Problem 1. More dilemmas for the prisoners.
Consider the following prisoners' dilemma game.
Suppose that this game is repeated infinitely and consider the following "Tit-for-tat" strategy: "Choose C in period 1 and then do whatever your opponent did last period".
(a) For what values of δ, if any, both players choosing the "Tit-for-tat" strategy is a SPE? Consider now the "Pavlov" strategy: "Choose C in period 1. Choose C after any history in which the outcome in the last period is either (C, C) or (D, D). Choose D after any other history."
(b) For what values of δ, if any, both players choosing the "Pavlov" strategy is a SPE?
Problem 2. Swing voter 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 and B. Both citizens agree that candidate 1 is the best if the state is A and candidate 2 is the best if the state is B: both citizens obtain a payoff of 1 if the best candidate given the state wins (i.e., obtains more votes than the other candidate) and a payoff of 0 if the other candidate wins If candidates tie, then both citizens obtain a payoff of 1/2. Citizen 1 knows the state while citizen 2 believes it is A with probability 0.9 and B with probability 0.1. Each citizen can either vote for candidate 1, vote for candidate 2, or not vote (abstain). (a) Find the set of players, action, types and beliefs. Find the payoff matrix if the state turns out to be A and the payoff matrix if the state turns out to be B. (b) Show that there are exactly two pure strategy Bayesian Nash Equilibria: in the first one citizen 2 does not vote and in the second she votes for candidate 1. Find the payoffs for the citizens in each equilibrium. (c) Show that one of the player's actions in the second equilibrium is weakly dominated. (d) Why doesn't citizen 2 vote in the first equilibrium?