Weighted Average-convexity and Cooperative Games

TL;DR

This paper introduces weighted average-convexity, explores its relation to cooperative game solutions like the weighted Shapley value, and characterizes conditions under which these properties hold in specific graph structures.

Contribution

It generalizes convexity concepts to weighted average-convexity and establishes new links between this property and core allocations in cooperative games.

Findings

Weighted average-convex games have their weighted Shapley value in the core.

Necessary conditions are identified for preserving weighted average-convexity in communication games.

Complete characterization of weighted average-convexity in priority decreasing tree graphs.

Abstract

We generalize the notion of convexity and average-convexity to the notion of weighted average-convexity. We show several results on the relation between weighted average-convexity and cooperative games. First, we prove that if a game is weighted average-convex, then the corresponding weighted Shapley value is in the core. Second, we exhibit necessary conditions for a communication TU-game to preserve the weighted average-convexity. Finally, we provide a complete characterization when the underlying graph is a priority decreasing tree.