WEBVTT 00:00:00.000 --> 00:00:02.412 align:middle line:90% [SQUEAKING] 00:00:02.412 --> 00:00:04.324 align:middle line:90% [RUSTLING] 00:00:04.324 --> 00:00:05.758 align:middle line:90% [CLICKING] 00:00:05.758 --> 00:00:08.887 align:middle line:90% 00:00:08.887 --> 00:00:09.720 align:middle line:90% IAN BALL: All right. 00:00:09.720 --> 00:00:11.220 align:middle line:84% So today, we're starting a new phase 00:00:11.220 --> 00:00:15.560 align:middle line:84% of the course, which is going to be about games 00:00:15.560 --> 00:00:19.433 align:middle line:90% with incomplete information. 00:00:19.433 --> 00:00:21.600 align:middle line:84% And I'll warn you up front that the terminology here 00:00:21.600 --> 00:00:22.420 align:middle line:90% is a bit confusing. 00:00:22.420 --> 00:00:24.640 align:middle line:84% We've been talking about imperfect and perfect 00:00:24.640 --> 00:00:26.200 align:middle line:84% information, and now we're talking 00:00:26.200 --> 00:00:28.620 align:middle line:84% about complete and incomplete information. 00:00:28.620 --> 00:00:31.220 align:middle line:84% Complete and perfect sound similar colloquially, 00:00:31.220 --> 00:00:33.470 align:middle line:84% but in game theory, they have very different meanings. 00:00:33.470 --> 00:00:37.075 align:middle line:84% So I want to be precise about what's going on. 00:00:37.075 --> 00:00:42.100 align:middle line:84% So far, we've always looked at complete information. 00:00:42.100 --> 00:00:48.640 align:middle line:90% 00:00:48.640 --> 00:00:51.260 align:middle line:84% And what's confusing is that under complete information, 00:00:51.260 --> 00:00:53.500 align:middle line:84% under this complete information umbrella, 00:00:53.500 --> 00:00:55.200 align:middle line:84% there are two special cases that we've 00:00:55.200 --> 00:00:57.680 align:middle line:84% talked about or two distinct cases, which 00:00:57.680 --> 00:01:04.190 align:middle line:84% is perfect information and imperfect information. 00:01:04.190 --> 00:01:08.488 align:middle line:90% 00:01:08.488 --> 00:01:10.030 align:middle line:84% So I'll give some formal definitions, 00:01:10.030 --> 00:01:11.250 align:middle line:84% but then I'll also give some examples 00:01:11.250 --> 00:01:13.610 align:middle line:84% to keep in mind, because I think these definitions can 00:01:13.610 --> 00:01:14.790 align:middle line:90% seem a bit abstract. 00:01:14.790 --> 00:01:18.250 align:middle line:84% So in games of perfect information, what that meant 00:01:18.250 --> 00:01:22.470 align:middle line:84% was that every information set was a singleton. 00:01:22.470 --> 00:01:33.250 align:middle line:90% 00:01:33.250 --> 00:01:35.370 align:middle line:84% And I think the canonical game to keep in mind 00:01:35.370 --> 00:01:38.270 align:middle line:84% for games of perfect information would be games like chess. 00:01:38.270 --> 00:01:42.850 align:middle line:90% 00:01:42.850 --> 00:01:46.170 align:middle line:84% So when we play chess, whenever it's our turn to move, 00:01:46.170 --> 00:01:48.770 align:middle line:84% we can perfectly see how everyone else 00:01:48.770 --> 00:01:50.090 align:middle line:90% has moved so far in chess. 00:01:50.090 --> 00:01:51.465 align:middle line:84% We see the position of the board, 00:01:51.465 --> 00:01:54.330 align:middle line:84% and we remember each move that everyone else played. 00:01:54.330 --> 00:01:58.460 align:middle line:84% That's the sense in which every information set is a singleton. 00:01:58.460 --> 00:02:01.300 align:middle line:84% Then we also looked at games of imperfect information. 00:02:01.300 --> 00:02:04.540 align:middle line:84% What that meant was that some information sets were not 00:02:04.540 --> 00:02:05.780 align:middle line:90% singletons. 00:02:05.780 --> 00:02:12.800 align:middle line:84% So I'll say some non-singleton info sets. 00:02:12.800 --> 00:02:15.420 align:middle line:90% 00:02:15.420 --> 00:02:17.020 align:middle line:84% Now, economically, what does it mean 00:02:17.020 --> 00:02:19.122 align:middle line:84% when we had non-singleton information sets? 00:02:19.122 --> 00:02:21.580 align:middle line:84% It meant that there were times when someone was called upon 00:02:21.580 --> 00:02:24.640 align:middle line:84% to move, and there was some aspect of the game 00:02:24.640 --> 00:02:26.140 align:middle line:84% or of the previous moves they didn't 00:02:26.140 --> 00:02:27.700 align:middle line:84% know about-- either moved by nature 00:02:27.700 --> 00:02:29.120 align:middle line:90% or moved by a different player. 00:02:29.120 --> 00:02:31.200 align:middle line:84% And they knew which information set they were at, 00:02:31.200 --> 00:02:36.460 align:middle line:84% but they didn't know which node within that information 00:02:36.460 --> 00:02:37.660 align:middle line:90% set they were at. 00:02:37.660 --> 00:02:41.000 align:middle line:90% And this was the case-- 00:02:41.000 --> 00:02:43.460 align:middle line:84% there's a few leading examples here. 00:02:43.460 --> 00:02:45.760 align:middle line:84% Whenever we had a simultaneous move game-- 00:02:45.760 --> 00:02:49.813 align:middle line:90% 00:02:49.813 --> 00:02:51.980 align:middle line:84% whenever the players moved simultaneously, remember, 00:02:51.980 --> 00:02:54.920 align:middle line:84% the way we represented that in the extensive form was to say, 00:02:54.920 --> 00:02:57.210 align:middle line:84% well, technically one player has moved first. 00:02:57.210 --> 00:02:59.790 align:middle line:84% But then the other player moves without observing the move 00:02:59.790 --> 00:03:02.190 align:middle line:84% by the other player, and therefore that second player's 00:03:02.190 --> 00:03:03.870 align:middle line:84% information set is going to include 00:03:03.870 --> 00:03:08.790 align:middle line:84% different nodes that specify how the other player moved. 00:03:08.790 --> 00:03:11.110 align:middle line:84% So simultaneous moves-- the simple example here, 00:03:11.110 --> 00:03:14.270 align:middle line:90% I would say, is poker. 00:03:14.270 --> 00:03:17.270 align:middle line:84% Poker would be another case, because when I'm called upon 00:03:17.270 --> 00:03:20.510 align:middle line:84% to move in poker, I don't know the cards that the other players 00:03:20.510 --> 00:03:24.510 align:middle line:84% have been dealt. Even if we move sequentially in poker, 00:03:24.510 --> 00:03:26.790 align:middle line:84% my information set is still non-singleton 00:03:26.790 --> 00:03:30.030 align:middle line:84% because when I move, I don't know what cards the other player 00:03:30.030 --> 00:03:30.530 align:middle line:90% has. 00:03:30.530 --> 00:03:32.230 align:middle line:84% So these are two distinct ways we 00:03:32.230 --> 00:03:35.430 align:middle line:90% can have imperfect information. 00:03:35.430 --> 00:03:39.250 align:middle line:84% But why do I say both of these are complete information? 00:03:39.250 --> 00:03:43.430 align:middle line:90% 00:03:43.430 --> 00:03:46.550 align:middle line:84% So the defining feature, the defining property 00:03:46.550 --> 00:03:49.270 align:middle line:84% of these two classes of games is what makes 00:03:49.270 --> 00:03:50.730 align:middle line:90% them complete information. 00:03:50.730 --> 00:03:54.462 align:middle line:90% 00:03:54.462 --> 00:03:55.920 align:middle line:84% And I'll warn you, some people give 00:03:55.920 --> 00:03:57.580 align:middle line:84% slightly different definitions of this, 00:03:57.580 --> 00:04:00.038 align:middle line:84% and I think my definition is slightly different than what's 00:04:00.038 --> 00:04:01.000 align:middle line:90% given in the notes. 00:04:01.000 --> 00:04:03.280 align:middle line:84% But what I like to think of it is, at the beginning 00:04:03.280 --> 00:04:16.260 align:middle line:84% of the game, information is symmetric. 00:04:16.260 --> 00:04:24.320 align:middle line:90% 00:04:24.320 --> 00:04:26.480 align:middle line:84% So beginning of the game is crucial here. 00:04:26.480 --> 00:04:30.705 align:middle line:84% Once we start playing poker, we're going to be dealt hands. 00:04:30.705 --> 00:04:33.080 align:middle line:84% And one player won't know what cards the other player has 00:04:33.080 --> 00:04:35.800 align:middle line:84% been dealt. But when we start the game, when we sit down 00:04:35.800 --> 00:04:38.087 align:middle line:84% at the poker table, whatever one player knows, 00:04:38.087 --> 00:04:38.920 align:middle line:90% everyone else knows. 00:04:38.920 --> 00:04:40.580 align:middle line:84% No one's been dealt any cards yet. 00:04:40.580 --> 00:04:42.260 align:middle line:84% We all understand the rules of the game. 00:04:42.260 --> 00:04:43.677 align:middle line:84% We all understand that people want 00:04:43.677 --> 00:04:45.080 align:middle line:84% to win as much money as possible, 00:04:45.080 --> 00:04:47.760 align:middle line:84% and information is symmetric in the sense 00:04:47.760 --> 00:04:56.320 align:middle line:84% that anything known by one player is known by all players. 00:04:56.320 --> 00:05:00.215 align:middle line:90% 00:05:00.215 --> 00:05:01.840 align:middle line:84% And maybe if I want to be more precise, 00:05:01.840 --> 00:05:03.620 align:middle line:90% I'll say it's common knowledge. 00:05:03.620 --> 00:05:08.958 align:middle line:90% 00:05:08.958 --> 00:05:10.500 align:middle line:84% This doesn't mean we know everything. 00:05:10.500 --> 00:05:11.960 align:middle line:84% When we sit down at a poker table, 00:05:11.960 --> 00:05:14.940 align:middle line:84% we don't know what hand we're going to be dealt in the future. 00:05:14.940 --> 00:05:16.197 align:middle line:90% But none of us know that. 00:05:16.197 --> 00:05:17.780 align:middle line:84% There's no information that one player 00:05:17.780 --> 00:05:19.900 align:middle line:84% knows that other players don't know 00:05:19.900 --> 00:05:22.060 align:middle line:90% at the beginning of the game. 00:05:22.060 --> 00:05:26.020 align:middle line:84% And it's crucial here that we're at the beginning of the game. 00:05:26.020 --> 00:05:28.480 align:middle line:84% The distinction with incomplete information-- 00:05:28.480 --> 00:05:36.140 align:middle line:90% 00:05:36.140 --> 00:05:38.080 align:middle line:84% when I say information is symmetric, 00:05:38.080 --> 00:05:41.143 align:middle line:84% let me add here, relevant information. 00:05:41.143 --> 00:05:43.060 align:middle line:84% Of course, when you sit down at a poker table, 00:05:43.060 --> 00:05:44.840 align:middle line:84% you know your birthday and someone else doesn't. 00:05:44.840 --> 00:05:46.840 align:middle line:84% But that's not really relevant to the game. 00:05:46.840 --> 00:05:48.340 align:middle line:84% So in a technical sense, we never 00:05:48.340 --> 00:05:50.030 align:middle line:84% have complete information because there's always 00:05:50.030 --> 00:05:51.930 align:middle line:84% these unmodeled things that are irrelevant. 00:05:51.930 --> 00:05:54.390 align:middle line:84% But what I mean is relevant information 00:05:54.390 --> 00:05:56.090 align:middle line:90% with incomplete information. 00:05:56.090 --> 00:06:05.070 align:middle line:84% The difference is that at the beginning of the game, 00:06:05.070 --> 00:06:08.950 align:middle line:84% there is relevant information that some player knows 00:06:08.950 --> 00:06:11.950 align:middle line:90% but another player doesn't know. 00:06:11.950 --> 00:06:22.590 align:middle line:84% So there is information, or more precisely, relevant information 00:06:22.590 --> 00:06:24.810 align:middle line:84% that is known by some player, but not others. 00:06:24.810 --> 00:06:39.750 align:middle line:90% 00:06:39.750 --> 00:06:42.170 align:middle line:84% And let's give a few examples of this. 00:06:42.170 --> 00:06:44.750 align:middle line:90% 00:06:44.750 --> 00:06:46.970 align:middle line:84% So examples of incomplete information games. 00:06:46.970 --> 00:06:49.560 align:middle line:84% And what you'll see is that a lot 00:06:49.560 --> 00:06:52.000 align:middle line:84% of our examples of complete information games 00:06:52.000 --> 00:06:53.100 align:middle line:90% come from parlor games. 00:06:53.100 --> 00:06:54.905 align:middle line:84% They come from things like chess or poker. 00:06:54.905 --> 00:06:56.280 align:middle line:84% But in the real world, when we're 00:06:56.280 --> 00:06:58.200 align:middle line:84% analyzing economic situations, we're 00:06:58.200 --> 00:07:00.340 align:middle line:84% much more often in the incomplete information case. 00:07:00.340 --> 00:07:02.620 align:middle line:84% So let's give a few examples of that. 00:07:02.620 --> 00:07:12.745 align:middle line:90% 00:07:12.745 --> 00:07:13.620 align:middle line:90% Was there a question? 00:07:13.620 --> 00:07:14.560 align:middle line:90% Yeah. 00:07:14.560 --> 00:07:15.587 align:middle line:90% AUDIENCE: No, nevermind. 00:07:15.587 --> 00:07:16.920 align:middle line:90% IAN BALL: You're welcome to ask. 00:07:16.920 --> 00:07:18.260 align:middle line:84% AUDIENCE: Oh, I thought you were asking for example. 00:07:18.260 --> 00:07:19.052 align:middle line:90% IAN BALL: Oh, sure. 00:07:19.052 --> 00:07:20.040 align:middle line:90% Give an example, yeah. 00:07:20.040 --> 00:07:21.373 align:middle line:90% Sorry, sometimes I'm rhetorical. 00:07:21.373 --> 00:07:22.147 align:middle line:90% It's not clear. 00:07:22.147 --> 00:07:23.980 align:middle line:84% You can always answer any question you want. 00:07:23.980 --> 00:07:25.938 align:middle line:84% And if I didn't intend it, that's totally fine. 00:07:25.938 --> 00:07:27.302 align:middle line:90% Yeah, go ahead. 00:07:27.302 --> 00:07:28.760 align:middle line:84% AUDIENCE: Would one of them be when 00:07:28.760 --> 00:07:31.385 align:middle line:84% there's price haggling and they don't know each other's values? 00:07:31.385 --> 00:07:32.135 align:middle line:90% IAN BALL: Exactly. 00:07:32.135 --> 00:07:33.100 align:middle line:90% That's a great example. 00:07:33.100 --> 00:07:36.153 align:middle line:90% And actually, that fits in. 00:07:36.153 --> 00:07:38.820 align:middle line:84% The first example I was going to give is a special case of that. 00:07:38.820 --> 00:07:41.580 align:middle line:84% So let me give this is the kind of classic example. 00:07:41.580 --> 00:07:43.540 align:middle line:84% And it exactly fits into your setting. 00:07:43.540 --> 00:07:45.850 align:middle line:84% And this, I would say is buying a used car. 00:07:45.850 --> 00:07:48.110 align:middle line:84% And of course, price haggling over the used car. 00:07:48.110 --> 00:07:52.057 align:middle line:90% 00:07:52.057 --> 00:07:54.390 align:middle line:84% Let's say we're haggling over the price of the used car. 00:07:54.390 --> 00:07:55.215 align:middle line:90% What don't we know? 00:07:55.215 --> 00:07:56.590 align:middle line:84% Well, you brought up one example. 00:07:56.590 --> 00:07:59.350 align:middle line:84% We don't know each player's valuation for the car. 00:07:59.350 --> 00:08:02.070 align:middle line:84% And we can be a little more specific, right? 00:08:02.070 --> 00:08:04.502 align:middle line:84% Let's say a salesperson is selling you a used car. 00:08:04.502 --> 00:08:06.710 align:middle line:84% There's something they know that you don't, probably. 00:08:06.710 --> 00:08:09.770 align:middle line:84% What might they know that you don't know about the car? 00:08:09.770 --> 00:08:11.130 align:middle line:90% Yes, in the second row? 00:08:11.130 --> 00:08:12.450 align:middle line:90% AUDIENCE: The condition? 00:08:12.450 --> 00:08:13.908 align:middle line:84% IAN BALL: The condition of the car. 00:08:13.908 --> 00:08:16.250 align:middle line:84% So in this case, the seller, maybe 00:08:16.250 --> 00:08:18.225 align:middle line:84% they're an experienced mechanic or auto dealer. 00:08:18.225 --> 00:08:19.350 align:middle line:90% They've seen a lot of cars. 00:08:19.350 --> 00:08:21.370 align:middle line:84% They've examined the car closely. 00:08:21.370 --> 00:08:23.270 align:middle line:84% You maybe don't know a lot about cars. 00:08:23.270 --> 00:08:26.770 align:middle line:84% So often, the seller knows whether there's 00:08:26.770 --> 00:08:27.882 align:middle line:90% a problem with the engine. 00:08:27.882 --> 00:08:30.090 align:middle line:84% They know whether it's been making a sound every time 00:08:30.090 --> 00:08:30.790 align:middle line:90% they drive it. 00:08:30.790 --> 00:08:35.470 align:middle line:84% So the seller knows the condition of the car. 00:08:35.470 --> 00:08:40.289 align:middle line:90% 00:08:40.289 --> 00:08:41.970 align:middle line:84% And that's probably relevant, actually, 00:08:41.970 --> 00:08:43.580 align:middle line:90% to both of your valuations. 00:08:43.580 --> 00:08:45.200 align:middle line:84% If it's a high quality car, the seller 00:08:45.200 --> 00:08:46.700 align:middle line:84% probably values it more because they 00:08:46.700 --> 00:08:47.950 align:middle line:90% could sell it to someone else. 00:08:47.950 --> 00:08:50.100 align:middle line:84% The buyer probably values it more as well. 00:08:50.100 --> 00:08:51.820 align:middle line:84% I'd argue the buyer probably also 00:08:51.820 --> 00:08:53.700 align:middle line:90% has some private information. 