Show that (D ; d: (1)/(2) ; (1)/(2)) is the only perfect Bayesian equilibrium of the above game.

Step 1: u000ATo show that u005C((u005Cmathrm{D}u003B u005Cmathrm{d}: u005Cfrac{1}{2}u003B u005Cfrac{1}{2})u005C) is a perfect Bayes

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Show that $\left(\mathrm{D} ; \mathrm{d}: \frac{1}{2} ; \frac{1}{2}\right)$ is the only perfect Bayesian equilibrium of the above game. a) First show that ( $\mathrm{D} ; \mathrm{d}: \frac{1}{2} ; \frac{1}{2}$ ) is a perfect Bayesian equilibrium of the game. b) Show that $\mathrm{Q}$ strictly dominates $\mathrm{A}$ (and $\mathrm{q}$ strictly dominates a). Consequently, A and a can never appear in a perfect Bayesian equilibrium strategy. c) Show that once neither player ever attacks (play $\mathrm{A}$ or a), D strictly dominates $\mathrm{Q}$ (and d strictly dominates $\mathrm{q}$ ). Thus only the given belief-strategy pairing can be a perfect Bayesian equilibrium.

   Show that $\left(\mathrm{D} ; \mathrm{d}: \frac{1}{2} ; \frac{1}{2}\right)$ is the only perfect Bayesian equilibrium of the above game.
a) First show that ( $\mathrm{D} ; \mathrm{d}: \frac{1}{2} ; \frac{1}{2}$ ) is a perfect Bayesian equilibrium of the game.
b) Show that $\mathrm{Q}$ strictly dominates $\mathrm{A}$ (and $\mathrm{q}$ strictly dominates a). Consequently, A and a can never appear in a perfect Bayesian equilibrium strategy.
c) Show that once neither player ever attacks (play $\mathrm{A}$ or a), D strictly dominates $\mathrm{Q}$ (and d strictly dominates $\mathrm{q}$ ). Thus only the given belief-strategy pairing can be a perfect Bayesian equilibrium.

Game Theory for Political Scientists

James D. Morrow 1st Edition

Chapter 6, Problem 10

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Step 1

Step 1: To show that $(\mathrm{D}; \mathrm{d}: \frac{1}{2}; \frac{1}{2})$ is a perfect Bayesian equilibrium, we need to verify two conditions: (1) the strategies must be a best response given the beliefs, and (2) the beliefs must be updated correctly according Show more…

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Show that $\left(\mathrm{D} ; \mathrm{d}: \frac{1}{2} ; \frac{1}{2}\right)$ is the only perfect Bayesian equilibrium of the above game. a) First show that ( $\mathrm{D} ; \mathrm{d}: \frac{1}{2} ; \frac{1}{2}$ ) is a perfect Bayesian equilibrium of the game. b) Show that $\mathrm{Q}$ strictly dominates $\mathrm{A}$ (and $\mathrm{q}$ strictly dominates a). Consequently, A and a can never appear in a perfect Bayesian equilibrium strategy. c) Show that once neither player ever attacks (play $\mathrm{A}$ or a), D strictly dominates $\mathrm{Q}$ (and d strictly dominates $\mathrm{q}$ ). Thus only the given belief-strategy pairing can be a perfect Bayesian equilibrium.

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