WEBVTT 00:00:00.000 --> 00:00:00.985 align:middle line:90% 00:00:00.985 --> 00:00:02.440 align:middle line:90% [SQUEAKING] 00:00:02.440 --> 00:00:04.380 align:middle line:90% [RUSTLING] 00:00:04.380 --> 00:00:07.775 align:middle line:90% [CLICKING] 00:00:07.775 --> 00:00:09.240 align:middle line:90% 00:00:09.240 --> 00:00:11.160 align:middle line:84% IAN BALL: And today, we're going to discuss 00:00:11.160 --> 00:00:15.180 align:middle line:84% a new topic, which is bargaining with incomplete information. 00:00:15.180 --> 00:00:25.000 align:middle line:90% 00:00:25.000 --> 00:00:27.000 align:middle line:90% Incomplete information. 00:00:27.000 --> 00:00:29.180 align:middle line:84% And you recall that earlier in the course, 00:00:29.180 --> 00:00:32.439 align:middle line:84% we studied a model of bargaining, but in that model, 00:00:32.439 --> 00:00:34.020 align:middle line:90% we had complete information. 00:00:34.020 --> 00:00:36.680 align:middle line:84% And when we studied bargaining with complete information, 00:00:36.680 --> 00:00:45.880 align:middle line:84% we observed-- so let's recall that with complete info, 00:00:45.880 --> 00:00:54.400 align:middle line:84% we had no disagreement and no delay. 00:00:54.400 --> 00:00:56.360 align:middle line:84% So when the two sides were bargaining, 00:00:56.360 --> 00:00:58.420 align:middle line:84% whenever there was an agreement to be reached, 00:00:58.420 --> 00:01:00.590 align:middle line:84% whenever there was some possibility for agreement, 00:01:00.590 --> 00:01:03.150 align:middle line:84% it was always achieved, and it was always 00:01:03.150 --> 00:01:04.690 align:middle line:90% achieved in the first period. 00:01:04.690 --> 00:01:07.110 align:middle line:84% There was never any of this costly delay. 00:01:07.110 --> 00:01:10.350 align:middle line:84% But of course, in reality, we see a lot of disagreement 00:01:10.350 --> 00:01:11.550 align:middle line:90% and a lot of delay. 00:01:11.550 --> 00:01:14.030 align:middle line:84% When we look at real bargaining in practice. 00:01:14.030 --> 00:01:16.430 align:middle line:84% And we argued that maybe the missing piece 00:01:16.430 --> 00:01:19.045 align:middle line:84% of our previous model was this very strong assumption 00:01:19.045 --> 00:01:20.670 align:middle line:84% of complete information, the assumption 00:01:20.670 --> 00:01:23.310 align:middle line:84% that both sides exactly understand the preferences 00:01:23.310 --> 00:01:24.370 align:middle line:90% of the other side. 00:01:24.370 --> 00:01:26.290 align:middle line:84% And just to make this a little more concrete, 00:01:26.290 --> 00:01:28.030 align:middle line:90% let's look at an example-- 00:01:28.030 --> 00:01:29.330 align:middle line:90% let's say plea bargaining. 00:01:29.330 --> 00:01:30.330 align:middle line:90% So this is common. 00:01:30.330 --> 00:01:33.230 align:middle line:90% 00:01:33.230 --> 00:01:35.790 align:middle line:84% You might have a defendant who's accused 00:01:35.790 --> 00:01:41.150 align:middle line:90% of a crime and a prosecutor. 00:01:41.150 --> 00:01:42.830 align:middle line:84% And before potentially going to trial, 00:01:42.830 --> 00:01:45.197 align:middle line:84% they might negotiate some plea agreement which 00:01:45.197 --> 00:01:47.030 align:middle line:84% says you don't go to trial, and instead, you 00:01:47.030 --> 00:01:48.790 align:middle line:90% serve a certain amount of time. 00:01:48.790 --> 00:01:50.310 align:middle line:84% And here, I think there's definitely 00:01:50.310 --> 00:01:52.750 align:middle line:84% a role for incomplete information. 00:01:52.750 --> 00:01:55.005 align:middle line:84% So let's put yourself in the role of the defendant, 00:01:55.005 --> 00:01:57.130 align:middle line:84% and let's say you really want to reach a plea deal, 00:01:57.130 --> 00:01:59.690 align:middle line:84% you don't want to go to court, so you make a really generous 00:01:59.690 --> 00:02:01.630 align:middle line:90% offer to the prosecutor. 00:02:01.630 --> 00:02:04.490 align:middle line:84% You say, look, I'll plead guilty and I'll serve 30 years 00:02:04.490 --> 00:02:05.750 align:middle line:90% to avoid going to court. 00:02:05.750 --> 00:02:07.250 align:middle line:84% Now generally you would think making 00:02:07.250 --> 00:02:11.230 align:middle line:84% a very generous offer would help you get the offer approved, 00:02:11.230 --> 00:02:14.310 align:middle line:84% but what might the prosecutor think if you say, 00:02:14.310 --> 00:02:15.710 align:middle line:90% I'm happy to serve 30 years? 00:02:15.710 --> 00:02:16.493 align:middle line:90% Yes? 00:02:16.493 --> 00:02:18.410 align:middle line:84% That you're super guilty or maybe the evidence 00:02:18.410 --> 00:02:20.070 align:middle line:84% is really-- yeah, the evidence is really, really strong, 00:02:20.070 --> 00:02:21.910 align:middle line:84% it's going to be easy to show in court. 00:02:21.910 --> 00:02:24.970 align:middle line:84% So there's this perverse effect where normally a better offer is 00:02:24.970 --> 00:02:27.610 align:middle line:84% more likely to be accepted, but now a better offer might 00:02:27.610 --> 00:02:29.930 align:middle line:84% suggest, wait a second, there's a reason 00:02:29.930 --> 00:02:31.510 align:middle line:84% she's giving me a really good offer, 00:02:31.510 --> 00:02:32.910 align:middle line:84% and maybe I actually don't want to accept it 00:02:32.910 --> 00:02:34.035 align:middle line:90% and I'd rather go to court. 00:02:34.035 --> 00:02:36.730 align:middle line:84% So in this case, the defendant has private information 00:02:36.730 --> 00:02:40.170 align:middle line:84% about certainly their guilt, but maybe more relevantly, 00:02:40.170 --> 00:02:42.490 align:middle line:84% how easy it is to prove their guilt. 00:02:42.490 --> 00:02:46.370 align:middle line:84% And the defendant, their offer maybe conveys 00:02:46.370 --> 00:02:49.930 align:middle line:90% information about-- 00:02:49.930 --> 00:02:54.285 align:middle line:84% we'll just say guilt or evidence of guilt or other things. 00:02:54.285 --> 00:02:55.660 align:middle line:84% Now let's look at the other side. 00:02:55.660 --> 00:02:57.840 align:middle line:84% The prosecutor may also have private information. 00:02:57.840 --> 00:03:00.960 align:middle line:84% Let's say the prosecutor also really wants to get a deal. 00:03:00.960 --> 00:03:02.692 align:middle line:84% And the prosecutor says, look, I'll 00:03:02.692 --> 00:03:04.400 align:middle line:84% give you a really generous deal, you only 00:03:04.400 --> 00:03:07.180 align:middle line:84% have to serve six months in prison if you plead guilty, 00:03:07.180 --> 00:03:08.840 align:middle line:84% and then we don't have to go to court. 00:03:08.840 --> 00:03:10.320 align:middle line:90% That sounds like a great deal. 00:03:10.320 --> 00:03:12.520 align:middle line:84% Why might the defendant be hesitant 00:03:12.520 --> 00:03:14.150 align:middle line:84% if they received an offer like that? 00:03:14.150 --> 00:03:15.900 align:middle line:84% A really generous offer by the prosecutor? 00:03:15.900 --> 00:03:16.300 align:middle line:90% Yeah? 00:03:16.300 --> 00:03:17.740 align:middle line:84% AUDIENCE: If we assume the prosecutor doesn't 00:03:17.740 --> 00:03:19.500 align:middle line:84% have any evidence, so if they go to court, 00:03:19.500 --> 00:03:21.140 align:middle line:84% the defendant might be able to have-- 00:03:21.140 --> 00:03:21.997 align:middle line:90% IAN BALL: Exactly. 00:03:21.997 --> 00:03:23.580 align:middle line:84% So this is, I think, the key question. 00:03:23.580 --> 00:03:25.330 align:middle line:84% They don't just say how good is the offer, 00:03:25.330 --> 00:03:27.380 align:middle line:84% they say, what is the quality of the offer? 00:03:27.380 --> 00:03:29.180 align:middle line:84% Tell me about the other side's information. 00:03:29.180 --> 00:03:30.580 align:middle line:84% And maybe they just have no case. 00:03:30.580 --> 00:03:32.360 align:middle line:84% So maybe I should go to court even though this offer 00:03:32.360 --> 00:03:33.340 align:middle line:90% seems really good. 00:03:33.340 --> 00:03:37.200 align:middle line:84% So here, the prosecutor's offer might convey info 00:03:37.200 --> 00:03:41.240 align:middle line:84% about maybe the evidence they have. 00:03:41.240 --> 00:03:43.800 align:middle line:84% And I think-- legal scholars certainly 00:03:43.800 --> 00:03:47.000 align:middle line:84% think this is exactly why we often see disagreement and delay 00:03:47.000 --> 00:03:49.720 align:middle line:84% in bargaining because the kinds of offers that you might make 00:03:49.720 --> 00:03:53.190 align:middle line:84% to try to reach agreement can convey information that make it 00:03:53.190 --> 00:03:55.910 align:middle line:90% harder to reach an agreement. 00:03:55.910 --> 00:04:01.830 align:middle line:84% So today, we're going to look at two general classes 00:04:01.830 --> 00:04:05.030 align:middle line:90% of bargaining games. 00:04:05.030 --> 00:04:08.190 align:middle line:84% And I think it's helpful, at first, 00:04:08.190 --> 00:04:17.149 align:middle line:84% to distinguish two different situations, signaling 00:04:17.149 --> 00:04:18.789 align:middle line:90% and screening. 00:04:18.789 --> 00:04:21.450 align:middle line:84% So, so far, we already talked about signaling games 00:04:21.450 --> 00:04:24.710 align:middle line:84% last class where someone might choose a level of education 00:04:24.710 --> 00:04:27.830 align:middle line:84% to try to signal their ability as a worker. 00:04:27.830 --> 00:04:30.830 align:middle line:84% And more generally, in signaling games, what distinguishes them 00:04:30.830 --> 00:04:35.650 align:middle line:84% is that the first mover has private information. 00:04:35.650 --> 00:04:41.590 align:middle line:90% 00:04:41.590 --> 00:04:43.510 align:middle line:84% So when the first mover, the player 00:04:43.510 --> 00:04:46.410 align:middle line:84% who takes an action first, has private information, 00:04:46.410 --> 00:04:48.710 align:middle line:84% their action is going to convey something 00:04:48.710 --> 00:04:50.050 align:middle line:90% about their private information. 00:04:50.050 --> 00:04:52.750 align:middle line:84% So their action might reveal or convey information. 00:04:52.750 --> 00:04:59.527 align:middle line:90% 00:04:59.527 --> 00:05:01.110 align:middle line:84% And when the first mover is strategic, 00:05:01.110 --> 00:05:03.610 align:middle line:84% they're going to think through how their action might convey 00:05:03.610 --> 00:05:07.090 align:middle line:84% information, and that's going to affect their choice of action. 00:05:07.090 --> 00:05:09.170 align:middle line:90% In screening. 00:05:09.170 --> 00:05:13.350 align:middle line:84% It's the second mover that has private information. 00:05:13.350 --> 00:05:18.438 align:middle line:90% 00:05:18.438 --> 00:05:20.230 align:middle line:84% So you might say, oh, well the first mover, 00:05:20.230 --> 00:05:22.230 align:middle line:84% they're going first, they don't have any private information. 00:05:22.230 --> 00:05:24.230 align:middle line:84% Private information doesn't have much of a role. 00:05:24.230 --> 00:05:26.810 align:middle line:84% But it still has a role because what the first mover is 00:05:26.810 --> 00:05:29.210 align:middle line:84% potentially going to do is they're 00:05:29.210 --> 00:05:31.650 align:middle line:84% going to choose their action, anticipating 00:05:31.650 --> 00:05:34.170 align:middle line:84% that the response to their action conveys information 00:05:34.170 --> 00:05:35.650 align:middle line:90% that the other side has. 00:05:35.650 --> 00:05:39.370 align:middle line:84% So if I make an opening offer to you, 00:05:39.370 --> 00:05:41.330 align:middle line:84% and your choice whether to accept or reject 00:05:41.330 --> 00:05:44.617 align:middle line:84% that opening offer conveys to me information that you have. 00:05:44.617 --> 00:05:46.450 align:middle line:84% So here, they're signaling information here, 00:05:46.450 --> 00:05:48.750 align:middle line:84% you might call they're kind of fishing for information. 00:05:48.750 --> 00:05:52.120 align:middle line:84% They're choosing an action so that the response to that action 00:05:52.120 --> 00:05:54.040 align:middle line:84% conveys information from the other party. 00:05:54.040 --> 00:05:57.580 align:middle line:84% So maybe I'll say fishing for information. 00:05:57.580 --> 00:06:04.200 align:middle line:90% 00:06:04.200 --> 00:06:07.378 align:middle line:84% And the formal term screening-- oh, I guess the first mover, 00:06:07.378 --> 00:06:08.420 align:middle line:90% I should say, is the one. 00:06:08.420 --> 00:06:12.880 align:middle line:90% 00:06:12.880 --> 00:06:14.760 align:middle line:84% And the term screening comes with the idea 00:06:14.760 --> 00:06:16.100 align:middle line:90% that your response-- 00:06:16.100 --> 00:06:18.240 align:middle line:84% I'm the first mover, your response to my action 00:06:18.240 --> 00:06:21.162 align:middle line:84% screens you and distinguishes different players 00:06:21.162 --> 00:06:22.620 align:middle line:84% with different private information. 00:06:22.620 --> 00:06:23.870 align:middle line:90% So it's a screening mechanism. 00:06:23.870 --> 00:06:25.760 align:middle line:90% I might offer something to you. 00:06:25.760 --> 00:06:28.103 align:middle line:84% I might say, oh, you really want this? 00:06:28.103 --> 00:06:30.020 align:middle line:84% Well, let me offer it to you for a high price, 00:06:30.020 --> 00:06:31.520 align:middle line:84% and that will screen out people who actually 00:06:31.520 --> 00:06:33.600 align:middle line:84% want it and are willing to pay a high price from people 00:06:33.600 --> 00:06:34.180 align:middle line:90% who are not. 00:06:34.180 --> 00:06:36.360 align:middle line:84% That's where the term screening comes from. 00:06:36.360 --> 00:06:38.863 align:middle line:84% So today we're going to study bargaining 00:06:38.863 --> 00:06:40.280 align:middle line:84% with incomplete information, we're 00:06:40.280 --> 00:06:42.860 align:middle line:84% going to look at both the cases of signaling and screening, 00:06:42.860 --> 00:06:45.640 align:middle line:84% and we're going to do it all within a canonical framework 00:06:45.640 --> 00:06:47.808 align:middle line:84% of just a buyer and seller negotiating 00:06:47.808 --> 00:06:48.850 align:middle line:90% over the price of a good. 00:06:48.850 --> 00:06:51.650 align:middle line:84% Of course, these ideas can be applied much more generally, 00:06:51.650 --> 00:06:54.208 align:middle line:84% but these games are a bit harder to analyze than the ones 00:06:54.208 --> 00:06:56.750 align:middle line:84% we've done before, so we want to stick to a very simple model 00:06:56.750 --> 00:06:59.470 align:middle line:84% to begin with, but the lessons extend a lot more broadly. 00:06:59.470 --> 00:07:02.250 align:middle line:84% So today, we're going to look at a buyer and a seller. 00:07:02.250 --> 00:07:09.590 align:middle line:90% 00:07:09.590 --> 00:07:12.110 align:middle line:84% And they're going to be bargaining 00:07:12.110 --> 00:07:15.670 align:middle line:84% over the price at which they exchange the good. 00:07:15.670 --> 00:07:18.270 align:middle line:84% So we have our buyer and our seller. 00:07:18.270 --> 00:07:21.618 align:middle line:84% The buyer has some value v for the good. 00:07:21.618 --> 00:07:23.910 align:middle line:84% And we're going to make different assumptions about who 00:07:23.910 --> 00:07:25.620 align:middle line:84% knows v and who doesn't, but it's always 00:07:25.620 --> 00:07:27.870 align:middle line:84% going to be called v. So this is how much the buyer is 00:07:27.870 --> 00:07:29.550 align:middle line:90% willing to pay for the good. 00:07:29.550 --> 00:07:33.350 align:middle line:90% And the seller has some cost c. 00:07:33.350 --> 00:07:36.950 align:middle line:84% I should point out here that it's just nice to use v and c 00:07:36.950 --> 00:07:39.430 align:middle line:84% And think of c as a cost, but another interpretation 00:07:39.430 --> 00:07:42.590 align:middle line:84% of the cost of the seller is how much they value the item. 00:07:42.590 --> 00:07:45.770 align:middle line:84% Because remember, the cost c is what the seller loses 00:07:45.770 --> 00:07:47.290 align:middle line:90% when they give up the item. 00:07:47.290 --> 00:07:49.838 align:middle line:84% And we can interpret that utility loss in two ways. 00:07:49.838 --> 00:07:52.130 align:middle line:84% Either they actually have to produce the item at a cost 00:07:52.130 --> 00:07:53.690 align:middle line:84% in order to give it to the other player, 00:07:53.690 --> 00:07:55.190 align:middle line:84% or they actually like it themselves, 00:07:55.190 --> 00:07:56.670 align:middle line:84% and when they give it up, they lose that. 00:07:56.670 --> 00:07:58.570 align:middle line:84% So we'll call it a cost c, but you could also 00:07:58.570 --> 00:08:00.230 align:middle line:84% think of it as the seller's value, 00:08:00.230 --> 00:08:02.050 align:middle line:84% it's just, this way, when I say value, 00:08:02.050 --> 00:08:04.490 align:middle line:84% we know we're talking about the buyer. 00:08:04.490 --> 00:08:05.750 align:middle line:90% So what are their utilities? 00:08:05.750 --> 00:08:12.970 align:middle line:84% Well, if they exchange at a price of p, 00:08:12.970 --> 00:08:17.010 align:middle line:84% then the buyer's utility is v minus p. 00:08:17.010 --> 00:08:19.870 align:middle line:84% They get value v from the good because they're exchanging, 00:08:19.870 --> 00:08:22.370 align:middle line:84% but they have to pay a price p to the seller. 00:08:22.370 --> 00:08:25.090 align:middle line:84% And then the seller's utility is p 00:08:25.090 --> 00:08:28.470 align:middle line:84% minus c because the price p is what goes to the seller, 00:08:28.470 --> 00:08:30.971 align:middle line:84% the seller gets p dollars, but has 00:08:30.971 --> 00:08:32.929 align:middle line:84% to bear the cost c either of producing the good 00:08:32.929 --> 00:08:34.690 align:middle line:90% or of losing it. 00:08:34.690 --> 00:08:40.890 align:middle line:84% And then if there's no exchange, well now, we just 00:08:40.890 --> 00:08:44.440 align:middle line:84% say each player gets utility 0 because the good is not 00:08:44.440 --> 00:08:46.680 align:middle line:84% exchanged and money does not change hands. 00:08:46.680 --> 00:08:51.620 align:middle line:84% So here, we have uB equals 0 and uS equals 0. 00:08:51.620 --> 00:08:55.560 align:middle line:90% 00:08:55.560 --> 00:09:02.880 align:middle line:84% Now let's-- to make some progress here, at first, 00:09:02.880 --> 00:09:07.640 align:middle line:84% let's first think about, well, if we think we're going 00:09:07.640 --> 00:09:11.440 align:middle line:84% to-- if we might see exchange, what are the possible ranges 00:09:11.440 --> 00:09:14.880 align:middle line:84% of prices at which exchange takes place? 00:09:14.880 --> 00:09:17.625 align:middle line:84% Well, at least if v and c are known-- so let's 00:09:17.625 --> 00:09:18.500 align:middle line:90% start with this case. 00:09:18.500 --> 00:09:20.800 align:middle line:84% So case 1 is kind of the easy case. 00:09:20.800 --> 00:09:23.740 align:middle line:84% Case 1 is where v and c are publicly known. 00:09:23.740 --> 00:09:26.360 align:middle line:90% 00:09:26.360 --> 00:09:28.860 align:middle line:84% So this is the case that we studied earlier in the course. 00:09:28.860 --> 00:09:31.300 align:middle line:84% This is actually a case of complete information. 00:09:31.300 --> 00:09:39.260 align:middle line:90% 00:09:39.260 --> 00:09:40.760 align:middle line:84% So let's review this, and then we'll 00:09:40.760 --> 00:09:43.950 align:middle line:84% contrast it with the other cases. 00:09:43.950 --> 00:09:48.210 align:middle line:84% Well, if v and c are public, there's something we can say. 00:09:48.210 --> 00:09:53.430 align:middle line:84% If trade happens at p, we can say a lot 00:09:53.430 --> 00:09:55.570 align:middle line:90% about what p must satisfy. 