A Brief Primer on Auction Theory

About a month and a half ago now, I became extremely interested in auction theory and mechanism design. As I was an outsider looking in, I think I can concisely summarize what the important facts are, and how it might improve your thinking.

The Standard Types of Auctions:

We shall start by going over the three big types of auctions: first price, second price, and ascending (or English) auctions. A first price auction is when everyone submits a bid without knowing what other people have bid, and the highest bid wins, and pays the price that they bid. A second price auction is when everyone submits bits as before, and the highest bidder wins, but pays the price of the second highest bidder. An ascending auction is when bidding does not occur in one round, but over multiple, and the price is increased until all but one bidder drops out. 

Now, these auctions are more similar than they appear. In particular, the ascending auction is isomorphic to a second price auction. You go until all but one bidders drop out in an ascending auction, and the good is bought then and there — the auctioneer does not continue raising the price until the winning bidder has exhausted their willingness to pay. The second price auction has the advantage of requiring only one round of bidding, which makes the auction quicker, a not insubstantial advantage. (Google’s advertisement slots, for example, are sold by auction in the milliseconds before a page loads; an ascending auction might increase latency!) A Dutch auction — a variant where the price starts at some point far beyond what anyone would be willing to pay, and drops at a predetermined rate — is isomorphic to a first price auction, though it can be configured as a second price auction by not stopping the auction until two people have bid. 

And why would the auctioneer prefer for people to pay the second price, rather than the total they bid? They do this because it makes reporting your honest evaluation of its value weakly dominant. Suppose there is an auction with two buyers, you and the other guy. You value the good sold at ten dollars, but you bid at nine. This either: does not change the price you pay (they bid less than nine); leads you to not get the good when it was worthwhile (they bid between nine and ten); or does not affect what the outcome would have been either way (they bid greater than ten). In the other direction, increasing your bid would result only in either no change, or you purchasing an item at a price beyond what you think it is worth. Thus, no player benefits from bidding anything other than what they think the good is worth. I did say it was weakly dominant; and we will turn to what can cause reality to diverge from theory later.

And there is another reason why these auctions are very similar to each other — with some assumptions, all of these methods will allocate the good to the same people for the same reasons, and so it will yield the same price. This is the Revenue Equivalence Theorem, described first by Roger Myerson in 1981 in “Optimal Auction Design”. (That is the second half of the paper; the first half of the paper, equally important, will be discussed later). Stated formally, so long as the buyer with the highest reservation price wins, and everyone else doesn’t expect to get anything, revenue for the seller will be the same. The first price auction will, in the long run, become identical to the other auction forms. It being easier to assess your own private value, than to assess everyone else’s values for a good, and that it is better to do an auction in a short amount of time, are reasonable assumptions which still suggest a second price auction is best, but these are assumptions of realism which need not figure in. We shall assume them for the rest of this blog post, however.

Breaking the Assumptions:

Thus far, we have assumed that buyers know their own valuations, and are rational and play the optimal strategy; and that sellers are unable to charge different prices to different people or groups, and that they can credibly commit to a certain future course of action. We will see what happens if we relax certain assumptions, and how that changes what the optimal auction is. For example, suppose bidders aren’t certain about their own valuations. In that case, an ascending auction becomes superior to a second price auction, because bidders can observe other people’s bidding behavior and deduce information. This helps avert, in resources whose value is unknown but the same to all buyers, what is known as the “winner’s curse” – suppose many companies are bidding on the rights to drill oil on a plot of land. The value of the oil is the same to all buyers, but the person who buys it has the most optimistic assessment of all of them. If we assume that the average of all participants guesses are correct, then having an auction is going to create a lot of losers. People shade down their estimates in equilibrium.

The second price auction also suffers from a serious credibility problem. The auctioneer cannot be bound by the structure of the bidding in a second price auction — they could profit greatly by lying about the second lowest bid, and extract value from the buyer. Even if they cannot lie about the bids directly, confederates among the bidders could opportunistically increase the price. Shengwu Li and Muhammad Akbarpour codify this in “Credible Auctions: a Trilemma” in 2020. “Static” means that it requires only one round of bidding to resolve a winner.

We can also relax the rationality constraint. We are not saying that they no longer act in self-interest, but just the optimal strategy for an auction may be non-obvious. It would certainly be no accomplishment if auction theory were unable to describe anything which had not been in long practice! In these cases, obviously strategy proof mechanisms are better than those which are not – Li 2017 operationalizes this as any mechanism by which the worst case outcome of the dominant strategy is better than the best case outcome of any other strategy, no matter how stupidly other bidders might behave. This pushes us toward preferring simpler to more complex mechanisms, even if the more complex auctions might be more optimal. There are also, especially in cases where multiple goods are to be sold, bundled or unbundled, times when it becomes computationally infeasible to optimally allocate goods, in which case, again, simpler mechanisms are preferred. Thankfully, these have been shown to be tolerably close to optimal (although the math goes a bit beyond my ability to precisely explain why this is the case). See, for example, “Multi-parameter Mechanism Design and Sequential Posted Pricing”.

