Secret Reserve Prices
Imagine a seller that is poorly informed about his reservation value. By not committing to a reserve price prior to the auction, the seller gains an ability to learn (say, the resale value) from the bids and thus pick a better reserve price. Secret reserve prices give the seller a strategic advantage, but at the price of the lack of commitment.
Goal of the project: To test whether the informational advantage of secret reserve prices can out-weight its lack of commitment and whether it is sustainable in equilibrium.
Methodology: Since there are (as for 2020) no analytically tractable models, suitable for our needs, we choose a semi-applied approach. This means building a structural model flexible enough to allow learning from bids, and estimating it non-parametrically a-la-GPV, with deconvolution tricks for unobserved heterogeneity. We then run a large number of MC simulations to see how the auction performs under alternative configurations, and compare the results.
Theoretical background: Our intuition from the classic auction and mechanism design literature (which traditionally focuses on the symmetric IPV models), tells that keeping the reserve price secret is not efficient (nor it is optimal). A formal analysis shows that one wants to sell whenever a private value (or a virtual value) exceeds the reservation value of the seller. All stochastic solutions are therefore strictly inferior to the binary (bang-bang) ones. However, this is only true if the values are independent. In most contemporary auctions, whenever a good is durable, values are clearly interdependent, due to numerous resale opportunities.
Data: At the very least, the data should have certain variation in auctioneer behavior before and after the bids were submitted. This variation can be seller-to-auctioneer communication, post-auction re-negotiation, or any other reliable source. In a sense, the data should contain two different reserve prices: ex-ant and ex-post; and they have to disagree at least sometimes, otherwise there would be no learning.
In our empirical application (French timber auctions), we exploit an estimate of the seller value, referred to as "reserve price" in his internal documents, as an ex-ante reserve price. The ex-post reserve price is inferred indirectly, using the sellers final decision.
Identification: Since we aim at a model with unobserved heterogeneity, we need some assumption to aggregate a large number of diverse auctions. This assumption turns out to be very simple. If R_1(Y,B) is the ex-post reserve price, where B is the vector of bids and Y are all other appraisal channels (such as estimates, lot characteristics e.t.c.), then we want it to be homogenous degree one in both Y and B. While homogeneity binds only one degree of freedom, and so is a very weak assumption, it carries to powerful identification results and unlocks the estimation a-la-GPV with deconvolution.
Empirical puzzle: While the presence of a reserve price that is not enforced is a puzzle on its own, there is also a strong and unusual relationship between the observed variables: whenever a bid exceeds the ex-ante reserve price, it also exceeds the ex-post (that is the lot is sold), but the converse is not true, see Figure below.
We argue that this pattern, as strange as it may seem, is consistent with the ex-ante reserve price being a certain "conservatively high" predictor of the ex-post, ignorant of the distribution of bids. Denote R_0(Y) the ex-ante counterpart of R_1(Y, B). Let R_0 solve the following equation R_0=R_1(Y, R_0, ... , R_0), that is, the ex-ante reserve price is the pivotal value of the ex-post reserve price. Under homogeneity degree one, this implies that if the maximal bid exceeds R_0 it also exceeds R_1 and thus the lot should be sold.
Results: To set the scale, we first estimate the change in surplus and revenue under (sellers) first-best scenario. Letting the seller know ones true value (but not the bidders values) yields an increase in revenue by 8.41% and surplus by 6.73% compared to what appears in the data. However, the seller can not learn ones true value without forfeiting commitment, so we estimate the change in surplus and revenue under an ad-hoc linear updating rule. We show that even with the linearity constraint, there are weights that yield a significant increase in surplus (see Figure below), but always with a loss in revenue.
The intuition is that to gain flexibility necessary for a more efficient allocation, the seller forfeits the commitment power of a public reserve price, thus lowering competition. Therefore, there is a trade-off between revenue and surplus.
Conclusion: Learning from submitted bids improves allocative efficiency albeit with some revenue loss.