Minimum Winning Coalitions Reconsidered: A Rejoinder to Butterworth's “Comment”

Abstract

The purpose of my note, On the Size of Winning Coalitions, was two-fold. First, I wished to address the notion of competitive bribery, underscoring its conceptual relevance to a theory of side-payments and exploiting its theoretical power in a less arbitrary manner than had Butterworth in his Research Note.' Since there is still some apparent confusion on this point, additional details are presented below. The second purpose of my note was to present a general strategy for dealing with some of the problems surrounding the determination of winning coalition size, as well as to argue in behalf of a specific tactic. The general strategy still appears appropriate; the specific tactic, however, has been subjected to some valid criticisms, I believe, in Butterworth's Comment. It needs, therefore, to be re-examined. Butterworth's original contribution in this debate is the discovery of the existence of an individually rational bribe-a form of side-payment which allows an apparent loser to attempt to bribe his way into an otherwise minimal winning coalition so as to minimize his own losses. This process occurs at no apparent expense to the members of the formerly minimal winning coalition (MWC), and may, if for some reason the other losers do not respond appropriately, redound to their positive benefit. In the particular illustration employed by Butterworth (the characteristic function of this game appears in my paper elsewhere in this volume), in which a threeman winning coalition { abc } and a two-man losing coalition {de } are organized so as to receive the payoff