00:08:53.700 --> 00:08:55.727 align:middle line:90% What does the buyer know? 00:08:55.727 --> 00:08:58.060 align:middle line:84% And these are things-- when I say the seller knows this, 00:08:58.060 --> 00:08:59.040 align:middle line:90% but the buyer doesn't. 00:08:59.040 --> 00:09:05.510 align:middle line:90% 00:09:05.510 --> 00:09:07.260 align:middle line:84% And of course, these things aren't binary. 00:09:07.260 --> 00:09:09.302 align:middle line:84% The buyer probably knows something about the car. 00:09:09.302 --> 00:09:11.515 align:middle line:84% They can look at it and see if it's totally dented. 00:09:11.515 --> 00:09:14.140 align:middle line:84% They can see that, but there's probably some aspects of the car 00:09:14.140 --> 00:09:16.245 align:middle line:84% that they can't see or can't observe. 00:09:16.245 --> 00:09:17.120 align:middle line:90% What about the buyer? 00:09:17.120 --> 00:09:18.280 align:middle line:84% What is something the buyer may know? 00:09:18.280 --> 00:09:18.980 align:middle line:90% Yeah. 00:09:18.980 --> 00:09:22.580 align:middle line:84% AUDIENCE: The buyer knows how much they want a car. 00:09:22.580 --> 00:09:24.300 align:middle line:84% I feel like that informs how much they'd 00:09:24.300 --> 00:09:25.360 align:middle line:90% be willing to pay for it. 00:09:25.360 --> 00:09:25.740 align:middle line:90% IAN BALL: Exactly. 00:09:25.740 --> 00:09:27.560 align:middle line:84% So the buyer knows-- and this is exactly what you mentioned 00:09:27.560 --> 00:09:29.935 align:middle line:84% in the front row-- they know their valuation for the car. 00:09:29.935 --> 00:09:32.923 align:middle line:84% Buyer knows, or at least they know-- 00:09:32.923 --> 00:09:34.840 align:middle line:84% maybe they don't know their valuation exactly, 00:09:34.840 --> 00:09:36.140 align:middle line:90% but they know-- 00:09:36.140 --> 00:09:37.800 align:middle line:84% maybe I'll say how much they need it. 00:09:37.800 --> 00:09:40.060 align:middle line:90% 00:09:40.060 --> 00:09:42.310 align:middle line:84% The reason I don't say they don't know their valuation 00:09:42.310 --> 00:09:44.482 align:middle line:84% is if they don't know if the car is broken or not, 00:09:44.482 --> 00:09:46.190 align:middle line:84% then they probably don't know exactly how 00:09:46.190 --> 00:09:47.610 align:middle line:90% much it's really worth to them. 00:09:47.610 --> 00:09:50.910 align:middle line:84% But they know they have a job they need to get to tomorrow, 00:09:50.910 --> 00:09:52.270 align:middle line:90% and they have to get there. 00:09:52.270 --> 00:09:54.630 align:middle line:84% And they know they went to the last three dealerships, 00:09:54.630 --> 00:09:56.297 align:middle line:84% and they couldn't find a car they liked. 00:09:56.297 --> 00:09:58.297 align:middle line:84% Or maybe they know that they like the color red, 00:09:58.297 --> 00:10:00.422 align:middle line:84% and this car is red-- or they hate the color brown, 00:10:00.422 --> 00:10:01.850 align:middle line:84% and that's why they value it less. 00:10:01.850 --> 00:10:04.183 align:middle line:84% There's a lot of aspects of the buyer's preferences that 00:10:04.183 --> 00:10:05.890 align:middle line:84% the buyer knows, but the seller doesn't. 00:10:05.890 --> 00:10:14.150 align:middle line:90% 00:10:14.150 --> 00:10:15.670 align:middle line:84% And notice how this is fundamentally 00:10:15.670 --> 00:10:19.850 align:middle line:84% different from the imperfect information we see in poker. 00:10:19.850 --> 00:10:23.410 align:middle line:84% In poker, yes, I don't know if you have queens or kings. 00:10:23.410 --> 00:10:24.890 align:middle line:84% But when you sit down at the table, 00:10:24.890 --> 00:10:26.723 align:middle line:84% we basically all know the game we're playing 00:10:26.723 --> 00:10:30.310 align:middle line:84% and we all know the preferences of the other players. 00:10:30.310 --> 00:10:34.830 align:middle line:84% Another great example would be health insurance-- 00:10:34.830 --> 00:10:38.923 align:middle line:90% so also buying health insurance. 00:10:38.923 --> 00:10:40.840 align:middle line:84% I would argue in this case probably the seller 00:10:40.840 --> 00:10:44.640 align:middle line:84% and the buyer know some things that the other party doesn't 00:10:44.640 --> 00:10:47.520 align:middle line:90% know very well. 00:10:47.520 --> 00:10:48.643 align:middle line:90% Any thoughts here? 00:10:48.643 --> 00:10:50.560 align:middle line:84% Certainly the buyer probably knows some things 00:10:50.560 --> 00:10:53.420 align:middle line:90% the seller doesn't know. 00:10:53.420 --> 00:10:53.960 align:middle line:90% Yeah. 00:10:53.960 --> 00:10:57.230 align:middle line:84% AUDIENCE: The buyer knows his own medical history. 00:10:57.230 --> 00:10:57.980 align:middle line:90% IAN BALL: Exactly. 00:10:57.980 --> 00:10:59.620 align:middle line:84% So the buyer knows medical history. 00:10:59.620 --> 00:11:04.520 align:middle line:90% 00:11:04.520 --> 00:11:05.817 align:middle line:90% And again, this is not binary. 00:11:05.817 --> 00:11:07.900 align:middle line:84% If you buy health insurance in the private market, 00:11:07.900 --> 00:11:09.140 align:middle line:84% the seller is concerned about this. 00:11:09.140 --> 00:11:10.240 align:middle line:84% And they're going to have you fill out a form. 00:11:10.240 --> 00:11:11.800 align:middle line:84% And they're going to have a doctor go to your house and do 00:11:11.800 --> 00:11:12.773 align:middle line:90% a 30-minute evaluation. 00:11:12.773 --> 00:11:14.940 align:middle line:84% But of course, that's not going to cover everything. 00:11:14.940 --> 00:11:17.523 align:middle line:84% You might know that last week, you started feeling a bit sick, 00:11:17.523 --> 00:11:19.607 align:middle line:84% and that's why you're buying the health insurance. 00:11:19.607 --> 00:11:22.080 align:middle line:84% Or you might know about a family history of a condition 00:11:22.080 --> 00:11:24.440 align:middle line:84% that the health insurer doesn't know about. 00:11:24.440 --> 00:11:26.420 align:middle line:84% And this is certainly relevant to both of you. 00:11:26.420 --> 00:11:28.220 align:middle line:84% It's relevant to how much you value insurance, 00:11:28.220 --> 00:11:30.512 align:middle line:84% because the sicker you are, the more valuable insurance 00:11:30.512 --> 00:11:31.160 align:middle line:90% is to you. 00:11:31.160 --> 00:11:34.020 align:middle line:84% It's also relevant to how costly the insurance is to the seller. 00:11:34.020 --> 00:11:35.940 align:middle line:84% If the seller insures a very sick person, 00:11:35.940 --> 00:11:37.530 align:middle line:84% they're going to have to be on the hook for a lot 00:11:37.530 --> 00:11:38.697 align:middle line:90% of different doctor's bills. 00:11:38.697 --> 00:11:41.730 align:middle line:84% And this is one reason health insurance markets often 00:11:41.730 --> 00:11:42.750 align:middle line:90% don't run very well. 00:11:42.750 --> 00:11:47.010 align:middle line:84% I'd argue the seller probably, even in this case, 00:11:47.010 --> 00:11:50.850 align:middle line:90% maybe knows some things. 00:11:50.850 --> 00:11:55.330 align:middle line:84% This is not the main missing information in health insurance 00:11:55.330 --> 00:11:57.930 align:middle line:84% markets, but I think there's probably some things they know. 00:11:57.930 --> 00:12:00.013 align:middle line:84% AUDIENCE: Could the seller know that their company 00:12:00.013 --> 00:12:02.050 align:middle line:84% is about to do a discount next month? 00:12:02.050 --> 00:12:03.110 align:middle line:90% Does it tell the buyer? 00:12:03.110 --> 00:12:03.330 align:middle line:90% IAN BALL: Great. 00:12:03.330 --> 00:12:04.230 align:middle line:90% That's a good thing. 00:12:04.230 --> 00:12:06.070 align:middle line:84% So one thing-- that's in a dynamic context. 00:12:06.070 --> 00:12:07.710 align:middle line:84% But certainly in a dynamic context, 00:12:07.710 --> 00:12:10.325 align:middle line:84% the sellers know the prices next period. 00:12:10.325 --> 00:12:11.450 align:middle line:90% This is certainly relevant. 00:12:11.450 --> 00:12:13.990 align:middle line:84% These clothing stores always do these regular discounts. 00:12:13.990 --> 00:12:15.490 align:middle line:84% If you go and buy something-- if you 00:12:15.490 --> 00:12:16.330 align:middle line:84% knew that they were going to have 00:12:16.330 --> 00:12:18.970 align:middle line:84% a 15% discount the next day, you probably wouldn't buy. 00:12:18.970 --> 00:12:20.370 align:middle line:84% And they usually don't tell you that the discount 00:12:20.370 --> 00:12:21.210 align:middle line:90% is coming in the future. 00:12:21.210 --> 00:12:22.470 align:middle line:84% They announced it when it started. 00:12:22.470 --> 00:12:23.762 align:middle line:90% They don't tell you in advance. 00:12:23.762 --> 00:12:27.270 align:middle line:84% So the seller might know things about future pricing. 00:12:27.270 --> 00:12:30.330 align:middle line:90% 00:12:30.330 --> 00:12:32.910 align:middle line:84% And maybe this is a bit more of a behavioral, non-rational 00:12:32.910 --> 00:12:35.410 align:middle line:84% story, but I would argue that when you buy health insurance, 00:12:35.410 --> 00:12:37.030 align:middle line:90% it's very, very complicated. 00:12:37.030 --> 00:12:39.070 align:middle line:84% The exact quality of the network you have 00:12:39.070 --> 00:12:42.290 align:middle line:84% is very, very difficult to untangle. 00:12:42.290 --> 00:12:44.870 align:middle line:84% So technically, when you buy a health insurance plan, what's 00:12:44.870 --> 00:12:47.710 align:middle line:84% relevant to you is every single doctor 00:12:47.710 --> 00:12:50.370 align:middle line:84% in the world who's covered by the plan and who's not. 00:12:50.370 --> 00:12:52.150 align:middle line:84% And more precisely, exactly how much 00:12:52.150 --> 00:12:55.750 align:middle line:84% your copay is at every single doctor maybe in the world. 00:12:55.750 --> 00:12:57.490 align:middle line:84% You probably don't know that that well, 00:12:57.490 --> 00:12:59.490 align:middle line:84% even if you pore through the contract. 00:12:59.490 --> 00:13:02.510 align:middle line:84% The seller probably has a much better idea of, 00:13:02.510 --> 00:13:03.710 align:middle line:90% maybe what I'll say-- 00:13:03.710 --> 00:13:06.975 align:middle line:90% the network coverage. 00:13:06.975 --> 00:13:08.350 align:middle line:84% Or what they certainly know about 00:13:08.350 --> 00:13:13.070 align:middle line:84% is when you apply to get reimbursed for a treatment, how 00:13:13.070 --> 00:13:15.790 align:middle line:84% cumbersome that process is, and what the rejection 00:13:15.790 --> 00:13:16.850 align:middle line:90% rate is of the insurer. 00:13:16.850 --> 00:13:18.350 align:middle line:84% This is something the insurer knows. 00:13:18.350 --> 00:13:21.070 align:middle line:84% Some insurers might be a bit more lenient than others. 00:13:21.070 --> 00:13:22.230 align:middle line:90% They privately know that. 00:13:22.230 --> 00:13:24.050 align:middle line:84% Maybe you've gone on Reddit and read a bit about it, 00:13:24.050 --> 00:13:26.570 align:middle line:84% but you probably don't have full information about that. 00:13:26.570 --> 00:13:29.590 align:middle line:90% 00:13:29.590 --> 00:13:33.120 align:middle line:90% Let's do a few more examples. 00:13:33.120 --> 00:13:35.800 align:middle line:90% What about in an auction? 00:13:35.800 --> 00:13:37.080 align:middle line:90% Context? 00:13:37.080 --> 00:13:39.180 align:middle line:84% Well, here it's the classic case. 00:13:39.180 --> 00:13:40.900 align:middle line:84% When the bidders show up to an auction, 00:13:40.900 --> 00:13:43.560 align:middle line:84% the bidders generally know how much 00:13:43.560 --> 00:13:45.157 align:middle line:84% they value the item being auctioned, 00:13:45.157 --> 00:13:46.740 align:middle line:84% but the other bidders don't know that. 00:13:46.740 --> 00:13:52.620 align:middle line:84% So each bidder knows her value for the good. 00:13:52.620 --> 00:13:58.552 align:middle line:90% 00:13:58.552 --> 00:14:00.760 align:middle line:84% And of course, this value could reflect a few things. 00:14:00.760 --> 00:14:02.540 align:middle line:84% If I'm bidding against someone in auction, 00:14:02.540 --> 00:14:03.840 align:middle line:84% their value might be high because they just 00:14:03.840 --> 00:14:05.660 align:middle line:84% love this painting or maybe they're just really, 00:14:05.660 --> 00:14:06.220 align:middle line:90% really rich. 00:14:06.220 --> 00:14:07.360 align:middle line:84% Or maybe they just got all this money 00:14:07.360 --> 00:14:08.693 align:middle line:90% and they have to spend it today. 00:14:08.693 --> 00:14:10.540 align:middle line:84% There's all these stories that could go on, 00:14:10.540 --> 00:14:12.520 align:middle line:84% but their willingness to pay for the item 00:14:12.520 --> 00:14:16.000 align:middle line:84% is something that they know that I don't know. 00:14:16.000 --> 00:14:17.640 align:middle line:84% And then a classic case that we've also 00:14:17.640 --> 00:14:21.920 align:middle line:84% discussed so far is maybe bargaining negotiation 00:14:21.920 --> 00:14:22.940 align:middle line:90% more broadly. 00:14:22.940 --> 00:14:25.760 align:middle line:90% 00:14:25.760 --> 00:14:29.560 align:middle line:84% Let's say we gave the example of bargaining over a peace treaty. 00:14:29.560 --> 00:14:31.692 align:middle line:84% When both sides sit down at the table, 00:14:31.692 --> 00:14:33.650 align:middle line:84% they don't really know what terms of the treaty 00:14:33.650 --> 00:14:34.650 align:middle line:90% are acceptable. 00:14:34.650 --> 00:14:36.510 align:middle line:84% If I'm negotiating with a foreign leader-- 00:14:36.510 --> 00:14:38.050 align:middle line:90% I would never be doing that-- 00:14:38.050 --> 00:14:40.350 align:middle line:84% they know their political constraints at home. 00:14:40.350 --> 00:14:42.210 align:middle line:84% They know what kind of deal they can make that would get them 00:14:42.210 --> 00:14:42.970 align:middle line:90% in trouble or not. 00:14:42.970 --> 00:14:44.470 align:middle line:84% They might know facts on the ground. 00:14:44.470 --> 00:14:47.010 align:middle line:84% They might know things about the condition of their soldiers 00:14:47.010 --> 00:14:49.302 align:middle line:84% and their army that are all relevant to the negotiation 00:14:49.302 --> 00:14:51.370 align:middle line:84% that I, as the counterparty, don't know. 00:14:51.370 --> 00:14:55.650 align:middle line:84% So there's just tons of private information. 00:14:55.650 --> 00:15:00.870 align:middle line:84% I'll just say about really, what is it about? 00:15:00.870 --> 00:15:04.290 align:middle line:84% It's about their preferences over the terms 00:15:04.290 --> 00:15:06.310 align:middle line:84% of the negotiation, over terms of the deal. 00:15:06.310 --> 00:15:10.490 align:middle line:90% 00:15:10.490 --> 00:15:13.890 align:middle line:84% So I think it's nice to step back here and pause 00:15:13.890 --> 00:15:16.730 align:middle line:84% and look at the history of game theory. 00:15:16.730 --> 00:15:19.430 align:middle line:84% Game theory started-- as I said earlier in the course, 00:15:19.430 --> 00:15:21.070 align:middle line:84% it was studied by mathematicians. 00:15:21.070 --> 00:15:23.810 align:middle line:84% They were analyzing things like chess and checkers and poker, 00:15:23.810 --> 00:15:26.210 align:middle line:84% and they presented this theory of complete information 00:15:26.210 --> 00:15:30.020 align:middle line:84% that did a very good job of capturing games like chess 00:15:30.020 --> 00:15:31.140 align:middle line:90% and checkers and poker. 00:15:31.140 --> 00:15:32.515 align:middle line:90% But there was this folk wisdom. 00:15:32.515 --> 00:15:34.140 align:middle line:84% There was this view that we could never 00:15:34.140 --> 00:15:36.120 align:middle line:84% apply game theory to the real world, 00:15:36.120 --> 00:15:38.800 align:middle line:84% to things like this because the game isn't common knowledge. 00:15:38.800 --> 00:15:40.980 align:middle line:84% There's just so much information that we 00:15:40.980 --> 00:15:42.813 align:middle line:84% don't know that we can never apply the tools 00:15:42.813 --> 00:15:45.460 align:middle line:84% and the rigor of game theoretic reasoning. 00:15:45.460 --> 00:15:49.360 align:middle line:84% And there was a big breakthrough in the '60s. 00:15:49.360 --> 00:15:55.980 align:middle line:84% So the breakthrough came with Harsanyi 00:15:55.980 --> 00:16:01.420 align:middle line:84% in 1967, 1968, who presented a framework 00:16:01.420 --> 00:16:05.553 align:middle line:84% for analyzing situations like this using game theory. 