Q3 Guess 5/6 of the Average Game A hundred players are participating in this game (N = 100). Each player has to choose an integer between 1 and 100 in order to guess "5/6 of the average of the responses given by all players". Each player who guesses the integer closest to the 5/6 of the average of all the responses, wins.

(a) Find all weakly dominated strategies (if any).

(b) Find all strategies that survive the Iterative Elimination of Dominated Strategies (IEDS) (if any).

Q5 "Your strength can be your weakness" A three-person committee has to choose a winner for a national art prize. After some debate, there are three candidates still under consideration: a woman who draws antelope in urban settings, a man who makes rectangular lead boxes, and a woman who sculpts charcoal. Let's call these candidates a, b and c; and call the committee members 1, 2 and 3. The preferences of the committee members are 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, if they disagree, they should vote (secret ballot, one member one vote) and that, if and only if the vote is tied, the winner will be the candidate for whom member 1 voted. Thus, it might seem that member 1 has an advantage.

(a) Consider this voting game. Each voter has three strategies: a, b, or c. For each voter, which strategies are weakly or strictly dominated? [Hints. Be especially careful in the case of voter 1. To answer this question, you do not need to know the exact payoffs: any payoffs will do provided that they are consistent with the preference orders given above. To answer this question, you do not have to write out matrices].

(b) Now consider the reduced game in which all weakly and strictly dominated strategies have been removed. For each voter, which strategies are now weakly or strictly dominated? What is the predicted outcome of the vote? Compare this outcome to voter 1's preferences and comment.

Hint: When answering this question, it is useful to construct the best responses for each player. For player 1, the best response function will look like: s2 a a a b b b c c c s3 a b c a b c a b c BR1(s2, s3) [a,b,c] a ... ... ... ... ... ... ... For example, if Player 2 and Player 3 both chose a, then a is the winner no matter what Player 1 does, its best response is to chose anything. In contrast, if Player 2 votes for a and Player 3 votes for b, then Player 1 is better off by voting for a (voting for b or c is strictly worse for Player 1)