WEBVTT 00:00:00.000 --> 00:00:00.994 align:middle line:90% [SQUEAKING] 00:00:00.994 --> 00:00:01.982 align:middle line:90% [RUSTLING] 00:00:01.982 --> 00:00:04.452 align:middle line:90% [CLICKING] 00:00:04.452 --> 00:00:09.700 align:middle line:90% 00:00:09.700 --> 00:00:12.520 align:middle line:84% IAN BALL: So today we're going to consider an application 00:00:12.520 --> 00:00:13.840 align:middle line:90% of backward induction. 00:00:13.840 --> 00:00:18.300 align:middle line:84% Today is going to be about negotiation. 00:00:18.300 --> 00:00:23.360 align:middle line:90% 00:00:23.360 --> 00:00:26.920 align:middle line:84% And the basic idea is that when two parties are bargaining 00:00:26.920 --> 00:00:30.040 align:middle line:84% or negotiating, they often-- one party might make an offer 00:00:30.040 --> 00:00:31.180 align:middle line:90% and say, let's do this. 00:00:31.180 --> 00:00:33.680 align:middle line:84% And then the other party might respond with a counteroffer, 00:00:33.680 --> 00:00:34.820 align:middle line:84% and they might go back and forth. 00:00:34.820 --> 00:00:36.700 align:middle line:84% So right now the government just shut down. 00:00:36.700 --> 00:00:38.800 align:middle line:84% That's exactly what's happening between Trump 00:00:38.800 --> 00:00:40.380 align:middle line:90% and the Democratic Party. 00:00:40.380 --> 00:00:43.360 align:middle line:84% They might be countering offers and go back and forth 00:00:43.360 --> 00:00:46.160 align:middle line:84% and see if they can eventually reach an agreement. 00:00:46.160 --> 00:00:48.360 align:middle line:84% So the first model-- we're going to discuss 00:00:48.360 --> 00:00:50.160 align:middle line:90% two models of negotiation. 00:00:50.160 --> 00:00:54.180 align:middle line:84% One is going to be about pretrial settlements. 00:00:54.180 --> 00:01:00.570 align:middle line:90% 00:01:00.570 --> 00:01:02.950 align:middle line:84% And the other is going to be about price haggling. 00:01:02.950 --> 00:01:07.097 align:middle line:90% 00:01:07.097 --> 00:01:09.430 align:middle line:84% And I think these are kind of illustrative applications. 00:01:09.430 --> 00:01:11.530 align:middle line:84% Of course, we're not going to include all the details 00:01:11.530 --> 00:01:12.447 align:middle line:90% of these applications. 00:01:12.447 --> 00:01:15.150 align:middle line:84% We're going to introduce foundational stylized models. 00:01:15.150 --> 00:01:17.610 align:middle line:84% But I think they can capture some important elements 00:01:17.610 --> 00:01:18.990 align:middle line:90% of these issues. 00:01:18.990 --> 00:01:22.450 align:middle line:90% So let's start with settlements. 00:01:22.450 --> 00:01:24.190 align:middle line:84% So in many cases, in criminal cases, 00:01:24.190 --> 00:01:29.050 align:middle line:84% we often have pretrial plea bargain bargaining. 00:01:29.050 --> 00:01:33.170 align:middle line:84% And in civil cases, we often have the two parties 00:01:33.170 --> 00:01:37.130 align:middle line:84% working together before the case and trying to reach a settlement 00:01:37.130 --> 00:01:38.770 align:middle line:90% to avoid trial. 00:01:38.770 --> 00:01:40.930 align:middle line:84% So today, let's think about civil cases. 00:01:40.930 --> 00:01:44.290 align:middle line:84% I think that fits the model best. 00:01:44.290 --> 00:01:48.010 align:middle line:84% And in practice, a lot of civil cases never make it to trial. 00:01:48.010 --> 00:01:49.810 align:middle line:84% The two parties reach some settlement 00:01:49.810 --> 00:01:51.590 align:middle line:90% before the case goes to trial. 00:01:51.590 --> 00:01:54.130 align:middle line:84% What are some reasons why cases like this 00:01:54.130 --> 00:01:55.510 align:middle line:90% often don't go to trial? 00:01:55.510 --> 00:01:58.970 align:middle line:84% Why do we see so many pretrial settlements? 00:01:58.970 --> 00:02:00.210 align:middle line:90% Any thoughts? 00:02:00.210 --> 00:02:01.050 align:middle line:90% Yeah? 00:02:01.050 --> 00:02:03.235 align:middle line:84% STUDENT: Both sides want to avoid having legal fees. 00:02:03.235 --> 00:02:04.110 align:middle line:90% IAN BALL: Legal fees. 00:02:04.110 --> 00:02:06.710 align:middle line:84% So legal costs are a big part of it. 00:02:06.710 --> 00:02:08.650 align:middle line:84% And that's really what our model today 00:02:08.650 --> 00:02:11.170 align:middle line:84% is going to focus on, that having a drawn-out trial 00:02:11.170 --> 00:02:12.010 align:middle line:90% is very costly. 00:02:12.010 --> 00:02:14.370 align:middle line:84% Did you have the same idea or any other-- 00:02:14.370 --> 00:02:15.250 align:middle line:90% same idea. 00:02:15.250 --> 00:02:15.870 align:middle line:90% Anyone else? 00:02:15.870 --> 00:02:19.130 align:middle line:90% Any other reasons? 00:02:19.130 --> 00:02:20.590 align:middle line:90% It could be legal costs. 00:02:20.590 --> 00:02:22.570 align:middle line:84% It could just be the cost of delay, which is 00:02:22.570 --> 00:02:23.910 align:middle line:90% similar but slightly different. 00:02:23.910 --> 00:02:27.170 align:middle line:84% It could be that even aside from paying the lawyers, 00:02:27.170 --> 00:02:29.170 align:middle line:84% if we're going to be stuck in court for so long, 00:02:29.170 --> 00:02:32.570 align:middle line:84% and it's going to delay our productive activity, that 00:02:32.570 --> 00:02:35.290 align:middle line:90% might be costly. 00:02:35.290 --> 00:02:39.270 align:middle line:84% Another big one could be risk aversion. 00:02:39.270 --> 00:02:41.250 align:middle line:84% This is not going to be in our model today. 00:02:41.250 --> 00:02:43.990 align:middle line:84% But it might be that I think that if we go to court, 00:02:43.990 --> 00:02:46.050 align:middle line:84% there's a chance I win tons of money. 00:02:46.050 --> 00:02:47.770 align:middle line:84% There's a chance I don't win anything. 00:02:47.770 --> 00:02:49.570 align:middle line:84% So I'd rather accept a settlement 00:02:49.570 --> 00:02:52.890 align:middle line:84% where I win some small intermediate amount of money 00:02:52.890 --> 00:02:54.210 align:middle line:90% with certainty. 00:02:54.210 --> 00:02:56.740 align:middle line:84% So risk aversion, and I should be clear here, 00:02:56.740 --> 00:02:58.120 align:middle line:90% over monetary payoffs. 00:02:58.120 --> 00:03:02.260 align:middle line:90% 00:03:02.260 --> 00:03:04.420 align:middle line:84% I think this is also true in criminal cases 00:03:04.420 --> 00:03:06.300 align:middle line:84% where someone might think, well, there's 00:03:06.300 --> 00:03:08.180 align:middle line:90% a chance I get a life sentence. 00:03:08.180 --> 00:03:10.540 align:middle line:84% I'd rather accept a plea bargain where 00:03:10.540 --> 00:03:12.720 align:middle line:84% I'm certain to go to prison for five years, 00:03:12.720 --> 00:03:16.100 align:middle line:84% rather than have a risk that I serve a life sentence. 00:03:16.100 --> 00:03:18.460 align:middle line:84% And then maybe one final one that we often see 00:03:18.460 --> 00:03:24.580 align:middle line:84% is about privacy and maybe embarrassing information 00:03:24.580 --> 00:03:29.280 align:middle line:84% coming out, or potentially trade secrets being revealed at trial. 00:03:29.280 --> 00:03:35.238 align:middle line:90% 00:03:35.238 --> 00:03:37.280 align:middle line:84% There are some provisions to try to prevent this, 00:03:37.280 --> 00:03:42.660 align:middle line:84% but I think there's still some risk of this. 00:03:42.660 --> 00:03:45.380 align:middle line:84% So today, we're going to introduce a formal model that 00:03:45.380 --> 00:03:49.420 align:middle line:84% can try to make some predictions about what kind of settlements 00:03:49.420 --> 00:03:52.140 align:middle line:84% we'll see and how long negotiations will last. 00:03:52.140 --> 00:03:54.480 align:middle line:84% Of course, this model isn't going to capture everything, 00:03:54.480 --> 00:03:56.500 align:middle line:84% but it's going to be a good starting point for today. 00:03:56.500 --> 00:03:58.000 align:middle line:84% So in our model, we're going to have 00:03:58.000 --> 00:04:01.400 align:middle line:84% two parties who we call the plaintiff and the defendant. 00:04:01.400 --> 00:04:04.900 align:middle line:90% 00:04:04.900 --> 00:04:07.498 align:middle line:84% So the plaintiff is the party that's suing someone else. 00:04:07.498 --> 00:04:09.540 align:middle line:84% So the plaintiff feels that they've been wronged, 00:04:09.540 --> 00:04:12.260 align:middle line:84% and they're seeking a monetary settlement. 00:04:12.260 --> 00:04:15.740 align:middle line:84% The defendant has been accused of wronging someone, 00:04:15.740 --> 00:04:17.300 align:middle line:84% and they're being potentially asked 00:04:17.300 --> 00:04:18.680 align:middle line:90% to pay a monetary settlement. 00:04:18.680 --> 00:04:19.524 align:middle line:90% So let's be clear. 00:04:19.524 --> 00:04:20.899 align:middle line:84% At the end, there's going to be-- 00:04:20.899 --> 00:04:25.540 align:middle line:84% if they reach some settlement, that's an amount of money that's 00:04:25.540 --> 00:04:29.940 align:middle line:84% going to go from the defendant to the plaintiff. 00:04:29.940 --> 00:04:32.980 align:middle line:84% So from-- maybe I'll just write it with arrows-- 00:04:32.980 --> 00:04:34.660 align:middle line:90% D to P. 00:04:34.660 --> 00:04:36.820 align:middle line:84% So just to make sure we're on the same page, 00:04:36.820 --> 00:04:41.380 align:middle line:84% the defendant is hoping to pay out a very small settlement. 00:04:41.380 --> 00:04:42.980 align:middle line:84% The defendant doesn't want to have 00:04:42.980 --> 00:04:47.060 align:middle line:84% to pay a big fine or a big punishment to the plaintiff. 00:04:47.060 --> 00:04:49.200 align:middle line:84% The plaintiff is seeking a very large settlement 00:04:49.200 --> 00:04:53.030 align:middle line:84% because they want to get paid out as much as possible. 00:04:53.030 --> 00:04:56.190 align:middle line:84% And in this case, we're going to make a few simplifying 00:04:56.190 --> 00:04:57.310 align:middle line:90% assumptions. 00:04:57.310 --> 00:04:59.910 align:middle line:84% One of them is that there's no uncertainty 00:04:59.910 --> 00:05:01.610 align:middle line:90% about the outcome of a trial. 00:05:01.610 --> 00:05:09.413 align:middle line:90% 00:05:09.413 --> 00:05:10.330 align:middle line:90% Why are we doing this? 00:05:10.330 --> 00:05:13.325 align:middle line:84% Well, we want to isolate the effect of legal costs. 00:05:13.325 --> 00:05:14.950 align:middle line:84% In this model, we're not going to worry 00:05:14.950 --> 00:05:17.390 align:middle line:84% about risk aversion and the potential uncertainties that 00:05:17.390 --> 00:05:18.290 align:middle line:90% come with trial. 00:05:18.290 --> 00:05:20.707 align:middle line:84% So what we're going to assume here, to make things simple, 00:05:20.707 --> 00:05:26.310 align:middle line:84% is that if they go to trial, the settlement amount 00:05:26.310 --> 00:05:28.890 align:middle line:84% is known to be-- will definitely be J. 00:05:28.890 --> 00:05:30.238 align:middle line:90% And we'll use to indicate judge. 00:05:30.238 --> 00:05:31.530 align:middle line:90% This is what the judge decides. 00:05:31.530 --> 00:05:35.190 align:middle line:84% So both parties know if we go to trial the amount that's 00:05:35.190 --> 00:05:36.370 align:middle line:90% going to be agreed upon. 00:05:36.370 --> 00:05:37.953 align:middle line:84% So here, we're not thinking, you might 00:05:37.953 --> 00:05:41.150 align:middle line:84% imagine the judge deciding whether the plaintiff-- 00:05:41.150 --> 00:05:43.170 align:middle line:84% whether the defendant did anything wrong at all. 00:05:43.170 --> 00:05:45.230 align:middle line:84% Here, we're imagining that yeah, the defendant 00:05:45.230 --> 00:05:46.688 align:middle line:84% has certainly done something wrong. 00:05:46.688 --> 00:05:49.680 align:middle line:84% The question is only how much they have to pay. 00:05:49.680 --> 00:05:52.260 align:middle line:84% And here, the assumption is that if they go to court, 00:05:52.260 --> 00:05:55.920 align:middle line:84% the amount of the settlement will be some fixed amount, J. 00:05:55.920 --> 00:05:59.000 align:middle line:84% And then we need to specify what the cost of going to court are. 00:05:59.000 --> 00:06:02.200 align:middle line:84% So maybe the trial costs are going 00:06:02.200 --> 00:06:07.937 align:middle line:84% to be Cp for the plaintiff and Cd for the defendant. 00:06:07.937 --> 00:06:10.020 align:middle line:84% And I think in the notes they use capital letters. 00:06:10.020 --> 00:06:11.480 align:middle line:84% But I find at the board it's hard to know what's 00:06:11.480 --> 00:06:12.938 align:middle line:84% capital and lowercase, so I'll just 00:06:12.938 --> 00:06:17.200 align:middle line:84% use a hat to indicate the costs if we go to trial. 00:06:17.200 --> 00:06:21.200 align:middle line:84% But then each period that we negotiate before the trial also 00:06:21.200 --> 00:06:21.840 align:middle line:90% has costs. 00:06:21.840 --> 00:06:23.860 align:middle line:84% You don't just pay the lawyers when you go to trial. 00:06:23.860 --> 00:06:26.277 align:middle line:84% If you want to try to reach a settlement, that, in itself, 00:06:26.277 --> 00:06:26.880 align:middle line:90% is costly. 00:06:26.880 --> 00:06:31.520 align:middle line:84% So we're also going to assume that there are per period-- 00:06:31.520 --> 00:06:33.760 align:middle line:84% and that will be clear when I describe the periods-- 00:06:33.760 --> 00:06:40.260 align:middle line:84% per period negotiation or pretrial negotiation costs. 00:06:40.260 --> 00:06:44.080 align:middle line:90% 00:06:44.080 --> 00:06:48.160 align:middle line:84% And these are just going to be cp and cd without hats. 00:06:48.160 --> 00:06:50.720 align:middle line:84% So each period that we negotiate for, 00:06:50.720 --> 00:06:55.140 align:middle line:84% the plaintiff has to pay a cost of cp to their lawyers, 00:06:55.140 --> 00:06:59.000 align:middle line:84% and the defendant pays a cost, cd, to their lawyers. 00:06:59.000 --> 00:07:01.920 align:middle line:84% So now let's describe the alternating offer structure 00:07:01.920 --> 00:07:02.920 align:middle line:90% of this game. 00:07:02.920 --> 00:07:06.300 align:middle line:84% This is a very common format that we see in reality. 00:07:06.300 --> 00:07:07.180 align:middle line:90% You make some offer. 00:07:07.180 --> 00:07:10.397 align:middle line:84% The other side will either say it's acceptable, 00:07:10.397 --> 00:07:12.480 align:middle line:84% or they might reject your offer and then come back 00:07:12.480 --> 00:07:15.680 align:middle line:84% with a counteroffer that's more favorable to them. 00:07:15.680 --> 00:07:20.880 align:middle line:84% So we're going to imagine that we have time going-- we have 0, 00:07:20.880 --> 00:07:25.640 align:middle line:84% period 1, period 2, period 3, all the way up to, let's say, 00:07:25.640 --> 00:07:33.120 align:middle line:84% 2n minus 1, 2n minus 1, and then 2n. 00:07:33.120 --> 00:07:34.680 align:middle line:84% And what we'll imagine is that it's 00:07:34.680 --> 00:07:37.440 align:middle line:84% known that the date of the trial is set. 00:07:37.440 --> 00:07:38.580 align:middle line:90% So this is the trial. 00:07:38.580 --> 00:07:42.160 align:middle line:90% 00:07:42.160 --> 00:07:47.130 align:middle line:84% So we know that the trial is going to occur in period 2n. 00:07:47.130 --> 00:07:48.588 align:middle line:90% And if we go to trial-- 00:07:48.588 --> 00:07:50.130 align:middle line:84% when I say it will occur, what I mean 00:07:50.130 --> 00:07:52.373 align:middle line:84% is if we don't reach a settlement before then, 00:07:52.373 --> 00:07:54.790 align:middle line:84% then the trial occurs, and we know what's going to happen. 00:07:54.790 --> 00:07:56.540 align:middle line:84% The settlement is going to be J, and we're 00:07:56.540 --> 00:07:58.730 align:middle line:84% going to have to pay all these legal costs. 00:07:58.730 --> 00:08:00.270 align:middle line:84% But in anticipation of the trial, 00:08:00.270 --> 00:08:05.170 align:middle line:84% what we're imagining is there's time to have n rounds, 00:08:05.170 --> 00:08:08.730 align:middle line:84% or 2n minus 2n periods, but n total rounds of alternating 00:08:08.730 --> 00:08:09.650 align:middle line:90% offers. 00:08:09.650 --> 00:08:14.410 align:middle line:84% So what we imagine is in the first period, 00:08:14.410 --> 00:08:19.410 align:middle line:84% the plaintiff is going to be the one who's going to propose. 