Hierarchy of players in swap robust voting games and minimal winning coalitions

Ordinarily, the process of decision making by a committee through voting is modelled by a monotonic game the range of whose characteristic function is restricted to f0; 1g: The decision rule that governs the collective action of a voting body induces a hierarchy in the set of players in terms of the a-priori in‡uence that the players have over the decision making process. In order to determine this hierarchy in a swap robust game, one has to either evaluate a number-based power index (e.g., the Shapley-Shubik index, the Banzhaf-Coleman index) for each player or conduct a pairwise comparison between players in order to …nd out whether there exists a coalition in which player i is desirable over another player j as a coalition partner. In this paper we outline a much simpler and more elegant mechanism to determine the ranking of players in terms of their a-priori power using only minimal winning coalitions, rather than the entire set of winning coalitions. Keywords Simple game, Swap robust game, Desirability, Weak desirability, Lexicographic ordering. JEL classi…cation C71, D71 Email: mbishnu@iastate.edu yEmail: sroy@iastate.edu