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Hi, Welcome back, In this set of lectures we are talking about mechanism design. And

Â the idea here, is that you want to use models to help us design institutions, and

Â to also how to think about which institution we might use. In this

Â particular lecture, we're going to talk about auctions, and how we auction things

Â off. Now auctions are used in a lot of settings, they're used to auction off air

Â waves, oil leases, there's even things like, wine auctions. And you go to these

Â auctions, sometimes they have ascending bids, where people call out prices, and

Â you keep bidding, until no one can bid anymore. Other times there's sealed bid

Â auctions, where you just write down an amount. There's a third type of auction,

Â that I'm going to talk about as well, called a. And price auction, which has a

Â slightly more complicated set of rules. And when you auction something off, your

Â objective is to get as much money as you can possibly get. And so that's what we'll

Â talk about here, we'll talk about auctions from the perspective, of the person who's

Â selling the thing off. And if you're selling the thing off, you want to think

Â about, how can I make as much money as I possibly can. So, we're gonna talk about

Â these three types of auctions, Ascending bid, second price and sealed price. Now

Â again, an ascending bid is an auction where, we just keep calling out prices

Â until an want, no one wants to stay in anymore. A second price auction is a

Â sealed auction, where each person writes down an amount. The highest bid gets it,

Â but they get it with the second highest bid. Okay, So it's sorta complicated. So,

Â the highest bidder wins the good, but they only pay the second highest bid. And then

Â finally a sealed bid auction, is everybody submits a bid and the highest bidder gets

Â it, but they pay it at the amount that they bid. 'Kay, so let's start with

Â ascending-bid auctions. Ascending-big auctions are pretty simple. Individuals

Â keep calling out bids or there's an auctioneer, until no one's willing to go

Â above a price, and whoever's bid the highest price gets it. So let's think

Â about how that would work. You gotta think about how that would work, you gotta think

Â about different types of behavioral models. [inaudible] said there's, sort of,

Â three ways we can think about modeling people. One is, we can think of people

Â being rational. The other is, we can thinking of people following psychological

Â rules, having maybe some biases. And the third thing we can think about people

Â being rule following. Having some sort of heuristic or rule of thumb that they use

Â in different situations. So if we think about an ascending bid auction where

Â somebody keeps calling out price. If I'm rational, I'm gonna [inaudible], if I've

Â got some value, if it's worth $100 to me, I'm gonna keep bidding Until it gets up to

Â $100. You know if, so if it's gonna sell for 90, that's where the 100 to be, then

Â I'll bid 91. So I'll bid pretty much up to my value. What about these other two? What

Â about psychological or rule following in this setting? Let's first do rule

Â following. In this setting, a rule following bidder might have some rule

Â like, I'm going to start off at half my value, and then I'm going to go up by five

Â dollars, or two dollars. And there are some that are you know, fairly sub-stated

Â [inaudible] about how they raise their bids. But at the end of the day, it seems

Â like they're probably only going to bid up to their value. They wouldn't bid above

Â their value, because then they're paying more then it's worth to them, and they

Â probably wouldn't stop bidding at less then their value. However, their rule

Â could determine how much they raise bids by, and things like that. Now for

Â psychological bidder, here, a bunch of things could come. It could be that their

Â initial bid is based on how much the previous goods sold for, or all sorts of

Â things. But again, it's hard to imagine in this ascending bid auction, someone not

Â bidding for something if it was going to sell for less than they wanted it for.

Â It's also hard to imagine someone bidding more than they value something for. Well,

Â maybe that's not that hard of it. Because you can image, in an ascending good

Â action, that people could get, you know, frenzy. They could really want to win. So,

Â even though something is valued at $100, to them, it might be that if they're

Â winning it at 95 and then somebody else get 105 that they did 110 just for, you

Â know, just for the thrill of winning. Once I was, I was talking to a real auctioneer

Â recently at a charity auction. He said that he feels he can raise the amount of

Â money, increase the amount of money that you get because he gets people all

Â excited, and they get excited about winning. And they forget that even though

Â they only want that vase for $100, they'll pay 150, just for the thrill of winning,

Â for the thrill of the chase. So psychological models in this setting could

Â actually lead to higher values. But let's start out by assuming that people are

Â rational. So what's the outcome in a rescinding bid auction. The good just goes

Â to whoever bid the most. So, fairly straight forward, how much is that person

Â gonna pay? Well, they're probably gonna pay the value of the second highest

Â bidder. Why's that? Because let's suppose that one person buys at 100 and another

Â person buys at 70. So it starts out the bidding is at 40, and then 50, and then

Â 60, and then 70. And at 70, this person is gonna drop out. So this person who buys at

Â 100 shouldn't pay any more than 70. Yeah, they could make a mistake, if they, if

Â their rule is to keep [inaudible] by ten, they could pay 80, but most of the time,

Â you'd expect them to pay only a little bit over 70, only a little bit over the value

Â of the second highest bidder. Okay, now let's look a second price auction. Totally

Â different auction mechanism and we can compare the two. In a second price

Â auction, everybody writes down a bid, whoever is the highest bid gets it but you

Â pay the second highest price. So let's suppose there's three bidders. One bidder

Â values it at, one puts in 90, one puts in 60, one puts in 70. So the winner is the

Â one that bids 90 but they only pay 70, be cause they pay the second highest price.

Â Totally straightforward Well, let's think about how you'd bid in this setting. You

Â could be a rational bidder, a psychological bidder, or a rule-following

Â bidder. Let's focus on the rational bidder to start with. Let's think about, how

Â would you rationally bid in this setting? So let's suppose your value's 80. And

Â let's for a moment suppose you bid your true value, you bid 80. But all we care

Â about is the highest other bid. So if the highest other bid is 60, and you bid 80,

Â you're gonna get it. Right? You're going to pay 60, so you're going to end up

Â winning, in a sense, $twenty. Because you paid 60 and it was worth 80. Suppose the

Â highest other bid is 75. If you bid 80, you get it, you pay 75, and so your net

Â gain is going to be five. So let's suppose the highest other bid is 85. So you're no

Â longer the highest bid. That means somebody else is going to get it. They're

Â going to pay 80. You don't get it. So your value is zero. So your values are 25 and

Â zero. Well let's suppose you think, maybe I should bid a little bit more, maybe I

Â should bid 90. Well if you bid 90, and the highest other bid is 60, you're gonna get

Â it for 60, and so your net is gonna be twenty. Now, why twenty? Twenty, because

Â you valued it at 80, and you paid 60. So bidding 90 didn't hurt you anyway. You

Â know, the fact, you did ten over your bid, your real value didn't cost you at all.

Â Well, suppose the highest other bid is 75. Again you bid 90, but you only pay 75, and

Â so your net is five just as it was before. But suppose the second the highest other

Â bid is 85 and now you bid 90. Well, now you're gonna pay 85. You only value it at

Â 80, so you're gonna lose five. Notice you're worse off than you were before. Cuz

Â before in that case, you didn't lose anything, and here you lose five. So it's

Â fairly [inaudible] in a second price auction you don't wanna overbid. But do

Â you want to under bid. Suppose you bid 70. Highest other bid is 60. You're gonna get

Â it for 60. So your net is gonna be twenty. But, if the s econd highest bid is 75. And

Â you bid 70. And you're not gonna get it, because they're gonna get it and they're

Â only gonna pay 70. And you're gonna say, oh I wish I'da bid 80, or at least 76. So,

Â you're only gonna get zero, whereas if you woulda bid 80, you'd have gotten it for 75

Â and you would've made $five. And finally if the other highest other bid is 85,

Â you're not gonna get it anyway and your payoff is zero. So what we see in the

Â second price auction, is if you tell the truth you get 25 zero, if you over bid you

