0:00

Hey folks. So let's talk now about optimal auctions.

Â And what do I mean by optimal auctions?

Â Basically, and up till this point in the course, usually when we've been analyzing

Â auctions, we've taken auctions as given and analyzed them.

Â Or we've talked a little bit about optimal efficient auctions,

Â in terms of Vickrey-Clarke-Groves mechanisms and so forth.

Â And here the idea is that we'll think about a seller designing an auction.

Â And the seller might have incentives, not necessarily to always do the efficient

Â thing, but instead to maximize the price that they get, maybe expected price.

Â So they're interested in maximizing expected revenue.

Â And what does that mean in terms of possible losses in efficiency.

Â It may mean that sometimes the seller doesn't sell the good, so

Â they may risk failing to sell it.

Â They might also even sell to a buyer that doesn't necessarily have the highest bid.

Â And that could happen in settings where there's some asymmetries among buyers and

Â you want to use one buyer, the threat of selling to one buyer to make sure that you

Â increase the bids by the other buyer.

Â And we'll see a little bit of that in a bit.

Â So here, we're going to think now about optimal auctions in this narrow sense of

Â trying to design an auction which is going to maximize the seller's revenue.

Â So, what kind of setting are we going to look at?

Â We'll keep things relatively simple, in terms of the structure,

Â so one good to be sold, private valuations.

Â So each buyer has some value for the good, they're each risk neutral.

Â The bidder's valuation is going to be drawn from a nice distribution that CDF

Â is the cumulative distribution function.

Â So we have a cumulative distribution function,

Â Fi with a pdf density function of Fi.

Â Which is going to be continuous and bounded below, so

Â nicely defined continuous distributions over values, so a uniform distribution,

Â or a series of other distributions that will keep things nicely shaped.

Â 2:11

We're going to allow for asymmetries among the individuals.

Â So, now we're going to allow one bidder to have one distribution and

Â another bidder to have a different distribution.

Â So, it could be that I think of some bidders as having deep pockets and

Â high values, and other bidders as being more thrifty and

Â not having as high a value for an object.

Â So we're going to have different distributions, and

Â the important thing is that the risk neutral seller here is going to

Â know the distributions and has no value for the object.

Â So zero value for the object, knows the distributions, but

Â doesn't know the actual values.

Â So has an idea of what range the values are and what relative probabilities

Â of different values are, but does not know the precise values, okay?

Â So the optimal auction here, we're going to

Â be looking at an auction that maximizes the seller's expected revenue,

Â subject to some form of individual rationality.

Â In this particular case it's not going to matter so

Â much which type of individual rationality we impose.

Â We can think about it as exposed, or interim,

Â you can always move things around, in terms of expected payments,

Â given all the risk neutrality, and the transferability here.

Â So, we want Bayesian incentive compatibility, and

Â make sure that the buyers don't want to walk away from the auction.

Â Okay, and we're going to think about optimal auctions.

Â 3:46

And so here,

Â let's start with two bidders, each has a value uniformly distributed on 0,1.

Â And let's think of changing the auction in a very simple format,

Â what we're going to do is stick in a reserve price, okay?

Â And this is quite common in auction.

Â So, often when people are selling things of,

Â there'll be a minimum bid, reserve price and

Â you're not allowed to enter into the bidding unless you exceed that threshold.

Â And so we get a set of reserve price in here,

Â we're going to have a second price auction to make life simple,

Â keep with the dominant strategies, but just stick in a reserve price.

Â Okay, so let's think about how this works.

Â Once we've put in the reserve price,

Â then if both people end up bidding below that we're not going to get anything so.

Â And here's the risk now, if we didn't have a reserve price,

Â we would always sell the object, now there's a possibility of no sale, so

Â we're not going to get anything.

Â If one bidder bids above the reserve and one below it, then the reserve price is

Â going to kick in, that's going to be the second highest price.

Â Second highest bid we'll think of the reserve price as if it was a bid, and

Â the second highest bid is now the reserve price and

Â so we'll sell at reserve if somebody bids above it and the other below.

Â So again, second price auction that's going to be the second highest.

Â And if both of them bid above it, if both bidders happen to bid above it

Â then it will just be second highest bid that will be the price.

Â Okay so that's the setting.

Â Which reserve price is going to maximize the expected revenue, so

Â let's take a look at that.

Â So first thing to note, it's still a dominant strategy to bid your true value,

Â and in fact,

Â you could think of the reserved prices just like the third bidder, in this case.

Â And so from many bidders' perspective,

Â you're still want to be bidding your true value if it's a second price auction

Â where you think of this reserved prices that was just affixed in out there.

Â You just happen to know what one of the bids are,

Â it's still the dominant strategy to bid your true value.

