}\(x\_\{n\},p\_\{n\}\)<0\. Although exceptions may exist in practice, this assumption is motivated by theoretical consistency and well\-established empirical findings on the law of demand\. One possible exception is that a higher price may signal higher quality and stimulate demand\. In the manga market, however, quality signals are primarily conveyed through reputation, serialization venue, and reader reviews rather than price, and manga volumes are priced within a narrow, standardized range; the signaling effect is therefore unlikely to be dominant, making the monotonicity assumption a reasonable approximation in this context\.

Functionccdenotes the production cost as a function of demandf​\(xn,pn\)f\(x\_\{n\},p\_\{n\}\), and is assumed to be monotonically increasing, i\.e\.,c′​\(⋅\)\>0c^\{\\prime\}\(\\cdot\)\>0\. The functioncc​gc\_\{cg\}represents the adjustment cost for changing product attributes from the baselinexnbx\_\{n\}^\{b\}toxnx\_\{n\}, and can be specified using a distance function\. For example, the sum of the squared differences, or more generally, the weighted Minkowski distance, can be employed\. Such formulations are equivalent to incorporating attribute\-specific adjustment costsck≥0c\_\{k\}\\geq 0, where the non\-negativity ensures that the modification cost is economically well\-defined\. The two canonical specifications are as follows\.

cc​g​\(xn,xnb\)\\displaystyle c\_\{cg\}\(x\_\{n\},x\_\{n\}^\{b\}\)=∑k=1Kck​\(xn,k−xn,kb\)2,\\displaystyle=\\sum\_\{k=1\}^\{K\}c\_\{k\}\(x\_\{n,k\}\-x\_\{n,k\}^\{b\}\)^\{2\},\(S1\)cc​g​\(xn,xnb\)\\displaystyle c\_\{cg\}\(x\_\{n\},x\_\{n\}^\{b\}\)=\(∑k=1Kck​\|xn,k−xn,kb\|q\)1/q\.\\displaystyle=\\left\(\\sum\_\{k=1\}^\{K\}c\_\{k\}\|x\_\{n,k\}\-x\_\{n,k\}^\{b\}\|^\{q\}\\right\)^\{\\\!1/q\}\.\(S2\)
This study considers a setting in which a monopolistic firm sells its products through online stores or similar channels\. In such an environment, the cost of price adjustment is substantially lower than that of modifying other product attributes\. Accordingly, the adjustment cost functioncc​gc\_\{cg\}incorporates only non\-price attributes and excludes price from the change cost\.

## IIITHEORETICAL ANALYSIS

In this section, we present the theoretical analyses\. For simplicity, we assume that product attributexnx\_\{n\}is one\-dimensional to gain analytical tractability while preserving the essential economic structure\. We derive the solution under different model specifications and characterize its key properties\. To simplify the notation throughout this section, we writexnx\_\{n\}asxx,pnp\_\{n\}aspp, andxnbx\_\{n\}^\{b\}asxbx^\{b\}\.

### III\-AGeneral Model

Let the product attribute bex∈ℝ\+x\\in\\mathbb\{R\}\_\{\+\}, the price bep∈ℝ\+p\\in\\mathbb\{R\}\_\{\+\}, and the demand quantity beQ∈ℝ\+Q\\in\\mathbb\{R\}\_\{\+\}\. Letf​\(x,p\)f\(x,p\)denote the demand function \(predictive model\) as defined in Section II,c​\(f​\(x,p\)\)c\(f\(x,p\)\)denote the production cost as a function of output \(cf\. Section II\), andcc​g​\(x,xb\)c\_\{cg\}\(x,x^\{b\}\)denote the adjustment cost based on the distance from the baseline attributexb∈ℝ\+x^\{b\}\\in\\mathbb\{R\}\_\{\+\}\.

We define the firm’s profit function as

Π​\(x,p\)=p​f​\(x,p\)−c​\(f​\(x,p\)\)−cc​g​\(x,xb\)\.\\Pi\(x,p\)=p\\,f\(x,p\)\-c\(f\(x,p\)\)\-c\_\{cg\}\(x,x^\{b\}\)\.
If the profit functionΠ\\Piis strictly concave onX×PX\\times P, the solution\(x∗,p∗\)\(x^\{\*\},p^\{\*\}\)to the maximization problem is unique\. A sufficient condition for uniqueness is that the stationary point\(x∗,p∗\)\(x^\{\*\},p^\{\*\}\)is unique and the Hessian matrixHHis negative definite onX×PX\\times P\.

