Shapley-like values without symmetry

TL;DR

This paper introduces a non-probabilistic approach to cooperative game value allocation, focusing on efficiency and reasonableness without requiring symmetry, and characterizes the structure of such allocations.

Contribution

It proposes an alternative to symmetric Shapley values by defining reasonable, efficient allocations and characterizes their structure using convex combinations and Krein-Milman theorem.

Findings

Identifies properties of linear maps for value allocations.

Defines a class of allocations for which all reasonable, efficient maps are convex combinations.

Provides a new framework for non-symmetric cooperative game solutions.

Abstract

Following the work of Lloyd Shapley on the Shapley value, and tangentially the work of Guillermo Owen, we offer an alternative non-probabilistic formulation of part of the work of Robert J. Weber in his 1978 paper "Probabilistic values for games." Specifically, we focus upon efficient but not symmetric allocations of value for cooperative games. We retain standard efficiency and linearity, and offer an alternative condition, "reasonableness," to replace the other usual axioms. In the pursuit of the result, we discover properties of the linear maps that describe the allocations. This culminates in a special class of games for which any other map that is "reasonable, efficient" can be written as a convex combination of members of this special class of allocations, via an application of the Krein-Milman theorem.