Exercise 1.12. Consider the following game. There is a club with three members: Ann, Bob and Carla. They have to choose which of the three is going to be president next year. Currently Ann is the president. Each member is both a candidate and a voter. Voting is as follows: each member votes for one candidate (voting for oneself is allowed); if two or more people vote for the same candidate then that person is chosen as the next president; if there is complete disagreement, in the sense that there is exactly one vote for each candidate, then the person from whom Ann voted is selected as the next president. (a) Represent this voting procedure as a game frame, indicating inside each cell of each table which candidate is elected. (b) Assume that the players' preferences are as follows: Ann Ann Carla Ann Bob, Carla Bob Bob Bob Ann, Bob Carla Ann Carla Carla. Using utility values 0, 1 and 2, convert the game frame into a game. (c) Apply the IDWDS to the game of part (b). Is there a weak iterated dominant- strategy equilibrium? (d) Does the extra power given to Ann (in the form of tie-breaking in case of complete disagreement) benefit Ann?

Match the following according to the criterion. If a candidate receives [ Choose ] more than half the first- place votes in an election, then that candidate should be declared the winner. If a candidate is favored [ Choose ] when compared separately with every other candidate in an election, then that candidate should be declared the winner. If a candidate wins an [ Choose] election and, in a reelection, the only changes are changes that favor the candidate, then that candidate should win the reelection. If a candidate wins an [ Choose ] election and, in a recount, the only changes are that one or more of the other candidates are removed from the ballot, then that candidate should still win the election.

Table 1.3: Erica and Fred's game. Fred Testify Remain silent Testify (-14, –14) (0, – 15) Erica Remain silent (-15,0) (-1, –1) What is the best global outcome, and how many years total prison time does it lead to? Both confess, 14 years in prison Both confess, 28 years in prison Both silent, 15 years in prison Both silent, 2 years in prison

Exercise 1.12. Consider the following game. There is a club with three members: Ann, Bob and Carla. They have to choose which of the three is going to be president next year. Currently Ann is the president. Each member is both a candidate and a voter. Voting is as follows: each member votes for one candidate (voting for oneself is allowed); if two or more people vote for the same candidate then that person is chosen as the next president; if there is complete disagreement, in the sense that there is exactly one vote for each candidate, then the person from whom Ann voted is selected as the next president. (a) Represent this voting procedure as a game frame, indicating inside each cell of each table which candidate is elected. (b) Assume that the players' preferences are as follows: AnnAm Carla Ann Bob, Carla Bob Ann, Bob Carla Ann Caria Carla. Using utility values 0, 1 and 2, convert the game frame into a game. (c) Apply the IDWDS to the game of part (b). Is there a weak iterated…

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)…

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How would this exercise be solved step by step?

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