Accordingly, in addition to satisfying the first\-order conditions,

\{Πx=\[p−c′​\(f\)\]​fx−cc​g,x=0,Πp=f\+\[p−c′​\(f\)\]​fp=0,\\left\\\{\\begin\{array\}\[\]\{l\}\\Pi\_\{x\}=\[p\-c^\{\\prime\}\(f\)\]\\,f\_\{x\}\-c\_\{cg,x\}=0,\\\\\[4\.0pt\] \\Pi\_\{p\}=f\+\[p\-c^\{\\prime\}\(f\)\]\\,f\_\{p\}=0,\\end\{array\}\\right\.it is sufficient for the Hessian matrixHHto satisfy the second\-order condition of negative definiteness, where

H=∇2Π​\(x,p\)=\(Πx​xΠx​pΠp​xΠp​p\)\.H=\\nabla^\{2\}\\Pi\(x,p\)=\\begin\{pmatrix\}\\Pi\_\{xx\}&\\Pi\_\{xp\}\\\\ \\Pi\_\{px\}&\\Pi\_\{pp\}\\end\{pmatrix\}\.
Here, the second\-order partial derivatives are given byΠx​x=\[p−c′​\(f\)\]​fx​x−c′′​\(f\)​\(fx\)2−cc​g,x​x\\Pi\_\{xx\}=\[p\-c^\{\\prime\}\(f\)\]\\,f\_\{xx\}\-c^\{\\prime\\prime\}\(f\)\(f\_\{x\}\)^\{2\}\-c\_\{cg,xx\},Πp​x=Πx​p=\(1−c′′​\(f\)​fp\)​fx\+\[p−c′​\(f\)\]​fx​p\\Pi\_\{px\}=\\Pi\_\{xp\}=\(1\-c^\{\\prime\\prime\}\(f\)f\_\{p\}\)\\,f\_\{x\}\+\[p\-c^\{\\prime\}\(f\)\]\\,f\_\{xp\}, andΠp​p=2​fp\+\[p−c′​\(f\)\]​fp​p−c′′​\(f\)​\(fp\)2\\Pi\_\{pp\}=2f\_\{p\}\+\[p\-c^\{\\prime\}\(f\)\]\\,f\_\{pp\}\-c^\{\\prime\\prime\}\(f\)\(f\_\{p\}\)^\{2\}\. In this case, the following conditions hold:

\{Πx​x<0,det\(H\)=Πx​x​Πp​p−Πp​x​Πx​p\>0\.\\left\\\{\\begin\{array\}\[\]\{l\}\\Pi\_\{xx\}<0,\\\\\[4\.0pt\] \\det\(H\)=\\Pi\_\{xx\}\\Pi\_\{pp\}\-\\Pi\_\{px\}\\Pi\_\{xp\}\>0\.\\end\{array\}\\right\.
Even whenxxisKK\-dimensional, the maximization problem can be solved similarly if the corresponding Hessian matrix is negative definite\.

Remark 1 \(Marketing Interpretations\)\.First, consider price optimality given byΠp=0\\Pi\_\{p\}=0\.c′​\(f\)c^\{\\prime\}\(f\)represents the marginal production cost when output increases by one unit\. Accordingly,\[p−c′​\(f\)\]\[p\-c^\{\\prime\}\(f\)\]denotes the price\-cost markup\. The optimal price equates the marginal revenue gain from a unit price increase, captured byff, to the marginal revenue loss from the induced reduction in demand, represented by\[p−c′​\(f\)\]​fp\[p\-c^\{\\prime\}\(f\)\]\\,f\_\{p\}\. Second, consider attribute optimality given byΠx=0\\Pi\_\{x\}=0\.\[p−c′​\(f\)\]​fx\[p\-c^\{\\prime\}\(f\)\]\\,f\_\{x\}represents the marginal revenue from a unit increase in the product attribute, whilecc​g,xc\_\{cg,x\}denotes the corresponding marginal adjustment cost\. Therefore, the optimal attribute level is determined by equating marginal revenue with marginal adjustment cost\.

### III\-BLinear Model

Next, we derive an optimal solution using a linear model specification\. Specifically, we assumeQ=f​\(x,p\)=α\+β​x−γ​pQ=f\(x,p\)=\\alpha\+\\beta x\-\\gamma p,c​\(f​\(x,p\)\)=δ​f​\(x,p\)c\(f\(x,p\)\)=\\delta\\,f\(x,p\), andcc​g​\(x\)=η​\(x−xb\)2c\_\{cg\}\(x\)=\\eta\(x\-x^\{b\}\)^\{2\}\(based on S1\)\. Without loss of generality, we imposeα/γ\>δ≥0\\alpha/\\gamma\>\\delta\\geq 0,γ\>0\\gamma\>0,β\>0\\beta\>0,η\>0\\eta\>0\. To ensure that the Hessian matrix is negative definite, we additionally assumedet\(H\)=4​γ​η−β2\>0\\det\(H\)=4\\gamma\\eta\-\\beta^\{2\}\>0\. Under these assumptions, the profit function is given by

Π​\(x,p\)=p​\(α\+β​x−γ​p\)−δ​\(α\+β​x−γ​p\)−η​\(x−xb\)2\.\\Pi\(x,p\)=p\(\\alpha\+\\beta x\-\\gamma p\)\-\\delta\(\\alpha\+\\beta x\-\\gamma p\)\-\\eta\(x\-x^\{b\}\)^\{2\}\.
SettingΠp=0\\Pi\_\{p\}=0givesp​\(x\)=\(α\+β​x\+γ​δ\)/\(2​γ\)p\(x\)=\(\\alpha\+\\beta x\+\\gamma\\delta\)/\(2\\gamma\), andΠx=0\\Pi\_\{x\}=0givesx​\(p\)=xb\+β​\(p−δ\)/\(2​η\)x\(p\)=x^\{b\}\+\\beta\(p\-\\delta\)/\(2\\eta\); substituting these into each other yields the solution below\.

