“Optimal Auction Design Under Non-Commitment,” V. Skreta (2011)

Consider a government selling an oil or timber tract (and pretend for a second that the relevant uncertainty is over how effectively the buyers can exploit the resource, not over some underlying correlated value of the tract; don’t blame me, I’m just using the example from the paper!). If buyers have independent private values, the revenue-maximizing auction is the Myerson auction, which you can think of as a second-price auction with a reserve price. But if a reserve price is set, then sometimes the auction will end with all bids below the reserve and the government still holding the resource. Can the government really commit not to rehold the auction in that case? Empirical evidence suggests not.

But without this type of commitment, what does the optimal sequence of auction mechanisms look like for the government? There is a literature beginning in the 80s about this problem. In any dynamic mechanism, the real difficulty comes down to the fact that, in any given stage, buyers do not want to reveal their actual type (where type, in our example, is their private value for the good). If they reveal in period 1, then the seller can fully extract their surplus in period 2 because private values are now public information. So there is a tradeoff often called the “ratchet effect”: mechanism designers can try to learn information in early stages of the dynamic mechanism to separate types, but will probably have to compensate agents in equilibrium for giving up this information. On a technical level, the big problem is that this effect means the revelation principle will not apply. This, you won’t be surprised, is a huge problem, since finding an “optimal” mechanism without the revelation principle means searching over any way to use messages that imply something about types revealed in some sequence, whereas a direct truthful mechanism means searching only over the class of mechanisms where every agent states his type and does not want to mimic some other type.

In a clever new paper, Vasiliki Skreta uses some tricks to actually solve for the optimal sequential auction mechanism. McAfee and Vincent (GEB 1997) proved that if an auctioneer is using a first price or second price sequence of auctions, revenue equivalence between the two still hold even without commitment, and if the discount rate goes to 1, seller revenue and optimal reserve prices converge to the static case. This result doesn’t tell us what the optimal mechanism is when discount rates matter, however; oil tract or cellular auctions may be years apart. Skreta shows that the highest perfect Bayesian equilibrium payoff is achieved with Myerson style reserve prices in each period and shows how to solve for these (often numerically due to their formula complexity). In the case of symmetric bidders, the optimal mechanism can be written as a series of first price or second price auctions, with reserve prices varying by period.

The actual result is probably less interesting than how we reach it. Two particular tricks are worth noting. First, Skreta lets the seller have total flexibility in how much information to reveal to buyers. That is, after period 1, if there is no sale the auctioneer can reveal everyone’s bids to each bidder, or not, or do so partially. Using a result from a great 2009 paper, also by Skreta, it turns out that with independent private values, it does not matter what information the seller reveals to the buyers in each period. That is, when looking for optimal mechanisms, we can just assume either than every bidder knows the full history of bids by every other bidder, or assume that every bidder knows only their own full history of bids. The proof is difficult, but the intuition is very much along the lines of Milgrom-Stokey no-trade: all agents have common priors about other agents’ types, and auctioneers can lie in their information disclosure, so if a seller is willing to offer some disclosure, it must be that this disclosure has negative expected value for bidders. For buyers to be willing to “offer their type” in an early stage to the auctioneer, they must be compensated positively. Since disclosure can’t be good for both buyer and seller by a type of Milgrom-Stokey result, any information disclosure rule that is able to gather good information has expected value zero for the seller. Obviously, I’m eliding the difficulties in the proof, but this is the basic idea.

The second useful result is about when you might want to separate types as the auctioneer. Imagine you are committing to holding the auction no more than 6 times. Should you separate early or late? It turns out you want to separate types as late as possible. That is, the optimal auction will, in period 1, sell to the highest bidder above some cutoff. The auction does want to separate types above that cutoff. But below that cutoff, buyers know that by not pooling their types, they are giving the auctioneer information about their private values which will be used in period 2 to extract rents. In equilibrium in period 1, then, the low-value buyers will need to be compensated to not choose actions that pool their types. This compensation turns out to be so expensive that the informational gains to the seller in period 2’s auction are not valuable enough to make it worth the cost.

A few comments: really what we care about here is what happens as the number of auctions which may occur goes to infinity. The principle is proven to be the same – cutoff rules are optimal – but how do those cutoff rules as the lack of commitment gets arbitrarily bad? What revenue properties are there? For instance, for various discount rates in the 2-bidder, uniform [0,1] private value auction, does seller revenue approach zero? Or does it asymptote to some fraction of the revenue in the auction with commitment? I think these types of applied theory questions are quite important, and should be answered here. My hunch is that even in simple cases the mathematics becomes too difficult as the number of periods grows large, and this is why Skreta didn’t include it. If the loss to the auctioneer from non-commitment turns out to be really large, we already know the “optimal” auction without commitment is to assign the rights to the auction to some second seller who, perhaps for contractual reasons, actually can commit not to resell. If the loss in some simple numerical examples is no more than 10%, then no big deal.

http://sites.google.com/site/vskreta/research/auctions_rev_3.pdf (WP version, Jan 2011)

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