3. (30 points) The following example is adapted from Zahavi (1975) and Grafen (1990). Many animal species are characterized by a morphology apparently unadapted to their environment. The peacock's tail for in- stance is useless, and dangerous as it makes the animal more noticeable to predators, but still it attracts peacock females. Other animal courtship rituals or sexual ornaments seem to similarly decrease an ani- mal's probability of survival, which seems contradictory with evolutionary laws. In the long-run, evolution will select morphological traits that maximize an individual's probability to survive and mate. Under some assumptions, the outcome of the evolutionary process can also be mod- elled as the outcome of a "game" where each individual values his probability to pass on his genes over several generations and can "choose" his morphology. Here is an example with the peacock. A male peacock can be either "strong" (type H) with probability q or weak (type L) with probability 1 -q. He can "choose" to grow a brightly colored tail (strategy T) or not (NT). A female peacock then observes the male's tail, and chooses whether to mate with the male peacock or not. The payoffs are the following: the female gets 0 if she mates with a weak male, 1 if she does not mate, and 2 if she mates with a strong male. A male's payoff is the sum of two components. If a male does not grow a tail, he first gets v, which reflects the positive value of escaping predators through not having a too visible tail. If he is strong and grows a tail he gets PHU v> pLutr. (a) Formalize the situation as a dynamic game of incomplete information and draw the tree of the game. (b) Solve for the weak perfect bayesian equilibria of this game. (c) What would change if we assumed that piotr > v? Interpret. (d) Find other applications of the "handicap principle"