^* I wish to thank Irwin Mann for stimulating my interest in the subject of this paper and Gordon W. Sensiba for research assistance.

¹ On the role of the representative, see, among other sources, Eulau, Heinz, “Changing Views of Representation,” in Pool, Ithiel de Sola (ed.), Contemporary Political Science: Toward Empirical Theory (New York: McGraw-Hill Book Company, Inc., 1967), pp. 53–85 Google Scholar; and Pitkin, Hanna Fenichel (ed.), Representation (New York: Atherton Press, 1969).Google Scholar

² This requirement is not satisfied by “list systems” of proportional representation in which the individual voter has no opportunity to express preferences for particular candidates of a party. Allowing the expression of intensities in a one-shot affair (e.g., election) is equivalent to what Buchanan and Tullock call “logrolling” – wherein a voter in a one-man, one-vote context trades his votes on measures where his preferences are weak for those of other voters on measurs where his preferences are strong – over a continuing sequence of votes. See Buchanan, James M. and Tullock, Gordon, The Calculus of Consent: Logical Foundations of Constitutional Democracy (Ann Arbor, Michigan: University of Michigan Press, 1962), chpt. 10.Google Scholar It also may be argued that non-participation in an election is one means, albeit a very limited one, that an individual has for expressing his intensities of preference — or lack thereof. For a general discussion of the “intensity” problem in democratic theory, see Kendall, Willmore and Carey, George W., “The ‘Intensity’ Problem and Democratic Theory,” American Political Science Review, 62 (March 1968), 5–24.CrossRefGoogle Scholar

³ For a discussion and critique, see Black, Duncan, The Theory of Committees and Elections (Cambridge, England: Cambridge University Press, 1958), pp. 59–66, 157–159, 180–183Google Scholar; also, Grazia, Alfred De, “Mathematical Derivation of an Election System,” ISIS, 44 (June 1953), 42–51.CrossRefGoogle Scholar The Hare system involving the single transferable vote also requires that the voter rank-order all candidates; if his first-choice candidate does not receive a “quota,” the lowest-ranking candidate is dropped and his second choices are allocated to the remaining candidates, with the process terminating when all seats are filled by candidates who have received a quota. Thus, second choices may be used to express intensities of preference in a limited way, but only for those voting for candidates dropped because they rank lowest in first choices.

⁴ Evidence for the “cube law” is summarized in March, James G., “Party Legislative Representation as a Function of Election Results,” in Lazarsfeld, Paul F. and Henry, Neil W. (eds.), Readings in Mathematical Social Science (Cambridge, Mass.: MIT Press, 1968), pp. 220–241.Google Scholar

⁵ In game-theoretic terms – a formulation we shall give later – the latter strategy (majority supports five) dominates the former (majority supports four), though both strategies yield the same payoff (four seats) against the optimal strategy of the minority (support two candidates). On the other hand, while not offering any opportunity to capitalize on the possibile mistakes of the minority, the majority's strategy of supporting only four is “safer” in the sense that it insures as small as a 58 percent majority with four seats while the majority's strategy of supporting five does not insure less than a 63 percent majority with four seats. For further details, see Appendix.

⁶ This was apparently first recognized by James Garth Marshall in a pamphlet entitled “Majorities and Minorities: Their Relative Rights” (1853). See Black, Duncan, “Lewis Carroll and the Theory of Games,” American Economic Review: Papers and Proceedings ol the Eighty-first Annual Meeting of the AEA, 59 (May 1969), 207.Google Scholar

