Symmetries of Quasi-Values

TL;DR

This paper explores how relaxing symmetry axioms in cooperative game theory affects the uniqueness and structure of quasi-values, providing classifications and algorithms for constructing symmetric quasi-values under various permutation groups.

Contribution

It classifies permutation groups ensuring unique G-symmetric quasi-values and develops algorithms for constructing these values.

Findings

Identifies permutation groups guaranteeing a unique G-symmetric quasi-value.

Provides a structural and dimensional analysis of G-symmetric quasi-values.

Develops algorithms to construct G-symmetric quasi-values by averaging basic quasi-values.

Abstract

According to Shapley's game-theoretical result, there exists a unique game value of finite cooperative games that satisfies axioms on additivity, efficiency, null-player property and symmetry. The original setting requires symmetry with respect to arbitrary permutations of players. We analyze the consequences of weakening the symmetry axioms and study quasi-values that are symmetric with respect to permutations from a group $G\leq S_n$. We classify all the permutation groups $G$ that are large enough to assure a unique $G$-symmetric quasi-value, as well as the structure and dimension of the space of all such quasi-values for a general permutation group $G$. We show how to construct $G$-symmetric quasi-values algorithmically by averaging certain basic quasi-values (marginal operators).