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Simon Fraser University Department of Economics Economics ECON - 383-3 Daniel Monte Selected Topics in Game Theory Spring 2008 Problem Set 6: Solutions 1. Consider the two-player game: p2 a b c d A 3; 1 0; 0 0; 0 5; 0 p1 B 0; 0 1; 3 0; 0 0; 0 C 0; 0 0; 0 2; 2 0; 0 D 0; 0 0; 5 0; 0 4; 4 a) Find all pure strategy Nash Equilibria of this game. (A; a) (B; b) (C; c) b) Suppose that this game is played twice (i.e., played once, outcomes are revealed, then played again). Assume that players do not discount payo¤s (i.e., = 1). Construct a pure-strategy SPNE in which (D; d) is played in the …rst stage. Player 1: play D in the …rst period play A in the second period if the outcome in the …rst period was such that player 2 played a2 2 fa; b; cg and player 1 played D play B in the second period if the outcome in the …rst period was such that player 1 played a1 2 fA; B; Cg and player 2 played d play C in the second period if the outcome in the …rst period was (D; d) or if a1 6= D AND a2 6= d. Player 2: play d in the …rst period play a in the second period if the outcome in the …rst period was such that player 2 played a2 2 fa; b; cg and player 1 played D: play b in the second period if the outcome in the …rst period was such that player 1 played a1 2 fA; B; Cg and player 2 played d play c in the second period if the outcome in the …rst period was (D; d) or if a1 6= D AND a2 6= d. In words: both players play (D; d) in the …rst period. If nobody deviates, the next stage game is a Nash Equilibrium that is good for both players. 1 If only player deviates, the next stage game is a bad Nash equilibrium for the player that deviates. If both players deviate, then the next stage game is a Nash Equilibrium that is good for both players. 2. Consider the Cournot oligopoly game that we have seen in class. The stage game is such that two …rms choose quantities simultaneously and have zero marginal cost. The demand is given by: p = a q1 q2 . Consider a trigger strategy de…ned as play qi = a if it is the initial history or if everyone has 4 always played q = a . Play qi = a forever if anyone has ever played q 6= a in any previous round. 4 3 4 a) What is the best response in the stage game for p1 if p2 is playing a ? 4 a maxq1 q1 a q1 4 F.O.C. a 2q1 a = 0 4 q1 = 3a 8 The pro…t that …rm 1 makes with this deviation is: 1 = q1 (a q1 q2 ) 2 D = 3a a 3a a 3a 8a 3a 2a = 3a 3a = 9a 1 8 8 4 = 8 8 8 8 64 The pro…t from collusion is: M = a a a a aa a2 1 4 4 4 = 42 = 8 C a a a aa a2 The pro…t from Cournot is 1 = 3 a 3 3 = 33 = 9 b) What is the minimum such that both players playing a trigger strategy is a SPNE? On equilibrium path: 2 2 2 2 following trigger strategy: v T S = a + a + 2 a + ::: = a 1 1 8 8 8 8 9a2 a2 2 a2 9a2 a2 1 One-step deviation: v D = 64 + 9 + 9 + ::: = 64 + 9 1 v T S v D if: a2 1 9a2 a2 1 + 8 1 64 9 1 64:9 8 81 (1 ) + 64 72 81 (1 ) + 64 17 9 9 17 O¤ equilibrium path: a2 a2 2 a2 a2 1 following trigger strategy: v T S = 9 + 9 + 9 + ::: = 9 1 2 2 2 a2 a2 1 One-step deviation: v D = a + a + 9 9 9 + ::: = 9 1 Thus, trivially satis…ed for any . 