¹ Riker, William, The Theory of Political Coalitions (New Haven: Yale University Press, 1962), p. 32 Google Scholar.

² Butterworth, Robert Lyle, “A Research Note on the Size of Winning Coalitions,” American Political Science Review, 65 (09, 1971), 741–745 CrossRefGoogle Scholar, and Russell Hardin, “Hollow Victory: The Minimum Winning Coalition,” Fels Center of Government, Discussion Paper #30, Butterworth calls Riker's findings into question via a theoretical argument, whereas Hardin presents a different theoretical argument as well as a discussion of the empirical problems inherent in any attempt to test Riker's thesis. Although both make telling points the argument here concentrates on extending Butterworth's line of reasoning.

³ Shepsle, Kenneth, “On the Size of Winning Coalitions,” American Political Science Review, 68 (06, 1974), 505–518 CrossRefGoogle Scholar, Robert Lyle Butterworth, “Comment on Shepsle's ‘On the Size of Winning Coalitions,’” and Shepsle, “Minimum Winning Coalitions Reconsidered: A Rejoinder to Butterworth's ‘Comment’,” Ibid., 519–524.

⁴ Riker's argument was articulated in terms of symmetric games, as can be seen from his use of a graphical representation of the characteristic function. This paper discusses only symmetric games of the type treated by Riker.

⁵ In this paper I examine only games involving five or more players. This allows us to examine competitive bribery when there are at least two potential losers. All of the results not involving competitive bribery can be easily demonstrated for games with fewer than five members.

⁶ Here for the sake of convenience I treat the division of payoffs between S and R as if S appropriated to it-self all of the winnings of v(s + r) and in addition got a bribe of amount “a” from R for the formation of S ∪ R. The use of the term “bribe,” however, may be misleading. If, for example, the game in question were of positive slope over the range s% s + r then the members of S might actually offer the members of R a “bribe” to secure the formation of S ∪ R. In that event “a” would be a negative quantity. In either event other ways of representing the division of payoffs are possible. One could posit that the members of S ∪ R divided the payoffs symmetrically and that in addition the members of R gave the members of S a bribe of size “b”. The analysis would proceed in the same way as it does in this representation. Only the algebra would be different. The conclusions would be the same.

⁷ The assumption that the “losers” initially expect to share the losses symmetrically is not necessary for the argument which follows. It is a convenience, adopted for ease of exposition. All the conclusions that follow can be derived without that assumption.

⁸ Differences in the payoff structures in legislatures in different countries might explain some of the variance in cabinet size found by researchers such as Dodd, Lawrence, “Party Coalitions in Multiparty Parliaments: A Game-Theoretic Analysis,” American Political Science Review, 68 (09, 1974), 1093–1117 CrossRefGoogle Scholar.