1. The town of Sunnydale, CA is inhabited by two vampires, Spike and Anya. Each night Spike and Anya independently hunt for food, which each one finds with probability 1/2 . Because of a telepathic link, each vampire knows whether the other vampire has found food. A food source contains 3/2 liters of blood, but in one session a vampire can only eat 1 liter of blood. If each vampire finds food they both eat 1 liter, giving each vampire a payoff of 1. If neither vampire finds food, both vampires have a payoff of 0. If one vampire finds food and the other doesn't, the vampire who found food may choose to share his or her food with the other vampire. If the vampire chooses to share, both vampires experience a payoff of 3/4 . If the vampire with food does not choose to share, the vampire who found food receives a payoff of 1 and the other vampire receives a payoff of 0. A vampire only has to make a decision if one of them finds food and the other doesn't.
(a) Suppose that the situation described above happens only one time. If Spike finds food but Anya doesn't, will he share? Would Anya share with Spike if she found food but he didn't? Does a vampire who found food have a dominant strategy?
(b) What is the expected payoff to each vampire if they follow the strategy in a.? Vampires live forever. Therefore, imagine that this situation is repeated an infinite number of times, and both vampires have the same discount factor. Suppose Spike and Anya agree to share food (if one finds food and the other doesn't) as long as in every past period in which one of them found food but the other didn't, the one who found food shared it. If at any point in the past a vampire found food but didn't share, they act as in part a.
(c) At the beginning of any particular night, before the vampires know whether they will find food, what is Spike's expected payoff if both he and Anya follow the agreement?
(Hint: with probability 1/4 both find food. With probability 1/2 at exactly one vampire finds food. With probability 1/4 neither vampire finds food.)
(d) Suppose Spike found food in period k but Anya didn't. What is his expected payoff from following the agreement in that period? (Hint: his payoff is composed of two pieces. First Spike experiences today's payoff of sharing food with Anya. By doing so, he also preserves the agreement forever after)
(e) What is Spike's expected payoff if he doesn't share his food with Anya? (Hint: he does better in the current period, but destroys the agreement from tomorrow on)
(f) For which values of is the agreement self-sustaining? (That is, acting according to the agreement is a sub-game perfect Nash Equilibrium)
(g) Speculate about what would happen if the probability of finding food were close to 1 or close to 0. Would the agreement be easier or more difficult to sustain?
(h) Now suppose that the telepathic link between Spike and Anya does not always operate. Thus, neither vampire knows for certain whether the other one has found food. Of course, if the other shares, the receiver learn that the other one found food. However, if a vampire finds food but chooses not to share, the other vampire only finds out through the telepathic link with probability 2/3. For which values of is the agreement self-sustaining? Explain the difference with f.