3. Consider the following game: 2 p2 L R p1 T 2; 1 1; 4 B 3; 3 0; 0 Assume that this game is played an in…nite number of times and that players discount their payo¤s using a discount factor < 1. a) Suppose that both players use a grim-trigger strategy described as follows. The strategy of player 1 is: play T if it is the initial period; play T if in the all previous round the outcome has been (T; L); play B if in any of all the previous rounds the outcome was di¤erent than (T; L). The strategy of player 2 is: play L if it is the initial period; play L if in the all previous round the outcome has been (T; L); play R if in any of all the previous rounds the outcome was di¤erent than (T; L). Find the minimum value of such that this is a subgame perfect Nash Equilibrium. (Remember to check on and o¤ the equilibrium path). First lets check for player 1: On equilibrium path: following trigger strategy: v T S = 2 + 2 + 2 2 + ::: = 2 1 1 One-step deviation: v D = 3 + 0 + 2 0 + ::: = 3 v T S v D if: 1 2 3 1 2 3 (1 ) 2 3 3 3 1 1 3 O¤ equilibrium path: following trigger strategy: v T S = 0 + 0 + 2 0 + ::: = 0 One-step deviation: v D = 0 + 0 + 2 0 + ::: = 0 Thus, trivially satis…ed for any . Now lets check for player 2: On equilibrium path: following trigger strategy: v T S = 1 + 1 + 2 1 + ::: = 1 1 1 One-step deviation: v D = 4 + 0 + 2 0 + ::: = 4 v T S v D if: 1 4 1 1 4 (1 ) 3 1 4 4 4 3 3 4 O¤ equilibrium path: following trigger strategy: v T S = 0 + 0 + 2 0 + ::: = 0 One-step deviation: v D = 0 + 0 + 2 0 + ::: = 0 Thus, trivially satis…ed for any . 3 1 Therefore we need 4: (Note that if this happens, then 3 as well). 4. Consider the following game: p2 L R p1 T 2; 2 3; 3 B 3; 3 0; 0 Assume that this game is played an in…nite number of times and that players discount their payo¤s using a discount factor < 1. Now suppose that the players follow the following strategies. The strategy of player 1 is: play T if the game is in state m1 and play B if it is in state m2 or state m3 . The strategy of player 2 states’are is: play L if in state m1 and play R if it is in states m2 or m3 . The transition between ‘ explained below. The initial state is m1 . State m1 : If the game is in state m1 , stay in that state when the outcome of the stage game is (T; L). If the outcome of the stage game in m1 is di¤erent than (T; L) move to m2 . State m2 : From state m2 , move to state m3 if the outcome of the stage game is (B; R). If the outcome of the stage game in m2 is di¤erent than (B; R) stay in m2 . State m3 : From state m3 , move to state m1 if the outcome of the stage game is (B; R). If the outcome of the stage game in m3 is di¤erent than (B; R) move to state m2 . This can be represented by the picture below: 4 a) Is this a strategy? Explain. Yes, this is a strategy, since for each player it speci…es an action after every possible history. Thus, it is a complete plan of action. b) Find the minimum value of such that both players using this strategy is a subgame perfect Nash Equilibrium. (Remember to check on and o¤ the equilibrium path). On equilibrium path: following suggested strategy: v S = 2 + 2 + 2 2 + ::: = 2 + 2 + 2 2 + 3 2 1 1 One-step deviation: v D = 3 + 0 + 2 0 + 3 2 + 4 2 + ::: = 3 + 2 3 1 1 v T S v D if: 2 + 2 + 22 3 p p 2 2+2 1 0, Solution is: 1 1; 2 3 1 [2 1 2 3 1 2; 1 Thus, any 1p 1 3 0:366; 2 2 is such that v S vD : O¤ equilibrium path (on histories at m2 ): following suggested strategy: v T S = 0 + 0 + 2 2 + ::: = 2 2 1 1 One-step deviation: v D = 0 + 0 + 2 2 + ::: = 2 2 1 1 Thus, trivially satis…ed for any . O¤ equilibrium path (on histories at m3 ): following suggested strategy: v T S = 0 + 2 + 2 2 + ::: = 2 1 1 One-step deviation: v D = 0 + 0 + 2 2 + ::: = 2 2 1 1 Thus, trivially satis…ed for any . 5

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