Hi folks it's Matt Jackson again. So we're here and now we're talking about an application of subgame perfect reasoning and we're looking at what's known as ultimatum bargaining. So let's have a peek. So ultimatum bargaining is, 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 and Popular folklore. So the idea here is, 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 and units. So we have say $10, 10 Euros, 10 whatever to split between two players and they have to agree in order to get anything. So, player 1 makes an offer, says okay look here you can have 6, I'll keep the rest. and then player 2 can accept or reject. And based on what happens, then if player 2 accepts the, the offer, so if x is the offer that's made, and player 2 accepts, then. Player 2 gets whatever was offered and player 1 gets the remainder. If it's rejected then everybody gets 0. So in this case you only reach something if it's in agreement and this is 1 shot bargaining in a sense that the, there's just 1 offered 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, okay. So let's analyze this game using sub game perfection. you can actually write it out, the tree for this. you know, it's still manageable. So, player 1 moves first. They could offer x=0, 1, 2, etcetera. All the way up to 10. Then player 2 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, right. So, the payoffs here if they reject, everybody gets zero in every case. If an offer is made of ten adn it's accepted, player two gets ten, player one gets zero. For an offer of six is made, and it's accepted Player 2 gets 6, player 1 gets 4, an offer of 1 is made, and it's accepted, player 2 gets 1, player 1 gets 9, and so forth, right. 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 2 They should accept any offer which is positive. Right? 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 offers. 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 0 they might say yes, they might say no they could mix, they could randomize. so we're, 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 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 1 to player 2, 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 2 is going to do here. But basically, once we've assumed the, or once we've deduced that player 2 is going to accept any positive offer. Then, given that, the, player 1 gets a higher payoff from the lowest possible amount. They're going to offer player 2, at most, 1. Right, so we get a pretty direct prediction. Player 2 accepts any positive thing. Player 1 is going to offer either 0 or 1 depending on 2's decision at 0. But in a sub-game perfect equilibrium we have a prediction that 2 or more would never be offered. Okay. So let's have a look. these are some data from online games played last year. In, in the game theory course. And, let's have a look at what actually. we're, we're played. So here are the offers. how much was offered to the second player. And, in fact, you can see that 5 was the modal offer offered more than 2,000 times. the next highest offer of 1, which is the. prediction or one of the predictions of the sub game perfect equilibrium was slightly less than a thousand. and we can also look the acceptance. so here the way this worked, is, is players were asked what's the minimal amount that you would be willing to accept And the theory predicts that everybody should should be saying either zero or one they should never be ru rejecting offers the of of at least one so the minimum amount they should be [INAUDIBLE] to 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 point for five. 50%. So in fact when we look at the data here, the data are not congruent with what subgame perfection is predicting. And you know, when you, when you think about 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 was what the population is doing. Suppose I know that this is the way people were, are acting in terms of what their going to accept. What should I offer? well, there's some chance I'm going to meet somebody who's only going to accept 5. At least 5. I have a pretty fair chance that if I offered something above 5 it would almost certainly be accepted, but whether it has to be 5 or whether I drop all the way down to 1, 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, to offer 5 given what the players are doing in terms of, of their acceptance, so when we go back to the play here, the fact that that that these players are playing 5 is consistent with what the players are doing in the second stage. So where sub-game perfection is missing things in here. So players here, are, a lot of them are best replying to the actual distribution that they're facing. It's really the acceptance/rejection part which is, which is contradicting what the sub-game perfect play would have. And you know, there's different explanations for this. we could think you know in, in, in terms of why we're seeing this particular play, it could be that players for instance are you know, 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 1, 2, 3, 4, 5, and so forth is not the actual payoff that people have. Maybe they have a disutility. Of, of getting less than another person, and that makes them feel really badly, and, 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. it, you know, it, there, there's a lot of alternative explanations. It could be that they, they always want to have more than the other player, or, you know, you could think of different kinds of things. which would govern different kinds of play. So there's some players who, who just seem to be taking whatever they can, other players who seem to be pushing for an equal split and somehow feeling that's what the, the minimum amount that they would be willing to accept. one other possible conjecture that people have brought up a number of times, is, is that maybe the stakes aren't large enough. So, you know, 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, you know, suppose you're now splitting, instead of ten fictional units, you're splitting $10 million, or 10 million Euros. [SOUND] are you going to reject an offer