¹ The previous ceteris paribus condition is quite important. We may find differences in public policy that, for example, we can attribute to legislative (e.g., coalition) differences, but not to electoral differences.

² Kramer, Gerald H., “A Decision-Theoretic Analysis of a Problem in Political Campaigning,” in Mathematical Applications in Political Science, II, Berndt, Joseph L., ed. (Dallas: Southern Methodist University Press, 1966), pp. 140–141 Google Scholar.

³ If only two candidates compete (n = 2), and ϕ _i (θ) denotes candidate i's expected plurality, the game is zero-sum and is a pure strategy minimax solution. If the game is nonzero-sum, but θ* is unique, then θ* is a Nash solution to the election game. See, for example, the discussion in Luce, R. Duncan and Raiffa, Howard, Games and Decisions: Introduction and Critical Survey (New York: Wiley, 1957)Google Scholar.

⁴ For the correspondence between spatial analysis and the theory of games, see Barr, James L. and Davis, Otto A., “An Elementary Political and Economic Theory of the Expenditures of Local Governments,” The Southern Economic Journal, 33 (October, 1966), 149–165 CrossRefGoogle Scholar; Shubik, Martin, “A Two-Party System, General Equilibrium, and the Voters' Paradox, “Zeitschrift für Nationalökonomie, 28 (1968), 341–354 CrossRefGoogle Scholar; Hinich, Melvin J., Ledyard, John O., and Ordeshook, Peter C., “A Theory of Electoral Equilibrium: A Spatial Analysis Based on the Theory of Games,” The Journal of Politics, 35 (February, 1973), 154–193 CrossRefGoogle Scholar and Ordeshook, Peter C., “Pareto Optimality and Election Competition,” American Political Science Review, 65 (December, 1971), 1141–1145 CrossRefGoogle Scholar. See also Kessel, John M., “A Game Theoretic Analysis of Campaign Strategies,” in The Electoral Process, ed. Jennings, M. Kent and Zeigler, L. Harmon (Englewood Cliffs, New Jersey: Prentice-Hall, 1966), pp. 290–304 Google Scholar.

⁵ Later in Section IV we offer a second definition of equivalence that does not require candidates to adopt equilibrium strategies. To avoid confusion, we ignore the second definition until that section.

⁶ A useful restatement of Definition 1 is: if θ* satisfies (1) for all j≠i, and are equivalen if . In the proofs of the theorems we show that two objective functions are equivalent by assuming that θ* satisfies (1) for all candidates except i, and by demonstrating that if θ* satisfies (1) with , it satisfies (1) with , and vice-versa.

⁷ Here we note simply that the condition of equilibrium we incorporate in the definition of equivalence has an internal prescriptive logic. That is, even if a candidate has no knowledge of his opponent's(') likely strategies, he might make a minimax-like calculation, especially in a competitive context, and assume that his opponents will “do their damnedest” by adopting their respective equilibria.

⁸ This is a property of most spatial analyses. The exceptions to this are in: Hinich, Ledyard, and Ordeshook; and Chapman, David, “Models of the Working of a Two-Party Electoral System,” Public Choice, 3 (Fall, 1967), 19–37, and 5 (Fall, 1968), 19–37CrossRefGoogle Scholar.

⁹ For an example of such an equilibrium, see Tullock, Gordon, Toward a Mathematics of Politics (Ann Arbor: The University of Michigan Press, 1967), pp. 53–55 Google Scholar.

¹⁰ See Hinich, Melvin J. and Ordeshook, Peter C., “Plurality Maximization vs Vote Maximization: A Spatial Analysis with Variable Participation,” American Political Science Review, 64 (September, 1970), 772–791 CrossRefGoogle Scholar, which cites examples of this ambiguity. Specifically, many political scientists treat vote, plurality, and proportion maximization as if they were equivalent.

¹¹ There are, of course, other alternative goals for candidates in proportional representation systems. Moreover, these goals are dependent upon postelection coalition rules. For a discussion of these dependencies, see Aranson, Peter H., “A Theory of the Calculus of Voting for Alternative Three-Contestant Election Systems” (Ph.D. dissertation, The University of Rochester, Rochester, New York, 1972), chap. iiiGoogle Scholar.

