The Hawk-Dove Game. This problem is based on an example developed by the biologist John Maynard Smith to illustrate the uses of game theory in the theory of evolution. Males of a certain species frequently come into conflict with other males over the opportunity to mate with females.  If a male runs into a situation of conflict, he has two alternative “strategies” . A male can play “Hawk” in which case he will fight the other male until he either wins or is badly hurt.  Or he can play “Dove”, in which case he makes a display of bravery but retreats if his opponent starts to fight. If an animal plays Hawk and meets another male who is playing Hawk, they both are seriously injured in battle. If he is playing Hawk and meets an animal who is playing Dove, the Hawk gets to mate with the female and the Dove slinks off to celibate contemplation. If an animal is playing Dove and meets another Dove, the both strut around for a while. Eventually the female either chooses one of them or gets bored and wanders off.  The expected payoffs to each of two males in a single encounter depend on which strategy each adopts. These payoffs are depicted in the box below.

Animal B

Hawk      Dove

Now while wandering through the forest, a male will encounter many conflict situations of this type. Suppose that he cannot tell in advance whether another animal that he meets will behave like a Hawk or like a Dove. The payoff to adopting either strategy oneself depends on the proportion of the other guys that is Hawks and the proportion that is Doves.

(a) Find the Nash equilibria of the Hawk-Dove Game in pure strategies.

(b) If strategies that are more profitable tend to be chosen over strategies that are less profitable, explain why there cannot be an equilibrium in which all males act like

Doves or all act like Hawks.

(c) Find the Nash equilibria of the Hawk-Dove Game in mixed strategies.

(d) Suppose that the fraction of a large male population that chooses the Hawk strategy is p. Then if one acts like a Hawk, the fraction of one’s encounters in which he meets another Hawk is about p and the fraction of one’s encounters in which he meets a Dove is about 1 - p.

• If the more profitable strategy tends to be adopted more frequently in future plays, then if the strategy proportions are out of equilibrium, will changes tend to move the proportions back toward equilibrium or further away from equilibrium?

2.  A group of n students go to a restaurant. It is common knowledge that each student will simultaneously choose his/her own meal, but all students will share the total bill equally. If a student gets a meal of price p and contributes x towards paying the bill, his/her payoff will be^p - x.

(a) Compute the Nash equilibrium of this game.

(b) Discuss the limiting cases: n = 1 and n → ,.