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Axiomatic and Probabilistic Foundations for the Hodge-Theoretic Shapley Value

Tongseok Lim

TL;DR

This work addresses extending the Shapley value to all coalition states by embedding the problem in combinatorial Hodge theory. It provides a complete axiomatic characterization with five axioms and a probabilistic diffusion-path interpretation, proving these viewpoints coincide. The main contributions are (i) a unique solution to the graph Poisson equation that extends fairness to every coalition, and (ii) a path-integral representation showing $\Phi=\Psi$, with the classic Shapley value recovered at the grand coalition $N$. The results position the Hodge-theoretic value as a canonical generalization of Shapley, offering a principled framework for analyzing contributions within partial coalitions and enabling broader applications in economics, politics, and machine learning.

Abstract

This paper establishes a complete theoretical foundation for the Hodge-theoretic extension of the Shapley value introduced by Stern and Tettenhorst (2019). We show that a set of five axioms--efficiency, linearity, symmetry, a modified null-player condition, and an independency principle--uniquely characterize this value across all coalitions, not just the grand coalition. In parallel, we derive a probabilistic representation interpreting each player's value as the expected cumulative marginal contribution along a random walk on the coalition graph. These dual axiomatic and probabilistic results unify fairness and stochastic interpretation, positioning the Hodge-theoretic value as a canonical generalization of Shapley's framework.

Axiomatic and Probabilistic Foundations for the Hodge-Theoretic Shapley Value

TL;DR

This work addresses extending the Shapley value to all coalition states by embedding the problem in combinatorial Hodge theory. It provides a complete axiomatic characterization with five axioms and a probabilistic diffusion-path interpretation, proving these viewpoints coincide. The main contributions are (i) a unique solution to the graph Poisson equation that extends fairness to every coalition, and (ii) a path-integral representation showing , with the classic Shapley value recovered at the grand coalition . The results position the Hodge-theoretic value as a canonical generalization of Shapley, offering a principled framework for analyzing contributions within partial coalitions and enabling broader applications in economics, politics, and machine learning.

Abstract

This paper establishes a complete theoretical foundation for the Hodge-theoretic extension of the Shapley value introduced by Stern and Tettenhorst (2019). We show that a set of five axioms--efficiency, linearity, symmetry, a modified null-player condition, and an independency principle--uniquely characterize this value across all coalitions, not just the grand coalition. In parallel, we derive a probabilistic representation interpreting each player's value as the expected cumulative marginal contribution along a random walk on the coalition graph. These dual axiomatic and probabilistic results unify fairness and stochastic interpretation, positioning the Hodge-theoretic value as a canonical generalization of Shapley's framework.

Paper Structure

This paper contains 7 sections, 4 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of Shapley axioms and the Shapley formula
  3. Stern-Tettenhorst's extension of the Shapley value via the graph Poisson's equation
  4. Characterization of $\Phi$
  5. Probabilistic representation of $\Phi$ via diffusion path integration
  6. Conclusion
  7. Proof of Theorem \ref{['main']} and Theorem \ref{['coincide']}

Key Result

Theorem 2.1

There exists a unique allocation $v \in {\mathcal{G}}_{N} \mapsto \bigl( \phi _i (v) \bigr) _{ i \in N }$ satisfying the following conditions: $\cdot$ efficiency: $\sum _{ i \in N } \phi _i (v) = v (N)$. $\cdot$ symmetry: $v \bigl( S \cup \{ i \} \bigr) = v \bigl( S \cup \{ j \} \bigr)$ for all $S \

Figures (2)

  • Figure 1: Coalition game graphs for $N=2$ and $N=3$. Each vertex of the cube corresponds to a coalition. The vertex $(1,0,1)$, for example, corresponds to the coalition $\{1,3\}$, and $(0,1,1)$ corresponds to $\{2,3\}$.
  • Figure 2: An increasing coalition path as in the Shapley formula (left) and a more general coalition path from our random walk model (right).

Theorems & Definitions (8)

  • Theorem 2.1: shapley1953value
  • Example 2.1: Glove game
  • Theorem 3.1: stern2019hodge, Theorem 3.4
  • Theorem 4.1
  • Theorem 5.1
  • Example 5.1
  • proof : Proof of Theorem \ref{['main']}
  • proof : Proof of Theorem \ref{['coincide']}