of 4 million of that if somebody says. Okay, you can have 4 million. I'll take, I'll keep 6 million. Are you going to say no? probably not. so one possibility is to, you know? To see whether the p-, the size of the pie matters. And so, here are some interesting experiments to try and test that hypothesis. Maybe it's just that, you know? We're not paying people enough to, to see that. The, 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, 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, they did 3 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, right? So, per month, you're getting 5,500, So when you're getting up to 1,500, you're talking about more than a, a week's wage. Right? So you're looking at, at offering somebody a week's salary to be split. So now arguably the money at stake is, is reasonably large, right? So the high stakes version is, is on an order of a week's wage. Okay.So what happened.Um, so what they did here is they had a 1,000 units. So instead of just putting things 1, 2, 3, 4, 5, up to 10, you could put it in units of 1,000 where the, you know, the full 1000 corresponded in the first game to the 60 crowns. In the 2nd game to 300 and so forth, right? So 1 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 1st 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, of subgame perfection. And when you look at the medians, they're are very similar, 465. 480, 450, so a little bit, people are shading a little bit below 50% but they're not pushing to much further than 50%. And when we look at the rejection offers, So let's look, just for instance, at when people offered less than 250 out of the 1000 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 re, 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, is, you know? [INAUDIBLE]. 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 the offers that are being made on average are, are. fairly high. Okay, so what, what do we learn from this? Well subgame perfection does not always match the data. and if you go back to this game and you think about the Nash equilibria Any 1 of the offers can be supported as part of a Nash equilibrium. Right, so it it it could be that I make that offer because I think that it's the only one that the other person's going to be willing to accept and indeed they accept it and I never know if they would have rejected the other one. So, there's lots of Nash equilibria to this game. And subgame perfection is picking a few of them out. And, in some cases, you know? These violate rationality. But rationality, where we believe that the payoffs are just exactly the monetary amounts. And not something else. Right? So it could be that we have the payoffs incorrect. It written down incorrectly. People could value equity. They could be feeling emotions. so there's a whole area of game theory which is, is basically expanded and, 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, it, it moves away from the very narrow definitions of rationality. Which are, that we just look directly at sum. Very specific monetary or, or simple payoff and are looking at, at either expanding the way in which payoffs are there or bringing in other kinds of, of biases and tendencies that people might have to understand things and that can expand and help. So overall when, you know we look at some game imperfection and what we've learned from it. the, the basic premise and I think the one of the important things is to take away from studying subgame 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, the incentives in the game better, and at least gives us some feeling for the game. so, the results of subgame perfection and, and backward induction, we, 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, right? 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 that the prescription of what players are going to do in all of these circumstances Is credible. one thing that, that it's very interesting to start thinking about when you think about subgame perfection. what about the game of chess? Chess is actually a game of complete information, right? 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, 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 is ever reached three times, the game ends. So, so there are ending rules which make sure that the game doesn't go on infinitely. So it's actually a, 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, and it's just such a large game that, that That solving the subgame perfect equilbria of that. seemed to be impossible. Maybe on another planet, they've solved chess. And they could, it could be that they think of it as, like our tic tac toe, which is a much simpler game to solve. And, and after you've played it a few times, you get pretty bored by it. One, one other thing that's important is, 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, it's not completely clear that everybody abides by the logic. And in particular, you need to believe in the rationality of others, right? So you need to, in order to really solve this think backwards. You have to think about, well, I, I think the other player's going to do this in a certain situation. And. And then you're back up and that you know the, the demands that are placed on players in that situation are, could be quite difficult to meet as the game becomes more complicated. Another thing to say about this, is there is some controversy in, in game theory about the ideas behind things like backward induction. And part of that is that you know, that according to the theory there's certain parts of the game that you should never see yourself in. And then you can begin to ask the question, okay well let's suppose we really did end up there, what should I believe about the other player? how did we get there? so there's, there's, it, it's not so easy actually to, to very carefully write down a foundation in terms of logical thinking which makes these predictions and that's an, an interesting area of research. [SOUND] So just to, to wrap up next time, we'll be thinking a little bit about incomplete information and bringing that into the study.
