This problem is an illustration of the Hawk-Dove game described in the text. The game was first
Question:
If a Hawk meets a Dove, the Hawk gets to mate with the female and the Dove slinks off to celibate contemplation. If a Dove meets another Dove, they both strut their stuff but neither chases the other away. Eventually the female may select one of them at random or may get bored and wander off. The expected payoffs to each male are shown in the box below.
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(a) 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 is a Hawk or a Dove. The payoff to adopting either strategy for oneself depends on the proportions of Hawks and Doves in the population at large. For example, that there is one Hawk in the forest and all of the other males are Doves. The Hawk would find that his rival always retreated and would therefore enjoy a payoff of ____________ on every encounter. Given that all other males are Doves, if the remaining male is a Dove, his payoff on each encounter would be _________
(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
(c) If all the other males are Hawks, then a male who adopts the Hawk strategy is sure to encounter another Hawk and would get a payoff of _____________ If instead, this male adoptled the Dove strategy, he would again be sure to encounter a Hawk, but his payoff would be __________
(d) Explain why there could not be an equilibrium where all of the animals acted like Hawks.
(e) Since there is not an equilibrium in which everybody chooses the same strategy, we look for an equilibrium in which some fraction of the males are Hawks and the rest are Doves. Suppose that there is a large male population and the fraction p are Hawks. Then the fraction of any player€™s encounters that are with Hawks is about p and the fraction that are with Doves is about 1 ˆ’ p. Therefore with probability p a Hawk meets another Hawk and gets a payoff of ˆ’5 and with probability 1 €“ p he meets a Dove and gets 10. It follows that the payoff to a Hawk when the fraction of Hawks in the population is p, is p × (ˆ’5) + (1 ˆ’ p) × 10 = 10 ˆ’ 15p. Similar calculations show that the average payoff to being a Dove when the proportion of Hawks in the population is p will be ______________
(f) Write an equation that states that when the proportion of Hawks in the population is p, the payoff to Hawks is the same as the payoffs to Doves
(g) Solve this equation for the value of p such that at this value Hawks do exactly as well as Doves. This requires that p =
(h) On the axes below, use blue ink to graph the average payoff to the strategy Dove when the proportion of Hawks in the male population who is p. Use red ink to graph the average payoff to the strategy, Hawk, when the proportion of the male population who are Hawks is p. Label the equilibrium proportion in your diagram by E.
(i) If the proportion of Hawks is slightly greater than E, which strategy does better? ________ If the proportion of Hawks is slightly less than E, which strategy does better? __________ 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? ____________
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a Now while wandering through the forest a male will encounter many conflict situations of this type Suppose that he cannot tell in advance whether an...View the full answer
