ECON 159: Game Theory

Lecture 17

 - Backward Induction: Ultimatums and Bargaining


We develop a simple model of bargaining, starting from an ultimatum game (one person makes the other a take it or leave it offer), and building up to alternating offer bargaining (where players can make counter-offers). On the way, we introduce discounting: a dollar tomorrow is worth less than a dollar today. We learn that, if players are equally patient, if offers can be in rapid succession, and if each side knows how much the game is worth to the other side, then the first offer is for an equal split of the pie and this offer is accepted. But this result depends on those assumptions; for example, bargaining power may depend on wealth.

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Game Theory

ECON 159 - Lecture 17 - Backward Induction: Ultimatums and Bargaining

Chapter 1. Ultimatum Games: Why Backward Induction Fails Here [00:00:00]

Professor Ben Polak: So today I want to look at two kinds of games and then we’ll change topic a bit. The games I want to look at are about ultimatums and bargaining. And we’ll start with ultimatums and we’ll move smoothly through, and we’ll see why that’s an easy transition in a while. So the game we’re going to play involves two players, 1 and 2 and the game is this. Player 1 is going to make a take it or leave it offer to Player 2 and this offer is going to concern a pie and let’s make the pie worth $1 for now. We’ll probably play this for real in a bit so we’ll play it for $1.

So it’s a split of a pie, we can think of the split as offering S to Player 1 and (1–S) to Player 2. Player 2 has two choices: 2 can accept this offer, and if 2 accepts the offer then they get exactly the offer. Player 1 gets S and Player 2 gets (1–S). Alternatively, Player 2 can reject, and if Player 2 rejects the offer then both players get 0, both players gets nothing. So a very simple game, there’s a dollar on the table–I’ll take it out in a minute–so there’s a dollar on the table, and our two players are going to bargain for this, but it isn’t much of a bargain. Player 1’s basically going to announce what the division’s going to be. 2 can either accept that division or no one gets anything. Everyone understand the game?

So I thought we’d start off by playing this game for real a couple of times, so why don’t I come down and do that. So why don’t we play with some people in this row. I’ve been playing with that row all the time, so Ale’s going to help me find somebody. You found somebody okay, so the person behind you? Your name is? You have to shout out, this doesn’t make a noise. Equia, thank you. So you’ll be Player 1, and Player 2 will be this gentleman here whose name is?

Student: Noah.

Professor Ben Polak: Noah, okay. So Equia, do you know each other? Okay, so Equia can make any offer she wants. We’ll play this for real money, so there’s a real dollar at stake. You can make any offer you want–it can be in fractions of pennies if you like–of the amount of the dollar that you’re going to offer to Noah, and the amount you’re going to keep yourself. There you go, shout it out so everyone can hear. Stand up, this is your moment in the lights all right. Will you stand up as well? There we go.

Student: I’m going to offer him a penny.

Professor Ben Polak: You’re going to offer him a penny, and Noah are you going to take a penny?

Student: No.

Professor Ben Polak: Noah’s not going to take a penny. So I didn’t lose any money on that: so no one made any money. I didn’t lose any money. that seems pretty good. Let’s try a different couple. Let’s move around the room a little bit. So Ale why don’t you take the guy behind here who name is? Shout out.

Student: Gary.

Professor Ben Polak: Gary, all right so Gary why don’t you stand up, and we’ll let Gary play with the gentleman in here, what’s your name in here?

Student: Anish.

Professor Ben Polak: Anish, all right stand up, do you know each other? So why don’t we let Anish be Player 1 this time. You understand the rule of the game? So make your offer.

Student: $.30. I’m offering him $.30.

Professor Ben Polak: You’re offering him $.30? He’s saying no as well. Okay, so let’s raise the stakes a bit. Let’s make it $10, since we’re not getting much acceptance here. Let me try a third couple, working my way back, so how about you, what’s your name?

Student: Courtney.

Professor Ben Polak: So why don’t you stand up as well. We’ll let Courtney be Player 1. Actually why don’t you sit down so I can do it from here, and you are?

Student: Danny.

Professor Ben Polak: Danny, all right so Courtney’s going to be Player 1 and Danny’s going to be Player 2. These weren’t our dating couple from earlier right? We’re safe on that? Okay good, so Courtney what are you going to offer?

Student: $5.

Professor Ben Polak: $5, which is half of the $10.

Student: Accept.

Professor Ben Polak: All right accept, all right. So it turned out in this game that a lot of people were rejecting offers. Let’s have a look at it on the board a second. Let’s think about it a second and we’ll come back, we’re not done with this couple. We’re not going to need new couples, we’re going to come back to you. So in this game–it’s pretty simple to analyze this game by backward induction. By backward induction, we’re going to start with the receiver of the offer and the receiver of the offer is choosing between the offer made to them which in our notation is (1-S). And in our three examples that was a penny, then it was $.30 and then it was half of it, so $.50 just to be consistent.

We actually saw two of those offers rejected, but according to backward induction, assuming that people are trying to maximize their dollar payoffs here, what should we see the receiver do? They should accept even the somewhat insulting offer of a penny that was made: even that somewhat insulting offer should be accepted by Noah. So Noah didn’t accept the offer of a penny and let’s come back over there. So Noah didn’t accept the offer a penny, and I forget who it was but our second player didn’t accept an offer of $.30, but we’ve just argued that they should have been accepted. In fact, when Equia made the offer of one penny to Noah, I think she assumed that Noah would accept it, is this right? Why did you think Noah would accept it?

Student: Because I felt he would be better off with a penny than nothing.

Professor Ben Polak: He’d be better off with a penny than nothing, right. Then we saw an offer. The offer came up a bit. Who was my second offerer? You were my second offerer, you offered a bit more. Why did you offer more?

Student: I felt $.30 was a pretty fair share. It’s a lot better than nothing.

Professor Ben Polak: Thirty cents is better than nothing, although not necessarily fair, but I guess it’s better than nothing. Where was my rejector of the second offer? Who rejected the second offer? Why did you reject $.30?

