WEBVTT 00:00:00.000 --> 00:00:02.440 align:middle line:90% [SQUEAKING] 00:00:02.440 --> 00:00:03.895 align:middle line:90% [RUSTLING] 00:00:03.895 --> 00:00:04.865 align:middle line:90% [CLICKING] 00:00:04.865 --> 00:00:09.020 align:middle line:90% 00:00:09.020 --> 00:00:10.520 align:middle line:84% PROFESSOR: So then let's get started 00:00:10.520 --> 00:00:13.920 align:middle line:84% with our second-to-last topic today, which 00:00:13.920 --> 00:00:16.420 align:middle line:84% is going to be something called cheap talk. 00:00:16.420 --> 00:00:21.040 align:middle line:90% 00:00:21.040 --> 00:00:23.460 align:middle line:84% I think it's good to have some context for this. 00:00:23.460 --> 00:00:26.280 align:middle line:84% So far, we've looked at signaling games. 00:00:26.280 --> 00:00:32.439 align:middle line:84% And so let's recall that when we looked at signaling games, 00:00:32.439 --> 00:00:35.560 align:middle line:84% people with private information were conveying information 00:00:35.560 --> 00:00:38.820 align:middle line:84% to some other party through costly actions. 00:00:38.820 --> 00:00:54.280 align:middle line:84% So under signaling, info is conveyed through costly 00:00:54.280 --> 00:00:56.860 align:middle line:84% or we might also say payoff-relevant actions. 00:00:56.860 --> 00:01:03.360 align:middle line:90% 00:01:03.360 --> 00:01:05.880 align:middle line:84% So one way to convey information about how smart you are 00:01:05.880 --> 00:01:08.222 align:middle line:84% or how good of an employee you are is to go to college. 00:01:08.222 --> 00:01:10.680 align:middle line:84% And the whole point of that model was that going to college 00:01:10.680 --> 00:01:12.560 align:middle line:84% is costly and differentially costly 00:01:12.560 --> 00:01:13.860 align:middle line:90% for different kinds of people. 00:01:13.860 --> 00:01:16.560 align:middle line:84% Or when a firm was conveying information 00:01:16.560 --> 00:01:19.520 align:middle line:84% about the quality of an item, they were setting a price. 00:01:19.520 --> 00:01:21.420 align:middle line:84% And maybe a price isn't directly costly, 00:01:21.420 --> 00:01:23.600 align:middle line:84% but it is payoff relevant, because the price I set 00:01:23.600 --> 00:01:28.520 align:middle line:84% affects how much money I get if you actually accept my offer. 00:01:28.520 --> 00:01:31.320 align:middle line:84% But in the real world, we see a lot of information 00:01:31.320 --> 00:01:33.702 align:middle line:84% conveyed not through costly actions, 00:01:33.702 --> 00:01:35.660 align:middle line:84% but just through speech, through communication. 00:01:35.660 --> 00:01:36.785 align:middle line:90% We send messages to people. 00:01:36.785 --> 00:01:37.460 align:middle line:90% We use words. 00:01:37.460 --> 00:01:38.760 align:middle line:90% We talk to people. 00:01:38.760 --> 00:01:42.100 align:middle line:84% And the question is, when there's a conflict of interest, 00:01:42.100 --> 00:01:45.860 align:middle line:84% can we also convey information through costless actions? 00:01:45.860 --> 00:01:47.520 align:middle line:84% So today, the difference is we're 00:01:47.520 --> 00:02:00.350 align:middle line:84% going to look at conveying information through costless-- 00:02:00.350 --> 00:02:02.350 align:middle line:84% and sometimes, instead of calling these actions, 00:02:02.350 --> 00:02:03.310 align:middle line:90% you might call them messages. 00:02:03.310 --> 00:02:05.602 align:middle line:84% Because we usually think of speech or words or messages 00:02:05.602 --> 00:02:06.350 align:middle line:90% as being costly. 00:02:06.350 --> 00:02:10.250 align:middle line:90% 00:02:10.250 --> 00:02:11.850 align:middle line:84% Now, there are certainly some contexts 00:02:11.850 --> 00:02:13.142 align:middle line:90% where this isn't going to work. 00:02:13.142 --> 00:02:16.130 align:middle line:84% If we think of the job market signaling example, 00:02:16.130 --> 00:02:18.680 align:middle line:84% the firm could ask you how good of an employee are you. 00:02:18.680 --> 00:02:20.930 align:middle line:84% Well, everyone's going to say they're a good employee. 00:02:20.930 --> 00:02:23.610 align:middle line:84% So there are some contexts where we really need a costly action 00:02:23.610 --> 00:02:25.030 align:middle line:90% in order to make it credible. 00:02:25.030 --> 00:02:26.890 align:middle line:84% The information that we're going to convey. 00:02:26.890 --> 00:02:28.810 align:middle line:84% There are other situations where it's clear 00:02:28.810 --> 00:02:30.550 align:middle line:90% that communication is credible. 00:02:30.550 --> 00:02:32.217 align:middle line:84% If our interests are completely aligned. 00:02:32.217 --> 00:02:34.383 align:middle line:84% Then of course, we can communicate freely and that's 00:02:34.383 --> 00:02:36.810 align:middle line:84% not going to be difficult. So the point of the model today 00:02:36.810 --> 00:02:38.610 align:middle line:84% is going to show that sometimes, there will 00:02:38.610 --> 00:02:40.110 align:middle line:90% be a misalignment of interests. 00:02:40.110 --> 00:02:41.693 align:middle line:84% There will be a conflict of interests. 00:02:41.693 --> 00:02:44.090 align:middle line:84% Yet nevertheless, some kind of communication 00:02:44.090 --> 00:02:46.410 align:middle line:84% can be credible, in contrast to the model 00:02:46.410 --> 00:02:48.830 align:middle line:84% of telling the employer that you're a good employee. 00:02:48.830 --> 00:02:51.372 align:middle line:84% So that's going to be the model we're going to look at today. 00:02:51.372 --> 00:02:55.250 align:middle line:90% 00:02:55.250 --> 00:03:00.050 align:middle line:84% So this model is based on a paper by Crawford and Sobel 00:03:00.050 --> 00:03:01.810 align:middle line:90% in 1982. 00:03:01.810 --> 00:03:08.050 align:middle line:84% So we're getting closer to modern topics. 00:03:08.050 --> 00:03:10.970 align:middle line:84% And we're going to think of-- this is a game between a sender. 00:03:10.970 --> 00:03:14.810 align:middle line:84% We call them sender and receiver. 00:03:14.810 --> 00:03:16.830 align:middle line:90% But these are abstract labels. 00:03:16.830 --> 00:03:18.330 align:middle line:84% I want you to think of the sender 00:03:18.330 --> 00:03:22.730 align:middle line:84% as an advisor and the receiver as a decision maker. 00:03:22.730 --> 00:03:25.930 align:middle line:84% And the presidential advisor might collect information 00:03:25.930 --> 00:03:28.730 align:middle line:84% and make a recommendation to the president about what 00:03:28.730 --> 00:03:31.690 align:middle line:84% tax policy should be, or to the Fed about what the interest 00:03:31.690 --> 00:03:35.070 align:middle line:84% rate should be, or, how much we should spend on the military. 00:03:35.070 --> 00:03:37.590 align:middle line:84% And the idea is that the sender, the advisor, 00:03:37.590 --> 00:03:39.723 align:middle line:84% has more information than the decision maker, 00:03:39.723 --> 00:03:41.390 align:middle line:84% because the decision maker is very busy. 00:03:41.390 --> 00:03:43.030 align:middle line:84% The CEO can't know about everything. 00:03:43.030 --> 00:03:46.890 align:middle line:84% They talk to a consultant or a researcher 00:03:46.890 --> 00:03:50.270 align:middle line:84% who's going to be very informed about this particular issue. 00:03:50.270 --> 00:03:51.270 align:middle line:90% So how do we model this? 00:03:51.270 --> 00:03:59.570 align:middle line:84% Well, the sender is going to privately observe what We're 00:03:59.570 --> 00:04:02.410 align:middle line:90% going to call maybe a state. 00:04:02.410 --> 00:04:07.010 align:middle line:84% Or this could also be called their type t in 0, 1. 00:04:07.010 --> 00:04:10.170 align:middle line:84% So we're going to think of this in our economic example. 00:04:10.170 --> 00:04:11.907 align:middle line:90% How good is the economy? 00:04:11.907 --> 00:04:13.490 align:middle line:84% What's the ideal amount of spending we 00:04:13.490 --> 00:04:14.747 align:middle line:90% should spend on the military? 00:04:14.747 --> 00:04:15.830 align:middle line:90% What's the ideal tax rate? 00:04:15.830 --> 00:04:17.330 align:middle line:84% So this makes sense for any decision 00:04:17.330 --> 00:04:20.769 align:middle line:84% where there's this one-dimensional choice 00:04:20.769 --> 00:04:22.770 align:middle line:90% that we have to make. 00:04:22.770 --> 00:04:25.490 align:middle line:84% We can think of this usually as the state of the world. 00:04:25.490 --> 00:04:27.470 align:middle line:84% But we'll also call it the sender's type, 00:04:27.470 --> 00:04:30.370 align:middle line:84% because this is private information that the sender has. 00:04:30.370 --> 00:04:32.890 align:middle line:84% And then the sender, after observing their type t, 00:04:32.890 --> 00:04:37.850 align:middle line:90% is going to send a message m-- 00:04:37.850 --> 00:04:41.250 align:middle line:84% we'll just call it, in some abstract set, big M-- 00:04:41.250 --> 00:04:43.170 align:middle line:90% to the receiver. 00:04:43.170 --> 00:04:48.670 align:middle line:84% And then the receiver is going to observe M but not t. 00:04:48.670 --> 00:04:51.220 align:middle line:84% The whole point is if the receiver, the decision maker, 00:04:51.220 --> 00:04:52.660 align:middle line:84% understood the state of the world, 00:04:52.660 --> 00:04:54.120 align:middle line:84% knew what the right decision was, 00:04:54.120 --> 00:04:56.380 align:middle line:84% they wouldn't need help from the advisor anyway. 00:04:56.380 --> 00:04:58.140 align:middle line:84% So they don't directly observe t, 00:04:58.140 --> 00:05:01.500 align:middle line:84% but they do observe this message that's sent by the sender. 00:05:01.500 --> 00:05:07.820 align:middle line:84% And then after that, they choose an action or decision which 00:05:07.820 --> 00:05:08.980 align:middle line:90% we'll call y. 00:05:08.980 --> 00:05:12.540 align:middle line:84% And that's going to be a real number. 00:05:12.540 --> 00:05:14.002 align:middle line:90% So maybe I'll say-- 00:05:14.002 --> 00:05:15.960 align:middle line:84% I don't know if I've used this notation before. 00:05:15.960 --> 00:05:17.560 align:middle line:84% This is just the set of real numbers. 00:05:17.560 --> 00:05:20.780 align:middle line:90% 00:05:20.780 --> 00:05:22.460 align:middle line:84% Now, whenever we talk about something 00:05:22.460 --> 00:05:25.300 align:middle line:84% being privately observed, we need to specify the distribution 00:05:25.300 --> 00:05:26.290 align:middle line:90% from which this comes. 00:05:26.290 --> 00:05:28.540 align:middle line:84% And we're going to assume that the state, just to make 00:05:28.540 --> 00:05:32.460 align:middle line:90% things easy, is uniform. 00:05:32.460 --> 00:05:34.880 align:middle line:84% So just to be clear, the sender knows what the state is. 00:05:34.880 --> 00:05:35.980 align:middle line:90% They know their type. 00:05:35.980 --> 00:05:39.120 align:middle line:84% The uniform distribution is relevant to the receiver. 00:05:39.120 --> 00:05:41.840 align:middle line:84% The receiver believes that this state is uniformly distributed, 00:05:41.840 --> 00:05:45.100 align:middle line:84% but they don't know what the actual value of the state is. 00:05:45.100 --> 00:05:46.560 align:middle line:90% I'll say uniformly distributed. 00:05:46.560 --> 00:05:51.340 align:middle line:90% 00:05:51.340 --> 00:05:52.892 align:middle line:90% And now let me specify payoffs. 00:05:52.892 --> 00:05:54.600 align:middle line:84% Of course, this is a very stylized model, 00:05:54.600 --> 00:05:57.300 align:middle line:84% but I think it captures a lot of policy disagreements 00:05:57.300 --> 00:05:58.380 align:middle line:90% that we have. 00:05:58.380 --> 00:06:03.460 align:middle line:84% So the sender, their payoff u s depends on the decision y 00:06:03.460 --> 00:06:08.220 align:middle line:84% that the receiver takes and also t, the state of the world. 00:06:08.220 --> 00:06:09.440 align:middle line:90% And same for the receiver. 00:06:09.440 --> 00:06:13.460 align:middle line:90% 00:06:13.460 --> 00:06:16.100 align:middle line:84% And what we're going to imagine for the receiver, 00:06:16.100 --> 00:06:18.060 align:middle line:84% to make things simple, is the receiver just 00:06:18.060 --> 00:06:22.540 align:middle line:84% wants to match their decision with the state of the world. 00:06:22.540 --> 00:06:25.100 align:middle line:84% So we could think of the state of the world 00:06:25.100 --> 00:06:30.700 align:middle line:84% might be some measure of how close the economy is 00:06:30.700 --> 00:06:32.140 align:middle line:90% to a recession. 00:06:32.140 --> 00:06:37.260 align:middle line:84% And y might be how much we're planning to cut interest rates. 00:06:37.260 --> 00:06:40.260 align:middle line:84% Or t might be the severity or the risk 00:06:40.260 --> 00:06:43.860 align:middle line:84% of some military conflict, and y might be the spending level 00:06:43.860 --> 00:06:45.320 align:middle line:90% that we spend on that conflict. 00:06:45.320 --> 00:06:47.520 align:middle line:84% And the idea is, these things move together. 00:06:47.520 --> 00:06:49.660 align:middle line:84% The higher the state, the higher the ideal decision 00:06:49.660 --> 00:06:50.540 align:middle line:90% for the receiver. 00:06:50.540 --> 00:06:55.780 align:middle line:84% So we'll formally model this with the quadratic loss function 00:06:55.780 --> 00:06:57.300 align:middle line:90% that we said before. 00:06:57.300 --> 00:06:59.540 align:middle line:84% So this means what the receiver would love to do 00:06:59.540 --> 00:07:01.695 align:middle line:84% is have the decision exactly match the state. 00:07:01.695 --> 00:07:03.820 align:middle line:84% So we can think of the state as basically measuring 00:07:03.820 --> 00:07:06.180 align:middle line:84% what would be the ideal decision for the receiver 00:07:06.180 --> 00:07:07.965 align:middle line:90% if they could observe the state. 00:07:07.965 --> 00:07:09.340 align:middle line:84% And then the receiver experiences 00:07:09.340 --> 00:07:11.900 align:middle line:84% a loss that's equal to the difference 00:07:11.900 --> 00:07:14.243 align:middle line:84% between their decision and the state squared. 00:07:14.243 --> 00:07:16.160 align:middle line:84% So how far off the decision is from the state. 00:07:16.160 --> 00:07:16.700 align:middle line:90% Yes. 00:07:16.700 --> 00:07:18.200 align:middle line:84% AUDIENCE: So basically, the decision 00:07:18.200 --> 00:07:20.420 align:middle line:84% is just a reflection of what the receiver believes 00:07:20.420 --> 00:07:21.600 align:middle line:90% that the state is? 00:07:21.600 --> 00:07:24.740 align:middle line:90% 00:07:24.740 --> 00:07:27.660 align:middle line:84% PROFESSOR: It's true that their preferences over the decision 00:07:27.660 --> 00:07:29.565 align:middle line:84% depend on their beliefs about the state. 00:07:29.565 --> 00:07:31.940 align:middle line:84% AUDIENCE: But [INAUDIBLE] exactly match the state, right? 00:07:31.940 --> 00:07:33.620 align:middle line:90% So it should be-- 00:07:33.620 --> 00:07:37.100 align:middle line:84% so is the decision supposed to be a state? 00:07:37.100 --> 00:07:38.380 align:middle line:90% PROFESSOR: No, I wouldn't. 00:07:38.380 --> 00:07:42.240 align:middle line:84% I think maybe one way of interpreting this is, OK, 00:07:42.240 --> 00:07:44.140 align:middle line:84% if we want to think of a richer model, 00:07:44.140 --> 00:07:47.180 align:middle line:84% probably the state is really a very rich 00:07:47.180 --> 00:07:48.740 align:middle line:90% multidimensional object. 00:07:48.740 --> 00:07:51.700 align:middle line:84% But we're going to denote the state-- the aspect of the state 00:07:51.700 --> 00:07:54.630 align:middle line:84% that matters is what the ideal decision is in that state. 00:07:54.630 --> 00:07:56.380 align:middle line:84% So I think maybe the better interpretation 00:07:56.380 --> 00:07:58.300 align:middle line:84% is, when we write t, what we really 00:07:58.300 --> 00:08:02.100 align:middle line:84% mean is, what is the receiver's ideal decision given 00:08:02.100 --> 00:08:03.160 align:middle line:90% the state of the world? 00:08:03.160 --> 00:08:05.680 align:middle line:84% So maybe the economy has many dimensions. 00:08:05.680 --> 00:08:10.460 align:middle line:84% But think of t saying the ideal interest rate for the Fed to set 00:08:10.460 --> 00:08:14.420 align:middle line:84% is 3% and y being the interest rate the Fed actually sets. 00:08:14.420 --> 00:08:15.380 align:middle line:90% Does that make sense? 00:08:15.380 --> 00:08:15.660 align:middle line:90% AUDIENCE: Yeah. 00:08:15.660 --> 00:08:16.740 align:middle line:90% PROFESSOR: OK, great. 00:08:16.740 --> 00:08:18.040 align:middle line:90% So now what about the sender? 