⁷ Because only a small minority could insure itself of the election of at least one seat, one effect of the proposed voting scheme might be to encourage the formation of several minority or splinter factions, whose influence would perhaps be less a function of their numerical size and more depend on their holding the balance of power between major factions in the elected body. For models of the dynamics of coalition-formation processes in voting bodies, with an empirical application to voting in multi-ballot U.S. national party conventions, see Steven J. Brams and William H. Riker, “Models of Coalition Formation in Voting Bodies” (paper to be delivered at the Sixty-sixth Annual Meeting of the American Political Science Association, Los Angeles, September 8–12, 1970, and published in Herndon, James F. (ed.), Mathematical Applications in Political Science, VI [Charlottesville, Virginia: University Press of Virginia, forthcoming 1971])Google Scholar; also, Brams, Steven J., “A Cost/Benefit Analysis of Coalition Formation in Voting Bodies,” in Niemi, Richard G. and Weisberg, Herbert F. (eds.), Probability Models ol Collective Decision-Making (Columbus, Ohio: Charles E. Merrill Publishing Company, forthcoming 1971)Google Scholar, and Brams, Steven J., “An Equilibrium Model of Coalition Formation in Voting Bodies” (unpublished paper, New York University, August 1970).Google Scholar

⁸ This was first demonstrated by Glasser, Gerald J. in “Game Theory and Cumulative Voting for Corporate Directors,” Management Science, 5 (Jan. 1959), 151–156.CrossRefGoogle Scholar

⁹ The extent to which political parties are rational according to the minimax criterion has been tested empirically for the one state legislature (Illinois General Assembly) which provides for “cumulative voting” – that is, voting in which members of the electorate in multimember constituencies are allowed to “cumulate” their votes on fewer than the number of candidates to be elected (as is proposed here, but for the entire electorate and not individual constituencies). In this case, both the Republican and Democratic parties employed optimal strategies in 69 percent of the 1,337 biennial elections in Illinois's 51 districts between 1902 and 1954, where “optimal” is defined as in the text (except for the majority in the 60+ to 75– percent range, as explained in footnote 26). See Sawyer, Jack and MacRae, Duncan Jr.,“Game Theory and Cumulative Voting in Illinois: 1902–1954,” American Political Science Review, 56 (Dec. 1962), 936–946.CrossRefGoogle Scholar

¹⁰ See his “The Central Argument in Lewis Carroll's The Principles of Parliamentary Representation,” Papers on Non-Market Decision Making (now Public Choice), 3 (Fall 1967), 1–17; and “Lewis Carroll and the Theory of Games,” op. cit., 206–210.

¹¹ See Kemeny, John G., Laurie Snell, J., and Thompson, Gerald L., Introduction to Finite Mathematics (2d ed.; Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1966), pp. 41–42.Google Scholar

¹² We say “in general,” because under special circumstances voting rules that do not allow for the full expression of intensities may still be able to insure proportionate representation. As we indicated in the text in the case of our example, if each voter can cast one vote for one candidate, or two votes for two candidates, the majority and minority can instruct their supporters to distribute their votes egually among five and two candidates, respectively, and the majority can insure the election of four of its five candidates and the minority both of its two candidates. The minority could not accomplish this feat, however, if each voter had three (or four or five or six) votes and could give no more than one vote to a candidate, for if each voter had, say, three votes, the minority could give at most votes to each of (three) candidates but the majority could give votes to each of five candidates and thus capture five of the six seats.