The Case of Monopoly:

We have also, up until now, assumed that there was no monopoly. It is rare that a perfect monopoly with unlimited powers of commitment arises, but the central results of the literature on this are applicable, with diminished strength, to any case where the demand curve is downward-sloping – in other words, almost always.

The first half of Myerson 1981 asks the question: what is the revenue-maximizing auction for a monopolist selling a single good, a single time? If the monopolist knew the exact amount every person would be willing to pay, they would offer the person with the highest value at that price, but it is very rare to know that with precision. Suppose that you didn’t know where the price is exactly, but you know the distribution in which it would be. In that case, you set a reserve price at the midpoint of the distribution of possible values. Suppose there are two buyers, who have a value drawn at random from between 0 and 1. The monopolist would bid at .5, with a total expected value of 5/12, and commit to never selling the good if that price is not paid (even though both of them would be willing to pay in excess of the seller’s valuation of the good). The seller is trying to get people’s true valuations out of them; retaining the second price mechanism with a floor does this. A second price auction with two buyers would yield a price of only ⅓, making the Myerson procedure obviously better. 

The really cool extension is when the distributions are not identical for every buyer. Suppose there are two buyers as before, but now one of them has a value between 0 and 10, and the other between 10 and 100. In a normal second price or ascending auction, the expected value would be only 5. In the Myerson auction, the seller gives a take-it-or-leave-it offer first to the higher distribution, at 50, and then to the lower distribution, at 5. The expected value here is 24.75 (45/2+5/4). Social loss, however, is baked in – there are times when it is optimal, not only to never sell at all, but to also sell to someone who’s valuation of the object is always strictly less than another’s!

The mechanism can be thought of as a type of price discrimination, specifically third degree price discrimination. Bulow and Roberts 1989 clearly and precisely (seriously, read it, it’s the best introduction) how it is implied by the normal microeconomic logic of economists. 

Various Non Pareils:

There are a few things which did not fit into the organizational scheme of this essay, but seemed worth telling about. Bulow and Klemperer 1996 combine the arguments in favor of simplicity with the Myerson auction by answering the following question: would you rather have market power, or more buyers? Stated formally, would you rather have N buyers, and have perfect market power, or have N+1 buyers, and have an ordinary ascending auction? No matter what, if we assume that all potential buyers have the same distribution, it is always better to have one more buyer than to have perfect market power. Imagine N is one, and all buyers have an expected value between 0 and 1. Then, your expected value under a Myerson auction is .25; and the expected value under a second price auction is ⅓. The second price auction stays slightly ahead of the Myerson auction at all points.

Another is the Coase Conjecture, which can be thought of as relating to the problem that the monopolist has in committing to not selling. Suppose one person owned a perfectly durable good – for example, all the land in the United States – and that they can only sell it at one price. If they wish to maximize revenue, then they will need to restrict the quantity sold, and sell at a higher price. So, this is done – and now they have some land, which they would want to sell. And so on, and so on, and so on. If everybody knows that the monopolist will eventually drop their price, and the monopolist cannot commit to a course of action, then the good will sell at the competitive price, even under a monopoly. The monopolist is, in essence, in competition with future versions of themselves!

A Brief Syllabus:

I err on the side of inclusion, but those papers which are absolutely essential to read will be preceded with a star. Many of the papers are worth reading even if only for the headline results – the math of proving it can be difficult and relatively unimportant for your day-to-day applications. 

“A Theory of Auctions and Competitive Bidding”, Milgrom and Weber 1982, https://www.jstor.org/stable/1911865 

“Obviously Strategy-Proof Mechanisms”, Li 2017, https://www.jstor.org/stable/44871788 

⭐“Credible Auctions: a Trilemma”, Akbarpour and Li 2020, https://web.stanford.edu/~mohamwad/Credible.pdf

Multi-parameter Mechanism Design and Sequential Posted Pricing

https://en.wikipedia.org/wiki/Coase_conjecture 

⭐“The Simple Economics of Optimal Auctions, Bulow and Roberts 1989, https://www.cs.princeton.edu/courses/archive/spr08/cos444/papers/bulow_roberts89

⭐“Optimal Auction Design”, Myerson 1981, https://www.eecs.harvard.edu/cs286r/courses/spring07/papers/myerson.pdf 

⭐“Auctions vs Negotiations”, Bulow and Klemperer 1996, https://www.jstor.org/stable/2118262 

“Counterspeculation, Auctions, and Competitive Sealed Tenders”, Vickrey 1961, https://cramton.umd.edu/market-design-papers/vickrey-counterspeculation-auctions-and-competitive-sealed-tenders.pdf