(Electoral competition for more general preferences) There is a finite number of positions and a…

(Electoral competition for more general preferences) There is a finite number of positions and a finite, odd, number of voters. For any positions x and y, each voter either prefers x to y or prefers y to x. (No voter regards any two positions as equally desirable.) We say that a position x∗ is a Condorcet winner if for every position y different from x∗, a majority of voters prefer x∗ to y.

a. Show that for any configuration of preferences there is at most one Condorcet winner.

b. Give an example in which no Condorcet winner exists. (Suppose there are three positions (x, y, and z) and three voters. Assume that voter 1 prefers x to y to z. Construct preferences for the other two voters such that one voter prefers x to y and the other prefers y to x, one prefers x to z and the other prefers z to x, and one prefers y to z and the other prefers z to y. The preferences you construct must, of course, satisfy the condition that a voter who prefers a to b and b to c also prefers a to c, where a, b, and c are any positions.)

c. Consider the strategic game in which two candidates simultaneously choose positions, as in Hotelling’s model. If the candidates choose different positions, each voter endorses the candidate whose position she prefers, and the candidate who receives the most votes wins. If the candidates choose the same position, they tie. Show that this game has a unique Nash equilibrium if the voters’ preferences are such that there is a Condorcet winner, and has no Nash equilibrium if the voters’ preferences are such that there is no Condorcet winner. A variant of Hotelling’s model of electoral competition can be used to analyze the choices of product characteristics by competing firms in situations in which price is not a significant variable. (Think of radio stations that offer different styles of music, for example.) The set of positions is the range of possible characteristics for the product, and the citizens are consumers rather than voters. Consumers’ tastes differ; each consumer buys (at a fixed price, possibly zero) one unit of the product she likes best. The model differs substantially from Hotelling’s model of electoral competition in that each firm’s objective is to maximize its market share, rather than to obtain a market share larger than that of any other firm. In the next exercise you are asked to show that the Nash equilibria of this game in the case of two or three firms are the same as those in Hotelling’s model of electoral competition.