Problem 4 Suppose a village of 100 people is deciding which of two candidates (a

Question

Problem 4 Suppose a village of 100 people is deciding which of two candidates (a Democrat and a Republican) to elect as mayor. Of the voting popun70try prefer the Democrat and the other 30 strictly prefer the Republican. Assume voting is costless (a) If voting is simultaneous and by secret ballot, either prove Nash equilibrium makes a unique prediction about who gets elected, or else argue that it does not. (b) If instead voting is by roll call (each person in turn casts her vote publicly, from the eldest eligible voter to the youngest), does subgame perfection allow you to make a unique prediction about who gets elected? Does it let you predict uniquely how each person will vote? Explain.

Explanation / Answer

a) A strategy profile in which each player is doing his or her best given what other players are doing is a Nash equilibrium. Here, 70 people strictly prefer the Democratic candidate and 30 people strictly prefer the Republican candidate. It is each player’s dominant strategy to vote for the candidate that they strictly prefer. And when everyone is playing their dominant strategy, the result is a Nash equilibrium. Therefore, the Nash equilibrium predicts that the Democratic candidate is selected.

b) In part a, the game was simultaneous. Now the game is sequential. But even in a sequential game, all the players will go by their dominant strategies. Therefore, the subgame perfection allows us to predict that 70 people will vote for the Democratic candidate and 30 people will vote for the Republican candidate. Each person will vote for the person that he or she strictly prefers.

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