A lot of times you just need more bidders
Observations on auction theory
Auction theorists are often concerned with an abstracted, and somewhat idealized, world. This is inevitable — how else will the problems be tractable — but can diverge from the real world. Entry and exit from an auction may not be free, or may require immense fixed costs. This may lead to a substantial divergence from optimal allocation. There is a substantial silver-lining, however — for most things, simply increasing the number of bidders is sufficient to remove the distortions.
To give a brief glossary of terms here at the beginning: a second-price auction is one in which everyone submits bids all at once, and the highest bidder wins, and pays the second highest bidder’s price. The great advantage of this is that it incentivizes truthful revelation of one’s value for the good. It is identical in outcome to an ascending auction — the usual sort of “I have five, gimme ten, I have ten, gimme 15” auction we are used to. The Myerson auction is a second-price auction where the seller themselves submits a bid at the midpoint of each buyer’s distribution of possible values for the good.
Let’s take the Myerson-Satterthwaite theorem, introduced here in 1983, and also discussed along with Myerson 1981 by Bulow and Roberts 1989. (I recommend the latter, beginning in section 7, as especially clear to the non-specialist). Suppose one person wants to sell a good to N bidders. They find that there is no mechanism can simultaneously make it:
Optimal for bidders to reveal their true valuations;
Be individually rational;
Optimally allocates the good;
Requires no outside subsidy.
Why? Let us suppose there are only two people, Bob and Sally. (Standing for, of course, buyer and seller). Sally does not know Bob’s value for the good. (If she did, this exercise would be easy — she would simply charge the highest value Bob is willing to pay, and never budge). Neither does Bob know Sally’s value. They do know the distribution of values from which it is drawn. Suppose it’s between 0 and 1. We will also assume that Sally’s value is 0, to keep the math simple. Since Sally is rational, the way for her to maximize profit is to give a take-it-or-leave-it offer at the midpoint of Bob’s distribution, or .5, with an expected value of .25. (Why? The expected value is the pay-off times the probability that they take it. Between 0 and 1 this turns into x(1-x), or -x^2 + x. The first derivative is -2x + 1, which means it peaks at ½. Any difference from this will lower your expected value). Bob will take it if his value is above .5, and not take it if it is below.
This means that there are a great deal of trades in which Bob’s value exceeds Sally’s value, but no trades occur. The only way to guarantee all mutually beneficial trades will occur would be if an outside broker stepped in to subsidize the trades.
The good news is that the degree of inefficiency is related to the total number of bidders for the good. Suppose there are two bidders. Now, the odds that neither have a value which exceeds .5 is only 1 in 4. Add another, and it’s 1 in 8. The chance that a trade does not occur is given by 2^-N, where N is the number of bidders. So just get more!
A similar result is that of Bulow and Klemperer, 1996. They ask: would a seller prefer to have N people to negotiate with, and perfect market power; or N+1 bidders, and no market power? In the first case, they would essentially run a Myerson auction (giving each person a take it or leave it offer at their midpoint). In the second, they would run a normal second price auction. All values are between 0 and 1 as before, and we assume the seller has a value of 0. With N as one, this comes out as a take it or leave it offer at .5, with an expected value of .25. With two bidders, the expected value is (n-1)/(n+1), or ⅓. This follows for every N. It is always preferable to have one more bidder, and no market power, than perfect market power. What’s more, the second case is socially optimal because the good will always be sold to the person who values it more! These results are, of course, dependent upon the bidders all either sharing the same values, or them being high enough to be meaningful. In the language of Bulow and Klemperer, they must be “credible”.
More bidders also solves problems of collusion. Paul Klemperer has a highly entertaining survey article of all the ways poor auction design or a constrained number of bidders has led to bad outcomes. In 1991, for example, England sold television franchises by auction. While other regions sold for around £10 per person, Midlands sold for one-twentieth of a penny per person, and Scotland for one-seventh of a penny. “What had happened was that bidders were required to provide very detailed region-specific programming plans. In each of these two regions, the only bidder figured out that no one else had developed such a plan.” (p. 172-173). Or another: the British originally planned to sell four mobile-phone licenses, but after noting that there were four main providers, increased the number of licenses to five. This encouraged other bidders to enter, such that there was genuine competition for all of the licenses, greatly increasing the prices paid. The Dutch imitated the British in form but not in intent, selling five licenses when they had five main entrants. As such, they sold for just a fraction per capita as the British licenses did.
You cannot collude with many buyers as easily as you would a few! So, if you’re ever trading off between optimizing the design of the auction, and expanding the number of bidders, choose the latter; and for that matter, do not underrate the value of advertisement.