Student: Just a pride thing.

Professor Ben Polak: It’s a pride thing. Pretty soon we converged onto $.50, which notice is no where near backward induction. So the third offer, which is Courtney, why did you offer half of it?

Student: Because half is better for me than nothing.

Professor Ben Polak: Half is better for you than nothing, and you figured he’s going to reject otherwise, and in fact he didn’t reject. Why didn’t you reject?

Student: Because $5 is better than nothing.

Professor Ben Polak: $5 is better than nothing. But the $5 is better than nothing argument would have argued against making any rejection in this game. Is that right? So here’s a game where backward induction is giving a very clear prediction. The clear prediction is, first of all, the second player will accept whatever’s given to them; and second, given that, the first player should offer them essentially nothing, should offer them just a penny. So backward induction predicts that the offer will be essentially, let’s say $.99 and $.01, or even virtually $1 and nothing. But in fact we don’t get that, we get a lot of rejection of these low offers, and often we get offers made much, much higher in the vicinity of half.

Now why? Why are we seeing a failure of backward induction in this game? I think this is not necessarily you guys. It’s a very reliable result in experimental data. So why do we see so many people in this ultimatum game both offer more, and, more importantly, reject less than small amounts. So let’s talk about it, so one person said it’s a pride thing. Let’s try the other aisle here, so what’s the smallest offer you would have accepted. What’s your name first of all?

Student: Jeff. A cent.

Professor Ben Polak: So there are some backward induction players in the room. Who would have rejected a cent? Who would have rejected $.10? We should be going down at least, who would have rejected $.30? Few people rejected $.30, not many actually. How many people would have rejected $.50? One person even would have rejected $.50, but essentially no one. So what’s happening here? Why do people think people are rejecting what is essentially money from my pocket, there’s nothing going on here, I’m just giving you money. Why are they rejecting being given money?

Student: Overall the stakes are really low, so if you have any value on sort of like pride, what people said, you know it’s not worth a penny or $.10.

Professor Ben Polak: All right, so it may be pride going on here, so certainly one thing is about pride. It turns out that people do this even in quite high stakes games, but you’re right, certainly that trade off is going to start to bite. What else is going on? So I agree, pride is part of what’s going on. What else is going on here? Let’s try and get some conversation going. Somebody in here, if I can get the mike in, shout out your name and really shout.

Student: Peter.

Professor Ben Polak: Go on.

Student: Change is cumbersome.

Professor Ben Polak: Change is cumbersome, you didn’t want the change, okay fair enough, but if the stakes go up that would get rid of that.

Student: Maybe people are tying their own outcomes to the other player’s outcomes as well.

Professor Ben Polak: Right, so maybe people have different payoffs here. Maybe people are comparing their payoff to the other person’s payoff that certainly seems like a plausible thing to be going on here. You might feel less happy about getting $.20 knowing the other person’s getting $.80 for example.

Student: You want to try to teach them a lesson to get them to offer more.

Professor Ben Polak: Right, you might be trying to set a sort of moral standard here. So there’s some notion of indignation or even teaching people that they really should offer people more. What else could be going on here?

Student: If people know I’m not going to accept less than $.50 then they should give me $.50 by backward induction.

Professor Ben Polak: Right, so part of what’s going on here–actually this game was a one shot game, we just played it once. We could have played it–in fact they often do play this this way in the lab–you could have played it without anybody knowing who the other player was. But particularly in this setting where everyone can see everyone else–even in the lab where people actually can’t see people, but they might imagine that the game is really repeated–you could imagine people trying to establish a reputation. Is that right? So there’s lots of these reasons, these sort of moral indignation reasons or teaching a lesson reasons, pride reasons. There’s also this basic reason that people might be thinking, I should play this game as I would a similar situation in life where I might want to be establishing reputation.

So there’s a certain amount of confusion going on in the game, and there’s also a certain amount of a lot of things make sense. Now notice that once we’ve established that people are going to reject small offers in this game, once we’ve established people are going to reject small offers, it makes perfectly good sense to offer a lot more than nothing. So it’s not that surprising that once we’ve established the idea that people are going to reject small offers, we’re going to see people making offers that are reasonably large, although not usually larger than $.50. Why is $.50 so focal here? Why is $.50 so focal?

It’s not a trick question. I’m just asking you why do people think $.50 is so focal? I think it’s typical that people end up offering around $.50, why? It sounds fair, it seems fair. There’s some notion of fairness. It’s not clear by the way, what ethical principal is involved here. It’s not clear that if you’re walking along the streets and you happen to find a dollar at your feet that you should pick it up and anyone else you’d happen to see at that moment you should give $.50 too. That’s essentially what the situation is, and you just chanced upon this dollar that I just gave you. It isn’t clear that there’s any particularly great moral claim to give it to someone else, but I think people read this as a situation about splitting a cake in an environment of distributional justice.

They view it as a larger picture. Is that right? So it turns out there’s a large literature on this. There’s a large experimental literature on the ultimatum game. And there’s an even larger literature on an even simpler game in which I give people a dollar, I say you can give whatever share you want to the other person, and they don’t even get a chance to accept or reject. That’s called a dictator game. In the dictator game, literally, you’re just simply given a dollar and you can give whatever share you want to the other person. It turns out that even in the dictator game, people give quite a lot of money and that suggests that there really is some notion of fairness or some notion of distributional justice going on in people’s heads here, rightly or wrongly.

So one thing this should tell us is, even in extremely simple games, we should be a little bit careful about reading backward induction into what’s going to happen in the real world. Part of this is because, as we mentioned the very first day of the class, people care about other things than just the obvious payoffs, and part of it is about more complicated things like reputation and so on. All right, having said that, let’s nevertheless for the purpose of today, act as if we are going to do backward induction, and let’s embed this into a slightly more complicated game. So the more complicated game is as follows.

