Mind Your Decisions

Why eBay and Craigslist are Similar in Theory: the Revenue Equivalence Theorem

Every Tuesday is a game theory article at Mind Your Decisions

I can be bought. How can I get a high selling price?

Imagine for a moment that I was for sale. Let’s say I was selling myself as a date to raise money for charity. There are many factors I might consider to increase my selling price.

How I describe myself is important. I’m not good at this, so I’d probably hire help to make it sexier, but a basic pitch would be:

“Buy a date with Presh. He’s a recent Stanford graduate and now a popular money writer. He has blue-green eyes, a clean-shaven look, and he loves cooking.”

Once I optimize my description, I’d move on to other considerations. I would want to inform all interested buyers through advertising. Then, it might be possible to team up with local businesses to include marketing tie-ins with the date, say, by including free flowers with every date.

All of these initial factors will affect my selling price. The more desirable I appear, the more interested bidders there are, and the more stuff a winner gets, will all increase my selling price.

What about the auction type?

You might see a factor missing from the list: the auction type. Would the auction type make a difference for revenue? At first it would surely seem so, as designers usually labor over specific rules and regulations to ensure a fair auction.

But the theory has an interesting answer. It says no; in a theoretical world, auction type will not affect the average seller revenue for certain kinds of auctions. This is the intuitive idea of the revenue equivalence theorem, the most famous result from auction theory.

The theory seems very odd at first, most notably because auction types do matter in practice. There are experiments to this effect, and anyone who has held an auction will tell you the rules matter. But that doesn’t mean theory is not worthwhile.

Why should I learn the theory?

It’s a question that has kept me thinking at times too. I’ll admit that I used to think revenue equivalence was a case of misguided economists. But that changed when I took an auction theory class and heard my professor was using revenue equivalence when pitching services to businesses like Google.

I surmised there are three reasons to learn about revenue equivalence. First, the theory provides insight into how economically modeled people might act, and that’s a starting benchmark before experiments take place. Second, the theory is a great introduction to the importance of bidding strategy. And third, the theory illustrates how auctions that look very different might produce similar results on average.

In short, theory will help you in practice, unlikely as it appears. Here’s an analogy that helps me. When learning theory, think about yourself as an athlete at the gym weight training. What you’re doing can’t be directly applied, but it will prepare you when you’re on the field vying for a championship.

The main difference between eBay versus Craigslist

Suppose the charity could not find a physical location and decided to go online. Since we’re doing a thought experiment, imagine it was legal to sell me as a date on eBay and Craigslist.

If the charity informed all interested buyers before the event, would the actual auction type make a difference?

Here are the main differences between the two venues:

eBay auctions are open bids.

It is an online version of the “going once, going twice, gone” auction houses. This kind of auction is known as an English auction or ascending open outcry.

Craigslist auctions are secret, or sealed bids.

Sellers typically offer goods with some price but then include an “or best offer” (OBO) option. Buyers secretly email bids to the seller, and the highest bid usually wins. This kind of auction is known as a first-price sealed bid auction.

There are other differences—eBay auctions have time dynamics—but I’ll save that discussion for a future article devoted to eBay.

With these thoughts in mind, let’s see how the auctions would run.

How the bidders would think

We’ll assume each bidder goes through the following process: they read my description, think about their own individual tastes, and then come up with a maximum amount they are willing to pay for me. A maximum value of $100 dollars means the woman is just different between having me for that price and dropping out.

An important restriction is the individual value doesn’t depend on what other people think about me. That is, if a woman sees her best friend outbid her at $101, she won’t start thinking I’m sexier, get jealous, and bid higher just for spite. Such a valuation is known as a private value—it depends only on personal tastes. (The case of common values is treated differently, and occurs in markets like bidding on an oil rig.)

How the auctions should end

Strangely, the logical place to start is at the end. Rather than try to figure out the dynamics of bidding and the strategies, let’s figure out who is likely to end up winning me and then reason how the players might have bid to get there.

Who will win me? Naturally it’s the person that’s willing to pay the most. This person will win me in any sensible auction.

