Threshold and median rank solutions for transitive abstract games

Abstract

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.