A seller has an indivisible asset to sell. Her reservation value for the asset is s, which she knows privately. A potential buyer thinks that the asset's value to him is b, which he privately knows. Assume that s and b are independently and uniformly drawn from [0, 1]. If the seller sells the asset to the buyer for a price of p, the seller's payoff is P -s and the buyer's payoff is b-p. 1. Suppose the buyer makes a take-it-or-leave-it offer p to the seller. What's the optimal offer if the buyer's value is b = = 1/? 2. Suppose simultaneously the buyer makes an offer p₁ and the seller makes an offer p2. A transaction occurs iff p₁ ≥ p2, and the transaction price is ½ (P₁ + P2). (a) Is the following strategy profile a Bayesian Nash equilibrium: the buyer chooses P₁ = ½ if b ≥ ½ and he chooses p₁ = 0 if b < ½; the seller chooses p2 = ½ if s ≤ ½ and she chooses p2 = 1 if s>. Why or why not? (b) Can there be a Bayesian Nash Equilibrium in which the transaction price is 0.9 whenever there is a transaction? Why or why not? (c) Suppose the buyer uses a strategy p₁(b) = 1/2 + b and the seller uses a strategy P2(s) = 1 + s. 1. Suppose the buyer's value is and the seller's valuation is 1. Will there be a transaction? Explain your answer. 2. Is this strategy profile a Bayesian Nash equilibrium? Explain your answer.

Managerial Economics: A Problem Solving Approach

5th Edition

ISBN:9781337106665

Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Publisher:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Chapter18: Auctions

Section: Chapter Questions

Problem 18.1IP

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Transcribed Image Text:Incomplete Information A seller has an indivisible asset to sell. Her reservation value for the asset is s, which she knows privately. A potential buyer thinks that the asset's value to him is b, which he privately knows. Assume that s and b are independently and uniformly drawn from [0, 1]. If the seller sells the asset to the buyer for a price of p, the seller's payoff is p – s and the buyer's payoff is b – p. 1. Suppose the buyer makes a take-it-or-leave-it offer p to the seller. What's the optimal offer if the buyer's value is b = ;? 2. Suppose simultaneously the buyer makes an offer pi and the seller makes an offer p2. A transaction occurs iff p1 > P2, and the transaction price is ; (P1 + P2). (a) Is the following strategy profile a Bayesian Nash equilibrium: the buyer chooses P1 = ; if b > and he chooses pi = 0 if b < ; the seller chooses p2 = ; if s < and she chooses p2 = 1 if s > . Why or why not? S (b) Can there be a Bayesian Nash Equilibrium in which the transaction price is 0.9 whenever there is a transaction? Why or why not? 5 + b and the seller uses a strategy (c) Suppose the buyer uses a strategy p1(b) P2(8) = } +s. 12 3 1. Suppose the buyer's value is and the seller's valuation is i. Will there be a transaction? Explain your answer. 2. Is this strategy profile a Bayesian Nash equilibrium? Explain your answer.

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