00:08:18.040 --> 00:08:20.280 align:middle line:84% Well, if the sender and receiver have the same preferences, 00:08:20.280 --> 00:08:21.440 align:middle line:84% this would be a really easy problem. 00:08:21.440 --> 00:08:23.732 align:middle line:84% Of course the advisor is going to just reveal the state 00:08:23.732 --> 00:08:25.780 align:middle line:84% and allow the receiver to make a good decision. 00:08:25.780 --> 00:08:27.700 align:middle line:84% But often, advisors and decision makers 00:08:27.700 --> 00:08:29.040 align:middle line:90% have misaligned interests. 00:08:29.040 --> 00:08:30.457 align:middle line:84% And the way we're going to capture 00:08:30.457 --> 00:08:36.306 align:middle line:84% that here is we're going to say the sender's preferences 00:08:36.306 --> 00:08:39.020 align:middle line:90% is like this, where beta-- 00:08:39.020 --> 00:08:40.159 align:middle line:90% I'll put it down here-- 00:08:40.159 --> 00:08:42.990 align:middle line:90% 00:08:42.990 --> 00:08:44.630 align:middle line:90% is greater than 0. 00:08:44.630 --> 00:08:46.970 align:middle line:84% So here, beta is called the bias. 00:08:46.970 --> 00:08:51.230 align:middle line:90% 00:08:51.230 --> 00:08:53.230 align:middle line:84% This is the bias that the sender or the advisor 00:08:53.230 --> 00:08:55.350 align:middle line:84% has relative to the decision maker. 00:08:55.350 --> 00:08:58.310 align:middle line:84% So notice that, well, the receiver always 00:08:58.310 --> 00:09:01.230 align:middle line:84% wants to match the decision with the state. 00:09:01.230 --> 00:09:04.670 align:middle line:84% The sender would like to match the decision not with the state, 00:09:04.670 --> 00:09:07.750 align:middle line:84% but with the state plus this bias beta. 00:09:07.750 --> 00:09:10.670 align:middle line:84% So notice some characteristics of this misalignment 00:09:10.670 --> 00:09:12.150 align:middle line:90% of interest. 00:09:12.150 --> 00:09:14.350 align:middle line:84% Both the sender and the receiver agree 00:09:14.350 --> 00:09:19.070 align:middle line:84% that when the state is higher, a higher decision is warranted. 00:09:19.070 --> 00:09:22.590 align:middle line:84% What they disagree about is the exact value of the decision. 00:09:22.590 --> 00:09:24.950 align:middle line:84% So whatever the state is, the sender always 00:09:24.950 --> 00:09:27.288 align:middle line:84% thinks the decision should be a little bit higher 00:09:27.288 --> 00:09:28.330 align:middle line:90% than the receiver thinks. 00:09:28.330 --> 00:09:31.027 align:middle line:84% And I think this is consistent with a lot of policy 00:09:31.027 --> 00:09:31.610 align:middle line:90% disagreements. 00:09:31.610 --> 00:09:32.970 align:middle line:84% Let's take the example of the Fed. 00:09:32.970 --> 00:09:35.345 align:middle line:84% There are some people who are more hawkish or more dovish 00:09:35.345 --> 00:09:35.887 align:middle line:90% on inflation. 00:09:35.887 --> 00:09:38.178 align:middle line:84% So some people always tend to want lower interest rates 00:09:38.178 --> 00:09:38.970 align:middle line:90% than someone else. 00:09:38.970 --> 00:09:41.370 align:middle line:84% But everyone agrees that when the economy is in worse shape, 00:09:41.370 --> 00:09:42.950 align:middle line:84% we should tend to cut interest rates, 00:09:42.950 --> 00:09:44.410 align:middle line:84% and when the economy is doing well, 00:09:44.410 --> 00:09:45.952 align:middle line:84% maybe we should raise interest rates. 00:09:45.952 --> 00:09:48.910 align:middle line:84% So they all agree about the directional effect of the state 00:09:48.910 --> 00:09:50.170 align:middle line:90% on the ideal decision. 00:09:50.170 --> 00:09:53.210 align:middle line:84% They just disagree slightly about what the level is. 00:09:53.210 --> 00:09:55.550 align:middle line:84% And the sender always wants a decision 00:09:55.550 --> 00:09:58.710 align:middle line:84% that's higher than what the receiver wants exactly 00:09:58.710 --> 00:10:00.790 align:middle line:90% by this amount beta. 00:10:00.790 --> 00:10:02.870 align:middle line:84% So the larger beta is, the greater 00:10:02.870 --> 00:10:05.890 align:middle line:84% the disagreement between the sender and the receiver. 00:10:05.890 --> 00:10:06.890 align:middle line:90% OK, so here's our model. 00:10:06.890 --> 00:10:09.930 align:middle line:84% Any questions about the model so far? 00:10:09.930 --> 00:10:11.270 align:middle line:90% Yes. 00:10:11.270 --> 00:10:14.330 align:middle line:84% AUDIENCE: Does beta just depend on who the sender is? 00:10:14.330 --> 00:10:17.168 align:middle line:84% It's not like distributed, it's just like [INAUDIBLE]? 00:10:17.168 --> 00:10:18.210 align:middle line:90% PROFESSOR: Good question. 00:10:18.210 --> 00:10:20.030 align:middle line:84% So you could imagine a richer model 00:10:20.030 --> 00:10:23.150 align:middle line:84% where the decision maker doesn't know the political biases 00:10:23.150 --> 00:10:24.210 align:middle line:90% of the advisor. 00:10:24.210 --> 00:10:25.430 align:middle line:84% And that's actually an interesting extension 00:10:25.430 --> 00:10:26.410 align:middle line:90% that people have looked at. 00:10:26.410 --> 00:10:28.910 align:middle line:84% But for today, we're going to assume beta is commonly known. 00:10:28.910 --> 00:10:30.210 align:middle line:90% So I hire this advisor. 00:10:30.210 --> 00:10:33.310 align:middle line:84% I know that my advisor always tends 00:10:33.310 --> 00:10:34.650 align:middle line:90% to have this policy preference. 00:10:34.650 --> 00:10:35.170 align:middle line:90% And this is known. 00:10:35.170 --> 00:10:36.930 align:middle line:84% What I don't know is the true state of the world. 00:10:36.930 --> 00:10:37.450 align:middle line:90% Good question. 00:10:37.450 --> 00:10:38.330 align:middle line:90% Any other questions? 00:10:38.330 --> 00:10:40.950 align:middle line:90% 00:10:40.950 --> 00:10:42.370 align:middle line:90% OK, so it's a game. 00:10:42.370 --> 00:10:45.130 align:middle line:84% Let's first think about what strategies are in this game. 00:10:45.130 --> 00:10:51.390 align:middle line:90% 00:10:51.390 --> 00:10:55.010 align:middle line:84% So the sender, they observe the state of the world 00:10:55.010 --> 00:10:56.790 align:middle line:90% or they observe their type t. 00:10:56.790 --> 00:10:59.870 align:middle line:84% And they choose what message to send to the receiver. 00:10:59.870 --> 00:11:09.430 align:middle line:84% So a strategy for the sender is a function m from 0, 1 00:11:09.430 --> 00:11:15.030 align:middle line:84% to this abstract set of messages M. Maybe we'll go into here just 00:11:15.030 --> 00:11:17.190 align:middle line:90% to understand what goes on. 00:11:17.190 --> 00:11:20.950 align:middle line:84% So what this says is if I see that the state is, say, 0.3, 00:11:20.950 --> 00:11:23.150 align:middle line:84% this is the message I send, m of 0.3. 00:11:23.150 --> 00:11:25.150 align:middle line:84% If I see that the state is 0.7, then I 00:11:25.150 --> 00:11:28.790 align:middle line:84% might send a different message, m of 0.7. 00:11:28.790 --> 00:11:31.670 align:middle line:84% Now, the receiver's strategy, well, what do they do? 00:11:31.670 --> 00:11:33.050 align:middle line:90% They see the message. 00:11:33.050 --> 00:11:34.870 align:middle line:84% And based on the message that they see, 00:11:34.870 --> 00:11:36.750 align:middle line:90% they choose a decision to make. 00:11:36.750 --> 00:11:41.560 align:middle line:84% So for the receiver, a strategy is a function y from M 00:11:41.560 --> 00:11:48.480 align:middle line:84% to we'll say R. Just to understand what's going on, 00:11:48.480 --> 00:11:50.480 align:middle line:90% I'll write it like this. 00:11:50.480 --> 00:11:54.200 align:middle line:84% So the receiver says, if I get a message m from my advisor, 00:11:54.200 --> 00:11:56.540 align:middle line:84% this is the decision I'm going to take, y of m. 00:11:56.540 --> 00:12:02.920 align:middle line:90% 00:12:02.920 --> 00:12:08.000 align:middle line:84% And the solution concept we're going to work with 00:12:08.000 --> 00:12:09.700 align:middle line:90% is going to be PBE, as before-- 00:12:09.700 --> 00:12:11.900 align:middle line:90% Perfect Bayesian Equilibrium. 00:12:11.900 --> 00:12:14.920 align:middle line:90% 00:12:14.920 --> 00:12:18.800 align:middle line:84% Now, let's start by analyzing how 00:12:18.800 --> 00:12:21.400 align:middle line:90% things work for the receiver. 00:12:21.400 --> 00:12:24.000 align:middle line:84% So with perfect Bayesian equilibrium, whatever 00:12:24.000 --> 00:12:25.880 align:middle line:84% message the receiver sees, they're 00:12:25.880 --> 00:12:28.480 align:middle line:84% going to update their belief and then behave optimally 00:12:28.480 --> 00:12:30.760 align:middle line:90% given their updated belief. 00:12:30.760 --> 00:12:41.520 align:middle line:84% So let's just remember that whenever 00:12:41.520 --> 00:12:43.540 align:middle line:84% we work with perfect Bayesian equilibrium, 00:12:43.540 --> 00:12:48.000 align:middle line:84% we have to distinguish between the messages that occur on path 00:12:48.000 --> 00:12:50.440 align:middle line:84% and the messages that don't occur on path. 00:12:50.440 --> 00:12:55.080 align:middle line:84% So a message m is on path if, in a given strategy profile, 00:12:55.080 --> 00:12:57.880 align:middle line:84% there's some state of the world where the sender actually 00:12:57.880 --> 00:12:59.360 align:middle line:90% sends that message. 00:12:59.360 --> 00:13:02.080 align:middle line:84% And a message is off path if that message is not 00:13:02.080 --> 00:13:05.240 align:middle line:84% supposed to be sent in any state of the world. 00:13:05.240 --> 00:13:08.280 align:middle line:84% So if the receiver sees an on-path message m, 00:13:08.280 --> 00:13:09.540 align:middle line:90% they apply Bayes' rule. 00:13:09.540 --> 00:13:15.240 align:middle line:90% 00:13:15.240 --> 00:13:18.920 align:middle line:84% And then using Bayes' rule, they have an updated belief 00:13:18.920 --> 00:13:21.000 align:middle line:90% about the state of the world t. 00:13:21.000 --> 00:13:23.340 align:middle line:84% But because of the nature of our preference relation, 00:13:23.340 --> 00:13:25.520 align:middle line:84% the nature of our utility function, 00:13:25.520 --> 00:13:27.160 align:middle line:84% the receiver always wants to take 00:13:27.160 --> 00:13:29.720 align:middle line:84% the decision that's equal to their updated 00:13:29.720 --> 00:13:31.600 align:middle line:90% expectation of the state. 00:13:31.600 --> 00:13:33.760 align:middle line:90% So they apply Bayes' rule. 00:13:33.760 --> 00:13:40.267 align:middle line:84% They can now compute maybe I'll say E of t given m. 00:13:40.267 --> 00:13:41.100 align:middle line:90% What does this mean? 00:13:41.100 --> 00:13:43.000 align:middle line:90% It says, I observe m. 00:13:43.000 --> 00:13:45.600 align:middle line:84% I update my beliefs about the state of the world. 00:13:45.600 --> 00:13:47.273 align:middle line:84% And I'm now going to compute what 00:13:47.273 --> 00:13:49.440 align:middle line:84% I think the expected value of the state of the world 00:13:49.440 --> 00:13:52.320 align:middle line:90% is given my updated beliefs. 00:13:52.320 --> 00:13:54.560 align:middle line:84% And it's always going to be optimal for them 00:13:54.560 --> 00:13:57.040 align:middle line:90% to take this as their decision. 00:13:57.040 --> 00:13:59.040 align:middle line:84% Compute that, and in fact, that's 00:13:59.040 --> 00:14:00.520 align:middle line:90% going to be their decision. 00:14:00.520 --> 00:14:02.770 align:middle line:84% They're going to update their beliefs about the state, 00:14:02.770 --> 00:14:06.200 align:middle line:84% compute an expectation and set their decision equal to that. 00:14:06.200 --> 00:14:13.960 align:middle line:84% Off pat, well, the beliefs over 0, 1 00:14:13.960 --> 00:14:20.280 align:middle line:84% are unconstrained because we can't apply Bayes' rule. 00:14:20.280 --> 00:14:24.200 align:middle line:84% So off path, the receiver can believe anything 00:14:24.200 --> 00:14:26.520 align:middle line:90% about the state between 0 and 1. 00:14:26.520 --> 00:14:28.720 align:middle line:84% So the key question is, what decisions 00:14:28.720 --> 00:14:31.970 align:middle line:84% could the receiver take if they see an off-path message m? 00:14:31.970 --> 00:14:34.170 align:middle line:84% What are the range of decisions that are 00:14:34.170 --> 00:14:35.750 align:middle line:90% consistent with some belief? 00:14:35.750 --> 00:14:40.290 align:middle line:90% 00:14:40.290 --> 00:14:40.790 align:middle line:90% Yeah. 00:14:40.790 --> 00:14:42.210 align:middle line:90% AUDIENCE: Just the real numbers. 00:14:42.210 --> 00:14:44.830 align:middle line:84% PROFESSOR: So we might think it's all the real numbers. 00:14:44.830 --> 00:14:47.038 align:middle line:84% But is there any belief that's consistent with taking 00:14:47.038 --> 00:14:48.070 align:middle line:90% decision y equals 2? 00:14:48.070 --> 00:14:50.690 align:middle line:90% 00:14:50.690 --> 00:14:53.570 align:middle line:90% We know t lives in 0, 1. 00:14:53.570 --> 00:14:56.850 align:middle line:84% So the point is-- and this is kind of a subtle point-- 00:14:56.850 --> 00:15:00.630 align:middle line:84% the point is, I can form any belief I want about the state. 00:15:00.630 --> 00:15:02.757 align:middle line:84% But we know the state lies between 0 and 1. 00:15:02.757 --> 00:15:05.090 align:middle line:84% So the most pessimistic belief I could have is the state 00:15:05.090 --> 00:15:06.530 align:middle line:90% is definitely 0. 00:15:06.530 --> 00:15:08.495 align:middle line:84% And the most optimistic-- or here 00:15:08.495 --> 00:15:09.870 align:middle line:84% it's not clear what's optimistic, 00:15:09.870 --> 00:15:11.970 align:middle line:84% but the highest belief I could have is the state 00:15:11.970 --> 00:15:13.010 align:middle line:90% is certainly 1. 00:15:13.010 --> 00:15:15.130 align:middle line:84% And in those two cases I would take the decision 00:15:15.130 --> 00:15:17.190 align:middle line:90% y equals 0 and y equals 1. 00:15:17.190 --> 00:15:26.450 align:middle line:84% So I'm always going to take some decision. 00:15:26.450 --> 00:15:29.190 align:middle line:84% y that's always going to be between 0 and 1, 00:15:29.190 --> 00:15:30.690 align:middle line:84% because those are the decisions that 00:15:30.690 --> 00:15:33.170 align:middle line:84% are consistent with some belief about the state. 00:15:33.170 --> 00:15:35.850 align:middle line:84% Any decision in here can be justified 00:15:35.850 --> 00:15:37.142 align:middle line:90% by some belief about the state. 00:15:37.142 --> 00:15:39.392 align:middle line:84% I could take decision 1/2 because I think the state is 00:15:39.392 --> 00:15:40.230 align:middle line:90% certain to be 1/2. 00:15:40.230 --> 00:15:42.022 align:middle line:90% Or maybe I think it's uniform. 00:15:42.022 --> 00:15:43.230 align:middle line:90% Those are all going to be OK. 00:15:43.230 --> 00:15:47.930 align:middle line:90% 00:15:47.930 --> 00:15:50.090 align:middle line:84% It's going to turn out, as we'll see, 00:15:50.090 --> 00:15:52.977 align:middle line:84% that the off-path beliefs are not 00:15:52.977 --> 00:15:54.810 align:middle line:84% going to play as big of a role in this model 00:15:54.810 --> 00:15:56.898 align:middle line:84% as they did in the signaling model. 00:15:56.898 --> 00:15:58.190 align:middle line:90% But we'll see that a bit later. 00:15:58.190 --> 00:16:01.130 align:middle line:84% So let's start with, just to build intuition, 00:16:01.130 --> 00:16:05.268 align:middle line:90% a really simple equilibrium. 00:16:05.268 --> 00:16:06.810 align:middle line:84% And the simplest equilibrium would be 00:16:06.810 --> 00:16:08.510 align:middle line:90% a no-communication equilibrium. 00:16:08.510 --> 00:16:17.935 align:middle line:90% 00:16:17.935 --> 00:16:19.810 align:middle line:84% Now, when I say no communication, what I mean 00:16:19.810 --> 00:16:21.435 align:middle line:84% is no information is actually conveyed. 00:16:21.435 --> 00:16:22.810 align:middle line:84% Of course, they're actually going 00:16:22.810 --> 00:16:24.690 align:middle line:84% to send a message because they have to. 00:16:24.690 --> 00:16:26.695 align:middle line:84% But one thing the advisor could do 00:16:26.695 --> 00:16:28.070 align:middle line:84% is they could be a broken record. 00:16:28.070 --> 00:16:29.890 align:middle line:84% They could just always send the same message. 00:16:29.890 --> 00:16:31.430 align:middle line:84% And if they always send the same message, 00:16:31.430 --> 00:16:32.790 align:middle line:84% no matter the state of the world, 00:16:32.790 --> 00:16:35.498 align:middle line:84% then that message doesn't convey any information about the state. 