Question

Whether candidate 1 or candidate 2 is elected depends on the votes of two citizens. The economy may be in one of two states, A or B. The citizens agree that candidate 1 is best if the state is A and candidate 2 is best if the state is B. Each citizen's preferences are represented by the expected value of a Bernoulli payoff function that assigns a payoff of 1 if the best candidate for the state wins (obtains more votes than the other candidate), a payoff of 0 if the other candidate wins, and payoff of 1/2 if the candidates tie. Citizen 1 is informed of the state, whereas citizen 2 believes it is A with probability 0.9 and B with probability 0.1 Each citizen may either vote for candidate 1, vote for candidate 2, or not vote.

a. Construct the table of payoffs for each state of the world and draw the rectangles to get a diagram that represents this game (Each player has three actions: vote for candidate 1, vote for candidate 2, or abstain).
b. Show that the game has exactly two pure strategy Bayesian Nash equilibria, in one of which citizen 2 does not vote and in the other she votes for 1.
c. Why is "swing voter's curse" an appropriate name for the determinant of citizen 2's decision in the equilibrium where he abstains?

Answer 1

Consider the following explanation

Step-by-step explanation:

Answer to Q (a)

Players: Citizens 1 and 2 along with 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 citizen 2 receives the same signal in both states.

Beliefs:

Each type of citizen 1 assigns probability 1 to the single state consistent with her signal.

The single type of citizen 2 assigns probability 0.9 to state A and probability 0.1 to state B.

Payoffs:

Both citizens Bernoulli payoffs are 1 if either the state is A and candidate 1 receives the most votes or the state is B and candidate 2 receives the most votes;

Their payoffs are 0 if either the state is B and candidate 1 receives the most votes or the state is A and candidate 2 receives the most votes;

Otherwise their payoffs are ½

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Answer to (b)

State A of citizen 1’s best action depends only on the action of citizen 2;

=> To vote for candidate 1 if citizen 2 votes for candidate 2 or does not vote, and either to vote for candidate 1 or not vote if citizen 2 votes for candidate 1.

Similarly, for state B of citizen 1’s best action is to vote for candidate 2 if citizen 2 votes for candidate 1 or does not vote, and either to vote for candidate 2 or not vote if citizen 2 votes for candidate 2.

Citizen 2’s best action is to vote for candidate 1 if state A -> citizen 1 either does not vote or votes for candidate 2 (regardless of how state B -> citizen 1 votes), not to vote if state A of citizen 1 votes for candidate 1 and state B of citizen 1 either votes for candidate 2 or does not vote, and either to vote for candidate 1 or not to vote if both types of citizen 1 vote for candidate 1.

Given the best responses of the two types of citizen 1, their only possible equilibrium actions are

(0, 0) (I.e. both do not vote), (0, 2), (1, 0), and (1, 2).

Checking citizen 2’s best responses we see that the only equilibria are

(i)==> (0, 2, 1) (Citizen 1 does not vote in state A and votes for candidate 2 in state B; Citizen 2 votes for candidate 1)

(ii)==> (1, 2, 0) (Citizen 1 votes for candidate 1 in state A and for candidate 2 in state B; citizen 2 does not vote).

(ii)

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Answer to (c)

Swing Voter’s curse:

Swing voter is an important player whose decision can affect the result of the election. Swing voters curse results into equilibrium as shown below.

In the equilibrium (1, 2, 0), Citizen 2 does not vote. This is because if she votes, then in the only case in which her vote affects the outcome (i.e. the only case in which she is a swing voter), it affects it adversely as shown below:

i. If she votes for candidate 1 then her vote makes no difference in state A, whereas it causes a tie, instead of a win for candidate 2 in state B

ii. If she votes for candidate 2 then her vote causes a tie, instead of a win for candidate 1 in state A, and makes no difference in state B.