Two citizens must vote to elect Candidate A or Candidate B to office. Both citizens agree that Candidate A is better if the world is in state alpha and Candidate B is better if the world is in state beta. If the better candidate for the state of the world wins, then each citizen gets a payoff of 1, versus a payoff of 0 if the worse candidate wins. If there is a tie, the citizens each get a payoff of 0.5. There are only two candidates and two states.

Suppose that Citizen 1 is informed of the state, whereas Citizen 2 believes the state is alpha with probability 0.9 and beta with probability 0.1. Each citizen may vote for A, vote for B, or abstain (not vote).

a) Depict this as a Bayesian game.

b) Find the Bayesian Nash Equilibria.

i) Show that the Bayesian game has two pure-strategy Bayesian Nash equilibria, one in which 2 votes for A, and one in which 2 abstains.

ii) Show that one of the player's actions is weakly dominated in the first of these equilibria. A common equilibrium selection criterion argues that equilibria in which some player is playing a (weakly) dominated strategy are less robust, and therefore less compelling predictions of behavior, than equilibria in which all players play undominated strategies.

c) The second equilibrium in (b), that is, the one that would be chosen by the selection criterion described in (b.ii), is often called "the swing voter's curse." Why is that an appropriate description? (Ask yourself when the uninformed voter's action changes the outcome, and whether she is pleased with the effect of her action.)

d) Think about this game and the jury game we did in class. In both games, it is assumed that voters/jurors have the same underlying preferences over the candidates/verdicts, and any differences in their behavior. are a result of differences in their beliefs about the world (whether they are different ex ante, as in the model above, or different because they receive different signals during the game, as in the Jury game we did in class.) What role does that assumption play in determining the results? To answer this question, it helps to think about how the results would be different if the players had different underlying preferences. Toward that end, write a variant of the game in which there is some difference in the players preferences—perhaps one player is biased toward choosing A for reasons that have nothing to do with the state— and find the equilibrium of that game. Describe some real-world situations in which people vote to choose an outcome for which this would be a good model, and explain why it is a good model of that situation. You may find it helpful to provide an example that would not fit the model well, and explain why, as a point of contrast.

e) Commentators often lament the fact that many voters take cues from candidate endorsements from (seemingly politically engaged) elites, celebrities, or local community leaders, rather than becoming better informed about the issues themselves. Based on the analysis you completed above, do you agree or disagree with this lamentation? In what circumstances would you find it troublesome or not?

f)Voters in the U.S. have historically voted (and largely continue to vote) along party lines, i.e. individual voters tend to have allegiance to a party and vote systematically for candidates of that party, across different offices and elections. Based on the analysis you completed above, under what circumstances might this be a good thing?

g)Sometimes people choose not to vote, and instead delegate decision-making power to an individual. Based on the analysis you completed above, what are key attributes of the circumstances that determine whether or not this is a desirable course of action? What attributes would they want that individual decision-maker to have?