The Core and the Selectope of Games
This second fundamental chapter addresses the following problem: Given a game or a capacity, does there exist an additive game dominating it on every subset, under the constraint that both coincide on the universal set? For normalized capacities, the problem amounts to finding probability measures dominating a given capacity, while in cooperative game theory, it amounts to the problem of sharing a cake so that no coalition of players is dissatisfied. The set of additive games (or equivalently vectors, if one represents additive games by their “distribution”) dominating a given game is called the core of that game. It is a convex polyhedron, whenever it is nonempty, and its properties have been studied in depth. In Sect. 3.2, we study the case where the game is defined on the whole power set, while Sect. 3.3 addresses the case of games on set systems. The latter case reveals to be much more complex, because the core is most often unbounded and may even have no vertices. Section 3.4 goes further in the analysis of the core of games on the power set, through the concept of exact games and large cores. Exact games are those which coincide with the lower envelope of their core and are of primary importance in decision under uncertainty. A game with large core has the property that for any vector y dominating it without constraint, it is possible to find a core element x smaller than y. The last topic addressed in this chapter is the selectope of a game (Sect. 3.5). It is the set of additive games (or vectors) obtained by sharing the Möbius transform of that game on every element. It always contains the core and equality holds if and only if the game has a nonnegative Möbius transform, except possibly on singletons.
KeywordsExtreme Point Maximal Chain Large Core Balance Game Cooperative Game Theory
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