Linear Core-Based Criterion for Testing Extreme Exact Games

The notion of a (discrete) coherent lower probability corresponds to a game-theoretical concept of an exact (cooperative) game. The collection of (standardized) exact games forms a pointed polyhedral cone and the paper is devoted to the extreme rays of that cone, known as extreme exact games. A criterion is introduced for testing whether an exact game is extreme. The criterion leads to solving simple linear equation systems determined by (the vertices of) the core polytope (of the game), which concept corresponds to the notion of an induced credal set in the context of imprecise probabilities. The criterion extends and modifies a former necessary and sufficient condition for the extremity of a supermodular game, which concept corresponds to the notion of a 2-monotone lower probability. The linear condition we give in this paper is shown to be necessary for an exact game to be extreme. We also know that the condition is sufficient for the extremity of an exact game in an important special case. The criterion has been implemented on a computer and we have made a few observations on basis of our computational experiments.