[Math] the optimal reserve price in a second price sealed bid auction

Question

Consider a seller who must sell a single private value
good. There are two potential buyers, each with a valuation that can take on one
of three values,θi∈{0,1,2}, each value occurring with an equal probability of 1/3.The players’ values are independently drawn. The seller will offer the good using a second-price sealed-bid auction, but he can set a “reserve price” of r≥0 that modifies the rules of the auction as follows. If both bids are below r then neither bidder obtains the good and it is destroyed. or above r then the regular auction rules prevail. If only one bid is at or above
r then that bidder obtains the good and pays r to the seller.

What is the optimal reserve price in a second price sealed bid auction?

Answer 1

First, we want to see which strategies the bidders will play: From a real bidders perspective, you can look at the reserve price as if it were the bid of a third bidder, regarding both allocation and payment rule. Therefore, as we know that the sealed bid second price auction is truthful, we can expect the bidders to bid their true value.

The probabilities of the bidders value profiles ($v_i, v_2$) are:

You can count the profiles if it helps you to understand the following case distinction:

1. Assume we set $r > 2$, then the expected revenue is 0 (as item is always destroyed).
2. Assume we set $2 \geq r > 1$, then the expected revenue is $$ E_{rev} = \frac{1}{9} * 2 + \frac{4}{9} * r + \frac{4}{9} * 0 $$ which is highest at r = 2 and takes the value $1.11$.
3. Assume we set $1 \geq r \geq 0$ , then the expected revenue is $$ E_{rev} = \frac{1}{9} * 2 + \frac{3}{9} * 1 + \frac{4}{9} * r + \frac{1}{9} * 0$$ which is highest at r = 1 and takes the value $1$.

Therefore, the expected revenue is maximized with $r = 2$.

Note that I assumed your "above r" is actually "greater or equal than r", otherwise, the case, where the highest bid equals the reserve price, would not be defined.

Best, miweiss