In this video, we're going to think about how to bid in second-price auction. Remember that second-price auction is one in which a single good is being sold, all of the bidders submit their bids in sealed envelopes, the auctioneer looks at all the bids, awards the good to the person who bid highest, and makes them pay the second highest amount. And what we want to think about here is how you should actually bid. What makes sense to do in a second-price auction. The main claim of this lecture today is that truth telling is a dominant strategy in a second-price auction. In other words, no matter how the other bidders are going to behave, the right thing for you to do in a second price auction is to bid the amount that the good is actually worth to you. Now, in fact, you should know this already. And the reason is, that a second price auction is a special case of the VCG mechanism. Remember that in VCG, every bidder is paid the amount of utility that all of the other bidders get in the world where they exist. And then is charged the amount of utility that all of the other bidders get in the world where they don't exist. In the case of a second price auction, where there's only one good, no other bidders get any utility in the case where I win. And one other bidder gets utility in the case where I lose. That's the person who would've won if I hadn't participated. So, I should be paid zero, and I should have to pay the second-highest price. And that's exactly what the second-price auction does. So, in a sense that's all you need to know from this lecture. But I'd like to show you a simpler, more direct proof that doesn't rely on VCG, because the VCG truthfulness proof is a bit abstract. Here, I can give a very intuitive proof that I think is an elegant way of seeing why truthfulness is the right thing to do in a second-price auction. So the proof is going to work as follows. We're going to assume that the other bidders bid arbitrarily. That's important because we want to show that it's a dominant strategy to build truthfully, so I can't make any assumption how the other bidders will behave. And I want to show that no matter what, the right thing for a bidder i to do is to respond by bidding truthfully. And I'm going to consider two cases. The case where if bidder i bid honestly he would win the auction, and the case where if bidding honestly he would lose the auction. So let's start by thinking about the case where, bidding honestly, bidder i wins the auction. In this case, we can sort of look at this picture here. So here we have i's true value at this level here. And then here we have the amount that I bid. Now, we're saying that i bid's truthfully initially and so that means that his bid is of this amount here. And of course in a second price auction, the amount that I actually pay is not his bid, but it's the amount of the second highest price, which we've shown here. All right, so let's do a kind of case analysis and think about what would have happened if I had bid differently. If I were to bid higher then he would still win and he would still pay the same amount. So nothing would change. If he were to bid just a bit lower again nothing would change as long as he's still higher than the next highest bidder. But if he bids sufficiently lower then he would stop being the highest bidder And he would lose the auction and instead the next highest bidder would win. So what we can see here is that in the case where i has the highest valuation and he changes his bid, four things can happen, three of them it doesn't make any difference to him and in one case something changes. And what's important here is in the case where something changes, things get worse for bidder i. Because he used to be getting positive utility, right? There was this utility difference here between how much he valued the good, how much utility he gets for consuming the good, and how much he has to pay. So he was getting positive utility here, and here when he loses, he's getting zero utility because he doesn't get the good. And bidder i would prefer to be in this situation with positive utility than this situation with zero utility. And so, it's the wrong thing for him in the case where he has highest valuation to bid any way other than truthfully. What about the case where he wouldn't be the winner if he were to bid honestly? Well in this case if he bids lower, he would still lose and pay nothing. If he bids a bit higher, he'll again still lose and pay nothing. And if he bids high enough, he'll eventually win. But in this case, if he wins, he's going to pay this amount, which we know is higher than his valuation. And so, by bidding untruthfully in the case where he isn't the highest bidder, he's either going to pay too much, pay more than the good is worth to him or he's going to not affect the outcome. And so again we can see that there's no way that better I can gain by changing his bid. And that's the whole proof. Notice that we didn't have to assume anything about the relationship between other people's bids and their own valuations, and because of that, we've shown it's a dominant strategy for bidder i to disclose truthfully to the auctioneer what his valuation is. Now I'd like to end by just making the remark about English and Japanese auctions. These are much more complicated auctions in that they define extensive form games rather than normal form games. So bidders are able to observe something about how other bidders have behave in the auction. They can condition their bids on this information. In the case of English auctions, bidders have a rich set of strategic options at every stage. They can decide not just whether to drop out or stay in, but they can decide to place what's called the jump in to make a. To bid from kind of $2 to go all the way to $10. But intuitively, in the case of independent private values, this revealed information doesn't make any difference. So we can show that in the independent private values model, It's a dominant strategy for bidders to bid up to and not beyond their valuations in both Japanese and English auctions. And so, in a sense Japanese, English and second price auctions are all kind of the same in the case of independent private values.
