Solve the Questions
1. To Infinity, and Beyond!
Consider the above game, played infinitely. Both Player A and Player B have the same
discount factor, δ.
(a) Suppose that both players follow a grim trigger strategy: play c as long as the
other player plays c – if the other player plays d, then subsequently play d forever.
Find the minimum δ such that this is a SPNE. (5 points)
(b) Assume that both A and B are playing some SPNE of the infinitely repeated
game. Let V
A and V
B be A and B’s discounted values. What are the
lowest values that V
A and V
B could have? (5 points)
(Hint: no calculations are needed. Apply the Folk Theorem.)
2. From Here to Eternity
Consider the following stage game, assuming the discount rate is δ = 0.99. Check
whether the strategy profiles are SPNE in the infinitely repeated game.
(a) Each player always plays c. (3 points)
(b) Each player always plays d. (3 points)
(c) Player A alternates between c and d, while player 2 always plays c; if any player
deviates from this scenario, then each player plays d thereafter. (3 points)
(d) Each player plays c, and if a player deviates, then he plays c, and the other player
plays d thereafter. (3 points)
3. Sharing Is Weak
Let us extend the Tragedy of the Commons problem we discussed during lectures. Consider N players, each of whom has the following utility function each period: log(x),
where x is the amount consumed. Suppose there is a common property resource of size
y > 0. The game has two stages. In stage one, player, i = 1, …N can withdraw ci
ci ≤ y. In stage two, each player consumes of the remaining quantity, y −
In the case that they attempt to consume in excess of the available amount, then each
player splits the resource equally. For the sake of simplicity, let P
ci = ¯c.
(a) Write down each player’s intertemporal utility function.1
(b) What is the Nash equilibrium in Stage Two? (3 points)
(c) What is this game’s SPNE? (6 points)
(d) What is the socially optimal level of consumption in Stage One? (4 points)
(e) How does the difference between the SPNE consumption in Stage One, and the
socially optimal consumption in Stage One, change as N becomes very large? What
is the intuition for this, and does this explain why ecologist Garrett Hardin suggested
sterilization as a solution to this problem? [Hint: take the limit as N approaches
infinity.] (5 points)