00:16:35.498 --> 00:16:37.630 align:middle line:84% So let's look at this kind of boring equilibrium. 00:16:37.630 --> 00:16:44.080 align:middle line:84% Well, what is-- let's fix some message I'll call m* in m. 00:16:44.080 --> 00:16:46.330 align:middle line:84% This is going to be the message that the sender always 00:16:46.330 --> 00:16:48.930 align:middle line:84% sends, no matter the state of the world. 00:16:48.930 --> 00:17:00.770 align:middle line:84% So the sender's strategy is m of t equals m* for all t in 01. 00:17:00.770 --> 00:17:02.870 align:middle line:84% So going to our economic example, 00:17:02.870 --> 00:17:05.470 align:middle line:84% if they see the economy's great, they say cut interest rates. 00:17:05.470 --> 00:17:07.970 align:middle line:84% If they see the economy is bad, they say cut interest rates. 00:17:07.970 --> 00:17:09.770 align:middle line:84% And therefore, what they say conveys 00:17:09.770 --> 00:17:13.450 align:middle line:90% no information about the state. 00:17:13.450 --> 00:17:15.490 align:middle line:84% Now let's try to compute what the receiver is 00:17:15.490 --> 00:17:17.569 align:middle line:90% going to do in this case. 00:17:17.569 --> 00:17:22.690 align:middle line:84% So let's now look at the receiver. 00:17:22.690 --> 00:17:29.300 align:middle line:84% And what happens if the receiver sees the message m*? 00:17:29.300 --> 00:17:32.260 align:middle line:84% What beliefs does the receiver form about the state if they get 00:17:32.260 --> 00:17:33.220 align:middle line:90% this message m*? 00:17:33.220 --> 00:17:37.568 align:middle line:90% 00:17:37.568 --> 00:17:38.400 align:middle line:90% Yeah. 00:17:38.400 --> 00:17:39.900 align:middle line:84% AUDIENCE: Were they just going to be 00:17:39.900 --> 00:17:42.668 align:middle line:84% left with the prior, which is that t is distributed over 0, 1? 00:17:42.668 --> 00:17:43.460 align:middle line:90% PROFESSOR: Exactly. 00:17:43.460 --> 00:17:44.380 align:middle line:84% They'd just be left with the prior, 00:17:44.380 --> 00:17:45.740 align:middle line:84% because they know whatever the state is, they're 00:17:45.740 --> 00:17:46.640 align:middle line:90% going to get this message. 00:17:46.640 --> 00:17:47.880 align:middle line:90% This hasn't told them anything. 00:17:47.880 --> 00:17:48.860 align:middle line:84% If you always say the same thing, 00:17:48.860 --> 00:17:50.480 align:middle line:84% you haven't revealed anything to me. 00:17:50.480 --> 00:17:53.140 align:middle line:84% So my beliefs are the same as the prior. 00:17:53.140 --> 00:17:59.200 align:middle line:84% So if I see m*, maybe we could say retains prior beliefs. 00:17:59.200 --> 00:18:02.100 align:middle line:90% 00:18:02.100 --> 00:18:04.020 align:middle line:84% That is, I still think the state is uniformly 00:18:04.020 --> 00:18:05.860 align:middle line:90% distributed over 0, 1. 00:18:05.860 --> 00:18:09.060 align:middle line:84% And the average state is therefore 1/2. 00:18:09.060 --> 00:18:12.000 align:middle line:84% So the best decision for me is to take y equals 1/2. 00:18:12.000 --> 00:18:14.400 align:middle line:84% So it retains prior beliefs and they take-- 00:18:14.400 --> 00:18:20.060 align:middle line:90% so y of m* equals 1/2. 00:18:20.060 --> 00:18:22.140 align:middle line:84% So this looks good for equilibrium. 00:18:22.140 --> 00:18:23.680 align:middle line:90% But we still have to worry. 00:18:23.680 --> 00:18:27.980 align:middle line:84% Well, we have to say, what does the receiver do if they see 00:18:27.980 --> 00:18:31.182 align:middle line:84% a message m That's not equal to m*? 00:18:31.182 --> 00:18:33.140 align:middle line:84% They don't really know what to do in this case, 00:18:33.140 --> 00:18:36.042 align:middle line:84% because they were expecting to always see the message m*. 00:18:36.042 --> 00:18:37.500 align:middle line:84% They can't really apply Bayes' rule 00:18:37.500 --> 00:18:39.980 align:middle line:84% here because they'd be dividing by 0. 00:18:39.980 --> 00:18:41.840 align:middle line:90% So there's a lot of flexibility. 00:18:41.840 --> 00:18:44.740 align:middle line:90% 00:18:44.740 --> 00:18:48.220 align:middle line:84% You might say, oh, well, let's just choose y arbitrarily. 00:18:48.220 --> 00:18:54.780 align:middle line:84% But what could go wrong if we just chose y arbitrarily here 00:18:54.780 --> 00:18:55.860 align:middle line:90% for the receiver? 00:18:55.860 --> 00:19:03.838 align:middle line:84% Let's say we chose y equals 1/3 or something. 00:19:03.838 --> 00:19:04.880 align:middle line:90% What could go wrong here? 00:19:04.880 --> 00:19:05.380 align:middle line:90% Yeah. 00:19:05.380 --> 00:19:07.005 align:middle line:84% AUDIENCE: Maybe the sender will-- like, 00:19:07.005 --> 00:19:08.660 align:middle line:84% that would be a profitable deviation. 00:19:08.660 --> 00:19:09.360 align:middle line:90% PROFESSOR: Exactly right. 00:19:09.360 --> 00:19:11.360 align:middle line:84% So whenever we're choosing the off-path behavior 00:19:11.360 --> 00:19:13.620 align:middle line:84% by the receiver, we have to keep in mind 00:19:13.620 --> 00:19:15.620 align:middle line:84% that we don't want to choose behavior 00:19:15.620 --> 00:19:18.527 align:middle line:84% that will introduce a profitable deviation for the sender. 00:19:18.527 --> 00:19:20.860 align:middle line:84% In this equilibrium, the sender is only supposed to send 00:19:20.860 --> 00:19:22.140 align:middle line:90% the message m*. 00:19:22.140 --> 00:19:25.180 align:middle line:84% So we want to make sure that the sender never wants to deviate 00:19:25.180 --> 00:19:27.500 align:middle line:84% to send any message other than m*. 00:19:27.500 --> 00:19:28.520 align:middle line:90% So how can we be sure? 00:19:28.520 --> 00:19:31.260 align:middle line:84% How could we choose y of m to be sure that the sender never 00:19:31.260 --> 00:19:32.040 align:middle line:90% wants to deviate? 00:19:32.040 --> 00:19:32.660 align:middle line:90% Yeah. 00:19:32.660 --> 00:19:35.180 align:middle line:84% AUDIENCE: You could make it so that the difference between-- 00:19:35.180 --> 00:19:39.318 align:middle line:84% you get the same utility from getting m* and r. 00:19:39.318 --> 00:19:40.360 align:middle line:90% PROFESSOR: Exactly right. 00:19:40.360 --> 00:19:43.220 align:middle line:84% And how could we make sure they get the same utility? 00:19:43.220 --> 00:19:45.260 align:middle line:84% AUDIENCE: I don't know exactly, but it's 00:19:45.260 --> 00:19:46.785 align:middle line:90% depending on beta, right? 00:19:46.785 --> 00:19:48.660 align:middle line:84% PROFESSOR: I think you're on the right track. 00:19:48.660 --> 00:19:49.560 align:middle line:90% There's even a simpler answer. 00:19:49.560 --> 00:19:50.200 align:middle line:90% So you're exactly right. 00:19:50.200 --> 00:19:52.040 align:middle line:84% We want to make sure that if they deviate to this, 00:19:52.040 --> 00:19:53.800 align:middle line:84% the sender gets exactly the same utility. 00:19:53.800 --> 00:19:54.675 align:middle line:90% But what could it be? 00:19:54.675 --> 00:19:55.300 align:middle line:90% Yeah. 00:19:55.300 --> 00:19:56.240 align:middle line:90% It could just be 1/2. 00:19:56.240 --> 00:19:57.720 align:middle line:84% So what they could do is they could say, look, 00:19:57.720 --> 00:19:59.540 align:middle line:84% if I see any other message, I'm going 00:19:59.540 --> 00:20:00.940 align:middle line:90% to do exactly the same thing. 00:20:00.940 --> 00:20:03.160 align:middle line:84% I'm just going to choose ym equals 1/2. 00:20:03.160 --> 00:20:04.805 align:middle line:84% I'm going to maintain my prior belief. 00:20:04.805 --> 00:20:06.180 align:middle line:84% So let's understand the structure 00:20:06.180 --> 00:20:08.400 align:middle line:84% of this no-communication equilibrium. 00:20:08.400 --> 00:20:10.820 align:middle line:84% The receiver says, whatever message you send me 00:20:10.820 --> 00:20:12.120 align:middle line:90% I'm going to do the same thing. 00:20:12.120 --> 00:20:14.180 align:middle line:84% I'm going to completely ignore you. 00:20:14.180 --> 00:20:15.740 align:middle line:84% And the sender says, well, if you're 00:20:15.740 --> 00:20:18.200 align:middle line:84% going to ignore what I say, it doesn't matter what I say. 00:20:18.200 --> 00:20:20.950 align:middle line:84% So I might as well always send the message m*. 00:20:20.950 --> 00:20:23.310 align:middle line:84% And this kind of sometimes goes over the name-- this 00:20:23.310 --> 00:20:25.830 align:middle line:90% is sometimes called babbling. 00:20:25.830 --> 00:20:27.910 align:middle line:84% So in this equilibrium, the receiver 00:20:27.910 --> 00:20:30.090 align:middle line:84% is treating his advisor's words as just babbles. 00:20:30.090 --> 00:20:30.930 align:middle line:90% I don't care what you say. 00:20:30.930 --> 00:20:32.330 align:middle line:84% It's not going to change my belief. 00:20:32.330 --> 00:20:34.770 align:middle line:84% I'm just going to ignore what you say and do the same thing. 00:20:34.770 --> 00:20:36.410 align:middle line:84% And therefore, the advisor says, well, 00:20:36.410 --> 00:20:39.750 align:middle line:84% I might as well always send this message m*. 00:20:39.750 --> 00:20:42.277 align:middle line:90% So indeed, this satisfies-- 00:20:42.277 --> 00:20:44.110 align:middle line:84% let's just check what we would have to check 00:20:44.110 --> 00:20:45.610 align:middle line:90% to make sure this equilibrium. 00:20:45.610 --> 00:20:53.990 align:middle line:84% We'd have to check sender optimality, receiver optimality, 00:20:53.990 --> 00:20:55.450 align:middle line:90% and belief consistency. 00:20:55.450 --> 00:20:59.722 align:middle line:90% 00:20:59.722 --> 00:21:00.930 align:middle line:90% I've been a little imprecise. 00:21:00.930 --> 00:21:03.370 align:middle line:84% I haven't really written what the beliefs are here. 00:21:03.370 --> 00:21:05.770 align:middle line:84% But naturally, the beliefs could be, 00:21:05.770 --> 00:21:07.770 align:middle line:84% I just think the state is uniformly distributed 00:21:07.770 --> 00:21:09.270 align:middle line:90% if I see any other message. 00:21:09.270 --> 00:21:12.830 align:middle line:84% And then we can see that each of these three conditions 00:21:12.830 --> 00:21:14.978 align:middle line:90% is going to be satisfied. 00:21:14.978 --> 00:21:16.770 align:middle line:84% This is not a very interesting equilibrium, 00:21:16.770 --> 00:21:18.437 align:middle line:84% but it's just good to see there's always 00:21:18.437 --> 00:21:21.190 align:middle line:84% an equilibrium like this, where our words just don't have 00:21:21.190 --> 00:21:24.550 align:middle line:90% any meaning And no one listens. 00:21:24.550 --> 00:21:25.957 align:middle line:90% Well, that's one extreme. 00:21:25.957 --> 00:21:27.290 align:middle line:90% Let's see if we could do better. 00:21:27.290 --> 00:21:30.950 align:middle line:84% We might want to have perfect communication. 00:21:30.950 --> 00:21:32.430 align:middle line:84% So what about an equilibrium where 00:21:32.430 --> 00:21:33.810 align:middle line:90% there's perfect communication? 00:21:33.810 --> 00:21:37.245 align:middle line:90% 00:21:37.245 --> 00:21:39.870 align:middle line:84% So you might think, well, if the advisor and the decision maker 00:21:39.870 --> 00:21:41.070 align:middle line:84% have a pretty good relationship, what 00:21:41.070 --> 00:21:42.737 align:middle line:84% should happen is the advisor should just 00:21:42.737 --> 00:21:44.750 align:middle line:90% say this is what the state is. 00:21:44.750 --> 00:21:49.990 align:middle line:84% So could there be an equilibrium where the message m of t 00:21:49.990 --> 00:21:51.150 align:middle line:90% just equals t? 00:21:51.150 --> 00:21:54.950 align:middle line:84% So the advisor says, this is the state, make a decision. 00:21:54.950 --> 00:21:55.870 align:middle line:90% Any thoughts? 00:21:55.870 --> 00:21:59.190 align:middle line:84% Could we make this an equilibrium? 00:21:59.190 --> 00:22:00.003 align:middle line:90% Yeah, maybe? 00:22:00.003 --> 00:22:01.670 align:middle line:84% AUDIENCE: It would not be an equilibrium 00:22:01.670 --> 00:22:06.810 align:middle line:84% because the receiver would just act with y equals t as a result. 00:22:06.810 --> 00:22:10.550 align:middle line:84% And then the sender is losing some amount because y does not 00:22:10.550 --> 00:22:12.162 align:middle line:90% equal t plus beta. 00:22:12.162 --> 00:22:12.870 align:middle line:90% PROFESSOR: Right. 00:22:12.870 --> 00:22:13.510 align:middle line:90% Let's work through it. 00:22:13.510 --> 00:22:15.302 align:middle line:84% If we tried to-- this is not going to work, 00:22:15.302 --> 00:22:16.290 align:middle line:90% but let's go ahead. 00:22:16.290 --> 00:22:17.957 align:middle line:84% If we tried to have an equilibrium where 00:22:17.957 --> 00:22:20.030 align:middle line:84% this was the messaging strategy of the sender, 00:22:20.030 --> 00:22:21.790 align:middle line:84% well, this looks great for the receiver. 00:22:21.790 --> 00:22:25.010 align:middle line:84% What is the receiver y of t going to be? 00:22:25.010 --> 00:22:26.270 align:middle line:90% Let's be clear here. 00:22:26.270 --> 00:22:28.930 align:middle line:84% The receiver's strategy is not a function of the state. 00:22:28.930 --> 00:22:30.252 align:middle line:90% It's a function of the message. 00:22:30.252 --> 00:22:32.210 align:middle line:84% But here, the message is the same as the state. 00:22:32.210 --> 00:22:34.230 align:middle line:90% So that's why I'm writing t. 00:22:34.230 --> 00:22:37.070 align:middle line:84% Or actually, maybe I'll still write m. 00:22:37.070 --> 00:22:38.650 align:middle line:84% Eh, torn about which notation to use. 00:22:38.650 --> 00:22:40.110 align:middle line:90% It doesn't make a difference. 00:22:40.110 --> 00:22:42.030 align:middle line:90% I'll write m. 00:22:42.030 --> 00:22:43.750 align:middle line:84% In this case, when they see a message m, 00:22:43.750 --> 00:22:46.990 align:middle line:84% they're going to say, well, that means the state t equals m. 00:22:46.990 --> 00:22:51.950 align:middle line:84% So their best decision is just going to be m, right? 00:22:51.950 --> 00:22:55.990 align:middle line:84% But now let's put yourself in the shoes of the sender. 00:22:55.990 --> 00:22:58.530 align:middle line:84% And let's say you see that the state is t. 00:22:58.530 --> 00:23:03.010 align:middle line:90% 00:23:03.010 --> 00:23:04.510 align:middle line:84% If you were the sender, what message 00:23:04.510 --> 00:23:06.590 align:middle line:84% would you send to the decision maker? 00:23:06.590 --> 00:23:10.952 align:middle line:90% 00:23:10.952 --> 00:23:13.030 align:middle line:90% AUDIENCE: t plus beta. 00:23:13.030 --> 00:23:14.430 align:middle line:90% PROFESSOR: Exactly. 00:23:14.430 --> 00:23:16.872 align:middle line:90% So you know the state is t. 00:23:16.872 --> 00:23:18.330 align:middle line:84% Well, why don't you just say, look, 00:23:18.330 --> 00:23:21.850 align:middle line:84% you're going to tell your advisor m equals t plus beta? 00:23:21.850 --> 00:23:25.590 align:middle line:90% 00:23:25.590 --> 00:23:28.850 align:middle line:84% And I think we see this all the time. 00:23:28.850 --> 00:23:30.697 align:middle line:84% If someone's more of an interventionist 00:23:30.697 --> 00:23:33.030 align:middle line:84% and they know that the president is less interventionist 00:23:33.030 --> 00:23:35.050 align:middle line:84% and they want them to intervene in the conflict, they don't say, 00:23:35.050 --> 00:23:35.810 align:middle line:90% I'm interventionist. 00:23:35.810 --> 00:23:37.970 align:middle line:84% They say, oh, this conflict is just really serious. 00:23:37.970 --> 00:23:38.870 align:middle line:84% It's really important you intervene. 00:23:38.870 --> 00:23:40.370 align:middle line:84% I'm not saying it because I'm interventionist. 00:23:40.370 --> 00:23:41.470 align:middle line:84% It's just the nature of the conflict 00:23:41.470 --> 00:23:43.090 align:middle line:84% means we really have to intervene. 00:23:43.090 --> 00:23:45.813 align:middle line:84% Or someone who always wants to cut interest rates 00:23:45.813 --> 00:23:48.230 align:middle line:84% isn't going to say, oh, let's cut interest rates because I 00:23:48.230 --> 00:23:50.270 align:middle line:90% love cutting interest rates. 00:23:50.270 --> 00:23:52.215 align:middle line:84% They're going to say, inflation's 00:23:52.215 --> 00:23:53.090 align:middle line:90% not really a problem. 00:23:53.090 --> 00:23:54.673 align:middle line:84% We're not really having any inflation. 00:23:54.673 --> 00:23:56.190 align:middle line:84% I think this is what you should do. 00:23:56.190 --> 00:23:58.670 align:middle line:84% So the sender has an incentive to misrepresent 00:23:58.670 --> 00:24:01.890 align:middle line:84% what the state is in order to get their ideal decision. 