Q1. There are four players: two women, Ann and Beth, and two men, Charlie and Dan. They independently (without any communication) decide whether to go to the bar downtown (D) or the bar on the riverside (It). They all have the same preferences, which are as follows: the best outcome is to be with just one other person, of the opposite sex; the second best outcome is to be with two other people, both of the opposite sex; the third best outcome is to be with everybody else; the fourth best outcome is to be with two other people, one of the same sex and the other of the opposite sex; the fifth best outcome is to be with just one other person, of the same sex; the worst outcome is to be alone in the bar.
(a) Using utility functions with values from the set {0, 1, 2, ... , 5}, represent this game by means of a set of tables. [Hint: you need four tables; have player 4 choose the top part or the bottom part of the sheet of paper.]
(b) Find all the Nash equilibria.
(c) Are there any Nash equilibria that are Pareto dominated?
Q2. There are four candidates: Andrea, Bob, Chris and Dan. Three voters, 1, 2 and 3 must choose one of the candidates. The voting procedure is organized as follows. First voter 1 vetoes (= eliminates) one of the candidates, then player 2 vetoes one of the remaining candidates and finally voter 3 vetoes one of the remaining candidates. The winning candidate is the one who was not vetoed by anybody. It is common knowledge among the voters that their ranking of the candidates is as follows:
|
Voter 1
|
Voter 2
|
Voter 3
|
|
first (= best)
|
Andrea
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Bob
|
Dan
|
|
second
|
Bob
|
Andrea
|
Bob
|
|
third
|
Chris
|
Dan
|
Andrea
|
|
fourth (= worst)
|
Dan
|
Chris
|
Chris
|
(a) Draw an extensive game of perfect information that represents this voting procedure. You don't need to write the payoffs: just show the choices and the final outcomes (you can write payoffs if you like).
(b) Find all the backward-induction solutions.