00:09:55.570 --> 00:09:58.230 align:middle line:84% If v and c are public-- let's suppose we see the players 00:09:58.230 --> 00:09:59.470 align:middle line:90% trade at price p. 00:09:59.470 --> 00:10:02.830 align:middle line:84% What do we know about how p compares to v and c? 00:10:02.830 --> 00:10:03.470 align:middle line:90% Yeah? 00:10:03.470 --> 00:10:06.308 align:middle line:84% AUDIENCE: So p has to be greater than c, but less than v. 00:10:06.308 --> 00:10:06.850 align:middle line:90% IAN BALL: OK. 00:10:06.850 --> 00:10:07.810 align:middle line:90% And can you explain why? 00:10:07.810 --> 00:10:08.350 align:middle line:90% Yeah? 00:10:08.350 --> 00:10:12.530 align:middle line:84% AUDIENCE: The reason is that we need the p to be at least 0, 00:10:12.530 --> 00:10:14.950 align:middle line:84% otherwise the buyer would choose not to exchange. 00:10:14.950 --> 00:10:15.950 align:middle line:90% IAN BALL: Exactly right. 00:10:15.950 --> 00:10:18.670 align:middle line:84% AUDIENCE: --p minus c could be greater than or equal to 0 00:10:18.670 --> 00:10:20.250 align:middle line:84% for the seller to want to exchange. 00:10:20.250 --> 00:10:21.250 align:middle line:90% IAN BALL: Exactly right. 00:10:21.250 --> 00:10:22.410 align:middle line:90% So we have two constraints. 00:10:22.410 --> 00:10:25.318 align:middle line:84% If they exchange at price p, this is voluntary exchange. 00:10:25.318 --> 00:10:27.110 align:middle line:84% It only happens if both players agree to it 00:10:27.110 --> 00:10:29.068 align:middle line:84% because either player can walk away and say no. 00:10:29.068 --> 00:10:31.630 align:middle line:84% So the only possible prices we could exchange 00:10:31.630 --> 00:10:34.190 align:middle line:84% are prices p that make each player weakly better 00:10:34.190 --> 00:10:36.870 align:middle line:84% off than their outside option of just walking away. 00:10:36.870 --> 00:10:40.137 align:middle line:84% So if we trade at price p, we must immediately have-- 00:10:40.137 --> 00:10:42.220 align:middle line:84% maybe I'll write it in two separate inequalities-- 00:10:42.220 --> 00:10:44.500 align:middle line:84% p greater than or equal to c and p less than 00:10:44.500 --> 00:10:48.520 align:middle line:84% or equal to v. This inequality is because of the seller, 00:10:48.520 --> 00:10:49.940 align:middle line:90% so maybe I'll put S here. 00:10:49.940 --> 00:10:51.780 align:middle line:84% If p were less than c, the seller 00:10:51.780 --> 00:10:54.220 align:middle line:84% would never want to trade at price p. 00:10:54.220 --> 00:10:56.620 align:middle line:90% And this comes from the buyer. 00:10:56.620 --> 00:10:58.320 align:middle line:84% If the price were greater than v, 00:10:58.320 --> 00:11:00.620 align:middle line:84% the buyer would simply walk away. 00:11:00.620 --> 00:11:04.940 align:middle line:84% So we immediately see from this that if trade is going to occur, 00:11:04.940 --> 00:11:07.140 align:middle line:84% we must have c smaller than p smaller 00:11:07.140 --> 00:11:11.340 align:middle line:84% than v, and in particular, that means c must be smaller than v. 00:11:11.340 --> 00:11:17.420 align:middle line:84% So let's assume that c is strictly smaller 00:11:17.420 --> 00:11:18.980 align:middle line:84% than v. This is a common assumption 00:11:18.980 --> 00:11:21.780 align:middle line:90% to rule out the trivial case. 00:11:21.780 --> 00:11:24.660 align:middle line:84% If v is strictly smaller than c, then there's 00:11:24.660 --> 00:11:26.900 align:middle line:84% no way we can ever have exchange. 00:11:26.900 --> 00:11:29.240 align:middle line:84% And if v equals c, well, then we could have exchange, 00:11:29.240 --> 00:11:31.800 align:middle line:84% but it's only going to be at the price p equals c equals v, 00:11:31.800 --> 00:11:33.480 align:middle line:84% and then there's no change in utility 00:11:33.480 --> 00:11:35.022 align:middle line:84% and it's not a very interesting case. 00:11:35.022 --> 00:11:36.980 align:middle line:84% So this is where sometimes what we say here 00:11:36.980 --> 00:11:38.460 align:middle line:90% is there are gains from trade. 00:11:38.460 --> 00:11:42.680 align:middle line:90% 00:11:42.680 --> 00:11:45.020 align:middle line:84% Some people say this is what economics is all about, 00:11:45.020 --> 00:11:45.883 align:middle line:90% gains from trade. 00:11:45.883 --> 00:11:47.300 align:middle line:84% Why are the gains from trade here? 00:11:47.300 --> 00:11:50.000 align:middle line:84% Well, when the good moves from the seller to the buyer, 00:11:50.000 --> 00:11:53.040 align:middle line:84% the buyer's utility goes up by v and the seller's utility 00:11:53.040 --> 00:11:54.200 align:middle line:90% goes down by c. 00:11:54.200 --> 00:11:56.080 align:middle line:84% And because v is bigger than c, it's 00:11:56.080 --> 00:11:57.818 align:middle line:84% possible for there to be a price such 00:11:57.818 --> 00:11:59.360 align:middle line:84% that both players are better off for. 00:11:59.360 --> 00:12:02.000 align:middle line:84% The gain to the buyer is bigger than the loss to the seller 00:12:02.000 --> 00:12:04.120 align:middle line:90% when we exchange. 00:12:04.120 --> 00:12:06.300 align:middle line:84% So in this case, we've done it before, 00:12:06.300 --> 00:12:08.900 align:middle line:84% but let's just go through the analysis. 00:12:08.900 --> 00:12:12.440 align:middle line:84% We need to make a choice, our modeling choice about the order 00:12:12.440 --> 00:12:13.630 align:middle line:90% in which the players move. 00:12:13.630 --> 00:12:15.880 align:middle line:84% So I think it's pretty standard, at least in practice, 00:12:15.880 --> 00:12:18.160 align:middle line:90% when we see-- 00:12:18.160 --> 00:12:20.280 align:middle line:84% certainly in stores, we see Sellers often 00:12:20.280 --> 00:12:21.378 align:middle line:90% post a price first. 00:12:21.378 --> 00:12:23.420 align:middle line:84% You could maybe walk in and offer your own price, 00:12:23.420 --> 00:12:25.500 align:middle line:84% and in one-off negotiations, that does happen, 00:12:25.500 --> 00:12:28.100 align:middle line:84% but let's think of the case where we have many goods, 00:12:28.100 --> 00:12:29.580 align:middle line:90% we tend to see it done this way. 00:12:29.580 --> 00:12:34.820 align:middle line:84% So let's make our timing be that the seller proposes a price. 00:12:34.820 --> 00:12:41.830 align:middle line:90% 00:12:41.830 --> 00:12:44.298 align:middle line:90% And then the buyer-- 00:12:44.298 --> 00:12:46.090 align:middle line:84% of course, it's always the potential buyer, 00:12:46.090 --> 00:12:47.550 align:middle line:84% they don't have to actually buy-- 00:12:47.550 --> 00:12:49.868 align:middle line:84% observes the price p that's offered, and then chooses 00:12:49.868 --> 00:12:50.910 align:middle line:90% whether to accept or not. 00:12:50.910 --> 00:12:53.290 align:middle line:84% Accept the offer or not, meaning buy or not buy. 00:12:53.290 --> 00:12:56.630 align:middle line:90% 00:12:56.630 --> 00:12:59.450 align:middle line:84% So maybe we'll say buy or not buy. 00:12:59.450 --> 00:13:03.470 align:middle line:90% 00:13:03.470 --> 00:13:05.680 align:middle line:84% And if we want to draw this in extensive form 00:13:05.680 --> 00:13:07.430 align:middle line:84% just as a-- so maybe this is a good review 00:13:07.430 --> 00:13:09.250 align:middle line:84% as we're getting closer to the exam. 00:13:09.250 --> 00:13:11.630 align:middle line:90% So we have the seller here. 00:13:11.630 --> 00:13:14.710 align:middle line:84% Now it's a little tricky because the price can take any value, 00:13:14.710 --> 00:13:16.330 align:middle line:90% say any non-negative value. 00:13:16.330 --> 00:13:19.050 align:middle line:84% So there's infinitely many prices the seller can offer. 00:13:19.050 --> 00:13:21.230 align:middle line:84% It's always a little hard to write that down, 00:13:21.230 --> 00:13:23.670 align:middle line:84% but maybe we'll write it something like this. 00:13:23.670 --> 00:13:26.310 align:middle line:90% I'll call this the price p. 00:13:26.310 --> 00:13:29.430 align:middle line:84% I'm visualizing this as the full range of prices 00:13:29.430 --> 00:13:31.630 align:middle line:90% the seller can offer. 00:13:31.630 --> 00:13:34.460 align:middle line:84% Maybe I'll put a distinguished one here. 00:13:34.460 --> 00:13:40.140 align:middle line:84% And then the buyer sees the price and makes a binary choice, 00:13:40.140 --> 00:13:42.020 align:middle line:84% I'll call it A and R, accept or reject. 00:13:42.020 --> 00:13:45.060 align:middle line:84% Accept means buy, reject means not buy. 00:13:45.060 --> 00:13:47.340 align:middle line:84% And then let's make sure we get the payoffs here. 00:13:47.340 --> 00:13:49.500 align:middle line:84% Remember, v and c are known, these are just 00:13:49.500 --> 00:13:50.820 align:middle line:90% parameters of the game. 00:13:50.820 --> 00:13:53.580 align:middle line:84% So if the price p is accepted by the buyer, 00:13:53.580 --> 00:13:55.840 align:middle line:84% then I think we put buyer first-- 00:13:55.840 --> 00:13:57.740 align:middle line:84% so let's do buyer utilities first. 00:13:57.740 --> 00:14:01.380 align:middle line:84% The buyer is going to get v minus p, the seller is 00:14:01.380 --> 00:14:05.115 align:middle line:90% going to get p minus c. 00:14:05.115 --> 00:14:06.740 align:middle line:84% And over here, it's just going to be 0, 00:14:06.740 --> 00:14:08.880 align:middle line:90% 0 because both sides reject. 00:14:08.880 --> 00:14:12.380 align:middle line:90% 00:14:12.380 --> 00:14:16.700 align:middle line:84% So now, let's try to understand, let's try 00:14:16.700 --> 00:14:20.620 align:middle line:84% to solve for a subgame perfect Nash equilibrium of this game. 00:14:20.620 --> 00:14:24.370 align:middle line:84% Notice, we can use the solution concept of subgame perfect Nash 00:14:24.370 --> 00:14:26.620 align:middle line:84% because there's no private information because this is 00:14:26.620 --> 00:14:28.120 align:middle line:90% a game of complete information. 00:14:28.120 --> 00:14:29.820 align:middle line:84% Once we move to incomplete information, 00:14:29.820 --> 00:14:32.100 align:middle line:84% we have to use perfect Bayesian equilibrium, 00:14:32.100 --> 00:14:34.440 align:middle line:84% but for this, we can just use subgame perfect Nash. 00:14:34.440 --> 00:14:38.520 align:middle line:84% So let's look for subgame perfect Nash equilibrium. 00:14:38.520 --> 00:14:39.920 align:middle line:90% What we want to do? 00:14:39.920 --> 00:14:41.533 align:middle line:84% Well, the trick with a game like this 00:14:41.533 --> 00:14:42.700 align:middle line:90% is we always work backwards. 00:14:42.700 --> 00:14:45.340 align:middle line:84% In fact with this game, we can apply backward induction. 00:14:45.340 --> 00:14:48.160 align:middle line:84% So let's start with the second player. 00:14:48.160 --> 00:14:50.920 align:middle line:90% This is the buyer. 00:14:50.920 --> 00:14:55.640 align:middle line:84% And we can ask, well, what offers 00:14:55.640 --> 00:14:58.000 align:middle line:90% is the buyer going to accept? 00:14:58.000 --> 00:15:01.502 align:middle line:84% Well, the buyer can either get 0 or take the offer. 00:15:01.502 --> 00:15:02.960 align:middle line:84% So they're going to accept an offer 00:15:02.960 --> 00:15:05.080 align:middle line:84% if their total utility from accepting the offer 00:15:05.080 --> 00:15:06.720 align:middle line:90% is at least 0. 00:15:06.720 --> 00:15:10.120 align:middle line:84% So the buyer is going to accept-- 00:15:10.120 --> 00:15:12.440 align:middle line:84% maybe I'll use a double arrow to say if and only 00:15:12.440 --> 00:15:18.000 align:middle line:84% if v minus p is greater than or equal to 0. 00:15:18.000 --> 00:15:20.600 align:middle line:84% This is the utility of accepting the offer, 00:15:20.600 --> 00:15:22.740 align:middle line:84% this is their utility of rejecting the offer, 00:15:22.740 --> 00:15:25.280 align:middle line:84% and they're going to accept if the utility from accepting 00:15:25.280 --> 00:15:27.320 align:middle line:84% is better than the utility of rejecting. 00:15:27.320 --> 00:15:30.230 align:middle line:84% There's a little flexibility if this equals exactly 0, 00:15:30.230 --> 00:15:34.090 align:middle line:84% but a common theme, when you're looking for equilibria, 00:15:34.090 --> 00:15:35.790 align:middle line:84% it's often better to break ties this way 00:15:35.790 --> 00:15:37.582 align:middle line:84% and say that when indifferent, they accept. 00:15:37.582 --> 00:15:42.270 align:middle line:84% It tends to make sure the sender deals with some existence 00:15:42.270 --> 00:15:44.887 align:middle line:84% problems for the sender's choice. 00:15:44.887 --> 00:15:47.470 align:middle line:84% Well, another way of saying this is they're going to accept if 00:15:47.470 --> 00:15:50.710 align:middle line:84% and only if v is greater than or equal to p. 00:15:50.710 --> 00:15:53.710 align:middle line:84% And anticipating that, we can now 00:15:53.710 --> 00:15:56.450 align:middle line:84% go to the first mover, the sender-- the seller. 00:15:56.450 --> 00:16:00.132 align:middle line:90% 00:16:00.132 --> 00:16:01.590 align:middle line:84% And what is the seller going to do? 00:16:01.590 --> 00:16:04.550 align:middle line:84% If they anticipate the buyer will accept if and only 00:16:04.550 --> 00:16:06.870 align:middle line:84% if v is greater than or equal to p, well, then, 00:16:06.870 --> 00:16:09.295 align:middle line:84% what price do they want to offer? 00:16:09.295 --> 00:16:10.670 align:middle line:84% AUDIENCE: Well, wouldn't you just 00:16:10.670 --> 00:16:11.950 align:middle line:84% offer the highest price possible? 00:16:11.950 --> 00:16:14.408 align:middle line:84% IAN BALL: The highest price possible that will be accepted. 00:16:14.408 --> 00:16:18.350 align:middle line:84% And in this case, that's just p equals v. Notice here, 00:16:18.350 --> 00:16:21.782 align:middle line:84% we're using our assumption that c is less than v. 00:16:21.782 --> 00:16:23.990 align:middle line:84% Because if v was less than c, they might not actually 00:16:23.990 --> 00:16:26.890 align:middle line:84% want to offer v because that wouldn't cover their costs, 00:16:26.890 --> 00:16:29.980 align:middle line:84% but under our assumption that v is at least as big as c, then 00:16:29.980 --> 00:16:32.660 align:middle line:84% the seller is going to make-- offer the highest price that 00:16:32.660 --> 00:16:34.980 align:middle line:84% will be accepted, that's p equals v, 00:16:34.980 --> 00:16:37.680 align:middle line:84% and then we can see what are the utilities going to be. 00:16:37.680 --> 00:16:41.060 align:middle line:90% 00:16:41.060 --> 00:16:44.180 align:middle line:84% Well, the buyer is going to get v minus v. The first v 00:16:44.180 --> 00:16:45.200 align:middle line:90% is their value. 00:16:45.200 --> 00:16:48.300 align:middle line:84% The second v is actually the price, but it equals the value, 00:16:48.300 --> 00:16:50.260 align:middle line:90% so that's going to be 0. 00:16:50.260 --> 00:16:57.160 align:middle line:84% And the sender is going to get v minus c. 00:16:57.160 --> 00:17:00.360 align:middle line:84% v here is playing the role of the price minus the cost, 00:17:00.360 --> 00:17:01.920 align:middle line:84% and this is strictly greater than 0. 00:17:01.920 --> 00:17:06.060 align:middle line:90% 00:17:06.060 --> 00:17:09.359 align:middle line:84% Sometimes terminology you might say, remember, we called v-- 00:17:09.359 --> 00:17:11.740 align:middle line:84% v minus c, we said there's gains from trade 00:17:11.740 --> 00:17:13.339 align:middle line:90% because v is greater than c. 00:17:13.339 --> 00:17:15.819 align:middle line:84% We sometimes think of v minus c as the potential gains 00:17:15.819 --> 00:17:16.640 align:middle line:90% from trade. 00:17:16.640 --> 00:17:19.900 align:middle line:84% This is the amount of utility that can go up if they trade. 00:17:19.900 --> 00:17:22.380 align:middle line:84% And here, we sometimes say the seller has fully 00:17:22.380 --> 00:17:23.880 align:middle line:90% extracted the gains from trade. 00:17:23.880 --> 00:17:25.338 align:middle line:84% Maybe I'll say this-- or maybe I'll 00:17:25.338 --> 00:17:29.180 align:middle line:84% say the seller gets all gains from trade. 00:17:29.180 --> 00:17:33.643 align:middle line:90% 00:17:33.643 --> 00:17:35.560 align:middle line:84% In this game, we can think of this-- remember, 00:17:35.560 --> 00:17:38.240 align:middle line:84% before we studied this game, split the dollar. 00:17:38.240 --> 00:17:39.780 align:middle line:84% v minus c is basically the dollar. 00:17:39.780 --> 00:17:42.160 align:middle line:84% This is the potential gains that are on the table. 00:17:42.160 --> 00:17:44.560 align:middle line:84% It's going to be split in some way between the buyer 00:17:44.560 --> 00:17:48.280 align:middle line:84% and the seller, and in this case, the buyer gets none of it 00:17:48.280 --> 00:17:49.940 align:middle line:90% and the seller gets all of this. 00:17:49.940 --> 00:17:52.295 align:middle line:84% And this is because the seller gets to move first, 00:17:52.295 --> 00:17:54.420 align:middle line:84% anticipating the way the buyer is going to respond. 00:17:54.420 --> 00:18:03.400 align:middle line:90% 00:18:03.400 --> 00:18:06.260 align:middle line:84% So now that we've reviewed this case, 00:18:06.260 --> 00:18:09.020 align:middle line:84% let's now introduce incomplete information. 00:18:09.020 --> 00:18:11.240 align:middle line:84% So let's make different assumptions about v and c, 00:18:11.240 --> 00:18:14.920 align:middle line:84% unless there's any questions on this. 00:18:14.920 --> 00:18:16.892 align:middle line:90% So we're going to start with-- 00:18:16.892 --> 00:18:18.600 align:middle line:84% I mentioned over here, there's two cases, 00:18:18.600 --> 00:18:19.700 align:middle line:90% signaling and screening. 00:18:19.700 --> 00:18:21.140 align:middle line:90% And let's be really clear here. 00:18:21.140 --> 00:18:24.030 align:middle line:84% This game didn't involve any signaling or screening 00:18:24.030 --> 00:18:26.450 align:middle line:84% because the seller and the buyer-- or in this case, 00:18:26.450 --> 00:18:28.870 align:middle line:84% the seller didn't have any private information. 00:18:28.870 --> 00:18:31.185 align:middle line:84% So the price they offered didn't convey anything, 00:18:31.185 --> 00:18:33.310 align:middle line:84% and this didn't fit into our framework of signaling 00:18:33.310 --> 00:18:36.110 align:middle line:84% and screening, but now we're going to move into that case. 00:18:36.110 --> 00:18:39.450 align:middle line:90% So let's start with screening. 00:18:39.450 --> 00:18:46.790 align:middle line:90% 00:18:46.790 --> 00:18:52.430 align:middle line:84% So what we want to imagine is that the buyer-- 00:18:52.430 --> 00:18:59.670 align:middle line:84% we're still going to assume that the seller makes an offer, 00:18:59.670 --> 00:19:01.670 align:middle line:84% but now we're going to assume that the buyer has 00:19:01.670 --> 00:19:02.910 align:middle line:90% private information. 00:19:02.910 --> 00:19:04.410 align:middle line:90% Buyer, but not seller. 00:19:04.410 --> 00:19:14.910 align:middle line:90% 00:19:14.910 --> 00:19:16.990 align:middle line:84% So let's be a little more precise. 00:19:16.990 --> 00:19:19.410 align:middle line:84% The seller doesn't have any private information, 00:19:19.410 --> 00:19:22.682 align:middle line:84% so let's assume that the cost c of the seller is publicly known. 00:19:22.682 --> 00:19:24.640 align:middle line:84% This is commonly known, everyone observes this. 00:19:24.640 --> 00:19:30.300 align:middle line:84% So we'll say the cost c is known-- 00:19:30.300 --> 00:19:31.280 align:middle line:90% or is public. 