Chapter 2. Bargaining Games: Generalization [00:14:15]

So we’re going to have a two period bargaining game. In this two period, bargaining game, the beginning of the game is exactly the same. So there’s a dollar on the table and Player 1 makes an offer to 2. And once again we can call this offer S and 1 - S, just the same as before. And, just as before, Player 2 can accept the offer, and if they accept the offer then this is indeed what they get. But now if 2 rejects, which is 2’s other alternative–if Player 2 rejects the offer then we flip the roles. We play the game again but we flip the roles. So we go to stage two, everything we’ve said up here is stage one. And down here we go to stage two, and in stage two, Player 2 gets to make an offer to 1. And once again we can call this–we better be careful. Let’s put ones here just to indicate we’re in the first round and twos to indicate we’re in the second.

So they make an offer is S2 and (1-S2) where S2 is that that goes to Player 1 and (1-S2) is that that goes to Player 2. So 2 gets to make an offer to Player 1 and now 1 can accept or reject. If she accepts then she gets her share from here. The offer is accepted, and if she rejects then we get nothing. So this game is exactly the same as playing the previous game, except we flip roles, but we’re going to add one catch. The catch is this, in the first round the money on the table is $1, but if we end up going into the second round, so the first offer is not accepted and we go into the second round, then part of the money is lost.

In particular, we’ll assume that the money on the table is δ. If you think of δ as being just some number less than 1–so if you want a concrete example think of this as being $.90. So the idea here is if you get into the second round, time has past, it’s costly, and so money in the second round is–I think of it as money in the second round as being worth less or could actually think of this cake being eaten up, some of it’s thrown away, some of it’s wasted. Everyone understand the game? So this is very similar to the previous game, but we’ve got this second stage coming in, and we’ve gotdiscounting. So this is the idea of discounting.

How many of you have heard the term discounting before? You probably saw it in a finance class or a macro class where we think about there being a value of time. Money today is worth more than money tomorrow, partly because you could put the money today into the bank and it could earn interest, partly because you’re simply impatient to get that money and go and have lunch, particularly on the day in which the clocks changed.

Okay, so let’s try this game again and let’s just play it for real, so let’s come down again. Everyone understand the game? Basically the same rules except we’re just flipping around and with the possibility that the cake may shrink. Let’s see what people have learned, so who were our first pair? Our first pair were Equia and Noah all right. So Equia, what are you going to offer? You’re Player 1 here but if your offer is rejected Noah’s going to get to make an offer to you. All right so what are you going to offer this time?

Student: $25.99, so $.25, $.46.

Professor Ben Polak: $.46 okay, so he gets $.46 if he accepts the offer, is that right? Student: Yes.

Professor Ben Polak: $.46 if he accepts the offer.

Student: I accept that.

Professor Ben Polak: He accepts that, okay that was easy. So Equia got $.54 and Noah got $.46. Who was our second pair? So that was, I’ve forgotten, Anish right and?

Student: Gary.

Professor Ben Polak: Gary. So Anish what are you going to offer this time?

Student: I’ll offer $.43.

Professor Ben Polak: $.43, you’re going to push the envelope a little bit.

Student: All right.

Professor Ben Polak: All right, that one got accepted as well. Okay, so people are converging here. What about our third offerer, receiver it was Courtney and?

Student: Danny.

Professor Ben Polak: Danny. So Courtney?

Student: $.30.

Professor Ben Polak: $.30, so it’s $.30 for him.

Student: I’ll accept.

Professor Ben Polak: Three acceptances, all right. Let’s find out something here, so I was hoping to get into the second round. Okay so you accepted, that’s fine–no chicken sounds around the room. So, it’s Danny right?

Student: Yeah.

Professor Ben Polak: So Danny had you rejected–you acceptedbut had you accepted what would you have offered in the second round?

Student: $.45.

Professor Ben Polak: $.45, all right, and would you have accepted that in the second round if $.30 hadn’t been accepted?

Courtney: Yes.

Professor Ben Polak: Okay, so you might have done better it turns out. Let’s go back through to the other rejections, to the other acceptances. So my second couple you offered $.43, is that right? You said yes to $.43. Had you rejected $.43 what would you have offered back in return?

Student: $.43.

Professor Ben Polak: $.43, the same thing back. Would you have accepted it?

Student: He gets $.47?

Professor Ben Polak: He would have got $.47 in that case.

Student: Yeah.

Professor Ben Polak: You would have accepted that, okay. Equia went first and she offered $.45 to Noah. And Noah had, in fact, you rejected, what would you have offered back?

Student: I would have also done $.45.

Professor Ben Polak: Same thing back and would you have accepted it?

Student: Yes.

Professor Ben Polak: Okay. So we can see here that the decision to accept or reject partly depends on what you think the other side is going to do in the second round, is that right? So here you are, if you’re in the middle of this game. If you’re Player 2 you’ve received an offer. None of these offers sounded crazy. $.30 was the lowest one, but none of them sounded crazy. And you’re trying to decide whether you should accept or reject this offer. And one thing you should have in mind is what would I offer if I reject. And will that offer that I then offer in the next round be accepted or rejected, is that right?

So if we just work backwards we can see what you should offer in the first round should be just enough to make sure it’s accepted knowing that the person who’s receiving the offer in the first round is going to think about the offer they’re going to make in the second round, and they’re going to think about whether you’re going to accept or reject in the second round, is that right? So that sounds like a bit of a mouthful but that mouthful of reasoning is exactly backward induction.

It’s exactly backward induction. It’s saying: to figure out what I should do in the first round or what I should offer in the first round, I need to figure out whether Player 2 is going to accept or reject. And to figure out whether he or she is going to accept or reject, and I have to put myself in his or her shoes, and figure out what he or she would offer if she did reject, and what he or she thinks I would do if I got that second round offer. Is that right? All right, so let’s try and analyze this as if backward induction was going to work here, as if we didn’t have to worry about things like pride here.

