 This video will introduce you to the revenue equivalence theorem, which is one of the most important results in auction theory. I'll begin by stating the theorem and explaining why it's important. I'll show how we can use it to think about first price auctions. And then at the end of the video, I'll prove the theorem to you, which is important because there's actually a lot we can learn from the way that the proof works. So to begin, what is this revenue equivalence theorem? Well, most importantly, it's an answer to the question of which auction we should choose. So far, we've seen all kinds of different auctions. First price auction, second price, English, Dutch. It's kind of confusing. You wonder how you should actually, as an auctioneer, decide which auction you should use. Well, strangely enough, the revenue equivalence theorem says that to a large extent, it actually doesn't matter which one you pick. So let's see formally what it says. Let's assume that we have n different risk neutral agents, each of whom has an independent private valuation for the single good that's at auction, and each of which is drawn from a cumulative distribution, F. Now we're going to consider two different auction mechanisms for these two similar settings. And we want to assume two things about these auction mechanisms. First of all, that in equilibrium, they allocate the good in the same way all of the time. So for all of the different types that agents might actually have, they have the same allocations. And secondly, that an agent with valuation zero, an agent with the lowest possible type, would have an expected utility of zero in the auction. If these two properties hold across both of these auctions, then the revenue equivalence theorem tells us that both auctions yield the same expected revenue, and indeed that both result in any bidder with the same valuation making the same expected payment. So in other words, as long as two mechanisms allocate in the same way, and as long as they charge an agent with the lowest possible valuation, the same amount, zero, then the rest of their payment functions have to be the same as well. So you can't get extra money out of agents without changing the allocation function or the payment of the lowest valued agent. Now, in fact, I've stated the revenue equivalence theorem here in a way that's a bit narrower than I have to, just to make it a bit easier to say to you. So in fact, I can get beyond the independent private values assumption that I've made, and I can also get a bit beyond the single good assumption. So stating it in its full generality would have been just a little bit more notationally clumsy, although the proof that I'll give you actually holds more generally. Okay, now let's see why the revenue equivalence theorem is useful for thinking about first and second price auctions. To do this, we need to think about a concept called the KTH order statistic of a distribution. So the KTH order statistic is the expected value of the KTH largest of N draws. So imagine that I was to independently draw five times from a distribution. What is the expected value of the biggest of those draws? Or what is the expected value of the second biggest of those draws? Those are questions about the first order and second order statistics of that distribution. For K equals one and two and for N equals five. Well, here's a useful formula to know for what we're gonna think about just now. If I make N IIAD, which is to say independent and identically distributed draws from the uniform distribution over the range zero to V max, the KTH order statistic is what you see here. It's this fraction multiplied by V max. Now, using this fact directly, I can conclude that in a second price auction, the seller's expected revenue is this fraction. And that's because I know that in a second price auction, the seller's expected revenue is the same as the expected value of the second highest bidder because the seller always gets the revenue of the second highest bid in a second price auction. So if I just substitute in, that's this is the expression that I get and you can confirm that. Now, first and second price auctions both satisfy the requirements of the revenue equivalence theorem. It's easy to see for second price auctions. For first price auctions, it's not quite as easy to see that they allocate goods in the same way, but as a second price auction does. But basically what I want to reason is that in a first price auction, the bidder with the highest value gets the good. And I can argue this essentially by saying every symmetric game has a symmetric equilibrium. The first price auction Bayesian game is a symmetric game because everyone's IID. And so in a symmetric equilibrium of this game, a higher valuation is gonna happen if and only if a bidder has a higher bid. And that's gonna mean that the person with the highest valuation wins the auction. So then I conclude that a bidder in a first price auction is going to have to pay on expectation the same amount as the same bidder would pay on expectation in a second price auction. And because only the winner has to pay anything in a first price auction, that means everyone should bid their expected payment conditional on being the winner of a second price auction. So each bidder in a first price auction should imagine that they're the highest bidder and then bid the expected second highest bid conditional on their value being the highest sample from the distribution. And you might think this is kind of strange because of all of the end bidders, N minus one of them will be wrong when they do this conditioning. Only one of them actually will be the highest bidder. So why should they all bid as though they think they're the highest bidder? Well, the reason this makes sense is that the only bidder that