00:08:19.410 --> 00:08:20.930 align:middle line:90% So it's a little confusing. 00:08:20.930 --> 00:08:22.350 align:middle line:90% P is for plaintiff. 00:08:22.350 --> 00:08:23.250 align:middle line:90% You're not proposer. 00:08:23.250 --> 00:08:31.290 align:middle line:84% But the plaintiff, maybe I'll say offers, I'll call this S0. 00:08:31.290 --> 00:08:35.122 align:middle line:84% So in period 0, they're going to say, look, let's just settle. 00:08:35.122 --> 00:08:37.289 align:middle line:84% And the amount that you're going to have to pay me-- 00:08:37.289 --> 00:08:38.990 align:middle line:84% I'm the plaintiff, you're the defendant. 00:08:38.990 --> 00:08:41.250 align:middle line:84% The amount that you're going to have to pay me is S0. 00:08:41.250 --> 00:08:43.690 align:middle line:84% Do you want to accept this or not? 00:08:43.690 --> 00:08:46.630 align:middle line:84% And if it's accepted, then you pay me this amount. 00:08:46.630 --> 00:08:47.330 align:middle line:90% We're done. 00:08:47.330 --> 00:08:49.130 align:middle line:84% And if it's rejected, we continue. 00:08:49.130 --> 00:08:56.670 align:middle line:84% So then the defendant can choose either to accept or reject. 00:08:56.670 --> 00:09:00.570 align:middle line:90% 00:09:00.570 --> 00:09:02.690 align:middle line:84% And if the defendant accepts this offer, 00:09:02.690 --> 00:09:03.930 align:middle line:90% well, now we're done. 00:09:03.930 --> 00:09:05.650 align:middle line:90% The defendant pays S0. 00:09:05.650 --> 00:09:06.790 align:middle line:90% We never go to court. 00:09:06.790 --> 00:09:08.330 align:middle line:90% The game is over. 00:09:08.330 --> 00:09:10.163 align:middle line:84% But if they reject this offer, well, they're 00:09:10.163 --> 00:09:11.205 align:middle line:90% not just going to reject. 00:09:11.205 --> 00:09:13.170 align:middle line:84% They're going to come back tomorrow and propose 00:09:13.170 --> 00:09:14.410 align:middle line:90% a counteroffer. 00:09:14.410 --> 00:09:16.290 align:middle line:84% So in the next period, the defendant 00:09:16.290 --> 00:09:18.512 align:middle line:90% is going to make an offer. 00:09:18.512 --> 00:09:20.970 align:middle line:84% And now I'll call that offer-- they can make any offer they 00:09:20.970 --> 00:09:21.510 align:middle line:90% want-- 00:09:21.510 --> 00:09:23.130 align:middle line:90% S1. 00:09:23.130 --> 00:09:26.490 align:middle line:84% And this is going to say, let's settle now, 00:09:26.490 --> 00:09:29.810 align:middle line:84% and I'll pay you S1 dollars if you accept it. 00:09:29.810 --> 00:09:32.150 align:middle line:84% And again now it's the plaintiff's turn 00:09:32.150 --> 00:09:33.570 align:middle line:90% who's going to say-- 00:09:33.570 --> 00:09:41.420 align:middle line:84% who's going to choose either to accept or reject. 00:09:41.420 --> 00:09:46.400 align:middle line:84% And we can see how this is going to keep going until-- 00:09:46.400 --> 00:09:46.900 align:middle line:90% let's see. 00:09:46.900 --> 00:09:50.260 align:middle line:84% In even periods, the plaintiff is making the offers. 00:09:50.260 --> 00:09:52.440 align:middle line:90% So 2n minus 2 is an even number. 00:09:52.440 --> 00:09:54.420 align:middle line:84% So that means here, the plaintiff, 00:09:54.420 --> 00:10:00.140 align:middle line:84% if we get to this point, would make an offer, 00:10:00.140 --> 00:10:03.020 align:middle line:90% let's say, s2n minus 2. 00:10:03.020 --> 00:10:04.700 align:middle line:84% And again, in this case, the defendant 00:10:04.700 --> 00:10:07.460 align:middle line:84% is going to choose to accept or reject. 00:10:07.460 --> 00:10:09.740 align:middle line:90% If it's rejected, we move here. 00:10:09.740 --> 00:10:11.300 align:middle line:90% Now it's the defendant's turn. 00:10:11.300 --> 00:10:14.020 align:middle line:84% They're going to make this last offer-- 00:10:14.020 --> 00:10:18.340 align:middle line:84% S2n minus 1, and the plaintiff is 00:10:18.340 --> 00:10:22.700 align:middle line:84% going to choose whether to accept or reject that offer. 00:10:22.700 --> 00:10:24.700 align:middle line:84% And this is only00 and only in this last period 00:10:24.700 --> 00:10:25.575 align:middle line:90% are things different. 00:10:25.575 --> 00:10:29.320 align:middle line:84% So in this last period, if the offer is accepted, great. 00:10:29.320 --> 00:10:30.620 align:middle line:90% It's over as before. 00:10:30.620 --> 00:10:33.360 align:middle line:84% But if it's rejected, well, now the trial is here. 00:10:33.360 --> 00:10:36.240 align:middle line:84% So instead of going to one more round of negotiation, 00:10:36.240 --> 00:10:38.100 align:middle line:84% we just go to trial, and we see what 00:10:38.100 --> 00:10:39.440 align:middle line:90% the outcomes are going to be. 00:10:39.440 --> 00:10:42.070 align:middle line:90% 00:10:42.070 --> 00:10:48.870 align:middle line:84% So just so we understand the costs, the Cp hat and Cd hat 00:10:48.870 --> 00:10:51.310 align:middle line:84% cost, the trial costs, are only realized 00:10:51.310 --> 00:10:53.630 align:middle line:90% if we actually go to trial. 00:10:53.630 --> 00:10:58.190 align:middle line:84% Whereas, the per period negotiation costs 00:10:58.190 --> 00:11:01.470 align:middle line:84% are realized in every period until we reach a settlement. 00:11:01.470 --> 00:11:04.790 align:middle line:90% So here, we have Cp Cd. 00:11:04.790 --> 00:11:08.990 align:middle line:90% Here, we have Cp Cd and so on. 00:11:08.990 --> 00:11:13.825 align:middle line:84% So just so we understand, if we were to settle in this period 0, 00:11:13.825 --> 00:11:15.950 align:middle line:84% we would have only had to negotiate for exactly one 00:11:15.950 --> 00:11:16.770 align:middle line:90% period. 00:11:16.770 --> 00:11:19.030 align:middle line:84% So we'd each bear these costs only once. 00:11:19.030 --> 00:11:22.030 align:middle line:84% But if we settled in, say, the third period, period three, 00:11:22.030 --> 00:11:24.010 align:middle line:84% well, it's tricky because we started at 0. 00:11:24.010 --> 00:11:25.630 align:middle line:84% So if we settled here, then we've 00:11:25.630 --> 00:11:29.050 align:middle line:84% had to negotiate 1, 2, 3, 4 periods. 00:11:29.050 --> 00:11:31.750 align:middle line:84% And that means we'll each have to pay four times our 00:11:31.750 --> 00:11:34.830 align:middle line:90% per period negotiation costs. 00:11:34.830 --> 00:11:36.950 align:middle line:84% If we were to all the way go to trial, 00:11:36.950 --> 00:11:39.250 align:middle line:84% we're going to have to pay our trial costs, 00:11:39.250 --> 00:11:42.250 align:middle line:84% and we've negotiated each of these periods. 00:11:42.250 --> 00:11:46.870 align:middle line:84% So we'll each have to pay 2n for the 2n negotiation periods times 00:11:46.870 --> 00:11:48.810 align:middle line:84% our per period negotiation costs, 00:11:48.810 --> 00:11:53.390 align:middle line:90% plus the cost of going to trial. 00:11:53.390 --> 00:11:54.750 align:middle line:90% So this is the game. 00:11:54.750 --> 00:11:58.470 align:middle line:84% And now what we'd like to do is analyze this game 00:11:58.470 --> 00:12:01.110 align:middle line:90% using backward induction. 00:12:01.110 --> 00:12:05.030 align:middle line:84% Any questions on the setup of the game? 00:12:05.030 --> 00:12:09.150 align:middle line:84% So let's first just recall what backward induction 00:12:09.150 --> 00:12:11.990 align:middle line:90% is going to deliver for us. 00:12:11.990 --> 00:12:14.750 align:middle line:84% So we're going to apply backward induction. 00:12:14.750 --> 00:12:20.410 align:middle line:84% What this is going to give us is a special Nash equilibrium. 00:12:20.410 --> 00:12:24.710 align:middle line:90% 00:12:24.710 --> 00:12:26.810 align:middle line:84% Remember, first we have to think, 00:12:26.810 --> 00:12:29.270 align:middle line:84% are we allowed to apply backward induction? 00:12:29.270 --> 00:12:30.830 align:middle line:84% Well, backward induction only can 00:12:30.830 --> 00:12:34.550 align:middle line:84% be applied to games with extensive form games 00:12:34.550 --> 00:12:36.600 align:middle line:90% with perfect information. 00:12:36.600 --> 00:12:38.558 align:middle line:84% And here, we have perfect information 00:12:38.558 --> 00:12:40.600 align:middle line:84% because there's no point in the negotiation where 00:12:40.600 --> 00:12:43.640 align:middle line:84% someone doesn't know what someone else did previously. 00:12:43.640 --> 00:12:45.460 align:middle line:84% Everything is observed in this game. 00:12:45.460 --> 00:12:48.000 align:middle line:84% So we're allowed to apply backward induction. 00:12:48.000 --> 00:12:49.680 align:middle line:84% The problem with games like this is 00:12:49.680 --> 00:12:52.200 align:middle line:84% that there are many, many Nash equilibria that may 00:12:52.200 --> 00:12:55.320 align:middle line:90% involve non-credible threats. 00:12:55.320 --> 00:12:58.080 align:middle line:84% So backward induction is going to give us this very special 00:12:58.080 --> 00:12:59.380 align:middle line:90% Nash equilibrium. 00:12:59.380 --> 00:13:02.960 align:middle line:84% And sometimes the term we use-- maybe another way of saying 00:13:02.960 --> 00:13:05.500 align:middle line:84% special is it's going to be a Nash equilibrium, 00:13:05.500 --> 00:13:07.840 align:middle line:84% but it's also going to have this property that's often 00:13:07.840 --> 00:13:09.660 align:middle line:90% called sequential optimality. 00:13:09.660 --> 00:13:15.440 align:middle line:90% 00:13:15.440 --> 00:13:17.760 align:middle line:84% Remember, our key observation from last class 00:13:17.760 --> 00:13:20.960 align:middle line:84% was that Nash equilibrium by itself 00:13:20.960 --> 00:13:23.940 align:middle line:84% doesn't necessarily imply sequential optimality. 00:13:23.940 --> 00:13:27.200 align:middle line:84% That is, it may be under some Nash equilibria 00:13:27.200 --> 00:13:30.760 align:middle line:84% that players are behaving suboptimally at contingencies 00:13:30.760 --> 00:13:33.080 align:middle line:90% that are not reached. 00:13:33.080 --> 00:13:36.520 align:middle line:84% And sequential optimality imposes the stronger condition 00:13:36.520 --> 00:13:42.380 align:middle line:84% that we have optimal play at every contingency. 00:13:42.380 --> 00:13:47.160 align:middle line:90% 00:13:47.160 --> 00:13:49.280 align:middle line:84% Particular, even those contingencies 00:13:49.280 --> 00:13:50.340 align:middle line:90% that are not reached. 00:13:50.340 --> 00:13:58.760 align:middle line:90% 00:13:58.760 --> 00:14:00.360 align:middle line:84% So Nash equilibrium is already going 00:14:00.360 --> 00:14:02.760 align:middle line:84% to take care of the contingencies that we do reach. 00:14:02.760 --> 00:14:05.880 align:middle line:84% Nash equilibrium is already going to require optimal play 00:14:05.880 --> 00:14:07.480 align:middle line:90% at those contingencies. 00:14:07.480 --> 00:14:09.240 align:middle line:84% But Nash equilibrium doesn't actually 00:14:09.240 --> 00:14:13.240 align:middle line:84% impose many constraints on these unreached contingencies. 00:14:13.240 --> 00:14:15.840 align:middle line:84% And what sequential optimality does, or requires, 00:14:15.840 --> 00:14:18.520 align:middle line:84% is that players are playing optimal even there. 00:14:18.520 --> 00:14:22.040 align:middle line:84% Because if players were not playing optimal 00:14:22.040 --> 00:14:24.560 align:middle line:84% at some contingency, that would capture 00:14:24.560 --> 00:14:26.373 align:middle line:84% what we think of as a noncredible threat. 00:14:26.373 --> 00:14:28.040 align:middle line:84% That would mean a player was threatening 00:14:28.040 --> 00:14:31.260 align:middle line:84% to behave in such a way that would not be optimal for them 00:14:31.260 --> 00:14:33.090 align:middle line:84% or credible for them to follow through 00:14:33.090 --> 00:14:35.950 align:middle line:84% on if that contingency were actually reached. 00:14:35.950 --> 00:14:38.570 align:middle line:90% 00:14:38.570 --> 00:14:41.210 align:middle line:84% Now, a key thing to keep in mind here 00:14:41.210 --> 00:14:44.530 align:middle line:84% is that what backward induction gives us is a Nash equilibrium. 00:14:44.530 --> 00:14:47.050 align:middle line:84% So it's going to give us a strategy profile. 00:14:47.050 --> 00:14:49.930 align:middle line:84% So often, when people analyze a game like this, 00:14:49.930 --> 00:14:54.590 align:middle line:84% it's tempting to say, oh the answer is we settle in period 5, 00:14:54.590 --> 00:14:56.610 align:middle line:84% and this is the amount of payments. 00:14:56.610 --> 00:14:58.770 align:middle line:84% That's an outcome of the game, but that's not 00:14:58.770 --> 00:15:00.830 align:middle line:84% a complete description of the strategies. 00:15:00.830 --> 00:15:02.770 align:middle line:84% So a Nash equilibrium is going to have 00:15:02.770 --> 00:15:05.650 align:middle line:84% to specify the player's complete contingent plans 00:15:05.650 --> 00:15:06.710 align:middle line:90% throughout the game. 00:15:06.710 --> 00:15:08.930 align:middle line:84% And that means that the strategies in games like this 00:15:08.930 --> 00:15:11.610 align:middle line:84% are much, much more complicated than the actual outcome that's 00:15:11.610 --> 00:15:13.210 align:middle line:84% reached, which might just be we settle 00:15:13.210 --> 00:15:15.610 align:middle line:90% for this amount in this period. 00:15:15.610 --> 00:15:17.075 align:middle line:90% OK, so let's get into it. 00:15:17.075 --> 00:15:18.450 align:middle line:84% It's backward induction, so we're 00:15:18.450 --> 00:15:21.170 align:middle line:84% going to work backwards, not surprisingly. 00:15:21.170 --> 00:15:24.610 align:middle line:84% And let's start with-- well, you might 00:15:24.610 --> 00:15:26.030 align:middle line:90% say we're working backwards. 00:15:26.030 --> 00:15:28.770 align:middle line:84% Let's start in period 2n minus 1. 00:15:28.770 --> 00:15:31.630 align:middle line:84% But actually, even within period 2n minus 1, 00:15:31.630 --> 00:15:33.410 align:middle line:90% we have sequential moves. 00:15:33.410 --> 00:15:36.330 align:middle line:84% First, the defendant makes an offer, 00:15:36.330 --> 00:15:38.012 align:middle line:90% and then the proposer responds. 00:15:38.012 --> 00:15:39.970 align:middle line:84% So if we want to work backwards, we're actually 00:15:39.970 --> 00:15:42.030 align:middle line:84% going to split this last period up into two, 00:15:42.030 --> 00:15:46.290 align:middle line:84% and we're going to work in the second part of the last period. 00:15:46.290 --> 00:15:56.210 align:middle line:84% So we're going to say suppose we're in period 2n minus 1. 00:15:56.210 --> 00:16:00.750 align:middle line:84% And the contingency we're at as the plaintiff, 00:16:00.750 --> 00:16:05.650 align:middle line:84% is that we're facing some offer from the defendant. 00:16:05.650 --> 00:16:12.970 align:middle line:84% So given some offer, S2n minus 1-- 00:16:12.970 --> 00:16:14.990 align:middle line:84% this is an offer by the defendant-- 00:16:14.990 --> 00:16:17.530 align:middle line:90% 00:16:17.530 --> 00:16:19.190 align:middle line:90% the proposer has two choices. 00:16:19.190 --> 00:16:22.010 align:middle line:90% 00:16:22.010 --> 00:16:26.850 align:middle line:84% They can either accept the offer, 00:16:26.850 --> 00:16:29.860 align:middle line:90% or they can reject the offer. 00:16:29.860 --> 00:16:33.340 align:middle line:84% I should point out here that I'm being a little sloppy. 00:16:33.340 --> 00:16:36.460 align:middle line:84% I'm saying suppose the proposer receives 00:16:36.460 --> 00:16:38.820 align:middle line:90% an offer of S2n minus 1. 00:16:38.820 --> 00:16:42.000 align:middle line:84% But technically, this doesn't specify the full contingency. 00:16:42.000 --> 00:16:44.580 align:middle line:84% What would I need to specify if I wanted to say-- 00:16:44.580 --> 00:16:46.620 align:middle line:84% if I were to write out this whole game tree, 00:16:46.620 --> 00:16:48.840 align:middle line:84% and I want to figure out where I am in the game tree, 00:16:48.840 --> 00:16:51.840 align:middle line:84% what is this contingency going to specify within the tree? 00:16:51.840 --> 00:16:56.140 align:middle line:90% 00:16:56.140 --> 00:17:00.860 align:middle line:84% It's not just enough to specify the offer in the last period. 00:17:00.860 --> 00:17:01.580 align:middle line:90% Yeah? 00:17:01.580 --> 00:17:03.380 align:middle line:84% STUDENT: Maybe that you need to know everyone before? 