Binary Relations from Tournament Solutions, and Back Again

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 alternatives 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 is given as well.