00:19:31.280 --> 00:19:33.980 align:middle line:90% 00:19:33.980 --> 00:19:36.640 align:middle line:84% What's not known is the buyer's valuation, 00:19:36.640 --> 00:19:39.580 align:middle line:84% how much the buyer values the good. 00:19:39.580 --> 00:19:43.280 align:middle line:84% But v is private information of the buyer. 00:19:43.280 --> 00:19:53.700 align:middle line:90% 00:19:53.700 --> 00:19:55.380 align:middle line:84% So this fits into our screening setting 00:19:55.380 --> 00:19:58.340 align:middle line:84% because it's the second mover, the buyer who 00:19:58.340 --> 00:19:59.880 align:middle line:90% has the private information. 00:19:59.880 --> 00:20:01.580 align:middle line:84% The first mover, the seller, doesn't 00:20:01.580 --> 00:20:03.080 align:middle line:90% have any private information. 00:20:03.080 --> 00:20:06.140 align:middle line:84% But as we said before, the second mover's response 00:20:06.140 --> 00:20:09.160 align:middle line:84% to the first mover's action is going to convey information. 00:20:09.160 --> 00:20:12.660 align:middle line:84% In this case, the first mover's action by the seller 00:20:12.660 --> 00:20:14.620 align:middle line:90% is to offer a price. 00:20:14.620 --> 00:20:17.020 align:middle line:84% The buyer, then, is going to choose whether or not 00:20:17.020 --> 00:20:18.420 align:middle line:90% to buy at that price. 00:20:18.420 --> 00:20:20.780 align:middle line:84% And that's going to reveal information about the buyer, 00:20:20.780 --> 00:20:22.680 align:middle line:84% and in particular, it's going to screen or separate 00:20:22.680 --> 00:20:24.360 align:middle line:84% different kinds of buyers because buyers 00:20:24.360 --> 00:20:25.760 align:middle line:84% who value the good more are going 00:20:25.760 --> 00:20:27.593 align:middle line:84% to choose to accept the offer and buyers who 00:20:27.593 --> 00:20:31.160 align:middle line:84% value the good less are going to choose not to accept it. 00:20:31.160 --> 00:20:33.500 align:middle line:90% So let's try to model this. 00:20:33.500 --> 00:20:37.283 align:middle line:84% Well, If c is known, that's just a number that everyone knows. 00:20:37.283 --> 00:20:39.200 align:middle line:84% We don't have to say anything more about that. 00:20:39.200 --> 00:20:41.600 align:middle line:84% But v, because it's private information of the buyer, 00:20:41.600 --> 00:20:45.120 align:middle line:84% we have to specify the seller's beliefs about this quantity v. 00:20:45.120 --> 00:20:47.580 align:middle line:84% So let's make our standard assumption. 00:20:47.580 --> 00:20:50.580 align:middle line:84% Let's say that v is uniformly distributed-- 00:20:50.580 --> 00:20:52.360 align:middle line:84% this just makes the algebra a bit easier-- 00:20:52.360 --> 00:20:58.120 align:middle line:90% is uniform over 0, 1. 00:20:58.120 --> 00:21:01.760 align:middle line:84% It's a random variable, it's uniformly distributed over 0, 1. 00:21:01.760 --> 00:21:03.600 align:middle line:84% And now to make things interesting, 00:21:03.600 --> 00:21:06.080 align:middle line:84% let's assume that c, this public number, 00:21:06.080 --> 00:21:07.780 align:middle line:90% is somewhere between 0 and 1. 00:21:07.780 --> 00:21:10.528 align:middle line:84% Because if c was greater than 1, well, we'd never have trade, 00:21:10.528 --> 00:21:12.820 align:middle line:84% and if c was less than 0, well, we'd always have trade, 00:21:12.820 --> 00:21:13.960 align:middle line:90% it's not so interesting. 00:21:13.960 --> 00:21:19.002 align:middle line:84% So let's assume c greater than 0 is known. 00:21:19.002 --> 00:21:20.210 align:middle line:90% And I want to highlight this. 00:21:20.210 --> 00:21:21.918 align:middle line:84% I think it's often a source of confusion, 00:21:21.918 --> 00:21:23.833 align:middle line:84% when people see a variable, a letter, 00:21:23.833 --> 00:21:25.250 align:middle line:84% they think somehow it's not known. 00:21:25.250 --> 00:21:26.490 align:middle line:84% It's just playing the role of a number. 00:21:26.490 --> 00:21:27.240 align:middle line:90% Everyone knows it. 00:21:27.240 --> 00:21:29.990 align:middle line:84% Maybe it's 0.3, maybe it's 0.7, but we're just 00:21:29.990 --> 00:21:32.390 align:middle line:84% using a abstract symbol c to make 00:21:32.390 --> 00:21:33.730 align:middle line:90% it things a little bit cleaner. 00:21:33.730 --> 00:21:34.322 align:middle line:90% Yes? 00:21:34.322 --> 00:21:36.530 align:middle line:84% AUDIENCE: Does the seller note the distribution of v? 00:21:36.530 --> 00:21:37.113 align:middle line:90% IAN BALL: Yes. 00:21:37.113 --> 00:21:39.390 align:middle line:84% So when we say that v is uniform, 00:21:39.390 --> 00:21:41.510 align:middle line:84% everything we're saying here is common knowledge 00:21:41.510 --> 00:21:42.290 align:middle line:90% among the players. 00:21:42.290 --> 00:21:43.915 align:middle line:84% So not only is it distributed according 00:21:43.915 --> 00:21:46.630 align:middle line:84% to the uniform distribution, but if we're really precise, 00:21:46.630 --> 00:21:48.670 align:middle line:84% the seller knows that, the buyer knows that the seller knows 00:21:48.670 --> 00:21:51.295 align:middle line:84% that, and so on, everything I've said here is common knowledge. 00:21:51.295 --> 00:21:54.610 align:middle line:90% Yeah, good question OK. 00:21:54.610 --> 00:21:59.850 align:middle line:90% 00:21:59.850 --> 00:22:00.350 align:middle line:90% Great. 00:22:00.350 --> 00:22:03.670 align:middle line:84% So now let's-- ah, c greater than 0, and I also see-- 00:22:03.670 --> 00:22:07.650 align:middle line:84% let me be more precise, it's 0 less than c less than 1. 00:22:07.650 --> 00:22:10.990 align:middle line:90% 00:22:10.990 --> 00:22:13.790 align:middle line:84% OK, so now let's look at the seller's problem. 00:22:13.790 --> 00:22:16.400 align:middle line:84% Let's suppose the seller offers a price of p. 00:22:16.400 --> 00:22:20.980 align:middle line:90% 00:22:20.980 --> 00:22:23.788 align:middle line:84% Well, if they offer a price of p, who's going to buy? 00:22:23.788 --> 00:22:25.580 align:middle line:84% Well, the buyer is going to buy if and only 00:22:25.580 --> 00:22:28.220 align:middle line:90% if their value is at least p. 00:22:28.220 --> 00:22:33.300 align:middle line:84% So then buyer accepts or buys-- kind of use those 00:22:33.300 --> 00:22:35.780 align:middle line:90% interchangeably-- 00:22:35.780 --> 00:22:41.780 align:middle line:84% if and only if v is greater than or equal to p. 00:22:41.780 --> 00:22:43.900 align:middle line:84% And let's be a little more precise here. 00:22:43.900 --> 00:22:46.500 align:middle line:84% What kind of price might the seller reasonably offer? 00:22:46.500 --> 00:22:48.760 align:middle line:84% It doesn't make sense for them to offer more than 1. 00:22:48.760 --> 00:22:50.240 align:middle line:84% If the price they charge is more than 1, 00:22:50.240 --> 00:22:51.620 align:middle line:90% it's never going to be accepted. 00:22:51.620 --> 00:22:53.020 align:middle line:84% And if it's less than 0, well, that 00:22:53.020 --> 00:22:54.280 align:middle line:84% means they're paying you to get the good, 00:22:54.280 --> 00:22:55.280 align:middle line:90% that doesn't make sense. 00:22:55.280 --> 00:22:59.038 align:middle line:84% So let's focus on a price offer that's between 0 and 1. 00:22:59.038 --> 00:23:00.580 align:middle line:84% And we can show that they would never 00:23:00.580 --> 00:23:04.080 align:middle line:84% want to offer a price outside that range. 00:23:04.080 --> 00:23:07.940 align:middle line:84% Now, the buyer is going to accept or buy-- 00:23:07.940 --> 00:23:10.780 align:middle line:84% slash, buy-- if and only if v is greater than or equal to p. 00:23:10.780 --> 00:23:12.500 align:middle line:84% Well, let's think about how likely this 00:23:12.500 --> 00:23:14.320 align:middle line:84% is, because as the seller, all I care about 00:23:14.320 --> 00:23:17.080 align:middle line:84% is when I offer you a price of p, what is the probability 00:23:17.080 --> 00:23:18.960 align:middle line:84% that you actually buy at that price? 00:23:18.960 --> 00:23:23.000 align:middle line:84% So we can think-- we can draw a little graph-- 00:23:23.000 --> 00:23:26.720 align:middle line:84% here's 0, here's 1, the buyer's value 00:23:26.720 --> 00:23:30.280 align:middle line:84% is uniformly distributed over this interval, 00:23:30.280 --> 00:23:34.200 align:middle line:84% the seller is choosing some price p in this interval, 00:23:34.200 --> 00:23:36.240 align:middle line:84% and the buyer is going to buy if and only 00:23:36.240 --> 00:23:38.820 align:middle line:90% if their valuation lies above p. 00:23:38.820 --> 00:23:41.840 align:middle line:90% So lies in this segment. 00:23:41.840 --> 00:23:44.360 align:middle line:84% Because we're working with the uniform distribution, 00:23:44.360 --> 00:23:47.003 align:middle line:84% the probability that the buyer's value lies in this segment 00:23:47.003 --> 00:23:48.420 align:middle line:84% is just the length of the segment. 00:23:48.420 --> 00:23:50.600 align:middle line:84% That's a nice, convenient property of the uniform. 00:23:50.600 --> 00:23:55.220 align:middle line:84% So the probability here is just 1 minus p. 00:23:55.220 --> 00:23:58.520 align:middle line:90% 00:23:58.520 --> 00:24:04.480 align:middle line:84% So now, we can compute what is the seller's profit 00:24:04.480 --> 00:24:06.400 align:middle line:90% from offering a price of p? 00:24:06.400 --> 00:24:10.160 align:middle line:84% Well, as usual, profit is the number or the probability 00:24:10.160 --> 00:24:12.350 align:middle line:84% that you sell the item times the profit 00:24:12.350 --> 00:24:13.750 align:middle line:90% you get when you do sell it. 00:24:13.750 --> 00:24:16.070 align:middle line:84% So if they do sell it, their profit 00:24:16.070 --> 00:24:20.790 align:middle line:84% is going to be p minus c because the buyer buys at price p, 00:24:20.790 --> 00:24:22.510 align:middle line:84% pays p to the seller, but remember, 00:24:22.510 --> 00:24:25.470 align:middle line:84% the seller has to pay a cost c to sell it. 00:24:25.470 --> 00:24:27.670 align:middle line:84% But we have to multiply that by the probability 00:24:27.670 --> 00:24:30.430 align:middle line:84% that the buyer actually sells it-- 00:24:30.430 --> 00:24:32.910 align:middle line:84% or that the buyer actually buys it, sorry, 00:24:32.910 --> 00:24:35.990 align:middle line:84% which is, here, the probability that v 00:24:35.990 --> 00:24:37.750 align:middle line:90% is greater than or equal to p. 00:24:37.750 --> 00:24:43.130 align:middle line:84% And from what we said down here, this is just 1 minus p. 00:24:43.130 --> 00:24:46.870 align:middle line:90% 00:24:46.870 --> 00:24:49.670 align:middle line:84% So as usual, we have the two counteracting forces. 00:24:49.670 --> 00:24:54.110 align:middle line:84% When we raise the price p, "we" being the seller, 00:24:54.110 --> 00:24:58.510 align:middle line:84% we earn more per item, but we reduce the probability that 00:24:58.510 --> 00:24:59.910 align:middle line:90% the item is purchased. 00:24:59.910 --> 00:25:01.530 align:middle line:84% And that's the fundamental trade-off. 00:25:01.530 --> 00:25:02.690 align:middle line:84% If you set the price really high, 00:25:02.690 --> 00:25:04.565 align:middle line:84% it's great when someone buys, but they're not 00:25:04.565 --> 00:25:05.790 align:middle line:90% very likely to buy. 00:25:05.790 --> 00:25:08.430 align:middle line:84% And if we just use calculus or use our trick here, 00:25:08.430 --> 00:25:15.120 align:middle line:84% we can see that the solution is going to be 1 plus c over 2. 00:25:15.120 --> 00:25:22.180 align:middle line:90% 00:25:22.180 --> 00:25:26.420 align:middle line:84% Now I've framed everything here, and let's just do a sanity check 00:25:26.420 --> 00:25:27.500 align:middle line:90% as we always do. 00:25:27.500 --> 00:25:29.800 align:middle line:84% If c increases, the price is increasing, 00:25:29.800 --> 00:25:31.180 align:middle line:90% and that should make sense. 00:25:31.180 --> 00:25:33.880 align:middle line:84% If the cost if something is more expensive to produce, 00:25:33.880 --> 00:25:36.500 align:middle line:84% the price of producing it-- the price that the seller charges 00:25:36.500 --> 00:25:38.540 align:middle line:90% is going to be higher. 00:25:38.540 --> 00:25:40.660 align:middle line:90% Now this all seemed maybe-- 00:25:40.660 --> 00:25:43.480 align:middle line:84% we talked-- we used some fancy words, incomplete information, 00:25:43.480 --> 00:25:46.140 align:middle line:84% but it turns out, this is really equivalent to a very 00:25:46.140 --> 00:25:48.440 align:middle line:84% classical kind of Econ 101 problem, 00:25:48.440 --> 00:25:51.340 align:middle line:84% and I want to show the connection here. 00:25:51.340 --> 00:25:56.380 align:middle line:84% So I know some of you have taken econ, maybe some of you 00:25:56.380 --> 00:25:59.820 align:middle line:84% haven't, but if we just look at the classic supply and demand 00:25:59.820 --> 00:26:07.330 align:middle line:84% curve, we have price on this axis and quantity on this axis. 00:26:07.330 --> 00:26:11.330 align:middle line:84% Well, we can think of a seller facing a buyer like this 00:26:11.330 --> 00:26:19.190 align:middle line:84% as facing a demand curve that looks like this. 00:26:19.190 --> 00:26:23.770 align:middle line:90% 00:26:23.770 --> 00:26:27.610 align:middle line:84% Because if I'm the seller, whatever price p I offer, 00:26:27.610 --> 00:26:30.370 align:middle line:84% we already said from our calculation over here 00:26:30.370 --> 00:26:34.050 align:middle line:84% that the buyer is going to buy with probability 1 minus p. 00:26:34.050 --> 00:26:37.890 align:middle line:84% So I'm effectively facing a demand curve 00:26:37.890 --> 00:26:39.450 align:middle line:90% that looks like this. 00:26:39.450 --> 00:26:42.730 align:middle line:84% If I charge a price of 1, the buyer never buys, 00:26:42.730 --> 00:26:44.570 align:middle line:90% so demand is 0. 00:26:44.570 --> 00:26:47.770 align:middle line:84% If I set a price of 0, the buyer always 00:26:47.770 --> 00:26:50.910 align:middle line:84% buys, which means their probability of buying is just 1. 00:26:50.910 --> 00:26:52.410 align:middle line:90% That's my demand here. 00:26:52.410 --> 00:26:56.630 align:middle line:84% And if I set an intermediate price of p, well, 00:26:56.630 --> 00:26:59.510 align:middle line:84% then the buyer buys with probability 1 minus p, 00:26:59.510 --> 00:27:01.310 align:middle line:84% and it's this point along the demand curve. 00:27:01.310 --> 00:27:04.650 align:middle line:84% So as usual, in economics, we see a downward-sloping demand 00:27:04.650 --> 00:27:05.330 align:middle line:90% curve. 00:27:05.330 --> 00:27:08.230 align:middle line:84% The lower the price I offer as the seller, the more likely 00:27:08.230 --> 00:27:09.390 align:middle line:90% the buyer is to buy. 00:27:09.390 --> 00:27:12.190 align:middle line:84% Now I'll highlight-- I'll say up front that normally we 00:27:12.190 --> 00:27:15.290 align:middle line:84% think of q as the number of units purchased, 00:27:15.290 --> 00:27:16.910 align:middle line:90% not the probability of purchase. 00:27:16.910 --> 00:27:20.050 align:middle line:84% But if we're facing a population of, say, 100 people, well, 00:27:20.050 --> 00:27:22.430 align:middle line:84% we can reinterpret this probability where a 1/2 00:27:22.430 --> 00:27:25.390 align:middle line:84% probability of buying now means, in expectation, 00:27:25.390 --> 00:27:28.090 align:middle line:84% 50 out of 100 people in your town are going to buy. 00:27:28.090 --> 00:27:30.390 align:middle line:84% So we can reinterpret quantity as-- 00:27:30.390 --> 00:27:33.270 align:middle line:84% let's reinterpret it here as probability of buying 00:27:33.270 --> 00:27:38.410 align:middle line:90% times the number of people. 00:27:38.410 --> 00:27:47.190 align:middle line:90% 00:27:47.190 --> 00:27:49.293 align:middle line:84% Up here, I just assumed there was a single buyer, 00:27:49.293 --> 00:27:50.710 align:middle line:84% but the math doesn't really change 00:27:50.710 --> 00:27:54.170 align:middle line:84% if you just multiply everything by 100-- say there's 100 people. 00:27:54.170 --> 00:27:56.888 align:middle line:84% And here, we see the connection with-- if you've taken econ 00:27:56.888 --> 00:27:58.930 align:middle line:84% and we talk about the monopolist pricing problem, 00:27:58.930 --> 00:28:00.830 align:middle line:84% this is exactly what they're doing. 00:28:00.830 --> 00:28:02.970 align:middle line:84% Of course, I need one more thing on the figure, 00:28:02.970 --> 00:28:06.860 align:middle line:84% I need the cost, because that's also relevant to the monopolist. 00:28:06.860 --> 00:28:10.060 align:middle line:90% So let me put the cost c here. 00:28:10.060 --> 00:28:19.700 align:middle line:84% And what we can see is this box here represents 00:28:19.700 --> 00:28:22.020 align:middle line:90% the monopolist profit. 00:28:22.020 --> 00:28:23.120 align:middle line:90% Why? 00:28:23.120 --> 00:28:26.620 align:middle line:84% Well, the height of the box is p minus c. 00:28:26.620 --> 00:28:30.300 align:middle line:84% That's what we had up here for the profit per unit. 00:28:30.300 --> 00:28:34.100 align:middle line:84% The width of the box is 1 minus p. 00:28:34.100 --> 00:28:36.860 align:middle line:84% It's determined by the value of the demand curve 00:28:36.860 --> 00:28:39.780 align:middle line:84% at the price p that's offered, so that represents 00:28:39.780 --> 00:28:41.860 align:middle line:90% the probability of sale. 00:28:41.860 --> 00:28:43.940 align:middle line:84% And therefore, what's the seller's goal? 00:28:43.940 --> 00:28:45.860 align:middle line:84% They want to move this price up and down 00:28:45.860 --> 00:28:48.140 align:middle line:84% and choose the value that makes this box have 00:28:48.140 --> 00:28:50.540 align:middle line:90% the largest possible area. 00:28:50.540 --> 00:28:52.660 align:middle line:84% And it turns out that this is maximized 00:28:52.660 --> 00:28:54.837 align:middle line:90% when the box is a square. 00:28:54.837 --> 00:28:56.420 align:middle line:84% And that's actually what happens here. 00:28:56.420 --> 00:29:02.530 align:middle line:84% When p star equals 1 plus c over 2, then-- 00:29:02.530 --> 00:29:04.610 align:middle line:90% let's just observe this. 00:29:04.610 --> 00:29:09.850 align:middle line:84% If p star equals 1 plus c over 2, 00:29:09.850 --> 00:29:24.490 align:middle line:84% then that means p star minus c equals 1 minus c over 2. 00:29:24.490 --> 00:29:26.250 align:middle line:90% So I have 1 plus c over 2. 00:29:26.250 --> 00:29:29.170 align:middle line:84% I'm subtracting c, but since I'm dividing by 2, 00:29:29.170 --> 00:29:34.770 align:middle line:84% I can subtract 2c on the top, and then I get 1 minus c over 2. 00:29:34.770 --> 00:29:38.770 align:middle line:90% And then what is my demand? 00:29:38.770 --> 00:29:46.510 align:middle line:84% 1 minus-- oh, did I do something wrong here? 00:29:46.510 --> 00:29:56.350 align:middle line:90% 00:29:56.350 --> 00:29:56.850 align:middle line:90% Yes. 00:29:56.850 --> 00:29:59.790 align:middle line:90% 00:29:59.790 --> 00:30:01.830 align:middle line:84% We know what the answer has to be-- let's see. 00:30:01.830 --> 00:30:04.190 align:middle line:84% We'll write it out more carefully. 00:30:04.190 --> 00:30:14.110 align:middle line:84% Maybe we can write this as 2 over 2 minus 1 plus c over 2. 00:30:14.110 --> 00:30:15.050 align:middle line:90% And now it's clear. 00:30:15.050 --> 00:30:18.170 align:middle line:84% 2 minus 1 is 1, minus c, we get 1 minus c over 2. 00:30:18.170 --> 00:30:21.350 align:middle line:90% 00:30:21.350 --> 00:30:23.110 align:middle line:90% So this is at p star. 00:30:23.110 --> 00:30:26.110 align:middle line:84% The height of the box is 1 minus c over 2, 00:30:26.110 --> 00:30:28.390 align:middle line:84% and the width of the box is also 1 minus c over 2, 00:30:28.390 --> 00:30:30.240 align:middle line:90% and we get a square. 