Â get 25 minus five, and if you underbid you get twenty zero, zero. So the rational

Â bidder, In this case, should be your true value. What about the other types of

Â bitters? What if you're a rule following bitter? Well, the rule following bitter

Â here. Could do a lot of things. The rule-following bidder could've. Say, will

Â maybe I'm shade my bid, ten percent or they could over bid, it's hard to tell, so

Â rule following bidder, may not play the optimal rule. They could, play it, over

Â bid or under bid. The psychological bidder as well just going to be more variation we

Â don't know if people are going to tell the truth or not but the interesting thing

Â about this option is, is that weather or not the other people are irrational or not

Â it's still optimal for you. If your rational to bid your true value. So the

Â interesting thing here is, there's no sort of ratcheting up. Remember when you did

Â that race to the bottom game, if other people were rational, then you wanted to

Â start taking into account their irrationality. The interesting thing here

Â in the second price auction is, even if other people are psychological, or other

Â people are role based, you should still bid your true value. So what that's going

Â to mean is that's going to lead a general tendency towards people being more

Â rational. It doesn't mean we have to abandon the psychological and we're

Â following Rules for thinking about how people behave. But it does mean that

Â there's probability this general tendencies for people, over time at least

Â to make ration al bids. So, let's think about this for a second. What happens in

Â this auction? The outcome goes to the highest-valued bidder, and that person

Â pays the second-highest price. That's the exact same thing we got in the

Â ascending-bid auction. Okay, now let's go the, the sealed bid auction. This is, in

Â some ways, even though the simplest, it's the most complicated. So now everybody

Â puts in a bid. They're all sealed, and the highest bidder gets it, but they pay the

Â highest price. So if there's three bidders, bidder one bids 90, bidder two

Â bids 60, bidder three bids 70. Bidder one gets it at 90, but they, but she pays 90.

Â She doesn't pay 70 she pays 90. So in this setting, it makes sense to do what? To

Â shade, to bid a little bit less. So if we think about what a rational bidder should

Â do, that person should shade a little bit. If we think about a psychological bidder,

Â that person might also shade but they might, think, well other people are going

Â to bid even numbers like 75 some are going to bid $75 and one cent. Now a rule

Â following bidder in this case might shade by some fixed percentage. So think about a

Â rational bidder, how they should bid, it's gonna depend on a bunch of things

Â including the number of other people in the auction. Let's look at a simple case

Â where there's just two. And one thing we know right away is the higher you bid, the

Â more likely it is you're gonna win. So you wanna [inaudible] you wanna go under your

Â value. But you also wanna get, you know, somewhat higher bids, 'cause then you're

Â more likely to win. So we wanna think through how this logic plays out. So let's

Â do a two bidder model. And let's suppose the value of the other bidder is a uniform

Â distribution between zero and one. So remember, in a uniform distribution, it's

Â equally likely to be any value between zero and one. Let's suppose the other

Â bidder bids her true value. So if she bids her true value. What are the odds that you

Â win if you bid 60 cents? We are gonna win if our value happens to be less than 60

Â cents, and that's gonna happen 60 percent of the time. We can generalize this.

Â Suppose you bid some amount, B, which is between zero and one. What are the odds

Â that you win? Well again, you're probably, the odds that you win is just going to be

Â B. So let's formalize this. Let's suppose the other person is bidding virtually

Â badly, and think about what you should do. So V is your value, B is your bid. V minus

Â B is your surplus. That's how much you'd win, if you win. So if your value is 90.

