Problem Set 2, MIEG (Part II), Winter Term, 2015

Problem Set 2, MIEG (Part II), Winter Term, 2015

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 Nash equilibria if α <1/2 and if α >1/2.

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 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 show that in any Nash equilibrium the highest prize that either individual is will- ing to exchange is the smallest possible prize.

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 v_i , and each person knows only her own valuation. If good 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.

All individuals bid simultaneously. They bid either zero or c.

a) Find conditions under which for each value of i this game has a pure strategy Nash equi- librium in which each type v_i of player i withv_i ≥ ccontributes, 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 at least some cut off value v^∗.

4)Suppose there are two bidders, each bidder’s signal is uniformly distributed from 0 to 1, and the valuation of each bidder i is given by vi = ati +btj, where j is the other player, a≥b ≥0,t_i is type of player iand similarly for player j. Types are private information.

a) Show that second-price sealed-bid auction has a Nash equilibrium in which each type t_i of each player i bids (a+b)ti.

b) Show that when a = b = 1, for any value of z > 0, second-price sealed-bid auction has an (asymmetric) Nash equilibrium in which each type t₁ of player 1 bids (1 +z)t₁ and each type t2 of player 2 bids (1 + 1/z)t2.

c) Show that first-price sealed-bid auction has a Nash equilibrium in which each type t_i of each player i bids (a+b)t_i/2.

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d) Compare buyer’s revenue for above equilibria.

5) There are n bidders indexed by i = 1,2, . . . , n, the valuations v₁, . . . , v_n 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. Does it confirm toRevenue 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.

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