# Secret Reserve Prices

### What is this about?

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.

### Why are secret reserve prices understudied?

The first generation of auction models have solidified the rules according future models should be written. In particular, the seller's value is typically known (or even zero), and the solution concept is Bayes-Nash. It is not hard to show then, that secret reserve prices are inferior to public ones. This fact is so simple and (mathematically) transparent that the very idea of secret reserve prices would appear heretical to any junior researcher. Yet, secret reserve prices are being persistently used.

One of the most stubborn users of secret reserve prices is the French public office (ONF). It has provided data for some of the earliest papers on structural auction estimation:

yet these papers chose to interpret the secret reserve price as the seller (ONF) simply randomising over various reserve prices.

### What is the DGP behind reserve prices?

It never hurts to just ask.

In our private communication with ONF we learned that the seller is not just randomising, but really trying to infer something (presumably, the latent distribution) from the observed bids. While no written guidelines exist for how to take advantage of the observed bids, we believe that the data generating process could look something like this, in a situation where the seller is hesitating whether to sell or not:

if the winning bid is far from the losing bids, this indicates a buyer's private shock, thus my decision tilts towards selling

if the winning bid is close to the losing bids, this indicates a common shock, thus my decision tilts towards not selling

Unfortunately, in order to model this kind of reasoning, we have to give up the comfort of the classic auction models which means that we might not have a closed form solution.

### How to model such DGP?

The data comes to the rescue.

It turns out that while finding the equilibrium given model primitives is often quite difficult, recovering the primitives given the observed bid distribution is trivial most of the time. In other words, it is much easier to calibrate a convoluted auction model given data than solve an abstract model. However, this puts rather strong requirements on the data. Namely, we have to observe two things:

seller's appraisal before the bids were submitted

seller's appraisal (or binary decision) after the bids were submitted

Luckily, the very data (French timber) that motivated this research possesses exactly these two pieces of information.

### How to estimate such model?

The details of estimation are rather technical, but the main idea is that we have to achieve three goals

allow for the seller to be uninformed (or imperfectly informed) about her true value

modify the classic identification result a-la-GPV to allow for the seller to "respond" to bids

decide how to model seller's "learning" from bids

### How to model learning?

We chose to model this part of the DGP directly, so to not overcomplicate the model:

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.

### How to fix the identification?

We modified the classic non-parametric 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)

With secret reserve prices, 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)

### What is the main finding?

Our main finding is that secret reserve prices can, in fact, achieve higher efficiency than public reserve prices, albeit at some revenue loss. While we can not claim that we know what ONF was thinking when implementing secret reserve prices, our conjecture is that by looking at the position of losing bids relative to the winning bid, the seller could separate the common shock from the buyers private shock and thus take a more informed decision.