 Hi, folks. So let's talk a little bit about a first price auction and bidding in a first price auction. So when we look at second price auctions, there's this wonderful property that derives from victory style arguments that people have a dominant strategy to bid their actual true valuation. Since they can't affect the price, all they can do is affect whether they're getting the good or not, and they want to get the good when their value is above the next highest bid. In first price auctions, you're actually paying what you bid. And so if you tell people your valuation as you bid and you win, you're going to end up with no surplus because you're paying exactly what you think the item's worth. You get nothing out of it. So here, you're going to have an incentive to shade your bid. And that means that the bidding in first price auctions we're going to have to solve as an equilibrium, as a Bayesian equilibrium. We're not going to have dominant strategies and things are going to be more complicated. So let's start by just comparing the first price auction to another class of auctions, the Dutch auctions, and just noting that these things are strategically equivalent. And what does that mean in terms of strategically equivalent? It means that as a function of the valuations of the individuals, the outcomes are going to be the same in terms of the eventual payments and allocations of the objects. And in particular, there's a mapping between the strategies in these two games that say that for an equilibrium in one of them, we find an equivalent equilibrium in the other one. And in particular, just to understand what's going on here, in both of these auctions, you've got to make a decision on how much you're willing to bid, conditional on it being the highest bid, and not knowing anything else about the other individual's values. So Dutch auctions are an extensive form. So basically what's happening is that the price starts at a very high level, it starts dropping down, and then we keep going until somebody grabs it. Now once you grab it, what do you know? All you know is you grabbed it at some price, and everyone else is still below you. So beforehand, I could just start a state of price and say, OK, look, if nobody's grabbed it so far at this price, this is what I'm going to be willing to bid. So conditional on me being the highest, this is what I'm willing to put in. And so that strategy of figuring out where you want the clock to go down to before you grab it is equivalent to just writing down your value in a piece of paper and saying, OK, the highest person wins, because I have to make a decision on how much I'm willing to pay conditional on that being more than what everybody else wrote on their slips of paper. So we end up with the same strategic analysis, basically. Why do we see both of these different kinds of auctions still used in practice then? So one nice thing about first price auctions, seal bid auctions, is you can have people bid asynchronously. So a lot of procurement auctions might be done this way. So for instance, the government might say you can bid on a certain contract, put your bid in an envelope, and send it to us. Actually, in procurement auctions, if you're trying to sell something to government, usually it's the lowest bid that wins. But those can be held asynchronously. Everybody can just mail their things in, then you open up the things, and whichever is the winning bid is then picked out of the envelopes. Dutch auctions, in contrast, you have to have the people together and sitting there watching the clock go down. But the nice thing about those auctions is you don't need as much communication in the sense that all you need to do is have this thing drop, and then somebody to say yes at some point. So only one bid needs to be transmitted from the bidders to the auctioneer. So it's very efficient in terms of information communication. You have the clock dropping down, and then somebody saying give it to me. So there are different auctions in terms of the actual implementation. OK, how should people bid in these auctions? Well, as we mentioned just a few minutes ago, bid less in your valuation. Because in basically deciding how low to bid is going to be the tricky part. Because you've got to trade off. The lower that you make your bid, the lower the amount you pay, but also the lower the probability that you win. So you're trading off probability of winning against the amount that you are having to pay. And so now you don't have a dominant strategy. How low you want to go actually depends on what other people are doing. So if I think other people are going to bid very, very low, then I'm going to be willing to lower my bid a lot. If I think other people are going to be bidding fairly aggressively, I might have to keep my bid higher. So I can't make my decision of how I should bid without knowing exactly what the bids of others are. And so now we don't have a dominant strategy. I don't have one thing which is best to do regardless of what other people are doing. I have to think about what they're doing and calculating my bid. OK, so let's have a quick look at an equilibrium of one of these and see how we can at least verify that something's in equilibrium. And then we'll talk a little bit about how you might find equilibrium in these auctions, which is not going to be always extremely easy. So let's think of a first price auction. So let's start with a very simple case to analyze first. Two bidders, both risk-neutral, and they get an independent draw from a uniform distribution on 0, 1. So each of these two bidders, so we've got two bidders. Each person uniforms 0, 1 valuations, and those are drawn independently. And the claim here is that if we look at we want to get a Bayesian Nash equilibrium of this game, what are we going to have? The bids are just going to be each person drops their value by half. So if my value is 3 quarters, I'll say 3 eighths. If my value is a half, I'll say a quarter. So I just take whatever my value is and I just shade it by a half, and that's my bid. And so let's go through and just verify that that's a Bayesian Nash equilibrium. So we want to check whether this is an equilibrium. So given the symmetry here, we can just check for one of the bidders. So let's suppose that