Facets of the Cone of Totally Balanced Games

TL;DR

This paper characterizes the structure of totally balanced games by describing the facets of their cone, linking them to irreducible min-balanced systems, and proposes a conjecture about the cone of exact games.

Contribution

It provides a complete characterization of the facets of the cone of totally balanced games through irreducible min-balanced systems, advancing the understanding of their geometric structure.

Findings

Facet-defining inequalities correspond to irreducible min-balanced systems.

The cone of totally balanced games is described by these facets.

A conjecture about the facets of the cone of exact games is proposed.

Abstract

The class of totally balanced games is a class of transferable-utility coalitional games providing important models of cooperative behavior used in mathematical economics. They coincide with market games of Shapley and Shubik and every totally balanced game is also representable as the minimum of a finite set of additive games. In this paper we characterize the polyhedral cone of totally balanced games by describing its facets. Our main result is that there is a correspondence between facet-defining inequalities for the cone and the class of special balanced systems of coalitions, the so-called irreducible min-balanced systems. Our method is based on refining the notion of balancedness introduced by Shapley. We also formulate a conjecture about what are the facets of the cone of exact games, which addresses an open problem appearing in the literature.