So this is the game we’re actually playing, so let’s keep that one and actually analyze it on the board. I want to walk us from a largely mundane game of take it or leave it offers to a more complicated game in which there can be several rounds of offers. But we’re going to go slowly so we’ll start just with two rounds. So first of all let’s just look at the stage one game, and let’s keep in track what the offer is and what the receiver. This is the offerer and this is the receiver. In the one stage game, the game only has one stage, then we know from backward induction what the results should be. It isn’t what we’d find in the lab, it isn’t what we find in the classroom, but we know what we should get. The offerer should offer to keep everything essentially or maybe $.99 but let’s call it $1 and the receiver gets nothing. So again I’m approximating a little bit because it could be $.99 and a penny but who cares. Let’s just call it a $1 or nothing if it’s a one stage game.

So now let’s consider a two stage game. In the two stage game the person who’s making the offer in the first stage needs to look forward, anticipate what would happen if her offer was rejected by Player 2 and Player 2 went forward into the second stage. Is that right? So in the two stage game, in the first stage of the two stage game, the person making the offer wants to anticipate what the receiver would offer in the second round were the receiver to reject her offer.

But we can do that by backward induction. We know that in the second round if the receiver rejects the offer, then the second round of the two stage game is what? It’s a one stage game, and we’ve just argued, at least if we believe in backward induction, in that case, Player 2 who is then the offerer, will offer $1 and Player 1, who is now the receiver, will accept it and get nothing. So Player 1 in the first round of the two stage game wants to make an offer that’s just enough to get Player 2 to accept it now. So let’s think about this. So if Player 1 offers 2 something more than what? Tomorrow Player 2 can get $1 but that’s $1 tomorrow. So $1 tomorrow is worth how much today if we’re discounting? It’s just worth δ right. It’s just worth δ. So if Player 1 offers Player 2 more than δ x $1, which is what Player 2 can get tomorrow, then 2 will accept.

If Player 1 offers 2 less than δ x $1–because you can get a $1 tomorrow but that’s only worth $δ–a $1 tomorrow is worth just $δ today–then 2 will reject. So the offer has to be exactly enough to get accepted, which is exactly $δ. So Player 2 knows that she can get $1 tomorrow so you need to offer her at least $δ today to make it as good for her as getting $1 tomorrow. So we know the receiver must be offered at least $δ tomorrow, which means the offerer is going to keep $[1 - δ]. So in the first round of the two stage game, Player 1 should offer $[1 - δ] for herself and $δ for Player 2 and Player 2 should accept that because $δ dollars today is as good as $1 tomorrow.

Now, another way to see that is in a picture, so let’s just draw a picture. Let’s put the payoff of Player 1 here and the payoff of Player 2 on this axis. And we’re going to assume that they’re just going to maximize dollars where there’s no pride in here. And if we just look at the one stage game, we’re simply looking at this line. The offers in the one stage game: it could be that Player 1 gets everything herself and gives nothing to Player 2, it could be that Player 2 keeps everything, ends up getting everything and Player 1 gets nothing, and it could be any combination in between. We argued by backward induction–although not in reality–in backward induction, in the one stage game, Player 1 makes an offer to Player 2 which is kind of an insulting offer. Player 1 says I get everything and you get a $.01. So this is the one stage game.

In the two stage, if things are settled in the first stage this line represents the possible divisions between Player 1 and Player 2. But if we end up going into the second stage, the pie is shrunk. The pie is shrunk, instead of going from $1 to $1, it goes from $δ1 to $δ1. Or if you like, if δ is .9 it goes from $.9 to $.9. So let’s draw that line in. So if we head into the second stage, we’ll end up being here, and this goes from $δ here to $δ here where these dollars are being evaluated at time one. All right so the pie has shrunk. If we get into the second stage then, by backward induction, Player 2 is in an ultimatum game, Player 2 will be making the offer and Player 2 says: whatever cake is left I’m going to take all of it and you’re just going to get a $.01.

So if we get into the second stage then Player 2 will make this offer to Player 1. Player 2 will say I’m going to keep the whole of the pie, which in first period dollars is worth $δ. So I’m going to end up with a payoff of δ and you’re going to end up with a payoff of essentially nothing. Player 2 knows that they can therefore get at least $δ–or $δ in current day dollars worth–from rejecting your offer. Since they can get at least $δ current day offers from rejecting your offer, the lowest offer you can make to them is an offer that gives them at least $δ.

So the offer you’re going to make is this offer: this is the two stage offer. It happens in the first stage. Player 1 makes an offer that gives Player 2 what Player 2 could get in the second round, so gives Player 2 $δ and keeps $[1 - δ] for herself. Everyone understand the picture? So this picture is just corresponding to this table. The thing people tend to get confused about here, I think, is they get confused between current dollars and discounted dollars, so we’re going to do all the analysis here in terms of the first period dollars, dollars tomorrow are going to be worth δ. There’s a hand up, can I get a shouting out? Yeah?

Student: [Inaudible]

Professor Ben Polak: Yes, sorry. So this is the outcome if it was a one stage game and this is the outcome if it was a two stage game. The offer is made and accepted. Let’s roll it forward, let’s look at a three stage game. Let’s keep this picture handy and think about a three stage game. So the beginning of the game is the same. We’re going to look at three stage bargaining, and the rules in three stage bargaining are pretty much the same as in two stage bargaining, but now there’s two possible flips. In three stage bargaining, in the first period, in period one, 1 makes the offer and if it’s accepted the game is over. In period two, if we reject, then we go to period two when 2 makes the offer and if it’s rejected now, this time by Player 1, then we go to period three where once again 1 makes the offer.

So you can see where we’re heading, we’re heading towards an alternate offer bargaining model. I’m going to make an offer, Jake’s going to either accept or reject, then he’ll make an offer and we’ll flip to and fro. There’s a question, let me try and get a mike out to the question. Yeah?

Student: I have a question about the two player game, if δ is the best that Player 2 can get tomorrow then why wouldn’t 1 offer Player 2 δ discounted by δ today?

