Problem set 6
Advanced Microeconomics,
Ec 992, Spring 2024
Games of incomplete information II and sequential rationality
1. There are N > 1 players and there is one indivisible object which one of them may purchase. Player n’s value for the object is denoted vn, n = 1, . . . , N and is known only to player n. are identically and independently distributed according to the uniform. distribution over [0, 1]. The set of possible bids (ac-tions) for each player is [0,∞). The object is allocated to the player with the highest bid. In case of a tie, the object is allocated with equal probability to one of the players with the highest bid.
(a) Suppose the payment made by the player who gets the object is the highest bid. Consider the pure strategy Bayes Nash equilibrium where players use a symmetric linear strategy. What is the expected payment that type v1 of player 1 would make in this equilibrium? What is the ex-ante expected payment that player 1 would make in this equilibrium? Explain your answers.
(b) Suppose the payment made by the player who gets the object is the second-highest bid.
i. Does there exist a Bayes Nash equilibrium where the strategy of player 1 is b1(v1) = 1 for all v1 ∈ [0, 1] and the strategy of player i ≥ 2 is bi(vi) = 0 for all vi ∈ [0, 1]? Explain your answer.
ii. Show that there exists a pure strategy Bayes Nash equilibrium where the bid of type vn is vn, n = 1, . . . , N and answer the following.
A. Consider the random variable y = max{v2, v3, . . . , vN }. Obtain an expression for the cumulative distribution function and the proba-bility density function of y (you may restrict attention to the case y ∈ [0, 1]). Using this or otherwise, obtain an expression for the expected value of y conditional on y < ¯v1 for some ¯v1 ∈ [0, 1].
B. What is the expected payment made by type v1 of player 1 in the equilibrium where the bid of type vn is vn, n = 1, . . . , N? What is the ex-ante expected payment that player 1 would make in this equilibrium? How do these compare to the answers you obtained in part (a)? Explain.
(c) Suppose N = 2 and that the payment made by the player who gets the object is a weighted average of the highest and second-highest bid, with weight α ∈ (0, 1) on the highest bid. Does there exist a pure strategy Bayes Nash equilibrium where the players use a symmetric linear strategy? Explain.
2. Consider the following sealed-bid first-price auction for a single object. There are two bidders. Bidder i privately observes a signal ti
, i = 1, 2. The signals t1 and t2 are identically and independently distributed over [0, 1] according to the uniform. distribution. The value of the object to bidder 1 is v1 = αt1 + γt2 and to bidder 2 is v2 = αt2 + γt1 where α > 0 and γ > 0. Bidders submit their bids simultaneously. The bidder with the highest bid wins the object and pays the highest bid. In case of a tie, each bidder wins the object with equal probability. Does there exist a Bayes Nash equilibrium of this game in which the bid of player i who observes signal ti
is 0.5(α + γ)ti? Explain.
[Note: It may be useful consider (a) the probability of winning for player i who observes signal ti and submits bid bi
, (b) the expected value of t−i
if player i who observes signal ti and submits bid bi wins, and (c) the expected payoff to player i who observes signal ti and submits bid bi.]
3. Exercise 9.C.1 of the MWG textbook.
4. Exercise 9.C.2 of the MWG textbook.
5. Find all weak Perfect Bayesian Equilibria of the game in Figure 1.
6. Find a weak Perfect Bayesian Equilibrium of the game in Figure 2 when p = 0.5 and when p = 0.1.
Figure 1: Game tree I for question 5.
Figure 2: Game tree II for question 6.