Â And your bid was 30, and you won, you'd get 60. Right, that's how much you sort

Â of, it's the difference between your value for the object and how much you bid, In

Â this case though, these values are going to be at the interval 0,1, As are your

Â bids. Now B, if you bid point six, is also your probability of winning. So if you bid

Â a half, the probability of winning is a half. If you bid a quarter, the

Â probability of winning is a quarter. So your expected winnings are just the

Â probability of winning times your surplus. So that's just B times V minus D. All you

Â want to do is maximize D times V minus B. Well if I multiply that out at B V minus B

Â squared. Now, if you've had calculus, all you have to do is take the derivative to

Â this, with respect to D, and that's gonna give you V minus two B equals zero, which

Â we've got right here, and your optimal bid is to bid half your value. So if you think

Â the other person's bidding her true value, you should bid half your value. And so

Â let's think about this one again. So you're a rational bidder and you think, if

Â the other person's bidding her true value, I should bid half my value. But that means

Â the other person should probably also be bidding half of her value. If I'm bidding

Â half my value and I'm rational, she should be bidding half her value. So let's think

Â of, let's suppose she's bidding half her value, what should you do? If the other

Â bidder bids half her value, now if I bid B the probability that I win is going to be

Â 2B. Why is that? Let's think about it. So suppose that I bid .25, If I did .25 I'm

Â gonna win as long a s her value is less than .25 times two, because she's bidding

Â half her value. So that means I'm gonna l, win half the time. So what we get is, the

Â probability that I win if I get B is gonna be 2B, given that she's bidding half her

Â value. So now we just do the same calculation. V is my value, B is my bid.

Â So the difference between those two is how much I win. And now 2B is my probability

Â of winning. So my winnings are gonna be 2B times VB. So if I write down that, and set

Â the derivative again equal to zero, what I'm gonna get is that I again should bid V

Â over two. Now if you don't know how to take derivatives, don't worry about it.

Â All we're doing here is we're just using a little math to show that my [inaudible]

Â bid again is gonna be V over two. Well this is great, because it says, if I bid

Â half my value. And she bids half her value, we're both doing the optimal thing.

Â So the optimal thing for each of us in this case, the rational thing to do, would

Â be to bid half her value. So what's gonna happen is the highest value bidder is

Â gonna get it. And they're gonna get it at half their value. Great, So let's look

Â through all three of our auctions. In the sealed bit auction, the highest bidder

Â gets it at half her value. In the ascending big auction, the highest value

Â bidder gets it at the second highest value. And at the second price auction,

Â the highest value bidder gets it, also at the second highest value. Notice this,

Â though. Half of the highest bidder's value. If the highest bidder's value is

Â 60, half of their value is 30, and that's the expected value of the second highest

Â bidder. Why's that, remember, let's think about it, we got this distribution of

Â values and their uniform. So if I bid.6 and I win, that means that the other

Â person's bid is somewhere in here. So what's, if it's somewhere in there, the

Â expected value that would be halfway in between, which would be my bid over two.

Â So half of my value, if I've got a uniform distribution, is excatly equal to the

Â expected value of the second highest bidder. So wha t we get is, all three of

Â these auctions seem to work about the same way. The highest value bidder gets it, and

Â they get it at either the exact value of the second highest bidder, or at the

Â expected value of the second highest bidder. If since, if you're auctioning it

Â off, you don't know the exact value of the second Hindspitter. All you can expect to

Â get is the expected value of the second Hindspitter, so it looks like all three of

Â these things are the same. And in fact they are. So there's a theorem proven by

Â Roger Myerson, and he, incidentally, he won a Nobel Prize for this work. So it's a

Â fairly sophisticated theorem. That says if you have rational bidders, there's a Y

Â class of auction mechanisms that includes sealed bids, second price, and ascending

Â bid auctions, such that they get identical expected outcomes. So the expected

Â outcomes in all three of those cases were highest bidder gets it at the expected

Â value of the second highest bidder. And that's what we got and it's called the

Â revenue equivalents here. So what the model tells us is, it doesn't matter how

Â we auction things off, if voters are rational. So this is a really powerful

Â theory, and here we see the value, one of the values of models. Because we might sit

Â around and think, oh boy, ascending bid auctions are better. Seal bid auctions are

