# How to win big in eBay auctions, Part II: Winning stuff for free

So I promised I’d show how there’s a perfect Bayesian equilibrium in the eBay setup where all people pay only the price at which the bidding started, $\pi_0$. How’s that, you say? Won’t people want to bid up the price, if they’re willing to pay more?

The key to this equilibrium is, obviously, to remove the incentive to bid any higher. Since, in eBay, the highest bid up to any given point is not observed, we can describe perfect Bayesian equilibria based on beliefs for those situations if they were to come up. Thus, if any bidder bids something other than $\pi_0$ at any time except for the last moment, the other bidders can plausibly believe that this bidder has bid something ridiculously high, and so will respond by bidding up the price of the item, knowing that they have nothing to lose by doing so. This way, they can punish any bidder who deviates by bidding higher than $\pi_0$, so no one will do so.

For this equilibrium to work, we must also consider the timing and tie-breakers. eBay has structured its auction mechanism so that, once there is at least one bid on an item, all bids must be greater than the current price. This means that at most one person can bid $\pi_0$; the rest, if they will bid, must bid more. We can resolve this issue by stipulating that all bidders try to bid $\pi_0$ at some time $\tau$, and one of these, chosen randomly, does so successfully. Meanwhile, anyone who bids at some other time (rather than the last minute) is believed to have again bid something ridiculously high, and punished in the same way as the one who bids something different from $\pi_0$. Again, this will ensure that everyone bids only at $\tau$, except for maybe at the last moment when no one has time to respond to their bid.

We have to (finally) deal with what happens at the last moment. Suppose bidder $i$ is the lucky guy who successfully submitted his bid earlier at time $\tau$. At the last moment, to discourage others from trying to outbid him, now he is the one who bids ridiculously high. Knowing that they cannot win the item, no one else tries to submit a bid at the last moment.

One can formally check that this is indeed a perfect Bayesian equilibrium. Though this is unlikely to ever happen in reality, this shows the lack of uniqueness of symmetric equilibria (in the sense that all people’s strategies are ex ante the same) in eBay’s setup, and that we can get a pretty sweet outcome given the right beliefs. Pretty cool, huh?