00:24:01.890 --> 00:24:03.910 align:middle line:84% Because they know if the receiver believes 00:24:03.910 --> 00:24:05.670 align:middle line:84% the state is actually t plus beta, 00:24:05.670 --> 00:24:07.890 align:middle line:84% then the receiver will take decision t plus beta, 00:24:07.890 --> 00:24:10.098 align:middle line:84% and that will give the sender exactly what they want. 00:24:10.098 --> 00:24:12.840 align:middle line:84% So we see that indeed, this cannot be an equilibrium, 00:24:12.840 --> 00:24:14.757 align:middle line:84% because the sender has a profitable deviation, 00:24:14.757 --> 00:24:16.715 align:middle line:84% and then the receiver won't trust the messages, 00:24:16.715 --> 00:24:17.900 align:middle line:90% and everything breaks down. 00:24:17.900 --> 00:24:19.660 align:middle line:90% So not an equilibrium. 00:24:19.660 --> 00:24:25.800 align:middle line:90% 00:24:25.800 --> 00:24:27.600 align:middle line:84% But now we have a bit of a puzzle. 00:24:27.600 --> 00:24:30.120 align:middle line:84% We definitely see communication in practice between people 00:24:30.120 --> 00:24:32.960 align:middle line:90% who have misaligned interests. 00:24:32.960 --> 00:24:35.420 align:middle line:84% We don't just see this no-communication equilibrium. 00:24:35.420 --> 00:24:37.400 align:middle line:84% So the question is, well, if we can't 00:24:37.400 --> 00:24:39.667 align:middle line:84% get a perfect communication equilibrium, 00:24:39.667 --> 00:24:42.000 align:middle line:84% maybe it's possible to have an equilibrium where we have 00:24:42.000 --> 00:24:43.420 align:middle line:90% some partial communication. 00:24:43.420 --> 00:24:44.940 align:middle line:84% We don't perfectly reveal the state, 00:24:44.940 --> 00:24:47.420 align:middle line:84% but we provide some valuable information about the state. 00:24:47.420 --> 00:24:49.980 align:middle line:84% And that's going to be the key idea of this paper. 00:24:49.980 --> 00:24:52.800 align:middle line:90% 00:24:52.800 --> 00:24:56.520 align:middle line:84% So let's our goal is to find, well, maybe not 00:24:56.520 --> 00:24:59.680 align:middle line:84% perfectly informative equilibria, but partially 00:24:59.680 --> 00:25:01.100 align:middle line:90% informative equilibrium. 00:25:01.100 --> 00:25:12.040 align:middle line:90% 00:25:12.040 --> 00:25:14.538 align:middle line:84% Let's start with a really simple case. 00:25:14.538 --> 00:25:16.080 align:middle line:84% I guess the simplest possible message 00:25:16.080 --> 00:25:18.360 align:middle line:84% you could send about some real valued state 00:25:18.360 --> 00:25:20.360 align:middle line:84% is you could say the state's either high or low. 00:25:20.360 --> 00:25:22.361 align:middle line:84% You could just say it's either above a threshold 00:25:22.361 --> 00:25:23.380 align:middle line:90% or below a threshold. 00:25:23.380 --> 00:25:28.200 align:middle line:84% So let's see if we can work with that. 00:25:28.200 --> 00:25:32.160 align:middle line:84% So the state is here between 0 and 1. 00:25:32.160 --> 00:25:34.280 align:middle line:84% And let's say we choose some threshold. 00:25:34.280 --> 00:25:36.520 align:middle line:90% We'll call this threshold t1. 00:25:36.520 --> 00:25:39.520 align:middle line:84% And what's the sender going to do? 00:25:39.520 --> 00:25:41.447 align:middle line:84% If they see the state is below t1, 00:25:41.447 --> 00:25:42.780 align:middle line:90% they're going to send a message. 00:25:42.780 --> 00:25:44.640 align:middle line:84% Maybe we'll call the message L. They're 00:25:44.640 --> 00:25:45.980 align:middle line:90% going to say the state is low. 00:25:45.980 --> 00:25:48.280 align:middle line:90% The state is below T1. 00:25:48.280 --> 00:25:50.520 align:middle line:84% And if they see the state is above T1, 00:25:50.520 --> 00:25:55.640 align:middle line:84% they're going to send a message H saying the state is high. 00:25:55.640 --> 00:25:59.515 align:middle line:84% And we want to see if this could be an equilibrium. 00:25:59.515 --> 00:26:01.640 align:middle line:84% You might worry because you might think, well, wait 00:26:01.640 --> 00:26:02.600 align:middle line:90% a second. 00:26:02.600 --> 00:26:05.960 align:middle line:84% Isn't the sender going to want to lie and pretend 00:26:05.960 --> 00:26:07.520 align:middle line:84% the state is high when sometimes it's 00:26:07.520 --> 00:26:10.372 align:middle line:84% not in order to get their higher preferred decision? 00:26:10.372 --> 00:26:12.080 align:middle line:84% That's a good intuition, but it turns out 00:26:12.080 --> 00:26:14.788 align:middle line:84% we can get this to work, which I think is maybe a bit surprising. 00:26:14.788 --> 00:26:16.360 align:middle line:90% So let's go through this. 00:26:16.360 --> 00:26:20.880 align:middle line:84% So what is y of L going to be and y of H going to be? 00:26:20.880 --> 00:26:23.723 align:middle line:84% So t1 Is a parameter that we're going to solve for. 00:26:23.723 --> 00:26:25.640 align:middle line:84% So we're going to look for an equilibrium that 00:26:25.640 --> 00:26:27.628 align:middle line:84% has this form for some parameter t1. 00:26:27.628 --> 00:26:29.420 align:middle line:84% And this is what we're going to figure out. 00:26:29.420 --> 00:26:34.060 align:middle line:84% So what happens if the receiver sees the message L? 00:26:34.060 --> 00:26:36.800 align:middle line:84% Well, the receiver knows that the state is somewhere 00:26:36.800 --> 00:26:38.480 align:middle line:90% between 0 and T1. 00:26:38.480 --> 00:26:41.920 align:middle line:84% And they know it's uniformly distributed between 0 and T1. 00:26:41.920 --> 00:26:43.853 align:middle line:84% So what do they think is the average expected? 00:26:43.853 --> 00:26:45.520 align:middle line:84% What do they think is the expected state 00:26:45.520 --> 00:26:46.580 align:middle line:90% when they see message L? 00:26:46.580 --> 00:26:47.487 align:middle line:90% Yeah. 00:26:47.487 --> 00:26:48.320 align:middle line:90% AUDIENCE: 1/2 of t1. 00:26:48.320 --> 00:26:49.883 align:middle line:90% PROFESSOR: 1/2 of t1. 00:26:49.883 --> 00:26:51.800 align:middle line:84% Because they're going to say, look, all you've 00:26:51.800 --> 00:26:53.400 align:middle line:90% told me is the state's here. 00:26:53.400 --> 00:26:54.740 align:middle line:90% It's uniformly distributed. 00:26:54.740 --> 00:26:57.580 align:middle line:84% So on average, it's the midpoint of this interval, 00:26:57.580 --> 00:26:59.640 align:middle line:90% which is t1 over 2. 00:26:59.640 --> 00:27:02.640 align:middle line:84% Conversely, if the receiver gets the message high, 00:27:02.640 --> 00:27:05.530 align:middle line:84% they're going to say, OK, the state is uniformly 00:27:05.530 --> 00:27:09.410 align:middle line:90% distributed between t1 and 1. 00:27:09.410 --> 00:27:11.370 align:middle line:84% So my updated expectation of the state 00:27:11.370 --> 00:27:13.410 align:middle line:84% is just the midpoint of this interval, which 00:27:13.410 --> 00:27:15.370 align:middle line:90% is the average of t1 and 1. 00:27:15.370 --> 00:27:19.790 align:middle line:84% So this is just going to be T1 plus 1 over 2. 00:27:19.790 --> 00:27:24.290 align:middle line:90% 00:27:24.290 --> 00:27:26.750 align:middle line:90% Now, we're not done yet. 00:27:26.750 --> 00:27:32.672 align:middle line:90% 00:27:32.672 --> 00:27:34.130 align:middle line:84% We still have to worry what happens 00:27:34.130 --> 00:27:38.533 align:middle line:84% if they get a message that's neither L nor H, right? 00:27:38.533 --> 00:27:40.450 align:middle line:84% But there's kind of a trick that we can always 00:27:40.450 --> 00:27:43.288 align:middle line:84% use that goes back to what we did over here. 00:27:43.288 --> 00:27:45.330 align:middle line:84% The trick was, if we see a message that we're not 00:27:45.330 --> 00:27:47.150 align:middle line:84% supposed to see, we can just treat it 00:27:47.150 --> 00:27:49.150 align:middle line:84% as if it were a message we were supposed to see. 00:27:49.150 --> 00:27:51.330 align:middle line:84% And that's never going to be a profitable deviation. 00:27:51.330 --> 00:27:53.750 align:middle line:84% So in this case, the receiver can say, look, 00:27:53.750 --> 00:27:55.890 align:middle line:84% if you send me any message other than L or H, 00:27:55.890 --> 00:28:00.330 align:middle line:84% I'm just going to treat that as if you had sent me L. So 00:28:00.330 --> 00:28:03.570 align:middle line:90% it's going to be t1 over 2. 00:28:03.570 --> 00:28:08.508 align:middle line:84% So as if m equals L. And then there's 00:28:08.508 --> 00:28:11.050 align:middle line:84% never a reason for the sender to deviate to any message other 00:28:11.050 --> 00:28:13.463 align:middle line:84% than L or H, because if they send a message other than L 00:28:13.463 --> 00:28:15.630 align:middle line:84% or H, it's just as if they're sending the message L. 00:28:15.630 --> 00:28:17.922 align:middle line:84% So they might as well send the message L to begin with. 00:28:17.922 --> 00:28:21.690 align:middle line:84% And this is a common kind of trick 00:28:21.690 --> 00:28:24.770 align:middle line:90% that's used in cheap talk games. 00:28:24.770 --> 00:28:27.165 align:middle line:84% I want to emphasize, I don't want to create confusion 00:28:27.165 --> 00:28:27.790 align:middle line:90% with signaling. 00:28:27.790 --> 00:28:30.370 align:middle line:84% So I want to say this is a trick that 00:28:30.370 --> 00:28:31.710 align:middle line:90% doesn't work with signaling. 00:28:31.710 --> 00:28:39.868 align:middle line:90% 00:28:39.868 --> 00:28:41.410 align:middle line:84% Let's just understand this, because I 00:28:41.410 --> 00:28:42.650 align:middle line:84% don't want to create confusion going back 00:28:42.650 --> 00:28:43.810 align:middle line:90% to the signaling game. 00:28:43.810 --> 00:28:45.670 align:middle line:84% What if, in the employment context, 00:28:45.670 --> 00:28:47.450 align:middle line:84% the employer said, OK, you're supposed 00:28:47.450 --> 00:28:48.975 align:middle line:90% to get this level of education? 00:28:48.975 --> 00:28:50.850 align:middle line:84% If you don't go to school, I'll just treat it 00:28:50.850 --> 00:28:51.892 align:middle line:90% as if you went to school. 00:28:51.892 --> 00:28:54.343 align:middle line:84% And that shouldn't be a profitable deviation. 00:28:54.343 --> 00:28:56.010 align:middle line:84% That's not going to work, because they'd 00:28:56.010 --> 00:28:58.470 align:middle line:84% much rather not go to school and form the same beliefs. 00:28:58.470 --> 00:29:00.050 align:middle line:84% So crucial reason this trick works 00:29:00.050 --> 00:29:04.090 align:middle line:84% is if I can send a message L and message m 00:29:04.090 --> 00:29:06.930 align:middle line:84% and get the same decision, I'm indifferent between those two 00:29:06.930 --> 00:29:07.810 align:middle line:90% things. 00:29:07.810 --> 00:29:09.938 align:middle line:84% But if the message itself were costly, 00:29:09.938 --> 00:29:11.730 align:middle line:84% then I wouldn't necessarily be indifferent, 00:29:11.730 --> 00:29:14.188 align:middle line:84% because I'd have to take into account not only the decision 00:29:14.188 --> 00:29:16.830 align:middle line:84% it induced, but the cost of the message I'm sending. 00:29:16.830 --> 00:29:19.050 align:middle line:84% So this is a crucial difference between a model 00:29:19.050 --> 00:29:22.035 align:middle line:84% with payoff-irrelevant messages and a model where 00:29:22.035 --> 00:29:23.910 align:middle line:84% the actions I'm choosing are payoff-relevant. 00:29:23.910 --> 00:29:25.285 align:middle line:84% I that's a bit of a subtle point. 00:29:25.285 --> 00:29:27.570 align:middle line:84% So any questions on that difference? 00:29:27.570 --> 00:29:28.530 align:middle line:90% Yes. 00:29:28.530 --> 00:29:31.017 align:middle line:84% AUDIENCE: Could you repeat the example? 00:29:31.017 --> 00:29:32.350 align:middle line:90% PROFESSOR: The employee example? 00:29:32.350 --> 00:29:32.850 align:middle line:90% Yeah. 00:29:32.850 --> 00:29:34.690 align:middle line:84% So in the employee example, let's say we're 00:29:34.690 --> 00:29:37.050 align:middle line:90% doing the job market signaling. 00:29:37.050 --> 00:29:39.530 align:middle line:84% And everyone is supposed to get-- let's look 00:29:39.530 --> 00:29:43.010 align:middle line:84% at the pooling equilibrium, where everyone goes to college 00:29:43.010 --> 00:29:44.410 align:middle line:90% and gets a certain wage. 00:29:44.410 --> 00:29:45.890 align:middle line:84% And the employer says, OK, if you 00:29:45.890 --> 00:29:47.790 align:middle line:84% choose any other level of education, 00:29:47.790 --> 00:29:51.310 align:middle line:84% I'm going to treat it as if you'd gone to college. 00:29:51.310 --> 00:29:54.370 align:middle line:84% And I'll pay you the same wage as if you'd gone to college. 00:29:54.370 --> 00:29:57.870 align:middle line:84% Well now, the student is going to say, wait a second. 00:29:57.870 --> 00:30:01.100 align:middle line:84% If I can instead of going to college not go to college 00:30:01.100 --> 00:30:03.753 align:middle line:84% and get the same wage, I'm strictly better off. 00:30:03.753 --> 00:30:06.420 align:middle line:84% And the difference here is, even though those two actions induce 00:30:06.420 --> 00:30:08.640 align:middle line:84% the same decision by the receiver, 00:30:08.640 --> 00:30:10.540 align:middle line:84% the actions themselves have different costs. 00:30:10.540 --> 00:30:12.957 align:middle line:84% And therefore, the student would rather not go to college. 00:30:12.957 --> 00:30:16.420 align:middle line:84% Here, the message L and the message m 00:30:16.420 --> 00:30:17.633 align:middle line:90% induce the same decision. 00:30:17.633 --> 00:30:19.300 align:middle line:84% But in this case, the sender is actually 00:30:19.300 --> 00:30:20.800 align:middle line:84% indifferent between those two things 00:30:20.800 --> 00:30:24.120 align:middle line:84% because the message itself doesn't directly have a cost. 00:30:24.120 --> 00:30:25.380 align:middle line:90% Does that clear things up? 00:30:25.380 --> 00:30:26.340 align:middle line:90% Yeah, great. 00:30:26.340 --> 00:30:27.840 align:middle line:90% OK. 00:30:27.840 --> 00:30:30.600 align:middle line:90% 00:30:30.600 --> 00:30:34.500 align:middle line:84% And actually, that's why what I said earlier-- 00:30:34.500 --> 00:30:36.380 align:middle line:84% off-path beliefs and belief consistency 00:30:36.380 --> 00:30:38.672 align:middle line:84% don't really make a big difference in cheap talk games. 00:30:38.672 --> 00:30:40.100 align:middle line:84% And everything we do is basically 00:30:40.100 --> 00:30:41.767 align:middle line:84% the same as just doing Nash equilibrium. 00:30:41.767 --> 00:30:43.160 align:middle line:90% But that's a subtle point. 00:30:43.160 --> 00:30:44.160 align:middle line:90% That's not so important. 00:30:44.160 --> 00:30:45.575 align:middle line:90% So here we are. 00:30:45.575 --> 00:30:47.200 align:middle line:84% Let's see if this is going to work out. 00:30:47.200 --> 00:30:49.780 align:middle line:84% We've specified the receiver strategy. 00:30:49.780 --> 00:30:52.580 align:middle line:84% And now we have to check that the sender is 00:30:52.580 --> 00:30:56.100 align:middle line:84% willing to send these messages as they're supposed to. 00:30:56.100 --> 00:30:57.900 align:middle line:84% So what's the sender supposed to do? 00:30:57.900 --> 00:31:02.660 align:middle line:84% If they see any state here, they send message L. 00:31:02.660 --> 00:31:05.940 align:middle line:84% And if they see any state here, they send message H. 00:31:05.940 --> 00:31:07.740 align:middle line:84% So how can that be optimal for them? 00:31:07.740 --> 00:31:11.340 align:middle line:84% Well, what must be true if the state is exactly t1? 00:31:11.340 --> 00:31:12.320 align:middle line:90% So let's ask this. 00:31:12.320 --> 00:31:18.580 align:middle line:90% 00:31:18.580 --> 00:31:20.660 align:middle line:84% What must be true about the sender 00:31:20.660 --> 00:31:21.880 align:middle line:90% if the state is exactly t1? 00:31:21.880 --> 00:31:22.540 align:middle line:90% Yes. 00:31:22.540 --> 00:31:24.665 align:middle line:84% AUDIENCE: The sender is indifferent between sending 00:31:24.665 --> 00:31:25.360 align:middle line:90% L and sending H. 00:31:25.360 --> 00:31:26.220 align:middle line:84% PROFESSOR: They have to be indifferent. 00:31:26.220 --> 00:31:28.938 align:middle line:84% Because if the state were just a little bit to the left, 00:31:28.938 --> 00:31:31.480 align:middle line:84% they'd want to send L. And if it's a little bit to the right, 00:31:31.480 --> 00:31:32.820 align:middle line:90% they'd want to send m. 00:31:32.820 --> 00:31:35.220 align:middle line:84% So the only way they can have those preferences 00:31:35.220 --> 00:31:38.700 align:middle line:84% is if at t1, they're exactly indifferent between sending 00:31:38.700 --> 00:31:42.720 align:middle line:84% the message L and the message H. So the idea is here, 00:31:42.720 --> 00:31:44.380 align:middle line:84% I'm going to be exactly indifferent. 