Problem Set 2, MIEG (Part II), Winter Term, 2016
1) Two people are involved in a dispute. Person 1 does not know whether person 2 is strong or weak; she assigns probabilityαto person 2 being strong. Person 2 is fully informed.
Each person can either fight or yield. Each person’s preferences are given by: payoff of 0 if she yields (regardless of the other person’s action) and a payoff of 1 if she fights and her opponent yields; if both people fight then their payoffs are (-1, 1)if person 2 is strong and (1, -1)if person 2 is weak. Formulate this situation as a Bayesian game and find its (Bayesian) Nash equilibria.
2) Each of two individuals receives a ticket on which there is an integer from 1 to m indi- cating the size of a prize she may receive. The individuals’ tickets are assigned randomly and independently; the probability of an individual’s receiving each possible number is positive.
Each individual is given the option to exchange her prize for the other individual’s prize;
the individuals are given this option simultaneously. If both individuals wish to exchange then only the prizes are exchanged; otherwise each individual receives her own prize. Each individual’s objective is to maximize her expected monetary payoff. Model this situation as a Bayesian game and find its (Bayesian) Nash equilibria.
3) Suppose a public good is provided if at least one person is willing to pay the cost c >0. There are n persons. They differ in their valuations of the good vi , and each person knows only her own valuation. If good is not provided, then everyone has payoff 0. Assume 0≤v < c < v. The probability of any individual’s valuation being less than v is F(v), which is independent of all other individual’s valuations. F is continuous and strictly increasing with support [v, v]. All individuals bid simultaneously. They bid either zero or c.
a) Find conditions under which for each value ofi this game has a pure strategy Nash equi- librium in which each type vi of player i with vi ≥ c contributes, whereas every other type of player i, and all types of every other player, do not contribute.
b) Find a symmetric BNE where every player contributes if and only if her valuation is atleast some cut off value v∗.
4) There are n bidders indexed by i = 1,2, . . . , n, the valuations v1, . . . , vn are indepen- dently and identically distributed with Uniform distribution over 0 to 1.
(i) Compute the symmetric BNE for first price auction.
(ii) Compare the revenue to the seller from first price action and second price auction. Check whether it confirms to Revenue Equivalence Theorem.
(iii) If seller can refuse to sell the object below a certain price which is called reserve price.
Find the optimal reserve price in second price auction.
1