00:30:30.240 --> 00:30:32.490 align:middle line:84% That's a special property of the uniform distribution. 00:30:32.490 --> 00:30:34.830 align:middle line:84% So one final comment is, when we drew 00:30:34.830 --> 00:30:37.070 align:middle line:84% the connection between the demand curve 00:30:37.070 --> 00:30:39.670 align:middle line:84% figure and the incomplete information figure, 00:30:39.670 --> 00:30:41.990 align:middle line:84% the distribution of the buyer's value 00:30:41.990 --> 00:30:44.710 align:middle line:84% exactly corresponds to the shape of the demand curve. 00:30:44.710 --> 00:30:47.150 align:middle line:84% So when we draw different demand curves in econ classes, 00:30:47.150 --> 00:30:49.190 align:middle line:84% one interpretation of that is it corresponds 00:30:49.190 --> 00:30:52.270 align:middle line:84% to different distributions of values in the population. 00:30:52.270 --> 00:30:54.740 align:middle line:84% A demand curve shifting out means 00:30:54.740 --> 00:30:56.990 align:middle line:84% more people are willing to buy at a given price, which 00:30:56.990 --> 00:30:59.260 align:middle line:84% means the distribution of values in the population 00:30:59.260 --> 00:31:00.700 align:middle line:90% tends to be higher. 00:31:00.700 --> 00:31:02.920 align:middle line:84% OK, but this was all still a review. 00:31:02.920 --> 00:31:07.820 align:middle line:84% Let's now move into the incomplete information case-- 00:31:07.820 --> 00:31:10.280 align:middle line:84% or the more complicated incomplete information case. 00:31:10.280 --> 00:31:12.960 align:middle line:90% So let's go back over here. 00:31:12.960 --> 00:31:28.940 align:middle line:90% 00:31:28.940 --> 00:31:32.500 align:middle line:84% So before, what we assumed is that-- 00:31:32.500 --> 00:31:35.520 align:middle line:84% well, one question here is to say, why does the buyer care? 00:31:35.520 --> 00:31:38.595 align:middle line:84% Why does the seller care about the buyer's valuation? 00:31:38.595 --> 00:31:40.220 align:middle line:84% They care because the buyer's valuation 00:31:40.220 --> 00:31:42.260 align:middle line:90% affects whether they buy. 00:31:42.260 --> 00:31:44.860 align:middle line:84% But it doesn't affect how costly it is for the seller 00:31:44.860 --> 00:31:46.300 align:middle line:90% to produce the item. 00:31:46.300 --> 00:31:49.820 align:middle line:84% But there's a lot of cases where the buyer's private 00:31:49.820 --> 00:31:51.620 align:middle line:90% information-- so let's say-- 00:31:51.620 --> 00:31:53.420 align:middle line:90% I'll say this is another case. 00:31:53.420 --> 00:31:55.140 align:middle line:84% It's the case of interdependent-- 00:31:55.140 --> 00:31:59.050 align:middle line:84% sometimes called interdependent values in the notes. 00:31:59.050 --> 00:32:05.650 align:middle line:90% Interdependent values. 00:32:05.650 --> 00:32:13.450 align:middle line:84% And this is the case where the buyer's private information 00:32:13.450 --> 00:32:16.690 align:middle line:84% is relevant not just to how much the buyer values the good, 00:32:16.690 --> 00:32:19.570 align:middle line:84% but the cost to the seller of providing the good. 00:32:19.570 --> 00:32:24.070 align:middle line:84% So the buyer's private information is about v and c. 00:32:24.070 --> 00:32:26.817 align:middle line:90% 00:32:26.817 --> 00:32:28.150 align:middle line:90% And this might be a bit strange. 00:32:28.150 --> 00:32:30.483 align:middle line:84% Why would the buyer know how costly it is for the seller 00:32:30.483 --> 00:32:31.650 align:middle line:90% to produce the item? 00:32:31.650 --> 00:32:35.050 align:middle line:84% But there's a few classic cases here. 00:32:35.050 --> 00:32:37.010 align:middle line:84% Does anyone know what's a classic case where 00:32:37.010 --> 00:32:39.130 align:middle line:84% the buyer might have private information that's 00:32:39.130 --> 00:32:41.090 align:middle line:84% relevant to both what the buyer gets out 00:32:41.090 --> 00:32:43.770 align:middle line:84% of the exchange and also the cost 00:32:43.770 --> 00:32:46.350 align:middle line:84% to the seller of making the exchange? 00:32:46.350 --> 00:32:50.290 align:middle line:90% 00:32:50.290 --> 00:32:51.050 align:middle line:90% Yeah? 00:32:51.050 --> 00:32:52.675 align:middle line:84% AUDIENCE: Is it possible that the buyer 00:32:52.675 --> 00:32:54.910 align:middle line:84% would value the good even more and know that it's 00:32:54.910 --> 00:32:57.150 align:middle line:90% costs more [INAUDIBLE]? 00:32:57.150 --> 00:32:59.430 align:middle line:90% IAN BALL: Yeah, that's-- 00:32:59.430 --> 00:33:03.270 align:middle line:90% I think that's possible, yes. 00:33:03.270 --> 00:33:06.070 align:middle line:84% Yeah, I think that's along the right track, yeah. 00:33:06.070 --> 00:33:08.230 align:middle line:84% There's one classical application 00:33:08.230 --> 00:33:10.470 align:middle line:90% that everyone talks about. 00:33:10.470 --> 00:33:11.630 align:middle line:90% So this is health care. 00:33:11.630 --> 00:33:14.092 align:middle line:90% Health insurance. 00:33:14.092 --> 00:33:15.550 align:middle line:84% And then I'll give one more example 00:33:15.550 --> 00:33:18.270 align:middle line:84% that's a bit closer to what you suggested. 00:33:18.270 --> 00:33:19.010 align:middle line:90% Health insurance. 00:33:19.010 --> 00:33:22.230 align:middle line:84% What does the buyer know that's relevant to both v and c 00:33:22.230 --> 00:33:24.610 align:middle line:84% in the context of health insurance? 00:33:24.610 --> 00:33:25.110 align:middle line:90% Yeah? 00:33:25.110 --> 00:33:26.030 align:middle line:90% AUDIENCE: Their own status? 00:33:26.030 --> 00:33:27.447 align:middle line:84% IAN BALL: Their own health status. 00:33:27.447 --> 00:33:29.070 align:middle line:84% The buyer is, here, an individual, 00:33:29.070 --> 00:33:30.470 align:middle line:90% they're buying insurance. 00:33:30.470 --> 00:33:33.030 align:middle line:84% This is one reason why we see the government quite involved 00:33:33.030 --> 00:33:34.550 align:middle line:84% in health insurance, and the reason 00:33:34.550 --> 00:33:37.250 align:middle line:84% why health insurance markets, unfettered private markets, 00:33:37.250 --> 00:33:39.670 align:middle line:84% tend not to work well, precisely because of this issue. 00:33:39.670 --> 00:33:42.950 align:middle line:84% Because the buyer has private information that's relevant 00:33:42.950 --> 00:33:45.170 align:middle line:84% both to how much they want the insurance, 00:33:45.170 --> 00:33:47.550 align:middle line:84% but also to how costly it is for the seller 00:33:47.550 --> 00:33:49.350 align:middle line:90% to provide insurance to them. 00:33:49.350 --> 00:33:59.160 align:middle line:84% So what the buyer knows their health status. 00:33:59.160 --> 00:34:01.860 align:middle line:90% 00:34:01.860 --> 00:34:03.680 align:middle line:84% Of course, this is a gross simplification, 00:34:03.680 --> 00:34:05.222 align:middle line:84% but in short, you might think someone 00:34:05.222 --> 00:34:07.960 align:middle line:84% who's healthy has a lower v and a lower c. 00:34:07.960 --> 00:34:10.780 align:middle line:84% If I'm healthy, I don't care about insurance as much. 00:34:10.780 --> 00:34:13.245 align:middle line:84% And also, insurance is cheaper to provide to me 00:34:13.245 --> 00:34:14.620 align:middle line:84% because I rarely go to the doctor 00:34:14.620 --> 00:34:17.219 align:middle line:84% and the insurance company doesn't have to pay much. 00:34:17.219 --> 00:34:20.540 align:middle line:84% On the other hand, someone who's quite unhealthy 00:34:20.540 --> 00:34:22.435 align:middle line:84% is going to value insurance a lot more. 00:34:22.435 --> 00:34:24.060 align:middle line:84% They're going to go to the doctor a lot 00:34:24.060 --> 00:34:25.520 align:middle line:84% and get a lot more out of the insurance, 00:34:25.520 --> 00:34:27.812 align:middle line:84% but that's also going to be more costly for the insurer 00:34:27.812 --> 00:34:31.020 align:middle line:84% because the insurer has to pay every time every time they go. 00:34:31.020 --> 00:34:33.580 align:middle line:84% Now, health insurance markets sometimes 00:34:33.580 --> 00:34:36.933 align:middle line:84% try to mitigate this effect by offering different prices based 00:34:36.933 --> 00:34:38.600 align:middle line:84% on different observable characteristics. 00:34:38.600 --> 00:34:40.400 align:middle line:84% It may be that based on your price, 00:34:40.400 --> 00:34:41.931 align:middle line:84% older people tend to charge more-- 00:34:41.931 --> 00:34:44.139 align:middle line:84% charge more for health insurance than younger people, 00:34:44.139 --> 00:34:46.179 align:middle line:84% but there's still usually additional private information 00:34:46.179 --> 00:34:46.679 align:middle line:90% you have. 00:34:46.679 --> 00:34:49.199 align:middle line:84% Not all people who are 75 have exactly the same health, 00:34:49.199 --> 00:34:50.907 align:middle line:84% and they usually have private information 00:34:50.907 --> 00:34:53.530 align:middle line:90% about how healthy they are. 00:34:53.530 --> 00:34:55.770 align:middle line:84% So this can create a problem, which is often 00:34:55.770 --> 00:34:58.940 align:middle line:90% known as adverse selection. 00:34:58.940 --> 00:35:01.190 align:middle line:84% And this term actually-- it's used in economics a lot, 00:35:01.190 --> 00:35:03.410 align:middle line:84% but it actually was first-- the term was first 00:35:03.410 --> 00:35:05.410 align:middle line:84% coined by people who worked in insurance 00:35:05.410 --> 00:35:08.010 align:middle line:84% and noted this phenomenon, and then economists subsequently 00:35:08.010 --> 00:35:10.050 align:middle line:90% studied it. 00:35:10.050 --> 00:35:16.210 align:middle line:84% So let's look at quite, maybe, a stylized model of this. 00:35:16.210 --> 00:35:19.030 align:middle line:84% So let's represent the population or the buyer. 00:35:19.030 --> 00:35:22.410 align:middle line:90% 00:35:22.410 --> 00:35:25.890 align:middle line:84% The cost of providing insurance to the buyer 00:35:25.890 --> 00:35:28.150 align:middle line:84% is going to be uniform over 0, 1. 00:35:28.150 --> 00:35:32.290 align:middle line:90% 00:35:32.290 --> 00:35:34.370 align:middle line:84% So we can think of c as being a measure 00:35:34.370 --> 00:35:36.770 align:middle line:90% of the health of the buyer. 00:35:36.770 --> 00:35:38.930 align:middle line:84% I guess a higher c is someone who 00:35:38.930 --> 00:35:41.370 align:middle line:84% is less healthy because it's more costly to provide them 00:35:41.370 --> 00:35:42.997 align:middle line:90% insurance. 00:35:42.997 --> 00:35:45.330 align:middle line:84% And then we want there to be some relationship between c 00:35:45.330 --> 00:35:50.990 align:middle line:84% and v. So let's assume that v is equal to alpha plus beta c 00:35:50.990 --> 00:35:56.710 align:middle line:84% where alpha is greater than or equal to 0 00:35:56.710 --> 00:36:00.250 align:middle line:84% and beta is strictly greater than 0. 00:36:00.250 --> 00:36:04.310 align:middle line:90% 00:36:04.310 --> 00:36:07.710 align:middle line:84% So maybe it's helpful to draw a picture here 00:36:07.710 --> 00:36:09.590 align:middle line:90% to see what's going on. 00:36:09.590 --> 00:36:13.230 align:middle line:84% We have c, we have v. c is uniformly 00:36:13.230 --> 00:36:21.670 align:middle line:84% distributed between 0 and 1, and v is alpha plus beta times c. 00:36:21.670 --> 00:36:28.590 align:middle line:84% So let's put alpha here and alpha plus beta here. 00:36:28.590 --> 00:36:33.110 align:middle line:84% And notice that when c equals 0, v equals alpha; 00:36:33.110 --> 00:36:37.070 align:middle line:84% and when c equals 1, v equals alpha plus beta. 00:36:37.070 --> 00:36:40.710 align:middle line:84% So we can think of people as lying along 00:36:40.710 --> 00:36:47.680 align:middle line:90% this line that looks like this. 00:36:47.680 --> 00:36:50.597 align:middle line:90% 00:36:50.597 --> 00:36:52.180 align:middle line:84% We could have a more complicated model 00:36:52.180 --> 00:36:55.642 align:middle line:84% where maybe your cost doesn't directly pin down your value 00:36:55.642 --> 00:36:57.600 align:middle line:84% and vice versa, there might be some noise here, 00:36:57.600 --> 00:36:59.360 align:middle line:90% some individual heterogeneity. 00:36:59.360 --> 00:37:01.000 align:middle line:84% But here in this really simple model, 00:37:01.000 --> 00:37:03.180 align:middle line:90% everyone lives along this line. 00:37:03.180 --> 00:37:05.980 align:middle line:84% Here, we have the healthiest people 00:37:05.980 --> 00:37:08.960 align:middle line:84% who only get a value of alpha from the insurance, 00:37:08.960 --> 00:37:10.580 align:middle line:90% but only cost the insurer 0. 00:37:10.580 --> 00:37:12.080 align:middle line:84% And then as you move along the line, 00:37:12.080 --> 00:37:15.380 align:middle line:84% we have the least healthy people who 00:37:15.380 --> 00:37:17.480 align:middle line:84% cost one to the insurer to insure, 00:37:17.480 --> 00:37:19.807 align:middle line:84% but get utility alpha plus beta from the insurer. 00:37:19.807 --> 00:37:22.140 align:middle line:84% In reality, we might need a two-dimensional model, maybe 00:37:22.140 --> 00:37:24.580 align:middle line:84% health and how rich you are, and maybe everything would 00:37:24.580 --> 00:37:26.160 align:middle line:84% be filled in and we'd have a distribution, 00:37:26.160 --> 00:37:28.035 align:middle line:84% but we're going to restrict just to this line 00:37:28.035 --> 00:37:30.160 align:middle line:90% to make things concrete. 00:37:30.160 --> 00:37:32.385 align:middle line:84% And because c is uniform, that means 00:37:32.385 --> 00:37:34.260 align:middle line:84% people-- we're imagining people are uniformly 00:37:34.260 --> 00:37:36.220 align:middle line:90% distributed along this line. 00:37:36.220 --> 00:37:36.720 align:middle line:90% Yes? 00:37:36.720 --> 00:37:38.803 align:middle line:84% AUDIENCE: Just trying to understand qualitatively. 00:37:38.803 --> 00:37:40.540 align:middle line:84% How can someone did a lot of value 00:37:40.540 --> 00:37:44.770 align:middle line:84% out of the insurer, but not cost a lot or vice versa? 00:37:44.770 --> 00:37:46.690 align:middle line:84% IAN BALL: It could be-- so wealth 00:37:46.690 --> 00:37:48.250 align:middle line:90% would be the easiest case. 00:37:48.250 --> 00:37:51.950 align:middle line:84% So someone who is really rich might be willing to pay a lot. 00:37:51.950 --> 00:37:54.790 align:middle line:84% So what is value-- yeah, it gets into what does value mean? 00:37:54.790 --> 00:37:57.490 align:middle line:84% Let's say I go to the doctor and fix my toothache. 00:37:57.490 --> 00:37:59.868 align:middle line:84% How much am I willing to pay to fix my toothache? 00:37:59.868 --> 00:38:01.910 align:middle line:84% I might be willing to pay less than someone else, 00:38:01.910 --> 00:38:03.830 align:middle line:84% even though it's fixing the toothache. 00:38:03.830 --> 00:38:05.830 align:middle line:84% So the cost to the insurer is just literally, 00:38:05.830 --> 00:38:09.030 align:middle line:84% how much does the doctor get paid, and the facility, 00:38:09.030 --> 00:38:10.088 align:middle line:90% to perform the operation? 00:38:10.088 --> 00:38:12.130 align:middle line:84% But certainly, different people in the population 00:38:12.130 --> 00:38:14.130 align:middle line:84% would be willing to pay very different amounts 00:38:14.130 --> 00:38:15.185 align:middle line:90% for the same operation. 00:38:15.185 --> 00:38:17.810 align:middle line:84% A billionaire would tend to be willing to pay more than someone 00:38:17.810 --> 00:38:19.810 align:middle line:90% who doesn't have any money. 00:38:19.810 --> 00:38:23.352 align:middle line:84% So in this model, we're not taking into account that effect. 00:38:23.352 --> 00:38:24.810 align:middle line:84% We're looking at a simplified model 00:38:24.810 --> 00:38:28.810 align:middle line:84% where there's a one-to-one relationship between cost 00:38:28.810 --> 00:38:30.533 align:middle line:90% and value. 00:38:30.533 --> 00:38:32.450 align:middle line:84% But yeah, I guess the way to think about value 00:38:32.450 --> 00:38:36.250 align:middle line:84% is value is about your individual willingness to pay, 00:38:36.250 --> 00:38:38.930 align:middle line:84% because remember, going back to here, 00:38:38.930 --> 00:38:40.970 align:middle line:84% this is almost the definition of value. 00:38:40.970 --> 00:38:42.550 align:middle line:84% If your value is v, it means you're 00:38:42.550 --> 00:38:45.412 align:middle line:84% willing to pay any amount up to v to get that item. 00:38:45.412 --> 00:38:47.870 align:middle line:84% And some people are willing to pay a lot more for something 00:38:47.870 --> 00:38:49.430 align:middle line:90% that has the same cost. 00:38:49.430 --> 00:38:50.910 align:middle line:90% Yes, question? 00:38:50.910 --> 00:38:54.270 align:middle line:84% AUDIENCE: It's, like, not necessarily true that a plus-- 00:38:54.270 --> 00:38:57.350 align:middle line:84% that alpha plus beta is greater than 1. 00:38:57.350 --> 00:39:00.765 align:middle line:84% So are there any constraints that would make alpha plus beta 00:39:00.765 --> 00:39:01.390 align:middle line:90% greater than 1? 00:39:01.390 --> 00:39:03.590 align:middle line:84% Or in the real world, is it possible 00:39:03.590 --> 00:39:06.270 align:middle line:84% that alpha plus beta is less than 1? 00:39:06.270 --> 00:39:08.756 align:middle line:90% IAN BALL: I see. 00:39:08.756 --> 00:39:18.050 align:middle line:84% Yeah, probably-- no, I think it could go either way. 00:39:18.050 --> 00:39:21.510 align:middle line:84% So I guess one example would be-- 00:39:21.510 --> 00:39:23.230 align:middle line:84% let's think of a-- there might be 00:39:23.230 --> 00:39:29.150 align:middle line:84% a very cheap surgery that really improves my quality of life. 00:39:29.150 --> 00:39:32.610 align:middle line:84% So in that case, the value that I get out of that surgery, 00:39:32.610 --> 00:39:34.710 align:middle line:84% my willingness to pay might be much, much higher 00:39:34.710 --> 00:39:36.790 align:middle line:84% than the cost of producing that surgery. 00:39:36.790 --> 00:39:38.730 align:middle line:84% On the other hand, it might go the other way. 00:39:38.730 --> 00:39:41.100 align:middle line:84% There could be a really expensive new treatment that 00:39:41.100 --> 00:39:42.665 align:middle line:84% doesn't really help me out very much, 00:39:42.665 --> 00:39:45.040 align:middle line:84% and it may be hugely expensive to provide this treatment, 00:39:45.040 --> 00:39:46.580 align:middle line:84% but the utility gain and my willingness 00:39:46.580 --> 00:39:48.640 align:middle line:84% to pay for that treatment might actually be quite low. 00:39:48.640 --> 00:39:50.140 align:middle line:84% So I think in general you could have 00:39:50.140 --> 00:39:53.135 align:middle line:84% either ordering between v and c depending on the context 00:39:53.135 --> 00:39:53.760 align:middle line:90% that you're in. 00:39:53.760 --> 00:39:55.600 align:middle line:84% I mean, this is a very stylized model. 00:39:55.600 --> 00:39:59.180 align:middle line:84% I'm not saying-- if we wanted to really directly conduct 00:39:59.180 --> 00:40:01.680 align:middle line:84% health care policy, we'd have to use a more complex model, 00:40:01.680 --> 00:40:03.160 align:middle line:84% but just to convey the ideas, I want to do that. 00:40:03.160 --> 00:40:04.535 align:middle line:84% But yes, I think there could be-- 00:40:04.535 --> 00:40:07.922 align:middle line:84% and in fact, this is an issue, when people have full insurance 00:40:07.922 --> 00:40:09.880 align:middle line:84% and they don't have to pay to go to the doctor, 00:40:09.880 --> 00:40:12.420 align:middle line:84% they sometimes go to the doctor and get procedures 00:40:12.420 --> 00:40:14.240 align:middle line:84% that are extremely expensive to provide, 00:40:14.240 --> 00:40:15.860 align:middle line:84% but don't actually help them that much because they don't 00:40:15.860 --> 00:40:17.900 align:middle line:84% have to pay, and that's one kind of issue 00:40:17.900 --> 00:40:19.600 align:middle line:90% with insurance markets, yeah. 00:40:19.600 --> 00:40:20.100 align:middle line:90% Good. 00:40:20.100 --> 00:40:22.580 align:middle line:90% Any other questions on this? 00:40:22.580 --> 00:40:26.340 align:middle line:84% OK, so now let's try to work through-- 00:40:26.340 --> 00:40:28.500 align:middle line:84% an analogous problem is over there. 