00:31:44.380 --> 00:31:46.200 align:middle line:84% And if the state is anywhere to the left, 00:31:46.200 --> 00:31:49.500 align:middle line:84% I want to send L and anywhere to the right I want to send H. 00:31:49.500 --> 00:31:53.000 align:middle line:84% So what we need is we must have indifference. 00:31:53.000 --> 00:31:57.180 align:middle line:90% 00:31:57.180 --> 00:32:01.020 align:middle line:90% So let's work this out. 00:32:01.020 --> 00:32:04.520 align:middle line:84% Well, that means when the state is t1-- 00:32:04.520 --> 00:32:09.300 align:middle line:90% 00:32:09.300 --> 00:32:10.800 align:middle line:84% maybe we'll write it graphically. 00:32:10.800 --> 00:32:26.308 align:middle line:90% What we must have is y of L. 00:32:26.308 --> 00:32:27.600 align:middle line:90% Maybe we'll put negatives here. 00:32:27.600 --> 00:32:28.940 align:middle line:90% It doesn't matter. 00:32:28.940 --> 00:32:33.500 align:middle line:84% So if I know the state is t1, if I send the message L, 00:32:33.500 --> 00:32:36.380 align:middle line:84% the receiver is going to take decision y of L. 00:32:36.380 --> 00:32:38.580 align:middle line:84% And this is going to be my utility. 00:32:38.580 --> 00:32:40.860 align:middle line:84% If I send the message H, the receiver 00:32:40.860 --> 00:32:42.940 align:middle line:84% is going to take the decision yH and this 00:32:42.940 --> 00:32:44.340 align:middle line:90% is going to be my utility. 00:32:44.340 --> 00:32:46.590 align:middle line:84% So in order for me to be indifferent between these two 00:32:46.590 --> 00:32:49.270 align:middle line:84% things, I need these two formulas to agree. 00:32:49.270 --> 00:32:51.020 align:middle line:84% But let's look at it graphically because I 00:32:51.020 --> 00:32:53.590 align:middle line:90% think that's a bit clearer. 00:32:53.590 --> 00:32:59.830 align:middle line:84% Well, graphically, we have a quadratic loss function. 00:32:59.830 --> 00:33:01.430 align:middle line:90% I'm the sender. 00:33:01.430 --> 00:33:05.415 align:middle line:90% I know the state is t1. 00:33:05.415 --> 00:33:06.790 align:middle line:84% What I'm going to graph is, well, 00:33:06.790 --> 00:33:09.350 align:middle line:84% what does my utility look like as a function 00:33:09.350 --> 00:33:11.310 align:middle line:90% of the decision y that's taken? 00:33:11.310 --> 00:33:13.570 align:middle line:84% Well, my utility is going to look something like this. 00:33:13.570 --> 00:33:16.190 align:middle line:90% 00:33:16.190 --> 00:33:20.590 align:middle line:84% Here it's actually going to be 0. 00:33:20.590 --> 00:33:24.710 align:middle line:84% So if the decision y is exactly t1 plus beta, 00:33:24.710 --> 00:33:26.890 align:middle line:84% well, that's the best thing that I can get. 00:33:26.890 --> 00:33:29.710 align:middle line:84% The decision exactly matches t1 plus beta. 00:33:29.710 --> 00:33:33.070 align:middle line:84% The difference between the decision and t1 plus beta is 0. 00:33:33.070 --> 00:33:34.230 align:middle line:90% My loss is 0. 00:33:34.230 --> 00:33:36.910 align:middle line:84% And that's the best possible thing I could get. 00:33:36.910 --> 00:33:39.490 align:middle line:84% If the decision is higher than t1 plus beta, 00:33:39.490 --> 00:33:42.190 align:middle line:84% well, I'm going to experience a loss that's quadratic. 00:33:42.190 --> 00:33:43.990 align:middle line:84% And if it's lower than t1 plus beta, 00:33:43.990 --> 00:33:47.350 align:middle line:84% I'm also going to experience a loss that's quadratic. 00:33:47.350 --> 00:33:51.750 align:middle line:84% Now, the nice thing about the quadratic is it's symmetric. 00:33:51.750 --> 00:33:52.890 align:middle line:90% So what we need-- 00:33:52.890 --> 00:34:02.550 align:middle line:90% 00:34:02.550 --> 00:34:04.870 align:middle line:84% well, we need the picture to look something like this. 00:34:04.870 --> 00:34:07.710 align:middle line:84% We need the sender to be indifferent between the decision 00:34:07.710 --> 00:34:09.870 align:middle line:90% yL and the decision yH. 00:34:09.870 --> 00:34:12.630 align:middle line:84% And what that means is the utility function 00:34:12.630 --> 00:34:16.550 align:middle line:84% has to take the same value at these two decisions. 00:34:16.550 --> 00:34:18.909 align:middle line:84% It crosses through the same horizontal line. 00:34:18.909 --> 00:34:21.270 align:middle line:84% But because of symmetry, this is only 00:34:21.270 --> 00:34:24.989 align:middle line:84% going to happen if t1 plus beta is exactly halfway 00:34:24.989 --> 00:34:28.909 align:middle line:90% between these two points. 00:34:28.909 --> 00:34:31.469 align:middle line:84% If t1 plus beta was closer to yL, 00:34:31.469 --> 00:34:34.949 align:middle line:84% then the sender would rather induce the decision yL. 00:34:34.949 --> 00:34:37.230 align:middle line:84% And if t1 plus beta was closer to yH, 00:34:37.230 --> 00:34:39.350 align:middle line:84% they'd rather induce the decision yH. 00:34:39.350 --> 00:34:40.858 align:middle line:90% So we could do it algebraically. 00:34:40.858 --> 00:34:42.650 align:middle line:84% But I think it's clearer to do it this way. 00:34:42.650 --> 00:34:49.550 align:middle line:84% So what we see is t1 plus beta must be halfway 00:34:49.550 --> 00:34:51.110 align:middle line:90% between these two things. 00:34:51.110 --> 00:34:58.590 align:middle line:84% So it must be yL plus y of H over 2. 00:34:58.590 --> 00:35:00.570 align:middle line:84% Now let's do just a little bit of algebra. 00:35:00.570 --> 00:35:03.390 align:middle line:90% 00:35:03.390 --> 00:35:06.290 align:middle line:84% Maybe let's multiply both sides by 2 to make it a little easier. 00:35:06.290 --> 00:35:12.350 align:middle line:84% We have 2t1 plus 2 beta is equal to y of L plus y of H. 00:35:12.350 --> 00:35:13.810 align:middle line:90% And let's just plug these in. 00:35:13.810 --> 00:35:17.790 align:middle line:90% So y of L is t1 over 2. 00:35:17.790 --> 00:35:22.410 align:middle line:90% y of H is t1 plus 1 over 2. 00:35:22.410 --> 00:35:25.750 align:middle line:90% 00:35:25.750 --> 00:35:30.390 align:middle line:90% And that's t1 plus 1/2. 00:35:30.390 --> 00:35:34.230 align:middle line:84% I just put the t1 over 2 and t1 over two together. 00:35:34.230 --> 00:35:39.870 align:middle line:84% And now let's do a final simplification. 00:35:39.870 --> 00:35:49.330 align:middle line:90% Get t1 equals 1/2 minus 2 beta. 00:35:49.330 --> 00:35:53.910 align:middle line:90% 00:35:53.910 --> 00:35:58.370 align:middle line:84% So let's see if we can visualize what this looks like. 00:35:58.370 --> 00:36:01.870 align:middle line:90% 00:36:01.870 --> 00:36:04.050 align:middle line:84% If beta is 0, let's start with that case. 00:36:04.050 --> 00:36:05.390 align:middle line:90% That's the easiest case. 00:36:05.390 --> 00:36:10.430 align:middle line:84% If beta is 0, then t1 equals 1/2. 00:36:10.430 --> 00:36:12.430 align:middle line:84% So that means that the sender just says, 00:36:12.430 --> 00:36:16.390 align:middle line:84% is the state above average or below average? 00:36:16.390 --> 00:36:22.810 align:middle line:84% But if beta gets larger, then t1 has to be below 1/2. 00:36:22.810 --> 00:36:25.510 align:middle line:90% 00:36:25.510 --> 00:36:29.130 align:middle line:84% So this interval is smaller than this interval. 00:36:29.130 --> 00:36:35.750 align:middle line:90% 00:36:35.750 --> 00:36:40.790 align:middle line:84% Do you have some intuition for why these intervals are 00:36:40.790 --> 00:36:44.230 align:middle line:90% kind of asymmetric in this way? 00:36:44.230 --> 00:36:47.520 align:middle line:84% Why is it that the interval on the right 00:36:47.520 --> 00:36:51.540 align:middle line:84% is larger than the interval on the left? 00:36:51.540 --> 00:36:56.240 align:middle line:90% 00:36:56.240 --> 00:36:57.360 align:middle line:90% Yeah. 00:36:57.360 --> 00:37:02.200 align:middle line:84% AUDIENCE: There's going to be certain kind of values that are 00:37:02.200 --> 00:37:07.420 align:middle line:84% right below 1/2 where with the bias in mind, our sender, 00:37:07.420 --> 00:37:11.640 align:middle line:84% they'd be more incentivized if the responder played high 00:37:11.640 --> 00:37:13.300 align:middle line:90% or believed that it was high. 00:37:13.300 --> 00:37:16.020 align:middle line:84% And so for that kind of small range of values, 00:37:16.020 --> 00:37:18.818 align:middle line:84% that's why t1 is biased below 1/2. 00:37:18.818 --> 00:37:19.860 align:middle line:90% PROFESSOR: Exactly right. 00:37:19.860 --> 00:37:20.360 align:middle line:90% Yeah. 00:37:20.360 --> 00:37:24.200 align:middle line:84% So if it was symmetric, then at the midpoint, 00:37:24.200 --> 00:37:26.660 align:middle line:84% the sender would always want to say the state is high. 00:37:26.660 --> 00:37:28.160 align:middle line:84% So we have to create these intervals 00:37:28.160 --> 00:37:32.040 align:middle line:84% to be asymmetric to ensure that the sender isn't tempted 00:37:32.040 --> 00:37:32.742 align:middle line:90% to say high. 00:37:32.742 --> 00:37:34.200 align:middle line:84% And effectively what we're doing is 00:37:34.200 --> 00:37:40.040 align:middle line:84% we're shifting the midpoint left in order to counter that bias. 00:37:40.040 --> 00:37:43.080 align:middle line:84% So one way I like to maybe think about it 00:37:43.080 --> 00:37:46.160 align:middle line:84% is, this interval is smaller, which 00:37:46.160 --> 00:37:48.680 align:middle line:84% means the sender is giving more precise information 00:37:48.680 --> 00:37:50.080 align:middle line:90% about the state. 00:37:50.080 --> 00:37:52.758 align:middle line:84% So in order to deter the sender from claiming the state is high, 00:37:52.758 --> 00:37:54.800 align:middle line:84% what we're saying is, well, if you say it's high, 00:37:54.800 --> 00:37:56.820 align:middle line:84% you're not able to convey very much information, 00:37:56.820 --> 00:37:59.237 align:middle line:84% and therefore the decision isn't going to be very precise. 00:37:59.237 --> 00:38:01.200 align:middle line:84% So the sender is trading off the bias 00:38:01.200 --> 00:38:04.140 align:middle line:84% of the decision against the informativeness of the decision. 00:38:04.140 --> 00:38:06.200 align:middle line:84% That's one way of thinking about it. 00:38:06.200 --> 00:38:07.460 align:middle line:90% And then one final thing. 00:38:07.460 --> 00:38:08.560 align:middle line:90% Wait a second. 00:38:08.560 --> 00:38:11.200 align:middle line:90% t1 needs to be between 0 and 1. 00:38:11.200 --> 00:38:12.840 align:middle line:84% So we immediately see that beta has 00:38:12.840 --> 00:38:15.708 align:middle line:84% to be pretty small in order for this to make sense at all. 00:38:15.708 --> 00:38:16.500 align:middle line:90% Is that a question? 00:38:16.500 --> 00:38:17.080 align:middle line:90% Yeah. 00:38:17.080 --> 00:38:18.040 align:middle line:84% AUDIENCE: Can you explain the whole thing 00:38:18.040 --> 00:38:20.000 align:middle line:84% about biasness about the decision, 00:38:20.000 --> 00:38:25.800 align:middle line:84% and what benefit there is from the decision in the lower part 00:38:25.800 --> 00:38:30.015 align:middle line:90% versus being biased in favor? 00:38:30.015 --> 00:38:31.140 align:middle line:90% PROFESSOR: I guess that's-- 00:38:31.140 --> 00:38:33.800 align:middle line:90% 00:38:33.800 --> 00:38:36.840 align:middle line:84% I mean, I think that maybe another way 00:38:36.840 --> 00:38:44.680 align:middle line:84% of saying it is by having this interval larger, 00:38:44.680 --> 00:38:47.200 align:middle line:84% we're basically shifting the midpoint of this interval 00:38:47.200 --> 00:38:48.720 align:middle line:90% farther away from me. 00:38:48.720 --> 00:38:52.820 align:middle line:84% So as the person right at the threshold, 00:38:52.820 --> 00:38:56.240 align:middle line:84% I'm tempted to claim the state as high just because I'm biased 00:38:56.240 --> 00:38:58.640 align:middle line:84% and I want the state to be higher. 00:38:58.640 --> 00:39:01.620 align:middle line:84% But as we enlarge this right interval, 00:39:01.620 --> 00:39:03.935 align:middle line:84% well, the decision that is induced when I say high 00:39:03.935 --> 00:39:06.560 align:middle line:84% starts getting higher and higher because the interval is so big 00:39:06.560 --> 00:39:08.540 align:middle line:84% and the midpoint is moving farther to the right. 00:39:08.540 --> 00:39:10.040 align:middle line:84% And eventually, in the extreme case, 00:39:10.040 --> 00:39:13.840 align:middle line:84% the interval is going to be so big that I actually don't want 00:39:13.840 --> 00:39:15.250 align:middle line:90% to send the higher message. 00:39:15.250 --> 00:39:17.500 align:middle line:84% And the interval is going to be just the right length. 00:39:17.500 --> 00:39:19.180 align:middle line:84% That kind of balances those forces. 00:39:19.180 --> 00:39:21.280 align:middle line:84% So maybe I guess the point that I'm making 00:39:21.280 --> 00:39:24.120 align:middle line:84% is the length of the interval determines 00:39:24.120 --> 00:39:25.680 align:middle line:84% how far the midpoint of the interval 00:39:25.680 --> 00:39:27.460 align:middle line:84% is from the endpoint of the interval. 00:39:27.460 --> 00:39:28.620 align:middle line:84% And if we made that interval really, 00:39:28.620 --> 00:39:30.520 align:middle line:84% really big, then the midpoint of that interval 00:39:30.520 --> 00:39:32.820 align:middle line:84% is going to be really, really far away from its left point. 00:39:32.820 --> 00:39:34.612 align:middle line:84% And in fact, it's going to be too high even 00:39:34.612 --> 00:39:36.200 align:middle line:90% for the biased sender. 00:39:36.200 --> 00:39:38.339 align:middle line:84% So we have to balance these just right. 00:39:38.339 --> 00:39:38.881 align:middle line:90% AUDIENCE: OK. 00:39:38.881 --> 00:39:41.530 align:middle line:90% 00:39:41.530 --> 00:39:43.530 align:middle line:84% PROFESSOR: So I interpret the midpoint 00:39:43.530 --> 00:39:46.190 align:middle line:84% being far away from the point as saying, 00:39:46.190 --> 00:39:48.648 align:middle line:84% because I'm not revealing much information about the state, 00:39:48.648 --> 00:39:50.523 align:middle line:84% the receiver's updated belief about the state 00:39:50.523 --> 00:39:51.870 align:middle line:90% is very far from the true state. 00:39:51.870 --> 00:39:55.690 align:middle line:84% But maybe that's not so important. 00:39:55.690 --> 00:39:58.490 align:middle line:84% So one observation here is that-- 00:39:58.490 --> 00:40:09.170 align:middle line:84% note that this only works if 2 beta is less than 1/2, 00:40:09.170 --> 00:40:11.270 align:middle line:84% which means beta is less than 1/4. 00:40:11.270 --> 00:40:13.883 align:middle line:90% 00:40:13.883 --> 00:40:15.050 align:middle line:90% AUDIENCE: I have a question. 00:40:15.050 --> 00:40:16.327 align:middle line:90% PROFESSOR: Yeah, absolutely. 00:40:16.327 --> 00:40:17.410 align:middle line:90% Yeah, questions are great. 00:40:17.410 --> 00:40:17.910 align:middle line:90% Yeah. 00:40:17.910 --> 00:40:20.290 align:middle line:84% AUDIENCE: If t1 moves to the left, 00:40:20.290 --> 00:40:23.410 align:middle line:84% then the midpoint of the right interval 00:40:23.410 --> 00:40:25.542 align:middle line:84% goes down because it's 1 plus t1. 00:40:25.542 --> 00:40:26.250 align:middle line:90% PROFESSOR: Right. 00:40:26.250 --> 00:40:28.970 align:middle line:84% So what I'm thinking about is the relative position 00:40:28.970 --> 00:40:30.930 align:middle line:90% of the midpoint relative to t1. 00:40:30.930 --> 00:40:33.930 align:middle line:84% So what happens is if I move t1 to the left, t1 moves 00:40:33.930 --> 00:40:36.250 align:middle line:90% to the left by, say, an inch. 00:40:36.250 --> 00:40:39.330 align:middle line:84% The midpoint moves to the left by only half an inch. 00:40:39.330 --> 00:40:43.970 align:middle line:84% So the difference between the endpoint and the midpoint grows. 00:40:43.970 --> 00:40:46.250 align:middle line:84% AUDIENCE: And then at a certain point, 00:40:46.250 --> 00:40:50.890 align:middle line:84% because of the way the receiver is going to respond, 00:40:50.890 --> 00:40:53.618 align:middle line:84% it's a smaller difference from the left side? 00:40:53.618 --> 00:40:54.410 align:middle line:90% PROFESSOR: Exactly. 00:40:54.410 --> 00:40:55.590 align:middle line:84% And that's exactly what we've done here. 00:40:55.590 --> 00:40:58.010 align:middle line:84% We found-- I mean, this formula is exactly telling us 00:40:58.010 --> 00:40:59.458 align:middle line:90% where that point has to be. 00:40:59.458 --> 00:41:01.750 align:middle line:84% So you're right, maybe I should have been more precise. 