¹² It might seem that maximizing expected proportion of the vote is an alternative to maximizing proportion of the expected vote. While the relationship between these two objectives depends on the specific stochastic properties of V_i (θ), we show in the proof of Theorem 5 (Appendix) that with certain stochastic assumptions they approximate each other. We note also that another possible objective for candidates is maximizing plurality of vote proportion. For two candidate contests, however, this objective is equivalent to O ₂. Specifically,

(since v ₂/(v ₁ + v ₂) = 1 − v ₁/(v ₁ + v ₂)), which is linearly monotonic with v ₁/(v ₁ + v ₂).

¹³ Riker, William H., The Theory of Political Coalitions (New Haven: Yale University Press, 1962)Google Scholar.

¹⁴ Observe that if c < 1, candidate i's plurality is a monotonically increasing function of v_i (θ), but candidate j's plurality is a monotonically decreasing function of v_j (θ). This example is interesting, therefore, because with it if O ₁, and O ₃ are equivalent for candidate i, they cannot be equivalent for candidate j, and vice versa.

¹⁵ We observe, though, that O ₂ and O ₃ can be equivalent. But, this emphasizes that Theorem 1 presents a sufficient condition for equivalence, and not a necessary one.

¹⁶ One technical difficulty arises with this assumption because the dependent variables are proportions (of citizens) and, therefore, are bounded between 0 and 1. And, for values of v_i (θ), etc., near 0 and 1, the error distribution must change so as not to exceed the bounds. A simple resolution of this problem is to ignore candidates whose expected vote is small and to assume that candidates cannot adopt strategies that guarantee them nearly all of the vote. In nearly all elections, of course, the set of conceivable outcomes is considerably smaller than the set of theoretically possible outcomes. In our theoretical models, then, we can simply restrict the candidates to this set and assume that Assumption 1 is satisfied over it.

¹⁷ A utility function, U, is linear in x if U(x) = ax + b, such that a and b are constants and a is positive. Note, then, that E[U(x)] = aE[x] + b, in which case if we let a = 1 and b = 0, a person's payoff, E[U(x)], equals E(x). Hence, the most general justification for denoting a person's payoff by E(x) is to say that utility is linear in x.

¹⁸ See, for example, Riker, William H. and Zavoina, William James, “Rational Behavior in Politics: Evidence from a Three Person Game,” American Political Science Review, 64 (March, 1970), 48–60 CrossRefGoogle Scholar.

¹⁹ For a general discussion of risk acceptance and risk aversion in a different context, as well as for a spatial analysis of citizens' attitudes toward risk, see Shepsle, Kenneth A., “The Strategy of Ambiguity: Uncertainty and Electoral Competition,” American Political Science Review, 66 (June, 1972), 555–568 CrossRefGoogle Scholar.

²⁰ For a discussion of information sources and of polling by candidates, see Kingdon, John W., Candidates for Office: Beliefs and Strategies (New York: Random House, 1968), pp. 90–93 Google Scholar. See also a series of articles devoted to polling in campaigns, in Public Opinion Quarterly, 17 (Spring, 1973)Google Scholar. They include: Janowitz, Morris, “Political Polling,” 1–2 Google Scholar; Harris, Louis, “Polls and Politics in the United States,” 3–8 Google Scholar; Abrams, Mark, “Public Opinion Polls and Political Parties,” 9–18 Google Scholar; and Meadows, Martin, “Public Opinion Polls and the 1961 Philippine Election,” 19–27 Google Scholar. These references support our conjectures in a limited manner.

²¹ For a discussion of the effects on a strategy if party members and the electorate hold dissimilar preferences and goals, see Peter H. Aranson and Peter C. Ordeshook, “Spatial Strategies for Sequential Elections,” and Coleman, James, “The Positions of Political Parties in Elections,” in Probability Models of Collective Decision Making, ed. Niemi, Richard G. and Weisberg, Herbert F. (Columbus, Ohio: Charles E. Merrill, 1972), pp. 298–357 Google Scholar.

²² There is the additional possibility, given the Democractic party's proportional representation allocations of convention seats, that a Democratic presidential nomination candidate will adjust λ _i , depending on the number of his already committed delegates, his expectations for future primaries, and the numerical strength of a state delegation. We thank Edwin Fogelman for pointing out this possibility to us.

²³ Summarily, we observe that candidates who maximize according to O ₃, O ₄, or O ₅ might adopt one or another of these goals because of the information or sequential election effects we discuss previously. Here, we see the acute importance of finding nonequivalences as well as equivalences among election goals.

²⁴ See Froman, Lewis A. Jr., “A Realistic Approach to Campaign Strategies and Tactics,” in The Electoral Process, ed. Jennings, and Ziegler, , p. 2 Google Scholar.