Homework Help is Here – Start Your Trial Now!
Business
Economics
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.

Principles of Microeconomics
Principles of Microeconomics
7th Edition
ISBN:9781305156050
Author:N. Gregory Mankiw
Publisher:N. Gregory Mankiw
Chapter22: Frontiers Of Microeconomics
Section: Chapter Questions
Problem 7PA
See similar textbooks
icon
Related questions
Question

PLEASE CHECK THIS  HOW TO SOLVE  PLEASE TEACH  EXPLAIN STEP BY STEP 

 

 

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.
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.
Expert Solution
steps

Step by step

Solved in 5 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Paradox Of Voting
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
Principles of Microeconomics
Principles of Microeconomics
Economics
ISBN:
9781305156050
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Principles of Microeconomics (MindTap Course List)
Principles of Microeconomics (MindTap Course List)
Economics
ISBN:
9781305971493
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Principles of Economics (MindTap Course List)
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Principles of Economics, 7th Edition (MindTap Cou…
Principles of Economics, 7th Edition (MindTap Cou…
Economics
ISBN:
9781285165875
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Microeconomic Theory
Microeconomic Theory
Economics
ISBN:
9781337517942
Author:
NICHOLSON
Publisher:
Cengage
Economics (MindTap Course List)
Economics (MindTap Course List)
Economics
ISBN:
9781337617383
Author:
Roger A. Arnold
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS