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Hi folks, so letâ€™s talk a little bit about a first-price auction and

Â bidding in a first-price auction.

Â So when we look at second-price auctions thereâ€™s this wonderful property that

Â derives from Vickery style arguments that people have a dominant strategy to be,

Â to bid their actual true valuation, since they can't affect the price.

Â All they can do is affect whether they're getting the good or not, and

Â they want to get the good when their value is above the next highest bid.

Â In first-price auctions, you're actually paying what you bid.

Â And so, if you tell people your valuation is your bid and you win,

Â you're going to end up with no surplus because you're paying exactly

Â what you think the item is worth, you get nothing out of it.

Â So, here you're going to have incentive to shade your bid.

Â And that means that the bidding in first price auctions we're going to have to

Â solve as an equilibrium as a base in equilibrium.

Â We're not going to have dominant strategies and

Â things are going to be more complicated.

Â So let's start by just comparing the first-price auction to another class of

Â auctions, the Dutch auctions, and

Â just noting that these things are strategically equivalent.

Â And what does that mean in terms of strategically equivalent?

Â It means that as a function of the evaluation of the individuals,

Â the outcomes are going to be the same in terms of

Â the eventual payments and allocation of the objects.

Â And in particular there is a mapping between the strategies on these

Â two games they say that for an equilibrium in one of them,

Â we find an equivalent equilibrium in the other one.

Â And in particular just to understand what is going on here.

Â In both of these auctions, you've gotta make a decision on how much

Â you're willing to bid, conditional on it being the highest bid.

Â And not knowing anything else about the other individual's values.

Â So, Dutch auctions are an extensive form so

Â basically what's happening is that the base starts at a very high level.

Â It starts dropping down and then we keep going until somebody grabs it, okay?

Â Now once you grab it, what do you know?

Â All you know is you grabbed it at some price and

Â everyone else is still below you.

Â So beforehand I could just start a state of price and say, look, if nobody's

Â grabbed it so far at this price, this is what I'm going to be willing to bid.

Â So conditional on me being the highest, this is what I'm willing to put in.

Â And so, that strategy of figuring out where you want the clock to down to before

Â you grab it Is the equivalent to just writing down your value on

Â a piece of paper and saying, okay, the highest person wins.

Â Because I have to make a decision on how much I'm going to pay,

Â conditional on that being more than what everybody

Â else wrote on their slips of paper.

Â So we end up with the same strategic analysis basically.

Â Why do we see both of these different kinds of auctions still use and

Â practice then.

Â 2:57

So, one nice thing about first-price auction sealed with options,

Â is you can have people bid asynchronously.

Â So a lot of procurement auctions might be done this way.

Â So, for instance, the government might say maybe no,

Â you've been on a certain contract.

Â Put your bid in an envelope and send it to us.

Â Actually in procurement auctions if you're trying to sell something,

Â the government usually is the lowest bid that wins.

Â But those can be held to asynchronously.

Â Everybody can just mail their things in then you open up this the things and

Â whichever is the winning bid is unpicked out of the envelopes.

Â Dutch auctions in contrast you get to have the people together and

Â sitting there watching the clock go down.

Â But the nice thing about those auctions is you don't need as much communication in

Â the sense that, all you need to do is have this thing drop and

Â then somebody just say yes.

Â So only one bit needs to be transmitted from the bidders of the auctioneer so

Â it's very efficient in terms of the information and communication.

Â You have the clock dropping down and someone saying give it to me.

Â So, you know, there's different auctions in term of actual implementation, okay.

Â How should people bid in these auctions?

Â Well, as we mentioned just a few minutes ago bid less than your valuation.

Â Because in basically deciding how low to bid is going to be the tricky part because

Â you've got a trade off.

Â The lower that you make your bid, the lower the amount you pay, but

Â also the lower the probability that you win.

Â So you're trading off probability of winning against amount that you're

Â having to pay.

Â And so now, you don't have a dominant strategy.

Â How low you want to go actually depends on what other people are doing.

Â So, if I think other people are going to bid very very low,

Â then I'm going to be willing to lower my bid a lot.

Â If I think other people are going to be bidding fairly aggressively,

Â I might have to keep my bid higher.

Â So I can't make my decision on how I should bid without knowing exactly what

Â the bid of others are.

Â And so, now we don't have a dominate strategy,

Â I don't have one thing which is best to do regardless of what other people are doing.

Â I have to think about what they're doing in calculating my bid, okay?

Â So let's have a quick look at an equilibrium of one of these and

Â see how we can at least verify something in the equilibrium and

Â then we'll talk a little bit about how you might find the equilibrium in these

Â auctions which is not going to be always extremely easy.

Â So let's think of a first price option, so

Â let's start with a very simple case to analyze first.

Â Two bidders, both risk neutral, and

Â they get an independent draw from a uniform distribution at 0,1, okay.

Â So each of these two bidder, so we've got two bidders,

Â each person uniforms year one valuations and those are drawn independently.

Â And the claim here is that, if we look at,

Â we want to get a Bayes-Nash equilibrium of this game, what are we going have?

Â The bids are just going to be each person drops their value by half, so

Â if my values three-quarters, I'll say three-eighths.

Â If my value is a half, I'll say a quarter.

Â So I just take whatever my value is and I just shade it by a half and that's my bid.

Â And so let's go through and just verify that that's the Bayes-Nash equilibrium so

Â we want to check whether this is equilibrium, so let's,

Â given the symmetry here, we can check for one of the bidders.

Â So let's suppose that the other bidder is actually, whatever their value is,

Â they bid a half of it.

