10.9 Exercises

Exercise 10.1:
For the hawk-dove game of Example 10.11, where D>0 and R>0, each agent is trying to maximize its utility. Is there a Nash equilibrium with a randomized strategy? What are the probabilities? What is the expected payoff to each agent? (These should be expressed as functions of R and D). Show your calculation.
Exercise 10.2:
In Example 10.12, what is the Nash equilibrium with randomized strategies? What is the expected value for each agent in this equilibrium?
Exercise 10.3:
In the sequential prisoner's dilemma, suppose there is a discount factor of γ, which means there is a probability γ of stopping at each stage. Is tit-for-tat a Nash equilibrium for all values of γ? If so, prove it. If not, for which values of γ is it a Nash equilibrium?