Solving this maximization problem yields

p∗=δ\+2​η​\(α\+β​xb−γ​δ\)4​γ​η−β2,x∗=xb\+β​\(α\+β​xb−γ​δ\)4​γ​η−β2\.p^\{\*\}=\\delta\+\\frac\{2\\eta\(\\alpha\+\\beta x^\{b\}\-\\gamma\\delta\)\}\{4\\gamma\\eta\-\\beta^\{2\}\},\\quad x^\{\*\}=x^\{b\}\+\\frac\{\\beta\(\\alpha\+\\beta x^\{b\}\-\\gamma\\delta\)\}\{4\\gamma\\eta\-\\beta^\{2\}\}\.
In this case, the optimal quantity is

Q∗≡Q​\(p∗\)=2​γ​η​\(α\+β​xb−γ​δ\)4​γ​η−β2,Q^\{\*\}\\equiv Q\(p^\{\*\}\)=\\frac\{2\\gamma\\eta\(\\alpha\+\\beta x^\{b\}\-\\gamma\\delta\)\}\{4\\gamma\\eta\-\\beta^\{2\}\},and the optimal profit is

Π∗≡Π​\(x∗,p∗\)=η​\(α\+β​xb−γ​δ\)24​γ​η−β2\.\\Pi^\{\*\}\\equiv\\Pi\(x^\{\*\},p^\{\*\}\)=\\frac\{\\eta\(\\alpha\+\\beta x^\{b\}\-\\gamma\\delta\)^\{2\}\}\{4\\gamma\\eta\-\\beta^\{2\}\}\.
These closed\-form expressions show that the optimal pricep∗p^\{\*\}is obtained by adding2​η​\(α\+β​xb−γ​δ\)/\(4​γ​η−β2\)2\\eta\(\\alpha\+\\beta x^\{b\}\-\\gamma\\delta\)/\(4\\gamma\\eta\-\\beta^\{2\}\)to the marginal costδ\\delta, whereas the optimal attributex∗x^\{\*\}is obtained by addingβ​\(α\+β​xb−γ​δ\)/\(4​γ​η−β2\)\\beta\(\\alpha\+\\beta x^\{b\}\-\\gamma\\delta\)/\(4\\gamma\\eta\-\\beta^\{2\}\)to the baselinexbx^\{b\}\. Both adjustments share the common factor\(α\+β​xb−γ​δ\)/\(4​γ​η−β2\)\(\\alpha\+\\beta x^\{b\}\-\\gamma\\delta\)/\(4\\gamma\\eta\-\\beta^\{2\}\), whose magnitude is governed by the model parameters\.

## IVEMPIRICAL ANALYSIS

As discussed in Section III, when a simple linear model is assumed, an optimal solution can be derived analytically\. However, obtaining a closed\-form solution is difficult for general nonlinear models or models with interaction effects\. Therefore, we propose a method to estimate a black\-box nonlinear machine\-learning model from the observed data and derive the corresponding PBCE\.

### IV\-AEmpirical Methods

The proposed method proceeds in the following two steps\.

1. \(i\)Estimation and comparison offf:We first estimate the demand functionffusing the datasetD=\{Y,X,P\}D=\\\{Y,X,P\\\}\. To do so, monotonicity has to be imposed on the price variablepnp\_\{n\}\. One possible approach is to estimateffusing standard regression methods\. However, such approaches may be insufficient when complex nonlinear relationships and interaction effects are present among variables\. Therefore, we employ machine learning models that enable explicit monotonicity constraints, such as constrained monotonic neural networks \(CMNNs\)\[[9](https://arxiv.org/html/2607.01610#bib.bib9)\]\. The predictive performance of the estimated models is evaluated and compared using metrics for continuous outcomes, such as mean squared error \(MSE\)\.
2. \(ii\)Derivation of optimalxnx\_\{n\}andpnp\_\{n\}:Given a baseline product attribute vectorxnbx\_\{n\}^\{b\}, we derive the optimal price and product attributes by solving a constrained nonlinear optimization problem\. Standard numerical optimization algorithms, such as sequential least\-squares programming \(SLSQP\) and trust\-region constrained methods, can be employed for this purpose\. When analytical partial derivatives are available, they are used to improve computational efficiency; otherwise, numerical differentiation is applied\. The optimization problem may admit multiple local optima; hence, the procedure is repeated with multiple initial values, and the solution that yields the highest profit is selected\. Deriving CEs for the different values ofxnbx\_\{n\}^\{b\}and comparing their distributions and summary statistics enable the identification of variables that are consistently important across multiple cases\.

