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Fair Indivisible Payoffs through Shapley Value

Mikołaj Czarnecki, Michał Korniak, Oskar Skibski, Piotr Skowron

TL;DR

This work introduces the Indivisible Shapley Value (ISV) to fairly allocate an integer grand-coalition value $v(N)$ among players in indivisible coalitional games. It develops theory for convex integer games, proving that ISV lies in the core and providing a constructive algorithm; it extends to general games and non-identical objects with matching-based and large-game approximations. The approach is validated through case studies in approval-based apportionment, identifying key image regions, and coalition formation in elections, showcasing transparency, determinism, and fairness in allocations. The framework relies on Lower/Upper Quota constraints and connects to the classical Shapley value while ensuring integer payoffs and core feasibility where possible, offering practical tools for fair division across domains.

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.

Fair Indivisible Payoffs through Shapley Value

TL;DR

This work introduces the Indivisible Shapley Value (ISV) to fairly allocate an integer grand-coalition value among players in indivisible coalitional games. It develops theory for convex integer games, proving that ISV lies in the core and providing a constructive algorithm; it extends to general games and non-identical objects with matching-based and large-game approximations. The approach is validated through case studies in approval-based apportionment, identifying key image regions, and coalition formation in elections, showcasing transparency, determinism, and fairness in allocations. The framework relies on Lower/Upper Quota constraints and connects to the classical Shapley value while ensuring integer payoffs and core feasibility where possible, offering practical tools for fair division across domains.

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.

Paper Structure

This paper contains 27 sections, 17 theorems, 32 equations, 3 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Indivisible Coalitional Games
  4. Indivisible Coalitional Games
  5. Indivisible Shapley Value for Convex Integer Games
  6. Games with Shapley Values from the Unit Interval
  7. General Convex Integer Games
  8. Indivisible Shapley Value for General Games
  9. ISV-Approach for Large Games
  10. Distinguishable Objects
  11. Step 1
  12. Step 2
  13. Case Studies
  14. Approval-Based Apportionment
  15. Key Regions in Image Classification
  16. ...and 12 more sections

Key Result

Lemma 4.1

For every integer indivisible game $(N,v)$, player $i \in N$ and $c \in \mathbb{N}$, the $c$-reduced game $(N \setminus \{i\}, \Psi^{i \rightarrow c}_{v})$ is an integer indivisible game which is

Figures (3)

  • Figure 1: Share of mandates computed by various apportionment rules for the 2002 French presidential elections experiment data.
  • Figure 2: On the left, our method with $\alpha=0.5$; in the middle, a continuous Shapley value heatmap overlay; and on the right, the baseline method which selects regions with the highest Shapley values.
  • Figure 3: Comparison of our method for different values of parameter $\alpha$ on four pictures from MiniImageNet. Red dots indicate key regions chosen by our method. Color scale on figures with Shapley Value goes from red for regions with highest value to blue for regions with lowest value. Scale is centered at 0.

Theorems & Definitions (39)

  • Example 1
  • Example 2
  • Example 3
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Example 4
  • Example 5
  • Theorem 7.1
  • proof : Proof of \ref{['lemma:reduced_game']}
  • ...and 29 more