1 Solutions to Homework 3

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1 1 Solutions to Homework (Nash equilibria of extensive games) (Subgames) Karl R E B H B H B H B H B H B H There are 6 proper subgames, beginning at every node where or chooses an action. 1

2 (Burning a bridge) 1 Don't Attack Attack (1,) Fight Retreat (0,0) (,1) Note that this has the same structure as the Entry Game. The SPNE is (Attack, Retreat). If it were possible for Army to burn a bridge (i.e. remove the Retreat action), then the optimal action in the subgame would be F ight, and the SPNE would be (Don tattack, F ight) which has a higher payoff for Army (Firm-union bargaining) Union w Firm L We can formulate the game as follows. First, the union chooses its wage offer w 0. Next, the firm chooses the amount of labor to hire, L 0 (if L = 0, that is equivalent to rejecting the wage offer). The union s payoff is wl. The firm s payoff is its profit, given by: { L(100 L) wl if L 50 Π(w, L) = 500 wl if L > 50 The profit function has a quadratic part and a linear part. We want to check if the maximum profit is always achieved on the quadratic part. The derivative of the quadratic with respect to L is 100 w L, which gives L = 100 w at the maximum. At w = 0, L = 50; if w

3 increases, then L decreases. This guarantees that the maximum profit is always achieved on the quadratic. Next, if profit is negative even when maximized, then the firm should choose L = 0. This occurs when L < 0, which is when w > 100. In summary, the best response of the firm is: If w = 0, any L 50 is a best response. If 0 < w 100, L = 100 w is the unique best response. If w > 100, profit will be negative even at the maximizing L, so L = 0 is the best response. Now, consider the union s decision. The union s payoff wl is only positive if 0 < w < 100. The union maximizes wl = w(100 w), which is maximized at w = 50. Then L = 5. This is the unique subgame perfect equilibrium. The firm s payoff is 65 and the union s payoff is 150. An outcome of the game that both players prefer to the SPNE is if profits are higher than 65 and the wage bill is higher than 150. We can find one possible outcome by maximizing the sum of the payoffs L(100 L) with respect to w and L, which occurs at L = 50. For example, if w = 30, then the firm s payoff is 1000 and the union s payoff is A Nash equilibrium other than the SPNE is when w > 100, L = 0. The firm has no incentive to deviate, since it will make a negative payoff for L > 0. The union has no incentive to deviate, because it will get a payoff of 0 for any choice of w (Nash equilibria of the ultimatum game) (Stackelberg s duopoly game with fixed costs) Assume c = 0, α = 1, f = 4. Suppose Firm 1 goes first and chooses q 1 ; then Firm chooses q. Taking q 1 as given, Firm s profit function is π (q 1, q ) = q (α q 1 q ) f cq = q (1 q 1 q ) 4 which is maximized at q = 1 q 1, resulting in a profit of π (q 1, q ) = 1 4 (α 4f αq + q ) = 1 4 (18 4q 1 + q 1) 3

4 This is an upwards-facing quadratic. We want to find the level of q 1 that guarantees Firm s profit will be positive. To do this, find the roots of the quadratic: α 4f αq 1 + q 1 = 18 4q 1 + q 1 = 0 which are at q 1 = α ± f = 1 ± 4. So, if 8 q 1 16, then Firm cannot make a positive profit and is better off producing q = 0. Now, consider Firm 1 s decision problem. If q1 < 8, Firm 1 and Firm are competing in Cournot duopoly, in which case Firm 1 s optimal choice is the Cournot outcome, q 1 = q = α c 3 = 4, and profit is 1. If 8 q 1 16, Firm will not compete, and Firm 1 will be in a monopoly. Firm 1 s profit function in this case is q 1 (α c q 1 ) f = q1 + 1q 1 4, which is a downwardfacing quadratic that is maximized at α c = 6. As q 1 increases past 6, profits decrease, so we only need to check monopoly profits at q 1 = 8, which is 8. Therefore, Firm 1 s optimal choice is q 1 = 8, and the unique SPNE is q 1 = 8, q = (Sequential positioning by two political candidates) Recall in Hotelling s model of electoral competition (Section 3.3), there is a unique Nash equilibrium where both candidates choose m, the median. We know that any SPNE is also a NE (but not vice versa), and that at least one SPNE always exists for a finite-horizon extensive game with perfect information. This game has one NE, so the SPNE must also be the NE (The race G1(,)) 4

5 (Two-period Prisoner s Dilemma) There are 4x4=16 terminal histories, one for each possible combination of actions in period 1 and period. The payoff for the entire game is the sum of the payoffs in each period. In the second period, the NE is (F, F ). In the first period, taking this as given, the NE is still (F, F ) since the actions in each period don t affect the game in the other period. Therefore, the SPNE is ((F, F ), (F, F )): both players will defect in both periods (Bertrand s duopoly game with entry) In Bertrand s duopoly game, the NE is when both firms set their price equal to marginal cost, making zero profits. If the entry cost f that the challenger must pay to enter is positive, then the challenger will choose Out, so (Out, c), c) is the unique SPNE (Nash equilibria of the centipede game) Consider the strategy profile in which the game ends at the kth stage, k > 1, and Player i is the first to choose S. Player j must therefore have chosen C in stage k 1. If Player j deviates by changing his action in stage k 1 to S, he can obtain a higher payoff. Therefore, none of these strategy profiles can be Nash equilibria, except when k = 1. 5