The Nash equilibrium of the stage game is (D, R), giving payoffs (0, 0). Consider the trigger strategies:

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1 Problem Set 4 1. (a). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: B L R U 1, 1 2, 5 A D 2, 0 0, 0 Sketch a graph of the players payoffs. Show what utilities could be suppted using the Nash threats folk theem. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to play (U, L) in every period. What is the minimum δ that achieves cooperation? The Nash equilibrium of the stage game is (D, R), giving payoffs (0, 0). Consider the trigger strategies: If the histy is (U, L), (U, L),..., (U, L), play (U, L) this period. After any other histy, play (D, R) Does either player have a profitable deviation? F the row player, following the proposed strategies gives a payoff of and deviating gives a payoff of so cooperating is better than deviating if 1 + δ1 + δ = δ0 + δ = δ 1 2 F the column player, following the proposed strategies gives a payoff of and deviating gives a payoff of So cooperating is better than deviating if 1 + δ1 + δ = δ0 + δ = δ 4 5 THEREFORE, as long as δ 4/5, the trigger strategies are a Subgame Perfect Nash Equilibrium

2 (b). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: B L R U 2, 1 0, 0 A D 0, 0 1, 2 Sketch a graph of the players payoffs. Show what utilities could be suppted using the Nash threats folk theem. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to play (U, L) in every even period, and (D, R) in every odd period. What is the minimum δ that achieves cooperation? (Hint: You might need to use mixed strategies) There are three equilibria in the stage game. There are two pure-strategy Nash equilibria, (U, L) and (D, R), and a mixed one with strategies This gives expected payoffs 2/3 to each player. Consider the trigger strategies: σ u = 2 3, σ d = 1 3, σ L = 1 3, σ R = 2 3 If the histy is (D, R), (U, L), (D, R), (U, L),..., (D, R), play (U, L) this period. If the histy is (D, R), (U, L), (D, R), (U, L),..., (U, L), play (D, R) this period. If the histy is anything else, play the mixed strategy equilibrium of the stage game. Does either player have a profitable deviation? When it s your turn to get the 2, the payoff from cooperating is 2 + δ + 2δ 2 + δ 3 + 2δ = (2 + δ)(1 + δ 2 + δ ) = (2 + δ) 2 and when it s your turn to get the 1, the payoff from cooperating is 1 + 2δ + 1δ 2 + 2δ = (1 + 2δ)(1 + δ 2 + δ ) = (1 + 2δ) 2 Well, the row player might deviate after a histy (D, R), (U, L), (D, R), (U, L),..., (D, R) from U to D, after a histy (D, R), (U, L), (D, R), (U, L),..., (U, L) to U, both give a payoff 0 + δ δ = 2 δ 3 Comparing the sums term-by-term, 2, 1 > 0 and 1, 2 > 2/3, and 2, 1 > 2/3, and so on, so the first two sums are always better than the third one. Therefe neither player ever has an incentive to deviate. Or if you look at the sums, (2+δ), (1+δ) > 2/3, and 1/(1 δ 2 ) > 1/(1 δ), so that the discounted sums of the first two payoffs are greater than the discounted sum of the third payoff. Therefe, deviating is never profitable. THEREFORE, as long as δ 0, the trigger strategies are a Subgame Perfect Nash Equilibrium

3 (c). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: 2 L C R U 4, 4 0, 5 5, 6 1 M 5, 0 1, 1 6, 0 D 6, 5 0, 6 7, 7 Sketch a graph of the players payoffs. Show what utilities could be suppted using the Nash threats folk theem. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to play (U, L) in every period. What is the minimum δ that achieves cooperation? Are there multiple ways to design the punishments f cheating? What is the lowest δ you can achieve? There are three Nash equilibria in the stage game: (M, C), (D, L), and (U, R). Consider the following trigger strategies: If the histy is (U, L), (U, L),..., (U, L), then play (U, L). After any other histy, play (M, C). Cooperating then gives both players a discounted payoff of 4 + δ4 + δ = 4 Then the most profitable deviation f the row player is to choose D and get a payoff of 6, followed by the punishment, and the most profitable deviation f the column player is to choose R and get a payoff of 6, followed by the punishment. This gives a discounted payoff of deviating to either player of 6 + δ1 + δ δ = 6 + δ δ Or 4 6() + δ δ 2 5 THEREFORE, if δ 2/5, the above trigger strategies are a subgame perfect Nash equilibrium Are there other ways to enfce cooperation? Think about these strategies: If the histy is (U, L), (U, L),..., (U, L), then play (U, L). After any histy in which the row player deviated first, play (U, R) After any histy in which the column player deviated first, play (D, L)

