Question

Problem 2: We consider an auction with two bidders. Each bidder's valuation v is drawn as follows: 

• With probability the valuation v is distributed uniformly between 0 and 2.

 • With probability 3 the valuation v is distributed uniformly between a and 1. 

1. If we run a first price auction, what is the optimal bid function of a bidder with valuation v?

 2. What is the expected revenue of the seller? 

3. Suppose now that there is only one bidder and that the seller runs a second price auction with a reserve price. Calculate the optimal reserve price.

Answer 1

$$Solution$$

1.Let $B_i$ be agent i's bid and $\pi's$ agent i's profit. If B_i\geq\ v_i,\ then\ \pi\leq0,\then assuming rationality $B_i\leq V_i$

Thus $\pi=0$

If you want to maximize your expected profit (hence your valuation of money is 

risk-neutral), then your maximum bid is $V-2B=0\implies\ B=\frac{v}{2}$ where $B$ is the bid.

2. Here there are two options in place. One, a second-price auction, the seller is committed to collecting less money, because of charging the second-highest bid. On the other hand, in a first price auction, the bidders reduce their bids, in return reducing what the seller can collect.

Therefore, the seller's expected revenue is

$=(\frac{n-2}{n})(\frac{n}{n+2})=\frac{n-2}{n+2}\\ Where\ n=2\\ =\frac{2-2}{2+2}=\frac{0}{4}\\ =0$

3.Due to the reserve price the item will be on the highest bidder, only if the highest bid is above $r$, otherwise the item should not be sold. In the second price auction, if there is a winner, he pays the maximum of the second-place bid and the reserve price $r$.

If the item is worth $v$ to the seller, then r=v with a probability of $2-r$, the bidders' value is above $r$ and this item will be sold at a price $r$. Given a probability $r$, the bidders value will be below $r$, and the seller will hold the item with a payoff of $v=0$. Therefore, the seller expects revenue which is $r(2-r),$ this is maximized at $r=\frac{1}{2}$