# Research

I work in the fields of applied and theoretical auction design and, more generally, mechanism design. Sometimes I would borrow an idea from applied work and try to frame it theoretically, or the other way round. Below are some of the topics I'm actively working on.

### Explaining secret reserve prices in French timber auctions

Quite often the seller would keep the reserve price secret so to have the last word in the auction.

Here is a quote from Orley Ashenfelter (1989, p25)

*... a**uctioneers are very secretive about whether and at what level a reserve price may have been set, and there is a real art in getting the bidding started on each **item without revealing the reserve price... If you sit through an auction you will find that every item is hammered down **and treated as if it were sold. Only after the auction does the auctioneer reveal whether and at what price an item may **have actually been sold. In short, the **auctioneers do not reveal the reserve **price and they make it as difficult as they **can for bidders to infer it. *

Here is another quote from Marty and Preguet (2007, p27)

*A remarkable fact about the secret reserve price in French public timber sales is that sellers do not commit to any price. The reserve price is usually defined before the sale, but the sellers may raise or lower it as the auction proceeds. A seller may lower the reserve prices if the appraisal values have been overestimated at a given market state and many lots are left unsold. On the other hand, a seller may raise reserve prices if the objectives of receipts are already met. In fact, the seller may even change the reserve price when he sees the bids. This way, he can use the bids to revise his estimation of a lot.*

The rationale for this is that by observing the bids the seller can learn more about ones true value and therefore make a more informed decision to sell or not to sell. In other words, the auctioneer can update her reserve price upon the submission of bids. While practitioners agree that this, indeed, makes sense, this phenomenon received little coverage in the theoretical literature.

We develop a model and an estimation procedure to show that in French timber auctions secret reserve prices can increase efficiency, albeit, at some revenue loss. Our paper revolves around the following three ideas:

we have to relax the classic assumption that the seller knows her own true value perfectly, otherwise secret reserve prices will be inferior to public ones

we have to model the seller's learning rule in reduced form, for example, such rule could be:

R_1 = alpha R_0 + beta b_(1) + gamma (b_(2) + b_(3) + b_(4))/3 (1)

where R_0 is the (what would be otherwise public) ex-ante reserve price, b_(1) is the strongest bid and b_(2), b_(3), b_(4) are the losing bids, R_1 is the (updated) ex-post reserve price. Of course, bidders recognise their price impact which would depress the bids in equilibrium.

Crucially, the learning rule has to satisfy homogeneity degree 1, to preserve the linear scalability of the model. Without homogeneity, we lose all hope to test this model with real data.

we have to modify the estimation routine from (2), (3) to a more abstract (2), (4):

v = b + F(b)/f(b) (2)

F(b) = Pr(b_i<b) (3)

F(b) = Pr(win|b) (4)

This identifying equation can be estimated via Bayes rule, with the denominator being the usual cdf of bids, and the numerator being the cdf of bids conditional on winning the auction (which includes exceeding the secret reserve price) times the share of auctions being won by a representative bidder.

Pr(win|b) = pdf(b_i=b|win)Pr(win)/pdf(b_i=b) (5)

This establishes the identification a-la-GPV in this model.

### Estimating first price auctions using bid spacings.

In the structural auction literature, a common approach to estimation of bidders rents is to use the first order condition:

v = b + F(b)/f(b) (1)

f(b) = sum_i K((b_i-b)/h)/nh (2)

to recover latent valuations from the observed bids. This requires a rather straightforward estimator of the cdf F(b) and the pdf f(b) of bids, which are then combined into F/f in a plug-in fashion, which is often referred to as the GPV approach.

While this works perfectly well on the paper, in reality, the F/f ratio is a very capricious estimator. In particular, when attempting to generate the estimator over a large grid, the following problems arise: risk of division by zero, very difficult uniform inference, poor (namely, quadratic) computational performance. It was, in fact, mentioned by Silverman (1982) that *... it is highly inefficient to use (2) directly for this purpose, a method using Fourier Transform is far more efficient.* Several competing estimators have been recently proposed, but neither of them has claimed superiority over the others.

We focus on a, somewhat less explored, quantile-based estimator

v(u) = Q(u)+q(u)u (3)

q(u) = sum_i K(u-i/n)s_i (4)

s_i = b_(i) - b_(i-1) (5)

where v(u) is the quantile function of values, and Q(u) and q(u) are the estimates of the quantile function and quantile density of bids.

It is worth noting that q(u) mechanically takes the form of a convolution of bid spacings s_i with a discrete kernel (filter in the signal processing literature), which can be expedited using Fourier Transform. While the (3), (4), (5) estimator is superior to (1), (2) computationally, it is also, in a sense, pivotal, a rather unexpected and powerful property which allows for a very simple uniform inference.

### Detecting corruption using bid timings.

In an open auction, a bidder may delay her bid until the very last second so to not give away her private information to other bidders. This is a well known phenomenon and, in the context of Ebay auctions, is called sniping. Apart from that, bid timings should be mostly irrelevant to the auction outcomes.

We focus on procurement auctions in Russia that have a similar sniping pattern, but, because they are sealed bid, the same explanation as in Ebay does not work. What could it be then? It is believed that, because public procurement is highly corrupt, in a sealed-bid auction, the officer overlooking the auction may secretly open the envelope and convey the bid to her favored bidder or in exchange for a bribe.

We want to document and measure the extent of this corruption.

Reduced analysis shows that those bids that were submitted just before the deadline have a higher chance of winning. This can be explained by either the favored bidder delaying her bid to be able to see other bids, or by another bidder delaying her bid to protect it from being seen. Separating these two effects is the main challenge.

Three methods to detect and measure apparent corruption are proposed.

Assuming orthogonality of bid and time we first construct a simple difference estimator, that compares the winning frequency of snipers against that of everybody else. Second, we we replace the strong orthogonality assumption with mere continuity, and use a regression-discontinuity approach. Namely, for every pair of bids in the auction, we use the time-distance as a running variable, zero as a cutoff, and the indicator of being the stronger bid as the outcome in the RD design. The size of the discontinuity can be related to the magnitude of corruption.

Thirdly, one can argue that the RD design would somewhat underrepresent corruption due to the fact that it is not physically possible to open an envelope and submit a new bid is less than X minutes, where X is small. It then makes sense to drop the observations that are too close to the discontinuity, as in the dohnut-RD design. Depending on the dohnut size X, our assessment of the share of corrupt auctions varies between 10% and 15%.

Finally, to evaluate the associated damages, we construct a two-step estimator to see how bids vary across the cutoff.