1. Consider a game of chicken: two players approach a narrow bridge with room only for one, and each chooses either to go through and hope the other guy backs off (call this action H, for "hawk"), or backs off to let the other go through (call this action D, for "dove"). If both players are normal, then each gets a payoff of -1 from a collision (i.e. if both choose H), 1 if they choose H against D, 0 if they choose D (regardless of what the other player chooses). Now perturb the game as follows: there is a chance p > 0 that P2 is a “violent type", who gets 1 from playing H regardless of Pl's action, and -1 from playing D, again regardless of P1's action. (a) Illustrate the game tree corresponding to the Harsanyi transformation of this game (i.e. the dynamic game of imperfect information in which nature moves first). (b) Illustrate the extended game (i.e. the normal form game corresponding to your game tree in part (a)). (c) Find all values of p for which there is a BNE in which P1 chooses each action with chance ½, violent P2 chooses H, and normal P2 uses a mixed strategy.
1. Consider a game of chicken: two players approach a narrow bridge with room only for one, and each chooses either to go through and hope the other guy backs off (call this action H, for "hawk"), or backs off to let the other go through (call this action D, for "dove"). If both players are normal, then each gets a payoff of -1 from a collision (i.e. if both choose H), 1 if they choose H against D, 0 if they choose D (regardless of what the other player chooses). Now perturb the game as follows: there is a chance p > 0 that P2 is a “violent type", who gets 1 from playing H regardless of Pl's action, and -1 from playing D, again regardless of P1's action. (a) Illustrate the game tree corresponding to the Harsanyi transformation of this game (i.e. the dynamic game of imperfect information in which nature moves first). (b) Illustrate the extended game (i.e. the normal form game corresponding to your game tree in part (a)). (c) Find all values of p for which there is a BNE in which P1 chooses each action with chance ½, violent P2 chooses H, and normal P2 uses a mixed strategy.
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.5P
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Transcribed Image Text:1. Consider a game of chicken: two players approach a narrow bridge with room
only for one, and each chooses either to go through and hope the other guy
backs off (call this action H, for "hawk"), or backs off to let the other go
through (call this action D, for "dove"). If both players are normal, then each
gets a payoff of -1 from a collision (i.e. if both choose H), 1 if they choose H
against D, 0 if they choose D (regardless of what the other player chooses).
Now perturb the game as follows: there is a chance p > 0 that P2 is a “violent
type", who gets 1 from playing H regardless of Pl's action, and -1 from playing
D, again regardless of P1's action.
(a) Illustrate the game tree corresponding to the Harsanyi transformation of
this game (i.e. the dynamic game of imperfect information in which nature
moves first).
(b) Illustrate the extended game (i.e. the normal form game corresponding to
your game tree in part (a)).
(c) Find all values of p for which there is a BNE in which P1 chooses each
action with chance ½, violent P2 chooses H, and normal P2 uses a mixed
strategy.
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