00:16:05.553 --> 00:16:07.720 align:middle line:84% And I think this was a turning point in game theory, 00:16:07.720 --> 00:16:09.140 align:middle line:84% where before that, it was largely 00:16:09.140 --> 00:16:10.223 align:middle line:90% studied by mathematicians. 00:16:10.223 --> 00:16:12.557 align:middle line:84% And then once people realized that this is the right way 00:16:12.557 --> 00:16:15.040 align:middle line:84% to think about health insurance markets or bargaining, 00:16:15.040 --> 00:16:16.940 align:middle line:84% all of a sudden economists became much more interested 00:16:16.940 --> 00:16:19.120 align:middle line:84% in it, and the nature of the field kind of shifted. 00:16:19.120 --> 00:16:22.740 align:middle line:84% And Harsanyi later won the Nobel Prize for his work here. 00:16:22.740 --> 00:16:27.830 align:middle line:84% So I think Harsanyi's key insight, which was later 00:16:27.830 --> 00:16:30.510 align:middle line:84% formalized by others, is that we can 00:16:30.510 --> 00:16:43.070 align:middle line:84% model incomplete information like we 00:16:43.070 --> 00:16:45.530 align:middle line:90% model imperfect information. 00:16:45.530 --> 00:16:49.990 align:middle line:90% 00:16:49.990 --> 00:16:53.350 align:middle line:84% So in other words, I think as a sentence, 00:16:53.350 --> 00:16:54.850 align:middle line:84% this doesn't really make much sense, 00:16:54.850 --> 00:16:55.933 align:middle line:90% so let's say what we mean. 00:16:55.933 --> 00:16:57.750 align:middle line:84% What Harsanyi said is maybe we can 00:16:57.750 --> 00:17:00.870 align:middle line:84% model negotiation and health insurance the way 00:17:00.870 --> 00:17:02.870 align:middle line:90% we model poker. 00:17:02.870 --> 00:17:05.589 align:middle line:84% So why don't we think about someone's health status 00:17:05.589 --> 00:17:11.470 align:middle line:84% in healthcare or a military leader's private information 00:17:11.470 --> 00:17:13.630 align:middle line:84% about their army the way we model 00:17:13.630 --> 00:17:16.829 align:middle line:90% your hand in a game of poker? 00:17:16.829 --> 00:17:18.470 align:middle line:84% So the key is that we don't literally 00:17:18.470 --> 00:17:21.710 align:middle line:84% have a dealer who's dealing you cards, 00:17:21.710 --> 00:17:24.190 align:middle line:84% but we can model this private information 00:17:24.190 --> 00:17:28.160 align:middle line:84% as the move of a metaphorical dealer, or metaphorical move 00:17:28.160 --> 00:17:31.040 align:middle line:84% by nature who deals people their private information. 00:17:31.040 --> 00:17:43.520 align:middle line:84% So Sonia's idea is that we can have a metaphorical first move 00:17:43.520 --> 00:17:44.402 align:middle line:90% by nature. 00:17:44.402 --> 00:17:45.860 align:middle line:84% And I want you to think about this. 00:17:45.860 --> 00:17:46.960 align:middle line:84% I think the best way to think about this 00:17:46.960 --> 00:17:49.002 align:middle line:84% is nature is playing the role of the card dealer. 00:17:49.002 --> 00:17:51.680 align:middle line:84% But what they're dealing you is your health status 00:17:51.680 --> 00:17:54.440 align:middle line:84% or how much you value the painting or the condition 00:17:54.440 --> 00:17:55.680 align:middle line:90% of your army. 00:17:55.680 --> 00:17:59.040 align:middle line:84% And after nature makes that deal, 00:17:59.040 --> 00:18:01.840 align:middle line:84% then we can analyze the game like a game 00:18:01.840 --> 00:18:03.500 align:middle line:90% of imperfect information. 00:18:03.500 --> 00:18:08.880 align:middle line:90% 00:18:08.880 --> 00:18:16.840 align:middle line:84% Now, I think a challenge here is that this is metaphorical. 00:18:16.840 --> 00:18:20.920 align:middle line:84% So I think in poker, it's very clear 00:18:20.920 --> 00:18:23.570 align:middle line:84% how we can reason through a situation 00:18:23.570 --> 00:18:25.790 align:middle line:84% before the cards are dealt. If I'm playing poker, 00:18:25.790 --> 00:18:28.390 align:middle line:84% I'm sitting at the table, I don't have the cards yet. 00:18:28.390 --> 00:18:29.750 align:middle line:90% The dealer deals me a card. 00:18:29.750 --> 00:18:31.000 align:middle line:90% I know that's going to happen. 00:18:31.000 --> 00:18:33.090 align:middle line:84% And I can think about when that's going on. 00:18:33.090 --> 00:18:36.610 align:middle line:84% It's a bit hard to think about you buying health insurance 00:18:36.610 --> 00:18:41.230 align:middle line:84% at the stage before nature dealt you your health status. 00:18:41.230 --> 00:18:43.610 align:middle line:84% There is this subtlety in how we interpret it, 00:18:43.610 --> 00:18:46.250 align:middle line:84% but despite the difficulty of interpretation-- so maybe 00:18:46.250 --> 00:18:49.590 align:middle line:84% I'll say this creates some interpretational difficulties. 00:18:49.590 --> 00:18:55.170 align:middle line:90% 00:18:55.170 --> 00:18:57.610 align:middle line:84% And we'll see those throughout the course. 00:18:57.610 --> 00:19:02.170 align:middle line:84% But nevertheless, we can analyze these things mathematically 00:19:02.170 --> 00:19:03.790 align:middle line:90% and we can make some progress. 00:19:03.790 --> 00:19:07.810 align:middle line:84% One difficulty, I guess, in the first group by nature 00:19:07.810 --> 00:19:10.730 align:middle line:84% is we have to specify the probability distribution 00:19:10.730 --> 00:19:12.167 align:middle line:90% over what nature chooses. 00:19:12.167 --> 00:19:14.750 align:middle line:84% That's a lot easier when nature is drawing a card from a deck. 00:19:14.750 --> 00:19:15.708 align:middle line:90% We all understand that. 00:19:15.708 --> 00:19:18.167 align:middle line:84% When nature is drawing your health status or your valuation 00:19:18.167 --> 00:19:20.190 align:middle line:84% for a good, that becomes a bit more challenging. 00:19:20.190 --> 00:19:24.460 align:middle line:84% So one challenge is that we have to specify the distribution 00:19:24.460 --> 00:19:29.000 align:middle line:90% over nature's moves. 00:19:29.000 --> 00:19:33.780 align:middle line:90% 00:19:33.780 --> 00:19:39.020 align:middle line:84% But let's see if we can get into this and start analyzing this. 00:19:39.020 --> 00:19:41.120 align:middle line:84% So I could present the whole general model, 00:19:41.120 --> 00:19:43.680 align:middle line:84% but I think it's very complicated at first. 00:19:43.680 --> 00:19:46.160 align:middle line:84% And you might say, why do we need this complicated model? 00:19:46.160 --> 00:19:47.702 align:middle line:84% So let's start with an example to try 00:19:47.702 --> 00:19:49.360 align:middle line:84% to motivate why things are subtle 00:19:49.360 --> 00:19:53.020 align:middle line:84% and why we actually need to do some work to model this. 00:19:53.020 --> 00:19:55.168 align:middle line:84% So let's think about a classic example. 00:19:55.168 --> 00:19:56.960 align:middle line:84% Let's say we're working on a group project. 00:19:56.960 --> 00:19:59.780 align:middle line:90% 00:19:59.780 --> 00:20:01.720 align:middle line:84% And let's say we have two players. 00:20:01.720 --> 00:20:05.860 align:middle line:90% 00:20:05.860 --> 00:20:10.780 align:middle line:84% And I'll call them i equals 1 and i equals 2. 00:20:10.780 --> 00:20:18.260 align:middle line:84% And each player, i, is going to choose effort on the project. 00:20:18.260 --> 00:20:23.670 align:middle line:90% I'm going to choose effort ei. 00:20:23.670 --> 00:20:25.510 align:middle line:84% The player i, they choose effort ei. 00:20:25.510 --> 00:20:28.270 align:middle line:84% And let's say that's a number between 0 and 1. 00:20:28.270 --> 00:20:29.870 align:middle line:84% So 0 is they don't work at all, 1 00:20:29.870 --> 00:20:33.230 align:middle line:84% is they work as hard as they possibly can. 00:20:33.230 --> 00:20:39.750 align:middle line:84% And now we're going to say the payoffs to player i-- 00:20:39.750 --> 00:20:42.150 align:middle line:84% well, player i's payoff depends on how well 00:20:42.150 --> 00:20:44.950 align:middle line:84% the project turns out minus the cost they 00:20:44.950 --> 00:20:47.290 align:middle line:90% have to bear to exert effort. 00:20:47.290 --> 00:20:50.550 align:middle line:90% Let me see if I had 1/2 in here. 00:20:50.550 --> 00:20:59.010 align:middle line:84% So let's say that their payoff is e1 times e2 minus ei squared. 00:20:59.010 --> 00:21:02.190 align:middle line:90% 00:21:02.190 --> 00:21:06.590 align:middle line:84% So what this says is e1 times e2-- the product of the effort 00:21:06.590 --> 00:21:08.190 align:middle line:84% that the two players put in, this 00:21:08.190 --> 00:21:12.470 align:middle line:84% is going to represent basically how good the project is, 00:21:12.470 --> 00:21:13.770 align:middle line:90% how good the output is. 00:21:13.770 --> 00:21:16.990 align:middle line:90% 00:21:16.990 --> 00:21:18.550 align:middle line:90% And this is the cost of effort. 00:21:18.550 --> 00:21:22.170 align:middle line:90% 00:21:22.170 --> 00:21:27.190 align:middle line:84% So notice, if I'm player i, my effort ei affects my cost. 00:21:27.190 --> 00:21:29.530 align:middle line:84% And it also affects the quality of the output. 00:21:29.530 --> 00:21:31.193 align:middle line:84% The other player's effort affects 00:21:31.193 --> 00:21:32.610 align:middle line:84% the quality of the output, but not 00:21:32.610 --> 00:21:34.890 align:middle line:84% my cost, because I don't bear the cost for the effort 00:21:34.890 --> 00:21:37.210 align:middle line:84% that the other player is putting in. 00:21:37.210 --> 00:21:39.170 align:middle line:84% This is a pretty simple game to analyze, 00:21:39.170 --> 00:21:42.570 align:middle line:84% but the trick is we're now going to put a theta up here, 00:21:42.570 --> 00:21:45.290 align:middle line:84% where theta is going to be a state that represents 00:21:45.290 --> 00:21:46.398 align:middle line:90% how hard the project is. 00:21:46.398 --> 00:21:48.190 align:middle line:84% When you get working on something together, 00:21:48.190 --> 00:21:53.410 align:middle line:84% you often don't know how hard this task is. 00:21:53.410 --> 00:21:58.150 align:middle line:84% So theta is going to represent the difficulty of the project. 00:21:58.150 --> 00:22:01.510 align:middle line:84% Actually, a higher theta means the project is easier. 00:22:01.510 --> 00:22:04.550 align:middle line:84% So maybe I'll think of theta as how easy the project is. 00:22:04.550 --> 00:22:11.130 align:middle line:90% 00:22:11.130 --> 00:22:17.900 align:middle line:84% So as a simple case, we could say maybe two values, theta H 00:22:17.900 --> 00:22:21.220 align:middle line:84% and theta L. So these are the two possible values 00:22:21.220 --> 00:22:22.620 align:middle line:90% theta could take. 00:22:22.620 --> 00:22:25.900 align:middle line:84% Theta H is a low value because if the project is hard, 00:22:25.900 --> 00:22:27.360 align:middle line:84% when you put in this much effort, 00:22:27.360 --> 00:22:29.180 align:middle line:90% you don't get very much output. 00:22:29.180 --> 00:22:32.160 align:middle line:84% If the project is easy, theta L-- 00:22:32.160 --> 00:22:34.060 align:middle line:84% and I guess I'm so used to high and low. 00:22:34.060 --> 00:22:35.920 align:middle line:90% So easy-- theta H, theta e. 00:22:35.920 --> 00:22:38.900 align:middle line:84% If the project is easy, then you get more output 00:22:38.900 --> 00:22:41.965 align:middle line:90% for the same level of effort. 00:22:41.965 --> 00:22:43.340 align:middle line:84% The problem is you probably don't 00:22:43.340 --> 00:22:46.120 align:middle line:90% know how hard the project is. 00:22:46.120 --> 00:22:49.420 align:middle line:90% 00:22:49.420 --> 00:22:52.320 align:middle line:84% Let's think through a few cases here. 00:22:52.320 --> 00:22:56.100 align:middle line:90% 00:22:56.100 --> 00:23:01.960 align:middle line:84% So the easy case, case one would be no one knows theta. 00:23:01.960 --> 00:23:06.180 align:middle line:90% 00:23:06.180 --> 00:23:07.860 align:middle line:84% If no one knows theta, and they just 00:23:07.860 --> 00:23:10.220 align:middle line:84% believe theta follows some distribution, 00:23:10.220 --> 00:23:13.150 align:middle line:84% then we can analyze this as a classic game, 00:23:13.150 --> 00:23:16.790 align:middle line:84% because at the beginning of the game, information is symmetric. 00:23:16.790 --> 00:23:18.810 align:middle line:84% So here, this would be symmetric info. 00:23:18.810 --> 00:23:23.030 align:middle line:90% 00:23:23.030 --> 00:23:24.830 align:middle line:84% If no one knows theta, then there's 00:23:24.830 --> 00:23:27.190 align:middle line:84% no information that one player knows that another player 00:23:27.190 --> 00:23:28.230 align:middle line:90% doesn't know. 00:23:28.230 --> 00:23:29.670 align:middle line:90% But the harder case-- 00:23:29.670 --> 00:23:33.610 align:middle line:84% let's look at case 2 where only player one knows theta. 00:23:33.610 --> 00:23:46.150 align:middle line:90% 00:23:46.150 --> 00:23:49.990 align:middle line:84% So if only player 1 knows theta, let's 00:23:49.990 --> 00:23:54.870 align:middle line:84% see if we can start analyzing this game. 00:23:54.870 --> 00:23:58.590 align:middle line:84% Well, in case two, player 1 is going 00:23:58.590 --> 00:24:00.910 align:middle line:84% to possibly make a different choice depending 00:24:00.910 --> 00:24:03.110 align:middle line:90% on how hard the project is. 00:24:03.110 --> 00:24:12.760 align:middle line:84% So player 1 is going to make maybe one choice, e1 of theta H, 00:24:12.760 --> 00:24:16.840 align:middle line:84% and a different choice, e2 of theta L. If player 1 knows 00:24:16.840 --> 00:24:19.287 align:middle line:84% how hard the project is, if the project is hard, 00:24:19.287 --> 00:24:21.120 align:middle line:84% they might find it not as worthwhile to work 00:24:21.120 --> 00:24:22.320 align:middle line:90% on the project. 00:24:22.320 --> 00:24:24.580 align:middle line:84% This should be E1, and this should be easy, 00:24:24.580 --> 00:24:27.338 align:middle line:90% so everything's wrong here. 00:24:27.338 --> 00:24:28.880 align:middle line:84% If they know the project's easy, they 00:24:28.880 --> 00:24:30.560 align:middle line:84% might choose a different level of effort. 00:24:30.560 --> 00:24:31.800 align:middle line:84% So we're going to have to capture that when 00:24:31.800 --> 00:24:33.960 align:middle line:84% we think about a player's strategy in the game. 00:24:33.960 --> 00:24:37.880 align:middle line:84% Player 1's strategy might depend on how hard the project is. 00:24:37.880 --> 00:24:40.880 align:middle line:90% But now let's look at player 2. 00:24:40.880 --> 00:24:41.900 align:middle line:90% What should player 2 do? 00:24:41.900 --> 00:24:44.400 align:middle line:84% Let's imagine you are in this situation. 00:24:44.400 --> 00:24:47.360 align:middle line:84% You don't know how hard the project is, 00:24:47.360 --> 00:24:50.480 align:middle line:84% and you're thinking about how hard to work. 00:24:50.480 --> 00:24:51.120 align:middle line:90% I don't know. 00:24:51.120 --> 00:24:53.328 align:middle line:84% I think there's some information that's missing here. 00:24:53.328 --> 00:24:55.320 align:middle line:84% There's something you need to know. 00:24:55.320 --> 00:24:58.580 align:middle line:84% So we've said they don't know how hard the project is. 00:24:58.580 --> 00:25:05.640 align:middle line:90% 00:25:05.640 --> 00:25:06.780 align:middle line:90% So they don't know theta. 00:25:06.780 --> 00:25:10.940 align:middle line:90% 00:25:10.940 --> 00:25:12.690 align:middle line:84% But I think we're missing some information 00:25:12.690 --> 00:25:16.570 align:middle line:90% to see how they should behave. 00:25:16.570 --> 00:25:17.448 align:middle line:90% I don't know theta. 00:25:17.448 --> 00:25:17.990 align:middle line:90% I don't know. 00:25:17.990 --> 00:25:22.210 align:middle line:84% How would you approach this if you were player 2? 00:25:22.210 --> 00:25:23.330 align:middle line:90% Yeah. 00:25:23.330 --> 00:25:26.810 align:middle line:84% AUDIENCE: Maybe in terms of a security strategy. 00:25:26.810 --> 00:25:28.970 align:middle line:90% IAN BALL: OK. 00:25:28.970 --> 00:25:31.450 align:middle line:84% AUDIENCE: Let's try to minimize the possibility that it 00:25:31.450 --> 00:25:33.250 align:middle line:90% goes wrong, given that-- 00:25:33.250 --> 00:25:36.130 align:middle line:90% I don't know if it's a-- 00:25:36.130 --> 00:25:36.930 align:middle line:90% I'm not sure. 00:25:36.930 --> 00:25:37.950 align:middle line:84% IAN BALL: So that would be one way. 00:25:37.950 --> 00:25:39.650 align:middle line:84% No, I think that would be a reasonable answer. 