 Okay, so we're back again, and we're going to talk a little bit about what's known as the Meyerson-Saturday theorem. So Mark Saturday, you've seen his name on several different theorems now, the Mueller Saturday, Gibbard Saturday, and now the Meyerson-Saturday theorem. And what this theorem refers to is trade, and it's basically going to say that it's going to be difficult to get efficient trade out with voluntary participation. And when we talked about strikes and so forth at the beginning of last week, one thing that's going to be important in understanding why we might have difficulties in getting trade out is that people can have private information about their utilities for various exchange of goods. So how much am I really willing to work? How much do I value a particular job? Can we design a mechanism that always results in efficient trade? Our strike's unavoidable. That's the basic essence here. And the conclusions that we're going to find out of this result are going to be that if it's not always completely obvious that a trade should be taking place, then it's going to be impossible to align incentives to get efficient trade out, even just in a Bayesian incentive-compatible way. And so we're going to look at a very, very simple trade setting. And even in that simple trade setting, we're going to see that there's going to be difficulties in getting efficient trade out. And that is really a very important and fundamental result that comes out of game theory and mechanism design in terms of understanding why it is that we're going to have breakdown in bargaining, why it is that we don't always get efficient outcomes. This is really a fundamental insight that came out of this area. So it'll be a fairly easy one to see, too. So we're going to look at an exchange of a single unit of an indivisible good. So I've got something, I've got a car, I want to sell it, and I know how much it's worth to me, you know how much it's worth to you, and we start haggling. So we're going to haggle over this. And in particular, let's think of the seller. The seller has some value in 01. So we're just going to normalize things to lie in the unit interval. But think of this as, say, the seller's value for this car. They're not going to give it away for nothing. They have some value for it. How much is it really worth before they're going to be willing to part with it? The buyer shows up. The buyer has some value for the car as well, theta B. And generally, we're going to think of this as, you know, sometimes the buyer is going to have a higher value than the seller. Sometimes they might not. Sometimes the buyer might think that the car is not worth as much as the seller thinks it's worth. Therefore, it won't be efficient to trade. And so we're in a setting where we've got these two different valuations, and now they start going through some mechanism, haggling, and we want to see what the outcome looks like. OK, let's do a very simple example, which will be useful in illustrating and actually proving the main theorem here. So let's think of a situation where the buyer's value can either be very low, 0.1, or high at 1. And the seller can either think that this thing's worthless, a value of 0, or they're really attached to it at 0.9. OK, so we've got these two values for the buyer and two values for the seller. And what's true is if we look at different combinations they could have, basically in all cases, except one, the buyer's value is higher than the seller's value. So in three out of the four cases that are possible, these combinations, so we can think of listing buyer's value first and seller's value. You could have 0.10, we could have 0.1, 0.9, we could have 1, 0, or 1.9. So those are the four combinations. And basically in three of these, the buyer's value is higher than the seller's, and we should have a transaction because now there's an efficiency gain in trading the good from the seller to the buyer. We get a higher utility. They can make a payment. There's gains from trade here. The only case where there shouldn't be trade is this case where the buyer ends up having a lower value, thinks it's not worth very much. The seller thinks it's worth a lot. In that case, it's better to leave it in the seller's hands. It's more valuable to the seller than the buyer. That's the setting. And let's think of a mechanism here. So let's suppose that the seller gets to name any price. So this is a simple take it or leave it offer. So the seller just puts out a price and says, here's my advertised price. I'm not going to listen to anything. You either say yes or no at this price. And the buyer either buys or not at that price. So let's think of this world. And in this case, let's think of the seller basically should either sell at a price of 0.1 or 1. I mean, selling at a price of 0.5 doesn't make sense if you're the seller. You could still charge 0.6. And the high value buyer would still want to buy at a 0.6 rather than a 0.5. And the low value buyer isn't going to want to buy either of those. So once you go above 0.1, you might as well push the price all the way up to 1. And you don't want to sit down near 0. You might as well push it all the way up to 0.1. So we'll presume that the buyer says yes, one and different, but modulo some epsilons. Basically, the seller should be saying a price that either pushes up to the 0.1 or all the way up to 1. And now when we think about the seller, so imagine that the seller actually thinks the car is worthless. But they're the only one who knows that. And the buyer is going to show up and now the seller can post a price. And if I post a price of 0.1, that's a sale for sure, expected utility 0.1. OK? So I charge the low price I'm going to sell to both buyers. Price of 1, I sell at the high price. What happens now if the buyer's types are equally likely, I'm going to sell only to the high type. And my expected utility is I'm going to sell at a value of 1 when the value is high. So the seller's value from a transaction in this