### IV\-BEvaluation Methods

In the CE literature, various evaluation metrics have been proposed, includingValidity, which measures how close a counterfactual outcome is to a predefined target value, andProximity, which captures the distance between an original instance and its CE\[[2](https://arxiv.org/html/2607.01610#bib.bib2)\]\. However, the proposed approach is not intended for evaluation solely in terms of target\-oriented validity\. Accordingly, for the PBCE, we introduce the following evaluation metrics and assess the proposed method based on them\.

Profit:We evaluate PBCE using the achieved profitπn\\pi\_\{n\}and its average across instances\. Higher values of the optimal profitπn\\pi\_\{n\}indicate better performance\. Additionally, we report the corresponding optimal product attributesxn∗x\_\{n\}^\{\*\}, pricespn∗p\_\{n\}^\{\*\}, and predicted demandyn∗y\_\{n\}^\{\*\}as reference values\.

Dissimilarity:Dissimilarity operationalizes Proximity as the squared Euclidean distance of the changes in non\-price attributes,𝑑𝑖𝑠𝑠=∑k\(xn,k∗−xn,kb\)2\\mathit\{diss\}=\\sum\_\{k\}\(x\_\{n,k\}^\{\*\}\-x\_\{n,k\}^\{b\}\)^\{2\}, reported without the cost coefficient and consistent withcc​gc\_\{cg\}; smaller values indicate more desirable \(lower\-cost\) CEs\. As reference information, we also examined the magnitude and direction of changes in the decision variables, includingxn∗−xnbx\_\{n\}^\{\*\}\-x\_\{n\}^\{b\}andpn∗−pnbp\_\{n\}^\{\*\}\-p\_\{n\}^\{b\}\.

Mean Absolute Error \(MAE\):In settings such as simulation studies, where the optimal values ofxn∗x\_\{n\}^\{\*\},pn∗p\_\{n\}^\{\*\}, andπn∗\\pi\_\{n\}^\{\*\}can be computed analytically, we evaluate the accuracy of the proposed numerical optimization method using the mean absolute error \(MAE\)\. This evaluation enables the assessment of the closeness of the numerical solutions to the theoretical optima\. Specifically, we define𝑑𝑖𝑓𝑓​\_​𝑜𝑝𝑡​x=\|x¯n∗−xn∗\|\\mathit\{diff\\\_opt\}\\;x=\|\\bar\{x\}\_\{n\}^\{\*\}\-x\_\{n\}^\{\*\}\|,𝑑𝑖𝑓𝑓​\_​𝑜𝑝𝑡​p=\|p¯n∗−pn∗\|\\mathit\{diff\\\_opt\}\\;p=\|\\bar\{p\}\_\{n\}^\{\*\}\-p\_\{n\}^\{\*\}\|, and𝑑𝑖𝑓𝑓​\_​𝑜𝑝𝑡​π=\|π¯n∗−πn∗\|\\mathit\{diff\\\_opt\}\\;\\pi=\|\\bar\{\\pi\}\_\{n\}^\{\*\}\-\\pi\_\{n\}^\{\*\}\|, where the overbar denotes the theoretical optimal values\.

### IV\-CExperiment

We validate the method proposed in Section IV\-A in two stages: a simulation study and a real\-data application\. In the simulations, we assess whether the method could recover the theoretical optimum\. We then apply it to real\-world data\.

#### IV\-C1Simulation Data

In this simulation, the explanatory variablesxnx\_\{n\}and the pricepn\(≥0\)p\_\{n\}\\ \(\\geq 0\)are generated from a uniform distribution on\[0,1\]\[0,1\]\. This normalization places all variables on a common scale, which facilitates the comparability of the adjustment magnitudes and distance functions in CEs\. Notably, this range is purely a scaling choice for data generation and does not represent economically feasible domains\. Therefore, optimal solutions may lie outside this interval\. The error termεn\\varepsilon\_\{n\}is assumed to follow a standard normal distribution,N​\(0,1\)N\(0,1\)\. Using the model specified below, we generate the outcome variableyny\_\{n\}and construct the datasetD=\{Y,X,P\}D=\\\{Y,X,P\\\}\. A total of 5,000 samples are generated\. We further assume that the firm does not optimizexnx\_\{n\}andpnp\_\{n\}with respect to profits prior to the simulation\.

yn=3\+2​xn−pn\+0\.05​εn\.y\_\{n\}=3\+2x\_\{n\}\-p\_\{n\}\+0\.05\\varepsilon\_\{n\}\.
Using datasetDD, we estimate the demand functions using both linear regression \(LR\) and machine learning \(ML\) approaches\. We then compare the predictive performances of LR and CMNNs with different hyperparameter settings \(e\.g\., CMNN\(3,2\): two hidden layers with three and two units\) using MSE \(Table I\)\. For reference, we also compute the MAE\. Predictive performance is evaluated using 5\-fold cross\-validation \(CV\) on the full dataset; the reported MSE and MAE values in Table I represent the averages across the five folds\. Among the ML models, the architecture with the lowest CV MSE is selected\.

