Suppose the reserve price is set for each content page (or for each seller, or for each category) at the time point (denoted by t) when a user lands on this page and the auction for showing recommended content pages begins.
1. At this time point t, compute the average number of buyers bidding on this content page (or this seller, or this category), the average bid, and the average standard deviation of the bids, where the average was taken over all biddings. That is, every time the content page is viewed and the auction is conducted, the three statistics were computed, and then the average over all biddings for this given content page was take. The bid of the highest bidder in every auction was excluded from the statistics, because the theory does not allow us to pin it down (every bid of the highest bidder above a certain value results in the same payoffs). It is assumed that bidders’ values are drawn from a lognormal distribution, and its mean and standard deviation can be estimated.
Example: before this auction begins, we have bidding historical data by buyers.
Number of bidders
|
Mean(bid)
excluding the highest bid
|
Sd(bid)
Excluding the highest bid
|
3
|
$0.15
|
0.02
|
5
|
$0.17
|
0.04
|
4
|
$0.14
|
0.03
|
…
|
…
|
…
|
6
|
$0.20
|
0.05
|
Average of the above historical bidding data
| ||
4.3
|
$0.16
|
0.04
|
2. These two statistics can be achieved in step 1. At time point t when the auction begins, we may observe the number of bidders as well as their bids. Denote the number of bidders for this auction as k. Simulate three moments (observed number of bidders (which is equal to k for this auction), average bid in positions 2 and below, and the standard deviation of the bids in positions 2 and below) for various true values of the mean and standard deviation of the lognormal distribution of values. To do that, 1000 draws of the vectors of bidder values (with k bidders) were drawn from the lognormal distribution estimated in step 1. Note that the number of bidders is irrelevant for setting the optimal reserve price, but it needs to be estimated in order to get an accurate estimate of the mean and the standard deviation of the distribution of values.
Example: suppose k=4, then simulate 1000 draws from the estimated lognormal distribution with mean $0.16 and standard deviation 0.04 made up from example in step 1.
Number of bidders
|
bids
|
Mean(bid)
excluding the highest bid
|
Sd(bid)
Excluding the highest bid
|
4
|
$0.25, $0.18, $0.16,$0.14
|
$0.16
|
0.02
|
4
|
$0.22, $0.16, $0.13,$0.1
|
$0.13
|
0.03
|
4
|
$0.18, $0.15, $0.12,$0.09
|
$0.12
|
0.03
|
…
|
…
|
…
|
…
|
Average of the above simulated bidding data
| |||
4
|
$0.14
|
0.04
| |
3. As long as the lognormal normal distribution is estimated for the case of k bidders, the theoretically optimal reserve price r is computed by solving the equation
r-[1-F(r)]/f(r)=0, (1)
where F is the cumulative distribution function and f is the probability density function estimated at step 2 for the case of k bidders.
Example: as in the made-up example in step 2, for the case of k=4 bidders, the corresponding lognormal distribution has mean $0.14 and standard deviation 0.04. Then the reserve price can be calculated by solving equation (1) for the current auction.
4. The above calculated reserve price is based on uniform CTRs for each buyer, and it needs to be converted into buyer-specific reserve prices that reflected the quality scores of the buyers: buyers with higher CTR face lower per-click reserves, and vice versa.
In the current auction at time point t, suppose the k bidders’ CTRs are CTR_1, CTR_2,…, CTR_k, and denote their average as CTR_average. Then the i-th buyer’s reserve price is r_i=r*CTR_average/CTR_i.
Disadvantages: for some auctions, the reserve price may push out some potential qualified bidders so that the spots may not be fully filled and thus results in a revenue loss.
Reference:
[1] Ostrovsky, M., Schwarz, M., 2010. Reserve Prices in Internet Advertising Auctions: A Field Experiment.
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