¹³ Strictly speaking, allowing for the expression of intensities may not permit a group to insure its exact proportionate representation. For example, if each person in the electorate is allowed to cast a total of five votes for up to five candidates for five seats, a one-third minority could insure itself of one seat (20 percent representation), and a two-thirds majority of three seats (60 percent representation), but neither side could insure itself of greater representation. Indeed, if the one-third minority distributed its votes equally between two (instead of one) candidates, and the two-thirds majority among four (instead of three) other candidates, all six candidates – for the five seats – would receive the same number of votes! (In reality, especially for a large electorate, the possibility of such a deadlock's occurring is so remote that the electoral rules usually do not even prescribe how it would be resolved, should it occur.) Exact proportionate representation can always in principle be approximated to a finer and finer degree by increasing the total number of seats (and votes of an individual) up to the number of voters in the electorate. At this extreme, exact proportionate representation of a group – or even an individual – is achievable because the elected body is as large as the electorate! For further numerical examples, see Appendix. Practically speaking, however, the number of seats (and votes of an individual) need not be unduly large since most groups, having only a rough idea of their degree of support among the electorate prior to an election, cannot make very exact calculations for the purpose of pursuing an optimal strategy, anyway. Furthermore, as shown in the Appendix, the “representativeness” of a body, which increases at a decreasing rate with more members, is quite high for bodies of even moderate size. The major goal in the design of any system of proportional representation is to allow the cleavages produced by groups expressing different viewpoints in the electorate to be reflected approximately in the division of seats in the elected body, and this our proposed voting scheme does. For a concise review of schemes of proportional representation extant today in Western democracies, see Rae, Douglas W., The Political Consequences of Electoral Laws (New Haven, Conn.: Yale University Press, 1967), pp. 28–39 Google Scholar; also, Milnor, Andrew, Elections and Political Stability (Boston: Little, Brown and Company, 1969), chpt. 4.Google Scholar The extent to which systems of proportional representation – as compared with plurality-majority systems – mitigate in practice the disproportionate advantage of large parties (or groups) is shown in Rae, supra, chpt. 5.

¹⁴ One of the Caucus's eight original candidates for a two-year Council seat (Tobe Johnson) also withdrew but remained a candidate (the only one, after a withdrawal from the one-year Council race) for the single one-year Council seat. Thus, both the Caucus and the Representative Slate fielded only six candidates for the seven contested two-year Council seats. See PS, 3 (Winter 1970), 31–33.

¹⁵ John E. Mueller's analysis of (he election ballots in this issue of PS, “The Political Scientist Decides: An Examination of the 1969 APSA Ballots,” supports this conclusion: “out of the entire collection of 7864 ballots only 14 were marked exactly according to the wishes of the APSA Nominating Committee … there is a strong suggestion that the influence of the APSA Nominating Committee on the vote was minimal.” Just prior to the election, in a letter sent to the entire APSA membership, Donald G. Herzberg, representing the Representative Slate, argued that “it would obviously have been absurd [for the Nominating Committee of the APSA] to award the Caucus – which has never claimed more than a few hundred of 13,000 individual members – two of eight nominees [Henry Kariel and Lewis Lipsitz] for the Council seats,” Before the election, probably not even most Caucus members would have wagered that “two of eight” would actually underestimate their electoral strength! The Herzberg letter is reprinted in PS, 2 (Fall 1969), 703–704.

¹⁶ The Caucus candidates which the Association did support, however, garnered an average of 1,167 more votes than Caucus candidates it did not support; on the other hand, the Representative Slate candidates not endorsed by the Association actually received an average of 102 votes more than those endorsed by the Association. Taken at face value, Association endorsement helped considerably Caucus candidates (but not, except for Karlel, by enough to win) and hurt slightly Representative Slate candidates.

¹⁷ The analysis of the APSA election as an n-person, zero-sum game would require us to distinguish between “essential” and “inessential” elections (see Glasser, op. cit., 155–156), but the aggregate election results provide us with insufficient information on the electoral support of the four contestants, and possible coalitions among them, to undertake this analysis. This might, however, be attempted with the ballot data analyzed by Mueller, op. cit.

¹⁸ Prior to the election, of course, Caucus leaders did not know how much support they could count on among the membership, but the results of the 1969 election should now make their planning easier, as well as the planning of leaders of the Representative Slate or a group with similar views.

¹⁹ Unless, perhaps, the small minority holds the balance of power, as explained in footnote 7. For a discussion of the relationship between the number of voters in an electoral district (in our terms, of a group) and their opportunities to be pivotal in a voting body through their representatives, see Riker, William H. and Shapley, Lloyd S., “Weighted Voting: A Mathematical Analysis for Instrumental Judgments,” in Roland Pennock, J. and Chapman, John W. (eds.), Representation: Nomos X (New York: Atherton Press, 1969), pp. 199–216 Google Scholar: and Banzhaf, John F. III, “Multimember Electoral Districts – Do They Violate the ‘One Man, One Vote’ Principle?”, Yale Law Journal, 75 (July 1966), 1309–1338.CrossRefGoogle Scholar

