Linear criterion for testing the extremity of an exact game based on its finest min-representation

Abstract

A game-theoretical concept of an exact (cooperative) game corresponds to the notion of a discrete coherent lower probability, used in the context of imprecise probabilities. The collection of (suitably standardized) exact games forms a pointed polyhedral cone and the paper is devoted to the recognition of extreme rays of that cone, whose generators are called extreme exact games. We give a necessary and sufficient condition for an exact game to be extreme. Our criterion leads to solving a simple linear equation system determined by a certain min-representation of the game. It has been implemented on a computer and a web-based platform for testing the extremity of an exact game is available, which works with a modest number of variables.The paper also deals with different min-representations of a fixed exact game μ, which can be compared with the help of the concept of a tightness structure (of a min-representation) introduced in the paper. The collection of tightness structures (of min-representations of μ) is shown to be a finite lattice with respect to a refinement relation. We give a method to obtain a min-representation with the finest tightness structure, which construction comes from the coarsest standard min-representation of μ given by the (complete) list of vertices of the core (polytope) of μ. The newly introduced criterion for exact extremity is based on the finest tightness structure.