00:25:39.650 --> 00:25:41.590 align:middle line:84% We're not going to pursue that too much in this course, 00:25:41.590 --> 00:25:42.750 align:middle line:84% but that's something that people do. 00:25:42.750 --> 00:25:44.292 align:middle line:84% And that's one thing you could study. 00:25:44.292 --> 00:25:46.450 align:middle line:90% Yeah? 00:25:46.450 --> 00:25:48.370 align:middle line:84% AUDIENCE: If you know how much player 1 cares 00:25:48.370 --> 00:25:51.670 align:middle line:84% about the project, you could use backwards induction. 00:25:51.670 --> 00:25:54.428 align:middle line:84% If player 1 knows theta and player 1 cares a lot, 00:25:54.428 --> 00:25:56.970 align:middle line:84% you will know that player 1 is just going to do the projects, 00:25:56.970 --> 00:25:58.110 align:middle line:84% and you don't have to do anything. 00:25:58.110 --> 00:25:59.050 align:middle line:84% IAN BALL: So I think the first thing you 00:25:59.050 --> 00:25:59.990 align:middle line:90% said there was really crucial. 00:25:59.990 --> 00:26:01.530 align:middle line:84% You said, if player 1 knows theta, 00:26:01.530 --> 00:26:03.370 align:middle line:90% and then you reason from that. 00:26:03.370 --> 00:26:05.640 align:middle line:84% But does player 2 know that player 1 knows theta? 00:26:05.640 --> 00:26:07.390 align:middle line:84% Well, we need to make a choice about that. 00:26:07.390 --> 00:26:08.640 align:middle line:90% So this is the crucial thing. 00:26:08.640 --> 00:26:10.580 align:middle line:84% It's not just enough to say, I've 00:26:10.580 --> 00:26:12.760 align:middle line:84% said player 2 doesn't know whether project is hard. 00:26:12.760 --> 00:26:14.160 align:middle line:84% But you immediately gave it away. 00:26:14.160 --> 00:26:15.600 align:middle line:84% When you're reasoning about this, 00:26:15.600 --> 00:26:19.000 align:middle line:84% I need to think through how hard is player 1 going to work? 00:26:19.000 --> 00:26:20.500 align:middle line:84% But how hard player 1 works is going 00:26:20.500 --> 00:26:23.880 align:middle line:84% to depend on whether player 1 knows how hard the project is. 00:26:23.880 --> 00:26:24.820 align:middle line:90% So I need to think. 00:26:24.820 --> 00:26:26.860 align:middle line:84% Here's a question for the analysis. 00:26:26.860 --> 00:26:32.880 align:middle line:84% Does player 2 know whether player one knows theta? 00:26:32.880 --> 00:26:41.660 align:middle line:90% 00:26:41.660 --> 00:26:45.340 align:middle line:84% Because that's going to affect the way that player 1 moves, 00:26:45.340 --> 00:26:49.020 align:middle line:84% and therefore, affect the best thing for player 2. 00:26:49.020 --> 00:26:51.480 align:middle line:84% But now I think we're getting into a regress. 00:26:51.480 --> 00:26:52.720 align:middle line:90% Let's say you're player 1. 00:26:52.720 --> 00:26:55.100 align:middle line:84% How are you thinking through this? 00:26:55.100 --> 00:26:58.820 align:middle line:84% What's relevant now for player 1 when they're trying to think? 00:26:58.820 --> 00:27:01.260 align:middle line:90% We know player 1 knows theta. 00:27:01.260 --> 00:27:02.680 align:middle line:84% But player 1 needs to think, well, 00:27:02.680 --> 00:27:04.780 align:middle line:84% how hard is player 2 going to work? 00:27:04.780 --> 00:27:07.070 align:middle line:84% But how hard player 2 works depends on 00:27:07.070 --> 00:27:10.390 align:middle line:84% whether player 2 knows whether player 1 knows theta. 00:27:10.390 --> 00:27:14.910 align:middle line:84% So now we have to think from player 1's perspective, 00:27:14.910 --> 00:27:21.150 align:middle line:84% does player 1 know whether player 2 knows 00:27:21.150 --> 00:27:22.770 align:middle line:90% whether player 1 knows theta? 00:27:22.770 --> 00:27:36.550 align:middle line:90% 00:27:36.550 --> 00:27:40.510 align:middle line:84% And I think you can see where this is going, right? 00:27:40.510 --> 00:27:42.870 align:middle line:84% We care about what the other player is going to do. 00:27:42.870 --> 00:27:45.350 align:middle line:84% What they do depends on their beliefs about what I know, 00:27:45.350 --> 00:27:47.990 align:middle line:84% but also their beliefs about what I know that they know 00:27:47.990 --> 00:27:50.615 align:middle line:84% and my beliefs about what they know that I know that they know, 00:27:50.615 --> 00:27:54.750 align:middle line:90% and it unravels here. 00:27:54.750 --> 00:27:59.098 align:middle line:84% So we need a way to get a handle on this. 00:27:59.098 --> 00:28:01.390 align:middle line:84% And I think this was what was seen as the key challenge 00:28:01.390 --> 00:28:02.130 align:middle line:90% to these games. 00:28:02.130 --> 00:28:03.505 align:middle line:84% At first, you might think, all we 00:28:03.505 --> 00:28:05.683 align:middle line:84% need to specify is what each player knows 00:28:05.683 --> 00:28:07.600 align:middle line:84% about theta, about the state of the world, how 00:28:07.600 --> 00:28:08.840 align:middle line:90% hard the project is. 00:28:08.840 --> 00:28:11.700 align:middle line:84% But we soon realize that to fully analyze the game, 00:28:11.700 --> 00:28:14.120 align:middle line:84% we need to take a stance on what each player knows 00:28:14.120 --> 00:28:16.200 align:middle line:84% about what other people know, and what 00:28:16.200 --> 00:28:19.320 align:middle line:84% other people know about what other people know, and so on. 00:28:19.320 --> 00:28:22.360 align:middle line:84% And so the key takeaway here is that we really 00:28:22.360 --> 00:28:26.400 align:middle line:84% need to distinguish two different aspects of the game. 00:28:26.400 --> 00:28:28.700 align:middle line:84% So I think this motivates a key distinction. 00:28:28.700 --> 00:28:32.120 align:middle line:90% 00:28:32.120 --> 00:28:34.920 align:middle line:90% I need another arrow here. 00:28:34.920 --> 00:28:45.000 align:middle line:84% Between the physical state theta, this is, in this example, 00:28:45.000 --> 00:28:46.420 align:middle line:90% how hard the project is. 00:28:46.420 --> 00:28:48.770 align:middle line:84% It's totally outside the minds of the players. 00:28:48.770 --> 00:28:50.020 align:middle line:90% It's either hard or it's easy. 00:28:50.020 --> 00:28:53.200 align:middle line:84% That's just some fact of the world, fact of nature. 00:28:53.200 --> 00:28:56.360 align:middle line:84% But then we also need to think about maybe the mental state 00:28:56.360 --> 00:29:08.347 align:middle line:84% of each player, which maybe before we saw this example, 00:29:08.347 --> 00:29:10.180 align:middle line:84% we might have thought, oh, mental state just 00:29:10.180 --> 00:29:12.080 align:middle line:90% means their belief about theta. 00:29:12.080 --> 00:29:14.500 align:middle line:84% But now it really means their belief 00:29:14.500 --> 00:29:16.403 align:middle line:84% about the mental state of the other player. 00:29:16.403 --> 00:29:17.820 align:middle line:84% And in general, we're going to try 00:29:17.820 --> 00:29:21.980 align:middle line:84% to have this mental state capture all the relevant beliefs 00:29:21.980 --> 00:29:24.460 align:middle line:84% about the physical state and also about the mental state 00:29:24.460 --> 00:29:25.980 align:middle line:90% of the other player. 00:29:25.980 --> 00:29:28.900 align:middle line:84% And that's the direction we're going to go now 00:29:28.900 --> 00:29:31.220 align:middle line:90% with this formal setup. 00:29:31.220 --> 00:29:33.400 align:middle line:90% Any questions about this? 00:29:33.400 --> 00:29:35.280 align:middle line:84% So I don't expect people to see the solution, 00:29:35.280 --> 00:29:36.420 align:middle line:84% but just to see that there's a problem 00:29:36.420 --> 00:29:37.880 align:middle line:90% is the goal of this example. 00:29:37.880 --> 00:30:13.727 align:middle line:90% 00:30:13.727 --> 00:30:15.310 align:middle line:84% So now we're going to formally present 00:30:15.310 --> 00:30:18.893 align:middle line:84% our model of a game that can capture incomplete information. 00:30:18.893 --> 00:30:20.810 align:middle line:84% And we're going to be using Bayes' rule a lot, 00:30:20.810 --> 00:30:22.890 align:middle line:84% so these games are often called Bayesian games. 00:30:22.890 --> 00:30:27.710 align:middle line:90% 00:30:27.710 --> 00:30:32.230 align:middle line:84% So how do we try to capture a situation like this? 00:30:32.230 --> 00:30:34.210 align:middle line:84% Well, first we're going to have the players. 00:30:34.210 --> 00:30:38.830 align:middle line:90% 00:30:38.830 --> 00:30:41.390 align:middle line:90% i equals 1 up to n. 00:30:41.390 --> 00:30:43.630 align:middle line:84% And then we're going to have the actions. 00:30:43.630 --> 00:30:49.390 align:middle line:84% So each player-- so player i is going 00:30:49.390 --> 00:30:51.730 align:middle line:84% to choose some action in some action set. 00:30:51.730 --> 00:31:01.560 align:middle line:90% 00:31:01.560 --> 00:31:04.080 align:middle line:84% So just like in repeated games, if we 00:31:04.080 --> 00:31:05.777 align:middle line:84% were to call this a strategy set, 00:31:05.777 --> 00:31:07.360 align:middle line:84% we would then run into issues, because 00:31:07.360 --> 00:31:08.560 align:middle line:84% in the meta-Bayesian game, there's 00:31:08.560 --> 00:31:10.020 align:middle line:84% going to be a different notion of strategy. 00:31:10.020 --> 00:31:12.228 align:middle line:84% So I want to use the term action instead of strategy. 00:31:12.228 --> 00:31:14.960 align:middle line:84% I think the notes talk about Bayesian games 00:31:14.960 --> 00:31:17.132 align:middle line:84% with Bayesian extensive form games at this part. 00:31:17.132 --> 00:31:19.340 align:middle line:84% We're not going to talk about that, at least for now. 00:31:19.340 --> 00:31:21.280 align:middle line:84% So we're going to think of this as just a simultaneous move 00:31:21.280 --> 00:31:21.440 align:middle line:90% game. 00:31:21.440 --> 00:31:23.898 align:middle line:84% We're each choosing an action, just like we did over there. 00:31:23.898 --> 00:31:26.960 align:middle line:90% We each choose how hard to work. 00:31:26.960 --> 00:31:32.360 align:middle line:84% But then there's going to be some state of nature, maybe 00:31:32.360 --> 00:31:39.620 align:middle line:84% our physical state or sometimes called a state of nature. 00:31:39.620 --> 00:31:45.480 align:middle line:90% 00:31:45.480 --> 00:31:49.833 align:middle line:84% theta, which is going to be in some set, Big Theta. 00:31:49.833 --> 00:31:51.500 align:middle line:84% I don't know if we've used theta before. 00:31:51.500 --> 00:31:54.723 align:middle line:84% When I write theta, it just looks like an H inside a circle. 00:31:54.723 --> 00:31:56.390 align:middle line:84% But that's so we can see the difference. 00:31:56.390 --> 00:31:58.550 align:middle line:84% Little theta is like this, big theta is like this. 00:31:58.550 --> 00:32:00.330 align:middle line:84% So I know it looks weird, but that's 00:32:00.330 --> 00:32:02.690 align:middle line:90% how I write it on the board. 00:32:02.690 --> 00:32:05.290 align:middle line:84% And just so we understand, how we map it 00:32:05.290 --> 00:32:08.530 align:middle line:84% into our example over there-- there we had two players. 00:32:08.530 --> 00:32:13.850 align:middle line:84% Each player's action was some effort level between 0 and 1. 00:32:13.850 --> 00:32:15.810 align:middle line:84% And the physical state was the difficulty 00:32:15.810 --> 00:32:18.430 align:middle line:84% of the project, which over there, there were two states. 00:32:18.430 --> 00:32:21.090 align:middle line:84% It could either be a difficult project; H, hard project; 00:32:21.090 --> 00:32:22.810 align:middle line:90% or an easy project, theta e-- 00:32:22.810 --> 00:32:25.928 align:middle line:90% an easy project. 00:32:25.928 --> 00:32:27.470 align:middle line:84% And then we're going to need payoffs, 00:32:27.470 --> 00:32:28.970 align:middle line:90% just like we had over there. 00:32:28.970 --> 00:32:35.450 align:middle line:84% So then we're going to have utility for player i. 00:32:35.450 --> 00:32:37.490 align:middle line:84% This is going to depend on the actions 00:32:37.490 --> 00:32:39.690 align:middle line:84% that every player takes, but also potentially 00:32:39.690 --> 00:32:40.910 align:middle line:90% on the state of the world. 00:32:40.910 --> 00:32:43.370 align:middle line:84% So just like over here, my payoff 00:32:43.370 --> 00:32:47.053 align:middle line:84% depended on my own effort level, my opponent's effort level, 00:32:47.053 --> 00:32:48.470 align:middle line:84% and also the state of the project, 00:32:48.470 --> 00:32:49.750 align:middle line:90% how hard the project was. 00:32:49.750 --> 00:32:52.170 align:middle line:84% So over here, in the more abstract context, 00:32:52.170 --> 00:32:56.300 align:middle line:84% UI is going to be a function of A1 all the way up 00:32:56.300 --> 00:32:58.460 align:middle line:84% to An, the actions of every player, 00:32:58.460 --> 00:33:02.620 align:middle line:90% times the set capital theta. 00:33:02.620 --> 00:33:09.580 align:middle line:84% And it's going to go to R for each L. So this is going 00:33:09.580 --> 00:33:12.420 align:middle line:84% to give me a real numbered payoff that depends on what 00:33:12.420 --> 00:33:14.940 align:middle line:84% player 1 does, what player 2 does, all 00:33:14.940 --> 00:33:16.480 align:middle line:84% the way up to what player n does, 00:33:16.480 --> 00:33:19.300 align:middle line:84% and also the value of the physical state-- 00:33:19.300 --> 00:33:22.360 align:middle line:84% in this example, whether the project is easy or hard. 00:33:22.360 --> 00:33:25.380 align:middle line:90% 00:33:25.380 --> 00:33:27.580 align:middle line:84% But what we see from that game is 00:33:27.580 --> 00:33:29.900 align:middle line:84% we haven't specified enough yet because we also 00:33:29.900 --> 00:33:32.460 align:middle line:84% have to think about the mental states, what players know 00:33:32.460 --> 00:33:34.060 align:middle line:90% about other players. 00:33:34.060 --> 00:33:38.200 align:middle line:84% And Harsanyi's key trick is to introduce this abstract object, 00:33:38.200 --> 00:33:39.780 align:middle line:90% which we call a type. 00:33:39.780 --> 00:33:41.260 align:middle line:84% And the type of a player is going 00:33:41.260 --> 00:33:43.548 align:middle line:84% to capture their complete mental state. 00:33:43.548 --> 00:33:45.840 align:middle line:84% It could be their beliefs about the state of the world. 00:33:45.840 --> 00:33:47.882 align:middle line:84% It could be their beliefs about the other player. 00:33:47.882 --> 00:33:49.340 align:middle line:90% It could be all these things. 00:33:49.340 --> 00:34:03.350 align:middle line:84% So we're going to say player i has a type, ti, which 00:34:03.350 --> 00:34:06.390 align:middle line:90% is in some set, capital TI. 00:34:06.390 --> 00:34:13.010 align:middle line:84% So this is called player i's type space. 00:34:13.010 --> 00:34:16.909 align:middle line:90% 00:34:16.909 --> 00:34:19.750 align:middle line:84% And type is a weird terminology, but you 00:34:19.750 --> 00:34:22.713 align:middle line:84% can say there's the type of the player who knows the state. 00:34:22.713 --> 00:34:25.130 align:middle line:84% There's the type of the player who doesn't know the state. 00:34:25.130 --> 00:34:26.150 align:middle line:84% There's the type of the player who 00:34:26.150 --> 00:34:27.889 align:middle line:84% knows the other player knows the state, 00:34:27.889 --> 00:34:29.292 align:middle line:90% but doesn't know the state. 00:34:29.292 --> 00:34:31.750 align:middle line:84% We just think of these players with different mental states 00:34:31.750 --> 00:34:34.130 align:middle line:90% as different types of players. 00:34:34.130 --> 00:34:38.389 align:middle line:90% 00:34:38.389 --> 00:34:40.230 align:middle line:84% And what we're missing here is what's 00:34:40.230 --> 00:34:45.290 align:middle line:84% the connection between the types and the state of the world. 00:34:45.290 --> 00:34:47.330 align:middle line:84% And we're going to capture that with a prior. 00:34:47.330 --> 00:34:58.440 align:middle line:84% So the last component here is going to be a distribution, P. 00:34:58.440 --> 00:35:00.800 align:middle line:84% But it's not enough just to say the distribution 00:35:00.800 --> 00:35:03.200 align:middle line:84% of the physical state, but we also 00:35:03.200 --> 00:35:05.880 align:middle line:84% have to specify the distribution of the players types. 00:35:05.880 --> 00:35:09.840 align:middle line:84% And more generally, we have to specify the distribution 00:35:09.840 --> 00:35:13.420 align:middle line:84% over the combination of natural states and types. 