¹ Guide to the Congress of the United States (Washington, D.C.: Congressional Quarterly Research Service, 1971), p. 607Google Scholar.

² Riker, William H., The Theory of Political Coalitions (New Haven: Yale University Press, 1962), p. 32Google Scholar. The original presentation of the proof of the principle, at pp. 247–278, is somewhat involved. I will therefore generally refer to the briefer, clearer presentation in Riker, William H. and Ordeshook, Peter C., An Introduction to Positive Political Theory (Englewood Cliffs, New Jersey: Prentice-Hall, 1973), pp. 176–187Google Scholar. This later presentation is eminently accessible to anyone who has even limited patience for game-theoretic algebra. See also Riker, , “A New Proof of the Size Principle,” in Mathematical Applications in Political Science, II, ed. Bernd, Joseph L. (Dallas: Southern Methodist University Press, 1966), pp. 167–174Google Scholar.

³ Dodd, Lawrence C., “Party Coalitions in Multiparty Parliaments: A Game-Theoretic Analysis,” American Political Science Review, 68 (September 1974), 1093–1117CrossRefGoogle Scholar; Haefele, Edwin T., “Coalitions, Minority Representation, and Vote-Trading Probabilities,” Public Choice, 8 (Spring 1970), 75–90CrossRefGoogle Scholar, and “A Utility Theory of Representative Government,” American Economic Review, 61 (June 1971), 350–367Google Scholar; Niemi, Richard G. and Weisberg, Herbert F., “The Effects of Group Size on Collective Decision Making,” in Probability Models of Collective Decision Making, ed. Niemi, and Weisberg, (Columbus, Ohio: Merrill, 1972), pp. 125–148Google Scholar; and Russett, Bruce M., “Components of an Operational Theory of International Alliance Formation,” Journal of Conflict Resolution, 12 (September 1968), 285–301CrossRefGoogle Scholar. See further the book length manuscript of Brams, Steven J., “Coalition Dynamics: Models of Coalition Formation in Voting Bodies” (New York University, 1971, mimeo.), esp. chapters 1 and 3Google Scholar.

⁴ Most of the contributions to The Study of Coalition Behavior, ed. Groennings, Sven, Kelley, E. W., and Leiserson, Michael (New York: Holt, Rinehart and Winston, 1970)Google Scholar, either apply the size principle or discuss its relevance to various empirical problems. Leiserson, Michael, “Factions and Coalitions in One-Party Japan,” American Political Science Review, 62 (September 1968), 770–787CrossRefGoogle Scholar, finds that the size principle fits coalition formation in the conservative Japanese Liberal-Democratic Party. Koehler, David H., “The Legislative Process and the Minimal Winning Coalition,” in Niemi, and Weisberg, , pp. 149–164Google Scholar, argues that the size principle explains roll-call voting behavior in the United States House of Representatives. Rohde, David W., “A Theory of the Formation of Opinion Coalitions in the U.S. Supreme Court,” in Niemi, and Weisberg, , pp. 165–178Google Scholar, applies the principle to Supreme Court decisions. Laakso, Markku, “Riker's ‘Size Principle’ and Its Application to Finnish Roll-Call Data,” Scandinavian Political Studies, 9 (1972), 107–118CrossRefGoogle Scholar, applies the principle to a multiparty parliament.

Further references are given in Riker, and Ordeshook, , Positive Political Theory, pp. 192–194Google Scholar. In addition, Riker presents anecdotal evidence in support of the size principle from the collapse after victory of the wartime grand coalitions of 1815, 1917, and 1945; from the American presidential elections of 1824 and 1964–1968; and from the collapse after victory of recent national independence movements in colonial states. See further, Positive Political Theory, pp. 188–191, 194–196, and 199–201; and Riker, Political Coalitions, chap. 3.

⁵ Hinckley, Barbara, “Coalitions in Congress: Size and Ideological Distance,” Midwest Journal of Political Science, 16 (May 1972), 197–207CrossRefGoogle Scholar, and “Coalitions in Congress: Size in a Series of Games,” American Politics Quarterly, 1 (July 1973), 339–359CrossRefGoogle Scholar. Butterworth, Robert Lyle, “A Research Note on the Size of Winning Coalitions,” American Political Science Review, 65 (September 1971), 741–745CrossRefGoogle Scholar. Riker, rejects Butterworth's, arguments in “Comment on Butterworth,” American Political Science Review, 65 (September 1971), 745–747CrossRefGoogle Scholar; and in Riker and Ordeshook, p. 186n. Frohlich, Norman, “The Instability of Minimum Winning Coalitions,” American Political Science Review, 69 (September 1975), 943–946CrossRefGoogle Scholar. Young, Oran R., “The Size of Winning Coalitions: A Note on Riker's Size Principle,” unpublished manuscript (University of Texas, Austin, 1972, typescript)Google Scholar.

⁶ All possible permutations can be generated by switching the labels of the players through all orderings. With five players, these number 5!, or 120 outcomes.

⁷ A more striking example of a game whose normal form, or matrix of payoffs, is distinctly asymmetric but whose characteristic function form is symmetric was proposed by McKinsey, J. C. C., Introduction to the Theory of Games (New York: McGraw-Hill, 1952), pp. 351–353Google Scholar. The example is discussed in Luce, R. Duncan and Raiffa, Howard, Games and Decisions (New York: Wiley, 1957), pp. 190–191Google Scholar.

⁸ Butterworth, p. 745.

⁹ Shepsle, Kenneth A., “On the Size of Winning Coalitions,” American Political Science Review, 68 (June 1974), 505–518CrossRefGoogle Scholar, at p. 506.