Â So that makes our life easy in terms of the analysis.

Â So, if both bids are below R, that's going to happen now

Â with probability R squared, because people are bidding truthfully.

Â So the chance that they're both below R with a uniformed distribution,

Â independent draws of a uniformed distribution,

Â probability that any one of them is below is R.

Â So, the probability the both of them are below is R squared, in that case,

Â the revenue is 0.

Â The chance that one's above, and one below?

Â Well, the chance that one's above is one minus R, the other one's below is R, and

Â it could happen in two different ways, in terms of which bidder is above,

Â and which bidder's below.

Â In that case, the second highest price is the reserve price, that's the revenue.

Â And then, it indicates where they're both above,

Â that's going to happen with probability 1- R squared and in that case,

Â then we're going to get the expected minimum of the two values.

Â Conditional on both of the values being above R and if you just work out

Â the integral of what's the expected minimum of two uniformly distributed

Â random variables, conditional on those variables going between R to 1.

Â So now they fall both within R to 1.

Â Figure out what the expected minimum of that is I'll save you the algebra.

Â It's 1 +2R over 3.

Â Okay, so when we look at these,

Â then we do the overall expected revenue.

Â What do we get?

Â We've got this probability that we're going to get a revenue of R.

Â So multiply those out.

Â We get 2 R squared (1- R) here.

Â And then we've got this probability that both people are above and

Â then we have this expected revenue so

Â we end up with an overall expected revenue of this expression.

Â Because if they're both below, we don't give anything so

Â that's the overall expression as the expected revenue.

Â If we collect terms here, in terms of Rs, you can simplify this,

Â so the expected revenue here is given by this.

Â Expression 1 + 3R squared- 4R cubed over 3.

Â So now we have an expression for revenue.

Â Just maximize that with respect to R.

Â So let's take the derivative with respect to R and set that equal to 0.

Â What do you get?

Â 0 = 2R- 4R squared.

Â Solving that, R = a half.

Â So what does that tell us?

Â If you're facing two bidders in a second price auction,

Â uniform values between zero and one.

Â You set a reserve price of a half, that's going to maximize the expected revenue.

Â Okay, you can do calculations.

Â So supposed let's do our calculation, when we set the reserve price of a half,

Â what do we earn in terms of revenue?

Â Well, you can stick your half in here, do the calculation, you end up with 5/12.

Â If you set a reserve price of 0, then the expected revenue,

Â you get 0 for the R's, you get one-third here.

Â 9:02

So you end up with a lower, this is just 4/12, right?

Â So we end up with a higher expected revenue when we set reserve price at half.

Â So increasing at reserve price, increases the expected revenue for the seller.

Â The half was nicely chosen.

Â If we set it too high,

Â then you have no chance that you're going to end up with a sale.

Â And then your profits actually go down.

Â So it's hitting this sweet spot in terms of what the reserve price was.

Â So the trade-offs here, you lose sales when both bids were below a half by

Â setting this reserve price, but that was low revenue in any case and

Â it only has a probability of one-quarter of happening.

Â You increase the price when one of the bidder has low and

Â the other one was high and it's actually happen with probability of a half.

Â And so in this case.Increasing the result price lead to high

Â venue because the increase got from here more than the onset

Â the decrease you got by losing sales when both bids happen to be low.

Â Okay, so you can think of adding this reserve prices as if you're adding a third

Â bidder, a sort of increasing the competition in the auction, and

Â even though that players know exactly where that reserve is,

Â it's raising the overall revenue in the second price auction.

Â 14:09

If not, then we won't sell it to anybody if this isn't satisfied.

Â If the good is sold, then the winning person is going to be charged this

Â smallest valuation that, that person could have declared and still been winner.

Â So we look at the smallest valuation that has them

Â still being above their reserve price and still being the winner.

Â And think of that as sort of the benchmark of equivalent of

Â either the second highest price or the reserve price,

Â that's going to be the value that they have to pay, okay.

Â So we have an auction where now what we do is we,

Â people put in values, declare the values for the good, but

Â now we adjust them via these virtual valuations and have reserved prices.

Â And choose the person who has the highest virtual valuations subject to

Â that being above the reserve price and charge them the price that they could

Â have gotten in terms of the lowest valuation they could have announced and

Â still win the winner of this auction.

Â 15:21

Okay, a corollary of this where you can begin to simplify all of this.

Â If we're looking at a symmetric setting where everybody has exactly the same

Â distributions like we did earlier in the two better case and

Â the second price auction case.

Â What do we end up with?

Â It indeed, it turns out that the optimal auction is a second price auction with

Â the reserve price of r star where that reserve price satisfies this equation,

Â and in the case of a uniform distribution, the reserve price is going to be a half.