TABLE I:Comparison of Model Predictive AccuracyThe results show that LR achieved the best predictive performance overall, whereas among the CMNN models, architectures with four and three hidden layers yielded the lowest prediction errors\.

We next compute CEs\. The cost functions arec​\(Z\)=0\.5​Zc\(Z\)=0\.5Zandcc​g​\(xn,xnb\)=10​\(xn−xnb\)2c\_\{cg\}\(x\_\{n\},x\_\{n\}^\{b\}\)=10\(x\_\{n\}\-x\_\{n\}^\{b\}\)^\{2\}, and the CEs were derived based on these cost settings\. We use 15 random initializations with SLSQP; the maximum number of iterations \(30\) serves as the convergence criterion, with the default SLSQP tolerance otherwise applied\. No attribute bounds are imposed\.

We compare the distributions of the original estimated variablesxnx\_\{n\},pnp\_\{n\},yny\_\{n\}, andπn\\pi\_\{n\}with those of their counterfactual counterpartsxn∗x\_\{n\}^\{\*\},pn∗p\_\{n\}^\{\*\},yn∗y\_\{n\}^\{\*\}, andπn∗\\pi\_\{n\}^\{\*\}\. We also examine the differences from the baseline values, including\|xn∗−xnb\|\|x\_\{n\}^\{\*\}\-x\_\{n\}^\{b\}\|and\|pn∗−pnb\|\|p\_\{n\}^\{\*\}\-p\_\{n\}^\{b\}\|, to assess the magnitude of the changes induced by the counterfactual solutions \(Table II; LR\-based; cf\. Table III for ML\-based\)\.

TABLE II:Summary Statistics of CEs \(LR\)
The PBCE achieves profit improvements for many products with relatively limited attribute adjustments\. The average squared attribute change \(𝑑𝑖𝑠𝑠=0\.036\\mathit\{diss\}=0\.036\) is small, suggesting that profit gains can be realized without large modifications \(mean change inxx: 0\.186\)\. Profit improvements are achieved primarily through price adjustments, and profitπn\\pi\_\{n\}can vary considerably\. In many cases, the counterfactual profit satisfiesπn∗≥πn\\pi\_\{n\}^\{\*\}\\geq\\pi\_\{n\}, with the maximum counterfactual profit reachingπn∗=5\.51\\pi\_\{n\}^\{\*\}=5\.51\. Moreover, the results indicate a structure in which profits increase even when sales volumes decrease\. In many cases,yn∗y\_\{n\}^\{\*\}is lower than the original value\. This finding implies that firms that do not apply this CE\-based optimization tend to overproduce their quantities\. Instead, profit maximization can be achieved by reducing sales volumes and raising prices, thereby increasing the margin per unit\. From a marketing perspective, this indicates that profit maximization does not necessarily require sales maximization\. Although some firms pursue sales volume or market share maximization as their primary objective \(e\.g\.,\[[10](https://arxiv.org/html/2607.01610#bib.bib10)\]\), the present results suggest that such strategies may not always be consistent with profit maximization\. We discuss this point further in Section V\.

TABLE III:Summary Statistics of CEs \(ML: CMNN\)
Table III reports the summary statistics of the PBCE derived from the demand function estimated using the ML model\. The average attribute change induced by CE is moderate \(mean = 0\.207\), indicating that profit improvements require both price and attribute adjustments\. The average price change is 1\.476, suggesting that price remains the most influential decision variable\. The average squared attribute change \(𝑑𝑖𝑠𝑠\\mathit\{diss\}\) is small \(0\.066\), indicating that ML\-based CEs propose realistic and feasible levels of change\. When we examine the optimal solutions after applying CE, the minimum profit increases dramatically to 1\.157, compared with−1\.899\-1\.899in the original data, implying a substantial reduction in downside profit risk\. The maximum profit reaches 6\.260, exceeding the original maximum profit of 4\.679\. Although sales decrease on average, an increase in unit price contributes to higher profits\. Compared with the LR\-based CEs, the ML\-based CEs exhibit smaller price adjustments \(mean = 1\.476 vs\. 1\.813\)\. Finally, to assess deviations from the theoretical optimum, we compare the summary statistics of𝑑𝑖𝑓𝑓​\_​𝑜𝑝𝑡​x\\mathit\{diff\\\_opt\}\\;x,𝑑𝑖𝑓𝑓​\_​𝑜𝑝𝑡​p\\mathit\{diff\\\_opt\}\\;pand𝑑𝑖𝑓𝑓​\_​𝑜𝑝𝑡​π\\mathit\{diff\\\_opt\}\\;\\pi, which measure the absolute differences between the theoretical optimal values and the corresponding numerically obtained solutions \(Table IV\)\.