Professor Ben Polak: Good. Right so I think I was confusing about it, so let me make it clear. So tomorrow Player 2 can get everything, everything that there is. So whatever pie is left tomorrow Player 2 can get all of it. So call that pie tomorrow 1 and evaluate it in period one dollars as being worth $δ. Does that make sense? Okay, so I think I misspoke on that, so let me say it again. So every period there’s this pie and every period, if it was the last period of the game, the person making the offer is going to get the whole pie, but if I view that pie tomorrow from today, a pie of $1 tomorrow is only worth $δ today. A pie of $1 the day after tomorrow is only worth $δ tomorrow and $δ² today and so on. So that’s the way in which we’re going to do discounting here.

Good, all right. So in this game, if 1 makes an offer, if it’s accepted it’s over. If it’s accepted then we’re done. And if this offer’s accepted then we’re done. And if this offer’s accepted then we’re done. And in the third round, if it’s rejected then both players get nothing. Once again we’re going to assume that the players are discounting. So what does it mean to say they’re discounting? It means that dollars in period one are worth dollars, dollars in period two are discounted by δ, and dollars in period three are discounted by δ x δ, or if you like by δ². Just to put this into real notions of money, if you think of δ as being 90%, then $1 in period one is worth $1, a $1 in period two viewed from period one is worth $.90, and a $1 in period three viewed from period one is worth $.81.

Okay, so what do we think is going to happen here? Well, once again we can do backward induction. Here we are in our picture. Let’s look at the three-stage game. Once again, when we analyze, as always when we analyze these games using backward induction, we want to start at the end. If we start at the end, we know that the last stage, that’s the third stage of the three stage game looks like what? It looks like a one stage game. In the one stage game we know the offerer will get everything. Say it again, so the last stage of the three stage game, we know the person who makes the offer who this time will be Player 1 will get everything. However, that everything is only worth δ in period two dollars and it’s only worth δ² in period one dollars.

So in period one, in the first period of this game, we know that if their offer is rejected we know what’s going to happen. Say it again, in the first period of this three stage game, if the offer is rejected then we’ll go into a two stage game, and we already know what happens in a two stage game. In a two stage game, the person who gets to make an offer gets $[1 - δ] and the person who receives the offer gets $δ. So we know in the first stage of the game that the person who receives the offer always has the outside option of saying, no I reject. And we know that that person tomorrow will get $[1 - δ]. But $[1 - δ] tomorrow is worth how much today? It’s worth $δ[1 - δ]. Tomorrow they’re going to get $[1 - δ], so today that’s worth $δ[1–δ].

So the offer I have to make in the first round to make sure that the other person accepts it has to be just better in discounted dollars, than what they’re going to get tomorrow. They’re going to get $[1 - δ] tomorrow, so I have to give them $δ[1–δ] today, which means I keep for myself $[1 - δ[1–δ]]. And if you don’t like the algebra let’s look at the picture. In the picture, in the one stage game, this is the offer. In the two stage game, we know if we get to the second round, Player 2 gets everything so we have to give him that much today. And if we get into the third round, now we’re looking at δ² here and δ² here. We know that if we get into the third round the person who makes the offer in the third round will get everything, so we can actually work our way along. In the third round, the person who makes the third offer will get everything, so in the second round you’d have to give them that much, so in the first round you’d have to give Player 2 that much.

Say it again, in period three, the person making the offer can get everything. So in period two, they must be getting δ times that, so in period one you have to give them at least this much. And this here is the offer you’d make in the three stage game. So in the picture we’re just doing a little zigzag; on the chart we’re also always working across the diagonal. So we’ve done the one stage game: the one stage game, the person making the offer gets everything. In the two stage game, the person making the offer offers just enough to get the offer accepted which is $δ because that’s what $1 is worth tomorrow. In the three stage game, the person making the offer makes just enough to get the offer accepted, which is δ times what the receiver would get tomorrow. What they get tomorrow is $[1 - δ], so they get $δ[1–δ] today.

How about the four stage game? Let’s see if we can do that. So if we go to the four stage game now, in the four stage game if the person receiving the offer rejects the offer, then tomorrow they can get $[1 - δ[1–δ]]. So I need to offer them enough now in current dollars so they will prefer that to getting $[1–δ[1 - δ]] tomorrow, so how much must I offer them? I have to offer them at least δ times that much, so I have to offer them $[δ x [1 - δ x [1–δ]]]. Again, I’ll keep the rest for myself so I’ll get $[1 - δ[1–δ[1–δ]]]. And so the principle is always give people just enough today so they’ll accept the offer, and just enough today is whatever they get tomorrow discounted by δ.

So actually this backward induction isn’t so bad. What makes it a little bit easier is you don’t actually, when you go through an extra stage of this you don’t actually have to go all the way to the beginning, you could actually start where you were last time and just discount once more by δ. Let’s see if we can see any kind of pattern emerging in this algebra, so let’s just multiply out these brackets. In the four stage game, this thing is actually equal to–just multiplying through–it’s 1 - δ + δ² - δ³–I hope it is anyway. That’s what this is. And this thing is equal to δ - δ² + δ³, just multiplying out the brackets.

Does anyone see a pattern emerging here in these offers? We had offers of 1 and then 1–δ. We could also multiply out this one. It might be helpful to do so: This is 1 - δ + δ². Anyone see a pattern what these offers look like? They kind of alternate. So let’s have a look, rather than do every stage. Should I do one more stage to see if we can see a pattern emerging or should I jump straight to ten stages and see what happens? Go straight to ten people say? Let’s do one more, nah, let’s jump through to ten.

So imagine that this game actually had ten rounds, so this is a ten stage game, and let’s just continue our chart down here. So here’s–need a bit more space here–ten stages: ten stage game. I’m going to continue my chart and my chart says in the ten stage game what am I going to get? So the offer is going to be–it’s going to be the same pattern–1 - δ + δ² - δ³… + δ8 - δ9, everyone see that? So what I’m doing is I’m continuing the pattern from above. So if I had ten stages, I always start with a 1, the positive and negative terms just alternate, and I have as many terms as 1 minus the stage I’m in. So in the four stage game, I ended up with δ³, so in the ten stage game I’ll end up at δ9. Everyone happy with that?