00:17:03.380 --> 00:17:04.339 align:middle line:84% IAN BALL: Everything that happened before. 00:17:04.339 --> 00:17:07.079 align:middle line:84% Because we know in a tree, when we're at a node in the tree, 00:17:07.079 --> 00:17:09.380 align:middle line:84% that node tells us the entire path through the tree. 00:17:09.380 --> 00:17:12.060 align:middle line:84% So technically, a contingency would be something 00:17:12.060 --> 00:17:15.819 align:middle line:84% like, the plaintiff made this offer in the zeroth period, 00:17:15.819 --> 00:17:17.118 align:middle line:90% and it was rejected. 00:17:17.118 --> 00:17:19.160 align:middle line:84% The defendant made this offer in the next period. 00:17:19.160 --> 00:17:20.000 align:middle line:90% It was rejected. 00:17:20.000 --> 00:17:22.660 align:middle line:84% The plaintiff made this offer, and it was rejected, and so on. 00:17:22.660 --> 00:17:25.500 align:middle line:84% But what I implicitly mean here is 00:17:25.500 --> 00:17:27.589 align:middle line:84% there's a lot of different contingencies, 00:17:27.589 --> 00:17:29.470 align:middle line:84% but the incentives of the plaintiff 00:17:29.470 --> 00:17:32.070 align:middle line:84% are going to be the same at all the contingencies where 00:17:32.070 --> 00:17:33.370 align:middle line:90% this is the offer that's made. 00:17:33.370 --> 00:17:35.328 align:middle line:84% So I'm basically grouping all the contingencies 00:17:35.328 --> 00:17:37.430 align:middle line:84% in my head and reasoning about all of them 00:17:37.430 --> 00:17:38.850 align:middle line:90% together at the same time. 00:17:38.850 --> 00:17:42.633 align:middle line:84% Because at this point, if you're offering me S2n minus 1, 00:17:42.633 --> 00:17:45.050 align:middle line:84% I don't really care what you offered me three periods ago. 00:17:45.050 --> 00:17:46.050 align:middle line:90% That's not on the table. 00:17:46.050 --> 00:17:48.050 align:middle line:84% So it's not relevant to my optimal decision. 00:17:48.050 --> 00:17:49.790 align:middle line:84% But if I wanted to be really precise, 00:17:49.790 --> 00:17:52.110 align:middle line:84% I could, in principle, behave differently 00:17:52.110 --> 00:17:54.950 align:middle line:90% at different contingencies. 00:17:54.950 --> 00:17:58.030 align:middle line:84% So now let's carefully write down my payoffs. 00:17:58.030 --> 00:18:03.110 align:middle line:84% If I accept this offer as the plaintiff, 00:18:03.110 --> 00:18:05.870 align:middle line:84% what are my payoffs going to be, my total payoffs, 00:18:05.870 --> 00:18:08.170 align:middle line:84% taking into account the settlement, my legal costs, 00:18:08.170 --> 00:18:08.930 align:middle line:90% all these things? 00:18:08.930 --> 00:18:15.970 align:middle line:90% 00:18:15.970 --> 00:18:16.470 align:middle line:90% Yeah? 00:18:16.470 --> 00:18:24.785 align:middle line:84% STUDENT: [INAUDIBLE] be s2n minus 1 minus Cp times 2n S2. 00:18:24.785 --> 00:18:25.410 align:middle line:90% IAN BALL: Yeah. 00:18:25.410 --> 00:18:29.150 align:middle line:84% I think 2n minus 1 because we started at 0. 00:18:29.150 --> 00:18:30.730 align:middle line:90% Well, let's go over it. 00:18:30.730 --> 00:18:31.230 align:middle line:90% Yeah? 00:18:31.230 --> 00:18:34.810 align:middle line:84% STUDENT: That would be-- the 2n minus 2 is the 2n minus 00:18:34.810 --> 00:18:35.725 align:middle line:90% [INAUDIBLE]. 00:18:35.725 --> 00:18:36.350 align:middle line:90% IAN BALL: Yeah. 00:18:36.350 --> 00:18:38.610 align:middle line:84% So it'll be-- so I'll just write it, and then I'll explain. 00:18:38.610 --> 00:18:39.443 align:middle line:90% But yeah, it's easy. 00:18:39.443 --> 00:18:41.750 align:middle line:84% And I might get confused with some of these fence posts 00:18:41.750 --> 00:18:42.250 align:middle line:90% issues. 00:18:42.250 --> 00:18:46.690 align:middle line:84% But I think it's going to be Cp times 2n. 00:18:46.690 --> 00:18:50.270 align:middle line:90% 00:18:50.270 --> 00:18:50.930 align:middle line:90% Why is this? 00:18:50.930 --> 00:18:53.350 align:middle line:90% Well, we're in period n minus 1. 00:18:53.350 --> 00:18:56.510 align:middle line:84% If I accept the offer, then I'm going 00:18:56.510 --> 00:18:58.950 align:middle line:84% to get a settlement of S2n and minus 1. 00:18:58.950 --> 00:19:01.890 align:middle line:84% But I've also had to pay my legal fees. 00:19:01.890 --> 00:19:05.070 align:middle line:84% I'm the plaintiff, so my legal fees are Cp per period. 00:19:05.070 --> 00:19:08.390 align:middle line:84% And we negotiated in period 0 and period 1 and period 2 00:19:08.390 --> 00:19:11.570 align:middle line:84% and period 3, all the way up to period 2n minus 1. 00:19:11.570 --> 00:19:13.490 align:middle line:84% You might say that's 2n minus 1 periods. 00:19:13.490 --> 00:19:14.850 align:middle line:90% But remember, we started at 0. 00:19:14.850 --> 00:19:22.040 align:middle line:84% So we actually have 2n periods of negotiation costs. 00:19:22.040 --> 00:19:24.760 align:middle line:84% And then what if I reject the offer? 00:19:24.760 --> 00:19:25.780 align:middle line:90% What's going to happen? 00:19:25.780 --> 00:19:29.000 align:middle line:90% 00:19:29.000 --> 00:19:29.640 align:middle line:90% Yeah? 00:19:29.640 --> 00:19:30.813 align:middle line:90% STUDENT: J minus p hat. 00:19:30.813 --> 00:19:31.480 align:middle line:90% IAN BALL: Right. 00:19:31.480 --> 00:19:33.273 align:middle line:90% And we can't forget-- 00:19:33.273 --> 00:19:34.940 align:middle line:84% I've already-- I mean, it's a sunk cost, 00:19:34.940 --> 00:19:36.190 align:middle line:90% but then we have this as well. 00:19:36.190 --> 00:19:42.400 align:middle line:84% So it's going to be J minus Cp hat minus cp times 2n. 00:19:42.400 --> 00:19:43.260 align:middle line:90% What does this mean? 00:19:43.260 --> 00:19:47.600 align:middle line:84% Well, if I reject the offer, we're going to go to court. 00:19:47.600 --> 00:19:50.360 align:middle line:84% I get a judgment of J, as the plaintiff. 00:19:50.360 --> 00:19:52.860 align:middle line:84% I have to additionally pay all the court costs. 00:19:52.860 --> 00:19:54.780 align:middle line:84% And then I have all these costs from before. 00:19:54.780 --> 00:19:56.822 align:middle line:84% But I think what people are kind of jumping ahead 00:19:56.822 --> 00:19:58.140 align:middle line:90% is these don't really matter. 00:19:58.140 --> 00:20:00.200 align:middle line:90% So we can circle these. 00:20:00.200 --> 00:20:03.920 align:middle line:84% At this point, these are what are called sunk costs. 00:20:03.920 --> 00:20:06.380 align:middle line:84% I've already paid all of these costs. 00:20:06.380 --> 00:20:09.140 align:middle line:84% So in my decision today of whether to accept or reject, 00:20:09.140 --> 00:20:11.640 align:middle line:84% these aren't actually going to have any bearing on my choice 00:20:11.640 --> 00:20:12.940 align:middle line:90% because I pay them either way. 00:20:12.940 --> 00:20:13.900 align:middle line:90% I've already paid them. 00:20:13.900 --> 00:20:15.150 align:middle line:90% I've already paid the lawyers. 00:20:15.150 --> 00:20:16.760 align:middle line:84% The only choice I really have is, 00:20:16.760 --> 00:20:23.800 align:middle line:84% am I going to pay this additional trial fee, Cp hat? 00:20:23.800 --> 00:20:25.120 align:middle line:90% Great. 00:20:25.120 --> 00:20:29.600 align:middle line:84% So now if we compare these two payouts, 00:20:29.600 --> 00:20:32.300 align:middle line:84% when should the plaintiff accept the offer? 00:20:32.300 --> 00:20:36.512 align:middle line:84% What kind of offers should the plaintiff accept? 00:20:36.512 --> 00:20:38.220 align:middle line:84% Well, intuitively, they're the plaintiff. 00:20:38.220 --> 00:20:39.020 align:middle line:90% They want more money. 00:20:39.020 --> 00:20:41.478 align:middle line:84% So we always think the plaintiff should accept-- let's just 00:20:41.478 --> 00:20:42.380 align:middle line:90% write this down here. 00:20:42.380 --> 00:20:44.380 align:middle line:84% It's always going to take this form that they're 00:20:44.380 --> 00:20:50.560 align:middle line:84% going to accept, if and only if, or maybe I'll say use 00:20:50.560 --> 00:20:55.480 align:middle line:84% arrows-- if and only if S2n minus 1 00:20:55.480 --> 00:20:57.700 align:middle line:84% is greater than or equal to some threshold. 00:20:57.700 --> 00:21:01.160 align:middle line:84% Let me denote this threshold as S2n minus 1 star. 00:21:01.160 --> 00:21:04.000 align:middle line:84% And this is just-- we know this is generally how things are. 00:21:04.000 --> 00:21:06.787 align:middle line:84% If you're the plaintiff, this is how much you get paid. 00:21:06.787 --> 00:21:08.620 align:middle line:84% Getting offered more money is always better. 00:21:08.620 --> 00:21:10.078 align:middle line:84% So it makes sense that you're going 00:21:10.078 --> 00:21:12.160 align:middle line:84% to accept if the offer is high enough, 00:21:12.160 --> 00:21:14.520 align:middle line:84% and you're going to reject if the offer is too low. 00:21:14.520 --> 00:21:17.970 align:middle line:84% The question is, what is this critical threshold? 00:21:17.970 --> 00:21:19.770 align:middle line:84% What's the amount that you have to offer me 00:21:19.770 --> 00:21:24.930 align:middle line:84% as the plaintiff for me to accept your offer? 00:21:24.930 --> 00:21:25.550 align:middle line:90% Yeah? 00:21:25.550 --> 00:21:27.270 align:middle line:90% STUDENT: J minus Cp hat. 00:21:27.270 --> 00:21:28.690 align:middle line:90% IAN BALL: Right. 00:21:28.690 --> 00:21:30.050 align:middle line:90% J minus Cp hat. 00:21:30.050 --> 00:21:32.347 align:middle line:90% 00:21:32.347 --> 00:21:33.930 align:middle line:84% And you might say, well, we have to be 00:21:33.930 --> 00:21:37.200 align:middle line:84% a little careful if the offer is exactly equal to this. 00:21:37.200 --> 00:21:37.950 align:middle line:90% Maybe I'll accept. 00:21:37.950 --> 00:21:38.537 align:middle line:90% Maybe I won't. 00:21:38.537 --> 00:21:40.370 align:middle line:84% We're going to make the convention that when 00:21:40.370 --> 00:21:41.590 align:middle line:90% indifferent, I always accept. 00:21:41.590 --> 00:21:43.382 align:middle line:84% And this is what we have to do to make sure 00:21:43.382 --> 00:21:44.890 align:middle line:90% that equilibrium exists. 00:21:44.890 --> 00:21:46.890 align:middle line:84% But notice what's interesting here. 00:21:46.890 --> 00:21:48.750 align:middle line:84% What is the plaintiff willing to accept? 00:21:48.750 --> 00:21:50.530 align:middle line:84% They're willing to accept some offers that 00:21:50.530 --> 00:21:52.250 align:middle line:84% are strictly less than the amount 00:21:52.250 --> 00:21:53.930 align:middle line:90% that they would get in court. 00:21:53.930 --> 00:21:55.350 align:middle line:90% So what does this reflect? 00:21:55.350 --> 00:21:57.370 align:middle line:84% This reflects the cost of going to court. 00:21:57.370 --> 00:21:59.730 align:middle line:84% If you say, look, I'm not going to pay you 00:21:59.730 --> 00:22:01.330 align:middle line:90% as much as you'll get in court. 00:22:01.330 --> 00:22:02.413 align:middle line:90% I'll pay you a bit less. 00:22:02.413 --> 00:22:04.330 align:middle line:84% But it's still worth it for you because if you 00:22:04.330 --> 00:22:07.170 align:middle line:84% accept my lowball offer, you get to avoid 00:22:07.170 --> 00:22:09.130 align:middle line:90% the cost of going to court. 00:22:09.130 --> 00:22:11.890 align:middle line:84% So we can see the higher that Cp hat 00:22:11.890 --> 00:22:14.850 align:middle line:84% is, the weaker the bargaining position 00:22:14.850 --> 00:22:17.157 align:middle line:84% of the plaintiff, the lower the offer 00:22:17.157 --> 00:22:19.740 align:middle line:84% that they're going to be willing to accept because they really 00:22:19.740 --> 00:22:23.060 align:middle line:84% don't want to go to court and pay this enormous fee. 00:22:23.060 --> 00:22:28.020 align:middle line:84% So now we go back one step earlier. 00:22:28.020 --> 00:22:30.080 align:middle line:84% We're still in period 2n minus 1. 00:22:30.080 --> 00:22:35.600 align:middle line:84% But now let's think about the defendant's choice. 00:22:35.600 --> 00:22:39.752 align:middle line:90% 00:22:39.752 --> 00:22:41.460 align:middle line:84% Now the question is, how much should they 00:22:41.460 --> 00:22:46.660 align:middle line:84% offer, anticipating that the plaintiff will 00:22:46.660 --> 00:22:50.840 align:middle line:84% accept their offer exactly under these conditions? 00:22:50.840 --> 00:22:54.180 align:middle line:90% 00:22:54.180 --> 00:22:58.220 align:middle line:84% Well, intuitively, they want to make the smallest 00:22:58.220 --> 00:23:00.360 align:middle line:90% offer that will be accepted. 00:23:00.360 --> 00:23:02.920 align:middle line:84% They want to pay the least amount that they have to. 00:23:02.920 --> 00:23:06.260 align:middle line:84% So it's going to be optimal for them to choose S2n minus 1 00:23:06.260 --> 00:23:08.920 align:middle line:90% equals S2n minus 1 star. 00:23:08.920 --> 00:23:12.740 align:middle line:90% 00:23:12.740 --> 00:23:15.850 align:middle line:84% They could offer to pay strictly more than this. 00:23:15.850 --> 00:23:18.100 align:middle line:84% That will still be accepted, but that's worse for them 00:23:18.100 --> 00:23:21.060 align:middle line:84% because they don't want to pay out very much. 00:23:21.060 --> 00:23:23.945 align:middle line:84% I think it's a little tricky, though, because they might also 00:23:23.945 --> 00:23:25.320 align:middle line:84% want their offer to get rejected. 00:23:25.320 --> 00:23:26.940 align:middle line:84% So we still have to be a little careful. 00:23:26.940 --> 00:23:27.440 align:middle line:90% Let's see. 00:23:27.440 --> 00:23:29.000 align:middle line:90% What if the offer gets rejected? 00:23:29.000 --> 00:23:31.415 align:middle line:90% 00:23:31.415 --> 00:23:33.540 align:middle line:84% We're claiming that they want to make an offer that 00:23:33.540 --> 00:23:34.915 align:middle line:84% will be accepted, but we actually 00:23:34.915 --> 00:23:38.180 align:middle line:84% have to see what happens if it gets rejected. 00:23:38.180 --> 00:23:40.100 align:middle line:84% It turns out, if it gets rejected-- 00:23:40.100 --> 00:23:42.740 align:middle line:84% and let's just work this through-- 00:23:42.740 --> 00:23:51.780 align:middle line:84% this will be accepted, and then they will get minus S2n minus 1 00:23:51.780 --> 00:23:55.880 align:middle line:90% star minus cD 2n. 00:23:55.880 --> 00:23:58.608 align:middle line:90% 00:23:58.608 --> 00:24:00.900 align:middle line:84% And what if they were to make some different offer that 00:24:00.900 --> 00:24:02.620 align:middle line:90% were rejected? 00:24:02.620 --> 00:24:05.840 align:middle line:90% What would their payout-- 00:24:05.840 --> 00:24:08.530 align:middle line:90% what would their payoff be? 00:24:08.530 --> 00:24:09.030 align:middle line:90% Yeah? 00:24:09.030 --> 00:24:14.600 align:middle line:84% STUDENT: Minus J [INAUDIBLE] 2n minus C hat. 00:24:14.600 --> 00:24:15.350 align:middle line:90% IAN BALL: Exactly. 00:24:15.350 --> 00:24:17.610 align:middle line:84% So if they make an offer that's rejected, 00:24:17.610 --> 00:24:19.070 align:middle line:84% then they're going to go to court. 00:24:19.070 --> 00:24:21.950 align:middle line:84% And that means they're going to lose J. 00:24:21.950 --> 00:24:37.070 align:middle line:84% They're going to lose CD hat, and they're going to lose CD 2n 00:24:37.070 --> 00:24:39.270 align:middle line:90% But this is clearly worse. 00:24:39.270 --> 00:24:42.270 align:middle line:84% Because if they go to court, they have to pay more. 00:24:42.270 --> 00:24:44.850 align:middle line:84% They pay J rather than J minus Cp hat, 00:24:44.850 --> 00:24:46.530 align:middle line:84% and they have to pay the cost of court. 00:24:46.530 --> 00:24:48.870 align:middle line:90% So this is kind of extra bad. 00:24:48.870 --> 00:24:54.850 align:middle line:84% So they pay more, and they pay the court costs. 00:24:54.850 --> 00:24:59.590 align:middle line:90% 00:24:59.590 --> 00:25:03.285 align:middle line:84% And we'll see throughout that it's never really going 00:25:03.285 --> 00:25:04.910 align:middle line:84% to be optimal for people to make offers 00:25:04.910 --> 00:25:06.190 align:middle line:90% that are going to be rejected. 