TABLE IV:Summary Statistics of Differences Between Theoretical and Numerical Optimal CE SolutionsThe LR model yields near\-zero average differences from the theoretical optima \(𝑑𝑖𝑓𝑓​\_​p¯=0\.0014\\overline\{\\mathit\{diff\}\\\_p\}=0\.0014,𝑑𝑖𝑓𝑓​\_​x¯≈0\\overline\{\\mathit\{diff\}\\\_x\}\\approx 0,𝑑𝑖𝑓𝑓​\_​π¯=0\.0028\\overline\{\\mathit\{diff\}\\\_\\pi\}=0\.0028\), confirming that the proposed numerical optimization accurately recovers the analytical solution\. By contrast, the ML \(CMNN\) model exhibits larger differences \(𝑑𝑖𝑓𝑓​\_​p¯=0\.468\\overline\{\\mathit\{diff\}\\\_p\}=0\.468,𝑑𝑖𝑓𝑓​\_​π¯=0\.547\\overline\{\\mathit\{diff\}\\\_\\pi\}=0\.547\)\. This is expected: the theoretical optimum is derived from the linear model parameters, whereas the CMNN numerical solution reflects a different nonlinear model structure\. Accordingly, the ML differences quantify model divergence rather than optimization error\.

Overall, these results demonstrate that the proposed PBCE accurately recovers the analytical optimum when a correctly specified demand model is used\. The near\-zero LR differences confirm the validity of the numerical procedure, while the larger ML differences reflect inherent model misspecification relative to the linear theoretical benchmark\.

#### IV\-C2Real Data

Next, using data on manga \(Japanese comics\) sales volumes, prices, and other attributes in Japan, we examine how each manga title can be modified to achieve higher profits\. We assume that product attributes remain constant over the publication period of each manga series\. For empirical data, we collected the top 206 manga titles in terms of cumulative sales in Japan as of 2023\[[11](https://arxiv.org/html/2607.01610#bib.bib11)\]\. Two titles were excluded due to missing attribute data, yielding a final sample of 204 observations\. The sample size is limited by the availability of publicly accessible sales information, and the dataset is relatively small for machine learning analysis; therefore, the results should be interpreted with caution because of the potential risks of overfitting and estimation instability\. From these data, we computed an approximate per\-volume sales figure by dividing the cumulative sales of each manga by the number of volumes published and using this value asYY\. This procedure implicitly assumes that sales are uniformly distributed across volumes\. In practice, earlier volumes tend to sell more copies than later volumes\. However, this is a simplifying assumption, providing a tractable and consistent measure of average commercial performance at the series level, which is the unit of analysis in this study\. Next, based on the total price of all newly published print volumes and the number of volumes, we calculate the approximate average price per volume, which is used asPP\. Finally, as an explanatory variableXX, we employ the eight attributes listed below\. Manga inherently contains rich information at the episode level and exhibits complex sequential structures across episodes\. However, as quantifying such information in a consistent and comparable manner is difficult, this study focused on work\-level attributes related to the overall visual elements and story characteristics of each manga title\. This issue will be discussed in the following section\. The eight attributes are as follows\.

\(1\) Number of years since first publication\. This variable is constructed based on the year the manga was first published\. \(2\) Serialization of the manga in theWeekly Shonen Jump\. TheWeekly Shonen Jumpis the most popular weekly manga magazine in Japan; a large share of top\-selling manga titles have been serialized in this magazine\. Therefore, we include this attribute\. \(3–5\) Three feature vectors related to manga’s visual characteristics\. For each manga title, five images are collected using an image search engine, excluding non\-manga images\. These images are then embedded using a Vision Transformer \(ViT\), resulting in 768\-dimensional feature vectors\[[12](https://arxiv.org/html/2607.01610#bib.bib12)\]\. We subsequently apply principal component analysis \(PCA\) and reduce the dimensionality to three components selected based on scree plots and semantic interpretability\. The resulting visual components are interpreted as follows \(Table V\):vit1represents a spectrum from dramatic \(high values: cinematic and intense visual styles\) to gentle \(low values: everyday and soft visual styles\)\.vit2captures a range from realistic \(high values: greater realism and stronger shading\) to cartoonish \(low values: deformation and a simplified style\)\.vit3reflects a continuum ranging from a stylized appearance \(high values: mature and fashionable style\) to youthful characteristics \(low values: youthfulness and cuteness\)\. \(6–8\) Three feature vectors related to the story and building characteristics\. Using textual data, we construct three features based on story descriptions\. Specifically, we extract the English\-language summaries of each manga from MyAnimeList\[[13](https://arxiv.org/html/2607.01610#bib.bib13)\]\. These texts are embedded using BERT, yielding 768\-dimensional vectors\[[14](https://arxiv.org/html/2607.01610#bib.bib14)\]\. We then apply PCA to these embeddings and compress them into three components that are selected based on scree plots and semantic interpretability\. Based on the interpretation of high\-loading examples, the textual components are defined as follows \(Table VI\):bert1represents a spectrum ranging from heroic \(high values: battles, death, fate, and extraordinary events\) to mundane \(low values: everyday life, humor, and family\)\.bert2captures a continuum from masculine \(high values: social roles and occupations\) to emotional \(low values: emotions and aesthetics\)\.bert3reflects a range from epic \(high values: history, fantasy, and mythology\) to contemporary \(low values: modern settings, youth, and school life\)\. PCA is conducted independently for the visual and textual features\. As these features belong to different representational spaces, their variance structures are not directly comparable, reflecting the differences in the amount and nature of information captured by each modality\.

