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Abstract Games Admitting Stable Solutions



Reflexive and complete binary relations are also referred to as abstract games. An ordered pair comprising a non-empty subset of the universal set and an abstract game is referred to as a subgame. A (game) solution is a function which associates to all subgames of a given (nonempty) set of games, a nonempty subset of the set in the subgame. In this paper we obtain conditions which are either necessary or sufficient for an abstract game to admit a solution which always selects a von Neumann-Morgenstern stable set from every admissible subgame. It is proved in this paper that a sufficient condition for an abstract game to satisfy this property is that it does not admit any strict preference cycle of length three and satisfy what we call stable five element set property. We show by an example that these two properties are logically independent.

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