Threshold of Median Rank Solutions

In this paper we consider solutions defined on the class of transitive tournaments. Such solutions are essentially rank solutions i.e. solutions which depend on the ranks of the alternatives and not on any other physical characteristic. A solution is said to be a threshold solution, if for every feasible set of alternatives there exists an alternative such that the solution set coincides with the set of feasible alternatives which are not worse than the assigned alternative. We provide an axiomatic characterization of such solutions using two properties. The first property is functional acyclicity. The second property requires that given any set containing just two alternatives only the alternative only the alternative with the higher rank is selected. In order to make the presentation self contained we also provide a simple proof of an extension theorem, which is used to prove two the above mentioned axiomatic characterization. Subsequently, we provide two theorems which characterizes the median choice function when the universal set has atleast three alternatives. Several examples are provided to highlight the relationship between the axioms emphasised in this paper. It is also noted here that our second axiomatic characterization breaks down if the universal set contains precisely two elements. Following our discussion of the median rank solution, we provide two more axiomatic characterization. The first is a simultaneous axiomatic characterization of two solutions: one being that which always chooses the element with the highest rank form a set and the other being that which always selects the element with the lowest rank from a set. The second is also a simultaneous axiomatic characterization of two solutions: one being that which always chooses the greatest element from the median choice set of a set and the other being that which always selects the least element from the median choice set of a set.