Hi, Welcome back, In this set of lectures we are talking about mechanism design. And the idea here, is that you want to use models to help us design institutions, and to also how to think about which institution we might use. In this particular lecture, we're going to talk about auctions, and how we auction things off. Now auctions are used in a lot of settings, they're used to auction off air waves, oil leases, there's even things like, wine auctions. And you go to these auctions, sometimes they have ascending bids, where people call out prices, and you keep bidding, until no one can bid anymore. Other times there's sealed bid auctions, where you just write down an amount. There's a third type of auction, that I'm going to talk about as well, called a. And price auction, which has a slightly more complicated set of rules. And when you auction something off, your objective is to get as much money as you can possibly get. And so that's what we'll talk about here, we'll talk about auctions from the perspective, of the person who's selling the thing off. And if you're selling the thing off, you want to think about, how can I make as much money as I possibly can. So, we're gonna talk about these three types of auctions, Ascending bid, second price and sealed price. Now again, an ascending bid is an auction where, we just keep calling out prices until an want, no one wants to stay in anymore. A second price auction is a sealed auction, where each person writes down an amount. The highest bid gets it, but they get it with the second highest bid. Okay, So it's sorta complicated. So, the highest bidder wins the good, but they only pay the second highest bid. And then finally a sealed bid auction, is everybody submits a bid and the highest bidder gets it, but they pay it at the amount that they bid. 'Kay, so let's start with ascending-bid auctions. Ascending-big auctions are pretty simple. Individuals keep calling out bids or there's an auctioneer, until no one's willing to go above a price, and whoever's bid the highest price gets it. So let's think about how that would work. You gotta think about how that would work, you gotta think about different types of behavioral models. [inaudible] said there's, sort of, three ways we can think about modeling people. One is, we can think of people being rational. The other is, we can thinking of people following psychological rules, having maybe some biases. And the third thing we can think about people being rule following. Having some sort of heuristic or rule of thumb that they use in different situations. So if we think about an ascending bid auction where somebody keeps calling out price. If I'm rational, I'm gonna [inaudible], if I've got some value, if it's worth $100 to me, I'm gonna keep bidding Until it gets up to $100. You know if, so if it's gonna sell for 90, that's where the 100 to be, then I'll bid 91. So I'll bid pretty much up to my value. What about these other two? What about psychological or rule following in this setting? Let's first do rule following. In this setting, a rule following bidder might have some rule like, I'm going to start off at half my value, and then I'm going to go up by five dollars, or two dollars. And there are some that are you know, fairly sub-stated [inaudible] about how they raise their bids. But at the end of the day, it seems like they're probably only going to bid up to their value. They wouldn't bid above their value, because then they're paying more then it's worth to them, and they probably wouldn't stop bidding at less then their value. However, their rule could determine how much they raise bids by, and things like that. Now for psychological bidder, here, a bunch of things could come. It could be that their initial bid is based on how much the previous goods sold for, or all sorts of things. But again, it's hard to imagine in this ascending bid auction, someone not bidding for something if it was going to sell for less than they wanted it for. It's also hard to imagine someone bidding more than they value something for. Well, maybe that's not that hard of it. Because you can image, in an ascending good action, that people could get, you know, frenzy. They could really want to win. So, even though something is valued at $100, to them, it might be that if they're winning it at 95 and then somebody else get 105 that they did 110 just for, you know, just for the thrill of winning. Once I was, I was talking to a real auctioneer recently at a charity auction. He said that he feels he can raise the amount of money, increase the amount of money that you get because he gets people all excited, and they get excited about winning. And they forget that even though they only want that vase for $100, they'll pay 150, just for the thrill of winning, for the thrill of the chase. So psychological models in this setting could actually lead to higher values. But let's start out by assuming that people are rational. So what's the outcome in a rescinding bid auction. The good just goes to whoever bid the most. So, fairly straight forward, how much is that person gonna pay? Well, they're probably gonna pay the value of the second highest bidder. Why's that? Because let's suppose that one person buys at 100 and another person buys at 70. So it starts out the bidding is at 40, and then 50, and then 60, and then 70. And at 70, this person is gonna drop out. So this person who buys at 100 shouldn't pay any more than 70. Yeah, they could make a mistake, if they, if their rule is to keep [inaudible] by ten, they could pay 80, but most of the