 Hi, folks. It's Matt Jackson again. So we're here, and now we're talking about an application of sub-game perfect reasoning. And we're looking at what's known as ultimatum bargaining. So let's have a peek. So ultimatum bargaining is probably one of the simplest bargaining games you could imagine. It's sort of the take it or leave it offer kind of bargaining that you might have heard of about in popular folklore. So the idea here is let's say that there's 10 units to be split between two players. And in particular, we'll take these to be, say, integer units. So we have, say, $10, 10 euros, 10 whatever's to split between two players. And they have to agree in order to get anything. So player one makes an offer, says, OK, look. Here, you can have six. I'll keep the rest. And then player two can accept or reject. And based on what happens, then if player two accepts the offer, so if x is the offer that's made and player two accepts, then player two gets whatever was offered. And player one gets the remainder. If it's rejected, then everybody gets zero. So in this case, you only reach something if it's an agreement. And this is one shot bargaining in the sense that there's just one offer made and then accept or reject. They can't go back and forth, so it doesn't alternate. They don't get the chance to come back to the table and so forth. It's just here. Take it or leave it if you don't want it. Forget it. So that's the idea. OK, so let's analyze this game. Using sub game perfection, you can actually write it out the tree for this. Still manageable. So player one moves first. They could offer x equals zero, one, two, et cetera, all the way up to 10. Then player two moves second. They can accept or reject the offer. So they can reject, they can accept. And based on what the offers are, they're going to get different payoffs. So the payoffs here, if they reject, everybody gets zero in each case. If an offer is made of 10 and it's accepted, player two gets 10, player one gets zero. If an offer of six is made and it's accepted, player two gets six, player one gets four. An offer of one is made and it's accepted. Player two gets one, player one gets nine, and so forth. So that's the structure of the game, very simple. And you can solve this directly by backward induction or sub game perfection. What's true? Well, when we think about the player two, they should accept any offer which is positive. So any offer which is positive, you get a positive payoff if you accept it, zero if you don't. They should go ahead and accept any one of those authors. At zero, in that case, it looks like one where the second player is actually indifferent. So what they do is up to them at that point. So if they're offered zero, they might say yes, they might say no. They could mix, they could randomize. So we're not sure exactly what's going to happen in that part of the tree. But what is true is that since they should be accepting all of these things, we do know that a best reply for player one, given what they anticipate happening in the second thing, should never be to make an offer which involves more than one to player two, right? So they can get nine by offering this, which they know will be accepted. Whether they want to go down here depends on what their beliefs are about what player two is going to do here. But basically, once we've deduced that player two is going to accept any positive offer, then given that the player one gets a higher payoff from the lowest possible amount, they're going to offer player two at most one, right? So we get a pretty direct prediction. Player two accepts any positive thing. Player one is going to offer either zero or one, depending on two's decision at zero. But in a sub game perfect equilibrium, we have a prediction that two or more would never be offered. OK. So let's have a look. These are some data from online games played last year in the game three course. And let's have a look at what actually were played. So here are the offers. How much was offered to the second player? And in fact, you can see that five was the modal offer offered more than 2,000 times. The next highest offer of one, which is the prediction or one of the predictions of the sub game perfect equilibrium was slightly less than 1,000. And we can also look at the acceptance. So here the way this worked is players were asked, what's the minimal amount that you would be willing to accept? And the theory predicts that everybody should be saying either zero or one. They should never be rejecting offers of at least one. So the minimal amount they should be able to accept should be at least, sorry, at most one. And here we see that actually a majority of the players are in fact setting their minimal acceptance at higher. And a lot of them hold out for five, 50%. So in fact, when we look at the data here, the data are not congruent with what sub game perfection is predicting. And when you think about what the best offer is for a given player, given the strategies that are being played here, so when we look at the acceptance rate, so if I knew that this is what the population is doing, suppose I know that this is the way people are acting in terms of what they're going to accept, what should I offer? Well, there's some chance I'm going to meet somebody who's only going to accept five, at least five. I have a pretty fair chance that if I offered something above five, it would almost certainly be accepted. But whether it has to be five or whether I drop all the way down to one, that's going to depend on who I happen to meet. And if you look at my expected payoff, my expected payoff, my best payoff is actually to offer five given what the players are doing in terms of their acceptance. And so when we go back to the play here, the fact that these players are playing five is consistent with what the players are doing in the second stage. So where subgame perfection is missing things in here, so players here, a lot of them are best replying to the actual distribution that they're facing. It's really the acceptance rejection part, which is contradicting what the subgame perfect play would have. And there's different explanations for this. We could think in terms of why we're seeing this particular play, it could be that players, for instance, have strong aversions to anything that's unequal. And what that means then is that the payoff that we've written into this matrix of one, two, three, four, five, and so forth is not the actual payoff that people have. Maybe they have a disutility of getting less than another person. And that makes them feel really badly, and they want to avoid that bad feeling. And so their utility might actually represent something which includes equity concerns, for instance. That's one possibility. There's a lot of alternative explanations. It could be that they always want to have more than the other player. Or you could think of different kinds of things which would govern different kinds of play. So there's some players who just seem to be taking whatever they can, other players who seem to be pushing for an equal split, and somehow feeling that that's the minimal amount that they would be willing to accept. One other possible conjecture that people have brought up a number of times is that maybe the stakes aren't large enough. So for instance, when we played this online, the players were playing this just in terms of a question they weren't actually paid for it. Maybe if we paid them, suppose you're now splitting instead of 10 fictional units, you're splitting $10 million, or 10 million euros. Are you going to reject an offer