¹⁰ In this respect, the interest of mathematical game theory is contrary to the interest of what are normally called games, such as basketball or chess. In chess, Boris Spassky will find it profoundly important to know whether it is Bobby Fisher or I who sits across from him. If the game theorists ever solve the game of chess (as tic-tac-toe has been solved), chess will lose its interest except among the ignorant (as tic-tactoe can be played with interest only by the very young and the daft). Of course, game theorists have long been trying, and they may yet succeed. (Also of course, the class of the ignorant and daft may be much larger in the case of chess than in the case of tic-tac-toe.)

¹¹ Shepsle, p. 515.

¹² Riker, and Ordeshook, , Positive Political Theory, p. 182Google Scholar.

¹³ Ibid., emphasis added.

¹⁴ Assume by hypothesis that v(S)–v(S+i) = Δ, where Δ>0. It is possible that the security level, q_i, of player i is less than −Δ. Hence, it is possible that i was making side payments in excess of Δ to the members of S. If S now evicts i, its members collectively lose this excess over Δ. In Game 1 for example, if coalition ABCD has formed with payoffs (26, 26, 26, −28), then ABC cannot afford to evict D, although ABC would be of winning size and v(ABC) > v(ABCD).

¹⁵ Norman Frohlich has presented a related but substantially more elegant and more general demonstration of this conclusion in “The Instability of Minimum Winning Coalitions.”

¹⁶ von Neumann, John and Morgenstern, Oskar, Theory of Games and Economic Behavior (Princeton: Princeton University Press, 1953, 3rd edition), p. 423Google Scholar. Equation 2 implies that S_n-r, the losing complement of S_r, is flat, that is, every member of S_n-r receives his security level. Every subset of a flat set is also flat. For a symmetric game it follows that, if the losing coalition S_n-m is flat, then S_p is also flat for every p < n − m.

¹⁷ In the terminology of the previous footnote, the necessary and sufficient condition for Riker's argument to guarantee that only minimum winning coalitions form in a symmetric game is that all losing coalitions in the game be flat (where losing coalitions do not include blocking coalitions).

¹⁸ The 50 units are the difference between the value of coalition BCD (which is 60) and the sum of the payoffs to B, C, and D in coalition ABCD (which is 10).

¹⁹ Butterworth, , “A Research Note,” p. 744Google Scholar.

²⁰ David Rohde argues that the size principle applies to the formation of “opinion coalitions” in the Supreme Court. Hence, although all nine members might agree in upholding or reversing a lower court ruling, the actual opinion of the majority is likely to be written through compromise among only five members (while the other members are left to write their own opinions, if they choose). The raw data Rohde presents do not support the size principle (as can be seen by combining his Tables 2 and 3 and comparing them to the random results of his Table 1). Therefore, he subjectively splits the cases according to whether they involve “threats” to the Court. The threatening cases yield greater than random occurrences of unanimous and overwhelming opinion coalitions and less than random occurrences of minimum coalitions, whereas the non-threatening cases yield greater than random occurrences of minimum “winning” opinion coalitions (as well as, oddly, greater than random occurrences of eight and nine member coalitions). See further Rohde, , “A Theory of the Formation of Opinion Coalitions in the U.S. Supreme Court,” pp. 176–177Google Scholar.

²¹ Shepsle, , “On the Size of Winning Coalitions,” pp. 508–509Google Scholar. The notion of the bargaining set is developed in Aumann, Robert and Maschler, Michael, “The Bargaining Set for Cooperative Games,” in Advances in Game Theory, ed. Dresher, Melvin, Shapley, L. S., and Tucker, A. W. (Princeton: Princeton University Press, 1964), pp. 443–476Google Scholar.

²² Shepsle, , “On the Size of Winning Coalitions,” p. 515Google Scholar.

²³ Ibid.

²⁴ Von Neumann, and Morgenstern, , Theory of Games and Economic Behavior, esp. pp. 330–332, 339, and 403Google Scholar.

²⁵ If the Senate does not confine itself to symmetric games, then it plays in a space of 10³⁰ dimensions — a number which would stymie even the appropriations committee. But see Shepsle's footnote 8 and the paragraph to which it is footed; Shepsle, , “On the Size of Winning Coalitions,” p. 506Google Scholar.

²⁶ Von Neumann, and Morgenstern, , Theory of Games and Economic Behavior, p. 431Google Scholar.

²⁷ Ibid., p. 423.

²⁸ See further, ibid., p. 428.

²⁹ Ibid., pp. 420–503.

³⁰ Riker, , Political Coalitions, p. 260Google Scholar.

³¹ Rapoport, Anatol, N-Person Game Theory: Concepts and Applications (Ann Arbor: University of Michigan Press, 1970), pp. 286–287Google Scholar.

³² Shepsle, , “On the Size of Winning Coalitions,” p. 516Google Scholar.

³³ Rohde, “A Theory of the Formation of Opinion Coalitions in the U.S. Supreme Court,” is an exception.

³⁴ I owe this characterization to an anonymous reader of an earlier draft of this paper.

³⁵ Koehler, , “The Legislative Process and the Minimal Winning Coalition,” p. 155Google Scholar. For a contrary interpretation of House roll-call voting, see Hinckley, “Coalitions in Congress: Size and Ideological Distance.”

³⁶ Note that more than 20 per cent of the House votes were nearly unanimous, and these votes are not included in Table 2.

³⁷ Koehler's data are reported in 5 per cent clusters. Hence, the mean cannot be calculated any closer, but a safe guess is probably near 74 per cent.

³⁸ Riker, and Ordeshook, , An Introduction to Positive Political Theory, p. 177Google Scholar, suggest on the contrary that the advantages of the size principle over other game theoretic solution notions lie “especially in games with very large numbers of players.”