00:35:13.420 --> 00:35:15.040 align:middle line:84% So this distribution, P, is going 00:35:15.040 --> 00:35:23.240 align:middle line:84% to be over theta cross T1 cross tn. 00:35:23.240 --> 00:35:24.640 align:middle line:84% So it's going to tell us, what is 00:35:24.640 --> 00:35:28.400 align:middle line:84% the probability that the physical state is this? 00:35:28.400 --> 00:35:31.320 align:middle line:84% Player one's mental state is something, all the way up 00:35:31.320 --> 00:35:34.280 align:middle line:84% to player N's mental state being something over here. 00:35:34.280 --> 00:35:38.720 align:middle line:84% I think this is a bit abstract, so let's try 00:35:38.720 --> 00:35:41.460 align:middle line:90% to go through our example. 00:35:41.460 --> 00:35:42.880 align:middle line:90% Maybe I'll give one. 00:35:42.880 --> 00:35:48.290 align:middle line:84% Yeah, so let's go try to capture this in our example, 00:35:48.290 --> 00:35:52.610 align:middle line:84% where I didn't specify everything. 00:35:52.610 --> 00:35:57.970 align:middle line:84% Let's say, does player 2 know whether player 1 knows theta? 00:35:57.970 --> 00:36:01.930 align:middle line:84% We're going to say for this example yes. 00:36:01.930 --> 00:36:04.290 align:middle line:84% And does player 1 know whether player 2 knows 00:36:04.290 --> 00:36:05.752 align:middle line:90% whether player 1 knows theta. 00:36:05.752 --> 00:36:06.710 align:middle line:90% We're going to say yes. 00:36:06.710 --> 00:36:07.990 align:middle line:84% So we could make it more complicated, 00:36:07.990 --> 00:36:10.330 align:middle line:84% but we're going to say all these things are known. 00:36:10.330 --> 00:36:15.050 align:middle line:84% So in this example, player 1 knows the state theta. 00:36:15.050 --> 00:36:18.250 align:middle line:84% Player 2 doesn't know the state theta, 00:36:18.250 --> 00:36:21.518 align:middle line:84% but they do know that player 1 knows the state theta. 00:36:21.518 --> 00:36:23.810 align:middle line:84% And we're going to try to capture that in our framework 00:36:23.810 --> 00:36:24.470 align:middle line:90% over here. 00:36:24.470 --> 00:36:27.530 align:middle line:84% So the first, I think, challenging step 00:36:27.530 --> 00:36:29.530 align:middle line:84% whenever we set up these models, the first thing 00:36:29.530 --> 00:36:30.790 align:middle line:90% is to specify the state. 00:36:30.790 --> 00:36:32.357 align:middle line:90% That's easy-- the state theta. 00:36:32.357 --> 00:36:34.190 align:middle line:84% And then we have to specify the type spaces, 00:36:34.190 --> 00:36:35.510 align:middle line:90% and that's always a bit harder. 00:36:35.510 --> 00:36:36.953 align:middle line:84% So let's go through this example. 00:36:36.953 --> 00:36:38.370 align:middle line:84% What is the safe space theta going 00:36:38.370 --> 00:36:41.570 align:middle line:84% to be if we want to capture our example of the group project 00:36:41.570 --> 00:36:44.530 align:middle line:90% over there? 00:36:44.530 --> 00:36:48.910 align:middle line:84% What were the physical states in that example? 00:36:48.910 --> 00:36:50.350 align:middle line:90% Yeah. 00:36:50.350 --> 00:36:51.010 align:middle line:90% Right. 00:36:51.010 --> 00:36:52.150 align:middle line:90% So we just have-- 00:36:52.150 --> 00:36:55.070 align:middle line:84% and actually, theta H and theta E. I wrote it wrong. 00:36:55.070 --> 00:36:57.868 align:middle line:84% So often we do high and low, but here, 00:36:57.868 --> 00:37:00.410 align:middle line:84% let's think of it the project is hard or the project is easy. 00:37:00.410 --> 00:37:01.827 align:middle line:84% These are the two physical states. 00:37:01.827 --> 00:37:03.828 align:middle line:90% Did I write-- 00:37:03.828 --> 00:37:04.370 align:middle line:90% I don't know. 00:37:04.370 --> 00:37:07.390 align:middle line:84% I did originally make a mistake, so that's on me. 00:37:07.390 --> 00:37:10.850 align:middle line:84% But now the hard thing is specifying the types. 00:37:10.850 --> 00:37:12.270 align:middle line:90% This is always the challenge. 00:37:12.270 --> 00:37:16.605 align:middle line:84% So the next step is to specify T1 and T2. 00:37:16.605 --> 00:37:18.230 align:middle line:84% So I think the question you have to ask 00:37:18.230 --> 00:37:22.950 align:middle line:84% yourself is, how many distinct mental states does 00:37:22.950 --> 00:37:26.070 align:middle line:90% each player have? 00:37:26.070 --> 00:37:26.650 align:middle line:90% Any thoughts? 00:37:26.650 --> 00:37:29.510 align:middle line:90% Let's think about player 1. 00:37:29.510 --> 00:37:32.670 align:middle line:90% So we know that player 1-- 00:37:32.670 --> 00:37:35.270 align:middle line:90% player 1 knows the state. 00:37:35.270 --> 00:37:42.550 align:middle line:84% So what are some distinct mental states that player 1 could have? 00:37:42.550 --> 00:37:44.860 align:middle line:90% Yeah. 00:37:44.860 --> 00:37:46.280 align:middle line:90% AUDIENCE: Player 2 [INAUDIBLE]. 00:37:46.280 --> 00:37:50.000 align:middle line:84% Player 1 can know about player 2 but doesn't 00:37:50.000 --> 00:37:52.280 align:middle line:84% know that player 1 knows [INAUDIBLE]. 00:37:52.280 --> 00:37:53.660 align:middle line:84% IAN BALL: So there's a lot of things that could happen, 00:37:53.660 --> 00:37:55.243 align:middle line:84% and this is why I tried to simplify it 00:37:55.243 --> 00:37:57.560 align:middle line:84% by saying we're going to assume that player 1 does 00:37:57.560 --> 00:38:00.313 align:middle line:84% know whether player two knows the state. 00:38:00.313 --> 00:38:00.980 align:middle line:90% So you're right. 00:38:00.980 --> 00:38:02.060 align:middle line:84% If we really wanted to get general, 00:38:02.060 --> 00:38:03.227 align:middle line:90% we'd have to deal with that. 00:38:03.227 --> 00:38:05.560 align:middle line:84% But we're going to assume player 1-- these higher order 00:38:05.560 --> 00:38:06.340 align:middle line:90% beliefs are known. 00:38:06.340 --> 00:38:07.980 align:middle line:84% We're going to make it as simple as possible. 00:38:07.980 --> 00:38:09.660 align:middle line:84% So player 1 does know all these things, 00:38:09.660 --> 00:38:11.640 align:middle line:84% but I would still argue there's certainly 00:38:11.640 --> 00:38:14.440 align:middle line:84% two different mental states player 1 could have. 00:38:14.440 --> 00:38:16.020 align:middle line:84% What are these two mental states? 00:38:16.020 --> 00:38:19.000 align:middle line:84% We said player one knows the state, but what does that mean? 00:38:19.000 --> 00:38:21.012 align:middle line:90% What could player 1 believe? 00:38:21.012 --> 00:38:22.720 align:middle line:84% If they know the state, well, either they 00:38:22.720 --> 00:38:25.357 align:middle line:84% know the state is easy or they know the state is hard. 00:38:25.357 --> 00:38:26.940 align:middle line:84% So there's at least two mental states. 00:38:26.940 --> 00:38:28.400 align:middle line:84% There's the type of player 1 where 00:38:28.400 --> 00:38:30.017 align:middle line:84% they know this is a hard project, 00:38:30.017 --> 00:38:31.600 align:middle line:84% and there's the other type of player 1 00:38:31.600 --> 00:38:33.720 align:middle line:84% where they know this is an easy project. 00:38:33.720 --> 00:38:36.220 align:middle line:84% So in this case, if we really went down this rabbit hole, 00:38:36.220 --> 00:38:37.500 align:middle line:90% we'd have to make it harder. 00:38:37.500 --> 00:38:42.130 align:middle line:84% But in this simplest example, there's actually only two types. 00:38:42.130 --> 00:38:44.930 align:middle line:84% And they exactly correspond to the mental state 00:38:44.930 --> 00:38:48.670 align:middle line:84% where player one sees or knows the project is easy 00:38:48.670 --> 00:38:52.370 align:middle line:84% and the mental state where they see or know the project is hard. 00:38:52.370 --> 00:38:54.270 align:middle line:84% There's a pretty close correspondence here, 00:38:54.270 --> 00:38:57.290 align:middle line:84% but to keep the distinction clear, let's call this tH 00:38:57.290 --> 00:38:59.170 align:middle line:90% and tE. 00:38:59.170 --> 00:39:01.930 align:middle line:84% So the interpretation is if player 1's type 00:39:01.930 --> 00:39:03.870 align:middle line:90% is tH, what does that mean? 00:39:03.870 --> 00:39:06.170 align:middle line:84% That means player 1 knows the project is hard. 00:39:06.170 --> 00:39:08.210 align:middle line:84% Notice that's different from saying the state 00:39:08.210 --> 00:39:11.050 align:middle line:84% is theta H. When I say the state is theta H, that's 00:39:11.050 --> 00:39:14.090 align:middle line:84% a statement about whether the project is actually hard or not. 00:39:14.090 --> 00:39:16.610 align:middle line:84% When I say that player 1's type is tH, 00:39:16.610 --> 00:39:20.690 align:middle line:84% that's a statement about what player 1 knows or believes 00:39:20.690 --> 00:39:21.830 align:middle line:90% about the state. 00:39:21.830 --> 00:39:24.570 align:middle line:90% 00:39:24.570 --> 00:39:27.050 align:middle line:90% What about player 2 here? 00:39:27.050 --> 00:39:29.070 align:middle line:84% It turns out in this really simple case, 00:39:29.070 --> 00:39:31.170 align:middle line:84% there's only one mental state for player 2, 00:39:31.170 --> 00:39:34.570 align:middle line:84% because they don't know whether the project is hard or easy. 00:39:34.570 --> 00:39:39.010 align:middle line:84% They do know that player 1 knows that, so they show up 00:39:39.010 --> 00:39:40.040 align:middle line:90% and they play the game. 00:39:40.040 --> 00:39:42.540 align:middle line:84% There's only one possible type, so maybe we'll 00:39:42.540 --> 00:39:49.732 align:middle line:84% just call this t0 to show They don't really know anything. 00:39:49.732 --> 00:39:52.100 align:middle line:84% So we now have specified the physical states 00:39:52.100 --> 00:39:53.280 align:middle line:90% and the mental states. 00:39:53.280 --> 00:39:54.800 align:middle line:84% We could have made it much more complicated. 00:39:54.800 --> 00:39:57.258 align:middle line:84% If there was uncertainty about what the other players knew, 00:39:57.258 --> 00:39:59.600 align:middle line:84% we'd have to model larger type spaces. 00:39:59.600 --> 00:40:03.140 align:middle line:84% But for the simple example, we've made it quite nice. 00:40:03.140 --> 00:40:10.660 align:middle line:84% And let's also assume that theta H and theta L and theta 00:40:10.660 --> 00:40:12.760 align:middle line:90% E are equally likely. 00:40:12.760 --> 00:40:17.180 align:middle line:90% 00:40:17.180 --> 00:40:19.300 align:middle line:84% And this is known among the players. 00:40:19.300 --> 00:40:23.220 align:middle line:84% So to be clear, player 2 knows that it's 00:40:23.220 --> 00:40:26.100 align:middle line:84% equally likely that the project is hard or easy. 00:40:26.100 --> 00:40:28.720 align:middle line:84% They know that player 1 knows that the project is hard 00:40:28.720 --> 00:40:31.220 align:middle line:84% or knows that it's easy, but they don't know anything else. 00:40:31.220 --> 00:40:36.500 align:middle line:84% So now the challenge is, can we describe this distribution, P? 00:40:36.500 --> 00:40:38.630 align:middle line:84% So this distribution, P, is going 00:40:38.630 --> 00:40:44.150 align:middle line:84% to be a distribution over theta cross T1 cross T2. 00:40:44.150 --> 00:40:48.610 align:middle line:84% So what we need to specify is four different probabilities. 00:40:48.610 --> 00:40:57.950 align:middle line:84% So we need to say, what's the probability of theta H, tH, t0; 00:40:57.950 --> 00:41:04.790 align:middle line:90% theta H, tL, t0; theta E-- 00:41:04.790 --> 00:41:07.110 align:middle line:90% I keep Messing this up. 00:41:07.110 --> 00:41:15.090 align:middle line:90% tH, t0, and theta E tE t0. 00:41:15.090 --> 00:41:17.670 align:middle line:90% 00:41:17.670 --> 00:41:20.490 align:middle line:84% So sometimes these are called states of the world. 00:41:20.490 --> 00:41:22.830 align:middle line:84% The distinction is theta is a state of nature 00:41:22.830 --> 00:41:24.190 align:middle line:90% or physical state. 00:41:24.190 --> 00:41:25.870 align:middle line:84% The state of the whole world specifies 00:41:25.870 --> 00:41:28.510 align:middle line:84% both the physical state and the mental states. 00:41:28.510 --> 00:41:30.890 align:middle line:84% So these are the four possible states of the world, 00:41:30.890 --> 00:41:32.150 align:middle line:90% and let's interpret them. 00:41:32.150 --> 00:41:34.250 align:middle line:84% Let's look at the second one-- just an example. 00:41:34.250 --> 00:41:37.160 align:middle line:84% This is the state of the world where the state of nature 00:41:37.160 --> 00:41:40.360 align:middle line:84% is theta H. So the project is hard. 00:41:40.360 --> 00:41:43.240 align:middle line:84% The player one's mental state is te. 00:41:43.240 --> 00:41:47.000 align:middle line:84% That means player 1 believes that the project is easy. 00:41:47.000 --> 00:41:49.560 align:middle line:84% And player two's mental state is just T0. 00:41:49.560 --> 00:41:51.180 align:middle line:84% They believe what they always believe. 00:41:51.180 --> 00:41:52.972 align:middle line:84% They don't really have anything to believe. 00:41:52.972 --> 00:41:54.880 align:middle line:84% They just have the one thing they know. 00:41:54.880 --> 00:41:58.983 align:middle line:84% So what we want to do is we need to assign probabilities 00:41:58.983 --> 00:42:00.400 align:middle line:84% to each of these four things based 00:42:00.400 --> 00:42:02.280 align:middle line:90% on the story we've told so far. 00:42:02.280 --> 00:42:07.040 align:middle line:84% So any thoughts on what this probability distribution, P, 00:42:07.040 --> 00:42:09.360 align:middle line:90% is going to be here? 00:42:09.360 --> 00:42:10.340 align:middle line:90% Yes, back here. 00:42:10.340 --> 00:42:14.380 align:middle line:84% AUDIENCE: Would it be 1/2, 0, 0, 1/2? 00:42:14.380 --> 00:42:15.380 align:middle line:90% IAN BALL: Exactly right. 00:42:15.380 --> 00:42:19.240 align:middle line:90% 00:42:19.240 --> 00:42:23.480 align:middle line:84% So can you explain a little bit how you got that? 00:42:23.480 --> 00:42:24.280 align:middle line:90% AUDIENCE: For sure. 00:42:24.280 --> 00:42:27.760 align:middle line:84% So if the state of the world theta H is-- 00:42:27.760 --> 00:42:30.577 align:middle line:84% let's just say theta is theta H. That means we know-- 00:42:30.577 --> 00:42:31.660 align:middle line:90% IAN BALL: State of nature. 00:42:31.660 --> 00:42:32.785 align:middle line:90% Sorry, It's very confusing. 00:42:32.785 --> 00:42:35.610 align:middle line:84% Yeah, if the state of nature is theta H. Yeah, exactly. 00:42:35.610 --> 00:42:37.420 align:middle line:84% I may, have said it wrong myself. 00:42:37.420 --> 00:42:38.150 align:middle line:90% Yeah. 00:42:38.150 --> 00:42:40.150 align:middle line:84% AUDIENCE: Anyway, then, with absolute certainty, 00:42:40.150 --> 00:42:42.570 align:middle line:84% we know that player 1 knows that the state of nature 00:42:42.570 --> 00:42:44.940 align:middle line:84% is theta H, which means they have tH. 00:42:44.940 --> 00:42:45.690 align:middle line:90% IAN BALL: Exactly. 00:42:45.690 --> 00:42:47.607 align:middle line:84% AUDIENCE: It's not possible for them to have t 00:42:47.607 --> 00:42:50.670 align:middle line:84% and it's this mismatch, which is why that gives probability 0. 00:42:50.670 --> 00:42:53.820 align:middle line:84% And then you flip the labels and everything else is symmetric. 00:42:53.820 --> 00:42:54.570 align:middle line:90% IAN BALL: Exactly. 00:42:54.570 --> 00:42:57.170 align:middle line:90% Exactly right. 00:42:57.170 --> 00:43:00.090 align:middle line:84% I guess it's a hidden implicit assumption that when we say 00:43:00.090 --> 00:43:03.170 align:middle line:84% player 1-- well, this is what it means-- 00:43:03.170 --> 00:43:05.590 align:middle line:84% to know the state means when you think the state is high, 00:43:05.590 --> 00:43:06.830 align:middle line:90% it's actually high or hard. 00:43:06.830 --> 00:43:09.205 align:middle line:84% And when you think the state is easy, it's actually easy. 00:43:09.205 --> 00:43:12.570 align:middle line:84% Otherwise, you wouldn't know the state at all. 00:43:12.570 --> 00:43:17.610 align:middle line:84% OK, let's do maybe one more example 00:43:17.610 --> 00:43:22.110 align:middle line:84% to make sure we can understand what's going on here. 00:43:22.110 --> 00:43:51.020 align:middle line:90% 00:43:51.020 --> 00:43:58.620 align:middle line:84% So let's consider maybe example two, where we'll try 00:43:58.620 --> 00:43:59.880 align:middle line:90% to be a little more realistic. 00:43:59.880 --> 00:44:01.940 align:middle line:84% Probably player 1 can't perfectly 00:44:01.940 --> 00:44:04.653 align:middle line:84% tell whether the project is easy or hard, 00:44:04.653 --> 00:44:06.320 align:middle line:84% but they have some information about it. 00:44:06.320 --> 00:44:09.300 align:middle line:84% So what we're going to imagine is 00:44:09.300 --> 00:44:14.420 align:middle line:84% player 1 gets a signal about the difficulty of the state. 