¹ “Voting, or a Price System in a Competitive Market Structure,” American Political Science Review, 64 (March, 1970), 179–181CrossRefGoogle Scholar.

² For a spatial analysis of risky strategies, see Kenneth Shepsle, “Parties, Voters, and the Risk Environment,” in Richard G. Niemi and Herbert F. Weisberg, eds., Probability Models of Collective Decision-Making (forthcoming). Pure strategies are a special case of risky ones: P(θ) = 1 for one value of θ and equals zero otherwise. It is useful, however, to maintain the distinction in this essay because of the apparent confusion between risky and mixed strategies.

³ An implicit third assumption is that if a citizen is indifferent between the candidates, he either abstains from voting or chooses randomly by tossing a fair coin.

⁴ If S is a non-Pareto optimal strategy (i.e., a proposed non-Pareto optimal state of society or probability distribution over states of society) then S′ is a Pareto optimal complement to S if (1) the utility or expected utility of S′ is greater than or equal to the utility or expected utility of S for all citizens, (2), the utility or expected utility of S′ is strictly greater than the utility or expected utility of S for at least one citizen, and (3), S′ is Pareto optimal. We can readily prove now that for either (iii) or (ii) in conjunction with our other assumptions, if S′ is put against S with the majority voting procedure in use, S′ defeats S. (Parenthetically, we must note that while we do not impose (i) with (iii), we assume throughout this analysis that if a candidate can adopt S, he can adopt at least one Pareto optimal complement to S.)

⁵ Melvin J. Hinich, John O. Ledyard, and Peter C. Ordeshook, “Non-Voting and the Existence of Equilibrium Under Majority Rule,” Journal of Economic Theory (forthcoming); “A Theory of Electoral Equilibbrium: A Spatial Analysis Based on the Theory of Games,” (Carnegie-Mellon University, SUPA working paper, 1970)Google Scholar, and; Hinich, Melvin J. and Ordeshook, Peter C., “Transitive Social Preference and Majority Rule Equilibrium with Separable Probabilistic Choice Functions” (Carnegie-Mellon University, SUPA working paper, 1971)Google Scholar.

⁶ Shubik suggests in his interpretation of a mixed strategy that “we may view the mixed strategy as being a ‘degree of belief’ in the mind of the voter” (p. 180). Note, however, that if voters are risk averse, there is no incentive for the candidates to have the electorate interpret strategies as risky. And it is with risk averse voters that Shubik attempts to construct a counter-example to an efficient election.

⁷ Assuming that there are only a finite number of admissable strategies rather than an infinite number (e.g., θ is a continuous variable) simplifies notation but does not affect our conclusion.

⁸ In “A Two-Party System, General Equilibrium and the Voter's Paradox,” Zeitschrift für Nationalökonomie, 28 (1968), 348–349Google Scholar, Shubik concludes, as we do, that the answer to this question for symmetric games is no. He does not, however, provide a general proof of his assertion.

⁹ Assuming that θ is finite does not affect our analysis, except that it guarantees that equilibrium strategies exist. If, however, θ is infinite or uncountable, some additional mathematical constraints must be imposed on the candidates' payoff functions before the existence of strategies that satisfy (2) is guaranteed. (See, for example, Owen, Guillermo, Game Theory [Philadelphia: W. B. Saunders, 1968], pp. 110–112)Google Scholar. Since little can be said about the properties of the strategies candidates adopt if equilibria do not exist, we assume that they do exist.

¹⁰ Our notation implies that the second candidate's equilibrium strategy is not a risky strategy. This notation is adopted for convenience, however, and our results follow if we substitute a risky strategy for q*.

¹¹ We might, of course, assume simply that for future elections, citizens do not know the particular pure strategies candidates will adopt. Hence, instead of assuming that pure strategies are selected according to the minimax criterion, we could define any probability function, say f, over θ. To conduct any analysis, however, we must say something about f, and so we let f be the minimax function.

¹² The expected utility denoted by expression (9) is less than the utility of the market allocation in Shubik's example. If public goods rather than private goods are being allocated, however, markets generally, are not Pareto optimal, in which case we cannot offer any conjectures as to the relative value of (9)—even if citizens are risk averse. Also, it should be emphasized that this analysis assumes that mixed strategies are in equilibrium. If pure or risky strategies are in equilibrium, the corresponding value of (9) will equal or exceed the utility of a market allocation. The extent to which pure or risky strategies are minimax in electoral competition, and the extent to which the approximate nature of these strategies is knowable, then, is an important variable in any normative analysis.