40.316-Homework 4 Solved

 
 
 

Question 1: There are two players and suppose that player 2 has complete information about it’s two types: type L and type H. Type L loves going out with player 1 whereas type H hates it. Player 1 has only one type and is uncertain about player 2’s type and believes the two types are equally likely. The two actions available to each player is: {F,B}. The payoff matrices are as follows:

Question 3: Consider the Cournot duopoly model with imperfect information about costs as discussed in class. That is, consider the situation where firm 1’s unit cost (c) is known by both firms, but only firm 2 knows its own unit cost. Firm 1 believes that firm 2’s cost is cL with probability θ and cH with probability 1 − θ, where 0 < θ < 1 and cL < cH. Now assume that the (inverse) demand in the market is given by: P(Q) = α − Q for Q ≤ α and P(Q) = 0 for Q α. Assuming that all outputs are positive in the Bayes Nash equilibrium, what is the equilibrium?



 

Question 4: Consider a variant of the Bayesian game of first-price auction discussed in class in which the players are risk averse. Specifically, suppose each of the n players’ preferences are represented by the expected value of the payoff function x1/m, where x is the player’s monetary payoff and m 1. Suppose also that each player’s valuation is distributed uniformly between 0 and 1. Show that the game has a symmetric Bayes-Nash equilibrium in which each type vi of each player i bids  .

Question 5: Consider the all-pay auction as discussed in class. There are n bidders and vi is independently and identically distributed according to the uniform distribution [0,1].

The payoff to a bidder i is vi − bi if bi maxj6=ibj and −bi if bi < maxj6=ibj. Find the symmetric (Bayes-Nash) equilibrium bidding function. (Hint: First write down the expected payoff for a player i assuming everyone bids according to the symmetric equilibrium. Then write down the first order condition and use inverse function theorem. Finally, solve the resulting differential equation.)

Question 6: Consider the following auction problem: a seller has a single item for sale worth nothing to him. There are two buyers, each draws a “type” independently with equal probability from the set {$8,$12}. Consider a sealed bid second price auction in which the only possible bids are $8 or $12. Let p denote the second price in the auction. What is the expected revenue to the seller if each buyer i values the item equal to his own type ti? That is, the winner gets ti − p and the loser gets 0.