TABLE V:Top and Bottom Values of Feature Dimensions \(ViT\)TABLE VI:Top and Bottom Values of Feature Dimensions \(BERT\)Tables VII and VIII report the summary statistics for the dependent and explanatory variables, respectively\.

TABLE VII:Summary Statistics of the Dependent VariableTABLE VIII:Summary Statistics of the Explanatory VariablesPredictive accuracy is evaluated using 5\-fold CV\. Prior to model fitting, all explanatory variables are standardized to zero mean and unit variance for model input only; the optimization and cost function operate on the original scale\. CMNN training uses a batch size of 16, a maximum of 100 epochs, and early stopping \(patience of 5 epochs\) to prevent overfitting\. The CMNN with the lowest CV MAE is selected as the base model for CE derivation\. Table IX reports the predictive accuracy of the LR and several CMNN models\. According to the MSE and MAE metrics, the LR exhibits the smallest errors, indicating a strong fit between the linear model and the data\. Among the CMNN models, CMNN\(5,5,5\) achieves the best performance \(MSE = 26382\.287, MAE = 94\.791\), with error levels close to those of LR\. This result suggests that the relationship between the variables is approximately linear, although the relatively small sample size \(N=204N=204\) may also limit the performance of more flexible machine learning models\. However, the CMNN models achieve comparable predictive accuracies\.

TABLE IX:Comparison of Model Predictive AccuracyIn the context of the PBCE, employing models that satisfy economic constraints, such as a monotonic decrease in demand with respect to price, is particularly important\. CMNNs can directly incorporate the monotonicity constraints\. Although their prediction errors are slightly larger than those of LR, CMNNs offer the advantage of balancing economic consistency and model flexibility\. Therefore, in this study, we adopt the CMNN\(5,5,5\) as the base model for CE, as it preserves theoretical consistency while maintaining a sufficiently high predictive accuracy\.

We now derive the CEs\. For simplicity, the cost functions are specified asc​\(Z\)=10​Zc\(Z\)=10Zandcc​gc\_\{cg\}based on S1 with a common coefficientck=500c\_\{k\}=500for allkk, givingcc​g​\(xn,xnb\)=500​∑k=1K\(xn,k−xn,kb\)2c\_\{cg\}\(x\_\{n\},x\_\{n\}^\{b\}\)=500\\sum\_\{k=1\}^\{K\}\(x\_\{n,k\}\-x\_\{n,k\}^\{b\}\)^\{2\}\. We use 5 random initializations with SLSQP; the maximum number of iterations \(30\) serves as the convergence criterion, with the default SLSQP tolerance otherwise applied\.yearandjumpare held fixed via equality constraints; PCA\-based features are bounded to\[−3,3\]\[\-3,3\]; and price is left effectively unconstrained\. We now turn to the results of the CEs \(Table X\)\. In Tables X and XI,basedenotes the baseline value,cfthe counterfactual value,diffthe difference \(cf−\-base\), anddemandthe predicted demand\.

TABLE X:Counterfactual Explanations for a Representative Case
Case 1: Naruto\.For Naruto, CE shifts the visual style toward realism and strong shading \(vit2\) and strengthens epic elements \(bert3\), increasing both demand and profit\. In practice, thevit2shift suggests enhancing shading and anatomical detail—adjustments feasible at the character\-design and panel\-composition stages\.

Case 2: Doraemon\.For Doraemon, CE adjusts visual features toward more dramatic and realistic styles \(vit1,vit2\) and shifts the narrative toward social and contemporary themes \(bert2,bert3\), also modifying price\. Compared with Naruto, Doraemon requires adjustments across a larger number of elements\. In practice, thebert2andbert3shifts suggest foregrounding everyday social interactions and contemporary settings—achievable through scenario selection\.

Across all titles, ViT\-based features shift toward greater realism \(vit2\) and stylized designs \(vit3\), while BERT\-based features strengthen heroic \(bert1\) and epic \(bert3\) themes \(Table XI\)\. Average demand increases and profit improves from 94,612 to 97,603, confirming that PBCEs identify profit\-improving solutions through content\-related adjustments rather than exogenous conditions such as price or publication information\.

TABLE XI:Average Counterfactual Adjustments Across All Titles

## VDISCUSSION

We propose a CE framework that reformulates profit maximization as its primary objective, with a method for deriving such explanations and tailored evaluation metrics\. Theoretical analysis provides analytical solutions characterizing the key properties of PBCE\. Simulation results confirm that the method recovers solutions close to the theoretical optima, and real\-data analysis yields practically meaningful managerial insights\.