this matters to is the bidder with the highest valuation. And so if the bidder turns out, in fact, to win the first price auction, then he was right to do this conditioning. And otherwise his bid doesn't matter anyway. And so the fact that he conditioned wrong is okay. So thinking now about the highest bidder, if his valuation VI is the high value, then there are N minus one other values which are below his. And so all he knows is that they're drawn from the uniform distribution on the interval between zero and his bid. I hear I'm thinking about first price auctions in the case of a uniform distribution. And thus the expected value of the second highest bid is going to be the first order statistic of N minus one draws from this distribution. We're thinking about the second highest bid but we've conditioned on the first highest bid. So we're now gonna think about the biggest of the draws among the remaining agents. That's gonna be the second highest bid. And I can just substitute in to this expression for the first order statistic of N minus one draws. And I can substitute in N minus one into that expression and simplify. And what I get is this expression here. And that's something you've seen before, right? We've claimed in the past that the symmetric equilibrium of a first price auction with the independent uniform valuations is for all of the bidders to bid N minus one over N times their valuation. And now you can see actually where that comes from that we can actually find that by reasoning through revenue equivalence. So essentially there's two ways of finding the equilibrium of an auction like a first price auction. The hard way is that you do a ton of calculus to actually figure out what the equilibrium is from first principles. That's really hard. Even in the case of a first price auction that's really hard. The easy way is that you guess the equilibrium by appealing to revenue equivalence and then you just verify that it's an equilibrium. And so we know from revenue equivalence by comparing to second price auctions that are easy to analyze that the expected revenue has to be the same. And then we work through to figure out how bidders would have to bid to get that expected revenue and we arrive at this expression. Okay, well all that remains for this video is for me to show you how we prove the revenue equivalence theorem. So let me now do that. This is gonna be the most technical and advanced part of the video. If you find that you bogged down and this is too much for you, you understand everything else you need to know about revenue equivalence without watching the proof. So you can stop watching the video now. But I encourage you to go on because I think this is a really cool proof actually. I actually worked really hard to make this part of the video for you because I really like this proof. So I hope you enjoy it too. So let's see how it works. We have to introduce the idea of an x-intram allocation function and an x-intram payment. So let's see what that means. Well, I'm gonna call, remember we have this choice function x. Well, I'm now gonna have the choice function xi. And what I'm gonna mean by that is the choice function from the point of view of agent i given that everybody else is following the equilibrium strategy profile s for some Bayes Nash equilibrium and that agent i has type vi. So what this xi is gonna give us is the probability that an agent i with type vi is going to be allocated the good given that everybody follows the equilibrium strategy s including i. And similarly, p sub i is going to be the expected payment that agent i has to make. Again, under the assumption that everyone is following the equilibrium strategy profile s and that agent i has type vi. So here's a stronger theorem than the revenue equivalence theorem, but it should be easy to see that this implies the revenue equivalence theorem directly. So once we have this theorem, we just basically get the revenue equivalence theorem for free. What the revenue equivalence theorem tells us is that mechanisms that allocate in the same way all have to have the same payment function. What this tells us is exactly what the payment function has to be a little bit about how the allocation has to work as well. It really shows us exactly what the equilibrium even needs to be. And so as a side effect, we'll see that any two mechanisms that allocate in the same way have to have the same payments. But let's look at what this theorem says. This is really a kind of crucial theorem for mechanism design. So here we're saying that when values are drawn from a continuous joint distribution f and agents are risk-neutral, a strategy profile s is in Bayes Nash equilibrium only if the two conditions that follow are true for all agents i. First of all, we need monotonicity. So each agent's x-interim allocation probability needs to be monotone non-decreasing in that agent's value. So in other words, as agents have higher and higher values for the good, their probability of getting the good averaged across all of the other agents' type, as well as potentially across randomness in the mechanism itself, needs to weekly increase. So as I have higher and higher values for the good, it can never reduce my chance on average of getting the good. So that's necessary for a Bayes Nash equilibrium. The other thing that's necessary for a Bayes Nash equilibrium is the following payment identity. And this probably looks kind of scary. It's just a whole lot of math, right? So I'm saying the payment function, this x-interim expected payment that an agent makes needs to look like this. It needs to be the agent's value for the good multiplied by the probability that they get the good minus this integral, the integral from zero to VI of this x-interim allocation probability plus the expected payment