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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.

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  • $\begingroup$ To help you start, for (a) a natural strategy for each seller would be to say yes if the bid is above cost and no if it is below cost (equality happens with probability $0$ so might be ignored). Is there a better seller strategy than this? $\endgroup$
    – Henry
    May 5 at 14:41
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I think you are a bit confused. The game has the following structure.

  1. nature draws $c_1, c_2 \sim U[0,1]$, which is only revealed to the sellers and not to the buyer.
  2. The buyer proposes a price $p \in [0,1]$
  3. Sellers decide to sell the good at price $p$. The good is sold if at least one seller agrees with the price.

You need to solve this using backwards induction.

The last stage is easy: each seller $i$ is willing to sell the good if $p \ge c_i$.

No let's go to stage 2. If the good is sold, the buyer receives value $v - p$. If the good is not sold, the value is $0$ if the good is not sold. Denoting by $\pi(p)$ the probability that the good is sold at price $p$, this gives: $$ \pi(p) \times (v - p) + (1- \pi(p)) \times 0 = \pi(p) \times (v - p). $$

The only thing left is to determine $\pi(p)$ and then to optimize this with respect to $p$.

To determine $\pi(p)$ it might be easier to first compute the probability that the good will not be sold. This will be the case if $p > c_1$ and $p > c_2$. For $p \in [0,1]$ the probability that this happens is $$ \Pr[p > c_1 \text{ and } p > c_2] = \Pr[p > c_1] \times \Pr[p > c_2] = (1-p) \times (1-p) = (1-p)^2. $$ This uses the fact that $c_1$ and $c_2$ are independent.

Given this, the probability that at least one seller is willing to sell is given by $$ \pi(p) = 1 - (1-p)^2. $$

As such, the utility of the buyer is given by: $$ (1 - (1-p)^2)\times (v - p). $$ This has to be maximized with respect to $p$.

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