00:40:28.500 --> 00:40:31.760 align:middle line:84% We still have the seller choosing to post a price, 00:40:31.760 --> 00:40:33.020 align:middle line:90% but now what's different? 00:40:33.020 --> 00:40:36.220 align:middle line:84% Now, it's only going to be the high-value people who 00:40:36.220 --> 00:40:37.732 align:middle line:90% buy at a given price. 00:40:37.732 --> 00:40:39.190 align:middle line:84% But what's the key difference here? 00:40:39.190 --> 00:40:40.450 align:middle line:84% What's going to be true about the people who 00:40:40.450 --> 00:40:41.617 align:middle line:90% choose to buy the insurance? 00:40:41.617 --> 00:40:43.010 align:middle line:84% If I set a price, only the people 00:40:43.010 --> 00:40:45.135 align:middle line:84% who value it more than that price are going to buy. 00:40:45.135 --> 00:40:48.010 align:middle line:84% What's going to be true about those people? 00:40:48.010 --> 00:40:49.650 align:middle line:90% Who's going to buy my insurance? 00:40:49.650 --> 00:40:50.407 align:middle line:90% Yeah? 00:40:50.407 --> 00:40:52.490 align:middle line:84% AUDIENCE: The people that have high costs as well. 00:40:52.490 --> 00:40:53.950 align:middle line:84% IAN BALL: The people that also have high costs. 00:40:53.950 --> 00:40:56.200 align:middle line:84% And this is the fundamental idea of adverse selection. 00:40:56.200 --> 00:40:57.730 align:middle line:84% If I go out into the market and I 00:40:57.730 --> 00:40:59.647 align:middle line:84% say this is the price I'm going to offer, 00:40:59.647 --> 00:41:01.730 align:middle line:84% then the people who are going to buy the insurance 00:41:01.730 --> 00:41:04.230 align:middle line:84% are going to be the people who are most expensive to insure, 00:41:04.230 --> 00:41:05.830 align:middle line:90% and that can create problems. 00:41:05.830 --> 00:41:09.295 align:middle line:84% My favorite example of adverse selection is in the 1990s-- 00:41:09.295 --> 00:41:10.670 align:middle line:84% I don't know if I told you this-- 00:41:10.670 --> 00:41:11.650 align:middle line:84% I tell this story every year, sorry, 00:41:11.650 --> 00:41:12.790 align:middle line:90% I forget which year I told it. 00:41:12.790 --> 00:41:14.915 align:middle line:84% In the 1990s, United or Delta, one of the airlines, 00:41:14.915 --> 00:41:17.210 align:middle line:84% offered an unlimited flying program where you could fly 00:41:17.210 --> 00:41:18.570 align:middle line:90% forever. 00:41:18.570 --> 00:41:22.250 align:middle line:84% And it cost, I think, like $200,000 in today's terms. 00:41:22.250 --> 00:41:24.830 align:middle line:84% And Delta thought, well, most people don't fly that much. 00:41:24.830 --> 00:41:26.498 align:middle line:90% $200,000, that's a lot. 00:41:26.498 --> 00:41:28.290 align:middle line:84% It was a disastrous program for the airline 00:41:28.290 --> 00:41:30.123 align:middle line:84% because there's some really weird people who 00:41:30.123 --> 00:41:30.930 align:middle line:90% just like to fly. 00:41:30.930 --> 00:41:32.690 align:middle line:84% And one guy just took the program 00:41:32.690 --> 00:41:34.570 align:middle line:84% and just flew first class around the world every single day 00:41:34.570 --> 00:41:35.770 align:middle line:90% for the rest of his life. 00:41:35.770 --> 00:41:37.395 align:middle line:84% That's an example of adverse selection. 00:41:37.395 --> 00:41:39.000 align:middle line:84% When you post a price, the person who 00:41:39.000 --> 00:41:40.622 align:middle line:84% buys it is not the average flyer, 00:41:40.622 --> 00:41:42.580 align:middle line:84% it's the person who really values flying a lot. 00:41:42.580 --> 00:41:44.920 align:middle line:84% And this was a person who was extremely costly to give 00:41:44.920 --> 00:41:46.220 align:middle line:90% an unlimited flying license to. 00:41:46.220 --> 00:41:48.780 align:middle line:84% For me, probably 200K, the airline would make money on me. 00:41:48.780 --> 00:41:50.960 align:middle line:84% Of course, my behavior would change once I got it. 00:41:50.960 --> 00:41:52.820 align:middle line:84% But in this guy's case, they lost a lot of money, 00:41:52.820 --> 00:41:54.612 align:middle line:84% and they ended up going to court and trying 00:41:54.612 --> 00:41:56.940 align:middle line:84% to nullify the contract, and it was this big, big mess. 00:41:56.940 --> 00:41:57.440 align:middle line:90% OK. 00:41:57.440 --> 00:41:59.680 align:middle line:84% I guess another adverse selection effect 00:41:59.680 --> 00:42:01.580 align:middle line:90% might be health insurance. 00:42:01.580 --> 00:42:06.940 align:middle line:84% Another might be in financial markets. 00:42:06.940 --> 00:42:13.640 align:middle line:90% 00:42:13.640 --> 00:42:16.418 align:middle line:84% If I'm selling a mortgage-backed security 00:42:16.418 --> 00:42:17.960 align:middle line:84% and I don't know much about the value 00:42:17.960 --> 00:42:19.480 align:middle line:84% of this mortgage-backed security and someone's 00:42:19.480 --> 00:42:21.120 align:middle line:84% willing to buy it from me, well, the fact 00:42:21.120 --> 00:42:22.440 align:middle line:84% that they're willing to buy it probably 00:42:22.440 --> 00:42:24.260 align:middle line:84% means it's very valuable and I'm not getting a good price. 00:42:24.260 --> 00:42:26.380 align:middle line:84% If they're not willing to buy it, then it goes the other way. 00:42:26.380 --> 00:42:27.480 align:middle line:84% So there's always the problem, you always 00:42:27.480 --> 00:42:29.905 align:middle line:84% have to think, why is this person willing to buy from me? 00:42:29.905 --> 00:42:31.780 align:middle line:84% And often, if they're willing to buy from me, 00:42:31.780 --> 00:42:33.100 align:middle line:84% it might mean I don't want to sell it to them, 00:42:33.100 --> 00:42:35.240 align:middle line:84% and this is the fundamental issue of adverse selection. 00:42:35.240 --> 00:42:37.490 align:middle line:84% If someone's willing to buy an unlimited flight ticket 00:42:37.490 --> 00:42:40.740 align:middle line:84% for 200K, maybe it's not a good idea to sell it to them. 00:42:40.740 --> 00:42:44.260 align:middle line:84% OK, but let's now get into the math here. 00:42:44.260 --> 00:42:48.340 align:middle line:84% So let's suppose I offer a price of p. 00:42:48.340 --> 00:42:57.300 align:middle line:84% So the seller offers a price p, then the buyer buys if and only 00:42:57.300 --> 00:42:59.660 align:middle line:84% if-- well, as before, their value is 00:42:59.660 --> 00:43:02.100 align:middle line:90% greater than or equal to p. 00:43:02.100 --> 00:43:05.880 align:middle line:84% And now, instead of saying p is between 0 and 1, 00:43:05.880 --> 00:43:10.140 align:middle line:84% let's say p is between alpha and alpha plus beta. 00:43:10.140 --> 00:43:12.163 align:middle line:84% Because, again, if I charge a price below alpha, 00:43:12.163 --> 00:43:14.580 align:middle line:84% no one's going to buy, and if I charge it above alpha plus 00:43:14.580 --> 00:43:17.180 align:middle line:84% beta-- sorry, below alpha, everyone's going to buy, 00:43:17.180 --> 00:43:19.600 align:middle line:84% and above alpha plus beta, no one's going to buy, 00:43:19.600 --> 00:43:22.260 align:middle line:84% so we can just focus on this range-- 00:43:22.260 --> 00:43:24.340 align:middle line:90% this range of value space. 00:43:24.340 --> 00:43:26.900 align:middle line:84% So c buys-- or the buyer buys if and only 00:43:26.900 --> 00:43:28.420 align:middle line:84% if v is greater than or equal to p. 00:43:28.420 --> 00:43:29.545 align:middle line:90% Well, let's write that out. 00:43:29.545 --> 00:43:35.410 align:middle line:84% That means alpha plus beta c is greater than or equal to p. 00:43:35.410 --> 00:43:37.790 align:middle line:84% Here, I've just written out what v means. 00:43:37.790 --> 00:43:40.170 align:middle line:84% And now let's solve for c, and we 00:43:40.170 --> 00:43:49.770 align:middle line:84% get c is greater than or equal to p minus alpha over B. 00:43:49.770 --> 00:43:51.850 align:middle line:90% So now we can compute-- 00:43:51.850 --> 00:43:54.070 align:middle line:84% let's actually go over to this board here. 00:43:54.070 --> 00:44:04.730 align:middle line:90% 00:44:04.730 --> 00:44:08.810 align:middle line:84% Let's compute pi of p, which is what 00:44:08.810 --> 00:44:12.850 align:middle line:84% is the seller's expected profit if they post a price of p? 00:44:12.850 --> 00:44:19.730 align:middle line:84% Now over here, we just wrote p minus c times the probability 00:44:19.730 --> 00:44:21.770 align:middle line:84% that v is greater than or equal to p. 00:44:21.770 --> 00:44:23.230 align:middle line:90% This part stays the same. 00:44:23.230 --> 00:44:25.130 align:middle line:84% This is the probability they buy, but now we 00:44:25.130 --> 00:44:27.250 align:middle line:90% have to adjust p minus c. 00:44:27.250 --> 00:44:31.160 align:middle line:84% Because now, there's not one c, there's a population of people. 00:44:31.160 --> 00:44:33.040 align:middle line:84% And the c that we care about is not 00:44:33.040 --> 00:44:34.900 align:middle line:84% the average sea in the population, 00:44:34.900 --> 00:44:38.160 align:middle line:84% but it's the average c among the people who buy. 00:44:38.160 --> 00:44:41.300 align:middle line:84% So we crucially need a conditional expectation. 00:44:41.300 --> 00:44:44.160 align:middle line:90% 00:44:44.160 --> 00:44:46.040 align:middle line:90% Maybe I'll write it this way. 00:44:46.040 --> 00:44:50.888 align:middle line:84% p minus e of c, given that someone buys, 00:44:50.888 --> 00:44:52.680 align:middle line:84% and that means given that c is greater than 00:44:52.680 --> 00:44:59.120 align:middle line:84% or equal to p minus alpha over B, over beta. 00:44:59.120 --> 00:45:02.997 align:middle line:84% And then I'm going to multiply that by the probability 00:45:02.997 --> 00:45:04.580 align:middle line:84% that v is greater than or equal to p-- 00:45:04.580 --> 00:45:06.080 align:middle line:84% or maybe I'll write it the same way. 00:45:06.080 --> 00:45:10.460 align:middle line:84% I'll write this as c just to make the math a bit easier. 00:45:10.460 --> 00:45:15.520 align:middle line:90% 00:45:15.520 --> 00:45:18.280 align:middle line:84% And now we can see this pretty clearly in our graph. 00:45:18.280 --> 00:45:24.080 align:middle line:84% Let's say I post a price of p here. 00:45:24.080 --> 00:45:25.720 align:middle line:90% Well, who's going to buy? 00:45:25.720 --> 00:45:32.220 align:middle line:84% It's going to be people along this segment of society. 00:45:32.220 --> 00:45:35.420 align:middle line:84% These are going to be the sickest people, the people who 00:45:35.420 --> 00:45:37.060 align:middle line:90% value insurance the most. 00:45:37.060 --> 00:45:41.940 align:middle line:84% And a naive mistake that I think historically some companies made 00:45:41.940 --> 00:45:44.580 align:middle line:84% is they did this calculation just taking the expected cost 00:45:44.580 --> 00:45:46.500 align:middle line:90% in the population. 00:45:46.500 --> 00:45:48.740 align:middle line:84% Not recognizing that, well, I shouldn't 00:45:48.740 --> 00:45:51.140 align:middle line:84% take the expectation over the entire line, 00:45:51.140 --> 00:45:52.750 align:middle line:84% I need to only take the expectation 00:45:52.750 --> 00:45:55.000 align:middle line:84% among the people who are actually buying my insurance. 00:45:55.000 --> 00:45:57.788 align:middle line:84% Or in the airline example, only among the people-- 00:45:57.788 --> 00:45:59.580 align:middle line:84% the flying frequency of people who actually 00:45:59.580 --> 00:46:02.820 align:middle line:90% buy an unlimited flying ticket. 00:46:02.820 --> 00:46:06.140 align:middle line:84% So now let's try to compute this expectation. 00:46:06.140 --> 00:46:10.300 align:middle line:84% Well, it turns out, c is uniformly distributed. 00:46:10.300 --> 00:46:17.740 align:middle line:84% Down here, this is actually p minus alpha over B, over beta. 00:46:17.740 --> 00:46:19.380 align:middle line:90% So I'm only focusing-- 00:46:19.380 --> 00:46:22.700 align:middle line:84% I'm only actually selling to people who lie in this range. 00:46:22.700 --> 00:46:24.900 align:middle line:84% And because c is uniformly distributed, 00:46:24.900 --> 00:46:26.930 align:middle line:84% the average cost among people in this range 00:46:26.930 --> 00:46:29.290 align:middle line:84% is just going to be the midpoint of this range. 00:46:29.290 --> 00:46:33.330 align:middle line:90% So this is going to be p minus-- 00:46:33.330 --> 00:46:43.490 align:middle line:84% well, the midpoint of this is going to be 1/2 times 1 plus p 00:46:43.490 --> 00:46:46.970 align:middle line:90% minus alpha over beta. 00:46:46.970 --> 00:46:49.990 align:middle line:84% So it's still price minus roughly cost, 00:46:49.990 --> 00:46:52.530 align:middle line:84% but now it's expected cost conditional on buying, 00:46:52.530 --> 00:46:55.010 align:middle line:84% and then I'm going to need to put this probability. 00:46:55.010 --> 00:46:58.770 align:middle line:84% Well, this probability is just the length-- 00:46:58.770 --> 00:47:03.050 align:middle line:84% or the length of this segment, which is 1 minus p minus 00:47:03.050 --> 00:47:03.990 align:middle line:90% alpha over beta. 00:47:03.990 --> 00:47:14.730 align:middle line:90% 00:47:14.730 --> 00:47:18.010 align:middle line:90% And let's-- I think it's-- 00:47:18.010 --> 00:47:19.690 align:middle line:84% we're interested in p, so let's try 00:47:19.690 --> 00:47:22.330 align:middle line:84% to do a little algebra to get p by itself here. 00:47:22.330 --> 00:47:27.200 align:middle line:84% So in this, we have p here, and we have minus 1/2 times 1 over 00:47:27.200 --> 00:47:27.920 align:middle line:90% beta. 00:47:27.920 --> 00:47:41.582 align:middle line:84% So we get p times 1 minus 1 over 2 beta minus 1/2 times alpha 00:47:41.582 --> 00:47:43.060 align:middle line:90% over beta. 00:47:43.060 --> 00:47:55.200 align:middle line:90% 00:47:55.200 --> 00:47:59.720 align:middle line:84% So now let's analyze what's going on here. 00:47:59.720 --> 00:48:02.940 align:middle line:84% This is the purchase probability. 00:48:02.940 --> 00:48:08.960 align:middle line:90% 00:48:08.960 --> 00:48:18.280 align:middle line:84% And this is the profit per purchase. 00:48:18.280 --> 00:48:21.320 align:middle line:84% Now, as usual, the purchase probability 00:48:21.320 --> 00:48:23.020 align:middle line:90% is decreasing in the price. 00:48:23.020 --> 00:48:24.420 align:middle line:84% This is the effect we always see. 00:48:24.420 --> 00:48:27.420 align:middle line:84% When I'm a firm and I charge a higher price, fewer people buy. 00:48:27.420 --> 00:48:29.020 align:middle line:90% That's very intuitive. 00:48:29.020 --> 00:48:31.300 align:middle line:90% The new effect is here. 00:48:31.300 --> 00:48:33.740 align:middle line:84% This is saying, what profit do I make 00:48:33.740 --> 00:48:36.420 align:middle line:90% per person who buys from me? 00:48:36.420 --> 00:48:40.620 align:middle line:84% And notice that the coefficient on p before, the coefficient 00:48:40.620 --> 00:48:43.820 align:middle line:84% on p was 1 When I charged a price, when I increased 00:48:43.820 --> 00:48:46.620 align:middle line:84% the price by $1, every time someone bought from me, 00:48:46.620 --> 00:48:49.460 align:middle line:84% the amount of revenue I generated-- the profit that I 00:48:49.460 --> 00:48:51.580 align:middle line:84% got from that person increased by $1. 00:48:51.580 --> 00:48:54.400 align:middle line:84% But now, it doesn't increase by $1. 00:48:54.400 --> 00:48:56.638 align:middle line:84% It increases by $1 minus 1 over 2 beta. 00:48:56.638 --> 00:48:57.680 align:middle line:90% So let's understand this. 00:48:57.680 --> 00:49:00.220 align:middle line:90% I raise my price by $1. 00:49:00.220 --> 00:49:02.640 align:middle line:84% I don't make a more per person I'm selling to, 00:49:02.640 --> 00:49:04.797 align:middle line:84% I only make one minus 1 over 2 beta more, 00:49:04.797 --> 00:49:05.880 align:middle line:90% which is strictly smaller. 00:49:05.880 --> 00:49:07.420 align:middle line:90% I don't get the full dollar. 00:49:07.420 --> 00:49:08.280 align:middle line:90% Why is that? 00:49:08.280 --> 00:49:10.600 align:middle line:90% So why is this less than 1? 00:49:10.600 --> 00:49:13.980 align:middle line:90% 00:49:13.980 --> 00:49:14.692 align:middle line:90% Yeah? 00:49:14.692 --> 00:49:16.400 align:middle line:84% AUDIENCE: Because as you raise the price, 00:49:16.400 --> 00:49:18.840 align:middle line:84% the people who will be buying it get more value out of it, 00:49:18.840 --> 00:49:20.510 align:middle line:84% so they'll cost more [INAUDIBLE]. 00:49:20.510 --> 00:49:21.510 align:middle line:90% IAN BALL: Exactly right. 00:49:21.510 --> 00:49:23.690 align:middle line:84% So this is reflecting two effects. 00:49:23.690 --> 00:49:25.930 align:middle line:84% The first effect is, yes, when I raise my price 00:49:25.930 --> 00:49:29.550 align:middle line:84% by $1, my revenue per patient increases by $1. 00:49:29.550 --> 00:49:30.970 align:middle line:90% That always happens. 00:49:30.970 --> 00:49:33.630 align:middle line:84% But the second effect is when I raise my price by $1, 00:49:33.630 --> 00:49:37.370 align:middle line:84% I change the composition of the people who buy from me. 00:49:37.370 --> 00:49:41.330 align:middle line:84% I shift the people who buy from me to a less healthy subset 00:49:41.330 --> 00:49:45.375 align:middle line:84% of people, and on average, those people are costlier to serve. 00:49:45.375 --> 00:49:47.250 align:middle line:84% And in fact, this is showing they're costlier 00:49:47.250 --> 00:49:48.950 align:middle line:90% by exactly 1 over 2 beta. 00:49:48.950 --> 00:49:51.730 align:middle line:90% 00:49:51.730 --> 00:49:53.290 align:middle line:90% Where does that come from? 00:49:53.290 --> 00:49:57.010 align:middle line:84% Well, we can see here, these are the segment of people I serve. 00:49:57.010 --> 00:50:00.690 align:middle line:84% When I raise my price by $1, I increase the left endpoint 00:50:00.690 --> 00:50:03.330 align:middle line:84% of this segment by 1 over beta, and therefore, I 00:50:03.330 --> 00:50:05.950 align:middle line:84% increase the average of this segment by 1 over 2 beta. 00:50:05.950 --> 00:50:07.970 align:middle line:90% So this is exactly the-- 00:50:07.970 --> 00:50:09.850 align:middle line:84% 1 over 2 beta is the marginal effect 00:50:09.850 --> 00:50:13.170 align:middle line:84% of my price on the average cost among the people 00:50:13.170 --> 00:50:16.130 align:middle line:90% who buy from me. 00:50:16.130 --> 00:50:19.320 align:middle line:84% More precisely, it's the derivative of this expression 00:50:19.320 --> 00:50:20.260 align:middle line:90% with respect to p. 00:50:20.260 --> 00:50:24.000 align:middle line:90% 00:50:24.000 --> 00:50:28.280 align:middle line:90% And we can see-- 00:50:28.280 --> 00:50:30.800 align:middle line:84% let's look at different cases of beta. 00:50:30.800 --> 00:50:38.240 align:middle line:84% So what if beta is large and what if beta is small? 00:50:38.240 --> 00:50:42.600 align:middle line:84% So if beta is very large, adverse selection 00:50:42.600 --> 00:50:45.120 align:middle line:90% is not much of a problem. 