Ex. 4 — Strength Can Be Weakness A three-person committee has to choose a winner for a prize. After some debate, there are three candidates still under consideration. Let's call these candidates a, b and c, and call those committee members 1, 2 and 3. The committee members only care about which candidate wins the prize, and their preferences as follows: member 1 prefers a to b and b to c; member 2 prefers c to a and a to b; and member 3 prefers b to c and c to a. The rules of the competition say that the committee should first apply majority vote (secret ballot, one member one vote) and the candidate with the most votes wins. If the vote is tied, that is, the majority rule select a unique winning candidate, then the winner will be the candidate for whom member 1 voted. Thus, it might seem that member 1 has an advantage. (1) Write down the strategic form of this voting game. [You may assign any number to the payoff of each voter, as long as it is consistent with her preference order.] (2) For each voter, which strategies are weakly or strictly dominated? [You may want to pay extra attention to voter 1.] (3) Now consider the reduced game in which all weakly and strictly dominated strate- gies have been removed from the original game. For each voter, which strategies are now weakly or strictly dominated? What is the outcome predicted by the process of iterated elimination of dominated strategies? Compare this outcome to voter 1's preferences and comment.
Ex. 4 — Strength Can Be Weakness A three-person committee has to choose a winner for a prize. After some debate, there are three candidates still under consideration. Let's call these candidates a, b and c, and call those committee members 1, 2 and 3. The committee members only care about which candidate wins the prize, and their preferences as follows: member 1 prefers a to b and b to c; member 2 prefers c to a and a to b; and member 3 prefers b to c and c to a. The rules of the competition say that the committee should first apply majority vote (secret ballot, one member one vote) and the candidate with the most votes wins. If the vote is tied, that is, the majority rule select a unique winning candidate, then the winner will be the candidate for whom member 1 voted. Thus, it might seem that member 1 has an advantage. (1) Write down the strategic form of this voting game. [You may assign any number to the payoff of each voter, as long as it is consistent with her preference order.] (2) For each voter, which strategies are weakly or strictly dominated? [You may want to pay extra attention to voter 1.] (3) Now consider the reduced game in which all weakly and strictly dominated strate- gies have been removed from the original game. For each voter, which strategies are now weakly or strictly dominated? What is the outcome predicted by the process of iterated elimination of dominated strategies? Compare this outcome to voter 1's preferences and comment.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Ex. 4 Strength Can Be Weakness
A three-person committee has to choose a winner for a prize. After some debate, there
are three candidates still under consideration. Let's call these candidates a, b and c, and
call those committee members 1, 2 and 3. The committee members only care about which
candidate wins the prize, and their preferences as follows: member 1 prefers a to b and
b to c; member 2 prefers c to a and a to b; and member 3 prefers b to c and c to a. The
rules of the competition say that the committee should first apply majority vote (secret
ballot, one member one vote) and the candidate with the most votes wins. If the vote is
tied, that is, the majority rule select a unique winning candidate, then the winner will
be the candidate for whom member 1 voted. Thus, it might seem that member 1 has an
advantage.
(1) Write down the strategic form of this voting game. [You may assign any number
to the payoff of each voter, as long as it is consistent with her preference order.]
(2) For each voter, which strategies are weakly or strictly dominated? [You may want
to pay extra attention to voter 1.]
(3) Now consider the reduced game in which all weakly and strictly dominated strate-
gies have been removed from the original game. For each voter, which strategies are
now weakly or strictly dominated? What is the outcome predicted by the process
of iterated elimination of dominated strategies? Compare this outcome to voter 1's
preferences and comment.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc37afede-b4e9-4127-8995-f197f16093ce%2Fdf3df4b1-4a6b-4c53-8414-20f0e87851db%2Fzj57vse_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Ex. 4 Strength Can Be Weakness
A three-person committee has to choose a winner for a prize. After some debate, there
are three candidates still under consideration. Let's call these candidates a, b and c, and
call those committee members 1, 2 and 3. The committee members only care about which
candidate wins the prize, and their preferences as follows: member 1 prefers a to b and
b to c; member 2 prefers c to a and a to b; and member 3 prefers b to c and c to a. The
rules of the competition say that the committee should first apply majority vote (secret
ballot, one member one vote) and the candidate with the most votes wins. If the vote is
tied, that is, the majority rule select a unique winning candidate, then the winner will
be the candidate for whom member 1 voted. Thus, it might seem that member 1 has an
advantage.
(1) Write down the strategic form of this voting game. [You may assign any number
to the payoff of each voter, as long as it is consistent with her preference order.]
(2) For each voter, which strategies are weakly or strictly dominated? [You may want
to pay extra attention to voter 1.]
(3) Now consider the reduced game in which all weakly and strictly dominated strate-
gies have been removed from the original game. For each voter, which strategies are
now weakly or strictly dominated? What is the outcome predicted by the process
of iterated elimination of dominated strategies? Compare this outcome to voter 1's
preferences and comment.
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