Some solutions for abstract games: axiomatic characterisations

Abstract

In this paper we consider binary relations which are reflexive and complete. Such binary relations are referred to in the literature as abstract games. Given an abstract game a (game) solution is a function which associates to each subset a non-empty collection of points of the subset. An important consequence of this framework is that often, a set may fail to have an element which is best with respect to the given binary relation. To circumvent this problem the concept of the top cycle set is introduced, which selects from among the feasible alternatives only those which are best with respect to the transitive closure of the given relation. The top cycle set is always non-empty and in this paper we provide an axiomatic characterization of the top-cycle solution. It is subsequently observed that the top cycle solution is the coarsest solution which satisfies two innocuous assumptions. In the final section of this paper we address the problem of axiomatically characterizing the uncovered solution (where covering is now defined as a menu-based concept).