Question

Demonstrate that $$ \text { (D;d: } \left.\frac{1}{2} ; \frac{1}{2}\right) $$ and $$ \left[\left(\frac{2 a_2}{a_2+r_2} A, \frac{r_2-a_2}{a_2+r_2} D\right) ;\left(\frac{2 a_1}{a_1+r_1} a, \frac{r_1-a_1}{a_1+r_1} d\right): \frac{a_1+r_1}{2 r_1} ; \frac{a_2+r_2}{2 r_2}\right] $$ are perfect Bayesian equilibria of the game in Figure 6.6.

   Demonstrate that
$$
\text { (D;d: } \left.\frac{1}{2} ; \frac{1}{2}\right)
$$
and
$$
\left[\left(\frac{2 a_2}{a_2+r_2} A, \frac{r_2-a_2}{a_2+r_2} D\right) ;\left(\frac{2 a_1}{a_1+r_1} a, \frac{r_1-a_1}{a_1+r_1} d\right): \frac{a_1+r_1}{2 r_1} ; \frac{a_2+r_2}{2 r_2}\right]
$$
are perfect Bayesian equilibria of the game in Figure 6.6.

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Game Theory for Political Scientists

Game Theory for Political Scientists

James D. Morrow 1st Edition

Chapter 6, Problem 8

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6. Understand the players, their strategies, and the payoffs associated with each strategy profile. This will help in analyzing the equilibria. Show more…

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Demonstrate that $$ \text { (D;d: } \left.\frac{1}{2} ; \frac{1}{2}\right) $$ and $$ \left[\left(\frac{2 a_2}{a_2+r_2} A, \frac{r_2-a_2}{a_2+r_2} D\right) ;\left(\frac{2 a_1}{a_1+r_1} a, \frac{r_1-a_1}{a_1+r_1} d\right): \frac{a_1+r_1}{2 r_1} ; \frac{a_2+r_2}{2 r_2}\right] $$ are perfect Bayesian equilibria of the game in Figure 6.6.

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