00:25:06.190 --> 00:25:09.550 align:middle line:84% It's always just going to make things worse for them. 00:25:09.550 --> 00:25:13.050 align:middle line:84% So indeed, this is going to be the best choice for CD. 00:25:13.050 --> 00:25:16.450 align:middle line:90% 00:25:16.450 --> 00:25:22.790 align:middle line:84% Now let's try to go one step back and see what's going 00:25:22.790 --> 00:25:26.830 align:middle line:90% to happen in period 2n minus 2. 00:25:26.830 --> 00:25:28.970 align:middle line:84% So maybe I'll come back over here. 00:25:28.970 --> 00:25:32.630 align:middle line:90% 00:25:32.630 --> 00:25:34.970 align:middle line:84% And then hopefully, we'll be able to see a pattern. 00:25:34.970 --> 00:25:41.390 align:middle line:90% 00:25:41.390 --> 00:25:47.530 align:middle line:84% So now let's go to period 2n minus 2. 00:25:47.530 --> 00:25:50.350 align:middle line:90% 00:25:50.350 --> 00:25:57.110 align:middle line:90% And our conjecture is that the-- 00:25:57.110 --> 00:26:00.230 align:middle line:84% I have to do it right here-- the defendant is 00:26:00.230 --> 00:26:07.150 align:middle line:84% going to accept an offer of S2n minus 2, if and only if, 00:26:07.150 --> 00:26:08.500 align:middle line:90% S2n minus 2. 00:26:08.500 --> 00:26:09.400 align:middle line:90% Well, let's see. 00:26:09.400 --> 00:26:13.940 align:middle line:84% I'm the defendant now, so I'm willing to accept low offers. 00:26:13.940 --> 00:26:17.280 align:middle line:84% The plaintiff was willing to accept high offers. 00:26:17.280 --> 00:26:21.560 align:middle line:84% The defendant is willing to accept low offers. 00:26:21.560 --> 00:26:25.080 align:middle line:84% So there's going to be some threshold, S2n minus 2 star, 00:26:25.080 --> 00:26:28.320 align:middle line:84% such that the defendant will accept if they're offered 00:26:28.320 --> 00:26:30.453 align:middle line:90% to pay a small enough amount. 00:26:30.453 --> 00:26:31.620 align:middle line:90% Is there an issue over here? 00:26:31.620 --> 00:26:31.840 align:middle line:90% Yeah? 00:26:31.840 --> 00:26:33.215 align:middle line:84% STUDENT: Yeah, I have a question. 00:26:33.215 --> 00:26:34.023 align:middle line:90% IAN BALL: Yes, 00:26:34.023 --> 00:26:35.440 align:middle line:84% STUDENT: Basically, the comparison 00:26:35.440 --> 00:26:38.620 align:middle line:84% between [INAUDIBLE] giving an offer and being rejected. 00:26:38.620 --> 00:26:42.280 align:middle line:84% So getting an offer, they have to pay S2m minus 1 star, which 00:26:42.280 --> 00:26:44.800 align:middle line:90% is J minus Cp versus-- 00:26:44.800 --> 00:26:46.655 align:middle line:90% oh, J plus. 00:26:46.655 --> 00:26:47.280 align:middle line:90% IAN BALL: Yeah. 00:26:47.280 --> 00:26:49.030 align:middle line:84% So we have to be careful with the minuses. 00:26:49.030 --> 00:26:56.140 align:middle line:84% So this is going to be minus J plus Cp hat. 00:26:56.140 --> 00:26:58.185 align:middle line:84% STUDENT: So does it depend on how Cp is? 00:26:58.185 --> 00:26:59.060 align:middle line:90% IAN BALL: It doesn't. 00:26:59.060 --> 00:26:59.920 align:middle line:84% And that's why-- and this is what 00:26:59.920 --> 00:27:02.040 align:middle line:84% I meant about it's kind of doubly bad for them. 00:27:02.040 --> 00:27:05.100 align:middle line:84% Because if their offer is rejected, 00:27:05.100 --> 00:27:08.230 align:middle line:84% they're going to have to pay the full settlement, J, in court. 00:27:08.230 --> 00:27:11.630 align:middle line:84% So going the defendant will have to pay more in court. 00:27:11.630 --> 00:27:14.472 align:middle line:84% And on top of that, they'll have to pay their legal fees. 00:27:14.472 --> 00:27:15.930 align:middle line:84% So the difference between these two 00:27:15.930 --> 00:27:18.510 align:middle line:84% is actually the sum of Cp hat and Cd hat. 00:27:18.510 --> 00:27:20.430 align:middle line:84% It's not about the relationship between them. 00:27:20.430 --> 00:27:20.650 align:middle line:90% Yeah. 00:27:20.650 --> 00:27:21.350 align:middle line:90% Good point. 00:27:21.350 --> 00:27:23.058 align:middle line:84% So just to be clear what the argument is. 00:27:23.058 --> 00:27:26.070 align:middle line:84% The argument is, I could make some offer that's rejected. 00:27:26.070 --> 00:27:28.630 align:middle line:84% If it's rejected, the amount that I offer doesn't matter. 00:27:28.630 --> 00:27:30.430 align:middle line:90% All that matters, it's rejected. 00:27:30.430 --> 00:27:33.450 align:middle line:84% And we've shown that that's worse than making this offer. 00:27:33.450 --> 00:27:36.478 align:middle line:84% The alternative is I can make an offer that will be accepted. 00:27:36.478 --> 00:27:38.270 align:middle line:84% But to make an offer that will be accepted, 00:27:38.270 --> 00:27:39.972 align:middle line:84% I'd have to offer even more than this. 00:27:39.972 --> 00:27:41.430 align:middle line:84% And that is certainly worse for me. 00:27:41.430 --> 00:27:44.490 align:middle line:84% So the best thing I can do is offer the smallest amount 00:27:44.490 --> 00:27:46.110 align:middle line:90% that will be accepted. 00:27:46.110 --> 00:27:53.150 align:middle line:90% 00:27:53.150 --> 00:28:07.970 align:middle line:84% So now what our guess is that the structure in period 2n minus 00:28:07.970 --> 00:28:09.690 align:middle line:84% 2 is going to be something like this. 00:28:09.690 --> 00:28:12.148 align:middle line:84% The defendant is going to have some threshold where they're 00:28:12.148 --> 00:28:14.450 align:middle line:84% going to say, look, I'll accept the offer if 00:28:14.450 --> 00:28:16.370 align:middle line:84% and only if it's small enough, if and only 00:28:16.370 --> 00:28:18.385 align:middle line:90% if it's below some threshold. 00:28:18.385 --> 00:28:20.510 align:middle line:84% And then let's think about the plaintiff's problem. 00:28:20.510 --> 00:28:22.218 align:middle line:84% We could go through the algebra to check. 00:28:22.218 --> 00:28:27.370 align:middle line:84% But you can see that the plaintiff wants the offer 00:28:27.370 --> 00:28:30.490 align:middle line:84% to be, the settlement to be as large as possible. 00:28:30.490 --> 00:28:33.010 align:middle line:84% But if it's too large, it won't be accepted. 00:28:33.010 --> 00:28:36.010 align:middle line:84% So they want to make the largest demand, the largest settlement 00:28:36.010 --> 00:28:37.250 align:middle line:90% that will be accepted. 00:28:37.250 --> 00:28:40.770 align:middle line:84% And that's precisely S2n minus 2 star. 00:28:40.770 --> 00:28:43.273 align:middle line:84% So all we have to do is find this number. 00:28:43.273 --> 00:28:44.690 align:middle line:84% And that's how these games reduce. 00:28:44.690 --> 00:28:47.570 align:middle line:84% At each stage, one side is going to accept if and only 00:28:47.570 --> 00:28:50.430 align:middle line:84% if the offer is above or below some threshold. 00:28:50.430 --> 00:28:52.830 align:middle line:84% And the name of the game is finding these thresholds. 00:28:52.830 --> 00:28:56.010 align:middle line:90% 00:28:56.010 --> 00:28:56.910 align:middle line:90% So let's see. 00:28:56.910 --> 00:29:03.140 align:middle line:84% Well, let's try to understand what this threshold needs to be, 00:29:03.140 --> 00:29:04.580 align:middle line:90% by let's-- 00:29:04.580 --> 00:29:06.020 align:middle line:90% what's going up here. 00:29:06.020 --> 00:29:07.500 align:middle line:84% So the threshold needs to be such 00:29:07.500 --> 00:29:11.620 align:middle line:84% that they're indifferent between accepting this offer 00:29:11.620 --> 00:29:12.822 align:middle line:90% and rejecting it. 00:29:12.822 --> 00:29:14.280 align:middle line:84% That has to be true at a threshold. 00:29:14.280 --> 00:29:15.560 align:middle line:84% Because if they weren't indifferent, 00:29:15.560 --> 00:29:17.500 align:middle line:84% there's no way that could be the threshold. 00:29:17.500 --> 00:29:26.580 align:middle line:84% So if they accept this offer, what are they going to get? 00:29:26.580 --> 00:29:29.180 align:middle line:84% They're the defendant, so they have to pay it. 00:29:29.180 --> 00:29:34.740 align:middle line:84% So they're going to get minus S2n minus 2 star 00:29:34.740 --> 00:29:38.660 align:middle line:84% minus all the legal fees that they have. 00:29:38.660 --> 00:29:44.660 align:middle line:84% So that's minus CD times 2n minus 1. 00:29:44.660 --> 00:29:46.120 align:middle line:90% So now we're one period earlier. 00:29:46.120 --> 00:29:47.660 align:middle line:84% So if we accept a settlement here, 00:29:47.660 --> 00:29:50.780 align:middle line:84% we don't have to pay legal fees in the last period. 00:29:50.780 --> 00:29:53.200 align:middle line:84% And this is how much we're going to have to pay. 00:29:53.200 --> 00:29:55.100 align:middle line:84% So we have a negative sign in front. 00:29:55.100 --> 00:29:58.208 align:middle line:84% What happens if we reject this offer? 00:29:58.208 --> 00:30:00.500 align:middle line:84% Well, this is where backward induction comes into play. 00:30:00.500 --> 00:30:02.260 align:middle line:84% Because in general, I don't know what 00:30:02.260 --> 00:30:04.720 align:middle line:84% happens, what my payoff is from rejecting this offer, 00:30:04.720 --> 00:30:06.620 align:middle line:84% because it depends on what's then offered 00:30:06.620 --> 00:30:07.800 align:middle line:90% to me in the next period. 00:30:07.800 --> 00:30:09.500 align:middle line:84% But because we've already used backward induction 00:30:09.500 --> 00:30:10.700 align:middle line:84% in the next period, we know what's 00:30:10.700 --> 00:30:12.158 align:middle line:84% going to happen in the next period. 00:30:12.158 --> 00:30:14.860 align:middle line:84% In the next period, the plaintiff 00:30:14.860 --> 00:30:17.700 align:middle line:90% is going to offer exactly this. 00:30:17.700 --> 00:30:19.500 align:middle line:84% And then it's going to be optimal for me, 00:30:19.500 --> 00:30:21.500 align:middle line:90% as the defendant, to accept it. 00:30:21.500 --> 00:30:23.820 align:middle line:84% So if I wait till the next period, 00:30:23.820 --> 00:30:28.660 align:middle line:84% I'm going to have to pay negative S2n minus 1 star 00:30:28.660 --> 00:30:30.500 align:middle line:84% minus-- what are my legal fees if I 00:30:30.500 --> 00:30:32.560 align:middle line:90% accept this in the next period? 00:30:32.560 --> 00:30:36.240 align:middle line:90% 00:30:36.240 --> 00:30:36.740 align:middle line:90% Yeah? 00:30:36.740 --> 00:30:38.397 align:middle line:84% STUDENT: [INAUDIBLE] CD times 2n. 00:30:38.397 --> 00:30:39.480 align:middle line:90% IAN BALL: Times 2n, right? 00:30:39.480 --> 00:30:41.522 align:middle line:84% Because if I reject today and wait till tomorrow, 00:30:41.522 --> 00:30:44.120 align:middle line:84% I have one more period of negotiation. 00:30:44.120 --> 00:30:46.960 align:middle line:84% So that means it's going to be CD times 2n. 00:30:46.960 --> 00:30:49.500 align:middle line:90% 00:30:49.500 --> 00:30:51.300 align:middle line:90% And now we see the pattern. 00:30:51.300 --> 00:30:52.860 align:middle line:90% What are we going to find? 00:30:52.860 --> 00:30:59.150 align:middle line:84% We're going to find that S2n minus 2 star 00:30:59.150 --> 00:31:13.390 align:middle line:90% equals S2n minus 1 star plus CD. 00:31:13.390 --> 00:31:14.170 align:middle line:90% That's right. 00:31:14.170 --> 00:31:25.150 align:middle line:90% 00:31:25.150 --> 00:31:35.230 align:middle line:84% Yes, that's right, which we can do our math over here. 00:31:35.230 --> 00:31:41.810 align:middle line:84% And this is J minus Cp hat plus CD. 00:31:41.810 --> 00:31:46.270 align:middle line:90% 00:31:46.270 --> 00:31:47.130 align:middle line:90% Why is it a plus? 00:31:47.130 --> 00:31:50.310 align:middle line:84% We could try to go through the algebra. 00:31:50.310 --> 00:31:56.280 align:middle line:84% But can we just reason through this, why we get a plus CD here? 00:31:56.280 --> 00:31:59.280 align:middle line:84% Why am I, as the defendant, willing to accept 00:31:59.280 --> 00:32:02.240 align:middle line:84% a settlement that's strictly higher than the settlement I 00:32:02.240 --> 00:32:03.580 align:middle line:90% would get tomorrow? 00:32:03.580 --> 00:32:04.080 align:middle line:90% Yeah? 00:32:04.080 --> 00:32:07.180 align:middle line:84% STUDENT: Because you're not going to tomorrow. 00:32:07.180 --> 00:32:10.180 align:middle line:84% So you're not going to have the CD from the next period. 00:32:10.180 --> 00:32:11.440 align:middle line:90% So you're adding it back. 00:32:11.440 --> 00:32:12.440 align:middle line:90% IAN BALL: Exactly right. 00:32:12.440 --> 00:32:14.100 align:middle line:90% So what am I saying? 00:32:14.100 --> 00:32:16.100 align:middle line:84% I'm saying, look, I can wait till tomorrow, 00:32:16.100 --> 00:32:17.720 align:middle line:84% and then I have to pay you this much. 00:32:17.720 --> 00:32:20.160 align:middle line:84% That's going to be the outcome of negotiation tomorrow. 00:32:20.160 --> 00:32:21.940 align:middle line:84% But if I pay you an amount today, 00:32:21.940 --> 00:32:25.100 align:middle line:84% I save one unit of negotiation costs. 00:32:25.100 --> 00:32:27.680 align:middle line:84% So I'm actually willing to pay more today, I'm 00:32:27.680 --> 00:32:29.320 align:middle line:84% willing to get a worse offer today 00:32:29.320 --> 00:32:33.960 align:middle line:84% because I'm saving the legal costs of going tomorrow. 00:32:33.960 --> 00:32:37.960 align:middle line:84% And you can see how these enter differently. 00:32:37.960 --> 00:32:40.880 align:middle line:84% The plaintiff was also willing to accept a worse 00:32:40.880 --> 00:32:42.760 align:middle line:90% offer to save on legal costs. 00:32:42.760 --> 00:32:45.040 align:middle line:84% But for the plaintiff, a worse offer 00:32:45.040 --> 00:32:48.000 align:middle line:84% is a smaller settlement because they get paid the settlement. 00:32:48.000 --> 00:32:50.560 align:middle line:84% For the defendant, they're willing to accept a worse 00:32:50.560 --> 00:32:51.900 align:middle line:90% offer to save on legal fees. 00:32:51.900 --> 00:32:53.823 align:middle line:84% But for them, a worse offer means paying more. 00:32:53.823 --> 00:32:55.740 align:middle line:84% So if I'm the defendant, I'm basically saying, 00:32:55.740 --> 00:32:59.000 align:middle line:84% look, I'll pay you off to save me the money of going to court. 00:32:59.000 --> 00:33:02.460 align:middle line:84% And the plaintiff is saying, you better pay me a lot. 00:33:02.460 --> 00:33:05.580 align:middle line:90% 00:33:05.580 --> 00:33:06.080 align:middle line:90% Sorry. 00:33:06.080 --> 00:33:07.640 align:middle line:84% I'll accept not paying you very much 00:33:07.640 --> 00:33:11.480 align:middle line:84% in order to avoid going to court. 00:33:11.480 --> 00:33:13.640 align:middle line:84% I'll accept you not paying me very much 00:33:13.640 --> 00:33:16.560 align:middle line:90% to avoid going to court. 00:33:16.560 --> 00:33:20.740 align:middle line:84% So now I think we can keep doing this all day. 00:33:20.740 --> 00:33:22.740 align:middle line:84% But I think we're going to see a pattern emerge. 00:33:22.740 --> 00:33:25.040 align:middle line:84% So let's see if we can guess what the pattern is going 00:33:25.040 --> 00:33:28.220 align:middle line:90% to be for these thresholds. 00:33:28.220 --> 00:33:33.240 align:middle line:90% 00:33:33.240 --> 00:33:35.880 align:middle line:84% So it's always easy to get off by one. 00:33:35.880 --> 00:33:36.960 align:middle line:90% Let's see. 00:33:36.960 --> 00:33:37.860 align:middle line:90% I've done it right. 00:33:37.860 --> 00:33:40.880 align:middle line:90% Let's look at the-- 00:33:40.880 --> 00:33:43.120 align:middle line:90% we started in an odd period. 00:33:43.120 --> 00:33:48.280 align:middle line:84% So we have the odd periods and the even periods. 00:33:48.280 --> 00:33:59.010 align:middle line:84% In the odd periods, the defendant is going to propose S. 00:33:59.010 --> 00:34:12.130 align:middle line:84% Let's say the odd period is 2n minus 2k minus 1. 00:34:12.130 --> 00:34:16.929 align:middle line:84% And in the even period, it's 2n minus 2k. 00:34:16.929 --> 00:34:22.389 align:middle line:84% And here, we're going to have k greater than or equal to 1. 00:34:22.389 --> 00:34:25.570 align:middle line:90% 00:34:25.570 --> 00:34:28.710 align:middle line:84% So here, let's just write down the formula 00:34:28.710 --> 00:34:30.850 align:middle line:84% and then check that we have it right. 00:34:30.850 --> 00:34:37.330 align:middle line:84% I think the defendant is going to propose J minus. 00:34:37.330 --> 00:34:48.010 align:middle line:84% Cp hat plus k minus 1 of k CD minus CP. 00:34:48.010 --> 00:35:00.290 align:middle line:84% And here, I think the plaintiff is going 00:35:00.290 --> 00:35:08.670 align:middle line:90% to propose one step earlier. 