Binary Relations from Tournament Solutions, and Back Again

Abstract

We present a generalization of an abstract model of group choice in which the process of collective choice is modeled as a cooperative process of consideration and reconsideration of alternatives. We develop a general framework for studying choice from a finite set of alternatives, using the idea that one alternative may challenge – or displace from consideration – another in the course of a group choosing. From this binary relation – the “challenges” relation – new tournament solutions are obtained, the limit of which is the central object of the present study. The model presented generalizes that of contestation [Schwartz, 1990] and we characterize the set of alter- natives that can be chosen in a collective choice setting when the process of collective choice is viewed as cooperatively considering and reconsidering alternatives. Basic properties of the family of tournament solutions studied are given as well.

Keywords: collective choice, tournament solution, contestation, Tournament Equilibrium Set