So the four [error: ten] stage offer is this slightly ugly thing, 1 - δ + δ² - δ³… + δ4 - δ5, etc., + δ8 - δ9. That’s a pretty ugly thing, but fortunately some point in high school you learned how to sum that thing. Do you remember what this is? What do you call objects like this in high school? Anyone remember? Objects like 1, δ, δ², δ³, δ4 what are they called? They’re called geometric series right, they’re called geometric series. Anyone remember how to sum them? We know that S is equal to this, this is what the offer is, if there is ten rounds. We know the offer is accepted. So the way to remember how to sum it, the trick for summing it is to multiple both sides of this equation by the common ratio, so multiply both sides by δ.

So if I multiply this side by δ, I’m going to multiply the other side by δ. And this 1 will become a δ, this δ will become a δ², this δ² will become a δ³, this δ³ will become a δ4…. There will be a -δ8 coming from the seventh term. There will be a +δ9coming from the δ8 term, and there will be a -δ10. Everyone okay with that? I just multiplied everything through by δ and I just shifted along one for convenience. What do I do now, anyone remember? Add the two lines together. So by summing this side I get [1 + δ] S10. On the other side, what’s kind of convenient is everything cancels. The 1 comes through, I get a 1. These two terms cancel and these two terms cancel, and these two terms cancel, and so on and so on, all the way up to the end where I get -δ10. Everyone okay. All the other terms have cancelled out. 

So now just sorting out my algebra a bit–I’m going to take it on the other side–I’m going to have that the offer you make–this is what you’re going to get to keep–so the amount I claim I should keep in the first round is [1 - δ10]/[1 + δ]. Just be a bit careful with the notation here because it may be a little bit confusing. The 10 here doesn’t mean to the tenth power, it’s just the offer in the tenth round [error: ten-round game], whereas the 10 here really does mean in the tenth power. So if we play this game for ten rounds, the offer you’d make would be [1 - δ10]/[1 + δ] which means the amount you’re offering to the other side, which is 1 - S would be [δ + δ10]/[1 + δ].

Chapter 3. Bargaining Games: Summary of Proof of Generalization [00:47:44]

So to summarize where we are: we started off by considering a very simple game, a one stage take it or leave it offer. We know that, in that take it or leave it offer, Player 1 is going to claim everything for herself and offer nothing to Player 2. Then we considered a two stage game which was the same as the one stage game except that if Player 2 rejects the offer, he–let’s call player 2, “he”–he gets to make an offer to Player 1 in the second period. We know that in the second period of that two stage game Player 2 can keep everything for himself. Everything for himself tomorrow is worth δ today, so you have to offer him at least δ today and keep 1 - δ for yourself.

Then we looked at a three stage game. In this three stage game, if Player 2 rejects in the first round, Player 2 can make you an offer in the second round, but now if you’re Player 1 and you reject in the second round you get to make an offer to Player 2 in the third round. We argued that in the second round of this game, if Player 2 rejects you in the first round and makes an offer in the second round, it’ll be in a two stage game and they’ll be able to keep 1 - δ of the pie for themselves. So you have to offer them at least δ x [1–δ] today for them to accept the offer, keeping the rest for yourself.

Then we looked at a four stage game. In the four stage game if Player 2 rejects your offer, he can make you an offer, but if you reject the offer you can make him an offer, but if he rejects that offer he can make you an offer. And once again we asked how much do I have to offer Player 2 now for him to accept the offer now? He knows that if he rejects the offer, he can get this amount 1–δ x [1–δ] tomorrow. So I have to offer him δ times that today, and once again I keep the same for myself. That’s just a summary of what we did. And then what we did was we cheated. We jumped to the tenth stage [error: ten stages], just noticing that a pattern had emerged, and we found that in the tenth stage [error: ten-stage game], this is the offer you’d make just according to the same pattern. And it was this horrible thing, and then we used a little bit of high school math to simply this thing. And it turns out this is the amount you keep for yourself and this is the offer you’ll make to Player 2.

In each case I’ve accepted–Did I make a math mistake? Thank you. Let’s put a superscript in here. good. So what do we observe here? So the first thing to observe is in the one stage game if we believe backward induction you certainly want to be the person making the offer. In the one stage game, in the ultimatum game, there’s a huge first-mover advantage. In the two stage game it’s not clear if you want to make the offer, it depends on how large δ is, but if δ is a big number like .9 you’d rather be the person receiving the offer. In the three stage game, it looks like you’d probably rather make the offer, but it’s not so clear. So where does it go to as we go down the path, as it goes down towards the ten stage game?

It looks like in the ten stage game you’d probably still prefer to make the offer than not, but they’re certainly much closer together than they were before. Some of that initial bargaining power has been washed out by the fact that there are ten stages. So let’s try and push this just a little bit harder. Instead of looking at the tenth stage offer, what if we look at the infinite stage offer. So in principle we look at the infinite stage of this game. So I can make you an offer, you can say no and make me an offer, and then I can reject and make you an offer, and then you can reject and make me an offer, and so on and so forth. So we look at this term. If in principle and you can make an infinite number of offers–so, what’s this term going to look like if I can make an infinite number of offers?

So I claim it’s going to look like this [1 - δ] / [1 + δ] and over here, at least it’s going to converge towards this. We’ll be a bit more formal, and over here we’ll have [δ + δ] / [1 + δ]. However now I get a little bit simpler. What is δ? It’s 0 right, so .9 x .9 x .9 x .9 x .9 x. 9 x .9 x .9 x .9 x … is 0. So this last term disappears as does this one, and we just get 1 / [1 + δ] and δ / [1 + δ]. So if we make alternating offer bargaining–a bargaining game where in each round I make you an offer, you can accept it or you can reject and make me an offer, and we imagine there’s no bound to this game, it just goes on arbitrarily long–then our prediction is that Player 1, the person who makes the first offer will get 1 / [1 + δ] of the initial pie and Player 2 will get δ / [1 + δ] of the initial pie.

