¹ von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior (Princeton University Press, 1944)Google Scholar.

² Kaplan, M., System and Process in International Politics (Wiley, 1957)Google Scholar.

³ Simon, H. A., “A Comparison of Game and Learning Theory,” Psychometrika, Vol. 21 (1956), pp. 267–72CrossRefGoogle Scholar.

⁴ Richard C. Snyder outlines conceptual parallels and several potentially fruitful areas of application in “Game Theory and the Analysis of Political Behavior,” in Bailey, Stephen K. et al. , Research Frontiers in Politics and Government (The Brookings Institution, 1955)Google Scholar. See also the broad collection edited by Shubik, Martin, Readings in Game Theory and Political Behavior (Doubleday, 1954)Google Scholar.

⁵ Luce, R. D. and Raiffa, H., Games and Decisions (Wiley, 1957)Google Scholar.

⁶ Shapley, L. S. and Shubik, M., “A Method for Evaluating the Distribution of Power in a Committee System,” this Review, Vol. 48 (1954), pp. 787–792 Google Scholar; Luce, R. D. and Rogow, A. A., “A Game Theoretic Analysis of Congressional Power Distributions for a Stable Two-Party System,” Behavioral Science, Vol. 1 (1956), pp. 83–95 CrossRefGoogle Scholar; and Riker, William H., “A Test of the Adequacy of the Power Index,” Behavioral Science, Vol. 4 (1959), pp. 120–131 CrossRefGoogle Scholar.

⁷ Cumulative voting is, however, common in voting for corporate boards of directors; for an application of the theory of games in this area, see Glasser, G. J., “Game Theory and Cumulative Voting for Corporate Directors,” Management Science, Vol. 5 (1959), pp. 151–156 CrossRefGoogle Scholar. In England, cumulative voting for school boards of from five to fifteen members was instituted through the Education Act of 1870, and continued until school boards were abolished in 1902; see Ross, J. F. S., Elections and Electors: Studies in Democratic Representation (Eyre and Spottiswoode, 1955), pp. 62–63 Google Scholar.

⁸ In 1870, when cumulative voting was adopted in Illinois by referendum as part of a new constitution, the General Assembly was sharply divided by region. In the Assembly of 1867, one could draw a line across the state about the latitude of Springfield such that districts north of the line returned 52 Republicans and no Democrats and those south of the line returned 8 Republicans and 24 Democrats. In the first election after the change to cumulative voting, however, all but one of the 51 newly created three-member districts returned members of poth parties. North of the same line referred to above, the Democrats had increased their representation from zero to 34%, and south of the line, the Republicans had increased their representation from 25% to 38%. For an account of the operation of cumulative voting in Illinois over the period 1872–1954, see Blair, George S., Cumulative Voting: An Effective Electoral Device in Illinois Politics (University of Illinois Press, 1960)Google Scholar.

⁹ Moore, B. F., “The history of cumulative voting and minority representation in Illinois, 1870–1919,” University of Illinois Studies in the Social Sciences, Vol. 8, No. 2 (1919)Google Scholar; Hyneman, C. S. and Morgan, J. D., “Cumulative Voting in Illinois,” Illinois Law Review, Vol. 32 (1937), pp. 12–31 Google Scholar; and George S. Blair, op. cit.

¹⁰ The theoretical alternative of nominating none may be eliminated from consideration since it is never optimal and never employed.

¹¹ Op. cit., pp. 50–51.

¹² Deutsch, K. W., “Game Theory and Politics: Some Problems of Application,” Canadian Journal of Economics and Political Science, Vol. 20 (1954), pp. 76–83 CrossRefGoogle Scholar.

¹³ Actually, as will be made apparent later, the most desirable situation—although difficult to attain—would be the ability to maintain a very small, but highly reliable, difference among the candidates.

¹⁴ This is an extremely rare event, occurring only when there is a strong third party—a case not covered by the present exposition. Technically, this feature makes the game non-zero-sum, in that the gains of A plus the gains of B do not sum to the constant three for this case. In a two-party situation, however, this combination of strategies is never optimal, and, in practice, never occurs.

¹⁵ In general, the minimax loss criterion calls for an individual to determine, for each of his possible strategies, the maximum loss which he could suffer (depending upon the choice made by the other party); he then chooses that strategy which minimizes his maximum loss. Equivalently, this may be considered as maximizing one's minimum gain, referred to by Luce and Raiffa, op. cit., pp. 218–219, as maximin.

¹⁶ In the more than 1500 district elections since 1902, this has indeed been the case: with 75% of the vote, the majority party has never nominated less than two.

¹⁷ The detrimental consequences of local atrophy of party organization are suggested in Key, V. O. Jr., American State Politics (Knopf, 1956), ch. 6Google Scholar.

¹⁸ Savage, L. J., The Foundations of Statistics (Wiley, 1954), pp. 163–164 Google Scholar.

¹⁹ Illinois. Secretary of State, Blue Book of the State of Illinois (State of Illinois, 1902–1954)Google Scholar.

²⁰ Fortunately for research purposes—although in disregard of the constitution—the Illinois Assembly had failed to redistrict during this entire period.

²¹ Such non-linear utility, of course, implies a non-zero-sum game, although the more complex model resulting would necessarily describe the data somewhat better. Another possible extension of the present model would formally incorporate stochastic properties; there could be determined an expected distribution of the variable, per cent Democratic, based upon past experience, and from this could be stated the probability of any given one of the six matrices obtaining in the next election. Choice of strategy would then be based upon a composite super-game, amalgating the six matrices and their respective probabilities.

²² The relevance of the latter variable may be taken as reflecting on the committee's use of estimates of the forthcoming vote—which correlates, however, .82 with the vote of the previous election.

²³ A related theory, taking into account the differential composition of the electorate in Presidential and off-years, has been proposed in Campbell's, A. “Surge and Decline: A Study of Electoral Change,” Public Opinion Quarterly, Vol. 24 (1960), pp. 397–418 CrossRefGoogle Scholar. See also Key, V. O. Jr., “Partisanship and County Office: The Case of Ohio,” this Review, Vol. 47 (1953), p. 526 Google Scholar; and Moos, Malcolm, Presidents, Politics, and Coattails (Baltimore, 1952), pp. 13n, 110 Google Scholar.

²⁴ The possibility of coalitions of two against one would seem to make job-security-initiated pressure against an additional nomination more likely against a third candidate when the vote is 75%, than against a second candidate when the vote is 50%.

²⁵ Op. cit.

²⁶ This system has some similarity to the French provision for apparentements in their legislative elections of 1951 and 1956. But in France it was only the order of candidates among parties in an alliance, and not (for practical purposes) within parties, that was determined by the votes. American party organizations, like the French, would undoubtedly prefer to retain control of the intraparty ordering; insofar as their assent is necessary to electoral reform, some compromise in the direction of a list system might be required, although this would, however, run contrary to the presumed intention of the present direct primary law.