Question

There is a second price auction with N bidders initially with i.i.d private values vi from uniform distribution [),1]. What is the expected revenue in initial settings i.e. no reserve prices.

Suppose N = 3. What is the optimal reserve price r the auctioneer should set? How much smaller/greater is the revenue compared to the auction with no reserve price?

Answer 1

In a second-price auction with N bidders, the expected revenue in initial settings with no reserve prices can be calculated using the PDF of the highest private value among the bidders. For N = 3, the optimal reserve price can be computed as the expected revenue when there is no reserve price. The difference in revenue between the auction with no reserve price and the auction with the optimal reserve price can also be determined.

Step-by-step explanation:

A second-price auction with N bidders initially with i.i.d private values vi from a uniform distribution [0,1]. The expected revenue in initial settings with no reserve prices is calculated as follows:

Let V be the random variable representing the highest private value among the bidders. The probability distribution function (PDF) of V is given by:

f(v) = Nv^(N-1)

The expected revenue is then calculated as:

ER = ∫[V * f(v) dv]

For N = 3, the optimal reserve price r can be calculated as the expected revenue when there is no reserve price. The algorithm to compute the optimal reserve price involves finding the solution to the equation:

r = ∫[V * f(v|r) dv]

The revenue in the auction with no reserve price can be compared to the revenue with the optimal reserve price to determine how much smaller or greater it is.