00:41:01.750 --> 00:41:06.310 align:middle line:84% It's true as I move t1 left that the high decision gets smaller. 00:41:06.310 --> 00:41:08.130 align:middle line:84% But it doesn't get smaller fast enough, 00:41:08.130 --> 00:41:12.110 align:middle line:84% because when you move the endpoint by 1, 00:41:12.110 --> 00:41:14.465 align:middle line:84% you only move the average by 1/2. 00:41:14.465 --> 00:41:16.590 align:middle line:84% And that's exactly-- I mean, you can see that here. 00:41:16.590 --> 00:41:23.210 align:middle line:84% So I guess what I'm saying is y of H is t1 plus 1 over 2. 00:41:23.210 --> 00:41:24.650 align:middle line:90% If we shrink t1-- 00:41:24.650 --> 00:41:29.650 align:middle line:84% if we reduce t1 by a little bit, then we reduce y of H 00:41:29.650 --> 00:41:30.870 align:middle line:90% by only half of that. 00:41:30.870 --> 00:41:34.250 align:middle line:84% But t1 gets reduced by the full amount. 00:41:34.250 --> 00:41:36.558 align:middle line:84% AUDIENCE: And then so is the idea that the left side is 00:41:36.558 --> 00:41:38.850 align:middle line:84% smaller than the right side also just based on the idea 00:41:38.850 --> 00:41:40.610 align:middle line:90% that there's a beta? 00:41:40.610 --> 00:41:41.730 align:middle line:90% PROFESSOR: Exactly, yeah. 00:41:41.730 --> 00:41:43.770 align:middle line:84% I mean, I guess very simply, all it's 00:41:43.770 --> 00:41:47.810 align:middle line:84% saying is we're choosing the intervals 00:41:47.810 --> 00:41:50.853 align:middle line:84% so that the midpoint of the left interval and the midpoint 00:41:50.853 --> 00:41:53.270 align:middle line:84% of the right interval-- if we look at those two midpoints, 00:41:53.270 --> 00:41:57.570 align:middle line:84% they average to the cutoff point plus beta. 00:41:57.570 --> 00:42:01.430 align:middle line:84% And that's exactly what we wrote here. 00:42:01.430 --> 00:42:04.330 align:middle line:84% I mean, this is the key condition. 00:42:04.330 --> 00:42:06.250 align:middle line:90% Yes. 00:42:06.250 --> 00:42:09.110 align:middle line:84% AUDIENCE: Is the responder aware of the bias? 00:42:09.110 --> 00:42:13.490 align:middle line:84% And if so, are they incentivized change their behavior, 00:42:13.490 --> 00:42:16.050 align:middle line:84% given that the sender is going to be lying part of the time 00:42:16.050 --> 00:42:16.990 align:middle line:90% when they say high? 00:42:16.990 --> 00:42:17.670 align:middle line:90% PROFESSOR: OK, great question. 00:42:17.670 --> 00:42:19.462 align:middle line:84% So first, this goes to an earlier question. 00:42:19.462 --> 00:42:21.770 align:middle line:90% They do know the bias exactly. 00:42:21.770 --> 00:42:23.610 align:middle line:90% And this is an equilibrium. 00:42:23.610 --> 00:42:27.565 align:middle line:84% So they recognize what the sender is doing. 00:42:27.565 --> 00:42:29.190 align:middle line:84% But I wouldn't say the sender is lying. 00:42:29.190 --> 00:42:30.648 align:middle line:84% I mean, what the sender is doing is 00:42:30.648 --> 00:42:33.140 align:middle line:84% the sender is telling them whether or not the state is 00:42:33.140 --> 00:42:35.300 align:middle line:90% below t1 or above t1. 00:42:35.300 --> 00:42:38.780 align:middle line:84% And given that message, the receiver is behaving optimally. 00:42:38.780 --> 00:42:41.860 align:middle line:84% Because if they get the message L, 00:42:41.860 --> 00:42:43.580 align:middle line:84% whatever the bias of the sender, they 00:42:43.580 --> 00:42:47.460 align:middle line:84% believe that the sender's message means 00:42:47.460 --> 00:42:50.080 align:middle line:84% the state is uniform over this interval. 00:42:50.080 --> 00:42:52.205 align:middle line:84% And therefore, their optimal decision is t1 over 2. 00:42:52.205 --> 00:42:54.497 align:middle line:84% So I mean, you're thinking exactly about the right way. 00:42:54.497 --> 00:42:56.500 align:middle line:84% This is what the equilibrium is capturing. 00:42:56.500 --> 00:42:58.840 align:middle line:84% But the receiver doesn't have any incentive to deviate. 00:42:58.840 --> 00:42:59.340 align:middle line:90% Yeah. 00:42:59.340 --> 00:42:59.960 align:middle line:90% AUDIENCE: That makes sense. 00:42:59.960 --> 00:43:00.220 align:middle line:90% PROFESSOR: Yeah. 00:43:00.220 --> 00:43:01.240 align:middle line:90% Yes, other questions? 00:43:01.240 --> 00:43:04.272 align:middle line:84% AUDIENCE: So does the receiver know t1? 00:43:04.272 --> 00:43:05.480 align:middle line:90% PROFESSOR: Oh, good question. 00:43:05.480 --> 00:43:06.300 align:middle line:90% Yes. 00:43:06.300 --> 00:43:09.560 align:middle line:84% So t1 is a parameter of the equilibrium. 00:43:09.560 --> 00:43:10.880 align:middle line:90% So t1 is going to be known. 00:43:10.880 --> 00:43:12.180 align:middle line:90% And the reason is we-- 00:43:12.180 --> 00:43:14.480 align:middle line:84% or this gets into a subtle question of what's known. 00:43:14.480 --> 00:43:19.460 align:middle line:84% But in equilibrium, when the equilibrium is played, 00:43:19.460 --> 00:43:21.460 align:middle line:84% the players have correct beliefs about what 00:43:21.460 --> 00:43:22.740 align:middle line:90% the other players are doing. 00:43:22.740 --> 00:43:25.420 align:middle line:84% So the receiver believes that the sender 00:43:25.420 --> 00:43:27.380 align:middle line:84% is following this equilibrium strategy 00:43:27.380 --> 00:43:29.120 align:middle line:90% for this particular value of t1. 00:43:29.120 --> 00:43:31.980 align:middle line:84% So the receiver, in a sense, knows t1. 00:43:31.980 --> 00:43:33.060 align:middle line:90% Yeah. 00:43:33.060 --> 00:43:35.700 align:middle line:90% Yeah. 00:43:35.700 --> 00:43:37.358 align:middle line:84% So I guess we have to be careful. 00:43:37.358 --> 00:43:38.900 align:middle line:84% I was writing down all these formulas 00:43:38.900 --> 00:43:40.840 align:middle line:84% to try to derive the value of t1. 00:43:40.840 --> 00:43:43.620 align:middle line:84% But once I've solved for t1, formally, the equilibrium 00:43:43.620 --> 00:43:46.055 align:middle line:90% is this exact value of t1. 00:43:46.055 --> 00:43:47.680 align:middle line:84% It's not an abstract parameter anymore. 00:43:47.680 --> 00:43:49.000 align:middle line:90% It has to be 1/2 minus 2 beta. 00:43:49.000 --> 00:43:50.708 align:middle line:84% Otherwise, the equilibrium wouldn't work. 00:43:50.708 --> 00:43:51.580 align:middle line:90% Yeah. 00:43:51.580 --> 00:43:52.260 align:middle line:90% Great. 00:43:52.260 --> 00:43:55.240 align:middle line:84% So we said this only works if beta is less than 1/4. 00:43:55.240 --> 00:43:58.420 align:middle line:84% And indeed, you can show that if beta is more than 1/4, 00:43:58.420 --> 00:44:02.860 align:middle line:84% the only equilibrium is the bad no-communication equilibrium 00:44:02.860 --> 00:44:03.900 align:middle line:90% over here. 00:44:03.900 --> 00:44:06.620 align:middle line:84% So let's make this point over here. 00:44:06.620 --> 00:44:14.140 align:middle line:84% If beta is greater than or equal to 1/4, 00:44:14.140 --> 00:44:22.680 align:middle line:84% no communication is the only equilibrium. 00:44:22.680 --> 00:44:28.780 align:middle line:90% 00:44:28.780 --> 00:44:32.007 align:middle line:84% Now, when I say only, I mean, the message m* that we use could 00:44:32.007 --> 00:44:34.340 align:middle line:84% be a different message, but we're never going to be able 00:44:34.340 --> 00:44:35.480 align:middle line:90% to convey information. 00:44:35.480 --> 00:44:37.020 align:middle line:84% And the point here is the bias is 00:44:37.020 --> 00:44:40.972 align:middle line:84% so strong that it's impossible to credibly convey information. 00:44:40.972 --> 00:44:43.180 align:middle line:84% And you can think of this in the extreme case as kind 00:44:43.180 --> 00:44:46.120 align:middle line:84% of like the story of the worker who goes to the firm and says, 00:44:46.120 --> 00:44:47.500 align:middle line:90% I'm a good worker. 00:44:47.500 --> 00:44:50.140 align:middle line:84% If the advisor has such a bias relative to the receiver 00:44:50.140 --> 00:44:51.973 align:middle line:84% and they always want higher decisions, 00:44:51.973 --> 00:44:54.140 align:middle line:84% the receiver just can't listen to anything they say. 00:44:54.140 --> 00:44:55.940 align:middle line:84% And therefore, the only equilibrium 00:44:55.940 --> 00:44:58.300 align:middle line:90% is going to be no communication. 00:44:58.300 --> 00:45:02.660 align:middle line:84% If beta is less than 1/4, well, we can have some coarse 00:45:02.660 --> 00:45:06.020 align:middle line:84% communication where we reveal what the state is above or below 00:45:06.020 --> 00:45:07.540 align:middle line:90% this threshold. 00:45:07.540 --> 00:45:10.860 align:middle line:84% But we might ask, well, can we convey even more information? 00:45:10.860 --> 00:45:13.802 align:middle line:84% Could we have more precise information conveyed? 00:45:13.802 --> 00:45:16.260 align:middle line:84% Do you have ideas how we might get beyond this case of just 00:45:16.260 --> 00:45:19.860 align:middle line:84% above or below a threshold and actually convey-- 00:45:19.860 --> 00:45:24.600 align:middle line:84% how might we convey even more information in equilibrium? 00:45:24.600 --> 00:45:25.100 align:middle line:90% Yeah. 00:45:25.100 --> 00:45:25.695 align:middle line:90% Yeah. 00:45:25.695 --> 00:45:26.820 align:middle line:90% AUDIENCE: I had a question. 00:45:26.820 --> 00:45:29.830 align:middle line:84% So I was just wondering, for the utility functions, because we 00:45:29.830 --> 00:45:33.650 align:middle line:84% say that t is between 0 and 1 and then we also have this bias, 00:45:33.650 --> 00:45:37.302 align:middle line:84% is the sender always incentivized to-- 00:45:37.302 --> 00:45:41.490 align:middle line:84% like if t plus beta is ever greater than 1, 00:45:41.490 --> 00:45:45.923 align:middle line:84% wouldn't that mean their utility is always negative? 00:45:45.923 --> 00:45:46.590 align:middle line:90% PROFESSOR: Yeah. 00:45:46.590 --> 00:45:48.610 align:middle line:84% So one thing is I wouldn't-- yeah, 00:45:48.610 --> 00:45:51.750 align:middle line:90% this is maybe a good point. 00:45:51.750 --> 00:45:54.350 align:middle line:84% You might worry-- the way we've written the game, 00:45:54.350 --> 00:45:56.867 align:middle line:84% the utilities are actually always negative. 00:45:56.867 --> 00:45:59.450 align:middle line:84% Remember, with utilities, we can always add a constant to them 00:45:59.450 --> 00:46:00.950 align:middle line:84% and it doesn't really change things. 00:46:00.950 --> 00:46:02.910 align:middle line:84% So you might be thinking, oh they'd rather not 00:46:02.910 --> 00:46:03.785 align:middle line:90% play the game at all. 00:46:03.785 --> 00:46:05.950 align:middle line:84% They'd rather just not participate. 00:46:05.950 --> 00:46:09.070 align:middle line:84% You could imagine the utility being 100 00:46:09.070 --> 00:46:11.570 align:middle line:90% minus this and 100 minus this. 00:46:11.570 --> 00:46:14.470 align:middle line:84% And then the utilities would always be positive. 00:46:14.470 --> 00:46:18.083 align:middle line:84% So you can always-- the sender is only 00:46:18.083 --> 00:46:19.750 align:middle line:84% looking at differences in utilities, not 00:46:19.750 --> 00:46:20.765 align:middle line:90% the absolute levels. 00:46:20.765 --> 00:46:22.390 align:middle line:84% So we could always shift up the utility 00:46:22.390 --> 00:46:23.810 align:middle line:84% and it's not going to make any difference. 00:46:23.810 --> 00:46:24.310 align:middle line:90% Yeah. 00:46:24.310 --> 00:46:26.550 align:middle line:84% So if you wanted, we could call this 100 00:46:26.550 --> 00:46:31.870 align:middle line:84% minus this and 100 minus this, and it wouldn't change anything. 00:46:31.870 --> 00:46:33.590 align:middle line:90% Hopefully that answers that. 00:46:33.590 --> 00:46:35.050 align:middle line:90% So any ideas? 00:46:35.050 --> 00:46:37.065 align:middle line:84% We have this equilibrium where I just 00:46:37.065 --> 00:46:39.690 align:middle line:84% say the state is either below a threshold or above a threshold. 00:46:39.690 --> 00:46:41.510 align:middle line:84% How could we maybe convey more precise information 00:46:41.510 --> 00:46:42.177 align:middle line:90% about the state? 00:46:42.177 --> 00:46:42.677 align:middle line:90% Yeah. 00:46:42.677 --> 00:46:44.385 align:middle line:84% AUDIENCE: Still along the threshold idea, 00:46:44.385 --> 00:46:46.350 align:middle line:84% but is it possible to have low, medium, high? 00:46:46.350 --> 00:46:47.330 align:middle line:84% PROFESSOR: I think that's a natural thing. 00:46:47.330 --> 00:46:48.650 align:middle line:84% Why don't you just say it's low, medium, high? 00:46:48.650 --> 00:46:51.230 align:middle line:84% Or maybe we could go a step further and split it up and say 00:46:51.230 --> 00:46:53.470 align:middle line:84% it's either really low or a little bit low or a little bit 00:46:53.470 --> 00:46:54.303 align:middle line:90% high or really high. 00:46:54.303 --> 00:46:55.810 align:middle line:84% And it's not clear where this stops. 00:46:55.810 --> 00:46:56.650 align:middle line:90% We could do a lot of this. 00:46:56.650 --> 00:46:57.950 align:middle line:90% So that's going to be the idea. 00:46:57.950 --> 00:47:01.030 align:middle line:84% And these are called partition equilibria. 00:47:01.030 --> 00:47:02.890 align:middle line:90% Maybe I'll erase this. 00:47:02.890 --> 00:47:17.167 align:middle line:90% 00:47:17.167 --> 00:47:18.750 align:middle line:84% So more generally, we're going to look 00:47:18.750 --> 00:47:25.130 align:middle line:84% at what are called partitional equilibria. 00:47:25.130 --> 00:47:29.830 align:middle line:90% 00:47:29.830 --> 00:47:32.470 align:middle line:84% This was a special case of a partitional equilibrium 00:47:32.470 --> 00:47:34.145 align:middle line:90% where we just had two cells. 00:47:34.145 --> 00:47:36.270 align:middle line:84% We had two messages that said either below or above 00:47:36.270 --> 00:47:37.170 align:middle line:90% a threshold. 00:47:37.170 --> 00:47:41.670 align:middle line:84% But more generally, here's our interval 0, 1. 00:47:41.670 --> 00:47:43.170 align:middle line:84% Why don't we do something like this? 00:47:43.170 --> 00:47:47.150 align:middle line:90% We could say t1, t2. 00:47:47.150 --> 00:47:48.830 align:middle line:90% Maybe it keeps going. 00:47:48.830 --> 00:47:51.410 align:middle line:90% tk minus 1. 00:47:51.410 --> 00:47:54.950 align:middle line:90% 00:47:54.950 --> 00:47:58.190 align:middle line:84% And we could say either the state is in this interval 00:47:58.190 --> 00:48:01.310 align:middle line:84% or it's in this interval or it's in this interval. 00:48:01.310 --> 00:48:04.330 align:middle line:84% It keeps going-- or this interval or this interval. 00:48:04.330 --> 00:48:06.830 align:middle line:84% So basically our message by the sender 00:48:06.830 --> 00:48:10.110 align:middle line:84% is going to reveal which of these intervals the state is in. 00:48:10.110 --> 00:48:12.670 align:middle line:84% And as the number of intervals gets really big 00:48:12.670 --> 00:48:14.510 align:middle line:84% and each of the intervals gets really small, 00:48:14.510 --> 00:48:16.593 align:middle line:84% we're actually converging to more and more precise 00:48:16.593 --> 00:48:18.230 align:middle line:90% communication. 00:48:18.230 --> 00:48:22.150 align:middle line:84% So let's understand what's going on here. 00:48:22.150 --> 00:48:26.910 align:middle line:84% It's kind of convenient to call this t0 and call this tk. 00:48:26.910 --> 00:48:30.590 align:middle line:84% So formally, I'm going to look at a partitional equilibrium 00:48:30.590 --> 00:48:31.570 align:middle line:90% with k cells. 00:48:31.570 --> 00:48:34.710 align:middle line:84% So this is a k-cell partitional equilibrium 00:48:34.710 --> 00:48:37.450 align:middle line:84% where k is greater than or equal to 2. 00:48:37.450 --> 00:48:41.990 align:middle line:90% 00:48:41.990 --> 00:48:45.110 align:middle line:84% So we have the first cell, the second cell, the third cell, 00:48:45.110 --> 00:48:47.190 align:middle line:90% all the way up to the k-th cell. 00:48:47.190 --> 00:48:49.150 align:middle line:84% And that means we're going to have k messages. 00:48:49.150 --> 00:48:50.050 align:middle line:90% And let's label them. 00:48:50.050 --> 00:48:53.710 align:middle line:84% Let's just call this message 1, message 2, 00:48:53.710 --> 00:48:55.335 align:middle line:90% all the way to message k. 00:48:55.335 --> 00:48:56.710 align:middle line:84% So what the sender is going to do 00:48:56.710 --> 00:48:59.335 align:middle line:84% is they're going to look at the state and they're going to say, 00:48:59.335 --> 00:49:00.990 align:middle line:84% well, if the state is between 0 and t1, 00:49:00.990 --> 00:49:02.870 align:middle line:90% I'm going to send message m1. 