00:44:14.420 --> 00:44:16.690 align:middle line:84% And so player 1 has some information 00:44:16.690 --> 00:44:18.440 align:middle line:84% about whether the project is easy or hard, 00:44:18.440 --> 00:44:21.100 align:middle line:90% but they don't know it for sure. 00:44:21.100 --> 00:44:25.500 align:middle line:84% And more specifically-- so let's say player 1-- sometimes 00:44:25.500 --> 00:44:29.910 align:middle line:90% we say gets a signal. 00:44:29.910 --> 00:44:34.948 align:middle line:84% And that's a statistical way of looking at it, but you sit down, 00:44:34.948 --> 00:44:36.490 align:middle line:84% you try to start doing your homework. 00:44:36.490 --> 00:44:39.470 align:middle line:84% You think, I think it's probably hard, but I'm not sure. 00:44:39.470 --> 00:44:42.205 align:middle line:84% And you maybe assign some probability to how hard it is. 00:44:42.205 --> 00:44:44.330 align:middle line:84% And that's what we're going to try to capture here. 00:44:44.330 --> 00:44:47.550 align:middle line:90% 00:44:47.550 --> 00:44:53.630 align:middle line:84% I'm going to say with accuracy, q, 00:44:53.630 --> 00:45:00.550 align:middle line:84% where q is going to be between 1/2 and 1. 00:45:00.550 --> 00:45:04.610 align:middle line:84% So what this means is-- let me just draw this mapping. 00:45:04.610 --> 00:45:19.570 align:middle line:90% 00:45:19.570 --> 00:45:20.890 align:middle line:90% I run out of arrows here. 00:45:20.890 --> 00:45:27.330 align:middle line:90% 00:45:27.330 --> 00:45:30.268 align:middle line:84% So what I'm representing here is I'm 00:45:30.268 --> 00:45:31.810 align:middle line:84% going to think of theta H and theta L 00:45:31.810 --> 00:45:33.650 align:middle line:84% as the true difficulty of the project. 00:45:33.650 --> 00:45:36.570 align:middle line:84% And I'm going to think of tH and tL as the signal 00:45:36.570 --> 00:45:40.170 align:middle line:84% that player 1 gets about the difficulty of the project. 00:45:40.170 --> 00:45:43.690 align:middle line:84% And if the project is hard, then most of the time, 00:45:43.690 --> 00:45:47.450 align:middle line:84% player 1 gets this signal that's indicative of it being hard-- 00:45:47.450 --> 00:45:48.930 align:middle line:90% indeed, with probability q. 00:45:48.930 --> 00:45:52.510 align:middle line:84% But with probability 1 minus q, it's deceptively hard. 00:45:52.510 --> 00:45:54.150 align:middle line:84% It's actually hard, but it looks easy. 00:45:54.150 --> 00:45:55.470 align:middle line:90% And they get the wrong signal. 00:45:55.470 --> 00:45:57.790 align:middle line:84% And then we have the same thing over here. 00:45:57.790 --> 00:46:01.330 align:middle line:84% So let's just understand-- if I allow q to be 1, 00:46:01.330 --> 00:46:03.530 align:middle line:84% then if the project is hard, they always 00:46:03.530 --> 00:46:04.813 align:middle line:90% get the signal that it's hard. 00:46:04.813 --> 00:46:07.230 align:middle line:84% And I'm just going to say all day long, they get it wrong. 00:46:07.230 --> 00:46:11.050 align:middle line:90% 00:46:11.050 --> 00:46:12.550 align:middle line:84% Similarly, when the project is easy, 00:46:12.550 --> 00:46:15.090 align:middle line:84% they always get this signal that suggests it's easy. 00:46:15.090 --> 00:46:17.850 align:middle line:84% So the case of q equals 1 is exactly what we already 00:46:17.850 --> 00:46:19.290 align:middle line:90% studied over here. 00:46:19.290 --> 00:46:22.150 align:middle line:84% But now we're going to generalize to this case, 00:46:22.150 --> 00:46:23.220 align:middle line:90% what if q was 1/2? 00:46:23.220 --> 00:46:24.470 align:middle line:90% What would that correspond to? 00:46:24.470 --> 00:46:27.060 align:middle line:90% 00:46:27.060 --> 00:46:27.718 align:middle line:90% Yeah. 00:46:27.718 --> 00:46:29.260 align:middle line:84% AUDIENCE: You're right half the time. 00:46:29.260 --> 00:46:30.140 align:middle line:90% IAN BALL: Right. 00:46:30.140 --> 00:46:32.420 align:middle line:84% And if the prior-- since we said it's 00:46:32.420 --> 00:46:34.720 align:middle line:84% equally likely to be hard and easy, 00:46:34.720 --> 00:46:36.900 align:middle line:84% it actually means you know nothing. 00:46:36.900 --> 00:46:41.932 align:middle line:84% If you have a signal when the state is hard, 00:46:41.932 --> 00:46:44.140 align:middle line:84% it gives you each signal realization with probability 00:46:44.140 --> 00:46:44.400 align:middle line:90% 1/2. 00:46:44.400 --> 00:46:46.560 align:middle line:84% And when it's easy, you get each signal with probability 1/2, 00:46:46.560 --> 00:46:48.560 align:middle line:84% then it's not giving you any information at all. 00:46:48.560 --> 00:46:50.100 align:middle line:84% So you're just totally uninformed. 00:46:50.100 --> 00:46:53.860 align:middle line:84% Your gut is totally useless, which is maybe 00:46:53.860 --> 00:46:55.760 align:middle line:90% for retail stock traders. 00:46:55.760 --> 00:46:58.880 align:middle line:84% That's a pretty good model of how they trade stocks. 00:46:58.880 --> 00:47:03.660 align:middle line:84% So let's now try to capture this model, this example 00:47:03.660 --> 00:47:05.040 align:middle line:90% within our model over here. 00:47:05.040 --> 00:47:07.800 align:middle line:84% So it turns out the type space are going to be the same, 00:47:07.800 --> 00:47:09.400 align:middle line:84% but the interpretation is different. 00:47:09.400 --> 00:47:13.460 align:middle line:84% So we still have, for player 1, two types-- 00:47:13.460 --> 00:47:15.360 align:middle line:90% the hard type and the easy type. 00:47:15.360 --> 00:47:17.460 align:middle line:84% But now the hard type, we don't mean 00:47:17.460 --> 00:47:19.760 align:middle line:84% you know for certain the project is hard. 00:47:19.760 --> 00:47:22.100 align:middle line:84% What we mean is based on the evidence you see, 00:47:22.100 --> 00:47:25.190 align:middle line:84% it's more likely than not that the project is hard. 00:47:25.190 --> 00:47:29.270 align:middle line:84% And similarly, the type tE suggests 00:47:29.270 --> 00:47:31.410 align:middle line:84% that player 1 looks at the project, 00:47:31.410 --> 00:47:33.790 align:middle line:84% they think it's probably easy, but they're not certain 00:47:33.790 --> 00:47:34.530 align:middle line:90% that it's easy. 00:47:34.530 --> 00:47:37.750 align:middle line:84% And they can actually compute the probability that it's easy. 00:47:37.750 --> 00:47:39.990 align:middle line:90% So now let's represent this. 00:47:39.990 --> 00:47:51.190 align:middle line:84% We're going to represent this with the distribution, P, 00:47:51.190 --> 00:47:55.850 align:middle line:90% over theta cross t1 cross t2. 00:47:55.850 --> 00:47:58.470 align:middle line:90% 00:47:58.470 --> 00:47:59.790 align:middle line:90% So what is it going to be? 00:47:59.790 --> 00:48:01.030 align:middle line:90% Let's write it down. 00:48:01.030 --> 00:48:05.710 align:middle line:90% Again, we have theta H, tH, t0. 00:48:05.710 --> 00:48:12.430 align:middle line:90% We have theta H, tE, t0. 00:48:12.430 --> 00:48:23.200 align:middle line:84% We have theta E, tH, t0, and we have theta E, tE, t0. 00:48:23.200 --> 00:48:25.700 align:middle line:84% I guess it got progressively smaller as I ran out of space. 00:48:25.700 --> 00:48:28.610 align:middle line:90% 00:48:28.610 --> 00:48:30.240 align:middle line:84% So now what are the probabilities 00:48:30.240 --> 00:48:35.540 align:middle line:84% that we're going to assign to each of these? 00:48:35.540 --> 00:48:39.880 align:middle line:90% 00:48:39.880 --> 00:48:40.520 align:middle line:90% Yeah. 00:48:40.520 --> 00:48:42.812 align:middle line:84% AUDIENCE: Just to make sure there is equal probability, 00:48:42.812 --> 00:48:44.000 align:middle line:90% that it actually is hard. 00:48:44.000 --> 00:48:45.310 align:middle line:84% IAN BALL: We're going to maintain that assumption, so 00:48:45.310 --> 00:48:46.440 align:middle line:90% a good question. 00:48:46.440 --> 00:48:47.840 align:middle line:90% Let's maintain this assumption. 00:48:47.840 --> 00:48:50.080 align:middle line:90% I didn't say that. 00:48:50.080 --> 00:48:51.080 align:middle line:90% Second row, yeah. 00:48:51.080 --> 00:48:53.800 align:middle line:84% AUDIENCE: So for the first one, there's a 1/2 chance that 00:48:53.800 --> 00:48:54.920 align:middle line:90% it's hard. 00:48:54.920 --> 00:49:00.320 align:middle line:84% And then as tdH, that's going to be q. 00:49:00.320 --> 00:49:02.960 align:middle line:84% IAN BALL: So what's the probability of this? 00:49:02.960 --> 00:49:04.400 align:middle line:90% AUDIENCE: That's 1/2 q. 00:49:04.400 --> 00:49:06.440 align:middle line:90% IAN BALL: 1/2 q, exactly. 00:49:06.440 --> 00:49:09.337 align:middle line:90% AUDIENCE: [INAUDIBLE] 1 minus q. 00:49:09.337 --> 00:49:09.920 align:middle line:90% IAN BALL: Yes. 00:49:09.920 --> 00:49:12.893 align:middle line:84% AUDIENCE: And then 1/2 minus q and then [INAUDIBLE]. 00:49:12.893 --> 00:49:13.560 align:middle line:90% IAN BALL: Great. 00:49:13.560 --> 00:49:19.440 align:middle line:90% 00:49:19.440 --> 00:49:21.250 align:middle line:84% So it's going to be like this, because what 00:49:21.250 --> 00:49:23.417 align:middle line:84% has to happen for this to be the state of the world? 00:49:23.417 --> 00:49:25.610 align:middle line:84% Well, first, the project has to be hard. 00:49:25.610 --> 00:49:27.930 align:middle line:84% And we know that happens with probability 1/2. 00:49:27.930 --> 00:49:30.550 align:middle line:84% But further, not only is the project hard, 00:49:30.550 --> 00:49:32.730 align:middle line:84% but player 1 gets the signal that's 00:49:32.730 --> 00:49:34.610 align:middle line:90% indicative of it being hard. 00:49:34.610 --> 00:49:37.530 align:middle line:84% And when the state's actually hard, how likely is that? 00:49:37.530 --> 00:49:41.890 align:middle line:90% That's probability q. 00:49:41.890 --> 00:49:45.390 align:middle line:84% And then down here, well, we're going to say, well, 00:49:45.390 --> 00:49:48.990 align:middle line:84% there's still a probability 1/2 that it's actually hard. 00:49:48.990 --> 00:49:51.497 align:middle line:84% But now what's the probability they get the easy signal? 00:49:51.497 --> 00:49:53.330 align:middle line:84% So they get misled, they get the signal that 00:49:53.330 --> 00:49:55.510 align:middle line:84% suggests the project's easy even when it's hard. 00:49:55.510 --> 00:49:56.710 align:middle line:90% That's still possible. 00:49:56.710 --> 00:49:58.250 align:middle line:84% It's less likely, but it's possible. 00:49:58.250 --> 00:50:00.810 align:middle line:84% That has probability 1 minus q, and we have to multiply that 00:50:00.810 --> 00:50:01.310 align:middle line:90% by 1/2. 00:50:01.310 --> 00:50:03.930 align:middle line:90% 00:50:03.930 --> 00:50:08.210 align:middle line:84% Notice here if we sum the first two, we get 1/2, 00:50:08.210 --> 00:50:11.090 align:middle line:84% which makes sense because the sum of the first two is exactly 00:50:11.090 --> 00:50:13.130 align:middle line:84% the probability that the state is hard, 00:50:13.130 --> 00:50:16.610 align:middle line:84% because given the state is hard, we're summing over 00:50:16.610 --> 00:50:18.580 align:middle line:90% the additional types. 00:50:18.580 --> 00:50:20.260 align:middle line:90% Let's look at the extreme cases. 00:50:20.260 --> 00:50:24.340 align:middle line:84% If q goes to 1, what happens here? 00:50:24.340 --> 00:50:25.840 align:middle line:84% Well, now we're just back over here. 00:50:25.840 --> 00:50:26.715 align:middle line:90% This is what we said. 00:50:26.715 --> 00:50:29.340 align:middle line:84% This is the special case where player 1 perfectly observes 00:50:29.340 --> 00:50:30.760 align:middle line:90% the difficulty of the project. 00:50:30.760 --> 00:50:31.760 align:middle line:90% What if q goes to 1/2? 00:50:31.760 --> 00:50:36.700 align:middle line:90% 00:50:36.700 --> 00:50:37.500 align:middle line:90% Yeah. 00:50:37.500 --> 00:50:40.040 align:middle line:84% AUDIENCE: q goes to half, then each of the pairs, 00:50:40.040 --> 00:50:44.602 align:middle line:84% like the data pairs are approximately going to be 1. 00:50:44.602 --> 00:50:46.060 align:middle line:84% IAN BALL: So basically everything's 00:50:46.060 --> 00:50:48.393 align:middle line:84% a fourth because now your signal doesn't really give you 00:50:48.393 --> 00:50:49.860 align:middle line:90% any information about this. 00:50:49.860 --> 00:50:51.660 align:middle line:84% If the project is actually hard, you're 00:50:51.660 --> 00:50:53.327 align:middle line:84% equally likely to get these two signals. 00:50:53.327 --> 00:50:55.420 align:middle line:84% And just everything is going to be 1/4. 00:50:55.420 --> 00:50:56.260 align:middle line:90% Yes. 00:50:56.260 --> 00:50:59.020 align:middle line:84% AUDIENCE: Is player 1 aware of their own, 00:50:59.020 --> 00:51:01.340 align:middle line:90% or does that even matter? 00:51:01.340 --> 00:51:02.880 align:middle line:84% IAN BALL: No, it really does matter. 00:51:02.880 --> 00:51:06.220 align:middle line:84% So this is the subtle thing here that this 00:51:06.220 --> 00:51:09.900 align:middle line:84% is where the implicit assumption here-- and this is where it gets 00:51:09.900 --> 00:51:12.820 align:middle line:84% a little subtle-- is that the distribution, P, is 00:51:12.820 --> 00:51:14.260 align:middle line:90% common knowledge. 00:51:14.260 --> 00:51:17.150 align:middle line:84% So player 1 knows their own type, 00:51:17.150 --> 00:51:22.430 align:middle line:84% and they know the set of possible types the players have. 00:51:22.430 --> 00:51:25.310 align:middle line:84% And they know the distribution over the types. 00:51:25.310 --> 00:51:27.643 align:middle line:84% Now, you might say, is that a restriction? 00:51:27.643 --> 00:51:30.310 align:middle line:84% It turns out there's some really subtle mathematical points that 00:51:30.310 --> 00:51:35.630 align:middle line:84% argue that, as long as you make the type space big enough, 00:51:35.630 --> 00:51:36.730 align:middle line:90% the P is a little tricky. 00:51:36.730 --> 00:51:38.470 align:middle line:84% Basically, you can make the type space big enough 00:51:38.470 --> 00:51:39.950 align:middle line:84% and you can capture everything, but it's 00:51:39.950 --> 00:51:41.550 align:middle line:84% quite a subtle point about what's known 00:51:41.550 --> 00:51:42.425 align:middle line:90% and what's not known. 00:51:42.425 --> 00:51:45.550 align:middle line:84% And we're getting into this very abstract area of epistemic game 00:51:45.550 --> 00:51:49.870 align:middle line:84% theory that gets into logic and really subtleties about what 00:51:49.870 --> 00:51:51.230 align:middle line:90% belief and what knowledge is. 00:51:51.230 --> 00:51:56.670 align:middle line:84% But the assumption here is that P is known by everyone, yeah. 00:51:56.670 --> 00:51:57.170 align:middle line:90% OK. 00:51:57.170 --> 00:52:01.510 align:middle line:84% So the next thing we want to do is 00:52:01.510 --> 00:52:09.330 align:middle line:84% let's now think about behavior in this game. 00:52:09.330 --> 00:52:11.410 align:middle line:84% So far, I've just defined the game. 00:52:11.410 --> 00:52:12.732 align:middle line:90% We now have to define-- 00:52:12.732 --> 00:52:14.440 align:middle line:84% the remaining time, we're going to define 00:52:14.440 --> 00:52:15.780 align:middle line:90% an equilibrium of this game. 00:52:15.780 --> 00:52:23.640 align:middle line:90% 00:52:23.640 --> 00:52:26.200 align:middle line:84% So far, I've just defined what a Bayesian game is. 00:52:26.200 --> 00:52:29.000 align:middle line:84% Let's now define what an equilibrium of a Bayesian game 00:52:29.000 --> 00:52:29.500 align:middle line:90% is. 00:52:29.500 --> 00:52:31.220 align:middle line:84% The first step is to define equilibrium. 00:52:31.220 --> 00:52:33.160 align:middle line:84% You always have to define what a strategy is. 00:52:33.160 --> 00:52:38.280 align:middle line:84% So now let's define a strategy in a Bayesian game. 00:52:38.280 --> 00:52:40.600 align:middle line:84% Now a strategy, remember, is always 00:52:40.600 --> 00:52:42.640 align:middle line:90% a complete contingent plan. 00:52:42.640 --> 00:52:45.960 align:middle line:84% It specifies what action you take as a function of things 00:52:45.960 --> 00:52:48.320 align:middle line:84% that you can observe, things that you know. 00:52:48.320 --> 00:52:55.600 align:middle line:84% So in this case, for player i, a strategy 00:52:55.600 --> 00:53:03.260 align:middle line:84% is a function, si, from ti to Ai. 00:53:03.260 --> 00:53:05.840 align:middle line:90% 00:53:05.840 --> 00:53:07.280 align:middle line:90% So let's visualize it down here. 00:53:07.280 --> 00:53:12.200 align:middle line:84% We have some type, ti, and that gets mapped to si of t. 00:53:12.200 --> 00:53:15.180 align:middle line:90% 00:53:15.180 --> 00:53:17.380 align:middle line:84% So what this complete contingent plan says 00:53:17.380 --> 00:53:21.220 align:middle line:84% is it says if I'm player i, if my mental state is ti, 00:53:21.220 --> 00:53:24.580 align:middle line:84% this is the action I'm going to take. 00:53:24.580 --> 00:53:26.040 align:middle line:84% If I have a different mental state, 00:53:26.040 --> 00:53:29.940 align:middle line:84% I can potentially take a different action. 