00:35:08.670 --> 00:35:14.570 align:middle line:90% 00:35:14.570 --> 00:35:16.610 align:middle line:90% Let's check if I did this right. 00:35:16.610 --> 00:35:21.450 align:middle line:90% So let's first try k equals 1. 00:35:21.450 --> 00:35:25.090 align:middle line:84% If k equals 1, we're in period 2n minus 1. 00:35:25.090 --> 00:35:31.090 align:middle line:84% So we better get J minus CP hat, And that's exactly what we get. 00:35:31.090 --> 00:35:41.170 align:middle line:84% If k equals 1 down here, we're in 2n minus 2, 00:35:41.170 --> 00:35:45.070 align:middle line:84% and we better get J minus CP hat plus CD. 00:35:45.070 --> 00:35:47.630 align:middle line:90% So think that looks correct. 00:35:47.630 --> 00:35:54.500 align:middle line:90% 00:35:54.500 --> 00:35:58.620 align:middle line:84% And then let's just check what's happening here-- 00:35:58.620 --> 00:36:03.980 align:middle line:84% accepts if and only if, and here the defendant 00:36:03.980 --> 00:36:07.300 align:middle line:90% accepts if and only if. 00:36:07.300 --> 00:36:10.940 align:middle line:84% Well, in these periods they're going to accept if and only if. 00:36:10.940 --> 00:36:14.780 align:middle line:84% Well, the plaintiff is the one who receives the money, 00:36:14.780 --> 00:36:17.500 align:middle line:84% is only going to accept it if it's large enough, if 00:36:17.500 --> 00:36:19.500 align:middle line:84% and only if the settlement is greater than 00:36:19.500 --> 00:36:22.340 align:middle line:90% or equal to this number here. 00:36:22.340 --> 00:36:24.340 align:middle line:84% And the defendant is going to accept if and only 00:36:24.340 --> 00:36:27.160 align:middle line:84% if f is less than or equal to this number here. 00:36:27.160 --> 00:36:30.940 align:middle line:90% 00:36:30.940 --> 00:36:35.180 align:middle line:84% So the defendant is proposing the smallest settlement amount 00:36:35.180 --> 00:36:38.420 align:middle line:84% that will be accepted because they like low settlement 00:36:38.420 --> 00:36:40.100 align:middle line:90% amounts. 00:36:40.100 --> 00:36:43.420 align:middle line:84% And here, the plaintiff is proposing the largest settlement 00:36:43.420 --> 00:36:46.260 align:middle line:84% amount that will be accepted because they like 00:36:46.260 --> 00:36:48.320 align:middle line:90% to receive high settlements. 00:36:48.320 --> 00:36:52.900 align:middle line:90% 00:36:52.900 --> 00:36:56.620 align:middle line:90% So this is the strategies. 00:36:56.620 --> 00:36:59.820 align:middle line:84% We specified all the strategies at every contingency. 00:36:59.820 --> 00:37:02.220 align:middle line:84% But let's now compute what actually happens. 00:37:02.220 --> 00:37:06.040 align:middle line:84% The strategies say, in every contingency what do people do? 00:37:06.040 --> 00:37:08.870 align:middle line:84% But what actually happens in this big game? 00:37:08.870 --> 00:37:10.120 align:middle line:90% We made it really complicated. 00:37:10.120 --> 00:37:13.800 align:middle line:84% But what happens is actually going to be pretty simple. 00:37:13.800 --> 00:37:18.940 align:middle line:90% 00:37:18.940 --> 00:37:24.040 align:middle line:84% You want to see what is actually going to happen? 00:37:24.040 --> 00:37:28.420 align:middle line:90% 00:37:28.420 --> 00:37:29.958 align:middle line:84% So when are they going to settle? 00:37:29.958 --> 00:37:31.000 align:middle line:90% Are they going to settle? 00:37:31.000 --> 00:37:37.900 align:middle line:90% 00:37:37.900 --> 00:37:38.680 align:middle line:90% Do we go to trial? 00:37:38.680 --> 00:37:39.160 align:middle line:90% What happens? 00:37:39.160 --> 00:37:40.243 align:middle line:90% We have the strategy here. 00:37:40.243 --> 00:37:41.797 align:middle line:84% We just have to figure out what's 00:37:41.797 --> 00:37:42.880 align:middle line:90% actually going to be done. 00:37:42.880 --> 00:37:52.510 align:middle line:90% 00:37:52.510 --> 00:37:55.190 align:middle line:84% Well, let's start in the first period, and let's go through it. 00:37:55.190 --> 00:37:59.170 align:middle line:84% In period 0, what is going to happen? 00:37:59.170 --> 00:38:03.310 align:middle line:90% Well, period 0 means k equals n. 00:38:03.310 --> 00:38:06.790 align:middle line:84% So the plaintiff is going to propose this settlement amount, 00:38:06.790 --> 00:38:10.770 align:middle line:84% J minus Cp hat plus n minus 1 times this plus Cd. 00:38:10.770 --> 00:38:14.830 align:middle line:90% 00:38:14.830 --> 00:38:17.830 align:middle line:84% And then the defendant's going to accept it. 00:38:17.830 --> 00:38:19.390 align:middle line:90% So that's it. 00:38:19.390 --> 00:38:21.230 align:middle line:84% So the outcome is actually very simple. 00:38:21.230 --> 00:38:28.670 align:middle line:84% The outcome is we settle in period 0 for this amount 00:38:28.670 --> 00:38:29.290 align:middle line:90% down here. 00:38:29.290 --> 00:38:50.662 align:middle line:90% 00:38:50.662 --> 00:38:51.620 align:middle line:90% Yeah, this looks right. 00:38:51.620 --> 00:38:54.640 align:middle line:84% I'm just making sure I'm not off by one index. 00:38:54.640 --> 00:38:57.580 align:middle line:84% So can we try to interpret this amount? 00:38:57.580 --> 00:39:00.200 align:middle line:90% 00:39:00.200 --> 00:39:02.380 align:middle line:84% How does this amount compare to J? 00:39:02.380 --> 00:39:16.320 align:middle line:90% 00:39:16.320 --> 00:39:19.200 align:middle line:84% Well, we see that the settlement is 00:39:19.200 --> 00:39:25.080 align:middle line:84% going to be smaller if Cp hat is really big. 00:39:25.080 --> 00:39:27.400 align:middle line:84% So this settlement is going to tend 00:39:27.400 --> 00:39:35.200 align:middle line:90% to be smaller if Cp hat is big. 00:39:35.200 --> 00:39:42.360 align:middle line:84% And it's going to tend to be bigger if Cd minus Cp is large. 00:39:42.360 --> 00:39:53.440 align:middle line:90% 00:39:53.440 --> 00:39:55.060 align:middle line:84% So do we have intuition for this? 00:39:55.060 --> 00:39:58.040 align:middle line:90% 00:39:58.040 --> 00:39:59.160 align:middle line:90% Well, let's see. 00:39:59.160 --> 00:40:03.760 align:middle line:84% If Cd minus Cp is large, that means the defendant 00:40:03.760 --> 00:40:07.400 align:middle line:90% has higher negotiation costs. 00:40:07.400 --> 00:40:11.520 align:middle line:84% And that means the defendant is at a strategic disadvantage. 00:40:11.520 --> 00:40:14.840 align:middle line:84% The fact that they have high negotiation costs 00:40:14.840 --> 00:40:18.343 align:middle line:84% means the other side can drive a hard bargain. 00:40:18.343 --> 00:40:19.760 align:middle line:84% And because of that, the defendant 00:40:19.760 --> 00:40:22.800 align:middle line:84% is going to have to end up paying a larger settlement. 00:40:22.800 --> 00:40:26.180 align:middle line:84% Notice that it's not only big if Cd minus Cp is big, 00:40:26.180 --> 00:40:28.600 align:middle line:84% but it also depends on how far away the trial is. 00:40:28.600 --> 00:40:31.120 align:middle line:84% So if the trial is really far away, 00:40:31.120 --> 00:40:34.360 align:middle line:84% then the defendant is really in trouble 00:40:34.360 --> 00:40:37.323 align:middle line:84% because the defendant's higher legal fees per period 00:40:37.323 --> 00:40:39.490 align:middle line:84% could really add up over the many, many periods that 00:40:39.490 --> 00:40:41.330 align:middle line:90% going to trial. 00:40:41.330 --> 00:40:44.330 align:middle line:84% On the other hand, if the plaintiffs cost 00:40:44.330 --> 00:40:47.050 align:middle line:84% of going to trial is really, really high, 00:40:47.050 --> 00:40:50.050 align:middle line:84% then the settlement is going to be smaller because this is going 00:40:50.050 --> 00:40:52.430 align:middle line:90% to advantage the defendant. 00:40:52.430 --> 00:40:53.930 align:middle line:84% They're not going to have to pay out 00:40:53.930 --> 00:40:57.090 align:middle line:84% as much if they can drive a hard bargain to the plaintiff 00:40:57.090 --> 00:40:58.770 align:middle line:90% in the last period. 00:40:58.770 --> 00:41:00.250 align:middle line:90% But here's one kind of puzzle. 00:41:00.250 --> 00:41:01.790 align:middle line:90% What happened to Cd hat? 00:41:01.790 --> 00:41:08.490 align:middle line:90% 00:41:08.490 --> 00:41:09.690 align:middle line:90% It doesn't appear at all. 00:41:09.690 --> 00:41:15.030 align:middle line:84% So why does it not depend at all on Cd hat? 00:41:15.030 --> 00:41:19.850 align:middle line:90% 00:41:19.850 --> 00:41:20.490 align:middle line:90% Yeah? 00:41:20.490 --> 00:41:22.365 align:middle line:84% STUDENT: I'd say that the plaintiff is always 00:41:22.365 --> 00:41:24.810 align:middle line:84% going to accept on the last day, so the cost of trial 00:41:24.810 --> 00:41:26.792 align:middle line:90% is never realized. 00:41:26.792 --> 00:41:27.750 align:middle line:90% IAN BALL: That's right. 00:41:27.750 --> 00:41:31.130 align:middle line:84% But we also never get to the fifth period. 00:41:31.130 --> 00:41:33.998 align:middle line:84% But the costs that we would incur 00:41:33.998 --> 00:41:35.790 align:middle line:84% if we get to the fifth period also show up. 00:41:35.790 --> 00:41:37.830 align:middle line:84% So I think that's part of the way there. 00:41:37.830 --> 00:41:38.650 align:middle line:90% Yeah? 00:41:38.650 --> 00:41:41.030 align:middle line:84% STUDENT: Because the defendant is the last proposer. 00:41:41.030 --> 00:41:43.990 align:middle line:84% So they can base it on what the plaintiff will accept. 00:41:43.990 --> 00:41:44.990 align:middle line:90% IAN BALL: Exactly right. 00:41:44.990 --> 00:41:46.950 align:middle line:84% So if we actually change the timing here-- 00:41:46.950 --> 00:41:48.370 align:middle line:84% and this is one issue with these bargaining games 00:41:48.370 --> 00:41:50.037 align:middle line:84% that sometimes the details matter a lot. 00:41:50.037 --> 00:41:53.270 align:middle line:84% If we had an odd number of periods before trial, 00:41:53.270 --> 00:41:54.270 align:middle line:90% this would kind of flip. 00:41:54.270 --> 00:41:57.770 align:middle line:84% And we would care about Cd hat, not Cp hat. 00:41:57.770 --> 00:42:01.650 align:middle line:84% And the reason is that the defendant 00:42:01.650 --> 00:42:05.150 align:middle line:84% is the one able to exploit the plaintiff's cost of trial 00:42:05.150 --> 00:42:06.570 align:middle line:84% because the defendant gets to make 00:42:06.570 --> 00:42:08.505 align:middle line:90% the last offer before trial. 00:42:08.505 --> 00:42:10.130 align:middle line:84% So because the defendant is the one who 00:42:10.130 --> 00:42:13.050 align:middle line:84% makes the last offer before trial, the defendant can say, 00:42:13.050 --> 00:42:14.950 align:middle line:84% you better give me a really good deal. 00:42:14.950 --> 00:42:16.690 align:middle line:84% Otherwise, we're going to have to pay 00:42:16.690 --> 00:42:18.028 align:middle line:90% this high cost to go to trial. 00:42:18.028 --> 00:42:19.070 align:middle line:90% It's interesting, though. 00:42:19.070 --> 00:42:21.770 align:middle line:84% I mean, the defendant also pays a high cost to go to trial. 00:42:21.770 --> 00:42:26.232 align:middle line:84% But they're able to exploit the rationality of the plaintiff 00:42:26.232 --> 00:42:27.690 align:middle line:84% and make them an offer that they're 00:42:27.690 --> 00:42:30.010 align:middle line:84% going to have to accept, and therefore, take advantage 00:42:30.010 --> 00:42:31.110 align:middle line:90% of their bargaining power. 00:42:31.110 --> 00:42:34.010 align:middle line:90% 00:42:34.010 --> 00:42:36.100 align:middle line:90% Great. 00:42:36.100 --> 00:42:38.260 align:middle line:90% So I guess a few takeaways-- 00:42:38.260 --> 00:42:39.820 align:middle line:84% this model suggests that we shouldn't 00:42:39.820 --> 00:42:42.700 align:middle line:84% have drawn-out negotiations, that the settlement should 00:42:42.700 --> 00:42:43.820 align:middle line:90% end right away. 00:42:43.820 --> 00:42:47.420 align:middle line:84% But notice that the potential length of the negotiation 00:42:47.420 --> 00:42:48.640 align:middle line:90% affects the outcome. 00:42:48.640 --> 00:42:50.980 align:middle line:84% So even though we settle in the very first period, 00:42:50.980 --> 00:42:54.420 align:middle line:84% the potential to have a very long, drawn-out negotiation 00:42:54.420 --> 00:42:58.033 align:middle line:84% drives the amount that we settle for in this first period. 00:42:58.033 --> 00:42:58.700 align:middle line:90% One final thing. 00:42:58.700 --> 00:43:00.860 align:middle line:84% If on an exam you're asked about this, 00:43:00.860 --> 00:43:03.640 align:middle line:84% the strategy is not we settle in the first period. 00:43:03.640 --> 00:43:05.720 align:middle line:84% The strategy is this really complicated thing. 00:43:05.720 --> 00:43:08.460 align:middle line:84% The outcome is that we settle in the first period. 00:43:08.460 --> 00:43:11.920 align:middle line:84% Now, in reality, we often see long drawn-out negotiations. 00:43:11.920 --> 00:43:14.380 align:middle line:84% So I think it's often very helpful in economic models 00:43:14.380 --> 00:43:15.780 align:middle line:90% to say, what's missing? 00:43:15.780 --> 00:43:17.680 align:middle line:84% Why is it that in reality-- let's 00:43:17.680 --> 00:43:19.180 align:middle line:84% say Trump and the Democrats, I think 00:43:19.180 --> 00:43:21.680 align:middle line:84% they're going to be negotiating for a pretty long time would 00:43:21.680 --> 00:43:23.220 align:middle line:90% be my guess. 00:43:23.220 --> 00:43:25.220 align:middle line:84% That's completely consistent with the prediction 00:43:25.220 --> 00:43:26.040 align:middle line:90% of this model. 00:43:26.040 --> 00:43:28.580 align:middle line:84% So what do you think is missing in this model that 00:43:28.580 --> 00:43:30.800 align:middle line:84% explains why everything ends in the first period? 00:43:30.800 --> 00:43:31.540 align:middle line:90% Yeah? 00:43:31.540 --> 00:43:33.480 align:middle line:84% STUDENT: Uncertainty about what J actually is. 00:43:33.480 --> 00:43:34.230 align:middle line:90% IAN BALL: Exactly. 00:43:34.230 --> 00:43:36.578 align:middle line:84% An uncertainty about the preferences of the other side, 00:43:36.578 --> 00:43:37.870 align:middle line:90% what they're willing to accept. 00:43:37.870 --> 00:43:39.990 align:middle line:84% Here, we all know exactly what the other side 00:43:39.990 --> 00:43:42.670 align:middle line:84% is going to accept, and we can push them just 00:43:42.670 --> 00:43:44.643 align:middle line:84% to the brink of acceptance at every round. 00:43:44.643 --> 00:43:47.310 align:middle line:84% And therefore, there's no reason for things to evolve over time. 00:43:47.310 --> 00:43:51.070 align:middle line:84% The reason that we often have delay and impasses in bargaining 00:43:51.070 --> 00:43:55.150 align:middle line:84% is that we don't know what the other side is willing to accept, 00:43:55.150 --> 00:43:57.150 align:middle line:84% and we might want to make them a really bad deal 00:43:57.150 --> 00:44:01.550 align:middle line:84% early on on the off chance that they end up not 00:44:01.550 --> 00:44:03.100 align:middle line:90% being who we thought they were. 00:44:03.100 --> 00:44:05.350 align:middle line:84% And we'll see later in the course models of bargaining 00:44:05.350 --> 00:44:07.230 align:middle line:84% with incomplete information or uncertainty 00:44:07.230 --> 00:44:08.607 align:middle line:84% about the opponent's preferences. 00:44:08.607 --> 00:44:10.690 align:middle line:84% And that will exactly lead to delay in bargaining. 00:44:10.690 --> 00:44:12.440 align:middle line:84% Imagine you were negotiating with someone, 00:44:12.440 --> 00:44:14.565 align:middle line:84% and you didn't know if they were a tough negotiator 00:44:14.565 --> 00:44:15.470 align:middle line:90% or a soft negotiator. 00:44:15.470 --> 00:44:17.470 align:middle line:84% You might want to make a really ridiculous offer 00:44:17.470 --> 00:44:19.762 align:middle line:84% in the first period that only the soft negotiator would 00:44:19.762 --> 00:44:22.053 align:middle line:84% accept, but that you know the hard negotiator would not 00:44:22.053 --> 00:44:22.590 align:middle line:90% accept. 00:44:22.590 --> 00:44:25.007 align:middle line:84% And then move to the next period and change your strategy. 