Let’s try and get a handle about what those numbers are. So if you imagine these offers can be made fairly rapidly, for example, I can make offer today, you can make an offer back to me in half an hour’s time, and then I can make an offer back to you in half an hour’s time, then it’s reasonable to assume that the pie is not shrinking very fast. The discount factor is not a big deal here. So these offers can be made in rapid succession, but we might think that δ itself is approximately 1: the pie isn’t shrinking very fast. If δ is approximately 1, and if we take δ to 1 here–the time isn’t that valuable given how rapidly we can make offers to and fro–then what does this make this equal? In the case where δ is equal to 1 what do we get? We get 1/2, which means this will also be 1/2.

Chapter 4. Bargaining Games: Assumptions and Conclusions [00:54:29]

So we learned something from this which is kind of surprising. If you do alternating offers–the sort of standard, very natural game of bargaining–sort of the kind of bargaining you might do in the bazaar, in a market, or the kind of bargaining you might imagine going on in negotiations between baseball players or their agents and teams, or general managers of teams–in which offers just go to and fro and they go to and fro fairly rapidly, and in principle they could make lots and lots of offers. In principle–what this moral tells us is, in principle, we’re going to end up with each side splitting whatever the pie was equally. Very, very different from the ultimatum game where all the bargaining power was on the person who made the first offer.

So what are the lessons here? What can we conclude from this? We’ve looked at alternating offer bargaining, and we’ve concluded, under special conditions, we’ve concluded that you get an even split. You get an even share, an even split, a fifty-fifty split if three things are true. The first thing is there’s potentially infinitely many offers: potentially can bargain forever. And if discounting is not a big deal. What does discounting really not being a big deal means? It means those offers can be made in rapid succession. So no discounting, or if you like, rapid offers. If you have to wait a year between every offer then that discount factor would be a big deal.

But I actually made a third assumption, and I made it without telling you. What was the third assumption I made? I snuck the third assumption past you without telling you. What was that third assumption? Let me get the mike here. So I claim I snuck a third assumption. There’s somebody, let me start over here. What was the third assumption?

Student: I don’t know if this is what you’re looking for but they know how big the pie is.

Professor Ben Polak: They know how big the pie is, that’s true, that’s a big deal actually. That’s true, but there’s something else going on here. What is it?

Student: We assume that both players were rational.

Professor Ben Polak: We assumed that. That’s true, but we’ve kind of been assuming that throughout backward induction. You’re right none of this backward induction would apply so cleanly if we didn’t assume that. What else do I assume? It’s hidden actually, I snuck it in. We assume the discount factor is a constant, that’s true, but not just constant but something else. You’re on the right lines. They’re the same. I’ve assumed here, implicitly, I’ve assumed that both people are equally impatient, they have the same discount factor, δ1 = δ2. Why does that matter? Well let’s just think about it intuitively a second.

Suppose that one of these players is very, very impatient. They need the money now. If it’s cake, they need the cake now. They’re very impatient, and the other person is very patient. They can wait forever to get this bargain to come across. Who do you think is going to do better, the patient player or the impatient player? The patient player is going to do better. The way we ended up, the way we did all this analysis is we assumed that those discount factors were the same. We assumed that each person was discounting time at the same rate, perhaps because they were facing the same bank with the same interest rate. But in practice, often one side is going to be in a hurry to get the dispute resolved, and the other side can sit around forever. In that world, the side who could sit around forever is going to do much better.

Now, we’re going to look at relaxing this assumption and this assumption in particular we’re going to relax it on a homework exercise. So in your homework exercise you’re going to try redoing part of this analysis–good practice anyway–but doing it in a setting where the discount factors are different. So one thing we learned here was: yes, you get an even split, but it depends on these three assumptions. It’s kind of important because when you think about bargaining, I think a lot of people simply assume intuitively that whatever the bargain is about, people will eventually split in the middle. When you’re bargaining about a house, or the price you’re going to pay for a house, all these things, you kind of implicitly have this assumption you’re going to end up splitting the difference.

What I’m arguing here is you will split the difference in this natural bargaining game, but only under very special assumptions, and in particular, the assumption of patience is critical. There’s another remarkable thing here though, it’s also hidden. So not only did we end up with an even split, but something else remarkable happened in this bargaining game. What was the other thing? Somewhat amazingly, a very unrealistic thing that occurred in this bargaining game? See if you can spot it. So one thing was an assumption I made and the other is actually a prediction. Yeah.

Student: The first offer will be the offer that’s accepted.

Professor Ben Polak: Good. Did everyone see that? So in this bargaining game, I set it up as alternating offer bargaining, so the image you had in your mind was of haggling. I made an offer, you guys thought about this offer. Should I take this offer or not. Maybe I won’t take this offer. You make an offer back to me, and we kind of haggled to and fro. But actually, in the equilibrium of the game, none of that happened. That all happened in our mind. We thought about what offer we would make, and we thought about what offer you would make back to me if I made you this offer and you rejected it and so on. We did this backward induction exercise but it was all in our heads.

In this game the actual prediction is: the very first offer is accepted. There’s no haggling, there’s no bargaining. Now that doesn’t seem very realistic, there’s no haggling. Backward induction suggests that we should never see bargaining, never see the actual process of bargaining. What you should see is an offer is made and it’s accepted. Now what is it about the real world that allows for haggling to take place? If this was a model of the real world and we believed in backward induction, then we’re done. So why is it in fact in the real world we see people make offers to and fro? What’s different about the real world than this model? Let’s talk about it a bit.