that an agent of type zero gets. Now, it's often the case that we just want to have auctions in which an agent with the lowest possible valuation doesn't pay anything. And if that's true, then this just goes to zero in the payment identity and we get a slightly simpler expression. But nevertheless, this integral probably seems pretty mysterious to you. And what's cool about the proof is that I can give you a really intuitive idea of where this integral comes from. The final part of the proof that is also really interesting is this last statement here. So it says if S is onto, then the converse holds. What does that mean? Well, here we've said that a strategy profile S is in base Nash equilibrium only if these two conditions hold. Here we're saying if we add the requirement that S is onto, which I'll explain in a second, then it's also sufficient. Then these two conditions imply base Nash equilibrium. So what does it mean for S to be onto? It means there's some type you could have that would lead you to make any given bid. So I can always simulate any action that I want to take in the game by pretending to have a certain valuation. So that means that I can get to any action in the game by having a valuation. Formally, we say the strategy profile is onto. That's all that means. So this proof proceeds in three parts. Let me first say this proof follows a really elegant proof that I like very much by Jason Hartline. Here's a URL to his book in which he introduces this proof. And I also use figures adapted from his proof with his permission. So this proof has three different parts. I'm going to start with the last thing. I'm going to say that the strategy profile is a base Nash equilibrium if the monasticity and payment identity properties hold and furthermore S is onto. So I'm going to show that those things suffice to give us a base Nash equilibrium. Then I'm going to show that those two things are necessary. That S is a base Nash equilibrium only when monasticity holds and that S is a base Nash equilibrium only when the payment identity holds. I'll make a couple of simplifying assumptions in this proof just to make it a little bit less annoying to follow. I'm going to consider the special case where the support of every agent's valuation function is between zero and infinity. So we just have kind of arbitrary non-negative real valued valuations. And I'm going to assume that the payment of an agent with the lowest possible type is zero as I've mentioned before. Okay. So moving on to this first part where I'm going to show that the characterization and the assumption that S is onto implies that S is a base Nash equilibrium. So to begin, if agent I deviates from taking action from the strategy profile S and instead takes the action that the strategy profile S would have him take if his type was not VI but instead with some other type VI hat then what would agent I's utility be? Well, we can write an expression here. So I'm going to, first of all, introduce some notation. I'm going to say, I'm going to use this notation U sub I comma VI to mean the utility of an agent I whose real type is VI when he reports the type of VI hat I and when everyone including him then follows the strategy profile S. And what I mean by him following the strategy profile S is that he kind of lies to himself. He applies the strategy profile S to his incorrect type VI hat I. So this is kind of the exentrum expected utility of agent I for kind of misreporting his type in the strategy profile. And so what is that expected utility? Well, it's the utility that he gets from getting the good minus the payment that he has to make. So what is the utility that he gets from consuming the good? Well, here I use his real valuation because if he actually gets the good he values it according to his real valuation not according to the lie that he told. But the probability that he gets the good is going to depend on his declaration. So he's going to have the allocation probability of an agent with type VI hat I but then he's going to have the utility for getting the good of an agent of type VI. And his payment is of course gonna be the payment of an agent of type VI hat I because that's what he will have declared to the mechanism. And note that as I was just saying before it's possible for agent I to play any action by behaving in this way because of our assumption that S is onto. So he can find some VI hat I which will correspond to any action in the game any one of the real values in the interval. So what does it mean then for a strategy profile S to be in base Nash equilibrium? Well, it means that agent I is best responding by reporting his true type to his own strategy profile rather than kind of lying to himself sort of like we saw in the revelation principle. So in other words, his expected utility for actually using his true type has to be at least as big as his expected utility for lying to himself and reporting a different type to the strategy profile. If that's true for all agents and for all values that the agent might have and all lies that he might tell then we have a base Nash equilibrium. Okay, well, that was the fun part. It turns out we can prove this almost entirely using pictures. So let's start by considering some arbitrary and monotonic allocation rule. I'm assuming it's monotonic because recall the assumption of our proof is that the characterization holds and half of the characterization is that the allocation rule is monotonic. So let's think about what this picture says. Here on the X axis, I have the valuation that an agent might have. So this is the lowest possible valuation and this is infinitely big valuation as we go far enough out. And this is the probability of getting the good. So if the agent reports the lowest possible valuation then he doesn't win the good. And