Threshold and Median Rank Solutions for Transitive Abstract Games

The idea of a function which associates with each set and a binary relation a non-empty subset of the given set has a long history whose exact origin is very difficult to specify and in any case is unknown to the author. In Laslier (1997) can be found a very exhaustive survey of the related theory when binary relations are reflexive, complete and anti-symmetric. In a related paper (Lahiri [200b]) we extend the above set of binary relations to include those which are not necessarily anti-symmetric. Such binary relations which are reflexive and compete are referred to in the literature as abstract games. An ordered pair comprising a non-empty subset of the universal set and an abstract game is referred to as a subgames. 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. Lucas (1992) has a discussion of abstract games and related solution concepts, particularly in the context of cooperative games. Moulin (1986), is really the rigorous starting point of the axiomatic analysis of game solutions defined on tournaments, i.e. anti-symmetric abstract games. Much of what is discussed in Laslier (1997) and references therein carry through into this framework. In Lahiri (2000c), we obtain necessary and sufficient conditions that an abstract game needs to satisfy so that every subgame has atleast one von Neumann-Morgenstern stable set. In this paper we consider solutions defined on the class of transitive games. A solution is said to be a threshold solution, if for every subgame there exists an alternative such that the solution set for the subgame coincides with the set of feasible alternatives which are no worse than the assigned alternative. Such solutions are closely related to the threshold choice functions of Aizerman and Aleskerov (1995). We provide an axiomatic characterisation of such solutions using three properties. The first property says that if one alternative is strictly superior to another, then given a choice between the two, the inferior alternative is never chosen. The second property is functional acyclicity due to Aizerman and Aleskerov (1995). The third property requires that if two feasible alternatives are indifferent to each other, then either they are both chosen or they are both rejected. In order to make the presentation self contained we also provide a simple proof of an extension theorem due to Suzumura (1983), which is used to prove the above mentioned axiomatic characterization.