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Cooperative networks and Hodge-Shapley value

Tongseok Lim

TL;DR

This work reframes the classical Shapley value as an expected total marginal contribution along random coalition paths, enabling value allocations at every partial coalition state on a coalition graph. It introduces the Hodge-Shapley value by using edge flows and path integration, extending allocations to general cooperative networks via $Hodge$ calculus and Poisson-type equations. A five-axiom characterization (A1–A5) yields a unique allocation operator $\Phi$ that coincides with Shapley at the grand coalition while reflecting broader network dynamics. The approach unifies stochastic coalition processes, graph-theoretic marginal values, and reversible Markov chains, providing tractable computation through Poisson equations and linear systems with broad potential for applications beyond traditional coalition games.

Abstract

Lloyd Shapley's cooperative value allocation theory stands as a central concept in game theory, extensively utilized across various domains to distribute resources, evaluate individual contributions, and ensure fairness. The Shapley value formula and his four axioms that characterize it form the foundation of the theory. Traditionally, the Shapley value is assigned under the assumption that all players in a cooperative game will ultimately form the grand coalition. In this paper, we reinterpret the Shapley value as an expectation of a certain stochastic path integral, with each path representing a general coalition formation process. As a result, the value allocation is naturally extended to all partial coalition states. In addition, we provide a set of five properties that extend the Shapley axioms and characterize the stochastic path integral. Finally, by integrating Hodge calculus, stochastic processes, and path integration of edge flows on graphs, we expand the cooperative value allocation theory beyond the standard coalition game structure to encompass a broader range of cooperative network configurations.

Cooperative networks and Hodge-Shapley value

TL;DR

This work reframes the classical Shapley value as an expected total marginal contribution along random coalition paths, enabling value allocations at every partial coalition state on a coalition graph. It introduces the Hodge-Shapley value by using edge flows and path integration, extending allocations to general cooperative networks via calculus and Poisson-type equations. A five-axiom characterization (A1–A5) yields a unique allocation operator that coincides with Shapley at the grand coalition while reflecting broader network dynamics. The approach unifies stochastic coalition processes, graph-theoretic marginal values, and reversible Markov chains, providing tractable computation through Poisson equations and linear systems with broad potential for applications beyond traditional coalition games.

Abstract

Lloyd Shapley's cooperative value allocation theory stands as a central concept in game theory, extensively utilized across various domains to distribute resources, evaluate individual contributions, and ensure fairness. The Shapley value formula and his four axioms that characterize it form the foundation of the theory. Traditionally, the Shapley value is assigned under the assumption that all players in a cooperative game will ultimately form the grand coalition. In this paper, we reinterpret the Shapley value as an expectation of a certain stochastic path integral, with each path representing a general coalition formation process. As a result, the value allocation is naturally extended to all partial coalition states. In addition, we provide a set of five properties that extend the Shapley axioms and characterize the stochastic path integral. Finally, by integrating Hodge calculus, stochastic processes, and path integration of edge flows on graphs, we expand the cooperative value allocation theory beyond the standard coalition game structure to encompass a broader range of cooperative network configurations.
Paper Structure (7 sections, 9 theorems, 63 equations, 4 figures)

This paper contains 7 sections, 9 theorems, 63 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of Shapley axioms and the Shapley formula
  3. Extension of the Shapley value via random coalition processes
  4. A characterization of the value $\Phi$
  5. Hodge-Shapley value
  6. General cooperative networks
  7. Reversible Markov chains and Poisson's equations on graphs

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 (4)

  • Figure 1: Coalition game graph for $n=2$ and $3$. Each vertex of the cube corresponds to a coalition. The vertex $(1,0,1)$, for example, corresponds to the coalition $\{1,3\}$, $(0,1,1)$ to $\{2,3\}$, and so on.
  • Figure 2: Examples of Shapley's coalition path and our general coalition path. The path in this example has no loops, but in general, our coalition path is allowed to have an arbitrary number of loops.
  • Figure 3: A coalition path from $S$ to $T$ and its reflection w.r.t. $i$.
  • Figure 4: A cooperative graph can have an arbitrary structure. Each edge has a forward and reverse direction associated to it.

Theorems & Definitions (25)

  • Theorem 2.1: shapley1953value
  • Example 2.1: Glove game
  • Theorem 3.1
  • Example 3.1
  • Theorem 4.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3: Characterization of the Shapley value via potential function approach
  • Theorem 4.2: hart1989potential
  • Theorem 5.1
  • ...and 15 more