You must have all bargained for something in the real world. None of you have probably bargained for a house yet, but you might have bargained for a car or something. In the real world you make offers go to and fro, right? What’s going on? Why are we getting offers in the real world whereas we don’t in this game? What are we missing?

Student: In the real world you don’t actually know what the other person’s discount factor is, therefore, you have uncertainly as to what your highest possible offer could be.

Professor Ben Polak: Good. So in the real world, unlike the model on the board, not only are those discount factors different but you probably don’t know how patient or impatient the other side is. You can get some ideas about how patient or impatient the other side is by looking at their characteristics, for example. For example, if you know that the person you’re bargaining with over their car–you’re trying to buy their car–and they’re a graduate student who’s just a got a job in, I don’t know, Uzbekistan or something, and they aren’t going to be taking their car with them, and they’re leaving next week, you know they’re in a hurry. So there’s times when you’re going to know something about other people’s discount factors, how patient they are, but lots of times you’re not going to know. So one thing is you just don’t know what the discount factor is. By the way, what else might you not know about the other side? What else might you not know?

Student: How big the surplus is they were splitting.

Professor Ben Polak: Good. You might not know this good that you’re selling. We’ve been talking about the big one pie which you’re carving up, and everyone knows the size of the pie. But in the real world, I might not know this object that’s being sold, I might not know how much this object is worth to the other side, and he or she may not know how much it’s worth to me. So that lack of information is going to change the game considerably. In particular, I might want to turn down some offers in this game in order to appear like a patient person. Why I might want to turn down some offers in the game in order to appear like somebody who doesn’t really value this all very much. And in so doing, I’m going to try and get you to make me a better offer.

So what’s going on in haggling and bargaining, according to this model, what’s missing in this model, is the idea that you don’t know who it is you’re bargaining with. You don’t know how much they value the objects in question., You don’t know how impatient they are to get away with the cash. So it’s a big assumption here, a very big assumption, is that everything is known. So both the size of the pie, let’s call it the value of the pie, and the value of time is assumed to be known, but in the real world you typically don’t know the value of the pie on the other side, and you typically don’t know how much they value time.

So that produces a whole literature on bargaining, none of which we really have time to do in this class, which is a pity because bargaining’s kind of important. So instead, I want to spend the last five minutes just introducing, is it really worth introducing a new topic in the last five minutes? No, I think it is, let me talk a little bit more about bargaining rather than that. So what does this suggest if we’re going out in the real world? I’m actually taking this to reality.

So one thing it suggests is people for whom it’s known that they’re going to be impatient–people for whom it’s known that they desperately need this deal to go through–are going to do less well in bargains. We already know people may do less well in bargaining because they’re less sophisticated players, but here it isn’t that they’re less sophisticated–they can be as sophisticated as you like–but they’re just going to be in a hurry. We already talked about the graduate student whose leaving for Uzbekistan, but who else typically in bargaining is going to need cash now? Who else is going to be in a weaker position in their bargaining, socially in our society?

Student: When labor management disputes labor.

Professor Ben Polak: So that’s a good question. That’s a good question in labor management disputes–there’s one going on right now in Hollywood–it’s not clear there. It could be, it could be the management side who’s in a hurry because they just need right now to get David Letterman’s script written. That would tend to favor labor, but it could be the labor side. Why might it be the labor side? Does everyone know this? There’s a writer’s strike going on in Hollywood right now, so the people on the management side who are in the weakest position are the people who are in the most hurry to get this resolved, and those are the guys with the fewest scripts in the pipeline, and that tends to be late night TV shows. So those guys really want this thing settled fast. They’re in a weak bargaining position.

On the other hand, there may be a reason why labor’s in a weak bargaining position. Why might labor be in a weak bargaining position relative to management? They have rents to pay. They have immediate demands on their cash. The typical worker is typically poorer than your typical manager, not always but typically, and they have to pay the rent. They have to feed their children. So there’s a more general idea there. More generally in bargaining, the people who are poorer, typically–it isn’t just poor in terms of income, it’s poor in terms of wealth–are going to be more impatient to get things resolved; and that’s going to put them in a weaker bargaining position. So in bilateral bargaining, having low wealth and being known to have low wealth puts you in a weaker position. And that means that typically people who are poorer are going to do less well, although the late night TV show may be an exception.

It makes you think a little bit about whether adjusting up a bargaining position makes things equal for everybody. Any other thoughts about who has strength and who has weakness in bargaining? What other kind of stunts do we see people do in bargaining? What else is kind of missing here when you think about a particular–what we want to do in this class is develop these ideas and take them to the real world, so what other real world things here are kind of missing?

Student: Usually people will make their first offer a lot higher than what they’re actually willing to accept.

Professor Ben Polak: Right, so typically, you’re right, typically bargaining isn’t just a series of random numbers, typically people start out high and then they concede towards the middle, is that right? So if I’m the buyer I start out with a low price and come up, and if you’re the seller you start off with a high price and come down. So again that seems to be about establishing reputation and trying to indicate how much I want this good. There’s something worth saying here which we haven’t got time to do in this class, and I hope you all have time to take the follow up class 156. We can actually show formally that in a setting in which buyers and sellers are bargaining, and buyers and sellers do not know how much the good is worth to the buyer or how much it costs to the seller, typically you cannot expect to get efficiency.

Let me say it again. So it’s kind of an important economic fact that’s missing from 115 and unfortunately missing from this class, but if we go to a more real world setting in which people’s values are not known, not only are the offers not accepted immediately and not only is there some inequity in that the poor tend to be more impatient and do less well, but also you get bad inefficiency. The inefficiency occurs essentially because the sellers want to seem like they’re hard and the buyers want to seem like they’re hard, and you get a failure for deals to be made. So some deals are actually going to be lost or take a long time in coming, you’re going to get some strikes before the deals occur and that’s all inefficient. So bargaining, not in this model, but in the real world tends to lead to inefficiency. So I’ll leave it there, we’ll have an earlyish lunch since we’re all starving because of the clock change anyway.

[end of transcript]

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