time, you'd expect them to pay only a little bit over 70, only a little bit over the value of the second highest bidder. Okay, now let's look a second price auction. Totally different auction mechanism and we can compare the two. In a second price auction, everybody writes down a bid, whoever is the highest bid gets it but you pay the second highest price. So let's suppose there's three bidders. One bidder values it at, one puts in 90, one puts in 60, one puts in 70. So the winner is the one that bids 90 but they only pay 70, be cause they pay the second highest price. Totally straightforward Well, let's think about how you'd bid in this setting. You could be a rational bidder, a psychological bidder, or a rule-following bidder. Let's focus on the rational bidder to start with. Let's think about, how would you rationally bid in this setting? So let's suppose your value's 80. And let's for a moment suppose you bid your true value, you bid 80. But all we care about is the highest other bid. So if the highest other bid is 60, and you bid 80, you're gonna get it. Right? You're going to pay 60, so you're going to end up winning, in a sense, $twenty. Because you paid 60 and it was worth 80. Suppose the highest other bid is 75. If you bid 80, you get it, you pay 75, and so your net gain is going to be five. So let's suppose the highest other bid is 85. So you're no longer the highest bid. That means somebody else is going to get it. They're going to pay 80. You don't get it. So your value is zero. So your values are 25 and zero. Well let's suppose you think, maybe I should bid a little bit more, maybe I should bid 90. Well if you bid 90, and the highest other bid is 60, you're gonna get it for 60, and so your net is gonna be twenty. Now, why twenty? Twenty, because you valued it at 80, and you paid 60. So bidding 90 didn't hurt you anyway. You know, the fact, you did ten over your bid, your real value didn't cost you at all. Well, suppose the highest other bid is 75. Again you bid 90, but you only pay 75, and so your net is five just as it was before. But suppose the second the highest other bid is 85 and now you bid 90. Well, now you're gonna pay 85. You only value it at 80, so you're gonna lose five. Notice you're worse off than you were before. Cuz before in that case, you didn't lose anything, and here you lose five. So it's fairly [inaudible] in a second price auction you don't wanna overbid. But do you want to under bid. Suppose you bid 70. Highest other bid is 60. You're gonna get it for 60. So your net is gonna be twenty. But, if the s econd highest bid is 75. And you bid 70. And you're not gonna get it, because they're gonna get it and they're only gonna pay 70. And you're gonna say, oh I wish I'da bid 80, or at least 76. So, you're only gonna get zero, whereas if you woulda bid 80, you'd have gotten it for 75 and you would've made $five. And finally if the other highest other bid is 85, you're not gonna get it anyway and your payoff is zero. So what we see in the second price auction, is if you tell the truth you get 25 zero, if you over bid you get 25 minus five, and if you underbid you get twenty zero, zero. So the rational bidder, In this case, should be your true value. What about the other types of bitters? What if you're a rule following bitter? Well, the rule following bitter here. Could do a lot of things. The rule-following bidder could've. Say, will maybe I'm shade my bid, ten percent or they could over bid, it's hard to tell, so rule following bidder, may not play the optimal rule. They could, play it, over bid or under bid. The psychological bidder as well just going to be more variation we don't know if people are going to tell the truth or not but the interesting thing about this option is, is that weather or not the other people are irrational or not it's still optimal for you. If your rational to bid your true value. So the interesting thing here is, there's no sort of ratcheting up. Remember when you did that race to the bottom game, if other people were rational, then you wanted to start taking into account their irrationality. The interesting thing here in the second price auction is, even if other people are psychological, or other people are role based, you should still bid your true value. So what that's going to mean is that's going to lead a general tendency towards people being more rational. It doesn't mean we have to abandon the psychological and we're following Rules for thinking about how people behave. But it does mean that there's probability this general tendencies for people, over time at least to make ration al bids. So, let's think about this for a second. What happens in this auction? The outcome goes to the highest-valued bidder, and that person pays the second-highest price. That's the exact same thing we got in the ascending-bid auction. Okay, now let's go the, the sealed bid auction. This is, in some ways, even though the simplest, it's the most complicated. So now everybody puts in a bid. They're all sealed, and the highest bidder gets it, but they pay the highest price. So if there's three bidders, bidder one bids 90, bidder two bids 60, bidder three bids 70. Bidder one gets it at 90, but they, but she pays 90. She doesn't pay 70 she pays 90. So in this setting, it makes sense to do what? To shade, to bid a little bit less. So if we think about what a rational bidder should do, that person should shade a little bit. If we think about a psychological bidder, that person might also shade but they might, think, well other people are going to bid even numbers like 75 some are going to bid $75 and one cent. Now a rule following bidder in this case might shade by some fixed percentage. So think about a rational bidder, how they should bid, it's gonna depend on a bunch of things including the number of other people in the auction. Let's look at a simple case where there's just two. And one thing we know right away is the higher you bid, the more likely it is you're gonna win. So you wanna [inaudible] you wanna go under your value. But you also wanna get, you know, somewhat higher bids, 'cause then you're more likely to win. So we wanna think through how this logic plays out. So let's do a two bidder model. And let's suppose the value of the other bidder is a uniform distribution between zero and one. So remember, in a uniform distribution, it's equally likely to be any value between zero and one. Let's suppose the other bidder bids her true value. So if she bids her true value. What are the odds that you win if you bid 60 cents? We are gonna win if our value happens to be less than 60 cents, and that's gonna happen 60 percent of the time. We can generalize this. Suppose you bid some amount, B, which is between zero and one. What are the odds that you win? Well again, you're probably, the odds that you win is just going to be B. So let's formalize this. Let's suppose the other person is bidding virtually badly, and think about what you should do. So V is your value, B is your bid. V minus B is your surplus. That's how much you'd win, if you win. So if your value is 90. And your bid was 30, and you won, you'd get 60. Right, that's how much you sort of, it's the difference between your value for the object and how much you bid, In this case though, these values are going to be at the interval 0,1, As are your bids. Now B, if you bid point six, is also your probability of winning. So if you bid a half, the probability of winning is a half. If you bid a quarter, the probability of winning is a quarter. So your expected winnings are just the probability of winning times your surplus. So that's just B times V minus D. All you want to do is maximize D times V minus B. Well if I multiply that out at B V minus B squared. Now, if you've had calculus, all you have to do is take the derivative to this, with respect to D, and that's gonna give you V minus two B equals zero, which we've got right here, and your optimal bid is to bid half your value. So if you think the other person's bidding her true value, you should bid half your value. And so let's think about this one again. So you're a rational bidder and you think, if the other person's bidding her true value, I should bid half my value. But that means the other person should probably also be bidding half of her value. If I'm bidding half my value and I'm rational, she should be bidding half her value. So let's think of, let's suppose she's bidding half her value, what should you do? If the other bidder bids half her value, now if I bid B the probability that I win is going to be 2B. Why is that? Let's think about it. So suppose that I bid .25, If I did .25 I'm gonna win as long a s her value is less than .25 times two, because she's bidding half her value. So that means I'm gonna l, win half the time. So what we get is, the probability that I win if I get B is gonna be 2B, given that she's bidding half her value. So now we just do the same calculation. V is my value, B is my bid. So the difference between those two is how much I win. And now 2B is my probability of winning. So my winnings are gonna be 2B times VB. So if I write down that, and set the derivative again equal to zero, what I'm gonna get is that I again should bid V over two. Now if you don't know how to take derivatives, don't worry about it. All we're doing here is we're just using a little math to show that my [inaudible] bid again is gonna be V over two. Well this is great, because it says, if I bid half my value. And she bids half her value, we're both doing the optimal thing. So the optimal thing for each of us in this case, the rational thing to do, would be to bid half her value. So what's gonna happen is the highest value bidder is gonna get it. And they're gonna get it at half their value. Great, So let's look through all three of our auctions. In the sealed bit auction, the highest bidder gets it at half her value. In the ascending big auction, the highest value bidder gets it at the second highest value. And at the second price auction, the highest value bidder gets it, also at the second highest value. Notice this, though. Half of the highest bidder's value. If the highest bidder's value is 60, half of their value is 30, and that's the expected value of the second highest bidder. Why's that, remember, let's think about it, we got this distribution of values and their uniform. So if I bid.6 and I win, that means that the other person's bid is somewhere in here. So what's, if it's somewhere in there, the expected value that would be halfway in between, which would be my bid over two. So half of my value, if I've got a uniform distribution, is excatly equal to the expected value of the second highest bidder. So wha t we get is, all three of these auctions seem to work about the same way. The highest value bidder gets it, and they get it at either the exact value of the second highest bidder, or at the expected value of the second highest bidder. If since, if you're auctioning it off, you don't know the exact value of the second Hindspitter. All you can expect to get is the expected value of the second Hindspitter, so it looks like all three of these things are the