Bibliography

(eng): 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, 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 a fixed game) 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 the game given by the (complete) list of vertices of the core (polytope) of the game. The newly introduced criterion for exact extremity is based on the finest tightness structure.