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Hi folks it's Matt Jackson again. So we're here and now we're talking about

Â an application of subgame perfect reasoning and we're looking at what's

Â known as ultimatum bargaining. So let's have a peek.

Â So ultimatum bargaining is, is probably one of the simplest bargaining games you

Â could imagine. it's sort of the take it or leave it

Â offer kind of bargaining that you might have heard of, about and Popular

Â folklore. So the idea here is, is let's say that

Â there's 10 units to be split between two players and in particular we'll take

Â these to be say integer and units. So we have say $10, 10 Euros, 10 whatever

Â to split between two players and they have to agree in order to get anything.

Â So, player 1 makes an offer, says okay look here you can have 6, I'll keep the

Â rest. and then player 2 can accept or reject.

Â And based on what happens, then if player 2 accepts the, the offer, so if x is the

Â offer that's made, and player 2 accepts, then.

Â Player 2 gets whatever was offered and player 1 gets the remainder.

Â If it's rejected then everybody gets 0. So in this case you only reach something

Â if it's in agreement and this is 1 shot bargaining in a sense that the, there's

Â just 1 offered made and then accept or reject.

Â They can't go back and forth, so it doesn't alternate.

Â They don't get the chance to come back to the table and so forth.

Â It's just here, take it or leave it, if you don't want it forget it.

Â So that's the idea, okay. So let's analyze this game using sub game

Â perfection. you can actually write it out, the tree

Â for this. you know, it's still manageable.

Â So, player 1 moves first. They could offer x=0, 1, 2, etcetera.

Â All the way up to 10. Then player 2 moves second.

Â They can accept or reject the offer. So they can reject, they can accept.

Â And based on What the offers are, they're going to get

Â different payoffs, right. So, the payoffs here if they reject,

Â everybody gets zero in every case. If an offer is made of ten adn it's

Â accepted, player two gets ten, player one gets zero.

Â For an offer of six is made, and it's accepted Player 2 gets 6, player 1 gets

Â 4, an offer of 1 is made, and it's accepted, player 2 gets 1, player 1 gets

Â 9, and so forth, right. So that's the structure of the game, very

Â simple. And you can solve this directly by

Â backward induction or sub game perfection.

Â What's true? Well when we think about the player 2 They should accept any offer

Â which is positive. Right? So, any offer which is positive

Â you get a positive payoff if you accept it.

Â Zero if you don't. They should go ahead and accept any one

Â of those offers. at zero in that case, it looks like one

Â where The second player is actually indifferent, so what they do is up to

Â them at that point. So if they're offered 0 they might say

Â yes, they might say no they could mix, they could randomize.

Â so we're, we're not sure exactly what's going to happen in that part of the tree,

Â But what is true, is that since they should be accepting all of these things

Â we do know that a best a best reply for player one.

Â Given what they anticipate happening in the second thing should never be to make

Â an offer which involves more than 1 to player 2, right? So they can get nine by

Â offering this which they know will be accepted whether they want to go down

Â here depends on what their beliefs are about what player 2 is going to do here.

Â But basically, once we've assumed the, or once we've deduced that player 2 is going

Â to accept any positive offer. Then, given that, the, player 1 gets a

Â higher payoff from the lowest possible amount.

Â They're going to offer player 2, at most, 1.

Â Right, so we get a pretty direct prediction.

Â Player 2 accepts any positive thing. Player 1 is going to offer either 0 or 1

Â depending on 2's decision at 0. But in a sub-game perfect equilibrium we

Â have a prediction that 2 or more would never be offered.

Â Okay. So let's have a look.

Â these are some data from online games played last year.

Â In, in the game theory course. And, let's have a look at what actually.

Â we're, we're played. So here are the offers.

Â how much was offered to the second player.

Â And, in fact, you can see that 5 was the modal offer offered more than 2,000

Â times. the next highest offer of 1, which is

Â the. prediction or one of the predictions of

Â the sub game perfect equilibrium was slightly less than a thousand.

Â and we can also look the acceptance. so here the way this worked, is, is

Â players were asked what's the minimal amount that you would be willing to

Â accept And the theory predicts that everybody should should be saying either

Â zero or one they should never be ru rejecting offers the of of at least one

Â so the minimum amount they should be [INAUDIBLE] to be at least sorry at most

Â one. And here, we see that actually a majority

Â of the players are in fact setting their minimal acceptance at higher.

Â And, a lot of them hold out for point for five.

Â 50%. So in fact when we look at the data here,

Â the data are not congruent with what subgame perfection is predicting.

Â And you know, when you, when you think about the best offer is for a given

Â player. Given the strategies that are being

Â played here. So when we look at the acceptance rate.

Â So if I knew that this was what the population is doing.

Â Suppose I know that this is the way people were, are acting in terms of what

Â their going to accept. What should I offer? well, there's some

Â chance I'm going to meet somebody who's only going to accept 5.

Â At least 5. I have a pretty fair chance that if I

Â offered something above 5 it would almost certainly be accepted, but whether it has

Â to be 5 or whether I drop all the way down to 1, that's going to depend on who

Â I happen to meet and if you look at my expected payoff.

Â My expected payoff, my best payoff, is actually.

Â to offer, to offer 5 given what the players are doing in terms of, of their

Â acceptance, so when we go back to the play here, the fact that that that these

Â players are playing 5 is consistent with what the players are doing in the second

Â stage. So where sub-game perfection is missing

Â things in here. So players here, are, a lot of them are

Â best replying to the actual distribution that they're facing.

Â It's really the acceptance/rejection part which is, which is contradicting what the

Â sub-game perfect play would have. And you know, there's different explanations for

Â this. we could think you know in, in, in terms

Â of why we're seeing this particular play, it could be that players for instance are

Â you know, have strong aversions to anything that's unequal.

Â And what that means then, is that the payoff that we've written into this

Â matrix of 1, 2, 3, 4, 5, and so forth is not the actual payoff that people have.

Â Maybe they have a disutility. Of, of getting less than another person,

Â and that makes them feel really badly, and, and they want to avoid that bad

Â feeling. And so, their utility might actually

Â represent something which includes equity concerns, for instance.

Â that's one possibility. it, you know, it, there, there's a lot of

Â alternative explanations. It could be that they, they always want

Â to have more than the other player, or, you know, you could think of different

Â kinds of things. which would govern different kinds of

Â play. So there's some players who, who just

Â seem to be taking whatever they can, other players who seem to be pushing for

Â an equal split and somehow feeling that's what the, the minimum amount that they

Â would be willing to accept. one other possible conjecture that people

Â have brought up a number of times, is, is that maybe the stakes aren't large

Â enough. So, you know, for instance, when we

Â played this online the players were playing this just in terms of a question.

Â They weren't actually paid for it. Maybe if we paid them, you know, suppose

Â you're now splitting, instead of ten fictional units, you're splitting $10

Â million, or 10 million Euros. [SOUND] are you going to reject an offer

Â of 4 million of that if somebody says. Okay, you can have 4 million.

Â I'll take, I'll keep 6 million. Are you going to say no? probably not.

Â so one possibility is to, you know? To see whether the p-, the size of the pie

Â matters. And so, here are some interesting

Â experiments to try and test that hypothesis.

Â Maybe it's just that, you know? We're not paying people enough to, to see that.

Â The, the, real rational behavior. So there's a nice paper by Robert Slonim

Â and Al Roth, and Al Roth just won a Nobel Prize this year.

Â and what they're looking at here is, is learning in high stakes ultimatum games.

Â And to make things high stakes, what they did, is they went to Slovakia, the Slovak

Â Republic and they, they did 3 different versions of

Â this. So one where people could split 60 Slovak

Â Crowns. One where they could split 300 Slovak

Â Crowns. And another where they could split 1,500.

Â And the average monthly wage, right? So, per month, you're getting 5,500,

Â So when you're getting up to 1,500, you're talking about more than a, a

Â week's wage. Right? So you're looking at, at offering

Â somebody a week's salary to be split. So now arguably the money at stake is, is

Â reasonably large, right? So the high stakes version is, is on an order of a

Â week's wage. Okay.So what happened.Um, so what they

Â did here is they had a 1,000 units. So instead of just putting things 1, 2,

Â 3, 4, 5, up to 10, you could put it in units of 1,000 where the, you know, the

Â full 1000 corresponded in the first game to the 60 crowns.

Â In the 2nd game to 300 and so forth, right? So 1 unit in this would be 1.5

Â Slovak crowns, in the 1,500 treatment. And so the question is then how much was

Â offered to the other player on average? Well 451 in the 1st game, the low stakes

Â game. 460 in the middle, 423 in the higher

Â stakes game. So it did go down a little bit.

Â But certainly not down to 1, which would be the prediction of, of subgame

Â perfection. And when you look at the medians, they're

Â are very similar, 465. 480, 450, so a little bit, people are

Â shading a little bit below 50% but they're not pushing to much further than

Â 50%. And when we look at the rejection offers,

Â So let's look, just for instance, at when people offered less than 250 out of the

Â 1000 to the second mover, how often was that rejected? It wasn't offered that

Â frequently in the low stakes game. It was only offered once and it was

Â rejected. but in the middle stakes game, it was re,

Â rejected about half the time, 10 out of 20 and in the higher stakes game, it was

Â on the order of a third 12 out of 32. So, a little more than a third but what

Â we do see is, is, you know? [INAUDIBLE]. Subject to statistical significance here

Â basically we get a comparison between these two.

Â We are seeing as we up the stakes people are pushing down and rejecting, really

Â lower offers less frequently, but it's not going all the way down and still the

Â the offers that are being made on average are, are.

Â fairly high. Okay, so what, what do we learn from

Â this? Well subgame perfection does not always match the data.

Â and if you go back to this game and you think about the Nash equilibria Any 1 of

Â the offers can be supported as part of a Nash equilibrium.

Â Right, so it it it could be that I make that offer because I think that it's the

Â only one that the other person's going to be willing to accept and indeed they

Â accept it and I never know if they would have rejected the other one.

Â So, there's lots of Nash equilibria to this game.

Â And subgame perfection is picking a few of them out.

Â And, in some cases, you know? These violate rationality.

Â But rationality, where we believe that the payoffs are just exactly the monetary

Â amounts. And not something else.

Â Right? So it could be that we have the payoffs incorrect.

Â It written down incorrectly. People could value equity.

Â They could be feeling emotions. so there's a whole area of game theory

Â which is, is basically expanded and, and more or less exploded in the last couple

Â of decades. where people begin to analyze,

Â motivations of players. other kinds of concerns that they might

Â have, called behavioral game theory. And it, it, it moves away from the very

Â narrow definitions of rationality. Which are, that we just look directly at

Â sum. Very specific monetary or, or simple

Â payoff and are looking at, at either expanding the way in which payoffs are

Â there or bringing in other kinds of, of biases and tendencies that people might

Â have to understand things and that can expand and help.

Â So overall when, you know we look at some game imperfection and what we've learned

Â from it. the, the basic premise and I think the

Â one of the important things is to take away from studying subgame perfection is

Â that it imposes sequential rationality. So it's a certain kind of logic and

Â whether or not people play that way, understanding the logic helps us

Â understand The, the incentives in the game better, and at least gives us some

Â feeling for the game. so, the results of subgame perfection

Â and, and backward induction, we, generally, we'll pick out a subset of the

Â Nash equilibria. And they're doing that by sort of

Â imposing a credibility in circumstances that are never reached, right? So there's

Â this idea of what's happening off the equilibrium path can actually be

Â important in determining what people are doing.

Â And you want to make sure that that the prescription of what players are going to

Â do in all of these circumstances Is credible.

Â one thing that, that it's very interesting to start thinking about when

Â you think about subgame perfection. what about the game of chess? Chess is

Â actually a game of complete information, right? So you could write down a tree for

Â chess if you had a lot of time on your hands.

Â the first player can make a bunch of moves.

Â The second player can then make a bunch of moves.

Â The third player. Or, sorry, the first player then, again,

Â gets to make a move. And, and so you've got a tree.

Â which can be written out. And it's actually a finite game, a very

Â big but a finite game, in the sense that if the same board is ever reached three

Â times, the game ends. So, so there are ending rules which make

Â sure that the game doesn't go on infinitely.

Â So it's actually a, a finite extensive form game of complete information.

Â So at least theoretically you could solve chess.

Â but obviously we haven't managed to do that.

Â And, and it's just such a large game that, that That solving the subgame

Â perfect equilbria of that. seemed to be impossible.

Â Maybe on another planet, they've solved chess.

Â And they could, it could be that they think of it as,

Â like our tic tac toe, which is a much simpler game to solve.

Â And, and after you've played it a few times, you get pretty bored by it.

Â One, one other thing that's important is, is even with a game where it's dominant

Â solvable and so forth. sorry, not dominant solvable but solvable

Â by backward induction or subgame perfection it's, it's not completely

Â clear that everybody abides by the logic. And in particular, you need to believe in

Â the rationality of others, right? So you need to, in order to really solve this

Â think backwards. You have to think about, well, I, I think

Â the other player's going to do this in a certain situation.

Â And. And then you're back up and that you know

Â the, the demands that are placed on players in that situation are, could be

Â quite difficult to meet as the game becomes more complicated.

Â Another thing to say about this, is there is some controversy in, in game theory

Â about the ideas behind things like backward induction.

Â And part of that is that you know, that according to the theory there's certain

Â parts of the game that you should never see yourself in.

Â And then you can begin to ask the question, okay well let's suppose we

Â really did end up there, what should I believe about the other player? how did

Â we get there? so there's, there's, it, it's not so easy actually to, to very

Â carefully write down a foundation in terms of logical thinking which makes

Â these predictions and that's an, an interesting area of research.

Â [SOUND] So just to, to wrap up next time, we'll be thinking a little bit about

Â incomplete information and bringing that into the study.

Â