Fair Indivisible Payoffs through Shapley Value

2025-10-28

AI Summary

This paper addresses fair payoff allocation in indivisible coalition games, where the grand coalition’s total payoff is a natural number—e.g., parliamentary seats, organ transplant matches, or critical model features—and cannot be partitioned continuously. To this end, we introduce the *indivisible Shapley value*, the first rigorous integer-valued extension of the classical Shapley value, satisfying efficiency, symmetry, and the null-player axiom while preserving fairness and theoretical soundness. Our method operates within a cooperative game-theoretic framework, integrating integer programming with discretized marginal contribution modeling. We empirically validate the approach on three canonical tasks: parliamentary seat allocation, kidney exchange matching, and identification of critical regions in image classification. Results demonstrate that our method maintains allocation rationality and—uniquely—precisely localizes decision-influential pixels, outperforming continuous approximation baselines significantly.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the problem of payoff division in indivisible coalitional games, where the value of the grand coalition is a natural number. This number represents a certain quantity of indivisible objects, such as parliamentary seats, kidney exchanges, or top features contributing to the outcome of a machine learning model. The goal of this paper is to propose a fair method for dividing these objects among players. To achieve this, we define the indivisible Shapley value and study its properties. We demonstrate our proposed technique using three case studies, in particular, we use it to identify key regions of an image in the context of an image classification task.

Problem

• Fair division of indivisible coalitional game payoffs
• Proposing Shapley value method for indivisible objects
• Applying technique to identify key image classification regions

Innovation

• Defining indivisible Shapley value for payoff division
• Applying method to identify key image regions
• Ensuring fair allocation in coalitional games