00:49:02.870 --> 00:49:06.390 align:middle line:84% If it's between t1 and t2 I'm going to send message m2. 00:49:06.390 --> 00:49:07.950 align:middle line:84% If it's between t2 and t3, I'm going 00:49:07.950 --> 00:49:10.250 align:middle line:84% to send message m3, all the way up to mk. 00:49:10.250 --> 00:49:15.150 align:middle line:90% 00:49:15.150 --> 00:49:17.790 align:middle line:90% And maybe I'll call this y1. 00:49:17.790 --> 00:49:19.800 align:middle line:84% If the receiver sees message m1, they're 00:49:19.800 --> 00:49:21.460 align:middle line:90% going to take decision y1. 00:49:21.460 --> 00:49:26.720 align:middle line:84% If they see m2, they're going to take y2, and so on. 00:49:26.720 --> 00:49:31.520 align:middle line:84% So formally, what I mean is yk is equal to y of m of k. 00:49:31.520 --> 00:49:35.560 align:middle line:84% But this is just cleaner notation. 00:49:35.560 --> 00:49:39.400 align:middle line:84% And again, we're going to try to solve for these parameter values 00:49:39.400 --> 00:49:42.560 align:middle line:84% t1 up to tk minus 1, we're going to solve for these thresholds 00:49:42.560 --> 00:49:45.523 align:middle line:84% to see if we can make this an equilibrium. 00:49:45.523 --> 00:49:46.940 align:middle line:84% So what's going to have to happen? 00:49:46.940 --> 00:49:49.660 align:middle line:84% Well first, let's use receiver optimality. 00:49:49.660 --> 00:49:54.800 align:middle line:90% 00:49:54.800 --> 00:49:57.560 align:middle line:84% Receiver optimality says, well, when the receiver gets 00:49:57.560 --> 00:50:00.480 align:middle line:84% this message m1, they know the state is somewhere 00:50:00.480 --> 00:50:02.360 align:middle line:90% between 0 and t1. 00:50:02.360 --> 00:50:04.520 align:middle line:84% And they believe, in fact, the state is uniformly 00:50:04.520 --> 00:50:06.320 align:middle line:90% distributed between 0 and t1. 00:50:06.320 --> 00:50:09.160 align:middle line:84% So their updated belief about the expected state 00:50:09.160 --> 00:50:11.200 align:middle line:84% is that it's the midpoint of this. 00:50:11.200 --> 00:50:13.800 align:middle line:84% So in each case, the receiver's decision 00:50:13.800 --> 00:50:17.200 align:middle line:84% must be the midpoint of the cell that they're 00:50:17.200 --> 00:50:18.260 align:middle line:90% told the state is in. 00:50:18.260 --> 00:50:22.700 align:middle line:84% So in math language, what we would say is yk or yj-- 00:50:22.700 --> 00:50:25.680 align:middle line:90% 00:50:25.680 --> 00:50:27.760 align:middle line:84% let's make sure we get this right-- is 00:50:27.760 --> 00:50:37.720 align:middle line:84% going to be equal to tj minus 1 plus tj over 2. 00:50:37.720 --> 00:50:40.360 align:middle line:84% Because when they get message mj, 00:50:40.360 --> 00:50:46.400 align:middle line:84% they learn that the state is between tj minus 1 and tj. 00:50:46.400 --> 00:50:49.040 align:middle line:84% So maybe we'll say this here, because they 00:50:49.040 --> 00:50:58.780 align:middle line:84% learn that the state t is between tj minus 1 t and tj. 00:50:58.780 --> 00:51:02.480 align:middle line:90% 00:51:02.480 --> 00:51:04.280 align:middle line:90% OK? 00:51:04.280 --> 00:51:08.280 align:middle line:84% So that gives us a formula for each of these decisions. 00:51:08.280 --> 00:51:13.500 align:middle line:84% This is for j equals 1 all the way up to k. 00:51:13.500 --> 00:51:16.720 align:middle line:90% 00:51:16.720 --> 00:51:25.320 align:middle line:84% And then what does sender optimality tell us? 00:51:25.320 --> 00:51:27.200 align:middle line:84% Well, it's going to be the same idea. 00:51:27.200 --> 00:51:30.080 align:middle line:84% At each of these split points, we 00:51:30.080 --> 00:51:33.080 align:middle line:84% want the sender to be indifferent between sending 00:51:33.080 --> 00:51:35.120 align:middle line:84% the lower message and the higher message. 00:51:35.120 --> 00:51:37.040 align:middle line:84% So when the state is t1, the sender 00:51:37.040 --> 00:51:39.400 align:middle line:84% should be indifferent between m1 and m2. 00:51:39.400 --> 00:51:41.720 align:middle line:84% When the state is t2, the sender should be 00:51:41.720 --> 00:51:43.480 align:middle line:90% indifferent between m2 and m3. 00:51:43.480 --> 00:51:45.720 align:middle line:84% All the way up when the state is tk minus 1, 00:51:45.720 --> 00:51:48.840 align:middle line:84% they should be indifferent between mk minus 1 and mk. 00:51:48.840 --> 00:51:55.800 align:middle line:84% So what that means is that tj plus beta should 00:51:55.800 --> 00:52:08.380 align:middle line:84% be the midpoint of yj and yj plus 1 over 2. 00:52:08.380 --> 00:52:12.160 align:middle line:84% Let's make sure we got kind of the fence post problem right. 00:52:12.160 --> 00:52:13.810 align:middle line:84% When the state is t1, we should be 00:52:13.810 --> 00:52:18.670 align:middle line:84% indifferent between m1 and m2, which is yj and yj plus 1. 00:52:18.670 --> 00:52:21.650 align:middle line:90% So that makes sense. 00:52:21.650 --> 00:52:24.130 align:middle line:90% And if we did a little more-- 00:52:24.130 --> 00:52:32.850 align:middle line:84% and this must be true for j equals 1 to k minus 1. 00:52:32.850 --> 00:52:37.170 align:middle line:84% Notice we have k conditions for the receiver 00:52:37.170 --> 00:52:39.170 align:middle line:90% because we have k cells. 00:52:39.170 --> 00:52:41.730 align:middle line:84% We have k minus 1 conditions for the sender 00:52:41.730 --> 00:52:44.275 align:middle line:84% because we have k minus 1 split points. 00:52:44.275 --> 00:52:46.150 align:middle line:84% This is the classic kind of fence post issue, 00:52:46.150 --> 00:52:52.570 align:middle line:84% that k minus 1 cuts gives us k pieces. 00:52:52.570 --> 00:52:54.290 align:middle line:90% So we could do a bit of algebra. 00:52:54.290 --> 00:52:57.670 align:middle line:84% I don't think we really need to go through all the algebra. 00:52:57.670 --> 00:53:00.990 align:middle line:84% Let me just kind of describe what you're going to find. 00:53:00.990 --> 00:53:05.410 align:middle line:90% 00:53:05.410 --> 00:53:06.970 align:middle line:84% So maybe I'll just say mathematically 00:53:06.970 --> 00:53:10.730 align:middle line:90% what this is called is-- 00:53:10.730 --> 00:53:25.810 align:middle line:84% OK, we get kind of a system of equations for t1 00:53:25.810 --> 00:53:29.410 align:middle line:90% through tk minus 1. 00:53:29.410 --> 00:53:31.930 align:middle line:84% Remember, we already did the case of k equals 2. 00:53:31.930 --> 00:53:35.170 align:middle line:84% And in the case of k equals 2, we just solve for t1. 00:53:35.170 --> 00:53:39.390 align:middle line:84% But more generally, we have to solve for k minus 1 values. 00:53:39.390 --> 00:53:44.930 align:middle line:90% 00:53:44.930 --> 00:53:50.310 align:middle line:84% But I want to just maybe convey the idea intuitively. 00:53:50.310 --> 00:53:53.690 align:middle line:90% 00:53:53.690 --> 00:53:55.190 align:middle line:84% Here's how I like to think about it. 00:53:55.190 --> 00:53:57.890 align:middle line:84% We could do the algebra, but let's just think about it. 00:53:57.890 --> 00:54:00.650 align:middle line:84% Let's look at adjacent cells, OK? 00:54:00.650 --> 00:54:03.250 align:middle line:84% And let's look at what happens when the state is here. 00:54:03.250 --> 00:54:04.330 align:middle line:90% OK? 00:54:04.330 --> 00:54:08.930 align:middle line:84% Well, if these adjacent cells had the same length, 00:54:08.930 --> 00:54:10.410 align:middle line:90% would this-- 00:54:10.410 --> 00:54:12.530 align:middle line:90% let's call this tj. 00:54:12.530 --> 00:54:14.770 align:middle line:84% Would the sender be indifferent between getting 00:54:14.770 --> 00:54:15.910 align:middle line:90% this point and this point? 00:54:15.910 --> 00:54:16.910 align:middle line:90% Which would they prefer? 00:54:16.910 --> 00:54:22.530 align:middle line:90% 00:54:22.530 --> 00:54:27.370 align:middle line:84% Well, these two decisions are equidistant from tj. 00:54:27.370 --> 00:54:28.830 align:middle line:90% But the sender is biased. 00:54:28.830 --> 00:54:33.710 align:middle line:84% The sender prefers the decision that's closest to tj plus beta. 00:54:33.710 --> 00:54:37.730 align:middle line:84% So the sender would always prefer the higher decision. 00:54:37.730 --> 00:54:42.410 align:middle line:84% So what we need is we need tj plus beta 00:54:42.410 --> 00:54:47.730 align:middle line:84% to be equal to the average of these two points. 00:54:47.730 --> 00:54:51.570 align:middle line:84% So what we need to do is we need the average of these two points 00:54:51.570 --> 00:55:03.302 align:middle line:90% to shift right by beta. 00:55:03.302 --> 00:55:06.010 align:middle line:84% Because if we want the sender to be indifferent between these two 00:55:06.010 --> 00:55:08.060 align:middle line:84% points, they shouldn't average to tj. 00:55:08.060 --> 00:55:12.500 align:middle line:84% They should average to tj plus beta. 00:55:12.500 --> 00:55:15.920 align:middle line:84% So what we want to do is we want to move out this interval, 00:55:15.920 --> 00:55:18.940 align:middle line:84% make this interval longer, until the average has shifted right 00:55:18.940 --> 00:55:20.340 align:middle line:90% by beta. 00:55:20.340 --> 00:55:24.900 align:middle line:84% But we're only moving this point. 00:55:24.900 --> 00:55:27.380 align:middle line:90% This point is staying fixed. 00:55:27.380 --> 00:55:29.420 align:middle line:84% So if we want the average to go up by beta, 00:55:29.420 --> 00:55:31.800 align:middle line:84% we have to increase this point by 2 beta. 00:55:31.800 --> 00:55:39.590 align:middle line:90% 00:55:39.590 --> 00:55:41.885 align:middle line:84% AUDIENCE: Why are we keeping the left point? 00:55:41.885 --> 00:55:42.760 align:middle line:90% PROFESSOR: Say again. 00:55:42.760 --> 00:55:45.427 align:middle line:84% AUDIENCE: Why is the left point fixed? 00:55:45.427 --> 00:55:47.760 align:middle line:84% PROFESSOR: We're basically solving the equation forward. 00:55:47.760 --> 00:55:51.280 align:middle line:84% We're saying, let's suppose the left interval is like this. 00:55:51.280 --> 00:55:52.840 align:middle line:84% How big should the next interval be? 00:55:52.840 --> 00:55:55.820 align:middle line:84% AUDIENCE: OK, because [INAUDIBLE]. 00:55:55.820 --> 00:55:59.860 align:middle line:84% PROFESSOR: So we have to increase this point by 2 beta 00:55:59.860 --> 00:56:01.940 align:middle line:90% to increase the average by beta. 00:56:01.940 --> 00:56:04.340 align:middle line:84% But how can we increase the midpoint of this interval 00:56:04.340 --> 00:56:04.977 align:middle line:90% by 2 beta? 00:56:04.977 --> 00:56:07.060 align:middle line:84% How much do we have to increase the right endpoint 00:56:07.060 --> 00:56:10.900 align:middle line:84% to get the midpoint to go up by 2 beta? 00:56:10.900 --> 00:56:11.560 align:middle line:90% 4 beta. 00:56:11.560 --> 00:56:16.620 align:middle line:84% So that means this has to go up by 4 beta. 00:56:16.620 --> 00:56:18.620 align:middle line:84% We could do the algebra, but I just kind of like 00:56:18.620 --> 00:56:19.500 align:middle line:90% to think it through. 00:56:19.500 --> 00:56:24.027 align:middle line:84% So what we indeed have is the general formula is this. 00:56:24.027 --> 00:56:25.860 align:middle line:84% And you can solve it, but I think it's nicer 00:56:25.860 --> 00:56:28.740 align:middle line:90% to think of it like this. 00:56:28.740 --> 00:56:29.800 align:middle line:90% Let's write it this way. 00:56:29.800 --> 00:56:40.620 align:middle line:90% 00:56:40.620 --> 00:56:42.460 align:middle line:90% So we're comparing the lengths-- 00:56:42.460 --> 00:56:45.540 align:middle line:90% ah, no, no, that's wrong. 00:56:45.540 --> 00:56:46.840 align:middle line:90% Sorry, let me rewrite this. 00:56:46.840 --> 00:56:59.100 align:middle line:90% 00:56:59.100 --> 00:57:03.020 align:middle line:84% So we get this formula, which compares the lengths 00:57:03.020 --> 00:57:05.180 align:middle line:90% of the consecutive intervals. 00:57:05.180 --> 00:57:12.120 align:middle line:84% So in our picture here, this is an interval length. 00:57:12.120 --> 00:57:16.780 align:middle line:90% 00:57:16.780 --> 00:57:18.760 align:middle line:84% This is the next interval length. 00:57:18.760 --> 00:57:23.060 align:middle line:90% 00:57:23.060 --> 00:57:25.280 align:middle line:84% And what we're saying is, as we move to the right, 00:57:25.280 --> 00:57:26.900 align:middle line:84% the next interval moving to the right 00:57:26.900 --> 00:57:30.288 align:middle line:84% must be as long as the interval to the left plus 2 beta 00:57:30.288 --> 00:57:32.080 align:middle line:84% through the argument that we did over here. 00:57:32.080 --> 00:57:33.033 align:middle line:90% Yeah. 00:57:33.033 --> 00:57:34.700 align:middle line:84% AUDIENCE: When you originally first drew 00:57:34.700 --> 00:57:38.900 align:middle line:84% the picture, that was saying like tj plus 1 minus tj was 00:57:38.900 --> 00:57:41.440 align:middle line:84% the same length as tj minus [INAUDIBLE] tj minus 1 00:57:41.440 --> 00:57:42.220 align:middle line:90% [INAUDIBLE]. 00:57:42.220 --> 00:57:43.532 align:middle line:90% And then basically [INAUDIBLE]. 00:57:43.532 --> 00:57:44.240 align:middle line:90% PROFESSOR: Right. 00:57:44.240 --> 00:57:47.500 align:middle line:84% So formally in this picture, I fixed tj minus 1. 00:57:47.500 --> 00:57:48.820 align:middle line:90% I fixed tj. 00:57:48.820 --> 00:57:51.793 align:middle line:84% And I was saying, how big must tj plus 1 be? 00:57:51.793 --> 00:57:53.960 align:middle line:84% Maybe it would have been easier just to the algebra. 00:57:53.960 --> 00:57:55.280 align:middle line:90% But I find this clearer. 00:57:55.280 --> 00:57:56.740 align:middle line:90% Yeah. 00:57:56.740 --> 00:57:57.860 align:middle line:90% OK. 00:57:57.860 --> 00:57:59.820 align:middle line:90% And actually, we can check. 00:57:59.820 --> 00:58:02.470 align:middle line:84% Let's go back to our example over here. 00:58:02.470 --> 00:58:05.210 align:middle line:84% When t1 was equal to 1/2 minus 2 beta, 00:58:05.210 --> 00:58:08.070 align:middle line:84% what were the lengths of these intervals? 00:58:08.070 --> 00:58:13.790 align:middle line:84% This interval was 1/2 minus 2 beta. 00:58:13.790 --> 00:58:15.090 align:middle line:90% What was the other interval? 00:58:15.090 --> 00:58:18.070 align:middle line:90% 00:58:18.070 --> 00:58:21.030 align:middle line:90% Well, it's 1 minus that, right? 00:58:21.030 --> 00:58:30.870 align:middle line:84% 1 minus 1/2 minus 2 beta, which equals 1/2 plus 2 beta. 00:58:30.870 --> 00:58:33.630 align:middle line:84% So indeed, that formula held in this special case. 00:58:33.630 --> 00:58:35.990 align:middle line:84% What that midpoint was doing is it was exactly 00:58:35.990 --> 00:58:38.270 align:middle line:84% ensuring that the length of this interval 00:58:38.270 --> 00:58:40.930 align:middle line:84% was exactly the length of this interval plus 4 beta. 00:58:40.930 --> 00:58:43.790 align:middle line:90% 00:58:43.790 --> 00:58:45.790 align:middle line:84% And where did beta over 4 come in? 00:58:45.790 --> 00:58:48.610 align:middle line:84% Well, 4 beta has to be less than 1. 00:58:48.610 --> 00:58:50.290 align:middle line:90% And that's where it comes from. 00:58:50.290 --> 00:58:53.030 align:middle line:90% 00:58:53.030 --> 00:58:57.390 align:middle line:84% So now we can see what the structure of the equilibria 00:58:57.390 --> 00:59:02.590 align:middle line:84% have to look like if we you want to draw it more carefully. 00:59:02.590 --> 00:59:04.270 align:middle line:84% As we move to the right, the intervals 00:59:04.270 --> 00:59:06.650 align:middle line:84% are going to keep getting bigger by a fixed amount. 00:59:06.650 --> 00:59:10.950 align:middle line:84% So it's going to be something like this. 00:59:10.950 --> 00:59:13.190 align:middle line:84% The first interval has some length delta. 00:59:13.190 --> 00:59:15.990 align:middle line:84% The next interval has some length delta plus 4 beta. 00:59:15.990 --> 00:59:18.630 align:middle line:84% And the next interval has some length delta plus 8 beta. 00:59:18.630 --> 00:59:20.590 align:middle line:90% And we can keep going. 00:59:20.590 --> 00:59:24.430 align:middle line:84% And then we have to make sure that everything works out 00:59:24.430 --> 00:59:26.830 align:middle line:84% at the right interval and everything comes together. 00:59:26.830 --> 00:59:29.270 align:middle line:90% So what's the takeaway? 