case is going to be plus p and then minus the value of the seller, which is 0. So they don't lose anything by giving it away to get plus p. So in this case, they're getting a price of 1, probability a half, they get a utility of a half. OK? So what's true here is if you allow the seller to make a take it or leave it offer, they should set the price at a high enough level that you're only going to get trade to the high type. And so now what we end up with is an inefficiency. Right? So even though the seller's value is zero, when the buyer's value is 0.1, we're not getting a trade. OK? That's so that's where the inefficiencies sometimes the buyer has a higher value than the seller and we're not getting trade. OK? So that's the basic inefficiency. So we're better off to set the high price in terms of the seller and we get inefficient trade. These don't trade even though they should. OK, what about other mechanisms? Well, let's suppose that we sort of forced the seller to somehow charge a lower price. Or the buyer said, OK, after the fact in that the seller sets this high price of one, the buyer comes back and says, you know, really, it's too bad. I was the low value buyer. You really should have sold it to me. I was willing to pay 0.1. You still walk away with something for this. If the seller actually sells when you tell him that story, then the high value buyer might as well tell that story. Right? So the high value buyer is going to want to pretend to be the low value buyer and drive the price down to 0.1. So the basic difficulty in getting this is if you're willing to sell at the low price to the low buyer, the high price buyer can try and pretend to be the low price buyer. And so the incentive compatibility condition is going to mean it's going to be very difficult to get efficient trade, always be getting the good to be sold when you want it to, and not have people try to pretend to be somebody else. So the mechanism is not going to work if we actually want to get efficient trade out. So what does the theorem say? It says that there exists distributions on the buyers and seller's valuations, such that there does not exist any mechanism, any mechanism which is going to be Bayesian incentive compatible, which is efficient, weekly budget balanced, and interim individually rational. So if you want to make the decision of, in this case, trading always when you should and not trading when you shouldn't, making sure that it's weekly budget balanced, so that the amount that the buyer pays is at least the amount that the seller gets. So we're not having to stick in extra money in order to make this thing work. And it's interim individually rational, so individuals don't want to walk away once they're told their value and told what they expect to get out of this mechanism. They don't want to walk away from the mechanism, OK? So that's the theorem. And before we go into the proof, let me say a little bit about why this first part says there exists distributions for which this is true. There are some situations where you could get efficient trade, OK? And let me just go through one to make clear what's going on here. Suppose that we're in a situation where the buyer's value is always above some value of V, and the seller's value is always below that, OK? So there's some level of V, and we know that buyers always value these cars more and sellers always value the cars less. Well, in that world, there's a simple mechanism. Always exchange the good and always exchange it at a price of V, OK? So the buyer pays V, the seller gets V. That's going to be strategy proof. It's going to be individually rational. It's going to satisfy budget balance, the payments add up to zero. So that's a setting where we get all the conditions we want, and it's because we know that there always should be trade and we know that there's a price that is always going to please all of the buyers and all of the sellers, OK? So it's not going to be that this theorem is true regardless of what distribution is. It's going to have to be that these distributions cross over and that sometimes we want trade and sometimes we don't. And we've looked at an example like that and, in fact, let's show the proof based on our example. So buyers' values are equally likely to be 0.1 or 1. Seller's values are equally likely to be 0 or 0.9. Trade should take place for all combinations of values except 0.1, 0.9 in terms of efficient trade. And so what we can do is just go through the conditions, try to satisfy them and show that we can't satisfy them in all those situations. What I'm going to do is I'm going to show the proof for full budget balance, not weak budget balance. So that's going to make the proof a little bit easier. It's going to be very easy to extend. So I will allow you to do that to the case where we don't require full budget balance. And so here what we're going to have is trade should take place for all the combinations except when the buyer and seller values are 0.1 and 0.9. And the nice part about that, just in terms of notation, is that means that we can write payments down in terms of our full budget balance just as a single price, which is this is the payment made by the buyer and received by the seller. So we can think of some price as a function of what the buyer and seller's value is. This is the price paid from the buyer to the seller. And weak budget balance, the thing is now you're going to have two prices, one paid by the buyer and the other received by the seller. And the payment paid by the buyer is always going to have to be at least the payment received by the seller. And you'll see for quite easily that that'll bound these things. So everything we say and approve will work through for that case. OK, so first thing. What has to be true if in a situation where the values are 1 and 0.9? So the seller has a value of 0.9, buyer has a value of 1. By the individual rationality of the seller, and here we're using X-post individual rationality in this part of the proof, it's 