Â better. Second price auctions are better. What this tells us is, if we have rational

Â bidders, all three are equally good. But we may not have rational bidders. We could

Â have psychological bidders, we could have rule following bidders, so here's where we

Â take. Our model results, the revenue equivalence term, and we then try to bring

Â our experience in and think something about the bidders in the auction. So let's

Â suppose that we've got a bunch of really sophisticated bidders, so these are

Â multinational firms bidding on oil leases. Well in that case, we can imagine they are

Â probably fairly close to rational. And now we know that. Pretty much any auction

Â method is going to give us the same revenue. And so we could s ay, well maybe

Â it doesn't matter. Well now we may care about things like transparency. So for

Â instance maybe we decide to have it be a sealed bid auction so we can actually see

Â exactly how much people bid. And we know that, none of the, since the, all of the

Â bidders are highly rational it's going to be okay, we're going to get the same

Â revenue, and that way we'll see it. Let's suppose instead of having some charity

Â auction we are auctioning off Something in the community, just for fun, and now, we

Â know people maybe haven't participated in auction before, and it's somewhat

Â confusing to them. And then, they may be suffering from some psychological biases,

Â or they may just be following some simple rules. Well, in those settings, when

Â you've got unsophisticated bidders. Let's think about the three auctions. In the

Â sealed bid auction, they've gotta think about what are the distribution of other

Â people's values. Well that could be really hard for them to do and they may make all

Â sorts of mistakes. They may follow rules that don't make sense. They may suffer

Â from psychological biases. What about the second price auction? Where you say that

Â the highest bidder gets it at the second highest price. That may be confusing to

Â people and they might not have any idea how it works. So what about. The ascending

Â bid auction. This makes a lot of sense in that setting, because even if people are

Â biased or if they are rule-following, it's still probably going to be the case that

Â if the bid is lower than what they value it, that they'll probably bid. So that

Â way, no one is going to. Do some silly thing and underbid, and not get something

Â they want. And in addition, if we're trying to make as much money as we can,

Â given that there could be some psychological bias in that people could

Â just wanna win, we might even make more money by having an ascending bid auction.

Â We're not gonna make more money in the sealed bid case. So what we see is if you

Â have highly sophisticated people, maybe we go with sealed bid. Or maybe we go to

Â second price cuz they can figure it out. If you got unsophisticated people maybe we

Â go to the sending bid. For a couple reasons. One it's easier, and the other is

Â maybe we get them sort of in a psychological frenzy, and we make more

Â money. We have a powerful theorem, the revenue equivalence theorem, and that

Â tells us it doesn't matter which auction mechanism we do, we use, If people are

Â rational. But if we think about how people actually behave, we could then start to

Â make some distinctions about what institution to auction things off might

Â work best. And in some cases we might want a sealed bid. And in some cases we might

Â want ascending. And in other cases we might want second price. So what we've

Â seen here is we can write down models of auctions and we can develop some really

Â profound results saying that it doesn't matter how you auction things off provided

Â some conditions are met. So that's really nice. It sort of frees us up to think

Â about other things. And it frees us up to think about how are people are actually

Â going to behave. How much information do they have? How sophisticated are they? How

Â many of them are there? And that can then, then we can use those criteria to decide

Â which auctions we're going to use. As opposed to spending our time thinking

Â about, well this auction is better than this auction on purely rational grounds.

Â So we talked about what, why do we model. But why do we assume even rational actors?

Â Remember, I said, benchmarks are good things. Remember I said Roger Myerson

Â says, the one who's got the revenue equivalent theorum, that, assuming

Â rational behavior's often a very good benchmark. Well, we saw that was the case

Â here in options, because we see. If people are rational, doesn't matter what

Â mechanism you use. Once we relax that assumption, then the mechanism may matter.

Â But, now we know what criteria to use to think about choosing among auction

Â mechanisms. So it's really useful. Models are really helpful. All right. Thank you.

Â