Â So would actually say that, the optimal auction in the case of

Â the symmetric setting would be set your, if uniform zero one, set your.

Â Reserve price to a half and

Â then have a situation where you do a second price auction above that, okay?

Â So the nice thing about this is it says in symmetric settings,

Â things simplify dramatically and you're running basically second price auctions

Â with reserve prices and the reserve price is going to depend on its distribution.

Â What's really going on now you can understand a little more about what this

Â virtual valuation is doing in terms of adjusting things.

Â It's adjusting things to capture relatively how much of

Â the distribution is at high values compared to lower values.

Â And the reserved price has been pushed up to a point where you want to push it so

Â that you're making sure that you're not going so

Â high that you're going above the bids and not going to sell the item.

Â But you want to move it up high enough to have some competition.

Â And that trade off is being captured by exactly this expression.

Â 17:03

Okay, so the optimal auction has some features that we would have seen

Â as in a Vickrey-Clarke-Groves mechanism.

Â And what's a little bit different here is that we have two things going on.

Â One is that the actual valuations

Â are being adjusted by these virtual valuation functions.

Â And also it's not looking like a Vickrey-Clarke-Groves

Â mechanism in the sense that the outcome isn't always efficient.

Â Sometimes we do not sell the good in these kinds of auctions.

Â And how should bidders really bid in this?

Â So the good news is that we still have a second-price auction

Â kind of format held in this virtual valuation space.

Â So the transformations mean that it looks slightly different in terms of what

Â the payments are going to be and when somebody ends up winning something, but

Â the proof that a second-price auction is dominant strategy instead of compatible or

Â truthful still works out here.

Â So you still get these nice properties in terms of the incentives of the auction,

Â but it's different in terms of what the specifics are in terms of the payment

Â rule, and we're not always doing efficient things in this world.

Â So here again, we see that trade-off that we've

Â seen before in the Myerson-Satterthwaite theorem.

Â Here, when we're trying in this case, to get incentive compatibility and

Â maximize things for the seller.

Â So the seller's trying to figure out how they can get the most out of this auction,

Â we end up with some inefficient trade.

Â So the seller's going to actually have to commit to doing this, right?

Â So you have to commit to abiding by the reserve price and if everybody's bid comes

Â in afterwards, it's not saying, well, sorry, I didn't really mean that.

Â I'm going to revise my reserve price and drop it down.

Â So this mechanism works if the seller can commit to not selling

Â the object if the bids end up below the reserve price.

Â Okay, why does this work?

Â What's going on here?

Â Well, reserve prices again, are like competitors.

Â They increase the payments of the winning bidders.

Â The virtual valuations,

Â why are they playing a role here in terms of deciding who's a winner?

Â Well, if you have strong asymmetries in this auction, if you didn't do anything,

Â and you were the seller, it could be that the really weak bidder basically

Â isn't really competing with the strong bidder.

Â So by changing things relative to these virtual valuations,

Â that can make weak bidders be more competitive with strong bidders.

Â Now that has a plus and a minus to it.

Â One is that sometimes you end up selling it for

Â a lower price than you would otherwise to the lower bidder.

Â But other times, it actually inflates,

Â it means that the higher bidder has to end up paying a higher price, and

Â it effectively increases the competition there,

Â in some sense makes the bidding more aggressive.

Â Okay, so that's been a quick look at optimal auctions.

Â The proofs of these results are actually quite involved and

Â involve some calculus of variations and some delicate arguments.

Â They are worth looking into, if you really enjoy this stuff, dig in to

Â Myerson's paper and a whole series of ones that have followed up on that.

Â They're interesting papers, and the techniques behind them have become used in

Â other kinds of mechanism design arguments more generally.

Â 20:26

But what we want to take away from this is a couple of things.

Â One, is that the problems are well defined and they're well defined

Â solutions to solving for mechanisms that satisfy certain properties.

Â The ones we've been more interested in and

Â are probably more interested in from a societal point of view is,

Â how do we maximize overall efficiency, total utilities?

Â But sometimes we're also interested in understanding how sellers might act in

Â a market, or they might want to maximize revenue, not do the efficient thing.

Â How are they going to act?

Â What kinds of mechanisms are they going to choose?

Â And so mechanism design, you can put down whatever objective you want in terms of

Â what you're trying to maximize.

Â Is it overall societal welfare?

Â Is it revenue?

Â It could be some combination.

Â And in those kinds of situations, once you're going to do that,

Â then you put in instead of compatibility constraints,

Â that's actually part of what goes into the proof of Myerson's theorem.

Â Put those things in and maximize subject to those constraints and

Â see what falls out.

Â And that can give you a very general recipe for optimal mechanism design.

Â