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The Instability of Minimum Winning Coalitions*

Published online by Cambridge University Press:  01 August 2014

Norman Frohlich
Norman Frohlich*
Affiliation:
University of Texas, Austin
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Abstract

This paper examines William Riker's thesis that only minimum winning coalitions form in n-person zero-sum symmetric games. It demonstrates that Riker's conclusion is false by identifying the conditions under which larger than minimum winning coalitions can form. Since these conditions are quite general it indicates that Riker's conclusion is valid only for a highly restricted class of games. This class of games is identified as those in which players not in a minimum winning coalition have no incentive to form any coalitions among themselves. These games are characterized as games inessential over coalitions of losers. Only in these games can minimum winning coalitions be expected to form exclusively. In all other games, larger than minimum winning coalitions are possible.

Type
Research Article
Information
American Political Science Review , Volume 69 , Issue 3 , September 1975 , pp. 943 - 946
Copyright
Copyright © American Political Science Association 1975

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Footnotes

*

I have been the beneficiary of a great deal of helpful inspiration and criticism in the preparation of this paper. The impetus for it was provided by an idea in a preliminary draft of Oran Young's paper, “The Size of Winning Coalitions: A Note on Riker's Size Principle.” The argument has been refined and expanded as a result of truly helpful comments from Lawrence Dodd, Russell Hardin, Joe Oppenheimer, Kenneth Shepsle, Jeff Smith, and Harrison Wagner. I am deeply grateful for their help. An earlier version of this paper was presented at the annual meeting of the Public Choice Society, New Haven, Connecticut, March 1974. Research on this paper was supported by National Science Foundation Grant #GS-33490.

References

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.

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