of $4 million of that? If somebody says, OK, you can have $4 million. I'll keep $6 million. Are you going to say no? Probably not. So one possibility is to see whether the size of the pie matters. And so here are some interesting experiments to try and test that hypothesis. Maybe it's just that we're not paying people enough to see the real rational behavior. So there's a nice paper by Robert Slonim and Al Roth. And Al Roth just won a Nobel Prize this year. And what they're looking at here is learning in high stakes ultimatum games. And to make things high stakes, what they did is they went to Slovakia, the Slovak Republic. And they did three different versions of this. So one where people could split 60 Slovak crowns, one where they could split 300 Slovak crowns, and another where they could split 1,500. And the average monthly wage, so per month, you're getting 5,500. So when you're getting up to 1,500, you're talking about more than a week's wage. So you're looking at offering somebody a week's salary to be split. So now, arguably, the money at stake is reasonably large. So the high stakes version is on the order of a week's wage. OK, so what happened? So what they did here is they had 1,000 units. So instead of just splitting things 1, 2, 3, 4, 5, to 10, you could split it in units of 1,000 where the full 1,000 corresponded in the first game to the 60 crowns in the second game to 300 and so forth. So one unit in this would be 1.5 Slovak crowns in the 1,500 treatment. And so the question is, then, how much was offered to the other player on average? Well, 451 in the first game, the low stakes game, 460 in the middle, 423 in the higher stakes game. So it did go down a little bit, but certainly not down to 1, which would be the prediction of sub-game perfection. And when you look at the medians, they're very similar, 465, 480, 450. So people are shading a little bit below 50%, but they're not pushing too much further than 50%. And when we look at the rejection offer, so let's look just for instance at when people offered less than 250 out of the 1,000 to the second mover, how often was that rejected? It wasn't offered that frequently in the low stakes game. It was only offered once, and it was rejected. But in the middle stakes game, it was rejected about half the time, 10 out of 20. And in the higher stakes game, it was on the order of a third, 12 out of 32, so a little more than a third. But what we do see is subject to statistical significance here, basically, we get a comparison between these two. We are seeing as we up the stakes, people are pushing down and rejecting really lower offers less frequently, but it's not going all the way down. And still the offers that are being made on average are fairly high. So what do we learn from this? Well, sub game perfection does not always match the data. And if you go back to this game and you think about the Nash Equilibria, any one of the offers can be supported as part of a Nash Equilibrium. So it could be that I make that offer, because I think it's the only one that the other person is going to be willing to accept. And indeed, they accept it. And I never know whether they would have rejected the other one. So there's lots of Nash Equilibria to this game. And sub game perfection is picking a few of them out. And in some cases, these violate rationality, but rationality where we believe that the payoffs are just exactly the monetary amounts and not something else. So it could be that we have the payoffs written down incorrectly. People could value equity. They could be feeling emotions. So there's a whole area of game theory, which is basically expanded and more or less exploded in the last couple of decades, where people begin to analyze motivations of players, other kinds of concerns that they might have called behavioral game theory. And it moves away from the very narrow definitions of rationality, which are that we just look directly at some very specific monetary or simple payoff and are looking at either expanding the way in which payoffs are there or bringing in other kinds of biases or tendencies that people might have to understand things. And that can expand and help games. So overall, when we look at sub game perfection and what we've learned from it, the basic premise, and I think one of the important things to take away from studying sub game perfection, is that it imposes sequential rationality. So it's a certain kind of logic. And whether or not people play that way, understanding the logic helps us understand the incentives in the game better and at least gives us some feeling for the game. So they result in sub game perfection and backward induction. Generally, we'll pick out a subset of the Nash equilibria. And they're doing that by sort of imposing a credibility in circumstances that are never reached. So there's this idea of what's happening off the equilibrium path can actually be important in determining what people are doing. And you want to make sure that the prescription of what players are going to do in all these circumstances is credible. One thing that's very interesting to start thinking about when you think about sub game perfection, what about the game of chess? Chess is actually a game of complete information. So you could write down a tree for chess if you had a lot of time on your hands. The first player can make a bunch of moves. The second player can then make a bunch of moves. The third player, or sorry, the first player then again gets to make a move. And so you've got a tree, which can be written out. And it's actually a finite game, a very big but a finite game, in the sense that if the same board has ever reached three times, the game ends. So there are ending rules which make sure that the game doesn't go on infinitely. So it's actually a finite extensive form game of complete information. So at least theoretically, you could solve chess. But obviously, we haven't managed to do that. And it's just such a large game that solving the subgame perfect equilibrium that seemed to be impossible. Maybe on another planet, they've solved chess. And it could be that they think of it as like our tic-tac-toe, which is a much simpler game to solve. And after you've played it a few times, you get pretty bored by it. One other thing that's important is even with a game where it's dominant solvable and so forth, sorry, not dominant solvable, but solvable by backward induction or subgame perfection, it's not completely clear that everybody abides by the logic. And in particular, you need to believe in the rationality of others. So you need to, in order to really solve this thing backwards, you have to think about, well, I think the other player is going to do this in a certain situation, and then you back up. And the demands that are placed on players in that situation can be quite difficult to meet as the game becomes more complicated. Another thing to say about this is there is some controversy in game theory about the ideas behind things like backward induction. And part of that is that according to the theory, there are certain parts of the game that you should never see yourself in. And then you can begin to ask the question, well, let's suppose we really did end up there. What should I believe about the other player? How did we get there? So it's not so easy actually to very carefully write down a foundation in terms of logical thinking, which makes these predictions. And that's an interesting area of research. So just to wrap up, next time we'll be thinking a little bit about incomplete information and bringing that into the study of games.