Equivalent N-Person Games and the Null Space of the Shapley Value

Abstract

Two n-person games may be said to be equivalent if they have the same Shapley value. In this paper, we attempt to simplify the determination of the equivalence of two games. We do this by recognizing that a pair of games are equivalent if and only if their difference lies in the null space of the Shapley value. Using the representation theory of the symmetric groups we construct a direct-sum decomposition of this null space into invariant subspaces. We then use this same theory to derive a characterization of a very general type of value, of which the Shapley value is one particular example.

Axioms of invariance for TU-games

“…The axioms of associated consistency in Hamiache [7] and B-consistency in Driessen [5] require invariance of a rule if the worth of every coalition is adjusted in particularly specific ways, which allow much less freedom than our axiom of uniform addition invariance. Finally, our research can be connected to Kleinberg and Weiss [12] in which the set of TU-games for which the Shapley value recommends the null payoff vector is characterized. The link with our article is that requiring invariance for an additive rule between two TU-games is equivalent to impose that the rule specifies the null payoff vector in the TU-game obtained by taking the difference between the two starting TU-games.…”

A decomposition for the space of games with externalities

“…(x 1 ; x 2 ; :::; x n ) = (x (1) ; x (2) ; :::; x (n) ) 2 As noted by Kleinberg and Weiss (1985), for the space of TU games.…”

“…Kleinberg and Weiss (1985) used the representation theory of the symmetric group to construct a direct sum decomposition of the null space of the Shapley value for games in characteristic function form (TU games). In Kleinberg and Weiss (1986), the authors followed the same line of reasoning to characterize the space of linear and symmetric values for TU games.…”

Associated Consistency Characterization of Two Linear Values for Tu Games by Matrix Approach

“…The algebraic representation and matrix approach to cooperative game theory have appeared to be natural as well as powerful. Kleinberg and Weiss (1985) constructed a direct sum decomposition of the null space and studied equivalent classes of games with respect of the Shapley value, i.e., two games are in the same class if they have the same Shapley value. Dragan (1991Dragan ( , 1996 introduced the potential basis to study the weighted Shapley value and the Banzhaf value.…”