end-of-chapter-exercises-and-solutions

end-of-chapter-exercises-and-solutions - Chapter Backward...

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Chapter: Backward Induction and Subgame Per- fection (Questions) 1
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Chapter: Backward Induction and Subgame Per- fection - (Solutions) 5
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15 Backward Induction and Subgame Perfection 1. (a) (I, C, X) (b) (AF, C) (c) (BHJKN, CE) 2. (a) The subgame perfect equilibria are (WY, AC) and (ZX, BC). The Nash equilibria are (WY, AC), (ZX, BC), (WY, AC), (ZY, BC), and (WX, BD). (b) The subgame perfect equilibria are (UE, BD) and (DE, BC). The Nash equilibria are (UE, BD), (DE, BC), (UF, BD), and (DE, AC). 3. (a) A B 2 A B A B 1 O I x , 1 3, 1 0, 0 0, 0 1, 3 1 x , 1 x , 1 x , 1 x , 1 OA OB A B 2 1 3, 1 0, 0 IA IB 1, 3 0, 0 (b) If x > 3, the equilibria are (OA,A), (OB,A), (OA,B), (OB,B). If X = 3, add (IA, A) to this list. If 1 < x < 3, the equilibria are (IA,A), (OA,B), (OB,B). If x = 1, add (IB, B) to this list. If x < 1, the equilibria are (IA,A), (IB,B). (c) If x > 3 any mixture with positive probabilities over OA and OB for player 1, and over A and B for player 2. If 1 < x < 3, then IB is dominated. Any mixture (with positive proba- bilities) over OA and OB will make player 2 indi ff erent. Player 2 plays A with probability x/ 3, and plays B with probability 1 x/ 3. 110
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BACKWARD INDUCTION AND SUBGAME PERFECTION 111 For 3 / 4 x 1, let 1 p q denote the probability with which player 1 plays OA or OB. Let p denote the probability with which player 1 plays IA, and q denotes the probability with which she plays IB. Then any p and q (where both are positive and sum to not more than 1) such that p = 3 q will make player 2 indi ff erent between A and B. Player 2 plays A with probability 1 / 4. For x < 3 / 4 OA and OB are dominated. In equilibrium, player 1 chooses IA with probability 3 / 4 and IB with probability 1 / 4. In equilibrium, player 2 chooses A with probability 1 / 4, and B with probability 3 / 4. (d) 1, 3 0, 0 0, 0 3, 1 A B A B 2 1 The pure strategy equilibria are (A, A) and (B, B). There is also a mixed equilibrium (3 / 4 , 1 / 4; 1 / 4 , 3 / 4). (e) The Nash equilibria that are not subgame perfect include (OB, A), (OA, B), and the above mixed equilibria in which, once the proper sub- game is reached, player 1 does not play A with probability 3 / 4 and/or player 2 does not play A with probability 1 / 4. (f) The subgame perfect mixed equilibria are those in which, once the proper subgame is reached, player 1 does plays A with probability 3 / 4 and player 2 does plays A with probability 1 / 4. 4. (a) 1 C 2 C 1 S S S 1, 1 0, 3 2, 2 C S C S 3, 3 1, 4 C 0, 0 2 1 (b) Working backward, it is easy to see that in round 5 player 1 will choose S. Thus, in round 4 player 2 will choose S. Continuing in this fashion, we
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BACKWARD INDUCTION AND SUBGAME PERFECTION 112 fi nd that, in equilibrium, each player will choose S any time he is on the move. (c) For any fi nite k , the backward induction outcome is that player 1 chooses S in the fi rst round and each player receives one dollar. 5. Payo ff s in the extensive form representation are in the order RBC, CBC, and MBC.
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