00:50:45.120 --> 00:50:53.480 align:middle line:84% If beta is very large, this curve is very, very steep. 00:50:53.480 --> 00:50:57.400 align:middle line:84% And if beta is large, it says that there's 00:50:57.400 --> 00:51:02.440 align:middle line:84% not a huge effect on the average cost when I change the price. 00:51:02.440 --> 00:51:04.160 align:middle line:84% Because if beta is large, it says 00:51:04.160 --> 00:51:09.062 align:middle line:84% that c doesn't change much with v. Another way-- 00:51:09.062 --> 00:51:10.520 align:middle line:84% I think the easiest way to see this 00:51:10.520 --> 00:51:13.680 align:middle line:84% is to solve this expression for c as we've done. 00:51:13.680 --> 00:51:23.620 align:middle line:84% And so note, c equals v minus alpha over B. 00:51:23.620 --> 00:51:25.680 align:middle line:84% So when I change the price by a little bit, 00:51:25.680 --> 00:51:28.300 align:middle line:84% I'm changing by a little bit the value of people 00:51:28.300 --> 00:51:29.360 align:middle line:90% who are buying from me. 00:51:29.360 --> 00:51:32.117 align:middle line:84% And the question is, if someone's value changes a bit, 00:51:32.117 --> 00:51:34.200 align:middle line:84% someone's value of buying for me changes by a bit, 00:51:34.200 --> 00:51:36.460 align:middle line:84% how much does my cost of providing to them change? 00:51:36.460 --> 00:51:39.460 align:middle line:84% And we see, it's exactly proportional to 1 over beta. 00:51:39.460 --> 00:51:41.380 align:middle line:84% So when beta is really, really large, 00:51:41.380 --> 00:51:43.500 align:middle line:84% adverse selection is not much of a concern. 00:51:43.500 --> 00:51:45.000 align:middle line:90% So little adverse selection. 00:51:45.000 --> 00:51:50.860 align:middle line:90% 00:51:50.860 --> 00:51:53.380 align:middle line:84% When beta grows small, then people 00:51:53.380 --> 00:51:55.540 align:middle line:84% who value the item just a little bit more 00:51:55.540 --> 00:51:58.420 align:middle line:90% are extremely costly to insure. 00:51:58.420 --> 00:52:02.600 align:middle line:84% So we have severe adverse selection. 00:52:02.600 --> 00:52:08.260 align:middle line:90% 00:52:08.260 --> 00:52:15.530 align:middle line:84% And let's look at the case where beta is greater than 1/2. 00:52:15.530 --> 00:52:17.330 align:middle line:90% Or, sorry, less than 1/2. 00:52:17.330 --> 00:52:20.810 align:middle line:90% So this is the extreme case. 00:52:20.810 --> 00:52:22.590 align:middle line:84% What happens if beta is less than 1/2? 00:52:22.590 --> 00:52:24.030 align:middle line:84% What happens to this coefficient? 00:52:24.030 --> 00:52:27.470 align:middle line:90% 00:52:27.470 --> 00:52:27.970 align:middle line:90% Yeah? 00:52:27.970 --> 00:52:28.970 align:middle line:90% AUDIENCE: It's negative? 00:52:28.970 --> 00:52:30.090 align:middle line:90% IAN BALL: It's negative. 00:52:30.090 --> 00:52:31.048 align:middle line:90% So what does that mean? 00:52:31.048 --> 00:52:34.690 align:middle line:84% I raise prices and I actually earn less per person 00:52:34.690 --> 00:52:36.110 align:middle line:90% than I did with the lower price. 00:52:36.110 --> 00:52:39.410 align:middle line:90% 00:52:39.410 --> 00:52:43.050 align:middle line:84% What it says is if beta is less than 1/2, 00:52:43.050 --> 00:52:46.890 align:middle line:84% the shift in the composition of people who buy from me more than 00:52:46.890 --> 00:52:49.410 align:middle line:84% outweighs the fact that I'm charging a higher price. 00:52:49.410 --> 00:52:53.010 align:middle line:84% And I'm actually earning less per patient I serve. 00:52:53.010 --> 00:52:55.210 align:middle line:84% And what that means-- maybe we'll 00:52:55.210 --> 00:52:59.470 align:middle line:90% call this market breakdown. 00:52:59.470 --> 00:53:02.530 align:middle line:90% 00:53:02.530 --> 00:53:03.050 align:middle line:90% Why? 00:53:03.050 --> 00:53:08.370 align:middle line:84% Well, if beta is less than 1/2 then this is decreasing in p, 00:53:08.370 --> 00:53:11.330 align:middle line:84% and this is also decreasing in p. 00:53:11.330 --> 00:53:13.320 align:middle line:84% So that shows my profit is certainly 00:53:13.320 --> 00:53:21.180 align:middle line:84% decreasing in p over the range alpha to alpha plus beta. 00:53:21.180 --> 00:53:24.640 align:middle line:90% 00:53:24.640 --> 00:53:28.040 align:middle line:84% So when I say market breakdown, I 00:53:28.040 --> 00:53:30.020 align:middle line:90% guess the only possibility is-- 00:53:30.020 --> 00:53:37.040 align:middle line:90% 00:53:37.040 --> 00:53:40.300 align:middle line:84% maybe what I mean is screening is not possible. 00:53:40.300 --> 00:53:43.840 align:middle line:90% 00:53:43.840 --> 00:53:46.920 align:middle line:84% Because the optimal price is either p 00:53:46.920 --> 00:53:48.500 align:middle line:90% equals alpha or no sale. 00:53:48.500 --> 00:53:55.120 align:middle line:90% 00:53:55.120 --> 00:53:59.640 align:middle line:84% If my profit is decreasing in p, the only possible maximum 00:53:59.640 --> 00:54:01.520 align:middle line:84% of this expression would be at p equals alpha 00:54:01.520 --> 00:54:02.660 align:middle line:90% at the left endpoint. 00:54:02.660 --> 00:54:04.840 align:middle line:84% So I either have to serve everyone in the market 00:54:04.840 --> 00:54:05.850 align:middle line:90% or serve no one. 00:54:05.850 --> 00:54:08.100 align:middle line:84% And it may be optimal actually for me to serve no one. 00:54:08.100 --> 00:54:09.780 align:middle line:84% But what it says is there can't be a screening effect. 00:54:09.780 --> 00:54:12.405 align:middle line:84% I can't charge an intermediate price and get some people to buy 00:54:12.405 --> 00:54:13.680 align:middle line:90% and some people not to buy. 00:54:13.680 --> 00:54:16.140 align:middle line:84% The only possibility is I set the price low enough 00:54:16.140 --> 00:54:22.703 align:middle line:84% that everyone buys from me or I just don't sell at all. 00:54:22.703 --> 00:54:24.620 align:middle line:84% Maybe we can interpret everyone buying from me 00:54:24.620 --> 00:54:26.580 align:middle line:90% as kind of universal coverage. 00:54:26.580 --> 00:54:28.660 align:middle line:84% We can't have a premium insurance plan 00:54:28.660 --> 00:54:30.023 align:middle line:90% that only certain people buy. 00:54:30.023 --> 00:54:31.940 align:middle line:84% That can never be optimal if adverse selection 00:54:31.940 --> 00:54:32.648 align:middle line:90% is severe enough. 00:54:32.648 --> 00:54:33.340 align:middle line:90% Yes? 00:54:33.340 --> 00:54:36.800 align:middle line:84% AUDIENCE: And so if you sell to everyone at a price of alpha, 00:54:36.800 --> 00:54:38.480 align:middle line:90% is your profit supposed to be 0? 00:54:38.480 --> 00:54:39.900 align:middle line:90% Is that what the [INAUDIBLE]? 00:54:39.900 --> 00:54:42.150 align:middle line:84% IAN BALL: It's going to depend on the values of alpha. 00:54:42.150 --> 00:54:43.700 align:middle line:90% So in some cases-- 00:54:43.700 --> 00:54:45.220 align:middle line:84% so if you set the price to alpha, 00:54:45.220 --> 00:54:46.580 align:middle line:90% you're going to get some number. 00:54:46.580 --> 00:54:49.260 align:middle line:84% If that number is-- if your profit is greater than 0, 00:54:49.260 --> 00:54:50.160 align:middle line:90% that's the optimum. 00:54:50.160 --> 00:54:52.678 align:middle line:84% If it's less than 0, you're better off selling to no one. 00:54:52.678 --> 00:54:55.220 align:middle line:84% AUDIENCE: Oh, so your point is just that if beta is less than 00:54:55.220 --> 00:54:59.297 align:middle line:84% 1/2, then the only option-- like, you have to solve-- 00:54:59.297 --> 00:54:59.880 align:middle line:90% IAN BALL: Yes. 00:54:59.880 --> 00:55:01.120 align:middle line:90% There's no interior solution. 00:55:01.120 --> 00:55:03.078 align:middle line:84% Either I sell to everyone or I sell to no one. 00:55:03.078 --> 00:55:04.620 align:middle line:84% I mean, we could go more into details 00:55:04.620 --> 00:55:06.720 align:middle line:84% depending on the details of alpha and beta, 00:55:06.720 --> 00:55:10.930 align:middle line:84% but basically, we have a function, 00:55:10.930 --> 00:55:15.250 align:middle line:90% and it's going like this. 00:55:15.250 --> 00:55:22.982 align:middle line:84% So its maximum is at alpha equals 0, yeah. 00:55:22.982 --> 00:55:24.690 align:middle line:84% We have to be careful if sometimes things 00:55:24.690 --> 00:55:25.390 align:middle line:90% can be negative. 00:55:25.390 --> 00:55:26.927 align:middle line:90% So, yeah, when I say it's-- 00:55:26.927 --> 00:55:28.010 align:middle line:90% OK, we have to be careful. 00:55:28.010 --> 00:55:30.723 align:middle line:84% When we say it's decreasing in beta, that's not true 00:55:30.723 --> 00:55:32.390 align:middle line:84% if one of these expressions is negative, 00:55:32.390 --> 00:55:34.098 align:middle line:84% so let me make the more precise statement 00:55:34.098 --> 00:55:37.710 align:middle line:84% that the solution occurs either at alpha or beta-- 00:55:37.710 --> 00:55:38.710 align:middle line:90% at alpha plus beta. 00:55:38.710 --> 00:55:40.850 align:middle line:84% AUDIENCE: And so why does it make sense 00:55:40.850 --> 00:55:44.250 align:middle line:90% that you would sell to everyone? 00:55:44.250 --> 00:55:46.650 align:middle line:84% Like, what about beta makes it so 00:55:46.650 --> 00:55:50.557 align:middle line:84% that you could potentially make money from selling everyone? 00:55:50.557 --> 00:55:52.890 align:middle line:84% IAN BALL: So whether it's profitable to sell to everyone 00:55:52.890 --> 00:55:54.690 align:middle line:84% is really going to depend on alpha. 00:55:54.690 --> 00:55:57.170 align:middle line:84% What beta is controlling is what happens 00:55:57.170 --> 00:55:59.090 align:middle line:90% when I slightly raise my price. 00:55:59.090 --> 00:56:01.530 align:middle line:84% So when beta is very, very small, 00:56:01.530 --> 00:56:04.410 align:middle line:84% it says adverse selection is so severe that I would never 00:56:04.410 --> 00:56:06.840 align:middle line:84% want to sell to only some people in the population 00:56:06.840 --> 00:56:08.960 align:middle line:84% because by raising my price a little bit, 00:56:08.960 --> 00:56:11.160 align:middle line:84% I'm actually earning less per person 00:56:11.160 --> 00:56:13.280 align:middle line:84% because those people are so much sicker. 00:56:13.280 --> 00:56:15.095 align:middle line:84% So when that effect is really strong, 00:56:15.095 --> 00:56:17.220 align:middle line:84% I don't want to be vulnerable to adverse selection. 00:56:17.220 --> 00:56:19.640 align:middle line:84% I don't want to sell to the subset of the population who 00:56:19.640 --> 00:56:20.420 align:middle line:90% value it most. 00:56:20.420 --> 00:56:22.640 align:middle line:84% I either want to sell to everyone or sell to no one. 00:56:22.640 --> 00:56:24.000 align:middle line:84% And that's going to be controlled by alpha. 00:56:24.000 --> 00:56:25.360 align:middle line:90% Alpha could be enormous. 00:56:25.360 --> 00:56:28.060 align:middle line:84% If alpha is a million and c is pretty small, 00:56:28.060 --> 00:56:30.800 align:middle line:84% then I certainly want to sell to everyone. 00:56:30.800 --> 00:56:32.510 align:middle line:84% Alpha is kind of about a level effect. 00:56:32.510 --> 00:56:34.260 align:middle line:84% It's like, is this a good market to be in? 00:56:34.260 --> 00:56:35.840 align:middle line:84% Do I want to sell to people or not? 00:56:35.840 --> 00:56:38.000 align:middle line:84% Beta is a marginal effect, which is 00:56:38.000 --> 00:56:40.240 align:middle line:84% controlling what happens to my profit 00:56:40.240 --> 00:56:42.080 align:middle line:90% as I slightly change my price. 00:56:42.080 --> 00:56:44.480 align:middle line:84% And it's saying, if beta is small enough, 00:56:44.480 --> 00:56:47.760 align:middle line:84% the adverse selection effect is so strong 00:56:47.760 --> 00:56:49.807 align:middle line:84% that I never want to slightly shrink 00:56:49.807 --> 00:56:51.640 align:middle line:84% the set of people I sell to because the cost 00:56:51.640 --> 00:56:53.140 align:middle line:84% among those people is going to be so much higher. 00:56:53.140 --> 00:56:53.720 align:middle line:90% Yes, again. 00:56:53.720 --> 00:56:55.720 align:middle line:84% AUDIENCE: I'm thinking about real-world examples 00:56:55.720 --> 00:56:57.100 align:middle line:84% where you have insurance markets, 00:56:57.100 --> 00:56:58.860 align:middle line:84% I'm thinking like disaster insurance 00:56:58.860 --> 00:57:01.880 align:middle line:84% where if you live in a certain area, you have to buy insurance, 00:57:01.880 --> 00:57:07.670 align:middle line:84% or if you have health care given by your employer. 00:57:07.670 --> 00:57:10.830 align:middle line:84% Would that be a case where you just 00:57:10.830 --> 00:57:14.530 align:middle line:84% end up charging alpha because of the policy intervention? 00:57:14.530 --> 00:57:17.415 align:middle line:84% Like the insurance company charges alpha for everyone. 00:57:17.415 --> 00:57:19.790 align:middle line:84% IAN BALL: Yeah, so I guess a separate issue that comes up 00:57:19.790 --> 00:57:22.683 align:middle line:84% with disaster insurance is that often, disasters are correlated. 00:57:22.683 --> 00:57:24.350 align:middle line:84% So you have a heart attack, someone else 00:57:24.350 --> 00:57:25.730 align:middle line:84% is more likely to have a heart attack. 00:57:25.730 --> 00:57:27.030 align:middle line:84% If you get hit by a hurricane, everyone's 00:57:27.030 --> 00:57:28.210 align:middle line:90% often hit by a hurricane. 00:57:28.210 --> 00:57:30.710 align:middle line:84% So often there's an issue that these disasters 00:57:30.710 --> 00:57:32.350 align:middle line:90% are too big to insure. 00:57:32.350 --> 00:57:34.710 align:middle line:84% So often, the state steps in and provides the insurance 00:57:34.710 --> 00:57:36.252 align:middle line:84% because no private company can do it, 00:57:36.252 --> 00:57:40.390 align:middle line:84% and then the state puts in these regulations. 00:57:40.390 --> 00:57:44.550 align:middle line:84% I think really, what's going on in those cases-- there's 00:57:44.550 --> 00:57:47.070 align:middle line:84% a lot going on with disaster insurance. 00:57:47.070 --> 00:57:52.910 align:middle line:84% In this case, here, I think normally, 00:57:52.910 --> 00:57:54.483 align:middle line:84% with reasonable parameter values-- 00:57:54.483 --> 00:57:56.150 align:middle line:84% I mean, this is a very simplified model. 00:57:56.150 --> 00:57:57.530 align:middle line:84% With reasonable parameter values, 00:57:57.530 --> 00:58:00.110 align:middle line:84% when adverse selection is bad, the solution is probably going 00:58:00.110 --> 00:58:02.410 align:middle line:84% to be to sell to no one because you're going to have-- 00:58:02.410 --> 00:58:06.015 align:middle line:84% if you have any really healthy people who are willing to pay 00:58:06.015 --> 00:58:07.390 align:middle line:84% a very low amount for insurance-- 00:58:07.390 --> 00:58:08.350 align:middle line:90% I mean, think about it. 00:58:08.350 --> 00:58:11.610 align:middle line:84% Let's say-- if I'm supposed to insure everyone 00:58:11.610 --> 00:58:13.250 align:middle line:84% in the population, but I can only 00:58:13.250 --> 00:58:14.917 align:middle line:84% charge a price that makes the healthiest 00:58:14.917 --> 00:58:17.050 align:middle line:84% people willing to buy, then that's probably not 00:58:17.050 --> 00:58:18.300 align:middle line:90% going to be profitable for me. 00:58:18.300 --> 00:58:21.290 align:middle line:84% So I think what I would say is, in realistic settings, 00:58:21.290 --> 00:58:23.050 align:middle line:84% the alpha where markets break down, 00:58:23.050 --> 00:58:25.450 align:middle line:84% the alpha is going to be small enough so that there's not 00:58:25.450 --> 00:58:28.270 align:middle line:84% going to be a sustainable private insurance market. 00:58:28.270 --> 00:58:31.228 align:middle line:84% Of course, there's a separate issue here that we haven't-- 00:58:31.228 --> 00:58:32.770 align:middle line:84% there's a separate question, which is 00:58:32.770 --> 00:58:34.290 align:middle line:90% how well can we monitor people? 00:58:34.290 --> 00:58:35.707 align:middle line:84% So another thing that could happen 00:58:35.707 --> 00:58:37.750 align:middle line:84% is you might impose very strict health standards. 00:58:37.750 --> 00:58:40.333 align:middle line:84% You might say-- there might be a functioning insurance market, 00:58:40.333 --> 00:58:41.850 align:middle line:90% but only for people under 50. 00:58:41.850 --> 00:58:44.392 align:middle line:84% And then people over 50 are just excluded from the market. 00:58:44.392 --> 00:58:46.850 align:middle line:84% And that conditioning is not present, is not captured here, 00:58:46.850 --> 00:58:48.970 align:middle line:84% or at least the interpretation of this would 00:58:48.970 --> 00:58:51.850 align:middle line:84% have to be after we conduct medical tests and after we 00:58:51.850 --> 00:58:55.490 align:middle line:84% condition, and then beta might become bigger, yeah. 00:58:55.490 --> 00:58:58.050 align:middle line:90% Yeah, I think the-- 00:58:58.050 --> 00:58:59.470 align:middle line:84% yeah I think the insurance market, 00:58:59.470 --> 00:59:00.980 align:middle line:84% the disaster insurance is interesting, 00:59:00.980 --> 00:59:02.680 align:middle line:84% but I think it's really a separate issue because people 00:59:02.680 --> 00:59:04.400 align:middle line:84% don't really have private information about how 00:59:04.400 --> 00:59:05.840 align:middle line:90% likely their home is to flood. 00:59:05.840 --> 00:59:07.960 align:middle line:84% I mean, maybe if the government has to use really blanket 00:59:07.960 --> 00:59:09.900 align:middle line:84% policies and they're not fine-grained enough, 00:59:09.900 --> 00:59:11.380 align:middle line:84% but at least in the private market, 00:59:11.380 --> 00:59:13.588 align:middle line:84% I think that's not really an adverse selection issue, 00:59:13.588 --> 00:59:15.540 align:middle line:84% it's really a too-big-to-fail issue. 00:59:15.540 --> 00:59:17.165 align:middle line:84% Yeah, but it's an interesting question. 00:59:17.165 --> 00:59:19.280 align:middle line:84% And, yeah, actually one of my colleagues 00:59:19.280 --> 00:59:21.440 align:middle line:84% is just writing a paper about this 00:59:21.440 --> 00:59:23.960 align:middle line:84% in California wildfire insurance and looking 00:59:23.960 --> 00:59:25.220 align:middle line:90% at how insurers handle this. 00:59:25.220 --> 00:59:27.300 align:middle line:90% Yeah, it's interesting. 00:59:27.300 --> 00:59:27.800 align:middle line:90% OK. 00:59:27.800 --> 00:59:30.760 align:middle line:90% 00:59:30.760 --> 00:59:32.520 align:middle line:84% We've talked a lot about screening. 00:59:32.520 --> 00:59:34.880 align:middle line:90% Let's now talk about signaling. 00:59:34.880 --> 00:59:36.480 align:middle line:84% And this gets a little messy, so we're 00:59:36.480 --> 00:59:38.600 align:middle line:84% going to have to work with an even simpler model. 00:59:38.600 --> 00:59:41.700 align:middle line:84% And, yeah, we should have time just to finish this off. 00:59:41.700 --> 00:59:56.120 align:middle line:90% 00:59:56.120 --> 00:59:59.810 align:middle line:84% So now let's finally move to the case of signaling. 00:59:59.810 --> 01:00:02.590 align:middle line:90% 01:00:02.590 --> 01:00:04.330 align:middle line:84% So we're still with a seller and a buyer, 01:00:04.330 --> 01:00:06.372 align:middle line:84% we're still in incomplete information bargaining, 01:00:06.372 --> 01:00:10.750 align:middle line:84% but now it's the seller who has private information. 