00:44:25.007 --> 00:44:27.245 align:middle line:84% So that's the basic idea of what goes 00:44:27.245 --> 00:44:28.870 align:middle line:84% on when you have incomplete information 00:44:28.870 --> 00:44:30.590 align:middle line:90% about the other players. 00:44:30.590 --> 00:44:36.630 align:middle line:84% OK, let's now move to the second bargaining situation 00:44:36.630 --> 00:44:39.650 align:middle line:84% today, which is going to be haggling over the price. 00:44:39.650 --> 00:44:47.590 align:middle line:90% 00:44:47.590 --> 00:44:50.640 align:middle line:84% So now we're going to imagine that-- 00:44:50.640 --> 00:44:52.170 align:middle line:90% we'll call this price haggling. 00:44:52.170 --> 00:44:56.590 align:middle line:90% 00:44:56.590 --> 00:45:02.590 align:middle line:84% Let's suppose that we have a seller and a buyer. 00:45:02.590 --> 00:45:10.430 align:middle line:84% And the seller just wants to get paid. 00:45:10.430 --> 00:45:13.110 align:middle line:84% They're negotiating over the price of some item. 00:45:13.110 --> 00:45:16.550 align:middle line:84% So if they trade at price p, then the seller, 00:45:16.550 --> 00:45:19.510 align:middle line:84% their utility is going to be the price p that they get. 00:45:19.510 --> 00:45:22.830 align:middle line:84% The buyer's utility is going to be v minus p. 00:45:22.830 --> 00:45:27.630 align:middle line:84% So that means p is the price, and v is 00:45:27.630 --> 00:45:29.090 align:middle line:90% the buyer's value for the good. 00:45:29.090 --> 00:45:33.360 align:middle line:90% 00:45:33.360 --> 00:45:36.280 align:middle line:84% So notice here, we're assuming that it doesn't cost anything 00:45:36.280 --> 00:45:38.060 align:middle line:84% for the seller to produce the good. 00:45:38.060 --> 00:45:39.393 align:middle line:90% That's a simplifying assumption. 00:45:39.393 --> 00:45:40.518 align:middle line:90% It's already been produced. 00:45:40.518 --> 00:45:42.300 align:middle line:84% Maybe they can't resell it to anyone else. 00:45:42.300 --> 00:45:43.580 align:middle line:90% They're just stuck with it. 00:45:43.580 --> 00:45:46.160 align:middle line:84% So the seller values the good at 0. 00:45:46.160 --> 00:45:50.600 align:middle line:84% They're thinking of selling it to the buyer who values it at v, 00:45:50.600 --> 00:45:53.240 align:middle line:84% and they're trying to haggle over what the price should be. 00:45:53.240 --> 00:45:55.000 align:middle line:84% So first of all, let's just make sure we 00:45:55.000 --> 00:45:58.540 align:middle line:84% understand what's a reasonable range of what the price can be? 00:45:58.540 --> 00:46:00.940 align:middle line:84% So I think there's some price that we can rule out. 00:46:00.940 --> 00:46:04.240 align:middle line:90% How low could the price be? 00:46:04.240 --> 00:46:06.515 align:middle line:84% What's the lowest it could possibly be? 00:46:06.515 --> 00:46:08.140 align:middle line:84% Well, actually, it could be pretty low. 00:46:08.140 --> 00:46:10.473 align:middle line:84% I think it could go all the way down to 0, in principle. 00:46:10.473 --> 00:46:13.280 align:middle line:84% It's certainly not going to be less than 0, though. 00:46:13.280 --> 00:46:15.520 align:middle line:84% Because if it was less than 0, well, 00:46:15.520 --> 00:46:18.800 align:middle line:84% the seller would never accept being paid a negative price. 00:46:18.800 --> 00:46:20.520 align:middle line:84% What about the highest price that we 00:46:20.520 --> 00:46:23.780 align:middle line:90% would expect to see here? 00:46:23.780 --> 00:46:24.280 align:middle line:90% Yeah? 00:46:24.280 --> 00:46:24.960 align:middle line:90% STUDENT: v. 00:46:24.960 --> 00:46:26.877 align:middle line:84% IAN BALL: v. It certainly can't be more than v 00:46:26.877 --> 00:46:31.640 align:middle line:84% because the buyer would never accept a price above v. 00:46:31.640 --> 00:46:38.280 align:middle line:84% So we can, at first pass, just by basic individual rationality 00:46:38.280 --> 00:46:40.680 align:middle line:84% concerns, see that the price is going to be somewhere 00:46:40.680 --> 00:46:42.480 align:middle line:84% between 0 and v, and both players 00:46:42.480 --> 00:46:44.520 align:middle line:84% are going to get some non-negative utility. 00:46:44.520 --> 00:46:47.160 align:middle line:84% But the question is, where is it between 0 and v? 00:46:47.160 --> 00:46:50.000 align:middle line:84% The seller would love it to be very close to v, 00:46:50.000 --> 00:46:52.360 align:middle line:84% and basically leave the buyer with nothing 00:46:52.360 --> 00:46:54.560 align:middle line:84% and charge them their full valuation for the good. 00:46:54.560 --> 00:46:57.133 align:middle line:84% The buyer would love to pay a price that's 00:46:57.133 --> 00:46:58.800 align:middle line:84% really, really low, so that they can get 00:46:58.800 --> 00:47:00.240 align:middle line:90% all the value from the good. 00:47:00.240 --> 00:47:02.720 align:middle line:84% And we're going to try to make some predictions about where 00:47:02.720 --> 00:47:05.880 align:middle line:84% within this interval the price can be. 00:47:05.880 --> 00:47:08.000 align:middle line:84% Notice this is kind of a common situation 00:47:08.000 --> 00:47:11.440 align:middle line:84% that we generally only expect that goods 00:47:11.440 --> 00:47:13.720 align:middle line:84% are going to be exchanged when the seller values 00:47:13.720 --> 00:47:15.640 align:middle line:90% the good more than the buyer. 00:47:15.640 --> 00:47:18.820 align:middle line:84% And we generally expect the price to be somewhere between-- 00:47:18.820 --> 00:47:21.320 align:middle line:84% sorry-- when the seller values the good less than the buyer. 00:47:21.320 --> 00:47:22.945 align:middle line:84% And we expect the price to be somewhere 00:47:22.945 --> 00:47:24.720 align:middle line:84% between what the seller values it at 00:47:24.720 --> 00:47:26.370 align:middle line:90% and what the buyer values it at. 00:47:26.370 --> 00:47:29.290 align:middle line:84% And here, what the seller values it at is 0. 00:47:29.290 --> 00:47:33.450 align:middle line:84% You can think of this as the seller's value, 00:47:33.450 --> 00:47:36.810 align:middle line:90% and this is the buyer's value. 00:47:36.810 --> 00:47:38.435 align:middle line:90% So the buyer does value it more. 00:47:38.435 --> 00:47:40.310 align:middle line:84% That's why it makes sense for them to buy it. 00:47:40.310 --> 00:47:43.330 align:middle line:84% But where the price lies between these two valuations 00:47:43.330 --> 00:47:44.350 align:middle line:90% is the name of the game. 00:47:44.350 --> 00:47:46.668 align:middle line:84% That's what we're trying to figure out. 00:47:46.668 --> 00:47:48.210 align:middle line:84% Now, what we're also going to imagine 00:47:48.210 --> 00:47:52.210 align:middle line:90% is that there's a cost of delay. 00:47:52.210 --> 00:47:54.770 align:middle line:84% But we're going to model the cost of delay a bit differently. 00:47:54.770 --> 00:47:57.490 align:middle line:84% Here, we model the cost of delay as an additive cost. 00:47:57.490 --> 00:48:02.130 align:middle line:84% Each period we had to pay a lawyer some daily fee, 00:48:02.130 --> 00:48:03.610 align:middle line:90% and that had a cost. 00:48:03.610 --> 00:48:05.490 align:middle line:84% Here, we're going to say the cost of delay, 00:48:05.490 --> 00:48:08.750 align:middle line:84% we're going to use what's called discounting. 00:48:08.750 --> 00:48:11.930 align:middle line:90% 00:48:11.930 --> 00:48:20.570 align:middle line:84% So the idea is that if we trade in period t, 00:48:20.570 --> 00:48:22.970 align:middle line:84% then the utilities we actually get 00:48:22.970 --> 00:48:32.300 align:middle line:84% would be delta p for the seller, and delta v minus p 00:48:32.300 --> 00:48:41.600 align:middle line:84% for the buyer to the t, where delta is between 0 and 1. 00:48:41.600 --> 00:48:44.260 align:middle line:90% 00:48:44.260 --> 00:48:46.760 align:middle line:84% So here-- you may have seen this in other economics courses. 00:48:46.760 --> 00:48:49.540 align:middle line:84% Delta is called our discount factor. 00:48:49.540 --> 00:48:51.780 align:middle line:84% And what this says is each period 00:48:51.780 --> 00:48:55.540 align:middle line:84% we wait our utilities get multiplied by this number 00:48:55.540 --> 00:48:56.800 align:middle line:90% delta, which is less than 1. 00:48:56.800 --> 00:48:59.140 align:middle line:90% And therefore, they get smaller. 00:48:59.140 --> 00:49:02.820 align:middle line:84% So if t is really, really big, then the seller 00:49:02.820 --> 00:49:05.000 align:middle line:84% doesn't get the full utility from the price. 00:49:05.000 --> 00:49:07.080 align:middle line:84% They get that times delta to the t 00:49:07.080 --> 00:49:08.660 align:middle line:84% because each period they have to wait 00:49:08.660 --> 00:49:10.500 align:middle line:84% beyond the zeroth period they incur 00:49:10.500 --> 00:49:13.140 align:middle line:84% this multiplicative discount of delta. 00:49:13.140 --> 00:49:14.360 align:middle line:90% And similarly, for the buyer. 00:49:14.360 --> 00:49:17.740 align:middle line:84% They incur this multiplicative discount of delta. 00:49:17.740 --> 00:49:22.860 align:middle line:84% So let's understand if delta is very close to 1, 00:49:22.860 --> 00:49:27.420 align:middle line:84% then that would indicate that the players are quite patient. 00:49:27.420 --> 00:49:30.220 align:middle line:84% Because as time passes, delta to the t 00:49:30.220 --> 00:49:34.020 align:middle line:84% doesn't get smaller that quickly if delta is very close to 1. 00:49:34.020 --> 00:49:37.860 align:middle line:84% If delta is very close to 0, the players are extremely impatient. 00:49:37.860 --> 00:49:41.980 align:middle line:84% And as time passes, their utility is going to drop a lot. 00:49:41.980 --> 00:49:44.340 align:middle line:84% And therefore, the cost of delay becomes very high. 00:49:44.340 --> 00:49:49.820 align:middle line:84% So maybe I'll-- you can say this in a few ways. 00:49:49.820 --> 00:49:54.220 align:middle line:84% You might say low delta is another way of saying 00:49:54.220 --> 00:49:57.340 align:middle line:90% the players are impatient. 00:49:57.340 --> 00:49:59.840 align:middle line:84% Or another way of saying that is the cost of delay is high. 00:49:59.840 --> 00:50:04.220 align:middle line:90% 00:50:04.220 --> 00:50:06.620 align:middle line:84% And then conversely, if we have high delta, 00:50:06.620 --> 00:50:09.340 align:middle line:84% the players are very patient, and the cost of delay 00:50:09.340 --> 00:50:10.320 align:middle line:90% is not very high. 00:50:10.320 --> 00:50:14.900 align:middle line:90% 00:50:14.900 --> 00:50:16.460 align:middle line:84% Sometimes we think of delta as being 00:50:16.460 --> 00:50:17.668 align:middle line:90% related to the interest rate. 00:50:17.668 --> 00:50:20.372 align:middle line:84% So in a financial context, the cost of delay with money 00:50:20.372 --> 00:50:21.830 align:middle line:84% is the interest rate that you could 00:50:21.830 --> 00:50:24.310 align:middle line:84% have gotten by putting that money in a bank account 00:50:24.310 --> 00:50:26.410 align:middle line:84% or getting some return in equities. 00:50:26.410 --> 00:50:29.270 align:middle line:90% 00:50:29.270 --> 00:50:34.990 align:middle line:84% And now we're going to assume that time proceeds just as 00:50:34.990 --> 00:50:35.490 align:middle line:90% before. 00:50:35.490 --> 00:50:44.878 align:middle line:90% 00:50:44.878 --> 00:50:46.670 align:middle line:84% So I think this is kind of exactly what you 00:50:46.670 --> 00:50:48.087 align:middle line:84% see if you go a market, especially 00:50:48.087 --> 00:50:50.510 align:middle line:84% in developing countries and there aren't posted prices. 00:50:50.510 --> 00:50:51.770 align:middle line:84% You, as the buyer, go up and you say, 00:50:51.770 --> 00:50:52.690 align:middle line:90% I'm willing to pay you this. 00:50:52.690 --> 00:50:53.570 align:middle line:90% And they say, oh, no. 00:50:53.570 --> 00:50:54.490 align:middle line:90% You need to pay me this much. 00:50:54.490 --> 00:50:56.698 align:middle line:84% And then you counteroffer, and you go back and forth. 00:50:56.698 --> 00:50:59.670 align:middle line:84% And even in developed countries, this certainly 00:50:59.670 --> 00:51:01.950 align:middle line:84% happens if we're negotiating on some big deal. 00:51:01.950 --> 00:51:03.370 align:middle line:90% It doesn't happen at Starbucks. 00:51:03.370 --> 00:51:04.870 align:middle line:84% You're probably not going to be able to negotiate over 00:51:04.870 --> 00:51:05.912 align:middle line:90% the price of your coffee. 00:51:05.912 --> 00:51:07.590 align:middle line:84% But if you're getting hired at a firm, 00:51:07.590 --> 00:51:09.230 align:middle line:84% or if you're making a merger, people 00:51:09.230 --> 00:51:13.030 align:middle line:84% do negotiate all the time on prices. 00:51:13.030 --> 00:51:18.550 align:middle line:84% So 2n minus 2, 2n minus 1, and then 2n. 00:51:18.550 --> 00:51:21.190 align:middle line:84% And the order I think we're going to assume 00:51:21.190 --> 00:51:26.350 align:middle line:84% is the seller makes the first offer. 00:51:26.350 --> 00:51:30.958 align:middle line:84% So here, the seller offers a price, P0. 00:51:30.958 --> 00:51:32.750 align:middle line:84% So instead of offering a settlement amount, 00:51:32.750 --> 00:51:34.070 align:middle line:90% they're offering a price. 00:51:34.070 --> 00:51:37.650 align:middle line:84% The buyer can either accept or reject. 00:51:37.650 --> 00:51:42.550 align:middle line:90% 00:51:42.550 --> 00:51:44.010 align:middle line:90% If it's accepted, we're done. 00:51:44.010 --> 00:51:45.510 align:middle line:84% If it's rejected, now the buyer gets 00:51:45.510 --> 00:51:48.710 align:middle line:84% to make a counteroffer of P1, and the seller 00:51:48.710 --> 00:51:50.630 align:middle line:90% can accept or reject. 00:51:50.630 --> 00:51:55.070 align:middle line:84% And we're going to go on like this, all the way up to seller 00:51:55.070 --> 00:51:57.950 align:middle line:90% making an offer P2n minus 2. 00:51:57.950 --> 00:52:00.990 align:middle line:84% Buyer making an offer P2n minus 1. 00:52:00.990 --> 00:52:04.570 align:middle line:84% And then we're going to assume at period 2n, the world ends. 00:52:04.570 --> 00:52:05.410 align:middle line:90% Or just, that's it. 00:52:05.410 --> 00:52:06.630 align:middle line:90% We both get 0. 00:52:06.630 --> 00:52:14.590 align:middle line:84% So if we wait this long, we get payoffs of 0, 0. 00:52:14.590 --> 00:52:16.720 align:middle line:84% This seems like a strange assumption. 00:52:16.720 --> 00:52:18.417 align:middle line:84% The real reason is that we don't quite 00:52:18.417 --> 00:52:20.000 align:middle line:84% have the tools to analyze what happens 00:52:20.000 --> 00:52:21.880 align:middle line:90% if it could go on indefinitely. 00:52:21.880 --> 00:52:24.902 align:middle line:84% But it turns out that if you take n to be really, really big, 00:52:24.902 --> 00:52:26.360 align:middle line:84% the predictions that we'll get will 00:52:26.360 --> 00:52:27.777 align:middle line:84% be very similar to the predictions 00:52:27.777 --> 00:52:29.380 align:middle line:84% if there's an indefinite end date. 00:52:29.380 --> 00:52:32.320 align:middle line:84% So as I tell the story of the tourists get off the tour bus 00:52:32.320 --> 00:52:34.852 align:middle line:84% at the market, and the tour bus is leaving in 2n periods, 00:52:34.852 --> 00:52:37.060 align:middle line:84% and this is the last chance for them to make a trade. 00:52:37.060 --> 00:52:40.260 align:middle line:84% But are they really-- or people say the ice cream cone melts. 00:52:40.260 --> 00:52:43.040 align:middle line:84% But yeah, I don't know how compelling this exact end 00:52:43.040 --> 00:52:45.960 align:middle line:84% date is, and we'll talk about relaxing that assumption later 00:52:45.960 --> 00:52:46.460 align:middle line:90% on. 00:52:46.460 --> 00:52:49.900 align:middle line:90% 00:52:49.900 --> 00:52:56.780 align:middle line:84% So now let's try to start analyzing this. 00:52:56.780 --> 00:53:12.240 align:middle line:90% 00:53:12.240 --> 00:53:20.770 align:middle line:84% So let's start, as usual, in period 2n minus 1. 00:53:20.770 --> 00:53:25.590 align:middle line:84% And now what is the seller going to accept? 00:53:25.590 --> 00:53:28.410 align:middle line:90% 00:53:28.410 --> 00:53:30.630 align:middle line:90% So we're in period 2n minus 1. 00:53:30.630 --> 00:53:33.170 align:middle line:84% The world is going to end tomorrow. 00:53:33.170 --> 00:53:34.590 align:middle line:90% The buyer makes an offer. 