the other bidder is actually, whatever their value is, they bid a half of it. And now we think about bidder 1. And let's let bidder 1 choose a strategy, S1, of how high they're going to bid as a function of their value. So if I'm bidding S1, so let's suppose I put a bit of S1 in and the other bidder is bidding half of their value. So when am I going to win? I'm going to win when a half of V2 is less than S1 or V2 is less than 2S1. And what's my payoff then? My payoff is V1 minus S1. But I'm going to lose whenever the other individual bid, when V2 over 2 is bigger than S1 or V2 is bigger than 2S1. And then I get a utility of 0. So in terms of figuring out what my value is, there's cases where I win the auction. So I can integrate over those where V2 is up to 2 over S1. And then I'm getting this. And otherwise, if the value of the other person is bigger than 2S1, I'm going to get 0. So this is my expected utility, a simple integral. Integrate that thing, what do you get? You get 2V1 times S1 minus 2S1 squared. Very simple expression. So we have an expected utility as a function of what my S1 is, conditional on the other person following this prescribed strategy. So let's differentiate that with respect to S1, set it equal to 0. So when we do that, what do we end up with? We end up with S1 equals half V1 as being the solution to setting that equal to 0. So if you want to maximize this, and you can check the second order conditions, if you want to maximize this, indeed, here, we end up with a condition where my optimal bid, given that the other person is bidding half their value, is to bid half my value as well. So again, the calculation for the other person is exactly symmetric to this. So this is a Bayesian Nash equilibrium of this game. So what we've got is people bidding down and they're trading off a calculation implicitly in terms of what we're doing here. If we want to look at the calculation that we were doing, by decreasing your bid, the gain is that you pay less conditional on winning, but you also win less of the time. So the trade-off is coming that you're winning less of the time, but then you're paying less when you do win. And so that trade-off is exactly captured through this maximization problem, and you want to shade your bid by half. This is obviously a very narrow result. We did two bidders, uniform valuations. So we need to solve for this thing, because this is not incentive-compatible as a direct mechanism in terms of dominant strategies. So it's not a dominant. We have no dominant strategies here. We're not getting dominant strategy incentive compatibility. We need to solve for the equilibrium. And more generally, if we had n bidders instead of 2, what would the equilibrium look like then? It's going to look like instead of shading your bid by a half, the general formula is going to be n minus 1 over n. So when n is 2, this is a half. When n is 3, then it's going to be 2 thirds. When n is 4, you go up to 3 quarters and so forth. And so you just keep climbing in terms of how much you're bidding and what's happening. Well, it gets harder and harder to win with more and more people in the auction. You're going to have to bid closer and closer to your value to have a chance to win. And then the trade-off is just that as you lower your bid much beyond something that's very close to your bid, you have no chance of winning. And so you're going to end up having this trade-off be closer to the actual value. You can go through and do the calculation. It's going to be very similar to what we did. Just a different calculation in terms of integrals. Now you're going to win when everybody's below you, not just the other person below you. So that's a slightly more complicated integration. But basically, exactly the same logic we just did for the two-bitter case. So broader problem. What we did here is we only verified that this was an equilibrium. So we guessed the equilibrium and then verified it. How do you actually find an equilibrium in these cases? Well, if it's not a nice uniform distribution and guessing that it's going to be a linear function of your value, then you've got to guess more complicated functions. And there's going to be basically some integration problems, which will give you some ideas of how this works for certain distributions. For arbitrary distributions, the bidding function can be much more complicated functions. And basically, it will only be described up to some integration conditions. So there are papers that give some solutions to this. There's a paper by Milgram and Weber in 1982. There's some other papers out there which will give some background on solving these kinds of auctions. That first-price auction actually has been solved quite early on in the literature. But more generally, solving these ones when we don't have knowledge strategies is going to involve some guesses. And in some cases, there might be some integration conditions that will give us nice solutions, but they often won't exist in closed form. And especially here also, the symmetry helps a lot. So if you have asymmetric auctions, things can be difficult. And more generally, even finding whether equilibria exists in these worlds is not an easy problem. There's a literature, in fact, that I've worked on a bit, about existence of equilibria in auctions. So one thing about equilibria in auctions is they're discontinuous. I changed my bid a little, and I might be going from winning the object to not winning the object. And so suddenly, my payoff goes from being positive to being zero. And that discontinuity means that I might not always have nicely defined best responses that are going to move in ways that were used, that the implicit proofs of existence before made nice use of the best response correspondence. That it's not going to have the properties it used to have in these discontinuous games. And getting equilibrium and existence in these kinds of auction settings is quite tricky. And there's some things that are known about settings where there do exist equilibria. There are also some examples where there don't exist equilibria. So that's actually an interesting project on its own. So different kinds of auctions, different kinds of equilibria, they're related to each other, and we'll see more about that in a bit.