01:00:10.750 --> 01:00:16.110 align:middle line:84% And now we're going to say the seller has 01:00:16.110 --> 01:00:19.690 align:middle line:90% private info about v and c. 01:00:19.690 --> 01:00:21.890 align:middle line:84% So this is another canonical application 01:00:21.890 --> 01:00:23.390 align:middle line:84% where the seller-- we said insurance 01:00:23.390 --> 01:00:25.310 align:middle line:84% is the canonical case where the buyer has 01:00:25.310 --> 01:00:27.758 align:middle line:84% private information that's relevant to both v and c. 01:00:27.758 --> 01:00:29.550 align:middle line:84% Can you think of a setting where the seller 01:00:29.550 --> 01:00:33.270 align:middle line:84% has private information about v and c? 01:00:33.270 --> 01:00:38.130 align:middle line:84% It's kind of a classic story here. 01:00:38.130 --> 01:00:39.630 align:middle line:84% Used cars, kind of the classic case, 01:00:39.630 --> 01:00:40.880 align:middle line:90% or more generally, used goods. 01:00:40.880 --> 01:00:44.208 align:middle line:90% 01:00:44.208 --> 01:00:45.750 align:middle line:84% Here, I think it's important to think 01:00:45.750 --> 01:00:48.470 align:middle line:84% of c maybe not so much as a cost, but more as 01:00:48.470 --> 01:00:50.252 align:middle line:84% how much the seller values the good. 01:00:50.252 --> 01:00:52.210 align:middle line:84% And the idea is if I'm going to buy a used car, 01:00:52.210 --> 01:00:54.150 align:middle line:84% the seller of that used car knows 01:00:54.150 --> 01:00:56.138 align:middle line:84% whether it's making a bad sound, whether it's 01:00:56.138 --> 01:00:58.430 align:middle line:84% had a lot of accidents, whether there's these problems. 01:00:58.430 --> 01:01:01.930 align:middle line:84% Basically what they know is the quality of the good. 01:01:01.930 --> 01:01:04.410 align:middle line:84% And that quality, unobserved quality, 01:01:04.410 --> 01:01:06.410 align:middle line:84% is relevant both to how much the buyer values it 01:01:06.410 --> 01:01:07.870 align:middle line:84% and how much the seller values it. 01:01:07.870 --> 01:01:09.910 align:middle line:84% But because the seller has had the good for a while, 01:01:09.910 --> 01:01:11.493 align:middle line:84% they know a lot more about it than you 01:01:11.493 --> 01:01:15.490 align:middle line:84% as the buyer who come by and just try to inspect your good. 01:01:15.490 --> 01:01:18.630 align:middle line:84% And the classic case would be used cars. 01:01:18.630 --> 01:01:24.570 align:middle line:84% So there's a very famous paper called "The Market for Lemons" 01:01:24.570 --> 01:01:25.570 align:middle line:90% by Akerlof. 01:01:25.570 --> 01:01:28.010 align:middle line:84% He won the Nobel Prize for identifying 01:01:28.010 --> 01:01:31.010 align:middle line:84% this issue of why used car markets often don't work well, 01:01:31.010 --> 01:01:33.770 align:middle line:84% and it's the same classic thing that we said, going back 01:01:33.770 --> 01:01:35.770 align:middle line:84% to the defendant and the prosecutor, 01:01:35.770 --> 01:01:38.270 align:middle line:84% if someone comes to you and says, "I can't sell my used car, 01:01:38.270 --> 01:01:39.470 align:middle line:84% but I'm going to give you a really good deal. 01:01:39.470 --> 01:01:40.550 align:middle line:84% It's a really good deal, it's really cheap." 01:01:40.550 --> 01:01:41.110 align:middle line:90% What are you going to say? 01:01:41.110 --> 01:01:42.235 align:middle line:90% "Wait, why is it so cheap?" 01:01:42.235 --> 01:01:44.030 align:middle line:84% It's cheap because it must be a lemon. 01:01:44.030 --> 01:01:46.613 align:middle line:84% It must be-- that's another word for a car that doesn't really 01:01:46.613 --> 01:01:47.290 align:middle line:90% work. 01:01:47.290 --> 01:01:50.610 align:middle line:84% And for that reason, these markets can break down. 01:01:50.610 --> 01:01:53.030 align:middle line:84% I think my financial example was another good one. 01:01:53.030 --> 01:01:56.160 align:middle line:84% So mortgage-backed securities might 01:01:56.160 --> 01:01:57.680 align:middle line:84% be another case where it's not clear 01:01:57.680 --> 01:01:58.980 align:middle line:84% which side has more private information, 01:01:58.980 --> 01:02:00.900 align:middle line:84% it depends who they are, but in some cases, 01:02:00.900 --> 01:02:02.160 align:middle line:84% the holder of the mortgage-backed security 01:02:02.160 --> 01:02:03.720 align:middle line:84% may have private information that's 01:02:03.720 --> 01:02:05.660 align:middle line:84% relevant to the value of this asset, 01:02:05.660 --> 01:02:08.520 align:middle line:84% and both the buyer and the seller care about that. 01:02:08.520 --> 01:02:11.080 align:middle line:84% OK, so let's look at a really simplified model where we're 01:02:11.080 --> 01:02:19.040 align:middle line:84% going to say that the good has quality either L or H. 01:02:19.040 --> 01:02:21.660 align:middle line:84% And here, these are not numbers, these are just labels. 01:02:21.660 --> 01:02:23.910 align:middle line:84% It's either low quality or high quality. 01:02:23.910 --> 01:02:26.160 align:middle line:84% But the quality the good is relevant to both the buyer 01:02:26.160 --> 01:02:27.280 align:middle line:90% and the seller. 01:02:27.280 --> 01:02:29.700 align:middle line:84% So if it's a low-quality good, then-- 01:02:29.700 --> 01:02:38.920 align:middle line:90% 01:02:38.920 --> 01:02:40.760 align:middle line:90% and, I don't know. 01:02:40.760 --> 01:02:43.000 align:middle line:84% All these inequalities may be confusing. 01:02:43.000 --> 01:02:48.960 align:middle line:84% So if it's low quality, then the seller's cost is cL 01:02:48.960 --> 01:02:50.800 align:middle line:90% and the buyer's value is vL. 01:02:50.800 --> 01:02:53.430 align:middle line:84% If it's high quality, the seller's cost is cH 01:02:53.430 --> 01:02:56.630 align:middle line:84% and the buyer's value is vH, and we have all these inequalities. 01:02:56.630 --> 01:02:58.990 align:middle line:84% For the seller, a low-quality item 01:02:58.990 --> 01:03:01.350 align:middle line:84% is less costly than a high-quality item, 01:03:01.350 --> 01:03:03.350 align:middle line:84% but we also want the seller's cost 01:03:03.350 --> 01:03:06.910 align:middle line:84% to be lower than the buyer's value for a low-quality item. 01:03:06.910 --> 01:03:10.270 align:middle line:84% For a high-quality item, the seller's cost 01:03:10.270 --> 01:03:11.850 align:middle line:84% is smaller than the buyer's value, 01:03:11.850 --> 01:03:13.570 align:middle line:84% so there are potential gains from trade, 01:03:13.570 --> 01:03:15.230 align:middle line:84% but again, the buyer values it more 01:03:15.230 --> 01:03:17.510 align:middle line:84% when it's high quality than when it's low quality. 01:03:17.510 --> 01:03:20.870 align:middle line:84% So there's a string of inequalities here. 01:03:20.870 --> 01:03:23.890 align:middle line:84% And now what we're going to look at is price signaling. 01:03:23.890 --> 01:03:27.710 align:middle line:90% 01:03:27.710 --> 01:03:29.370 align:middle line:84% I'm going to set a price to the seller, 01:03:29.370 --> 01:03:31.110 align:middle line:84% the buyer is going to see that price and infer 01:03:31.110 --> 01:03:32.730 align:middle line:84% something about the value of the item, 01:03:32.730 --> 01:03:35.230 align:middle line:84% and then we want to try to solve for an equilibrium of this. 01:03:35.230 --> 01:03:37.790 align:middle line:84% And we're going to have to use perfect Bayesian equilibrium 01:03:37.790 --> 01:03:38.930 align:middle line:90% here to analyze this. 01:03:38.930 --> 01:03:50.150 align:middle line:90% 01:03:50.150 --> 01:03:54.010 align:middle line:84% Now we need to specify a prior distribution 01:03:54.010 --> 01:03:55.470 align:middle line:90% about the quality of the good. 01:03:55.470 --> 01:03:57.330 align:middle line:84% So let me say a little bit more here. 01:03:57.330 --> 01:04:01.030 align:middle line:90% The seller knows quality. 01:04:01.030 --> 01:04:03.492 align:middle line:90% 01:04:03.492 --> 01:04:04.950 align:middle line:84% The buyer doesn't know the quality, 01:04:04.950 --> 01:04:06.450 align:middle line:84% but they just know the distribution 01:04:06.450 --> 01:04:08.130 align:middle line:90% of quality in the population. 01:04:08.130 --> 01:04:09.930 align:middle line:90% And we'll say that-- 01:04:09.930 --> 01:04:14.290 align:middle line:84% we'll put pi here and 1 minus pi here. 01:04:14.290 --> 01:04:19.170 align:middle line:84% So the prior distribution is that fraction pi of things 01:04:19.170 --> 01:04:21.250 align:middle line:84% are high quality, fraction 1 minus pi of things 01:04:21.250 --> 01:04:23.710 align:middle line:84% are low quality, the seller actually knows the quality, 01:04:23.710 --> 01:04:24.590 align:middle line:90% the buyer does not. 01:04:24.590 --> 01:04:32.130 align:middle line:90% 01:04:32.130 --> 01:04:34.490 align:middle line:84% And after observing the price, the seller 01:04:34.490 --> 01:04:35.748 align:middle line:90% is going to set a price. 01:04:35.748 --> 01:04:37.290 align:middle line:84% The buyer sees the price and is going 01:04:37.290 --> 01:04:39.498 align:middle line:84% to update their beliefs about the quality of the item 01:04:39.498 --> 01:04:40.450 align:middle line:90% based upon the price. 01:04:40.450 --> 01:04:43.048 align:middle line:90% So let's try to construct-- 01:04:43.048 --> 01:04:45.090 align:middle line:84% and I think you'll have more practice doing these 01:04:45.090 --> 01:04:46.548 align:middle line:84% in the problem set, but we'll start 01:04:46.548 --> 01:04:50.280 align:middle line:90% constructing PBE in this game. 01:04:50.280 --> 01:04:51.680 align:middle line:90% Perfect Bayesian Equilibria. 01:04:51.680 --> 01:04:53.620 align:middle line:84% And as usual, we're going to separate into two cases. 01:04:53.620 --> 01:04:55.495 align:middle line:84% We're going to look for a pooling equilibrium 01:04:55.495 --> 01:04:57.120 align:middle line:90% and a separating equilibrium. 01:04:57.120 --> 01:05:00.960 align:middle line:84% So on the one hand, we'll do pooling, 01:05:00.960 --> 01:05:02.300 align:middle line:90% and the other is separating. 01:05:02.300 --> 01:05:09.960 align:middle line:90% 01:05:09.960 --> 01:05:12.800 align:middle line:84% A pooling equilibrium means sellers 01:05:12.800 --> 01:05:14.640 align:middle line:84% with high-quality products and sellers 01:05:14.640 --> 01:05:17.480 align:middle line:84% with low-quality products both sell at the same price. 01:05:17.480 --> 01:05:19.160 align:middle line:84% So the price alone doesn't give away 01:05:19.160 --> 01:05:21.040 align:middle line:84% whether it's high quality or low quality. 01:05:21.040 --> 01:05:24.060 align:middle line:84% So in a pooling equilibrium, the seller's strategy-- 01:05:24.060 --> 01:05:27.730 align:middle line:90% 01:05:27.730 --> 01:05:29.980 align:middle line:84% so let's make sure we understand what's going on here. 01:05:29.980 --> 01:05:31.780 align:middle line:84% What is the seller's strategy in this game? 01:05:31.780 --> 01:05:35.240 align:middle line:84% We need to say if the quality is low, what price do they set? 01:05:35.240 --> 01:05:38.120 align:middle line:84% And if the quality is high, what price do they set? 01:05:38.120 --> 01:05:40.820 align:middle line:84% In a pooling equilibrium, these are going to be the same price, 01:05:40.820 --> 01:05:42.350 align:middle line:84% and we'll call that price p star. 01:05:42.350 --> 01:05:44.600 align:middle line:84% And this is going to be a parameter of the equilibrium 01:05:44.600 --> 01:05:47.070 align:middle line:84% that we're going to try to solve for. 01:05:47.070 --> 01:05:48.290 align:middle line:90% What is the buyer's strategy? 01:05:48.290 --> 01:05:51.750 align:middle line:84% As I say, I think writing down the strategies correctly 01:05:51.750 --> 01:05:53.730 align:middle line:84% is often the hardest part of this. 01:05:53.730 --> 01:05:56.330 align:middle line:84% So what is the strategy for the buyer in this game? 01:05:56.330 --> 01:06:02.350 align:middle line:90% 01:06:02.350 --> 01:06:04.710 align:middle line:84% AUDIENCE: Can't you find an expected value 01:06:04.710 --> 01:06:07.710 align:middle line:90% of the quality in some way? 01:06:07.710 --> 01:06:08.350 align:middle line:90% Or-- 01:06:08.350 --> 01:06:09.750 align:middle line:90% IAN BALL: You could-- 01:06:09.750 --> 01:06:13.210 align:middle line:84% you could, but let's not even think about optimality. 01:06:13.210 --> 01:06:15.490 align:middle line:84% Let's just think what kind of object is a strategy? 01:06:15.490 --> 01:06:17.550 align:middle line:84% It's a number, is it a vector, is it a function? 01:06:17.550 --> 01:06:19.355 align:middle line:90% Just what thing is it? 01:06:19.355 --> 01:06:20.730 align:middle line:84% What kind of mathematical object? 01:06:20.730 --> 01:06:21.730 align:middle line:84% I think that's often the first step. 01:06:21.730 --> 01:06:22.480 align:middle line:90% Yeah, in the back. 01:06:22.480 --> 01:06:24.510 align:middle line:84% AUDIENCE: Would it be either accept or reject 01:06:24.510 --> 01:06:25.880 align:middle line:90% as a function of the price? 01:06:25.880 --> 01:06:26.630 align:middle line:90% IAN BALL: Exactly. 01:06:26.630 --> 01:06:28.210 align:middle line:84% As a function of the price of the seller offers. 01:06:28.210 --> 01:06:28.430 align:middle line:90% Exactly. 01:06:28.430 --> 01:06:30.110 align:middle line:84% So it's going to be a function-- maybe 01:06:30.110 --> 01:06:33.030 align:middle line:84% I'll call it q since we think of q as quantity of buying. 01:06:33.030 --> 01:06:38.897 align:middle line:84% q from maybe-- if we don't restrict the price at all, well, 01:06:38.897 --> 01:06:40.730 align:middle line:84% the seller can offer any non-negative price, 01:06:40.730 --> 01:06:46.930 align:middle line:84% say between 0 and infinity, not inclusive, to-- 01:06:46.930 --> 01:06:49.810 align:middle line:90% well, maybe we'll call 0, 1. 01:06:49.810 --> 01:06:52.230 align:middle line:84% So q, we can think of this as the quantity that they buy. 01:06:52.230 --> 01:06:54.570 align:middle line:84% They either buy 0, that means they don't buy or reject, 01:06:54.570 --> 01:06:57.110 align:middle line:84% or q equals 1, that means they buy or accept the offer. 01:06:57.110 --> 01:07:00.330 align:middle line:90% 01:07:00.330 --> 01:07:06.250 align:middle line:90% And let's put our graph here. 01:07:06.250 --> 01:07:09.870 align:middle line:90% We have vL, we have vH. 01:07:09.870 --> 01:07:12.450 align:middle line:90% 01:07:12.450 --> 01:07:16.250 align:middle line:84% And I want to try to work up gradually and try 01:07:16.250 --> 01:07:19.210 align:middle line:90% to build up an equilibrium. 01:07:19.210 --> 01:07:22.215 align:middle line:84% So the first, I think, useful observation 01:07:22.215 --> 01:07:24.090 align:middle line:84% when we're trying to pin down the equilibrium 01:07:24.090 --> 01:07:27.250 align:middle line:84% is to say, well, there are some prices where 01:07:27.250 --> 01:07:31.225 align:middle line:84% we know what the buyer has to do, whatever their belief is. 01:07:31.225 --> 01:07:31.850 align:middle line:90% So let's think. 01:07:31.850 --> 01:07:34.330 align:middle line:84% The buyer's belief is going to form a belief 01:07:34.330 --> 01:07:35.890 align:middle line:90% about the quality of the item. 01:07:35.890 --> 01:07:38.370 align:middle line:84% And if they're really pessimistic, 01:07:38.370 --> 01:07:40.548 align:middle line:84% they would think the item is worth vL. 01:07:40.548 --> 01:07:42.090 align:middle line:84% If they're really optimistic, they're 01:07:42.090 --> 01:07:44.120 align:middle line:84% going to think the item is worth vH. 01:07:44.120 --> 01:07:45.640 align:middle line:84% And if their beliefs vary, they're 01:07:45.640 --> 01:07:47.860 align:middle line:84% going to value the item somewhere between vL and vH 01:07:47.860 --> 01:07:49.943 align:middle line:84% depending on how likely they think it is that it's 01:07:49.943 --> 01:07:51.440 align:middle line:90% high quality or low quality. 01:07:51.440 --> 01:07:53.617 align:middle line:84% But if they get offered a price below vL, 01:07:53.617 --> 01:07:54.700 align:middle line:90% what are they going to do? 01:07:54.700 --> 01:07:58.120 align:middle line:90% 01:07:58.120 --> 01:08:00.280 align:middle line:84% They have to buy because even if they're 01:08:00.280 --> 01:08:03.600 align:middle line:84% certain the good is low quality, this price is still lower 01:08:03.600 --> 01:08:05.440 align:middle line:90% than their value for the good. 01:08:05.440 --> 01:08:08.190 align:middle line:84% What if they see a price above vH? 01:08:08.190 --> 01:08:09.440 align:middle line:90% They're never going to buy it. 01:08:09.440 --> 01:08:10.960 align:middle line:84% Doesn't matter-- even if they're certain the good is 01:08:10.960 --> 01:08:13.080 align:middle line:84% high quality, it's not worth buying because vH 01:08:13.080 --> 01:08:14.260 align:middle line:90% is smaller than the price. 01:08:14.260 --> 01:08:15.800 align:middle line:90% So we immediately know that-- 01:08:15.800 --> 01:08:17.920 align:middle line:90% let's think of q over here. 01:08:17.920 --> 01:08:24.520 align:middle line:84% We're immediately going to have 1 here and 0 here. 01:08:24.520 --> 01:08:27.200 align:middle line:84% And the flexibility is going to be between vL and vH. 01:08:27.200 --> 01:08:29.475 align:middle line:84% For a price between vL and vH, whether they 01:08:29.475 --> 01:08:31.100 align:middle line:84% buy is going to depend on their belief, 01:08:31.100 --> 01:08:32.760 align:middle line:84% because if they're certain it's high quality-- 01:08:32.760 --> 01:08:34.319 align:middle line:84% let's say there's a price in here. 01:08:34.319 --> 01:08:36.920 align:middle line:84% If they're certain it's high quality, they do want to buy, 01:08:36.920 --> 01:08:39.420 align:middle line:84% if they're certain it's low quality, they don't want to buy, 01:08:39.420 --> 01:08:43.870 align:middle line:84% and it's going to depend on what their belief is. 01:08:43.870 --> 01:08:46.830 align:middle line:90% OK. 01:08:46.830 --> 01:08:50.149 align:middle line:84% What else can we say when we're looking for an equilibrium? 01:08:50.149 --> 01:08:54.390 align:middle line:84% Well, remember, PBE, we have to have 01:08:54.390 --> 01:08:56.569 align:middle line:84% sequential optimality for the seller, 01:08:56.569 --> 01:08:58.149 align:middle line:84% sequential optimality for the buyer, 01:08:58.149 --> 01:09:00.550 align:middle line:84% and then Bayes' consistency for the buyer. 01:09:00.550 --> 01:09:03.612 align:middle line:84% Now Bayes' consistency only pins down the buyer's beliefs 01:09:03.612 --> 01:09:05.029 align:middle line:84% when they see a price that they're 01:09:05.029 --> 01:09:06.330 align:middle line:90% supposed to see in equilibrium. 01:09:06.330 --> 01:09:08.310 align:middle line:84% It only pins things down on-path. 01:09:08.310 --> 01:09:12.950 align:middle line:84% So what is going to be on-path in this equilibrium? 01:09:12.950 --> 01:09:17.310 align:middle line:84% What price are they expecting to see in this equilibrium? 01:09:17.310 --> 01:09:18.710 align:middle line:90% Just p star. 01:09:18.710 --> 01:09:20.910 align:middle line:84% That's the only price they should see. 01:09:20.910 --> 01:09:23.170 align:middle line:84% Because that's what the low type seller is offering 01:09:23.170 --> 01:09:25.212 align:middle line:84% and that's what the high type seller is offering. 01:09:25.212 --> 01:09:28.350 align:middle line:84% So if they see a price of p star-- 01:09:28.350 --> 01:09:33.492 align:middle line:84% so let's-- what belief do they form? 01:09:33.492 --> 01:09:34.450 align:middle line:90% They apply Bayes' rule. 01:09:34.450 --> 01:09:36.325 align:middle line:84% What happens when they see a price of p star? 01:09:36.325 --> 01:09:41.090 align:middle line:90% 01:09:41.090 --> 01:09:42.170 align:middle line:90% Yeah? 01:09:42.170 --> 01:09:44.270 align:middle line:84% AUDIENCE: We would have been given pi, 01:09:44.270 --> 01:09:46.970 align:middle line:84% we would have an intuition about is it 01:09:46.970 --> 01:09:49.010 align:middle line:84% low type or high type, and then-- like, 01:09:49.010 --> 01:09:52.569 align:middle line:84% we would know which one of those is higher probability mitigating 01:09:52.569 --> 01:09:54.943 align:middle line:84% whether the car is actually low type or high type. 01:09:54.943 --> 01:09:56.610 align:middle line:84% IAN BALL: Right, but here, we're looking 01:09:56.610 --> 01:09:57.750 align:middle line:90% for a pooling equilibrium. 