Â And now, we think about bidder one, and let's let bidder one choose

Â a strategy as to one of how high they're going to bid as a function of their value.

Â So, if I'm bidding s1, so let's suppose I put a bid of s1 in and

Â the other bidder is bidding half of their value.

Â So when am I going to win?

Â I'm going to win when half of v2 is less and s1 or v2 is less than 2s1.

Â Okay, and what's my payoff then?

Â My payoff is V1 minus S1.

Â But I'm going to lose whenever that the other individual bid

Â when v2 over 2 is bigger s1 or v2 is bigger than 2s1.

Â And then I get a utility of 0, okay.

Â So in terms of figuring out what my value is, there's cases where I win the auction.

Â So I can integrate over those, where v2 is up to 2 over s1, and

Â then I'm getting this, and otherwise,

Â if the value of the other person is bigger than 2s1, I'm going to get 0.

Â So this is my expected utility, the simple integral,

Â integrate that thing, what do you get?

Â You get 2v1 times s1- 2s1 squared, okay.

Â Very simple expression so

Â we have an expected utility as a function of what my s1 is.

Â Conditional on the other person following this prescribed strategy.

Â So let's differentiate that with respect to s1 set at equal to 0.

Â So when we do that what do we end up with?

Â We end up s1 = a half v1 as being the solution to setting that equal to 0.

Â So if you want to maximize this and you want to check the second order conditions,

Â you want to maximize this indeed here we ended up with a condition where my optimal

Â bid given that the other person is bidding half their value is to bid half my value

Â as well.

Â Okay, so and again, the calculation for

Â the other person is exactly symmetric to this.

Â So this is Bayesian mash equilibrium on this game.

Â So what we've got is people bidding down and they're trading off a calculation.

Â Implicitly, in terms of what we're doing here,

Â if we want to look at the calculation that we were doing, by increasing your or

Â by decreasing your bid, the gain is that you pay less conditional on winning but

Â also, win less of the times.

Â So the trade-off is coming at your winning less of the time but

Â then you're paying less when you do win.

Â And so that trade-off is exactly captured to this maximization problem and

Â you want to shade your bid by half.

Â Okay, this is obviously a very narrow result.

Â We did two bidders, uniform valuations.

Â So we need to solve for this thing because this is not incentive compatible.

Â It's a direct mechanism in terms of dominant strategies, right.

Â So it's not a dominant.

Â We have no dominant strategies here.

Â We're not getting dominant strategy incentive compatibility we need to solve

Â for the equilibrium.

Â And more generally you could solve for if we had n bidders instead of two,

Â what would the equilibrium look like then.

Â It's going to look like instead of shading your bid by a half,

Â the general formula is going to be n-1 over n right.

Â So when n is 2, this is a half.

Â When n is 3, then it's going to be two thirds.

Â When n is 4, you go up to three quarters and so forth and so,

Â you keep climbing in terms of how much your bidding and what's happening.

Â Well, it gets harder and harder to win with more and more people in the auction.

Â You going to have to bid closer and closer to your value to have a chance to win.

Â And then the trade off is just that, as you lower your bid much beyond

Â something that's very close to your bid you have no chance of winning.

Â And so you're going to end up having this trade-off be closer to the actual value.

Â You can go through and do the calculation.

Â It's going to be very similar to what we did,

Â just a different calculation in terms of integrals.

Â Now you're going to win when everybody's below you,

Â not just the other person below you.

Â So that's a slightly more complicated integration, but

Â basically exactly the same logic we just did for the two bidder case.

Â Okay, so broader problem.

Â What we did here is we only verified that this was an equilibrium.

Â So we guessed the equilibrium and then verified it.

Â How do you actually find an equilibrium in these cases?

Â Well if it's not a nice uniform distribution, and

Â guessing that it's going to be a linear function of your value, then you've

Â got to guess more complicated functions and there's going to be basically some

Â integration problems which will give you some ideas of how this works for

Â certain distributions, for arbitrary distributions the bidding functions can be

Â much more complicated functions and basically will only be described up to

Â some integration conditions, so there are papers that give some solutions to this.

Â There's a paper by Milgrom and Weber in 1982.

Â There's some other papers out there which will give some background on solving these

Â kinds of auctions.

Â The first price auction has actually been solved quite early on in literature.

Â More generally, solving any of these ones where we don't have

Â strategies is going to involve some guesses and in some cases,

Â there might be some integration conditions that will give us nice solutions,

Â but they often won't exist in closed form.

Â And especially, here also, the symmetry helps a lot.

Â So if you have asymmetric auctions things can be difficult.

Â And more generally,

Â even finding whether equilibria exists in these worlds is not an easy problem.

Â There's literature, in fact that I've worked on a bit,

Â about existence of equilibria in auctions.

Â So one thing about equilibria in auctions is they're discontinuous.

Â I change my bid a little, and

Â I might be going from winning the object to not winning the object and

Â so suddenly my pay off goes from being positive to being zero.

Â And that this continuity means that I might not always

Â have a nicely defined best responses that are going to

Â move in ways that were used implicit proofs of existence before.

Â Made nice use of the best response correspondence, with that, it's not

Â going to have the properties it used to have if this continues games and getting

Â equilibrium existence in these kinds of auction settings it's quite tricky and

Â there's somethings that are known about setting to do exist equilibria.

Â There are also some examples where they don't exist equilibria so

Â that's actually an interesting project on it's own.

Â So, different kinds of auctions, different kinds of equilibria,

Â they're related to each other and we'll see more about that in a bit.

Â