# How to win big in eBay auctions: Part I

OK, I lied a little bit: I can’t guarantee that you will win lots of really awesome things for dirt cheap. But I did stay at a Holiday Inn Express last night. What I can provide, though, is a perfect Bayesian equilibrium strategy that will mean you are bidding optimally (assuming others are also bidding in a similarly defined manner). Basically, I wrote my senior thesis at Princeton on eBay, so I know how it works pretty well.

I’m not going to go through the exact details of how eBay works, since I assume if you care, you already know more or less about that (you can look here for more information). I’ll also omit most of the gory details of how exactly to rigorously demonstrate that these strategies mathematically form an equilibrium, since I assume most readers will care mostly about the practical implications. I hope to upload a link to my thesis, so if you want, you can take a look at that yourselves.

At first glance, eBay seems to work exactly like a sealed-bid second-price auction. Indeed, eBay itself suggests that one should always bid exactly how much one values the item up for sale. But there are two potential issues. First, there are likely to be multiple items of the same type. For example, suppose I collect stamps: there are likely to be multiple copies of the same stamp (unless it is extremely rare). Thus, I might want to avoid bidding on an earlier auction of a stamp, if I could get the same one a little later for a better price (we’ll assume no discounting, since it doesn’t much change the reasoning). Second, eBay complicates the general second-price auction setup by showing the price history; that way, it might be possible to infer how badly others want, say, the stamp, and use that information to one’s advantage.[1]

Fortunately, in the single-item case, the reasoning for second-price auctions works for eBay auctions as well: one should always bid exactly the value of the stamp (though it’s no longer a dominant strategy). There is also a well-established result for multiple items, first shown by Robert Weber, which is that if there are $N$ items, and $M$ bidders (where $M>N$, and each person wants exactly one item), then we can rank the bidders’ valuations from high to low as $(V_{1},V_{2},...V_{M})$; it is then a subgame-perfect equilibrium for a person with valuation $V$ to bid
$b_{l}(V)=E[V_{N+1} |V_{l+1}=V]$ (a)
in the lth round of bidding; that is, one bids what one expects the $(N+1)^{th}$ highest valuation to be, if one were to have the $(l+1)^{th}$ highest valuation. For period $N$, this works out to bidding one’s valuation, so it fits nicely with the single-item case.

Can we generalize this to the eBay model? Fortunately, as I showed in my thesis, we can: all it takes is to construct a set of beliefs where the bidders will ignore the previous bids of others, and bid as they would in the sealed-bid case. The best way to break down the cases is to those of two items, and those of three or more.

In the former, given that, in the second period, everyone is just going to bid as in the single-item case anyway, they will bid as in the sealed-bid, two item case no matter what at the last moment. We just need to make sure that no one will screw things up by bidding earlier. We can ensure that this happens by, say, constructing an equilibrium in which bidders believe that only people who value the stamp very highly would bid earlier than the last moment; so, if someone else has bid earlier, the rest of the bidders would assume they had lost anyway. Since one can’t see the top bid (only the current stamp price, based on the second-highest bid up to then), this belief is plausible. And if they’ve lost, they might as well just bid what they would otherwise, as given by equation (a), since bids are costless. This solves the two-item case.

In the three-plus item case, we have to be a little more careful, since if everyone just blindly bid as in equation (a), you might learn other’s valuations in earlier rounds, realize that you weren’t going to win at all if you continued bidding as in (a), and outbid others who valued the stamp more highly to “steal” the stamp. To get around this, one could construct an equilibrium where the bids are staggered – those who want the stamp more bid earlier. Since equation (a) is monotonic in V, this means that no more than two bids can occur in any given period; those who are supposed to bid earlier cannot, since the item price is already greater than what they are willing to bid (eBay requires that all bids must be greater than the current item price, since otherwise they are not relevant for determining the price of the given item, given the mechanism eBay uses). By assuming some plausible beliefs so that bidders would ignore anyone who deviated from this strategy profile, we can ensure that all bidders will adhere to the strategy profile – they’ll have no incentive to deviate.

Do people actually bid like I described in the multiple-item cases? Obviously not – most people don’t even bid their valuations, let alone think things through as much as here. Nor does it seem that people time their bids as in the equilibria described here. Yet the basic idea, that if one can ignore the bids of other people in determining how much they want the item, and only focus on bidding up to the value suggested in equation [1], that continues to make sense. After all, really, does anyone actually try to memorize how much someone was willing to bid last time a stamp came up for auction? Come on. Since no one tracks bids like this, it’s safe to bid as described here in general.

Note that, unlike the sealed-bid case, the equilibrium described here is not the unique symmetric one. In fact, there is an equilibrium in which all items are sold for the start price, if sold at all. But we’ll get to that in part II.

[1]Indeed, this issue causes there to not be any pure-strategy, increasing bidding function for multiple-item auctions with bid revelations. See Cai, Wurman and Chao (2007).