Inheritance of Convexity for the $\tilde{\mathcal{P}}_{\min}$-Restricted Game

TL;DR

This paper investigates the inheritance of convexity in a restricted cooperative game on weighted graphs, using a novel partition based on minimum weight edges, and characterizes the graphs that preserve convexity.

Contribution

It introduces a new partition $ ilde{oldsymbol{ ext{P}}}_{ ext{min}}$ for restricted games and characterizes graphs where convexity is inherited from the original to the restricted game.

Findings

Characterization of graphs with inheritance of convexity.

Introduction of the $ ilde{oldsymbol{ ext{P}}}_{ ext{min}}$ partition.

Conditions under which convexity is preserved.

Abstract

We consider a restricted game on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the sub-components obtained with a specific partition $\tilde{\mathcal{P}}_{\min}$. This partition relies on the same principle as the partition $\mathcal{P}_{\min}$ introduced by Grabisch and Skoda (2012) but restricted to connected coalitions. More precisely, this new partition $\tilde{\mathcal{P}}_{\min}$ is induced by the deletion of the minimum weight edges in each connected component associated with a coalition. We provide a characterization of the graphs satisfying inheritance of convexity from the underlying game to the restricted game associated with $\tilde{\mathcal{P}}_{\min}$.