00:53:29.940 --> 00:53:33.220 align:middle line:84% My strategy can't depend on the mental states of anyone else 00:53:33.220 --> 00:53:35.440 align:middle line:84% or on the state of the world or the state of nature, 00:53:35.440 --> 00:53:36.982 align:middle line:84% because I don't observe those things. 00:53:36.982 --> 00:53:38.980 align:middle line:84% All I know is player i is my mental state, 00:53:38.980 --> 00:53:41.980 align:middle line:84% so the only thing I can contingent condition my action 00:53:41.980 --> 00:53:43.600 align:middle line:90% on is my own mental state. 00:53:43.600 --> 00:53:45.180 align:middle line:84% And that's exactly why this function 00:53:45.180 --> 00:53:48.040 align:middle line:84% depends on my mental state, but on no one else's mental state. 00:53:48.040 --> 00:53:51.420 align:middle line:90% 00:53:51.420 --> 00:53:56.380 align:middle line:84% I think there's one subtlety here in interpretation, 00:53:56.380 --> 00:53:58.220 align:middle line:90% which is that-- 00:53:58.220 --> 00:54:16.293 align:middle line:84% I'll say note, player i actually has one mental state slash type. 00:54:16.293 --> 00:54:18.710 align:middle line:84% So this is where I mentioned the interpretive difficulties 00:54:18.710 --> 00:54:19.830 align:middle line:90% we get into. 00:54:19.830 --> 00:54:22.363 align:middle line:84% I'm calling this a complete contingent plan, 00:54:22.363 --> 00:54:24.030 align:middle line:84% but it's never clear when you would make 00:54:24.030 --> 00:54:25.280 align:middle line:90% this complete contingent plan. 00:54:25.280 --> 00:54:27.390 align:middle line:84% You'd have to make this complete contingent plan 00:54:27.390 --> 00:54:29.710 align:middle line:84% before you know what your type is. 00:54:29.710 --> 00:54:32.910 align:middle line:84% And it's very hard with these intrinsic preferences. 00:54:32.910 --> 00:54:33.910 align:middle line:90% You're born. 00:54:33.910 --> 00:54:36.350 align:middle line:84% Let's say you like the color red in our car example. 00:54:36.350 --> 00:54:37.390 align:middle line:84% Formally, what you'd have to do here 00:54:37.390 --> 00:54:39.390 align:middle line:84% is make a strategy that said, if I were born 00:54:39.390 --> 00:54:41.230 align:middle line:84% and I liked the color blue, what would I do? 00:54:41.230 --> 00:54:42.930 align:middle line:84% If I were born differently and liked the color red, 00:54:42.930 --> 00:54:43.597 align:middle line:90% what would I do? 00:54:43.597 --> 00:54:46.830 align:middle line:84% It's a bit weird to think about, but I 00:54:46.830 --> 00:54:48.550 align:middle line:84% think the role of this assumption 00:54:48.550 --> 00:54:50.830 align:middle line:84% really is that, yes, player i is only 00:54:50.830 --> 00:54:52.872 align:middle line:84% going to have a single realized type, 00:54:52.872 --> 00:54:54.830 align:middle line:84% and they're going to take a single action based 00:54:54.830 --> 00:54:55.893 align:middle line:90% on that type. 00:54:55.893 --> 00:54:58.310 align:middle line:84% But the reason we have to specify this complete contingent 00:54:58.310 --> 00:55:00.590 align:middle line:84% plan is that the other players don't 00:55:00.590 --> 00:55:01.847 align:middle line:90% know what type player i has. 00:55:01.847 --> 00:55:03.430 align:middle line:84% So the other players are going to have 00:55:03.430 --> 00:55:05.710 align:middle line:84% to form beliefs about player i's type, 00:55:05.710 --> 00:55:08.720 align:middle line:84% and then form beliefs about what actions player i takes. 00:55:08.720 --> 00:55:10.680 align:middle line:90% And the other players need to-- 00:55:10.680 --> 00:55:13.420 align:middle line:84% so what this represents is what the other players believe 00:55:13.420 --> 00:55:16.160 align:middle line:84% player i will do when player i's type is ti, 00:55:16.160 --> 00:55:19.400 align:middle line:84% even if player i wakes up and their type is not ti. 00:55:19.400 --> 00:55:22.760 align:middle line:84% Maybe a little philosophical digression, but let's 00:55:22.760 --> 00:55:25.120 align:middle line:90% just stop there. 00:55:25.120 --> 00:55:25.680 align:middle line:90% Yes. 00:55:25.680 --> 00:55:29.720 align:middle line:84% AUDIENCE: So does that mean like the strategy is basically 00:55:29.720 --> 00:55:31.320 align:middle line:90% information that everybody has? 00:55:31.320 --> 00:55:35.200 align:middle line:84% Everyone knows everyone else's strategy. 00:55:35.200 --> 00:55:36.620 align:middle line:90% IAN BALL: So this is a subtlety. 00:55:36.620 --> 00:55:41.460 align:middle line:84% Normally in game theory, we make knowledge assumptions, 00:55:41.460 --> 00:55:44.320 align:middle line:84% not about what people do, but about what people know 00:55:44.320 --> 00:55:45.980 align:middle line:90% and about the circumstance. 00:55:45.980 --> 00:55:50.720 align:middle line:84% And then we define a solution concept, like an equilibrium. 00:55:50.720 --> 00:55:53.860 align:middle line:84% So yeah, do people know this is always a subtle thing? 00:55:53.860 --> 00:55:57.120 align:middle line:84% Do people know the strategies that are chosen in equilibrium? 00:55:57.120 --> 00:56:01.980 align:middle line:84% I don't think we would formally say people know them. 00:56:01.980 --> 00:56:07.570 align:middle line:84% We would say that equilibrium specifies the beliefs 00:56:07.570 --> 00:56:09.890 align:middle line:84% that players have, and a demand of equilibrium 00:56:09.890 --> 00:56:11.370 align:middle line:84% is that those beliefs are correct. 00:56:11.370 --> 00:56:13.930 align:middle line:84% But we don't literally-- if we wanted to model the game where 00:56:13.930 --> 00:56:16.130 align:middle line:84% you actually observed players strategies, that 00:56:16.130 --> 00:56:18.330 align:middle line:90% would be a different game. 00:56:18.330 --> 00:56:19.810 align:middle line:84% We generally only model knowledge 00:56:19.810 --> 00:56:21.830 align:middle line:84% about exogenous things from nature. 00:56:21.830 --> 00:56:23.490 align:middle line:84% We don't really think about knowledge 00:56:23.490 --> 00:56:28.890 align:middle line:84% about what other rational players do because that's just 00:56:28.890 --> 00:56:31.250 align:middle line:84% not a probabilistic process in the same way that's 00:56:31.250 --> 00:56:31.990 align:middle line:90% in their control. 00:56:31.990 --> 00:56:33.850 align:middle line:84% This gets into weird things about free will and all 00:56:33.850 --> 00:56:34.230 align:middle line:90% these things. 00:56:34.230 --> 00:56:36.022 align:middle line:84% But the point is, we don't assume knowledge 00:56:36.022 --> 00:56:39.090 align:middle line:90% of other people's strategies. 00:56:39.090 --> 00:56:41.930 align:middle line:84% In equilibrium, people's beliefs about other people's strategies 00:56:41.930 --> 00:56:42.630 align:middle line:90% will be correct. 00:56:42.630 --> 00:56:45.450 align:middle line:84% And that's very similar, but I would say slightly different. 00:56:45.450 --> 00:56:47.090 align:middle line:84% AUDIENCE: So where does the part come 00:56:47.090 --> 00:56:50.590 align:middle line:84% in where other players have to understand, oh, 00:56:50.590 --> 00:56:53.368 align:middle line:90% if player i is born like red-- 00:56:53.368 --> 00:56:54.910 align:middle line:84% IAN BALL: When we define equilibrium. 00:56:54.910 --> 00:56:56.150 align:middle line:84% So I haven't defined equilibrium yet, 00:56:56.150 --> 00:56:57.750 align:middle line:84% so we'll see when we define equilibrium. 00:56:57.750 --> 00:56:58.292 align:middle line:90% That's great. 00:56:58.292 --> 00:56:59.610 align:middle line:90% That's a great question. 00:56:59.610 --> 00:57:03.247 align:middle line:84% OK, so the final step when to define equilibrium is you 00:57:03.247 --> 00:57:05.080 align:middle line:84% have to say a little bit more about beliefs. 00:57:05.080 --> 00:57:07.700 align:middle line:84% So let's put ourselves in the position of player i. 00:57:07.700 --> 00:57:08.960 align:middle line:90% Suppose I'm player i. 00:57:08.960 --> 00:57:12.540 align:middle line:90% 00:57:12.540 --> 00:57:15.980 align:middle line:90% What I know is I know my type. 00:57:15.980 --> 00:57:19.140 align:middle line:90% So let's say my type is ti. 00:57:19.140 --> 00:57:20.660 align:middle line:84% Now I have to decide how to play. 00:57:20.660 --> 00:57:22.780 align:middle line:84% I have to decide what action to take. 00:57:22.780 --> 00:57:25.340 align:middle line:84% But the action for me is going to depend 00:57:25.340 --> 00:57:26.660 align:middle line:90% on a lot of other things. 00:57:26.660 --> 00:57:28.340 align:middle line:84% So the first step is what I need to do 00:57:28.340 --> 00:57:37.980 align:middle line:84% is I need to form beliefs about theta and about t negative i. 00:57:37.980 --> 00:57:42.420 align:middle line:84% I need to form beliefs about the state of nature theta 00:57:42.420 --> 00:57:46.220 align:middle line:84% and about the types of other players. 00:57:46.220 --> 00:57:49.440 align:middle line:84% In this model, why do I need to form beliefs about theta? 00:57:49.440 --> 00:57:50.880 align:middle line:90% Why do I care about theta? 00:57:50.880 --> 00:57:53.060 align:middle line:84% Well, theta directly affects my payoffs. 00:57:53.060 --> 00:57:54.960 align:middle line:84% So going back to the example over here, 00:57:54.960 --> 00:57:57.320 align:middle line:84% I need to form beliefs about how hard the project is. 00:57:57.320 --> 00:57:58.920 align:middle line:84% Because if the project is really hard, 00:57:58.920 --> 00:58:00.378 align:middle line:84% I might not want to work very hard. 00:58:00.378 --> 00:58:01.870 align:middle line:90% It may not be worth it. 00:58:01.870 --> 00:58:03.870 align:middle line:84% The second, subtler thing is, why do I 00:58:03.870 --> 00:58:06.792 align:middle line:84% need to form beliefs about other players' mental states? 00:58:06.792 --> 00:58:08.250 align:middle line:84% Let's think of the project example. 00:58:08.250 --> 00:58:10.550 align:middle line:84% Why do I care what another player thinks 00:58:10.550 --> 00:58:12.050 align:middle line:90% about how hard the project is? 00:58:12.050 --> 00:58:15.270 align:middle line:90% 00:58:15.270 --> 00:58:16.010 align:middle line:90% Yeah, up front. 00:58:16.010 --> 00:58:17.770 align:middle line:84% AUDIENCE: They'll probably adjust their effort for you. 00:58:17.770 --> 00:58:18.520 align:middle line:90% IAN BALL: Exactly. 00:58:18.520 --> 00:58:20.930 align:middle line:84% So I care about their mental states, 00:58:20.930 --> 00:58:23.498 align:middle line:84% not because their mental states directly affect my payoff, 00:58:23.498 --> 00:58:25.790 align:middle line:84% but because their mental states will affect the actions 00:58:25.790 --> 00:58:27.710 align:middle line:84% that they take, and the actions that they take 00:58:27.710 --> 00:58:29.470 align:middle line:90% will affect my payoff. 00:58:29.470 --> 00:58:31.390 align:middle line:84% And this is where, going back to your question 00:58:31.390 --> 00:58:34.350 align:middle line:84% about the red-blue thing, I'm forming beliefs 00:58:34.350 --> 00:58:36.750 align:middle line:84% about their mental states, and then their strategy 00:58:36.750 --> 00:58:37.990 align:middle line:84% is going to specify what they actually 00:58:37.990 --> 00:58:39.198 align:middle line:90% do under those mental states. 00:58:39.198 --> 00:58:41.870 align:middle line:84% So it's that combination that's going to give me information. 00:58:41.870 --> 00:58:47.230 align:middle line:84% So the first thing is we have to learn how to use Bayes' rule. 00:58:47.230 --> 00:58:48.770 align:middle line:90% To do this, we use Bayes' rule. 00:58:48.770 --> 00:59:05.560 align:middle line:90% 00:59:05.560 --> 00:59:07.000 align:middle line:90% We'll write something like this. 00:59:07.000 --> 00:59:08.320 align:middle line:90% What does this mean? 00:59:08.320 --> 00:59:14.080 align:middle line:84% Pi says I'm player i. ti says this is my mental state, ti. 00:59:14.080 --> 00:59:15.977 align:middle line:84% Given that this is my mental state, 00:59:15.977 --> 00:59:17.560 align:middle line:84% I want to say, what is the probability 00:59:17.560 --> 00:59:19.960 align:middle line:84% that the state of nature is theta 00:59:19.960 --> 00:59:23.160 align:middle line:84% and the mental states of all my opponents are ti? 00:59:23.160 --> 00:59:23.725 align:middle line:90% OK. 00:59:23.725 --> 00:59:25.600 align:middle line:84% And this is going to be a classic application 00:59:25.600 --> 00:59:26.460 align:middle line:90% of Bayes' rule. 00:59:26.460 --> 00:59:28.582 align:middle line:84% I hope people have seen this before. 00:59:28.582 --> 00:59:30.040 align:middle line:84% If you haven't, it may be something 00:59:30.040 --> 00:59:33.120 align:middle line:84% you have to brush up on a little bit. 00:59:33.120 --> 00:59:34.240 align:middle line:90% But what is this? 00:59:34.240 --> 00:59:38.600 align:middle line:84% Well, the probability of A given B is the probability of A and B, 00:59:38.600 --> 00:59:41.360 align:middle line:84% divided by the probability of B. This is what Bayes' rule says. 00:59:41.360 --> 00:59:44.520 align:middle line:84% And I'm going to go over it too fast, so if you haven't seen it, 00:59:44.520 --> 00:59:45.440 align:middle line:90% this won't make sense. 00:59:45.440 --> 00:59:46.660 align:middle line:84% I think you'll have to study it on your own. 00:59:46.660 --> 00:59:48.033 align:middle line:90% I'm happy to talk after class. 00:59:48.033 --> 00:59:49.200 align:middle line:90% So what are we going to get? 00:59:49.200 --> 00:59:53.080 align:middle line:84% On the top, we're going to have the probability that my type is 00:59:53.080 --> 00:59:55.180 align:middle line:84% ti and this is the state of the world, 00:59:55.180 --> 01:00:01.210 align:middle line:84% so we're going to get theta ti, t negative i. 01:00:01.210 --> 01:00:04.970 align:middle line:84% On the bottom, we want the probability of ti. 01:00:04.970 --> 01:00:06.230 align:middle line:90% This is a little more subtle. 01:00:06.230 --> 01:00:12.690 align:middle line:84% It's going to be a sum over all maybe theta prime, t negative i 01:00:12.690 --> 01:00:16.230 align:middle line:90% prime p of theta prime. 01:00:16.230 --> 01:00:25.410 align:middle line:90% 01:00:25.410 --> 01:00:30.450 align:middle line:84% And in words, this is the probability of ti. 01:00:30.450 --> 01:00:37.490 align:middle line:90% 01:00:37.490 --> 01:00:41.130 align:middle line:84% So Bayes' rule is what allows me to leverage 01:00:41.130 --> 01:00:43.410 align:middle line:84% the prior distribution, P, which, as we said, 01:00:43.410 --> 01:00:47.570 align:middle line:84% everyone knows to update my beliefs and form my beliefs 01:00:47.570 --> 01:00:51.210 align:middle line:84% and assign probabilities to the state of nature 01:00:51.210 --> 01:00:53.210 align:middle line:84% and to other people's mental states 01:00:53.210 --> 01:00:55.170 align:middle line:84% as a function of my mental state. 01:00:55.170 --> 01:00:57.340 align:middle line:84% And in general, this probability here 01:00:57.340 --> 01:00:58.760 align:middle line:90% could vary with my mental state. 01:00:58.760 --> 01:01:00.177 align:middle line:84% If I had a different mental state, 01:01:00.177 --> 01:01:02.540 align:middle line:84% I would form different beliefs about the other players. 01:01:02.540 --> 01:01:05.620 align:middle line:84% OK, so now I think we can finally write down 01:01:05.620 --> 01:01:09.180 align:middle line:90% the definition of equilibrium. 01:01:09.180 --> 01:01:13.500 align:middle line:90% So let's maybe go down here. 01:01:13.500 --> 01:01:15.540 align:middle line:84% It's a bit of a long definition, so I really 01:01:15.540 --> 01:01:17.725 align:middle line:84% just want to spend the last 15 minutes just 01:01:17.725 --> 01:01:19.100 align:middle line:84% writing down one equation to make 01:01:19.100 --> 01:01:20.400 align:middle line:90% sure we really understand it. 01:01:20.400 --> 01:01:28.260 align:middle line:90% 01:01:28.260 --> 01:01:31.660 align:middle line:84% So we have our Bayesian game over there. 01:01:31.660 --> 01:01:33.240 align:middle line:90% And here's our definition. 