00:53:34.590 --> 00:53:38.810 align:middle line:84% What prices will the seller be willing to accept? 00:53:38.810 --> 00:53:43.410 align:middle line:84% So they'll accept if and only if the offered price, P2n minus 1, 00:53:43.410 --> 00:53:45.230 align:middle line:84% is greater than or equal to what? 00:53:45.230 --> 00:53:54.330 align:middle line:90% 00:53:54.330 --> 00:53:55.110 align:middle line:90% Any thoughts? 00:53:55.110 --> 00:53:58.010 align:middle line:90% 00:53:58.010 --> 00:53:59.150 align:middle line:90% So you're the seller. 00:53:59.150 --> 00:54:01.510 align:middle line:84% The buyer says, I'm going to pay you this amount for it. 00:54:01.510 --> 00:54:03.177 align:middle line:84% You say, well, either I can accept this, 00:54:03.177 --> 00:54:05.450 align:middle line:84% or I can reject it and get a payoff of 0. 00:54:05.450 --> 00:54:07.110 align:middle line:90% So what am I willing to accept? 00:54:07.110 --> 00:54:07.610 align:middle line:90% Yeah? 00:54:07.610 --> 00:54:09.010 align:middle line:90% STUDENT: Won't it just be 0? 00:54:09.010 --> 00:54:11.650 align:middle line:84% IAN BALL: Yeah, anything that's above 0. 00:54:11.650 --> 00:54:13.350 align:middle line:90% Exactly 0, we can argue about. 00:54:13.350 --> 00:54:16.370 align:middle line:84% But we'll say if and only if it's greater than or equal to 0. 00:54:16.370 --> 00:54:18.970 align:middle line:84% So if you offer me any non-negative price, 00:54:18.970 --> 00:54:22.850 align:middle line:84% my choice is I accept it, or I wait till tomorrow and I get 0. 00:54:22.850 --> 00:54:24.650 align:middle line:84% So any positive amount you give me, I'm 00:54:24.650 --> 00:54:27.250 align:middle line:84% willing to accept because it's better than getting 0 tomorrow. 00:54:27.250 --> 00:54:29.610 align:middle line:84% And given that, what is the buyer going to offer? 00:54:29.610 --> 00:54:31.330 align:middle line:84% So again, we're working backwards. 00:54:31.330 --> 00:54:33.377 align:middle line:90% What will the buyer offer? 00:54:33.377 --> 00:54:34.710 align:middle line:90% How much will they offer to pay? 00:54:34.710 --> 00:54:40.410 align:middle line:90% 00:54:40.410 --> 00:54:42.417 align:middle line:84% It's a good position for the buyer, right? 00:54:42.417 --> 00:54:43.750 align:middle line:90% They're going to offer to pay 0. 00:54:43.750 --> 00:54:47.370 align:middle line:90% 00:54:47.370 --> 00:54:50.130 align:middle line:84% If I know that the seller will accept any price, 00:54:50.130 --> 00:54:54.090 align:middle line:84% I'll offer the smallest price they'll accept. 00:54:54.090 --> 00:54:56.810 align:middle line:90% What's the story here? 00:54:56.810 --> 00:54:58.975 align:middle line:84% I come at the end of the day, think about-- 00:54:58.975 --> 00:55:00.350 align:middle line:84% I mean, we see this all the time. 00:55:00.350 --> 00:55:00.892 align:middle line:90% It's a store. 00:55:00.892 --> 00:55:03.190 align:middle line:90% They have one croissant left. 00:55:03.190 --> 00:55:04.270 align:middle line:90% They're about to close. 00:55:04.270 --> 00:55:06.210 align:middle line:84% Or maybe they have tons of croissants left. 00:55:06.210 --> 00:55:07.725 align:middle line:84% I know if they don't sell it to me, 00:55:07.725 --> 00:55:09.350 align:middle line:84% they're not going to sell it to anyone. 00:55:09.350 --> 00:55:10.520 align:middle line:90% It'll go stale tomorrow. 00:55:10.520 --> 00:55:11.700 align:middle line:90% I can make a pretty good-- 00:55:11.700 --> 00:55:13.200 align:middle line:84% I'm in a pretty strong bargaining position. 00:55:13.200 --> 00:55:14.880 align:middle line:84% I can say, just give it to me for a price of 0. 00:55:14.880 --> 00:55:16.297 align:middle line:84% And we see bakeries have discounts 00:55:16.297 --> 00:55:19.220 align:middle line:84% at the end of the day for exactly this reason. 00:55:19.220 --> 00:55:22.660 align:middle line:84% But now let's go to period 2n minus 2. 00:55:22.660 --> 00:55:27.260 align:middle line:84% Now the buyer is making an offer. 00:55:27.260 --> 00:55:29.060 align:middle line:90% Sorry. 00:55:29.060 --> 00:55:30.380 align:middle line:90% The seller is making an offer. 00:55:30.380 --> 00:55:32.880 align:middle line:84% So the buyer is choosing whether to accept the offer or not. 00:55:32.880 --> 00:55:36.020 align:middle line:90% 00:55:36.020 --> 00:55:38.420 align:middle line:84% And what offers will the buyer accept? 00:55:38.420 --> 00:55:45.180 align:middle line:84% So the buyer will accept if the price, pt, offered by the seller 00:55:45.180 --> 00:55:48.620 align:middle line:90% is small enough. 00:55:48.620 --> 00:55:49.940 align:middle line:90% Yeah, question or-- yeah? 00:55:49.940 --> 00:55:52.573 align:middle line:84% STUDENT: Is the buyer's value of the good known to the seller? 00:55:52.573 --> 00:55:53.240 align:middle line:90% IAN BALL: It is. 00:55:53.240 --> 00:55:54.573 align:middle line:90% And that's a crucial assumption. 00:55:54.573 --> 00:55:56.140 align:middle line:90% So it is known. 00:55:56.140 --> 00:55:58.463 align:middle line:84% And in reality, you might think it wouldn't be known. 00:55:58.463 --> 00:55:59.880 align:middle line:84% And towards the end of the course, 00:55:59.880 --> 00:56:01.520 align:middle line:84% we'll talk about incomplete information bargaining. 00:56:01.520 --> 00:56:03.080 align:middle line:90% But it comes really messy. 00:56:03.080 --> 00:56:04.447 align:middle line:90% But yes, v is known. 00:56:04.447 --> 00:56:05.280 align:middle line:90% That's a good point. 00:56:05.280 --> 00:56:06.560 align:middle line:90% And delta is also known. 00:56:06.560 --> 00:56:11.060 align:middle line:90% 00:56:11.060 --> 00:56:15.220 align:middle line:84% So here, we're going to have, say, Pt star. 00:56:15.220 --> 00:56:20.020 align:middle line:84% The buyer will accept if the price is low enough. 00:56:20.020 --> 00:56:25.300 align:middle line:84% And let's try to compute what Pt star has to be. 00:56:25.300 --> 00:56:28.420 align:middle line:84% So as usual, Pt star is going to be defined by an indifference 00:56:28.420 --> 00:56:29.300 align:middle line:90% condition. 00:56:29.300 --> 00:56:31.540 align:middle line:84% The highest price they'll accept is the price 00:56:31.540 --> 00:56:34.180 align:middle line:84% where if they accept it, they do just as well 00:56:34.180 --> 00:56:37.540 align:middle line:84% as if they were to reject it and wait until tomorrow. 00:56:37.540 --> 00:56:42.660 align:middle line:84% So let's look at what happens if they were to accept a price 00:56:42.660 --> 00:56:50.940 align:middle line:84% of Pt star, then they're going to get delta to the 2n minus 2 00:56:50.940 --> 00:56:54.360 align:middle line:90% of v minus Pt star. 00:56:54.360 --> 00:57:00.900 align:middle line:90% 00:57:00.900 --> 00:57:02.160 align:middle line:90% So that's if they accept. 00:57:02.160 --> 00:57:06.190 align:middle line:90% 00:57:06.190 --> 00:57:07.610 align:middle line:90% What if they reject the offer? 00:57:07.610 --> 00:57:11.550 align:middle line:90% 00:57:11.550 --> 00:57:16.122 align:middle line:84% If they reject the offer, then tomorrow they get to propose, 00:57:16.122 --> 00:57:17.330 align:middle line:90% and they're in a great shape. 00:57:17.330 --> 00:57:21.350 align:middle line:84% We already said tomorrow they're going to offer a price of 0, 00:57:21.350 --> 00:57:23.030 align:middle line:90% and it's going to be accepted. 00:57:23.030 --> 00:57:26.590 align:middle line:84% So that means tomorrow they're going to get delta to the 2n 00:57:26.590 --> 00:57:34.590 align:middle line:84% minus 1 times v. Really, it's minus v PT plus 1 star. 00:57:34.590 --> 00:57:39.130 align:middle line:84% Or I guess I should say, here, t equals 2n minus 2. 00:57:39.130 --> 00:57:43.470 align:middle line:84% But I'm just writing t because it's a bit cleaner. 00:57:43.470 --> 00:57:47.270 align:middle line:84% And once again, we can see that we can simplify these discount 00:57:47.270 --> 00:57:48.430 align:middle line:90% factors a lot. 00:57:48.430 --> 00:57:51.950 align:middle line:84% Because we can factor out delta to the 2n minus 1 from both 00:57:51.950 --> 00:57:52.490 align:middle line:90% terms. 00:57:52.490 --> 00:57:55.590 align:middle line:84% And this makes sense because the question is, do I take it today, 00:57:55.590 --> 00:57:56.882 align:middle line:90% or do I wait till tomorrow? 00:57:56.882 --> 00:57:59.090 align:middle line:84% There's really only one discount factor that matters. 00:57:59.090 --> 00:58:01.090 align:middle line:84% It's the discounting between today and tomorrow. 00:58:01.090 --> 00:58:03.510 align:middle line:84% So we can equivalently-- we can cancel things out, 00:58:03.510 --> 00:58:06.720 align:middle line:90% and we can say-- 00:58:06.720 --> 00:58:08.960 align:middle line:90% and just put a delta here. 00:58:08.960 --> 00:58:11.980 align:middle line:84% So we divide by delta to the 2n minus 2. 00:58:11.980 --> 00:58:14.760 align:middle line:90% 00:58:14.760 --> 00:58:21.160 align:middle line:84% So what we get is v. And maybe I'll write the general formula-- 00:58:21.160 --> 00:58:25.540 align:middle line:90% v minus Pt star equals delta. 00:58:25.540 --> 00:58:36.600 align:middle line:90% 00:58:36.600 --> 00:58:41.400 align:middle line:84% In period t, in an even period where I'm the buyer, 00:58:41.400 --> 00:58:44.640 align:middle line:84% I can either get this utility today, 00:58:44.640 --> 00:58:47.025 align:middle line:84% or I can wait till tomorrow and get this utility 00:58:47.025 --> 00:58:48.400 align:middle line:84% where the price I get tomorrow is 00:58:48.400 --> 00:58:50.858 align:middle line:84% going to depend on what we've already computed for tomorrow 00:58:50.858 --> 00:58:51.980 align:middle line:90% using backward induction. 00:58:51.980 --> 00:58:55.680 align:middle line:90% 00:58:55.680 --> 00:58:59.620 align:middle line:84% And if we try to solve for this, we'll get Pt. 00:58:59.620 --> 00:59:19.000 align:middle line:90% 00:59:19.000 --> 00:59:23.680 align:middle line:84% Now, let's ask, how does this Pt star, this threshold price, 00:59:23.680 --> 00:59:25.460 align:middle line:90% compare to Pt plus 1? 00:59:25.460 --> 00:59:33.480 align:middle line:90% 00:59:33.480 --> 00:59:35.940 align:middle line:84% Is it bigger or smaller than Pt plus 1? 00:59:35.940 --> 00:59:44.080 align:middle line:90% 00:59:44.080 --> 00:59:49.760 align:middle line:84% Well, it's bigger because Pt is a weighted average of Pt 00:59:49.760 --> 00:59:52.480 align:middle line:84% plus 1 and the number v. But we know 00:59:52.480 --> 00:59:55.120 align:middle line:84% v is always bigger than or equal to Pt plus 1. 00:59:55.120 --> 00:59:57.960 align:middle line:84% So that means this is strictly greater than Pt plus 1. 00:59:57.960 --> 01:00:00.970 align:middle line:84% And let's try to think through the intuition for this. 01:00:00.970 --> 01:00:03.450 align:middle line:90% Again, it's about discounting. 01:00:03.450 --> 01:00:07.290 align:middle line:84% I'm willing to accept a price that's higher today 01:00:07.290 --> 01:00:09.810 align:middle line:84% than I would get tomorrow for the classic reason 01:00:09.810 --> 01:00:13.610 align:middle line:84% that if I wait till tomorrow, I'd have to pay a cost of delay. 01:00:13.610 --> 01:00:15.970 align:middle line:84% So I'm willing to accept a worse deal today 01:00:15.970 --> 01:00:17.530 align:middle line:84% than I would get tomorrow in order 01:00:17.530 --> 01:00:19.170 align:middle line:90% to avoid the cost of delay. 01:00:19.170 --> 01:00:23.570 align:middle line:84% For me, a worse deal is a higher price because I'm the buyer. 01:00:23.570 --> 01:00:26.010 align:middle line:84% So I started out with a case of 2n minus 2, 01:00:26.010 --> 01:00:28.890 align:middle line:84% but you can see that this formula actually holds for any 01:00:28.890 --> 01:00:29.870 align:middle line:90% even period. 01:00:29.870 --> 01:00:33.130 align:middle line:90% 01:00:33.130 --> 01:00:35.690 align:middle line:84% For any even period t, the buyer is 01:00:35.690 --> 01:00:38.330 align:middle line:84% going to accept if the price is below a threshold. 01:00:38.330 --> 01:00:43.490 align:middle line:84% The seller is then going to offer exactly that threshold. 01:00:43.490 --> 01:00:46.710 align:middle line:84% And the threshold is defined by this formula. 01:00:46.710 --> 01:00:52.650 align:middle line:90% 01:00:52.650 --> 01:00:56.850 align:middle line:84% Now, let's try to look at odd periods t. 01:00:56.850 --> 01:01:00.530 align:middle line:84% If we're in an odd period t, now it 01:01:00.530 --> 01:01:05.530 align:middle line:84% is the seller who's deciding whether to accept or reject. 01:01:05.530 --> 01:01:11.370 align:middle line:84% The seller is going to accept if Pt is greater 01:01:11.370 --> 01:01:15.810 align:middle line:90% than or equal to Pt star. 01:01:15.810 --> 01:01:19.890 align:middle line:84% And now can we get a formula for what Pt star is going to be? 01:01:19.890 --> 01:01:23.850 align:middle line:84% It has to say that the seller is indifferent between selling 01:01:23.850 --> 01:01:28.130 align:middle line:84% at a price of Pt star today or waiting until tomorrow. 01:01:28.130 --> 01:01:32.010 align:middle line:84% If they sell at Pt star today, they 01:01:32.010 --> 01:01:35.110 align:middle line:90% get Pt star in discounted terms. 01:01:35.110 --> 01:01:36.790 align:middle line:84% What happens if they wait till tomorrow? 01:01:36.790 --> 01:01:43.290 align:middle line:90% 01:01:43.290 --> 01:01:49.110 align:middle line:84% Well, they get delta times Pt plus 1 star. 01:01:49.110 --> 01:01:52.570 align:middle line:90% 01:01:52.570 --> 01:01:55.797 align:middle line:84% If I wait till tomorrow, we know from backward induction 01:01:55.797 --> 01:01:57.380 align:middle line:84% that the good is going to be exchanged 01:01:57.380 --> 01:01:59.260 align:middle line:90% at a price of Pt plus 1 star. 01:01:59.260 --> 01:02:01.120 align:middle line:84% But I have to discount that by delta, 01:02:01.120 --> 01:02:03.880 align:middle line:84% and I'm going to be exactly indifferent if this holds. 01:02:03.880 --> 01:02:06.540 align:middle line:90% 01:02:06.540 --> 01:02:12.520 align:middle line:84% So if you want to be kind of really mathy about it, 01:02:12.520 --> 01:02:14.760 align:middle line:84% we have a solution, but it's a bit of a mess. 01:02:14.760 --> 01:02:18.980 align:middle line:90% 01:02:18.980 --> 01:02:20.280 align:middle line:90% So what we know-- 01:02:20.280 --> 01:02:27.260 align:middle line:90% 01:02:27.260 --> 01:02:29.400 align:middle line:84% so how do we compute our solution? 01:02:29.400 --> 01:02:30.860 align:middle line:90% We just work backwards. 01:02:30.860 --> 01:02:34.940 align:middle line:84% We know-- we argued that the price that's going to be 01:02:34.940 --> 01:02:38.380 align:middle line:84% determined in period 2n minus 1 is exactly 0. 01:02:38.380 --> 01:02:40.580 align:middle line:84% And then we have a sequence of recursions 01:02:40.580 --> 01:02:45.660 align:middle line:84% that allow us to compute what Pt must be if we know Pt plus 1, 01:02:45.660 --> 01:02:48.220 align:middle line:84% what Pt is if we know Pt plus 1, and we have 01:02:48.220 --> 01:02:51.220 align:middle line:84% two different formulas depending on whether if t is even or odd. 01:02:51.220 --> 01:02:53.380 align:middle line:84% So technically, we could start at the end 01:02:53.380 --> 01:02:56.380 align:middle line:90% and work our way backwards. 01:02:56.380 --> 01:02:59.560 align:middle line:84% Then we're going to have some really messy formulas. 01:02:59.560 --> 01:03:00.760 align:middle line:90% It's going to get ugly. 01:03:00.760 --> 01:03:02.718 align:middle line:84% So I'm not going to go through all the algebra, 01:03:02.718 --> 01:03:05.600 align:middle line:84% but I just want to show you what we're going to find, 01:03:05.600 --> 01:03:08.680 align:middle line:84% so what's the limit of this game. 01:03:08.680 --> 01:03:20.900 align:middle line:90% 01:03:20.900 --> 01:03:23.620 align:middle line:84% So we can compute the full strategy. 01:03:23.620 --> 01:03:27.900 align:middle line:90% The outcome is going to be-- 01:03:27.900 --> 01:03:37.140 align:middle line:84% well, in period 0, the seller offers P0 star, 01:03:37.140 --> 01:03:41.940 align:middle line:84% and the buyer accepts, where P0 star 01:03:41.940 --> 01:03:43.980 align:middle line:84% is determined by starting at the end 01:03:43.980 --> 01:03:46.980 align:middle line:84% and working all the way backwards. 01:03:46.980 --> 01:03:49.160 align:middle line:90% So once again, we see no delay. 