01:09:57.750 --> 01:10:00.570 align:middle line:84% Where both the high-type seller and-- both the low-type seller 01:10:00.570 --> 01:10:02.610 align:middle line:84% and the high-type seller sell it at p star. 01:10:02.610 --> 01:10:06.330 align:middle line:84% So when I see the price of p star, what is my belief? 01:10:06.330 --> 01:10:07.170 align:middle line:90% Yeah? 01:10:07.170 --> 01:10:09.170 align:middle line:84% AUDIENCE: That it's high type of probability pi. 01:10:09.170 --> 01:10:09.430 align:middle line:90% IAN BALL: Right. 01:10:09.430 --> 01:10:10.950 align:middle line:90% It's just the same as the prior. 01:10:10.950 --> 01:10:13.970 align:middle line:84% So if everyone is charging a price of p star, 01:10:13.970 --> 01:10:15.630 align:middle line:90% I keep my beliefs. 01:10:15.630 --> 01:10:18.047 align:middle line:84% My beliefs stay exactly what they were at the prior. 01:10:18.047 --> 01:10:19.630 align:middle line:84% I have no reason to update my belief-- 01:10:19.630 --> 01:10:21.350 align:middle line:84% I mean, formally I'm applying Bayes' rule, 01:10:21.350 --> 01:10:22.850 align:middle line:84% but when I update my beliefs, they're 01:10:22.850 --> 01:10:24.690 align:middle line:84% the same as they were before because everyone 01:10:24.690 --> 01:10:25.815 align:middle line:90% offers the price of p star. 01:10:25.815 --> 01:10:27.570 align:middle line:84% The price doesn't really reveal anything. 01:10:27.570 --> 01:10:38.160 align:middle line:84% So maybe I'll say p star belief pi on H. 01:10:38.160 --> 01:10:43.480 align:middle line:84% Now, if I believe the item is high with probability pi, 01:10:43.480 --> 01:10:46.440 align:middle line:84% then the value I assign to it-- maybe I'll call the value v 01:10:46.440 --> 01:10:47.240 align:middle line:90% star-- 01:10:47.240 --> 01:10:54.080 align:middle line:84% is going to be pi times vH plus 1 minus pi times vL. 01:10:54.080 --> 01:10:57.600 align:middle line:84% That's my expected valuation of the item given 01:10:57.600 --> 01:10:59.140 align:middle line:90% that my belief here is pi. 01:10:59.140 --> 01:11:01.960 align:middle line:90% 01:11:01.960 --> 01:11:03.620 align:middle line:84% So we're kind of building this up. 01:11:03.620 --> 01:11:05.995 align:middle line:84% We're looking for an equilibrium where the seller charges 01:11:05.995 --> 01:11:06.760 align:middle line:90% a price of p star. 01:11:06.760 --> 01:11:09.600 align:middle line:84% We know that the buyer's valuation has to be this. 01:11:09.600 --> 01:11:12.520 align:middle line:84% And we want the seller to sell the item at a price of p star. 01:11:12.520 --> 01:11:14.600 align:middle line:84% So what must be true about p star 01:11:14.600 --> 01:11:18.360 align:middle line:84% relative to v star in this equilibrium? 01:11:18.360 --> 01:11:19.160 align:middle line:90% Yeah? 01:11:19.160 --> 01:11:20.480 align:middle line:90% AUDIENCE: p star equals v star? 01:11:20.480 --> 01:11:21.440 align:middle line:84% IAN BALL: It could equal v star, but it 01:11:21.440 --> 01:11:23.357 align:middle line:84% has to be at least weakly smaller than v star. 01:11:23.357 --> 01:11:24.980 align:middle line:90% You might-- it could be v star. 01:11:24.980 --> 01:11:27.920 align:middle line:90% So let's put v star in here. 01:11:27.920 --> 01:11:32.520 align:middle line:84% And p star is going to have to be 01:11:32.520 --> 01:11:36.350 align:middle line:90% somewhere smaller than v star. 01:11:36.350 --> 01:11:40.027 align:middle line:84% And we know that at a price of p star, 01:11:40.027 --> 01:11:42.110 align:middle line:84% let's look for an equilibrium where the buyer does 01:11:42.110 --> 01:11:44.790 align:middle line:90% buy at a price of p star. 01:11:44.790 --> 01:11:46.990 align:middle line:84% Now the trick for these equilibria 01:11:46.990 --> 01:11:49.430 align:middle line:84% is often you want to make deviations 01:11:49.430 --> 01:11:50.810 align:middle line:90% as unattractive as possible. 01:11:50.810 --> 01:11:52.510 align:middle line:84% So we want to make this an equilibrium. 01:11:52.510 --> 01:11:55.177 align:middle line:84% That means we want to make sure the seller doesn't want to offer 01:11:55.177 --> 01:11:56.850 align:middle line:90% a price different from p star. 01:11:56.850 --> 01:11:58.350 align:middle line:84% To make sure the seller doesn't want 01:11:58.350 --> 01:12:00.350 align:middle line:84% to offer a price different than p star, 01:12:00.350 --> 01:12:03.070 align:middle line:84% it better be the buyer never buys if a price is strictly 01:12:03.070 --> 01:12:04.847 align:middle line:90% higher than p star. 01:12:04.847 --> 01:12:06.930 align:middle line:84% So if I see a price higher than p star as a buyer, 01:12:06.930 --> 01:12:08.847 align:middle line:84% I'm not going to buy, and that way, the seller 01:12:08.847 --> 01:12:10.130 align:middle line:90% won't want to deviate to this. 01:12:10.130 --> 01:12:13.650 align:middle line:84% So let's look for an equilibrium that goes like this. 01:12:13.650 --> 01:12:17.790 align:middle line:90% 01:12:17.790 --> 01:12:20.630 align:middle line:84% Seems like a natural equilibrium. 01:12:20.630 --> 01:12:25.750 align:middle line:84% If I see any price that's weakly below p star as a buyer, I buy. 01:12:25.750 --> 01:12:30.750 align:middle line:84% And if I see any price above p star, I don't buy. 01:12:30.750 --> 01:12:32.090 align:middle line:90% Now we have to specify beliefs. 01:12:32.090 --> 01:12:35.460 align:middle line:84% I think the easiest thing is to say-- 01:12:35.460 --> 01:12:43.860 align:middle line:84% well, let's say-- it doesn't even-- 01:12:43.860 --> 01:12:46.360 align:middle line:84% it doesn't even matter, but sometimes this is a good trick. 01:12:46.360 --> 01:12:48.900 align:middle line:90% 01:12:48.900 --> 01:12:51.452 align:middle line:84% Let's say that if I see any price strictly below p star, 01:12:51.452 --> 01:12:53.660 align:middle line:84% here, I have a lot of flexibility in what the beliefs 01:12:53.660 --> 01:12:57.242 align:middle line:84% are because we're off-path, but I know that the buyer is buying, 01:12:57.242 --> 01:12:58.700 align:middle line:84% so to make that optimal, let's just 01:12:58.700 --> 01:13:01.040 align:middle line:84% say they believe the good is certain to be high quality. 01:13:01.040 --> 01:13:03.260 align:middle line:84% It doesn't really matter what I do here. 01:13:03.260 --> 01:13:06.060 align:middle line:84% If they see a price strictly greater than p star, 01:13:06.060 --> 01:13:08.740 align:middle line:84% they believe it's low quality and don't buy. 01:13:08.740 --> 01:13:11.953 align:middle line:84% And if they see a value-- a price of exactly p star, 01:13:11.953 --> 01:13:13.620 align:middle line:84% well, we know what they have to believe. 01:13:13.620 --> 01:13:16.860 align:middle line:84% They have to believe the good is high quality with probability pi 01:13:16.860 --> 01:13:19.500 align:middle line:84% because this price is on-path, and therefore, Bayes' rule 01:13:19.500 --> 01:13:20.980 align:middle line:90% can be applied. 01:13:20.980 --> 01:13:23.280 align:middle line:84% For these lower prices and these higher prices, 01:13:23.280 --> 01:13:25.620 align:middle line:84% I'm totally flexible in how I choose the buyer's beliefs 01:13:25.620 --> 01:13:28.240 align:middle line:84% because these are off-path, and if I try to apply Bayes' rule, 01:13:28.240 --> 01:13:30.198 align:middle line:84% I would divide by 0, and therefore, Bayes' rule 01:13:30.198 --> 01:13:31.180 align:middle line:90% doesn't have any bite. 01:13:31.180 --> 01:13:34.800 align:middle line:90% 01:13:34.800 --> 01:13:37.240 align:middle line:84% OK, so I think we're good, but I think 01:13:37.240 --> 01:13:38.580 align:middle line:90% we need one more constraint. 01:13:38.580 --> 01:13:41.800 align:middle line:90% So we've defined p star. 01:13:41.800 --> 01:13:45.160 align:middle line:84% We've said p star is between vL and v star. 01:13:45.160 --> 01:13:47.560 align:middle line:84% This means that the buyer is willing to buy. 01:13:47.560 --> 01:13:50.263 align:middle line:84% I think there's one more potential issue 01:13:50.263 --> 01:13:52.180 align:middle line:84% with this equilibrium, so let's go through it. 01:13:52.180 --> 01:13:55.800 align:middle line:84% We've checked-- maybe I'll just write this here. 01:13:55.800 --> 01:14:00.280 align:middle line:84% We've checked belief consistency because here, the buyer 01:14:00.280 --> 01:14:01.680 align:middle line:90% believes it's low quality. 01:14:01.680 --> 01:14:04.640 align:middle line:84% They believe it's high quality, they certainly want to buy. 01:14:04.640 --> 01:14:06.140 align:middle line:84% Here, they believe it's low quality, 01:14:06.140 --> 01:14:08.120 align:middle line:84% they certainly don't want to buy. 01:14:08.120 --> 01:14:11.640 align:middle line:84% Here, they believe its expected quality is v star 01:14:11.640 --> 01:14:14.720 align:middle line:84% and they're, therefore, willing to buy at a price of p star. 01:14:14.720 --> 01:14:17.360 align:middle line:90% So Bayes' consistency is right. 01:14:17.360 --> 01:14:21.840 align:middle line:84% Receiver optimality, RO-- ah, receiver-- buyer optimality 01:14:21.840 --> 01:14:23.520 align:middle line:84% is good, but I think there's one more 01:14:23.520 --> 01:14:25.180 align:middle line:90% issue with seller optimality. 01:14:25.180 --> 01:14:27.240 align:middle line:84% So we have to check, does the seller ever 01:14:27.240 --> 01:14:30.890 align:middle line:84% want to deviate to offering a different price? 01:14:30.890 --> 01:14:34.595 align:middle line:90% 01:14:34.595 --> 01:14:35.970 align:middle line:84% Well, the trick is always to say, 01:14:35.970 --> 01:14:41.570 align:middle line:84% what is the most attractive deviation? 01:14:41.570 --> 01:14:43.870 align:middle line:84% Would the seller ever want to charge a price-- 01:14:43.870 --> 01:14:45.200 align:middle line:90% a lower price here? 01:14:45.200 --> 01:14:47.450 align:middle line:84% No, because they're already selling it with certainty. 01:14:47.450 --> 01:14:48.992 align:middle line:84% If they offered a lower price, they'd 01:14:48.992 --> 01:14:50.588 align:middle line:90% still style it with certainty. 01:14:50.588 --> 01:14:52.130 align:middle line:84% But what might the seller want to do? 01:14:52.130 --> 01:14:55.310 align:middle line:90% What do we have to check? 01:14:55.310 --> 01:15:00.310 align:middle line:84% Well, does the seller actually want to sell it at price p star? 01:15:00.310 --> 01:15:02.322 align:middle line:84% Now the issue is, there's two types of sellers. 01:15:02.322 --> 01:15:04.530 align:middle line:84% There's the high-type seller and the low-type seller. 01:15:04.530 --> 01:15:07.343 align:middle line:84% We have to say, what is the more tempting deviation? 01:15:07.343 --> 01:15:09.510 align:middle line:84% The issue we have to worry about is-- and maybe I'll 01:15:09.510 --> 01:15:11.890 align:middle line:90% do separating on another page. 01:15:11.890 --> 01:15:16.390 align:middle line:90% 01:15:16.390 --> 01:15:19.510 align:middle line:90% Let's look at both cases. 01:15:19.510 --> 01:15:27.420 align:middle line:84% The high-type seller is comparing p star minus cH. 01:15:27.420 --> 01:15:32.980 align:middle line:90% 01:15:32.980 --> 01:15:36.420 align:middle line:90% These are the two constraints. 01:15:36.420 --> 01:15:40.500 align:middle line:84% If I'm a high-type seller, I could charge a price of p star, 01:15:40.500 --> 01:15:44.380 align:middle line:84% and I sell at p star, but I have to pay, but my cost is cH. 01:15:44.380 --> 01:15:47.020 align:middle line:84% Or I could set a price that no one will buy it, 01:15:47.020 --> 01:15:49.340 align:middle line:90% and therefore, I get 0. 01:15:49.340 --> 01:15:51.380 align:middle line:84% Conversely, if I'm the low-type seller, 01:15:51.380 --> 01:15:53.517 align:middle line:84% I could set a price of p star, pay my cost, 01:15:53.517 --> 01:15:55.100 align:middle line:84% or I could sell at a really high price 01:15:55.100 --> 01:15:56.540 align:middle line:90% that no one's going to buy it. 01:15:56.540 --> 01:15:58.280 align:middle line:84% Which of these is harder to satisfy? 01:15:58.280 --> 01:16:01.500 align:middle line:90% 01:16:01.500 --> 01:16:04.820 align:middle line:84% The top one, because the high-type seller finds 01:16:04.820 --> 01:16:05.920 align:middle line:90% it more costly to produce. 01:16:05.920 --> 01:16:07.420 align:middle line:84% So they might say this price p star, 01:16:07.420 --> 01:16:08.763 align:middle line:90% it's not really worth it for me. 01:16:08.763 --> 01:16:10.680 align:middle line:84% So this is going to be the binding constraint. 01:16:10.680 --> 01:16:13.300 align:middle line:84% This is the critical-- the important constraint. 01:16:13.300 --> 01:16:15.140 align:middle line:84% And this is going to be the final constraint 01:16:15.140 --> 01:16:18.020 align:middle line:90% that we need to impose. 01:16:18.020 --> 01:16:23.220 align:middle line:84% So what we need, to write it in words, 01:16:23.220 --> 01:16:36.200 align:middle line:84% is we need vL and cH less than or equal to p star less than 01:16:36.200 --> 01:16:38.140 align:middle line:90% or equal to v star. 01:16:38.140 --> 01:16:53.080 align:middle line:90% 01:16:53.080 --> 01:16:55.960 align:middle line:84% This inequality says the buyer is actually 01:16:55.960 --> 01:16:59.280 align:middle line:84% willing to buy at a price of p star. 01:16:59.280 --> 01:17:01.905 align:middle line:84% This inequality says even when the seller knows 01:17:01.905 --> 01:17:03.280 align:middle line:84% the good is high quality, they're 01:17:03.280 --> 01:17:06.400 align:middle line:84% still willing to sell at a price of p star. 01:17:06.400 --> 01:17:09.080 align:middle line:84% And the vL inequality is important 01:17:09.080 --> 01:17:13.080 align:middle line:84% because if p star were less than vL, then 01:17:13.080 --> 01:17:15.080 align:middle line:84% the seller could always slightly raise the price 01:17:15.080 --> 01:17:16.622 align:middle line:84% and the buyer would still always have 01:17:16.622 --> 01:17:18.880 align:middle line:84% to buy because we know the buyer always buys 01:17:18.880 --> 01:17:20.600 align:middle line:90% when the price is below vL. 01:17:20.600 --> 01:17:22.780 align:middle line:84% So these are the three inequalities that we get. 01:17:22.780 --> 01:17:25.230 align:middle line:84% And I think when you go through and try to solve for PBE, 01:17:25.230 --> 01:17:30.190 align:middle line:84% these are the steps you have to go through. 01:17:30.190 --> 01:17:33.630 align:middle line:84% Let me say just a little bit about the separating equilibrium 01:17:33.630 --> 01:17:36.070 align:middle line:90% in the last two minutes. 01:17:36.070 --> 01:17:37.350 align:middle line:90% So that was pooling. 01:17:37.350 --> 01:17:39.850 align:middle line:84% Let's go down here to separating. 01:17:39.850 --> 01:17:49.270 align:middle line:90% 01:17:49.270 --> 01:17:55.830 align:middle line:84% So for the separating equilibrium, 01:17:55.830 --> 01:17:59.790 align:middle line:90% we now have pL not equal to pH. 01:17:59.790 --> 01:18:01.910 align:middle line:84% So when the seller knows the good is low quality, 01:18:01.910 --> 01:18:03.022 align:middle line:90% they sell it at pL. 01:18:03.022 --> 01:18:05.230 align:middle line:84% When they know it's high quality, they sell it at pH. 01:18:05.230 --> 01:18:06.910 align:middle line:90% These are different. 01:18:06.910 --> 01:18:10.830 align:middle line:84% As before, the buyer is choosing a strategy q 01:18:10.830 --> 01:18:15.150 align:middle line:90% from 0, infinity to 0, 1. 01:18:15.150 --> 01:18:17.870 align:middle line:90% Now, which prices are on-path? 01:18:17.870 --> 01:18:19.790 align:middle line:90% It's both pL and pH. 01:18:19.790 --> 01:18:21.170 align:middle line:90% So now let's look at beliefs. 01:18:21.170 --> 01:18:26.420 align:middle line:90% 01:18:26.420 --> 01:18:28.620 align:middle line:84% If the buyer sees a price of pL, they 01:18:28.620 --> 01:18:31.660 align:middle line:84% know that the seller is low quality, so they 01:18:31.660 --> 01:18:36.460 align:middle line:90% assign probability 0 to h. 01:18:36.460 --> 01:18:41.340 align:middle line:84% If they see pH, they know the seller is high quality, so they 01:18:41.340 --> 01:18:46.260 align:middle line:90% assign probability 1 to h. 01:18:46.260 --> 01:18:51.792 align:middle line:84% And if they see any other price, well, 01:18:51.792 --> 01:18:53.500 align:middle line:84% now their beliefs are flexible because we 01:18:53.500 --> 01:18:56.440 align:middle line:84% can't apply Bayes' rule, this is off-path, so flexible. 01:18:56.440 --> 01:19:08.300 align:middle line:90% 01:19:08.300 --> 01:19:11.125 align:middle line:84% And we can go through and try to do the same analysis. 01:19:11.125 --> 01:19:13.000 align:middle line:84% Maybe we'll just say two kind of tricks here. 01:19:13.000 --> 01:19:16.540 align:middle line:90% 01:19:16.540 --> 01:19:18.600 align:middle line:84% You might look-- let me just say one final thing. 01:19:18.600 --> 01:19:21.080 align:middle line:84% You might try to find an equilibrium where the buyer buys 01:19:21.080 --> 01:19:22.940 align:middle line:90% at price pL and at pH. 01:19:22.940 --> 01:19:24.940 align:middle line:84% I'm going to argue that can't be in equilibrium. 01:19:24.940 --> 01:19:26.120 align:middle line:84% Why could there not be an equilibrium 01:19:26.120 --> 01:19:27.960 align:middle line:84% where the buyer buys when they see pL 01:19:27.960 --> 01:19:29.100 align:middle line:90% and buys when they see pH? 01:19:29.100 --> 01:19:30.892 align:middle line:84% Why can't there be a separating equilibrium 01:19:30.892 --> 01:19:32.140 align:middle line:90% where the buyer always buys? 01:19:32.140 --> 01:19:38.600 align:middle line:90% 01:19:38.600 --> 01:19:41.080 align:middle line:84% Think about the seller's incentives. 01:19:41.080 --> 01:19:45.080 align:middle line:84% There are two prices at which the seller always buys. 01:19:45.080 --> 01:19:46.420 align:middle line:90% What would you do as the seller? 01:19:46.420 --> 01:19:48.960 align:middle line:90% 01:19:48.960 --> 01:19:51.420 align:middle line:90% Well, one of these is higher. 01:19:51.420 --> 01:19:52.540 align:middle line:90% Let's say pH is higher. 01:19:52.540 --> 01:19:54.040 align:middle line:84% If you were the seller, you'd always 01:19:54.040 --> 01:19:56.160 align:middle line:84% sell at pH because you'd get more revenue 01:19:56.160 --> 01:19:57.663 align:middle line:90% and the buyer is willing to buy. 01:19:57.663 --> 01:19:59.080 align:middle line:84% So it turns out, there's not going 01:19:59.080 --> 01:20:01.538 align:middle line:84% to be a separating equilibrium where the buyer always buys. 01:20:01.538 --> 01:20:03.997 align:middle line:84% We're going to actually have to have a mixed strategy where 01:20:03.997 --> 01:20:06.620 align:middle line:84% the buyer buys with some probability following pH, 01:20:06.620 --> 01:20:09.465 align:middle line:84% but you'll see how to work through that in the problem set. 01:20:09.465 --> 01:20:10.840 align:middle line:84% But let's make sure we understand 01:20:10.840 --> 01:20:11.798 align:middle line:90% the intuition for that. 01:20:11.798 --> 01:20:12.800 align:middle line:90% OK, let me stop there. 01:20:12.800 --> 01:20:14.950 align:middle line:90% I'll see you on Thursday. 01:20:14.950 --> 01:20:18.000 align:middle line:90%