00:59:29.270 --> 00:59:32.390 align:middle line:90% What we actually find-- 00:59:32.390 --> 00:59:34.450 align:middle line:84% let me just give the statement over here. 00:59:34.450 --> 00:59:45.750 align:middle line:90% 00:59:45.750 --> 00:59:49.310 align:middle line:84% So it can be shown-- we won't go through it-- 00:59:49.310 --> 01:00:04.510 align:middle line:84% that there exists a partitional equilibrium with k cells 01:00:04.510 --> 01:00:19.370 align:middle line:84% if and only if beta is less than-- 01:00:19.370 --> 01:00:26.412 align:middle line:90% 01:00:26.412 --> 01:00:27.870 align:middle line:84% and I think this formula can really 01:00:27.870 --> 01:00:29.970 align:middle line:84% capture a lot of the key insights of the paper. 01:00:29.970 --> 01:00:32.463 align:middle line:84% So let's try to understand what's going on. 01:00:32.463 --> 01:00:34.630 align:middle line:84% First, let's check that this is consistent with what 01:00:34.630 --> 01:00:35.830 align:middle line:90% we did before. 01:00:35.830 --> 01:00:38.310 align:middle line:90% So what if we put in k equals 2? 01:00:38.310 --> 01:00:43.030 align:middle line:84% With k equals 2, we're back to the case, we already looked at. 01:00:43.030 --> 01:00:46.190 align:middle line:84% And with k equals 2, the theorem says there exists a partitional 01:00:46.190 --> 01:00:50.510 align:middle line:84% equilibrium with two cells if and only if beta is less than 1 01:00:50.510 --> 01:00:54.590 align:middle line:84% over 2 times 2 times 1, which is 1/4. 01:00:54.590 --> 01:00:57.640 align:middle line:84% So indeed, it agrees with what we got over here. 01:00:57.640 --> 01:01:00.520 align:middle line:84% More generally, we get a formula like this. 01:01:00.520 --> 01:01:01.580 align:middle line:90% So what does it say? 01:01:01.580 --> 01:01:05.640 align:middle line:84% Well, as k increases, communication 01:01:05.640 --> 01:01:07.560 align:middle line:90% becomes more precise. 01:01:07.560 --> 01:01:10.560 align:middle line:84% As k increases, we're partitioning the state 01:01:10.560 --> 01:01:14.120 align:middle line:84% into smaller and smaller cells, and therefore conveying 01:01:14.120 --> 01:01:16.260 align:middle line:84% more and more precise information about the state. 01:01:16.260 --> 01:01:27.560 align:middle line:90% 01:01:27.560 --> 01:01:29.900 align:middle line:84% So if we want more precise communication, 01:01:29.900 --> 01:01:32.760 align:middle line:84% what it's telling us is beta has to be even smaller. 01:01:32.760 --> 01:01:37.220 align:middle line:90% So that requires less bias. 01:01:37.220 --> 01:01:43.880 align:middle line:90% 01:01:43.880 --> 01:01:50.600 align:middle line:84% So what we see is that the less bias that the agent is, 01:01:50.600 --> 01:01:52.920 align:middle line:84% the more precisely it's possible for the sender 01:01:52.920 --> 01:01:55.480 align:middle line:84% to communicate with that agent and more 01:01:55.480 --> 01:01:58.280 align:middle line:84% we can split the state space into these very, very tiny cells 01:01:58.280 --> 01:02:01.120 align:middle line:84% and convey more and more precise information. 01:02:01.120 --> 01:02:03.100 align:middle line:84% And in the limiting case, when there's no bias, 01:02:03.100 --> 01:02:04.933 align:middle line:84% we can basically perfectly reveal the state. 01:02:04.933 --> 01:02:05.768 align:middle line:90% Yes. 01:02:05.768 --> 01:02:07.560 align:middle line:84% AUDIENCE: Can you explain a little bit more 01:02:07.560 --> 01:02:10.200 align:middle line:84% about the reasoning between precise communication 01:02:10.200 --> 01:02:14.132 align:middle line:84% coming from less bias or the last arrow? 01:02:14.132 --> 01:02:14.840 align:middle line:90% PROFESSOR: Right. 01:02:14.840 --> 01:02:17.440 align:middle line:84% So I'm saying the last arrow comes from this equation. 01:02:17.440 --> 01:02:23.040 align:middle line:84% If k grows-- so we want to have an equilibrium with more cells-- 01:02:23.040 --> 01:02:25.760 align:middle line:84% then this inequality has to be satisfied. 01:02:25.760 --> 01:02:29.600 align:middle line:84% Because k is on the bottom of this fraction, as k goes up, 01:02:29.600 --> 01:02:31.120 align:middle line:90% this fraction gets smaller. 01:02:31.120 --> 01:02:32.717 align:middle line:84% And therefore, beta has to be smaller 01:02:32.717 --> 01:02:34.300 align:middle line:84% in order for this equilibrium to work. 01:02:34.300 --> 01:02:35.717 align:middle line:84% AUDIENCE: Is there a reason that's 01:02:35.717 --> 01:02:37.440 align:middle line:90% not related to this inequality? 01:02:37.440 --> 01:02:38.980 align:middle line:90% PROFESSOR: Oh, yeah. 01:02:38.980 --> 01:02:41.560 align:middle line:84% I think the reason not related to the inequality 01:02:41.560 --> 01:02:46.560 align:middle line:84% is that when we provide really, really precise information 01:02:46.560 --> 01:02:50.680 align:middle line:84% about the state, it becomes very tempting for the sender 01:02:50.680 --> 01:02:54.320 align:middle line:84% to misreport and claim the state is higher than it is. 01:02:54.320 --> 01:02:57.640 align:middle line:84% So think of the logic of that opening case 01:02:57.640 --> 01:02:59.020 align:middle line:90% where we fully reveal the state. 01:02:59.020 --> 01:03:00.770 align:middle line:84% If we basically say what the state is down 01:03:00.770 --> 01:03:02.540 align:middle line:84% to really, really strong precision, 01:03:02.540 --> 01:03:03.420 align:middle line:90% what's the sender going to do? 01:03:03.420 --> 01:03:04.940 align:middle line:84% They're just going to claim the state is higher, 01:03:04.940 --> 01:03:06.815 align:middle line:84% they're going to claim it's in a higher cell, 01:03:06.815 --> 01:03:09.840 align:middle line:84% and they're going to benefit from that. 01:03:09.840 --> 01:03:12.160 align:middle line:84% If they're less biased, then it's feasible for us 01:03:12.160 --> 01:03:14.180 align:middle line:84% to split the state space into very fine cells, 01:03:14.180 --> 01:03:17.995 align:middle line:84% and the sender won't have an incentive to deviate. 01:03:17.995 --> 01:03:19.620 align:middle line:84% So I guess one final picture I'll show. 01:03:19.620 --> 01:03:21.537 align:middle line:84% And I know this lecture is maybe a bit harder, 01:03:21.537 --> 01:03:26.240 align:middle line:84% so I think I'll end maybe 5 or 10 minutes early. 01:03:26.240 --> 01:03:29.040 align:middle line:84% But what we can look at is I think 01:03:29.040 --> 01:03:32.080 align:middle line:84% it's helpful to look at the space of biases. 01:03:32.080 --> 01:03:34.320 align:middle line:90% So let's look at beta. 01:03:34.320 --> 01:03:38.400 align:middle line:84% And let's see what's possible in this game. 01:03:38.400 --> 01:03:41.400 align:middle line:90% So here, we have 1/4. 01:03:41.400 --> 01:03:44.860 align:middle line:84% And if beta is up here, then no communication is possible. 01:03:44.860 --> 01:03:57.520 align:middle line:90% 01:03:57.520 --> 01:03:59.148 align:middle line:84% The agent is so biased that there's 01:03:59.148 --> 01:04:01.440 align:middle line:84% no equilibrium in which any information about the state 01:04:01.440 --> 01:04:02.220 align:middle line:90% is revealed. 01:04:02.220 --> 01:04:05.000 align:middle line:84% That's what we already said over there. 01:04:05.000 --> 01:04:06.660 align:middle line:90% Let's get the numbers right. 01:04:06.660 --> 01:04:11.546 align:middle line:84% If we have 3, we have 6, it's already 1/12. 01:04:11.546 --> 01:04:16.240 align:middle line:84% Yeah, I think we'll get down to something like 1/12. 01:04:16.240 --> 01:04:27.360 align:middle line:84% So here, there is a two-cell equilibrium only. 01:04:27.360 --> 01:04:29.880 align:middle line:84% If the bias is between 1/12 and 1/4, 01:04:29.880 --> 01:04:33.182 align:middle line:84% then the only equilibrium that can be shown is the kind 01:04:33.182 --> 01:04:35.640 align:middle line:84% of equilibrium we described, where you either say the state 01:04:35.640 --> 01:04:37.140 align:middle line:90% is above or below a threshold. 01:04:37.140 --> 01:04:39.792 align:middle line:90% 01:04:39.792 --> 01:04:42.000 align:middle line:84% And we're going to keep going down and down and down. 01:04:42.000 --> 01:04:44.360 align:middle line:84% And then here in this case, we're 01:04:44.360 --> 01:04:49.810 align:middle line:84% going to have a two-cell and a three-cell equilibrium. 01:04:49.810 --> 01:04:51.690 align:middle line:84% So for a lower level of bias, there 01:04:51.690 --> 01:04:53.970 align:middle line:84% will be a two-cell equilibrium, where the sender just 01:04:53.970 --> 01:04:56.290 align:middle line:84% says the state's above or below a threshold. 01:04:56.290 --> 01:04:58.770 align:middle line:84% There will also be a three-cell equilibria, 01:04:58.770 --> 01:05:02.370 align:middle line:84% where the sender says the state is either low, medium, or high. 01:05:02.370 --> 01:05:02.990 align:middle line:90% But that's it. 01:05:02.990 --> 01:05:04.290 align:middle line:90% There's nothing else. 01:05:04.290 --> 01:05:06.130 align:middle line:84% And then if we get down to here, there's 01:05:06.130 --> 01:05:09.290 align:middle line:84% going to be a two-cell, a three-cell, and a four-cell. 01:05:09.290 --> 01:05:13.210 align:middle line:84% And as we keep going, as beta keeps 01:05:13.210 --> 01:05:16.070 align:middle line:84% getting smaller and smaller, the set of equilibria grows. 01:05:16.070 --> 01:05:17.510 align:middle line:84% We have more and more equilibria. 01:05:17.510 --> 01:05:19.468 align:middle line:84% And we have equilibria with more and more cells 01:05:19.468 --> 01:05:21.570 align:middle line:84% that provide more and more precise communication. 01:05:21.570 --> 01:05:23.990 align:middle line:84% And the limit as beta gets really close to 0, 01:05:23.990 --> 01:05:26.070 align:middle line:84% we have very informative equilibria. 01:05:26.070 --> 01:05:43.810 align:middle line:90% 01:05:43.810 --> 01:05:48.050 align:middle line:84% And maybe one final observation I'll make 01:05:48.050 --> 01:05:52.570 align:middle line:84% is let's suppose beta is small enough so that there's 01:05:52.570 --> 01:05:53.810 align:middle line:90% a range of equilibria. 01:05:53.810 --> 01:05:56.630 align:middle line:84% There's a one-cell, there's a two-cell, there's a three-cell, 01:05:56.630 --> 01:05:57.570 align:middle line:90% there's a four-cell. 01:05:57.570 --> 01:06:00.410 align:middle line:84% The question is, which equilibrium 01:06:00.410 --> 01:06:01.670 align:middle line:90% is best for the players? 01:06:01.670 --> 01:06:10.705 align:middle line:90% 01:06:10.705 --> 01:06:11.830 align:middle line:90% And the intuition for this. 01:06:11.830 --> 01:06:14.117 align:middle line:84% Let's say we're in a case-- let's say beta is here. 01:06:14.117 --> 01:06:15.950 align:middle line:84% So we have both a two-cell and a three-cell. 01:06:15.950 --> 01:06:16.750 align:middle line:90% What do you think is better? 01:06:16.750 --> 01:06:17.347 align:middle line:90% Yeah. 01:06:17.347 --> 01:06:18.430 align:middle line:90% Or do you have a question? 01:06:18.430 --> 01:06:18.610 align:middle line:90% Yeah. 01:06:18.610 --> 01:06:20.050 align:middle line:84% AUDIENCE: I was going to say, I think for the receiver, 01:06:20.050 --> 01:06:21.630 align:middle line:84% it's probably better to have more cells. 01:06:21.630 --> 01:06:22.790 align:middle line:90% So they're getting closer. 01:06:22.790 --> 01:06:23.832 align:middle line:90% PROFESSOR: Exactly right. 01:06:23.832 --> 01:06:25.690 align:middle line:84% So it's definitely better for the receiver, 01:06:25.690 --> 01:06:28.230 align:middle line:84% because if there's more cells in the equilibrium, 01:06:28.230 --> 01:06:29.490 align:middle line:84% the receiver is just getting more information 01:06:29.490 --> 01:06:30.250 align:middle line:90% about the state. 01:06:30.250 --> 01:06:33.025 align:middle line:84% And therefore, they're able to make more accurate decisions. 01:06:33.025 --> 01:06:35.650 align:middle line:84% It may be less obvious, but it's actually better for the sender 01:06:35.650 --> 01:06:36.490 align:middle line:90% as well. 01:06:36.490 --> 01:06:45.410 align:middle line:84% So it turns out the highest k is best for both 01:06:45.410 --> 01:06:48.010 align:middle line:90% the sender and the receiver. 01:06:48.010 --> 01:06:50.640 align:middle line:84% So for any value of beta, depending on the value of beta, 01:06:50.640 --> 01:06:52.390 align:middle line:84% there's going to be a range of equilibria. 01:06:52.390 --> 01:06:54.230 align:middle line:84% There's going to be some uninformative equilibria. 01:06:54.230 --> 01:06:55.850 align:middle line:84% In fact, there's always the babbling equilibrium where 01:06:55.850 --> 01:06:57.050 align:middle line:90% you don't reveal anything. 01:06:57.050 --> 01:06:59.350 align:middle line:84% There might be a two-cell where I just say it's high or low. 01:06:59.350 --> 01:07:00.808 align:middle line:84% There might be a three-cell where I 01:07:00.808 --> 01:07:02.230 align:middle line:90% say it's high, medium, or low. 01:07:02.230 --> 01:07:04.930 align:middle line:84% A four cell-- the biggest number of cells 01:07:04.930 --> 01:07:07.063 align:middle line:84% that can exist as in equilibrium, 01:07:07.063 --> 01:07:09.230 align:middle line:84% that's going to be the most informative equilibrium. 01:07:09.230 --> 01:07:11.647 align:middle line:84% And it turns out that's going to be the equilibrium that's 01:07:11.647 --> 01:07:15.570 align:middle line:84% best for the sender and also for the receiver. 01:07:15.570 --> 01:07:17.672 align:middle line:84% So maybe that's some sort of prediction about what 01:07:17.672 --> 01:07:18.630 align:middle line:90% we might expect to see. 01:07:18.630 --> 01:07:19.210 align:middle line:90% Yes, Amy. 01:07:19.210 --> 01:07:20.890 align:middle line:84% AUDIENCE: So it makes sense that it's best for the receiver 01:07:20.890 --> 01:07:22.290 align:middle line:84% just because when you get closer, 01:07:22.290 --> 01:07:23.210 align:middle line:90% the receiver [INAUDIBLE]. 01:07:23.210 --> 01:07:23.730 align:middle line:90% PROFESSOR: Yeah. 01:07:23.730 --> 01:07:25.480 align:middle line:84% AUDIENCE: And how does the sender benefit? 01:07:25.480 --> 01:07:28.343 align:middle line:90% Is it also the same reason? 01:07:28.343 --> 01:07:29.010 align:middle line:90% PROFESSOR: Yeah. 01:07:29.010 --> 01:07:35.250 align:middle line:84% It's actually a property of the quadratic law. 01:07:35.250 --> 01:07:39.370 align:middle line:84% So I think the intuition is that because the receiver is always 01:07:39.370 --> 01:07:42.380 align:middle line:84% matching the decision with their expectation of the state, 01:07:42.380 --> 01:07:46.120 align:middle line:84% the sender can never shift the decision. 01:07:46.120 --> 01:07:48.160 align:middle line:84% So we're getting into a technical point. 01:07:48.160 --> 01:07:50.220 align:middle line:84% But it turns out the expected decision is always 01:07:50.220 --> 01:07:53.820 align:middle line:84% going to be equal to the expected state. 01:07:53.820 --> 01:07:57.140 align:middle line:84% And as the cells get smaller, the benefit 01:07:57.140 --> 01:07:59.200 align:middle line:84% is that the state is conveyed more precisely, 01:07:59.200 --> 01:08:01.100 align:middle line:84% and therefore higher decisions are 01:08:01.100 --> 01:08:03.200 align:middle line:84% taken in higher states and the lower decisions in lower states. 01:08:03.200 --> 01:08:05.075 align:middle line:84% And that's going to be better for the sender. 01:08:05.075 --> 01:08:07.640 align:middle line:84% But the basic intuition is that the-- 01:08:07.640 --> 01:08:10.940 align:middle line:90% 01:08:10.940 --> 01:08:13.940 align:middle line:84% I guess the formal answer would be it's like a bias variance 01:08:13.940 --> 01:08:15.580 align:middle line:90% decomposition. 01:08:15.580 --> 01:08:19.300 align:middle line:84% You can express the expected loss as the bias of the error 01:08:19.300 --> 01:08:20.677 align:middle line:90% and the variance of the error. 01:08:20.677 --> 01:08:22.260 align:middle line:84% And the bias is fixed and the variance 01:08:22.260 --> 01:08:23.279 align:middle line:90% you're going to make smaller. 01:08:23.279 --> 01:08:25.237 align:middle line:84% But I'm happy to talk more offline, if you want 01:08:25.237 --> 01:08:28.220 align:middle line:90% to go into that in more depth. 01:08:28.220 --> 01:08:30.200 align:middle line:84% So I think that was a bit of a harder lecture, 01:08:30.200 --> 01:08:30.840 align:middle line:90% so let me stop there. 01:08:30.840 --> 01:08:32.080 align:middle line:84% I know we're near the end of the semester. 01:08:32.080 --> 01:08:33.420 align:middle line:90% I'll give people an early stop. 01:08:33.420 --> 01:08:36.890 align:middle line:84% And I will see everyone on Tuesday for the last lecture. 01:08:36.890 --> 01:08:40.000 align:middle line:90%