01:03:49.160 --> 01:03:52.990 align:middle line:90% 01:03:52.990 --> 01:03:54.410 align:middle line:84% And this already came up earlier. 01:03:54.410 --> 01:03:56.170 align:middle line:84% What's the reason that we have no delay? 01:03:56.170 --> 01:04:01.070 align:middle line:84% Why do we not see this protracted negotiation 01:04:01.070 --> 01:04:02.890 align:middle line:84% over the price that's going to be paid? 01:04:02.890 --> 01:04:08.027 align:middle line:90% 01:04:08.027 --> 01:04:09.610 align:middle line:84% What's kind of missing from the model? 01:04:09.610 --> 01:04:11.930 align:middle line:84% I think someone already brought it up. 01:04:11.930 --> 01:04:12.430 align:middle line:90% Yeah? 01:04:12.430 --> 01:04:14.390 align:middle line:84% STUDENT: They have perfect information so they 01:04:14.390 --> 01:04:15.740 align:middle line:90% can see what happens in there. 01:04:15.740 --> 01:04:16.490 align:middle line:90% IAN BALL: Exactly. 01:04:16.490 --> 01:04:17.948 align:middle line:84% In reality, when you're negotiating 01:04:17.948 --> 01:04:20.390 align:middle line:84% with a price for someone, you don't know their value, 01:04:20.390 --> 01:04:23.130 align:middle line:84% v. You don't know exactly how much they're willing to pay. 01:04:23.130 --> 01:04:25.710 align:middle line:84% We've made this really stylized assumption 01:04:25.710 --> 01:04:28.630 align:middle line:84% that everyone knows the buyer's willingness 01:04:28.630 --> 01:04:30.430 align:middle line:84% to pay-- exactly the most the buyer is 01:04:30.430 --> 01:04:31.440 align:middle line:90% willing to pay for this. 01:04:31.440 --> 01:04:33.690 align:middle line:84% That would make-- say you're negotiating over a house. 01:04:33.690 --> 01:04:36.390 align:middle line:84% It would be really easy if you understood 01:04:36.390 --> 01:04:38.710 align:middle line:84% exactly how much this family was willing to pay 01:04:38.710 --> 01:04:39.490 align:middle line:90% to buy this house. 01:04:39.490 --> 01:04:42.330 align:middle line:84% In reality, we don't know that, and that's why we see delay. 01:04:42.330 --> 01:04:45.730 align:middle line:84% But in this benchmark model, with complete information, 01:04:45.730 --> 01:04:46.775 align:middle line:90% we see no delay. 01:04:46.775 --> 01:04:49.150 align:middle line:84% I think this is an example of where an economic model can 01:04:49.150 --> 01:04:51.360 align:middle line:84% be useful, even if it gives us results 01:04:51.360 --> 01:04:52.620 align:middle line:90% that are counterintuitive. 01:04:52.620 --> 01:04:57.240 align:middle line:84% Because what it can show us is it can tell us-- 01:04:57.240 --> 01:05:00.100 align:middle line:84% demonstrate the importance of certain features. 01:05:00.100 --> 01:05:04.280 align:middle line:84% What this tells us is that the driver of delay in bargaining 01:05:04.280 --> 01:05:08.388 align:middle line:84% is really incomplete information about people's preferences. 01:05:08.388 --> 01:05:09.180 align:middle line:90% How do we see this? 01:05:09.180 --> 01:05:11.540 align:middle line:84% Well, we see if there were no incomplete information, 01:05:11.540 --> 01:05:12.580 align:middle line:90% we would have no delay. 01:05:12.580 --> 01:05:14.740 align:middle line:84% And this shows us that something must be missing from the model. 01:05:14.740 --> 01:05:16.820 align:middle line:84% And I guess we suspect it's incomplete information. 01:05:16.820 --> 01:05:18.070 align:middle line:90% And we can see that the later. 01:05:18.070 --> 01:05:19.720 align:middle line:90% We can model that later. 01:05:19.720 --> 01:05:24.220 align:middle line:84% I want to describe, though, briefly, 01:05:24.220 --> 01:05:33.520 align:middle line:84% what's the limiting case if n is very large? 01:05:33.520 --> 01:05:35.160 align:middle line:84% And I explained this is what we're 01:05:35.160 --> 01:05:37.960 align:middle line:84% going to converge to if we looked at the infinite horizon 01:05:37.960 --> 01:05:40.060 align:middle line:90% model. 01:05:40.060 --> 01:05:54.320 align:middle line:84% And it turns out that P0 star is going to be 1 over 1 01:05:54.320 --> 01:06:01.120 align:middle line:90% plus delta times v. 01:06:01.120 --> 01:06:03.383 align:middle line:90% Let's try to interpret this. 01:06:03.383 --> 01:06:04.300 align:middle line:90% And this is the case-- 01:06:04.300 --> 01:06:06.940 align:middle line:84% I should say in our model, the seller moves first. 01:06:06.940 --> 01:06:15.040 align:middle line:90% 01:06:15.040 --> 01:06:17.660 align:middle line:84% What if we had said had the buyer proposed first? 01:06:17.660 --> 01:06:24.000 align:middle line:90% 01:06:24.000 --> 01:06:26.440 align:middle line:84% We can also compute what P0 star is. 01:06:26.440 --> 01:06:27.820 align:middle line:90% Any guesses if the buyer-- 01:06:27.820 --> 01:06:29.440 align:middle line:84% So we're still looking at the limit 01:06:29.440 --> 01:06:33.560 align:middle line:84% where we could, in principle, negotiate for a very long time. 01:06:33.560 --> 01:06:35.320 align:middle line:84% This is the price that the seller is going 01:06:35.320 --> 01:06:38.000 align:middle line:90% to offer if they propose first. 01:06:38.000 --> 01:06:40.780 align:middle line:84% What if the buyer got to propose first? 01:06:40.780 --> 01:06:45.712 align:middle line:84% Any guesses on whether this price would be higher or lower? 01:06:45.712 --> 01:06:47.170 align:middle line:84% Again, we can do the algebra, but I 01:06:47.170 --> 01:06:49.930 align:middle line:84% think it's a lot better to think through these things 01:06:49.930 --> 01:06:52.570 align:middle line:90% intuitively. 01:06:52.570 --> 01:06:55.070 align:middle line:84% Would you rather go first or not in this negotiation? 01:06:55.070 --> 01:06:59.930 align:middle line:90% 01:06:59.930 --> 01:07:02.310 align:middle line:90% Any intuition? 01:07:02.310 --> 01:07:05.890 align:middle line:90% 01:07:05.890 --> 01:07:09.090 align:middle line:90% It's not so easy? 01:07:09.090 --> 01:07:13.450 align:middle line:84% It turns out going first is generally an advantage. 01:07:13.450 --> 01:07:15.570 align:middle line:84% And the reason is that if I go first, 01:07:15.570 --> 01:07:19.850 align:middle line:84% I can take advantage of the fact that, yes, we both experience 01:07:19.850 --> 01:07:20.950 align:middle line:90% a cost of delay. 01:07:20.950 --> 01:07:22.650 align:middle line:84% But, in particular, the other party 01:07:22.650 --> 01:07:24.270 align:middle line:90% experiences a cost of delay. 01:07:24.270 --> 01:07:26.450 align:middle line:84% And therefore, they must be willing to accept offers 01:07:26.450 --> 01:07:29.030 align:middle line:90% that aren't that good for them. 01:07:29.030 --> 01:07:30.942 align:middle line:84% So if I go first, I can say, look, 01:07:30.942 --> 01:07:32.650 align:middle line:84% I'm going to give you a pretty bad offer, 01:07:32.650 --> 01:07:34.450 align:middle line:84% and I know you're going to have to accept it. 01:07:34.450 --> 01:07:35.890 align:middle line:84% Because if you don't accept it, you're 01:07:35.890 --> 01:07:37.570 align:middle line:84% going to have to pay the cost of delay. 01:07:37.570 --> 01:07:39.715 align:middle line:84% And this gives kind of a first mover advantage. 01:07:39.715 --> 01:07:41.090 align:middle line:84% And it turns out what we're going 01:07:41.090 --> 01:07:46.890 align:middle line:84% to get here is P0 star is going to be delta over 1 01:07:46.890 --> 01:07:48.330 align:middle line:90% plus delta times v. 01:07:48.330 --> 01:07:52.250 align:middle line:84% So if the buyer goes first, the price the buyer is going to get 01:07:52.250 --> 01:07:55.235 align:middle line:84% is going to be strictly smaller by a factor of delta 01:07:55.235 --> 01:07:56.610 align:middle line:84% relative to what the seller would 01:07:56.610 --> 01:07:59.290 align:middle line:90% get if the seller went first. 01:07:59.290 --> 01:08:02.070 align:middle line:84% And here, we can see the role of discounting. 01:08:02.070 --> 01:08:04.270 align:middle line:90% What happens if delta goes to 1? 01:08:04.270 --> 01:08:08.050 align:middle line:90% 01:08:08.050 --> 01:08:12.450 align:middle line:84% If delta goes to 1, the price the seller 01:08:12.450 --> 01:08:17.170 align:middle line:84% gets if they go first converges to v over 2. 01:08:17.170 --> 01:08:19.770 align:middle line:84% And the price the buyer gets if they go first also 01:08:19.770 --> 01:08:21.609 align:middle line:90% converges to v over 2. 01:08:21.609 --> 01:08:24.250 align:middle line:84% So we see as delta goes to 1, and the players 01:08:24.250 --> 01:08:28.270 align:middle line:84% become very impatient, the first mover advantage disappears. 01:08:28.270 --> 01:08:43.560 align:middle line:90% 01:08:43.560 --> 01:08:45.560 align:middle line:84% Can someone see-- if they can provide intuition, 01:08:45.560 --> 01:08:50.260 align:middle line:84% why does the first mover advantage disappear 01:08:50.260 --> 01:08:52.080 align:middle line:90% if delta goes to 1? 01:08:52.080 --> 01:08:52.580 align:middle line:90% Yeah? 01:08:52.580 --> 01:08:54.497 align:middle line:84% STUDENT: Just to make sure, when delta goes 1, 01:08:54.497 --> 01:08:56.319 align:middle line:84% they both are very patient [INAUDIBLE]. 01:08:56.319 --> 01:08:57.319 align:middle line:90% IAN BALL: Exactly right. 01:08:57.319 --> 01:08:59.420 align:middle line:84% STUDENT: [INAUDIBLE] if they don't 01:08:59.420 --> 01:09:02.420 align:middle line:90% care about waiting [INAUDIBLE]. 01:09:02.420 --> 01:09:03.140 align:middle line:90% IAN BALL: Right. 01:09:03.140 --> 01:09:06.740 align:middle line:84% If we get very, very patient, then it shouldn't really 01:09:06.740 --> 01:09:08.660 align:middle line:90% matter who goes first. 01:09:08.660 --> 01:09:11.000 align:middle line:84% Because tomorrow the other person gets to go first. 01:09:11.000 --> 01:09:13.380 align:middle line:84% And tomorrow isn't that much different than today 01:09:13.380 --> 01:09:15.200 align:middle line:90% if delta gets very close to 1. 01:09:15.200 --> 01:09:17.805 align:middle line:84% Remember, another interpretation of delta going to 1, 01:09:17.805 --> 01:09:19.180 align:middle line:84% one interpretation is the players 01:09:19.180 --> 01:09:20.979 align:middle line:90% are literally more patient. 01:09:20.979 --> 01:09:23.420 align:middle line:84% Another interpretation is they can make offers 01:09:23.420 --> 01:09:25.300 align:middle line:90% in very rapid succession. 01:09:25.300 --> 01:09:28.700 align:middle line:84% So it could be that the reason period 1 is not very discounted 01:09:28.700 --> 01:09:31.649 align:middle line:84% relative to period 0 is that the offer made in period 1 01:09:31.649 --> 01:09:33.399 align:middle line:84% happens right after the offer in period 0. 01:09:33.399 --> 01:09:35.040 align:middle line:84% They happen in really quick succession. 01:09:35.040 --> 01:09:37.020 align:middle line:84% And intuitively, if we're making alternating 01:09:37.020 --> 01:09:39.020 align:middle line:84% offers every millisecond, it shouldn't really 01:09:39.020 --> 01:09:40.158 align:middle line:90% matter who goes first. 01:09:40.158 --> 01:09:41.450 align:middle line:90% And that's exactly what we see. 01:09:41.450 --> 01:09:44.069 align:middle line:84% The first mover advantage disappears. 01:09:44.069 --> 01:09:46.830 align:middle line:84% Let me just to conclude, try to derive, 01:09:46.830 --> 01:09:49.930 align:middle line:84% give a heuristic derivation of these limiting formulas, 01:09:49.930 --> 01:09:51.490 align:middle line:90% and then we'll conclude. 01:09:51.490 --> 01:10:07.230 align:middle line:84% So let's-- so how can we try to solve for, let's say, 01:10:07.230 --> 01:10:08.050 align:middle line:90% the first formula? 01:10:08.050 --> 01:10:12.650 align:middle line:84% So let's say the seller proposes. 01:10:12.650 --> 01:10:16.750 align:middle line:90% 01:10:16.750 --> 01:10:33.150 align:middle line:84% Then what we expect is if the seller proposes, and we're 01:10:33.150 --> 01:10:36.630 align:middle line:84% taking the limit as the number of periods 01:10:36.630 --> 01:10:40.550 align:middle line:84% get very large, then the difference between period 01:10:40.550 --> 01:10:44.950 align:middle line:84% 0 and period 2, we think that it should get very, very close. 01:10:44.950 --> 01:10:48.550 align:middle line:84% Because in period 2, we have a billion periods to go. 01:10:48.550 --> 01:10:51.510 align:middle line:84% In period 0, we have a billion plus 2 periods to go. 01:10:51.510 --> 01:10:54.630 align:middle line:84% Since it's a billion, they should be pretty similar. 01:10:54.630 --> 01:10:55.930 align:middle line:90% This is a heuristic derivation. 01:10:55.930 --> 01:10:59.230 align:middle line:84% And now let's plug this in and see what we can get. 01:10:59.230 --> 01:11:05.070 align:middle line:84% So we get P0 star from this recursion 01:11:05.070 --> 01:11:11.450 align:middle line:84% is equal to 1 minus delta times v plus delta times P1 star. 01:11:11.450 --> 01:11:14.790 align:middle line:90% 01:11:14.790 --> 01:11:21.230 align:middle line:84% But then using this formula, we know that P1 star 01:11:21.230 --> 01:11:26.850 align:middle line:90% equals delta times P2 star. 01:11:26.850 --> 01:11:30.710 align:middle line:84% I'm just applying this formula with t equals 0, 01:11:30.710 --> 01:11:33.470 align:middle line:84% and then I'm applying this formula with t equals 1. 01:11:33.470 --> 01:11:38.560 align:middle line:84% So now let's plug this into here, 01:11:38.560 --> 01:11:48.120 align:middle line:84% and we get P0 star equals 1 minus delta v plus delta squared 01:11:48.120 --> 01:11:48.980 align:middle line:90% P2 star. 01:11:48.980 --> 01:11:56.240 align:middle line:90% 01:11:56.240 --> 01:12:00.120 align:middle line:84% Now, in general, P0 star and P2 star are going to be different. 01:12:00.120 --> 01:12:03.592 align:middle line:84% But in the limit as n goes to infinity, P0 and P2 star 01:12:03.592 --> 01:12:05.300 align:middle line:84% are going to look really, really similar. 01:12:05.300 --> 01:12:07.240 align:middle line:84% And the reason is that, again, if there's 01:12:07.240 --> 01:12:10.180 align:middle line:84% many, many periods ahead of us, a billion periods and a billion 01:12:10.180 --> 01:12:12.180 align:middle line:84% plus 2 periods, it should be basically the same. 01:12:12.180 --> 01:12:15.780 align:middle line:84% So let's squint and make this P0 star. 01:12:15.780 --> 01:12:18.640 align:middle line:90% 01:12:18.640 --> 01:12:20.480 align:middle line:90% Now let's try to solve this. 01:12:20.480 --> 01:12:24.500 align:middle line:90% We bring this to the other side. 01:12:24.500 --> 01:12:34.120 align:middle line:84% So we get 1 minus delta squared P0 star equals 01:12:34.120 --> 01:12:40.840 align:middle line:84% 1 minus delta v. And now there's one kind of trick that I think 01:12:40.840 --> 01:12:43.220 align:middle line:84% is kind of good to remember on problem sets. 01:12:43.220 --> 01:12:46.740 align:middle line:84% It's not so clear if we solve for this, 01:12:46.740 --> 01:12:51.920 align:middle line:84% we get P0 star equals 1 minus delta over 1 minus delta 01:12:51.920 --> 01:12:54.960 align:middle line:90% squared times v. Uh-oh. 01:12:54.960 --> 01:12:58.500 align:middle line:84% That's not the formula we got, but it simplifies. 01:12:58.500 --> 01:12:59.000 align:middle line:90% Yeah? 01:12:59.000 --> 01:12:59.720 align:middle line:90% STUDENT: Factor. 01:12:59.720 --> 01:13:01.040 align:middle line:90% IAN BALL: We can factor, right? 01:13:01.040 --> 01:13:05.360 align:middle line:84% 1 minus delta squared is just 1 minus delta times 1 plus delta. 01:13:05.360 --> 01:13:08.840 align:middle line:84% The 1 minus deltas cancel, and we exactly get 1 over 1 01:13:08.840 --> 01:13:13.450 align:middle line:84% plus delta times v. So again, I don't 01:13:13.450 --> 01:13:15.200 align:middle line:84% want to make the problem set for the exams 01:13:15.200 --> 01:13:17.360 align:middle line:84% all about doing algebra, but this is one trick 01:13:17.360 --> 01:13:18.660 align:middle line:90% that I think is good to know. 01:13:18.660 --> 01:13:21.515 align:middle line:90% 01:13:21.515 --> 01:13:24.140 align:middle line:84% Let me stop there-- a little bit early, but I think we're good. 01:13:24.140 --> 01:13:28.000 align:middle line:84% And we'll continue on Tuesday with subgame perfect Nash 01:13:28.000 --> 01:13:28.840 align:middle line:90% equilibrium. 01:13:28.840 --> 01:13:30.710 align:middle line:90% I'll see you then. 01:13:30.710 --> 01:13:34.000 align:middle line:90%