same. And in fact they are. So there's a theorem proven by Roger Myerson, and he, incidentally, he won a Nobel Prize for this work. So it's a fairly sophisticated theorem. That says if you have rational bidders, there's a Y class of auction mechanisms that includes sealed bids, second price, and ascending bid auctions, such that they get identical expected outcomes. So the expected outcomes in all three of those cases were highest bidder gets it at the expected value of the second highest bidder. And that's what we got and it's called the revenue equivalents here. So what the model tells us is, it doesn't matter how we auction things off, if voters are rational. So this is a really powerful theory, and here we see the value, one of the values of models. Because we might sit around and think, oh boy, ascending bid auctions are better. Seal bid auctions are better. Second price auctions are better. What this tells us is, if we have rational bidders, all three are equally good. But we may not have rational bidders. We could have psychological bidders, we could have rule following bidders, so here's where we take. Our model results, the revenue equivalence term, and we then try to bring our experience in and think something about the bidders in the auction. So let's suppose that we've got a bunch of really sophisticated bidders, so these are multinational firms bidding on oil leases. Well in that case, we can imagine they are probably fairly close to rational. And now we know that. Pretty much any auction method is going to give us the same revenue. And so we could s ay, well maybe it doesn't matter. Well now we may care about things like transparency. So for instance maybe we decide to have it be a sealed bid auction so we can actually see exactly how much people bid. And we know that, none of the, since the, all of the bidders are highly rational it's going to be okay, we're going to get the same revenue, and that way we'll see it. Let's suppose instead of having some charity auction we are auctioning off Something in the community, just for fun, and now, we know people maybe haven't participated in auction before, and it's somewhat confusing to them. And then, they may be suffering from some psychological biases, or they may just be following some simple rules. Well, in those settings, when you've got unsophisticated bidders. Let's think about the three auctions. In the sealed bid auction, they've gotta think about what are the distribution of other people's values. Well that could be really hard for them to do and they may make all sorts of mistakes. They may follow rules that don't make sense. They may suffer from psychological biases. What about the second price auction? Where you say that the highest bidder gets it at the second highest price. That may be confusing to people and they might not have any idea how it works. So what about. The ascending bid auction. This makes a lot of sense in that setting, because even if people are biased or if they are rule-following, it's still probably going to be the case that if the bid is lower than what they value it, that they'll probably bid. So that way, no one is going to. Do some silly thing and underbid, and not get something they want. And in addition, if we're trying to make as much money as we can, given that there could be some psychological bias in that people could just wanna win, we might even make more money by having an ascending bid auction. We're not gonna make more money in the sealed bid case. So what we see is if you have highly sophisticated people, maybe we go with sealed bid. Or maybe we go to second price cuz they can figure it out. If you got unsophisticated people maybe we go to the sending bid. For a couple reasons. One it's easier, and the other is maybe we get them sort of in a psychological frenzy, and we make more money. We have a powerful theorem, the revenue equivalence theorem, and that tells us it doesn't matter which auction mechanism we do, we use, If people are rational. But if we think about how people actually behave, we could then start to make some distinctions about what institution to auction things off might work best. And in some cases we might want a sealed bid. And in some cases we might want ascending. And in other cases we might want second price. So what we've seen here is we can write down models of auctions and we can develop some really profound results saying that it doesn't matter how you auction things off provided some conditions are met. So that's really nice. It sort of frees us up to think about other things. And it frees us up to think about how are people are actually going to behave. How much information do they have? How sophisticated are they? How many of them are there? And that can then, then we can use those criteria to decide which auctions we're going to use. As opposed to spending our time thinking about, well this auction is better than this auction on purely rational grounds. So we talked about what, why do we model. But why do we assume even rational actors? Remember, I said, benchmarks are good things. Remember I said Roger Myerson says, the one who's got the revenue equivalent theorum, that, assuming rational behavior's often a very good benchmark. Well, we saw that was the case here in options, because we see. If people are rational, doesn't matter what mechanism you use. Once we relax that assumption, then the mechanism may matter. But, now we know what criteria to use to think about choosing among auction mechanisms. So it's really useful. Models are really helpful. All right. Thank you.