In the literature on CEs and algorithmic recourse, the dominant approach is to generate counterfactuals that achieve the desired output, while distances are either exogenously specified or interpreted as effort or adjustment costs\[[4](https://arxiv.org/html/2607.01610#bib.bib4),[8](https://arxiv.org/html/2607.01610#bib.bib8)\]\. Moreover, when regression or continuous outcomes are considered, the choice of target value is often determined exogenously \(e\.g\.,\[[2](https://arxiv.org/html/2607.01610#bib.bib2)\]\)\. By contrast, our approach directly maximizes profit as the primary managerial objective, thereby avoiding the problem of target specification\. Furthermore, by interpreting distances as costs associated with changes in products, prices, or managerial actions, the proposed profit\-based CE provides explanations based on the practical constraints of managerial decision making\. This finding resonates with the long\-standing debate in marketing that profit maximization does not necessarily require sales maximization\[[10](https://arxiv.org/html/2607.01610#bib.bib10),[15](https://arxiv.org/html/2607.01610#bib.bib15)\]\. The results provide a data\-driven illustration of this divergence, suggesting that firms oriented toward sales or market share maximization may be forgoing substantial profit gains\.

This study has several limitations\. First, there is the issue of the robustness of CEs\[[16](https://arxiv.org/html/2607.01610#bib.bib16)\], namely, how robust the derived CEs are to model misspecification, estimation errors, or changes in the data\-generating process\. In this study, CEs were extracted from a single predictive model; however, more robust explanations can be obtained by deriving CEs from multiple models and aggregating the results\. Second, the models employed do not guarantee causal relationships; PBCEs are derived from predictive models and do not directly represent intervention effects\. Although predictive models may capture certain causal patterns\[[17](https://arxiv.org/html/2607.01610#bib.bib17)\], explicit causal modeling would further improve the reliability of recommendations\. Third, there is an issue of data bias\. The manga data analyzed in this study include many popular titles and are concentrated in specific media outlets \(e\.g\., serialization inWeekly Shonen Jump\)\. Therefore, caution should be exercised when generalizing these findings to other media platforms or genres\. In addition, visual and textual features rely on data obtained from search engines and MyAnimeList, making it difficult to fully eliminate potential information bias\.

Finally, we conclude by discussing possible extensions and future research directions\. First, the proposed method applies to a wide range of goods \(e\.g\., consumer products\); more rigorous estimation of adjustment costs from historical data or expert judgment\[[18](https://arxiv.org/html/2607.01610#bib.bib18)\]would assess the feasibility of recommended changes\. A further extension would learn cost functions from decision\-making histories\. Second, in this study, no distance or adjustment cost was imposed on pricepnp\_\{n\}, based on the assumption that price changes are relatively easy to implement, particularly in online settings \(e\.g\.,\[[19](https://arxiv.org/html/2607.01610#bib.bib19)\]\)\. However, in contexts such as printed books, where prices are printed directly on products, or in brick\-and\-mortar retail environments where price changes incur operational costs, it is necessary to incorporate price adjustment costs\. This can be addressed by introducing a cost function of the formcc​g​\(\(xn,pn\),\(xnb,pnb\)\)c\_\{cg\}\\\!\\left\(\(x\_\{n\},p\_\{n\}\),\(x\_\{n\}^\{b\},p\_\{n\}^\{b\}\)\\right\)that explicitly incorporates price changes\. Third, the cost functions employed—linear production costs and quadratic attribute adjustment costs—represent one particular parametric specification\. Alternatives such as economies of scale in production or heterogeneous attribute\-specific adjustment costs would alter PBCE solutions\. Even under nonlinear production costs such as an S\-shaped cost curve, the qualitative result that a monopolist equates marginal revenue \(MR\) with marginal cost \(MC\) is preserved\. Heterogeneous attribute adjustment costs would concentrate recommended changes on lower\-cost attributes, yielding more targeted counterfactuals; examining such sensitivity is an important direction for future research\. Finally, while the present analysis treats each manga title as a single unit, future work could extend the framework to episode\-level analysis using text analysis and image features\. Extending the framework to oligopolistic or competitive market structures constitutes another promising direction\. Such an extension requires modifying the demand function to incorporate rivals’ prices and attributes, with Nash equilibrium replacing the monopoly optimum as the solution concept\. Competition would generally be expected to lower equilibrium prices and increase the distance between a firm’s own product attributes and those of its rivals relative to the monopoly benchmark, while the specific direction of product\-attribute adjustments would depend on the oligopoly model structure, altering both the direction and magnitude of PBCE\-recommended adjustments\.

## ACKNOWLEDGMENT

This study was supported by the JSPS KAKENHI Grants\-in\-Aid for Scientific Research \(C\) JP21K01468, JP25K05082, and JP25K05381\.

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