Consider the following game. There are two sellers, each of whom can produce one unit of an indivisible good. The cost of producing the unit for seller i is $c_i$ . There is a single buyer who wishes to buy at most one unit. The value of the unit to the buyer is v = 2. The buyer regards the costs of the two sellers as independent draws from the uniform distribution on [0, 1]. Consider the following two-stage game. The buyer proposes a price p ∈ [0, 1] to the two sellers. Each seller then responds simultaneously by indicating whether or not he is willing to sell at that price. If no seller is willing to sell, then trade does not occur and each seller and the buyer receives a utility of zero. If exactly one trader is willing to sell, then the buyer buys the item from that seller at the price p that the buyer proposed. If both sellers are willing to sell, then a fair coin is flipped to determine which one sells to the buyer, with the payment again being the price p proposed by the buyer.
a) What is each seller's optimal strategy?
b) Set up the buyer’s optimization problem and determine a Perfect Bayesian-Nash equilibrium of this game.
c) What is the buyer’s expected surplus in this equilibrium?
d) Alternatively, the buyer could arrange a second-price auction. That is, both sellers submit bids, and the price paid to the lower bidder is the higher bidder’s bid. Find the equilibrium in this auction. Briefly explain your answer.
e) Calculate the buyer’s expected surplus in the second price auction, and compare with the expected surplus in the game above where he sets a price.
f) Suppose you are a seller with type c. What is your expected surplus in the second price auction and under the posted price method?
For each seller, the natural strategy would be to sell if the bid is at or above cost, and not sell if the bid is below cost.
Seller's expected utility from offering p ∈ [0,1] is (1−p)p. This is maximised at p=1/2. So in the PBE the seller will offer p=1/2 and the buyer will accept. Is this correct? This would then give an expected surplus of 3/2 for the buyer in this equilibrium.