Axiomatic and Probabilistic Foundations for the Hodge-Theoretic Shapley Value

TL;DR

This paper provides a comprehensive axiomatic and probabilistic foundation for the Hodge-theoretic extension of the Shapley value, unifying fairness principles with stochastic interpretation.

Contribution

It introduces a complete set of axioms and a probabilistic representation that uniquely characterize the Hodge-theoretic Shapley value, extending its applicability beyond the grand coalition.

Findings

A set of five axioms uniquely characterizes the value.

A probabilistic interpretation as expected marginal contributions.

Unification of fairness and stochastic perspectives.

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.