Exercise 17.21

In the Prisoner’s Dilemma, consider the case where after each round, Alice and Bob have probability $X$ meeting again. Suppose both players choose the perpetual punishment strategy (where each will choose ${refuse}$ unless the other player has ever played ${testify}$). Assume neither player has played ${testify}$ thus far. What is the expected future total payoff for choosing to ${testify}$ versus ${refuse}$ when $X = .2$? How about when $X = .05$? For what value of $X$ is the expected future total payoff the same whether one chooses to ${testify}$ or ${refuse}$ in the current round?