0.9. And again, I'm going to leave it to you to extend this to an expected value of the interim individual rationality condition. Here I'll do the X-post version. Again, the extension is fairly straightforward. You'll just be working with expected payments instead of actual payments. So here the payment conditional on high-valued buyer and high-valued seller is at least 0.9. Otherwise, the seller doesn't want to sell it. When we see a low-valued buyer and a low-valued seller, then the price can't be more than 0.1. Otherwise, the buyer won't want to buy it. So individual rationality means the price in that case has to be below 0.1. When we don't have trade, so when there's no trade, the price exchange is going to have to be 0. And that's going to have to be true because of the individual rationality constraints of both the buyer and the seller. The buyer is not getting anything, so it can't be more than 0. And the seller's not giving anything away, so they can't be charged anything either. So the price is going to have to be 0 in a case where we have no trade. And so now we've got three conditions on three of the prices. And what we want to do is fill in the fourth price. So now what we need to do is we've got individual rationality satisfied for three of the prices. We want to make sure we're going to get an individual rationality here. Well, that's going to be easy. Any price between 0 and 1 will be individual rational. The difficulty is going to be to get the incentive compatibility condition to hold. So now we want to check what's going to be true so that the people want to tell us what their actual values are. So incentive compatibility for the seller of type 0, not wanting to pretend to be a 0.9 type. Let's go through that. If I actually truthfully say that I'm a type 0, what am I going to get? Well, half the time I'm going to be faced with a buyer of type 1. Half the time I'm faced with a buyer of type 0.1. So my expected utility is going to be the average of those two prices. And it has to be better than me pretending that I'm a 0.9 type. So alternatively, I could pretend that I'm a 0.9 type instead. And these would be the prices I would get. So in that situation, we can ask, and I have no value for the good. So it doesn't really matter in terms of where the good goes. So in this case, this turns out to be the incentive compatibility condition. Let's check what that implies. So by 1, 2, and 3, we have bounds on how low this price can be, how high this price can be, and we know exactly what this price is. So we know that this last one has to be 0. We know that this one is the 1.9 is at least 0.9. And this one is at most 0.1. And so in terms of an inequality, we know that this overall, in multiplying 2 by the 2, P10 plus 0.1 has to be greater than or equal to 0.9 plus 0. So we can take what we know from 1, 2, and 3, plug them in here in terms of bounds, given that we have the inequality pointing from the left to the right. And that tells us then that P10 has to be at least 0.8. So in order to make sure that the seller doesn't want to pretend they have the high value when they really have the low value, the price is going to have to be at least 0.8 when they have the low value and the buyer has the high value. OK. Do the same incentive compatibility with the buyer being a high value, not wanting to pretend that they have a low value, not trying to drive the price down. And basically, this is going to be a reverse kind of calculation of the same thing. And given all the symmetry here, it's not going to be surprising. You can work through the details, do exactly the same kind of calculation. And what do you end up with 1, 2, and 3? Now we're going to imply that this price has a cap of 0.2. So the seller has to be getting at least a price of 0.8. The buyer can't be seeing a price of more than 0.2. Well, it's going to be difficult to find a price which is at least 0.8 and also no more than 0.2 at the same time. That's impossible. So that's the end of the proof. Basically, once we put in the individual rationality conditions, then the incentive compatibility conditions are going to say the price has to be pretty high to keep the seller honest and the price is going to have to be very low to keep the buyer honest. Can't do that at the same time. No price is going to work. So incentive compatibility plus individual rationality plus having trade exactly in the right situations is going to be impossible. So in terms of summary, what have we learned? Private information about values necessitates some inefficiencies in voluntary trade, and it leads to a basic tension between incentives and efficiency. And this is really impossible to get around. And one thing to emphasize here, and this is the power of the revelation theorem, think of any mechanism you would want. Think of any mechanism. If that mechanism somehow has the equilibria that leads to efficient trade, we can write that mechanism as a direct mechanism. And so the proof here says, no matter what's going on, throughout this mechanism, it's going to be impossible to write anything down that's going to have the desired properties of always trading when you want to, and being instead of compatible. People are going to tend to lie, and that's going to lead to some inefficient trades and things not happen when you want them to. This begins to explain this question of why do we have strikes? Why do we have breakdowns? Why don't we always come to an agreement? And you can ask yourself, if you ever walked away from a bargain when you thought that a trade might really be possible, so sometimes we're going to be forced to leave things on the table, and that's a powerful result. And understanding this is really important and understanding why some inefficiency exists in the world and why we're going to have difficulties getting around them. To the extent that people have private information about their willingness to do something, that's going to be difficult to overcome in getting full efficiency.