²⁰ If the majority is indeed “benevolent,” or operates on a concept of noblesse oblige, it may reach an a priori compromise with opposition groups by offering them “side payments” (e.g., some of their own candidates on a compromise slate), which would not necessitate their running opposition slates. It is precisely the proposed rules for insuring proportionate representation to which opposition groups might resort, however, which would give these groups the bargaining power necessary to negotiate the composition of a compromise slate in line with their proportionate electoral support.

²¹ This figure is a maximum value, in contradistinction to Black's (Carroll's) “mathematical expectation of the percentage of the voters represented” (or unrepresented), which is a function of both the percentages of the electorate unrepresented (based on the differences between the minimum percentages necessary to elect n-1 versus n members, for all n seats on the body) and the probability distribution of party (or group) preferences. See Black, “The Central Argument in Lewis Carroll's The Principles of Parliamentary Representation,” op. cit., 10ff.

²² Cf. these theoretical results with Rae's (op. cit., pp. 117ff.) similar empirical findings for Western democracies.

²³ For a real-life example where the choice of a risky strategy by a majority group led to its losing control of the board of directors in a corporate election based on cumulative voting, see Glasser, op. cit., 154–155.

²⁴ For a procedure for incorporating subjective probabilities in one's decision analysis using Bayesian methods, see Ralffa, Howard, Decision Analysis: Introductory Lectures on Choices under Uncertainty (Reading, Mass.: Addison-Wesley, 1968)Google Scholar; Morgan, Bruce W., An introduction to Bayesian Statistical Decision Processes (Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1968)Google Scholar; and Harsanyi, J.C., “Games with Incomplete Information Played by ‘Bayesian’ Players,” Part I, Management Science, 14 (Nov. 1967), 159–189 CrossRefGoogle Scholar; Part II, ibid. (Jan. 1968), 320–334; Part III, ibid. (March 1968), 486–502.

²⁵ In general, a group supported by n voters (out of N in the electorate) can pursue a safe strategy that insures the election of K candidates for m seats (k<m) whenever a group can pursue a dominant strategy of running [k+1] candidates ([k+1] < m), and insure the election of k, whenever and a group can pursue a new-safe strategy that insures the election of [k+1] candidates whenever For a given n, the maximum value of k which satisfies each of these inequalities determines the optimal strategy for a group, except when the same maximum value satisfies the inequalities for both the safe and dominant strategies; in that case, the dominant strategy of running an additional candidate (i.e., [k+1] instead of k) is to be preferred (subject to the qualifications given in the text). Of course, when the same maximum value of k satisfies the inequalities for both the dominant and new-safe strategies, there is no necessity to choose between these two strategies since both involve running the same number of candidates (i.e., [k+1]).

²⁶ This is why Sawyer and MacRae argue that for a party with a 60+ to 75− percent majority, the “expected number” (which they imply to be the optimal number) of candidates it would nominate for three seats would be two rather than three, even though it could pursue a dominant strategy of nominating three candidates and insuring the election of two. See Sawyer and MacRae, op. cit., 939.

²⁷ Some strategic consequences of different voting procedures and types of voting in situations where preference scales of outcomes can be defined for all voters – as contrasted with our analysis, where the object is not for an individual to obtain a preferred outcome but for a group to maximize (or more accurately, minimax) its representation on a voting body – are masterfully developed in Farquharson, Robin, Theory ol Voting (New Haven, Conn.: Yale University Press, 1969).Google Scholar

²⁸ The concepts of “saddle point” and “pure” and “mixed” strategies are defined and illustrated in Rapoport, Anatol, Two-Person Game Theory: The Essential Ideas (Ann Arbor, Michigan: University of Michigan Press, 1966)Google Scholar, chpts, 5 and 6; also, Kemeny, Snell, and Thompson, op. cit., chpt. 6.