because this rule is monotonic as he reports higher and higher valuations he has an increasing chance of getting the good. It can flatten out and stay level. It can go up, it can go up gradually. It can go up discontinuously. And if it ever hits one, it has to stay at one because it's monotonic. And so there, of course it doesn't have to ever hit one. It could just gradually ramp up. But if it ever hits one as I've drawn it here then for all values beyond that point it has to stay at one. So I'm not gonna make any assumptions about what this function looks like except that it's monotonic. Okay, so now let's think about first of all agent i's value for consuming the good. Just what I talked about before. Remember that his utility has two parts. His value for consuming the good and the payment that he has to make. So first of all, let's think about his value. Well, as I argued before this is his value for telling the truth, for playing truthfully where he reports his own true value to the strategy profile. And this is his value for consuming the good when he has kind of lied to himself and so he gets a different allocation probability. So let's think about what those things look like on the picture. Well, notice that these expressions are products. It's a value multiplied by an allocation probability. So we can see that kind of in the picture as a rectangle where it's the area of this shape. Here's the valuation, right? This distance here is the valuation. It goes from zero to VI. And this distance here is the allocation probability. It goes from zero to XI of VI given as. And so this whole shaded region here is that product. So the area of that region is the product VI times XI of VI given as. What about in the case where he doesn't tell the truth to himself? Well, for this proof, I'm gonna assume that the agent is considering reporting a lower type than his true type. You, I'll leave it as an exercise for you to do the reverse proof where he considers reporting a higher type, but it's very much the same argument. So it should be easy for you to check your understanding by trying to run the proof in the other way. So when I assume that the value V hat I is lower, because of my assumption that the function is monotonic, that means that this point here is gonna be lower than this point here, the pointed V hat I as compared to the pointed VI. And so that means that this corresponding point on the Y axis is gonna be lower than this point was over here. And all of that means that the area of this rectangle here is gonna be smaller. And it also means that this corner of the rectangle is not gonna touch the line the way that it does here. That's fine. Okay, so what we've seen so far is we've kind of learned how to look at these pictures and we've seen that the area of these two rectangles is the expected utility that the agent gets for consuming the good. His true value in both cases VI multiplied by the allocation probability which depends on whether he's reporting his true type or reporting some lower type V hat I. Okay, so that was consuming the good. Now let's think about the payment that he has to make to the mechanism. Well, again, we're assuming that this characterization holds and that means that we're assuming that this payment identity is true. And remember the payment identity had a P zero in here but we don't have to worry about the P zero because we've assumed that it's zero. Oh, pardon me. Okay, so what is this? So now it's time to think about this kind of crazy integral. So let's actually think about what the payment identity is. Well, this first term here is the one that we've been thinking about all along, right? That's the area of the rectangle, right? VI multiplied by XI of VI given S, which is that. So it's that whole shaded region that we just looked at on the previous slide. Now, what are we subtracting here? This is the integral from zero to VI of XI of Z given S. So in other words, it's the area underneath this allocation curve. It's all of that. So the payment is the whole rectangle minus this stripy region here. And of course, that's the gray region that I'm showing you on the slide. So the agent's expected payment is the part of that rectangle that I showed you which lies above the curve. Well, that's kind of interesting. Whoops. Okay, so then I can put those two things together and figure out what the agent's expected utility is for playing as though his type was VI and playing as though his type was VI hat given that in fact, his type really is VI. And so, well, remember that the expression for this expected utility is simply the value for consuming the good, which as we saw before, whoops, is the rectangle minus the payment identity, which as we saw before is the part of the rectangle that lives above the line. And so this difference then is the area that lies below the line. So that's the amount of utility that the agent would get. And you can see that these two pictures are pretty similar. They're both this area under the curve. They both go up to VI. But the difference is that this one kind of gets clipped off at this lower allocation probability here, which we kind of correspond to about here, whereas this one goes a bit higher. And indeed, if I then think about the utility difference, the difference in utility between telling the truth, really playing as the true type, and the utility of playing as though you had a different type, that's gonna be this amount here. And what's important, the thing that we have done all of this in order to reason to ourselves is that this is a positive number. And the fact that it is an actual area in our graph that it has positive area goes to show that it's a positive number. And because of monotonicity, really the result that made this true is that this point here has to be weakly