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Problem Set 4 – Solutions
Exercise 1
Problem Set 4 - Solutions Exercise 1 Swing voter's curse a. The Bayesian game is defined as follows. Players Citizens 1 and 2. States {A,B}. Actions The set of actions of each player is {0,1,2} (where 0 means do not vote). Signals Citizen 1 receives different signals in states A and B, whereas cili- ven 2 receives the same signal in both states. Beliefs Each type of citizen 1 assigns probability 1 to the single state consis- tent with her signal. The single type of citizen 2 assigns probability ().9 to state A and probability 0.1 to state B. Payoffs Both citizens’ Bernoulli payoffs are 1 if either the state is A and can- didate 1 receives the most votes or the state is B and candidate 2 receives the most votes; their payofts are 0 if cither the state is B and candidate 1 receives the most votes or the state is A and candidate 2 receives the most votes; and otherwise their payoffs are 3. (These payoffs are shown in Figure 149.1.) 0 1 2 0 1 2 11 11 0) 44 1,1 | 0,0 o dh) 00) 41 11 11 1°11 = 115 3,4 10,0 0,0 5,5 11 11 20,0 | 5,5 | 0,0 2 911 535 > 21,1 State A State B Figure 149.1 ‘The payofls in the Bayesian game for Exercise 307.1. b, Type A of player 1’s best action depends only on the action of player 2; it is to vole for 1 if player 2 votes for 2 or does not vote, and either to vote for 1 or not vote if player 2 votes for 1. Similarly, type B of player 1’s best action is to vote for 2 if player 2 votes for 1 or does not vote, and either to vote for 2 or not vote if player 2 votes for 2. Player 2’s best action is to vote for 1 if type A of player 1 either does not vote or votes for 2 (regardless of how type B of player 1 votes), not to vote if type A of player 1 votes for 1 and type B of player 1 either votes for 2 or does not vote, and either to vote for 1 or not to vote if both types of player 1 vote for 1. 
