Hodge decomposition and the Shapley value of a cooperative game

Abstract

We show that a cooperative game may be decomposed into a sum of component games, one for each player, using the combinatorial Hodge decomposition on a graph. This decomposition is shown to satisfy certain efficiency, null-player, symmetry, and linearity properties. Consequently, we obtain a new characterization of the classical Shapley value as the value of the grand coalition in each player's component game. We also relate this decomposition to a least-squares problem involving inessential games (in a similar spirit to previous work on least-squares and minimum-norm solution concepts) and to the graph Laplacian. Finally, we generalize this approach to games with weights and/or constraints on coalition formation.

Cooperative trading of a price-maker wind power producer: A data-driven approach considering uncertainty

“…• 2) A cooperative bidding model for a price-maker wind power producers considering day-ahead electricity price uncertainty is formulated to maximize its profits. Besides, Shapley value [25] is used to distribute the profits for cooperative power producers.…”

Hodge theoretic reward allocation for generalized cooperative games on graphs

“…The four conditions listed above are often called the Shapley axioms. Quoted from [16], they say that [(i) efficiency] the value obtained by the grand coalition is fully distributed among the players, [(ii) symmetry] equivalent players receive equal amounts, [(iii) null-player] a player who contributes no marginal value to any coalition receives nothing, and [(iv) linearity] the allocation is linear in the game values.…”

“…(1.1) can be rewritten also quoted from [16]: Suppose the players form the grand coalition by joining, one-at-a-time, in the order defined by a permutation σ of [N]. That is, player i joins immediately after the coalition S σ,i = j ∈ [N] : σ(j) < σ(i) has formed, contributing marginal value v S σ,i ∪{i} −v(S σ,i ).…”

“…More recently, the combinatorial Hodge decomposition has been applied to game theory and various economic contexts, for instance Candogan et al [1], Jiang et al [5], Stern and Tettenhorst [16]. We refer to Lim [6] for an elementary introduction to the Hodge theory on graphs.…”

“…In particular, Stern and Tettenhorst [16] showed that, given a game v ∈ G(2 [N ] ), there exist component games v i ∈ G(2 [N ] ) for each player i ∈ [N] which are naturally defined via the combinatorial Hodge decomposition,…”