lower than this point here. It would have been possible for the function to be flat here in that region. And if it was, then this number would be zero rather than being positive, but it can never be negative. The only way that it could be negative would be if the function went down and the function is not allowed to go down by monotonicity. And so that tells us that this utility difference is always going to be non-negative. And remember we argued before that as long as this number is greater than or equal to this number, then S is a Bayes Nash equilibrium. And so what we've just argued is that indeed that's true. So the characterization plus our assumption that S is onto allows us to reason that S is a Bayes Nash equilibrium. So that's a really powerful thing. We've just seen something very structural about what the Bayes Nash equilibrium of a quasi-linear mechanism has to look like. Okay, now let's do the sufficient conditions. So now I want to say that S being a Bayes Nash equilibrium implies that monotonicity has to hold. Okay, so again, I'm gonna make the same kind of argument that I made before that having a Bayes Nash equilibrium implies that for all valuations that the agents could have and all other valuations that they could pretend to have, the utility that agent I gets on expectation from really having valuation VI and playing as though he has valuation VI is at least as big as the utility that he gets from really having valuation VI and playing like he has valuation VI prime. If I expand this out, I get this expression here. Here, all that I've done is just substituted in this expression that we already looked at in the previous part of the proof that says that my utility is the amount that I get by consuming the good minus the payment that I have to make in both cases, depending on the report that I really made. Okay, so you may have noticed I've switched from using V hat to V prime and that's because I wanna actually think that this has to be, it can be possible for the agent really to have valuation VI or really to have valuation VI prime and I can use these facts both ways. So I'm gonna consider two possible values, Z one and Z two that the agent really could have and I'm gonna substitute in to this expression here kind of both ways. So I'm gonna substitute in VI equals Z one, VI prime equals Z two and I'm also gonna substitute in the other way around and depending on which way I substitute, I get two different inequalities. So if I do the first substitution here, then I get this inequality which says, well, it says what you read, there's no point in my reading it out, but it gives an expression that has Z two here and Z one here for example, when I do things the other way around, everything switches. So I get Z ones and Z twos in the other way and both times I have the inequality pointing this way because that's what we always started out with. So basically what this is saying is an agent of type Z one wouldn't want to lie and pretend to have type Z two and similarly an agent of type Z two wouldn't want to lie and pretend to have type Z one either. Okay, well, I've got these two different inequalities here so I can just add them together and if I do that, then that causes these P terms to cancel out and so then I just get these two things added together and when I do that, I end up with this expression here and then I can just factor that expression and I end up with this simpler inequality here that doesn't talk about the payments at all and you can verify that for yourself. That's just some simple algebraic manipulation and really the mod ethnicity requirement just pops directly out of this expression here. So in particular, you can see that if Z two is bigger than Z one, meaning that this term here is positive, then in order for this inequality to be true, in order for this whole thing to be non-negative, we have to make sure that this is also not negative. So we can conclude that this being positive implies that this has to be positive. Oh, sorry, mistake on the slide I should say is positive. So I'll change that in the slides. So in other words, it means that the X function has to be monotone non-decreasing. It means that when this number is bigger than this number, this number has to be bigger than this number. So that means the function has to be weakly going up and that's exactly what mod ethnicity says. So that establishes the second part of the proof. All right, so now there's only one last thing to say and that is that this payment identity is required. So let's see what I wanna say here is that the strategy profile as a Bayes Nash equilibrium implies that we need the payment identity. The payment identity didn't just kind of suffice to give us a Bayes Nash equilibrium, but we actually need to have it. So I'm gonna start with those same two inequalities that we just talked about before and now I'm gonna do something different. I'm gonna solve each of them for this expression. So I'm gonna kind of rearrange them. You notice they both have a PI Z2 given S and a PI Z1 given S. And so if I rearrange things, then I can get them so they both have that on one side of the expression. And when I do that, I end up with one of the equations looking, one of the inequalities looking like this and the other one looking like that. And in total, I can think of that as upper and lower bounds on the differences in expected payments that have to be made between agents of type Z2 and agents of type Z1. So in other words, this difference in payments, whoops, this difference in payments here has to be not less than this amount and not more than this amount. Well, that's fine, but this just looks like a lot of messy math. It's not clear why these upper and lower bounds are very helpful to me. Well, here again, we wanna go back to our pictures. So