What price should be paid? It will not be the maximum value because there is a trick. The highest bidder knows that it’s not necessary to pay full price. All she has to do is pay just more than the next highest bidder.

If the second highest bidder would pay $100 for me, then the highest bidder could win at $101 (or whatever the next bid increment is).

In fact, that logic applies to any auction with private values and where only one person pays. The logic can be even extended to further auctions where more than one person pays (ex. everyone has to pay their bid). So long as the highest person gets the item, the lowest person has zero surplus, the seller cannot expect more than just above the second highest value. That’s the essence of revenue equivalence.

Regardless of the auction design, the average revenue in equilibrium for a single item with private values will be one bid increment above the maximum amount the second bidder would be willing to pay.

This equilibrium process has a statistical interpretation because of uncertainty. In the Craigslist posting, the highest bidder doesn’t know she’s the highest. She has to guess at the second highest bid, and so the bid might have an error in it. But if the auction were simulated millions of times, the errors would vanish and the selling price would be the same as in the eBay auction, where bids are seen by all.

The bidding strategies

Now that we know what should happen, let’s go a step back and think about the bidding strategies that would cause such an outcome.

The eBay auction has a simple bidding strategy: keep bidding until you reach the maximum you are wiling to pay. If you wanted to pay $100 before the auction started, then you keep bidding until the price exceeds that value. There is even an automatic bidding process on eBay to let you do this. This strategy is dominant for all players and the selling price will be one bid increment above the maximum amount the second highest person is willing to pay. (The practical strategy is a bit more complicated because eBay auctions have a time element to them.)

The Craigslist auction strategy is not as easy because you can’t view other bids. A starting point is realizing that you don’t want to bid higher than your value. If your value is $100, it does not make sense to bid $101. You might not win, but if you do, you are sure to overpay. The next step is realizing you don’t want to bid exactly your value. If you do that, then you’ll only break even when you win. So the idea is to bid some amount lower than your value. How much lower?

In this auction, you know that you only win if you’re the highest value, and you ideally would want to pay the least amount—one bid above the maximum for the second highest person. It’s impossible to know where your relative value is, so what you do is the following. You guess that you have the highest value, and then estimate the second highest value and bid just higher.

The practical value of revenue equivalence

There is experimental work that questions whether revenue equivalence actually happens. Here is a good discussion of the experiments.

What I take away is that actual auctions have strategies and psychological considerations that aren’t captured in the theory. Sellers on Craigslist might choose a lower bidder because the person seems more trustworthy. Buyers on eBay can use computer programs to exploit the hard timing of auctions.

But revenue equivalence can be useful because it reminds me to take care of all the non-auction elements. Rather than spend 20 hours debating where to list, I should instead spend that time making the product look appealing and establishing that I’m a reputable seller. And ultimately, that strategy has served me very well.

Appendix: An extension to two other auctions

There are several ways the auction could be held. Two closely related ways to eBay and Craigslist are:

Second-price sealed bid or Vickrey auction (similar to eBay):

This auction has a strange twist: after all bids are collected, the item is awarded to the highest bidder for the second highest bid. At first this auction sounds very strange. Why would you sell for less than the highest bid? Well, it’s because buyers take the rule into account when making the bid. Knowing they will only have to pay the second highest bid, buyers will bid higher than in the first-price sealed bid. These auctions are so rare that economists have credited William Vickrey in 1961 with suggesting this auction format. It turns out that second-price sealed bid auctions were actually quite common and can even be traced to 1797, years before Vickrey was born. These auctions are strategically similar to eBay—bid your value and the winner will pay just above the second price. I read about these stamp auctions in an interesting paper (need JSTOR access). The practical concern is trusting the seller to actually award the second highest bid rather than pump it up.

Dutch auction or descending open outcry (similar to Craigslist):

This auction is similar to the English auction but works in reverse. The auctioneer starts at a very high offer and keeps calling out lower and lower bids until somebody shows interest and says “I’ll take it.” The auction is named for the Dutch tulip auctions. These auctions are strategically similar to Craigslist—the highest bidder will jump in just before the second highest bidder would consider paying.