EXERCISE 74.1 (Electoral competition for more general preferences) There is a fi- nite 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 pref- erences 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 posi- tions, 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.
EXERCISE 74.1 (Electoral competition for more general preferences) There is a fi- nite 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 pref- erences 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 posi- tions, 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.
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Transcribed Image Text:EXERCISE 74.1 (Electoral competition for more general preferences) There is a fi-
nite 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 pref-
erences 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 posi-
tions, 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.
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