let's visualize this on the graphs that we were looking at before. So here I've just written the upper and lower bound expression for you again so that you can compare it. And here is the upper bound and here is the lower bound. And let's think about what those numbers mean. Well, here I'm multiplying by Z2. So here's the distance Z2. And here I have the allocation probability of an agent of type Z2 minus the allocation probability for an agent of type Z1. And that's this difference here. And so in total, the upper bound is this shaded region here. So the payment difference can be no more than that. The lower bound is Z1 multiplied by that same difference. So I have that same difference that I had before. Now I'm multiplied by Z1. So that gives me this shaded region. And the important thing to notice about these two shaded regions is that they intersect the curve at different points. So this one touches the curve right there and this one touches the curve right there. Well, I can argue this more formally using calculus but if you think about it, it's easy to get an intuition that makes it clear that the only payment rule that can satisfy these upper and lower bounds for every given pair of types Z1 and Z2 has payment difference that's equal exactly to the shaded region that's shown here. Basically it needs to touch the curve everywhere because if I change Z1 and Z2, I still need to have the property that it's never less, so it's never more than that amount and it's never less than that amount. And so as I kind of move around, I need to make sure that I always have those sort of points of contact with the curve. And the only way for that to work everywhere is to have contact with the curve everywhere. So kind of at an intuitive level that should be convincing to you and if you'd like to see the proof with calculus, you can look at the book citation that I gave you at the beginning. And that pretty much gives us the payment identity. In order to get the payment identity, we just want to set Z1 equals zero, so push this guy all the way down, set Z2 equals one and then that gives us this whole shaded region that we saw before. Okay, so that concludes our proof of the Bayes-Nash equilibrium characterization result and recall that that directly gives us the revenue equivalence theorem as a corollary because what the revenue equivalence theorem said is that having the same allocation probability implies having and also having the same payment for an agent with the lowest possible type implies having the same expected payments. What this characterization result said is something much stronger that having the same allocation probability implies exactly the payment identity that you saw. And the payment identity has a term in it for what the lowest possible type gets and so if that is set constant in both cases, then they have to be the same. So the overall claim that we want to make here that overall takeaway from all of this is that if two mechanisms have the same allocation rule, then they basically have the same payment rule too. And when I say basically essentially, all I really mean is that the only thing that can change is if I just offset everything by a constant. So if I pay you $100 for showing up in my auction and then we have an auction, then my expected revenue is different. You know, if I charge you $100 for walking in the door and you have no choice about whether to walk in the door and then we have an auction, I'll get a different amount of revenue but that's not very satisfying. That's why we have this requirement about the agent with the lowest possible type gets paid zero. As long as that's true, then there's really no other wiggle room in the payment rule. And so mechanisms with the same allocation rule just need to have the same payment rule. A key corollary is that all efficient auctions yield the same revenue and equilibrium because after all, efficiency is just one allocation rule. Efficiency means the agent with the highest valuation always gets the good. That's an example of a restriction on what the allocation rule needs to be. And so that tells us that it turns out most of the auctions we care about are efficient. And so for all of these efficient auctions, we know that there's actually nothing we can do to change the amount of revenue that we get in equilibrium by changing the auction within efficient auctions. And indeed, this applies not just to the auctions we've seen before, but to some pretty weird auctions. So you can think about not a first or second price auction, but a third price auction. And that actually turns out to be revenue equivalent. You can think about various auction designs in which not only the winners have to pay, but losers pay too. We didn't make any assumptions about these kinds of things in the proof, and so it turns out they're also revenue equivalent. One last caveat, risk neutrality did turn out to be important. So I've assumed everywhere in this video that agents are risk neutral, which means they treat expected payments the same as sure thing payments. They don't care about how much randomness there is in the environment. They just care about the expected payment they have to make. And if that isn't true, then these different auction mechanisms actually do turn out to be different. So that concludes this lecture. And you'll go on now in the next lecture to think about how to actually maximize revenue in an auction. And what we can see from today is that's going to have to happen by changing the allocation rule. There's no way of maximizing within efficiency because efficient auctions are all going to yield the same revenue. If what you want to do is to maximize your revenue, strangely enough, you are going to turn out not to want to sell to the person with the highest valuation all the time.