EBay and Second Price Auctions

In a second price auction, the winner pays, not his bid, but the second highest bid, or an amount just above it—the lowest price that would have won. In theory, this makes it in each bidder's interest to bid the highest amount he is willing to pay. Bidding less than that changes the outcome for him only if it puts him below what would otherwise have been the second bid, in which case it results in his losing an auction that he would rather have won,.

EBay uses a second price auction, and as far as I can tell by a little casual observation it fails to work in practice according to the theory, which is an interesting puzzle. There is probably published worked attempting to solve it, but I like puzzles and it isn't really my field, so I have been trying to think about this one.

My interest was spurred this morning when I lost an auction. The item was a kard, an indo-persian knife. I used to collect such things, although it's been years since I added to my collection. I happened across this one on ebay, it was a very attractive piece at a very reasonable (current second high bid) price, so I put in a bid on it. When I went to sleep last night mine was the high bid. When I got up this morning someone else had bought it for very slightly above my bid. My impression is that that is a not uncommon pattern—the winning bid coming in at the last minute. I have a bid currently up for a similar piece, in an auction that (unlike the first) terminates at an hour I am likely to be up, and I plan to watch the final minutes and see what happens.

Thinking about it this morning—I had not yet checked the outcome, but I thought it quite likely that I would be outbid—I came up with one theory, then discarded it. That theory started from the fact that, in a second price auction, there is a final bid—a bid made just before the auction closes, late enough so that other bidders cannot use the information provided by that bid to alter their own. Perhaps the optimal strategy involves putting in a low bid, or no bid, early, so that other bidders will think they are winning and so not realize they need to make relatively high bids to get the item, then put in a higher bid at the last minute.

The problem with that theory is that the same logic I started with implies that the last minute bid ought to represent your true value for the item—the highest price at which you would rather win the auction than lose it. If all bidders act that way, we are back with the original situation. And if each bidder expects the others to act that way, he might as well put in his high bid at the beginning, thus saving himself the trouble of getting up early in the morning, or staying up late at night, in order to cast a high bid at the last minute.

There is, however, a fancier version of the theory that might work. Up to this point, I have been assuming that each bidder knows what the item is worth to him. Perhaps that is the assumption that needs to be modified.

Suppose I would rather pay a hundred dollars for a kard like this than not have one, and further suppose I only want one for my collection. If there were only one kard in the world, it would be worth a hundred dollars to me. But if there is more than one, how much this one is worth to me today depends in part on what price, if I don't get it, I would be able to buy one of the others for tomorrow. That in turn depends in part on how much other buyers value such items at. It follows that observing that someone else has made a high bid for the item is a reason for me to reevaluate upward its value to me. A further reason is that high bids on this item signal that it will have a high resale value; since it is possible that I (or my heirs) will some day want to sell it, that also increases its current value to me. The argument is stronger still if I am, not a collector, but a reseller or a speculator, buying the item in order to later sell it.

This argument brings us back to the last minute strategy. Making a high bid early will drive up the auction price, either to the previous bidder's bid (if I am now the high bidder) or to just above my bid (if I am now the second highest bidder). That will provide information that will lead all other bidders to reevaluate the item up, increasing their (current or last minute) bid accordingly. So I postpone my high bid until the last minute of the auction, when it is too late for other bidders to react to the additional information.

This is, I think, a possible explanation of my initial puzzle, but I am not sure it is correct, or even that it explains all the available data. I have not looked carefully enough to be sure, but my impression is that the spread of prices for such items on ebay—antique indo-persian edged weapons—is larger than I would have expected, with prices for similar items ranging over an order of magnitude or so. That is based on a comparison of prices things are currently up for—a lower bound of their selling price in the case of auctions—with the price that the particular item I bid on actually sold for. I have not watched ebay long enough to be sure that is correct, and different items are not of course identical, but it is at least my impression.

That raises a second possibility for what is happening. Suppose the market for any single item of this sort is very thin. Suppose there are lots of people who value it at a hundred dollars, and a small and unknown number—say between zero and five—of enthusiasts who value it at a thousand. On our original model of how a second price auction works, the item would usually go for a thousand dollars—when at least two enthusiasts were bidding. But occasionally there would be one or zero enthusiasts bidding for the item, and it would go for a hundred dollars.

Now suppose you are the seller, and further suppose that you believe that if there is only one or zero enthusiasts in the market for your item today, you can probably do better next month or next year. You might decide to put in a bid yourself at, say, nine hundred and ninety dollars—either explicitly as a reservation price or covertly by pretending to be an independent bidder. The latter option isn't free, since you will have to pay EBay a commission if you end up buying the item from yourself, but it might still be worth doing.

Further suppose that, even for enthusiasts, watching auctions and bidding on them is costly, so that they will only do so if they think they have a good chance of getting the item they bid on for well under its value to them. The seller now has an incentive to use the covert bid strategy—at the last minute—thus enticing people to follow his auctions. And if you suspect, on the basis of bids made prior to the last minute, that there are no enthusiasts out there this time, you might put in your last minute bid at a hundred and one dollars. That not only reduces the commission you will have to pay to EBay, it also signals buyers looking at the auction after the fact that they have a chance of getting a very good deal on your auctions.

Do any of my readers know of published work exploring such theories or offering other and perhaps better ones? Empirical work? EBay provides, after all, a vast volume of real world data on the working of second price auctions, most of which (I presume) could be mined with suitable software at a relatively low cost.