¹ Downs, Anthony, An Economic Theory of Democracy (N.Y.: Harper & Row, 1957)Google Scholar..

² One exception to the use of this condition is Hinich, Melvin J., Ledyard, John O. and Ordeshook, Peter C., “A Theory of Electoral Equilibrium: A Spatial Analysis Based on the Theory of Games,” Journal of Politics, 35 (February, 1973), 154–193CrossRefGoogle Scholar. There, however, the restrictive assumption of continuous concave-convex payoff functions is used.

³ Briefly, a two-person zero-sum game is symmetric if the strategy sets of both players are identical, if the value of the game to both players is zero and if φ₁(x,y) = φ₂(y,x) for all x and y—i.e., if they interchange strategies, they interchange payoffs. A constant sum game, moreover, is equivalent to a zero-sum game since for any constant sum, there exists a positive monotonic transformation of the payoffs that renders the game zero-sum (e.g., substitute the payoffs, 1, 0, −1 for 1, 1/2, 0).

⁴ Plott, Charles R., “A Notion of Equilibrium and its Possibility Under Majority Rule,” American Economic Review, 57 (September, 1967), 787–806Google Scholar; and Sloss, Judith, “Stable Outcomes in Majority Rule Voting Games,” Public Choice, 15 (Summer, 1973), 19–48CrossRefGoogle Scholar.

⁵ See Davis, Otto A., DeGroot, M., and Hinich, Melvin J., “Social Preference Orderings and Majority Rule,” Econometrica, 40 (January, 1972)CrossRefGoogle Scholar; and Hoyer, R. W. and Mayer, Lawrence S., “Comparing Strategies in a Spatial Model of Electoral Competition,” American Journal of Political Science (1974)CrossRefGoogle Scholar.

⁶ Von Neumann, John and Morgenstern, Oskar, Theory of Games and Economic Behavior, 2nd ed. (Princeton: Princeton University Press, 1947)Google Scholar.

⁷ See, for example, Riker, William H. and Ordeshook, Peter C., An Introduction to Positive Political Theory (Englewood Cliffs, N.J.: Prentice-Hall, 1973), p. 340Google Scholar, and McKelvey, Richard D., “Policy Related Voting and Electoral Equilibrium,” Econometrica, 43 (September, 1975), 815–843CrossRefGoogle Scholar.

⁸ Formally, if B is a σ-algebra of sets in X, a probability measure P:B → R is a set function satisfying the usual probability axioms. I. e., for each A, B_εB, O < P(A) < 1 and A ⋂ B = Φ ⇒ P(A) + P(B) = P(A∪B). We use the notation P(x) = P({x}).

⁹ Note that Theorems 1 and 2 do not imply that there cannot exist mixed minimax strategies that give positive weight to positions not in admissible or weakly admissible sets (recall that minimax strategies need not be unique). With respect to Theorem 3, however, it is straightforward to show that such strategies cannot exist outside of the Pareto optimals, Q. Briefly, from the definition of Pareto optimality we see that the weak inequality in expression (1) holds for all zεX, while strict inequality holds for at least one zεX, namely x. Thus, any strategy, P, that gives positive weight to some position not in Q, can be strictly dominated by a strategy P′. In the case of a finite strategy space, for example, if P(y) ≠ 0 for some yεQ, we can define P′ thus,

P′(z) = P(z), zεQ, z ≠ x

P′(y) = 0

P′(x) = P(y) + P(x), xεQ, x ≻ y

By successively applying this transformation for all y Q to which P gives positive weight, we arrive at a strategy that dominates P but which gives weight only to Q. This result is established elsewhere by Ordeshook, Peter C., “Pareto Optimality in Electoral Competition,” American Political Science Review, 6 (December, 1971), 1141–1145CrossRefGoogle Scholar.

¹⁰ An important issue that Theorem 4 and its corollary fail to address is whether minimax strategy solutions exist in simple multidimensional elections. We know from the minimax theorem that if X is finite such strategies necessarily exist. From a practical perspective, moreover, this is perhaps the most reasonable context of election competition in the sense that the real constraints on candidate strategies and their perceptions of alternatives may render the game finite. For an infinite strategy space, however, establishing existence is more problematical. This is especially true in our analysis since φ is not continuous (for a proof that continuous zero-sum games over closed convex strategy spaces possess minimax solutions see Owen, Guillermo, Game Theory [Philadelphia: W. B. Saunders, 1968], pp. 76–78)Google Scholar. We conjecture, nevertheless, that the class of spatial games being considered here always possess minimax solutions.

¹¹ If the number of voters is finite and even, this assertion is not strictly true, given our definition of a median in terms of hyperplanes. Thus, if C is odd or infinite, there exists a unique median in every direction, whereas for C even, median hyperplanes are not necessarily unique. That is, for C even, a median in a particular direction can be a bounded region (e.g., for two voters on a line, the median is the set of all positions between the two ideal points). If we extend our notion of a median to bounded regions, the assertion follows. The definition of medians in terms of hyperplanes is simply a convenience for reducing notational complexity.

¹² The identification of I ^* in theory is simple in two-dimensions: identify all median lines and hence their intersections, and take the convex hull of these intersections. To generate Figure 8, however, a computer is essential to calculate and plot these median lines for the several values of λ. Identification of the weakly admissible set in Figure 5, on the other hand, is accomplished by trial and error. We note simply that a point at the center of the triangle dominates all of the excluded points within the triangle.

¹³ We note also that if we decrease the distance between the means of the two bivariate normals used to construct Figure 8 (or, equivalents, increase their variances), the site of I ^* decreases. A distance of 5σ, moreover, seems the extreme of opinion polarization.

¹⁴ The theorem is outlined in Owen, , Game Theory, p. 31, 36–37Google Scholar, and proved in Karlin, Samuel, Mathematical Methods and Theory in Games, Programming and Economics, Vol. 1 (Reading, Mass.: Addison-Wesley, 1959), pp. 179–189Google Scholar. As proved, the theorem applies, of course, only to finite games. Thus, we can only conjecture that it applies to infinite solvable games as well.

¹⁵ For a discussion of n-person majority games without side payments (the context of a spatial conceptualization of preferences) see Wilson, Robert, “A Game-Theoretic Analysis of Social Choice,” in Lieberman, B., ed., Social Choice (N.Y.: Gordon and Breach, 1971), pp. 393–408Google Scholar; and Wilson, , “Stable Coalition Proposals in Majority Rule Voting,” Journal of Economic Theory, 3 (September, 1971), 254–271CrossRefGoogle Scholar.

¹⁶ Usually, a bargaining set is defined in terms of a payoff configuration and a particular coalition structure. Here, however, exposition is eased if we consider only the payoff configuration x. Note also that payoffs are defined in game theory in terms of utility, whereas here x is a policy position. This transformation of terminology is of no consequence to our analysis owing to the 1 to 1 correspondence (given our assumptions about utility) between payoff configurations and spatial position.