4 After any histy in which both players deviated first at the same time, play (M, C). This punishes the deviat with a 5 payoff fever, and rewards the player who didn t cheat with a 6 fever. The payoff from deviating f either player is 6 5δ 5δ 2 5δ = 6 5 δ δ 4 6() 5δ δ 2 11 THEREFORE, if δ 2/11, the above trigger strategies are a subgame perfect Nash equilibrium This is the lowest you can get, since a payoff of 5 is the wst you can impose on a player, given that the player is trying to maximize his payoff. 2. Consider the N-player prisoners dilemma: If all players choose S, they each get a payoff of N If a single player chooses C while all others choose S, that player gets N 2 and the others get 1 If me than one player confesses, all players who confess get zero while any player who remains silent gets 1 i. What is the Nash equilibrium of the N-player prisoners dilemma? ii. Consider the infinitely repeated game with discount fact δ where the N-player prisoners dilemma is the stage game. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to play S in every period. What is the minimum δ that achieves cooperation? iii. What happens to the minimum δ as N increases? i. All players confess is the Nash equilibrium of the game. In fact, it is a strictly dominant strategy to confess. Suppose all other players are silent; by confessing, I get a payoff of N 2, and from silent, I get a payoff of N, so confessing is better. Suppose at least one player is confessing; by remaining silent, I get a payoff of 1, while by confessing I get a payoff of zero, so confessing is better. Therefe, no matter what my opponents are doing, I get a higher payoff by confessing, so it is a strictly dominant strategy. ii. Consider the following trigger strategies: After any histy in which all players chose silent in all previous periods, play silent this period.

5 After any other histy, confess. The discounted payoff of cooperating is Deviating and confessing gives a payoff N + δn + δ 2 N +... = N N 2 + δ0 + δ = N 2 N N 2 δ 1 1 N THEREFORE, if δ 1 1/N, the trigger strategies are a subgame perfect Nash equilibrium iii. As N increases, we get δ(2) = 1 1/2 = 1/1, δ(3) = 2/3, δ(4) = 3/4,..., which tends to 1. Since 1/N 0 as N, when the game is really big, the discount fact will be close to 1, making cooperating me difficult. 3. There s a market with two firms who are contemplating acting collusively. If they cut production so that each is making half the monopoly quantity, they each get π m = 2. If they act as Cournot competits, they both get π c = 1. If one player plays the monopoly quantity, the other player can take advantage of the situation and increase output, getting a larger share of the market f himself at the expense of his partner, who gets 0 if this occurs; call this value π d, f optimal deviation. Here s a strategic fm f the game: Collude F irmb Compete Collude 2, 2 0,π d F irma Compete π d, 0 1, 1 i. F what values of π d does this have the fm of a prisoner s dilemma, i.e., each player has a dominant strategy to play competitively? Assume this inequality holds f the rest of the problem. ii. Assume the game is repeatedly infinitely with discount fact δ. Show that f sufficiently patient players, it is a subgame perfect Nash equilibrium to collude in every period. What is the minimum δ that achieves cooperation? iii. How does the minimum δ change as π d increases? Does this make collusion me less likely to succeed? i. If π d > 2, the players have a dominant strategy to compete. ii. The Nash equilibrium of the stage game is (Compete, Compete), giving payoffs (1, 1). Colluding gives payoffs (2, 2). Consider the trigger strategies

6 If the histy is (Collude, Collude), (Collude, Collude),... Collude this period., (Collude, Collude), play If the histy is anything else, Compete this period. Then the payoff to cooperating is and the payoff to deviating is Solving f δ gives 2 + δ2 + δ δ = 2 π d + δ + δ 2 + δ = π d + 2 π d + δ 2 ()π d + δ δ π d 2 π d 1 δ THEREFORE, as long as δ π d 2, the above trigger strategies are a subgame perfect Nash π d 1 equilibrium iii. The minimum δ is δ = π d 2 π d 1 Differentiating with respect to π d gives so the minimum δ is increasing in π d. δ π d = (π d 1) (π d 2) (π d 1) 2 = 1 (π d 1) 2 > 0