01:01:33.240 --> 01:01:36.780 align:middle line:90% 01:01:36.780 --> 01:01:50.960 align:middle line:84% So a strategy profile, s star, which equals s1 star 2sn star-- 01:01:50.960 --> 01:01:52.960 align:middle line:84% so as usual, throughout the course, 01:01:52.960 --> 01:01:54.700 align:middle line:84% whenever we define an equilibrium, 01:01:54.700 --> 01:01:57.660 align:middle line:84% the object that an equilibrium is is always a strategy profile. 01:01:57.660 --> 01:02:00.040 align:middle line:84% It's a specification of a strategy for each player. 01:02:00.040 --> 01:02:03.775 align:middle line:90% So a strategy profile is a-- 01:02:03.775 --> 01:02:04.900 align:middle line:90% well, what do we call this? 01:02:04.900 --> 01:02:07.660 align:middle line:84% Well, it's a Nash equilibrium in a Bayesian game. 01:02:07.660 --> 01:02:09.660 align:middle line:84% So we call that a Bayesian Nash equilibrium. 01:02:09.660 --> 01:02:11.040 align:middle line:90% We just put the words together. 01:02:11.040 --> 01:02:12.900 align:middle line:84% It's a Bayesian Nash equilibrium. 01:02:12.900 --> 01:02:25.160 align:middle line:90% 01:02:25.160 --> 01:02:28.300 align:middle line:84% Well, what's the basic idea about equilibrium? 01:02:28.300 --> 01:02:32.520 align:middle line:84% In equilibrium, every player is choosing their action 01:02:32.520 --> 01:02:35.900 align:middle line:84% optimally given the beliefs they have about the other players 01:02:35.900 --> 01:02:37.220 align:middle line:90% and those beliefs are correct. 01:02:37.220 --> 01:02:39.840 align:middle line:84% So we want to make sure that everyone is behaving optimally, 01:02:39.840 --> 01:02:42.000 align:middle line:84% given the way everyone else is behaving. 01:02:42.000 --> 01:02:44.600 align:middle line:84% So I think the first step is to figure out 01:02:44.600 --> 01:02:46.400 align:middle line:90% what we need to quantify over. 01:02:46.400 --> 01:02:49.600 align:middle line:84% So what we want to consider is we want to say for every player 01:02:49.600 --> 01:02:51.930 align:middle line:84% and for every mental state that player could have, 01:02:51.930 --> 01:02:53.130 align:middle line:90% they're behaving optimally. 01:02:53.130 --> 01:03:11.850 align:middle line:84% So if for every player i and for every type ti in capital Ti, 01:03:11.850 --> 01:03:13.512 align:middle line:84% the following holds-- so just let's 01:03:13.512 --> 01:03:15.470 align:middle line:84% understand in words what we want to write here. 01:03:15.470 --> 01:03:16.845 align:middle line:84% What we want to capture down here 01:03:16.845 --> 01:03:21.970 align:middle line:84% is to say that when player i has mental state ti, 01:03:21.970 --> 01:03:25.645 align:middle line:84% the action they're choosing is optimal given their beliefs. 01:03:25.645 --> 01:03:27.770 align:middle line:84% And because we're requiring that for every player i 01:03:27.770 --> 01:03:32.170 align:middle line:84% and for every type ti, that's capturing what we want 01:03:32.170 --> 01:03:35.850 align:middle line:84% So first, let's compute what is the expected payoff 01:03:35.850 --> 01:03:37.953 align:middle line:90% of the player of type ti? 01:03:37.953 --> 01:03:39.370 align:middle line:84% So what we're going to do is we're 01:03:39.370 --> 01:03:40.830 align:middle line:90% going to take an expectation. 01:03:40.830 --> 01:03:44.690 align:middle line:90% 01:03:44.690 --> 01:03:47.050 align:middle line:84% And when I write this, I'm going to probably need 01:03:47.050 --> 01:03:47.910 align:middle line:90% a bit more space. 01:03:47.910 --> 01:03:51.300 align:middle line:90% 01:03:51.300 --> 01:03:53.380 align:middle line:84% I'm going to put something in here. 01:03:53.380 --> 01:03:56.060 align:middle line:84% But what I'm writing when I write the expectation given 01:03:56.060 --> 01:04:00.060 align:middle line:84% ti is I'm saying, given ti, I can 01:04:00.060 --> 01:04:02.100 align:middle line:84% compute conditional probabilities 01:04:02.100 --> 01:04:04.260 align:middle line:90% of these other things. 01:04:04.260 --> 01:04:06.380 align:middle line:84% And what I'm going to put in here 01:04:06.380 --> 01:04:09.345 align:middle line:84% is going to be a function of these things. 01:04:09.345 --> 01:04:11.220 align:middle line:84% And I'm computing the conditional expectation 01:04:11.220 --> 01:04:14.220 align:middle line:84% of that quantity given my type, ti. 01:04:14.220 --> 01:04:17.500 align:middle line:84% So what I want to put in here is my payoff when we all 01:04:17.500 --> 01:04:20.180 align:middle line:90% follow these strategies. 01:04:20.180 --> 01:04:21.500 align:middle line:90% So it's going to be my payoff. 01:04:21.500 --> 01:04:23.340 align:middle line:90% I'm player i. 01:04:23.340 --> 01:04:27.820 align:middle line:84% Now, my payoff depends on the actions that we all take 01:04:27.820 --> 01:04:31.420 align:middle line:84% and the state of the world, the state of nature. 01:04:31.420 --> 01:04:35.020 align:middle line:84% So my player 1's action is going to be what? 01:04:35.020 --> 01:04:42.378 align:middle line:84% Well, player 1's action is going to be s1 star of t1. 01:04:42.378 --> 01:04:44.420 align:middle line:84% And this is exactly reflecting the point we made. 01:04:44.420 --> 01:04:47.500 align:middle line:84% Why do I, as player i, care about player 1's mental state, 01:04:47.500 --> 01:04:48.690 align:middle line:90% about their t1? 01:04:48.690 --> 01:04:50.190 align:middle line:84% Because their t1 affects the action 01:04:50.190 --> 01:04:54.150 align:middle line:84% that they're taking under this equilibrium strategy profile. 01:04:54.150 --> 01:05:00.750 align:middle line:84% And then it's going to go all the way up to sn star of tn. 01:05:00.750 --> 01:05:02.410 align:middle line:90% Now, it's tricky. 01:05:02.410 --> 01:05:04.587 align:middle line:84% One of these things-- the i position here 01:05:04.587 --> 01:05:06.670 align:middle line:84% is actually something I'm controlling as player i. 01:05:06.670 --> 01:05:08.590 align:middle line:84% But for everyone other than player i, 01:05:08.590 --> 01:05:11.710 align:middle line:84% these are random things that I'm forming beliefs about. 01:05:11.710 --> 01:05:13.390 align:middle line:84% So here, we have the action profile. 01:05:13.390 --> 01:05:17.830 align:middle line:84% And then the final thing is the theta. 01:05:17.830 --> 01:05:22.980 align:middle line:84% So what you can think of in here is if the state of the world-- 01:05:22.980 --> 01:05:25.230 align:middle line:84% not the state of nature, but if the state of the world 01:05:25.230 --> 01:05:29.910 align:middle line:84% is theta T1 all the way up to tn, then under the strategy 01:05:29.910 --> 01:05:32.470 align:middle line:84% profile, these are the actions that all the players 01:05:32.470 --> 01:05:33.710 align:middle line:90% are going to take. 01:05:33.710 --> 01:05:35.510 align:middle line:90% This is the state of nature. 01:05:35.510 --> 01:05:37.930 align:middle line:84% And therefore, player i's payoff is this. 01:05:37.930 --> 01:05:40.690 align:middle line:84% And player i is simply computing expectations over that. 01:05:40.690 --> 01:05:43.230 align:middle line:90% 01:05:43.230 --> 01:05:44.310 align:middle line:90% Questions about that? 01:05:44.310 --> 01:05:48.360 align:middle line:84% So we want to compare that to what 01:05:48.360 --> 01:05:50.360 align:middle line:90% happens if player i deviates. 01:05:50.360 --> 01:05:54.460 align:middle line:84% So what I want is I want to now compute the same expected payoff 01:05:54.460 --> 01:05:57.503 align:middle line:84% if player i chooses some action-- let's say Ai prime. 01:05:57.503 --> 01:05:58.920 align:middle line:84% And we're going to want this to be 01:05:58.920 --> 01:06:06.360 align:middle line:90% true for all actions Ai prime. 01:06:06.360 --> 01:06:12.560 align:middle line:84% So it's also going to be a conditional expectation given 01:06:12.560 --> 01:06:14.240 align:middle line:90% ti. 01:06:14.240 --> 01:06:17.600 align:middle line:84% And it's a little tricky that the i component here 01:06:17.600 --> 01:06:19.160 align:middle line:84% is hidden amongst the rest, so I want 01:06:19.160 --> 01:06:21.200 align:middle line:90% to reorganize this a little bit. 01:06:21.200 --> 01:06:25.320 align:middle line:84% So another way of writing this which is useful 01:06:25.320 --> 01:06:34.738 align:middle line:84% is to write this as si star ti, s negative i star t negative i. 01:06:34.738 --> 01:06:36.280 align:middle line:84% This is just notation where I'm going 01:06:36.280 --> 01:06:40.720 align:middle line:84% to put player i first and put everyone else's grouped together 01:06:40.720 --> 01:06:42.040 align:middle line:90% over here. 01:06:42.040 --> 01:06:44.450 align:middle line:84% This is really, I think, misleading notation when I write 01:06:44.450 --> 01:06:47.218 align:middle line:84% s negative i of t negative i, because really, this 01:06:47.218 --> 01:06:49.010 align:middle line:84% is a collection of functions, each of which 01:06:49.010 --> 01:06:50.810 align:middle line:84% depends on a single player's type. 01:06:50.810 --> 01:06:53.150 align:middle line:90% So this is s2 star. 01:06:53.150 --> 01:06:55.010 align:middle line:90% Let's say player i is 1. 01:06:55.010 --> 01:06:59.750 align:middle line:84% This is s2 star of t2, s3 star of t3, s4 star of t4, and so on. 01:06:59.750 --> 01:07:02.370 align:middle line:84% It's a little subtle, but that's what's going on here. 01:07:02.370 --> 01:07:05.410 align:middle line:84% And then what do I have over here? 01:07:05.410 --> 01:07:06.210 align:middle line:90% Well, I'm player i. 01:07:06.210 --> 01:07:09.330 align:middle line:84% I'm unilaterally deviating, so instead 01:07:09.330 --> 01:07:12.950 align:middle line:84% of taking action si star of ti, which I'm supposed to take, 01:07:12.950 --> 01:07:15.370 align:middle line:84% I'm instead choosing action Ai prime. 01:07:15.370 --> 01:07:19.130 align:middle line:84% So we have Ai prime here, but everyone else's action 01:07:19.130 --> 01:07:20.410 align:middle line:90% is still the same. 01:07:20.410 --> 01:07:29.090 align:middle line:84% So they're still having s negative i star t negative i, 01:07:29.090 --> 01:07:32.170 align:middle line:84% but then my payoff also depends on the state. 01:07:32.170 --> 01:07:33.310 align:middle line:90% So we have the state here. 01:07:33.310 --> 01:07:36.570 align:middle line:90% 01:07:36.570 --> 01:07:39.830 align:middle line:90% And this is our definition. 01:07:39.830 --> 01:07:42.900 align:middle line:90% 01:07:42.900 --> 01:07:44.940 align:middle line:84% And this is to return to this question of do I 01:07:44.940 --> 01:07:46.020 align:middle line:90% like blue or red? 01:07:46.020 --> 01:07:50.363 align:middle line:84% When I'm player i, I'm forming beliefs about the other player. 01:07:50.363 --> 01:07:52.280 align:middle line:84% And I might not know if they like blue or red. 01:07:52.280 --> 01:07:53.620 align:middle line:84% So I'm taking into account there's 01:07:53.620 --> 01:07:55.328 align:middle line:84% some probability they like blue, and this 01:07:55.328 --> 01:07:56.380 align:middle line:90% is the action they take. 01:07:56.380 --> 01:07:57.720 align:middle line:84% There's some probability they like red, 01:07:57.720 --> 01:07:58.920 align:middle line:90% this is the action they take. 01:07:58.920 --> 01:08:00.480 align:middle line:84% Even though when they wake up in the morning, 01:08:00.480 --> 01:08:02.147 align:middle line:84% they either like blue or red and they're 01:08:02.147 --> 01:08:03.600 align:middle line:90% going to take a single action. 01:08:03.600 --> 01:08:04.100 align:middle line:90% Yes. 01:08:04.100 --> 01:08:06.940 align:middle line:84% AUDIENCE: Is there a reason why we have the condition that's 01:08:06.940 --> 01:08:11.860 align:middle line:84% given ti, based on the fact that it can only technically have 01:08:11.860 --> 01:08:14.860 align:middle line:90% your si star for a certain ti? 01:08:14.860 --> 01:08:19.979 align:middle line:84% And then you need to pin down your own mental state before you 01:08:19.979 --> 01:08:21.240 align:middle line:90% come up with a strategy? 01:08:21.240 --> 01:08:23.103 align:middle line:90% 01:08:23.103 --> 01:08:25.520 align:middle line:84% IAN BALL: So I think there's two separate questions there. 01:08:25.520 --> 01:08:31.300 align:middle line:84% One question is, why is the domain of s1 star t1? 01:08:31.300 --> 01:08:33.399 align:middle line:84% And why do I have this conditioning here? 01:08:33.399 --> 01:08:34.760 align:middle line:90% So it's two questions. 01:08:34.760 --> 01:08:36.380 align:middle line:84% One is, why is a strategy a function 01:08:36.380 --> 01:08:38.000 align:middle line:90% from my type to my action? 01:08:38.000 --> 01:08:41.970 align:middle line:84% It's because the only thing I know is my type, 01:08:41.970 --> 01:08:44.609 align:middle line:84% so the only thing that my action can depend on is my own type. 01:08:44.609 --> 01:08:46.090 align:middle line:84% It can't depend on things I don't observe. 01:08:46.090 --> 01:08:47.830 align:middle line:84% Just like in poker, my strategy can't 01:08:47.830 --> 01:08:50.380 align:middle line:84% be bet if my hand is higher than the other person, 01:08:50.380 --> 01:08:51.630 align:middle line:90% because I don't know his hand. 01:08:51.630 --> 01:08:52.930 align:middle line:84% So of course, that would be a great strategy. 01:08:52.930 --> 01:08:54.347 align:middle line:84% But it's not a legitimate strategy 01:08:54.347 --> 01:08:56.890 align:middle line:84% because I don't know what the other player's hand is. 01:08:56.890 --> 01:09:00.590 align:middle line:84% So that's why s1 star is defined as a function of player 1's 01:09:00.590 --> 01:09:01.710 align:middle line:90% type only. 01:09:01.710 --> 01:09:03.590 align:middle line:84% Now we go here, and this is let's 01:09:03.590 --> 01:09:05.930 align:middle line:90% think about the decision stage. 01:09:05.930 --> 01:09:08.770 align:middle line:84% When I wake up and make a decision, I know something. 01:09:08.770 --> 01:09:10.470 align:middle line:90% I know my own mental state. 01:09:10.470 --> 01:09:12.840 align:middle line:84% If I'm player i, I know my mental state, ti. 01:09:12.840 --> 01:09:14.090 align:middle line:90% And that gives me information. 01:09:14.090 --> 01:09:16.090 align:middle line:84% I'm going to update my beliefs based upon that. 01:09:16.090 --> 01:09:19.170 align:middle line:84% So in the same way, going back to our example here, 01:09:19.170 --> 01:09:20.975 align:middle line:84% when player 1 knows the project is hard, 01:09:20.975 --> 01:09:23.350 align:middle line:84% they have a different belief about whether the project is 01:09:23.350 --> 01:09:25.670 align:middle line:84% hard than if they wake up and they know the project is easy. 01:09:25.670 --> 01:09:27.810 align:middle line:84% And they're going to behave differently in those situations. 01:09:27.810 --> 01:09:29.229 align:middle line:90% This is exactly reflecting that. 01:09:29.229 --> 01:09:32.390 align:middle line:84% It's reflecting the fact that my mental state controls what 01:09:32.390 --> 01:09:34.430 align:middle line:90% I believe about this theta. 01:09:34.430 --> 01:09:39.090 align:middle line:84% So just to understand, if we put in a different ti here, 01:09:39.090 --> 01:09:40.729 align:middle line:84% this conditional expectation would 01:09:40.729 --> 01:09:44.729 align:middle line:84% change because the belief about theta and about everyone else's 01:09:44.729 --> 01:09:48.250 align:middle line:84% mental states could be different if this ti were a different ti-- 01:09:48.250 --> 01:09:51.250 align:middle line:90% if it were ti prime. 01:09:51.250 --> 01:09:53.290 align:middle line:84% AUDIENCE: ti and ti prime basically 01:09:53.290 --> 01:09:58.570 align:middle line:84% lead to different vectors of probabilities for other people's 01:09:58.570 --> 01:09:59.862 align:middle line:90% mental states as well as theta. 01:09:59.862 --> 01:10:00.862 align:middle line:90% IAN BALL: Exactly right. 01:10:00.862 --> 01:10:02.970 align:middle line:84% So remember we wrote down this formula here. 01:10:02.970 --> 01:10:05.930 align:middle line:84% This is really a whole vector because once I fix ti, 01:10:05.930 --> 01:10:08.810 align:middle line:84% I can vary the theta and the t negative I. 01:10:08.810 --> 01:10:10.870 align:middle line:84% So it's really a vector of probabilities. 01:10:10.870 --> 01:10:14.250 align:middle line:84% It's a lottery over everything I don't know, over all the theta t 01:10:14.250 --> 01:10:15.418 align:middle line:90% negative i pairs. 01:10:15.418 --> 01:10:17.210 align:middle line:84% AUDIENCE: So this is kind of similar to how 01:10:17.210 --> 01:10:19.310 align:middle line:90% we talked about belief. 01:10:19.310 --> 01:10:21.610 align:middle line:84% IAN BALL: No, it's exactly a belief. 01:10:21.610 --> 01:10:23.770 align:middle line:84% What we're talking about precisely is given my ti, 01:10:23.770 --> 01:10:24.655 align:middle line:90% I form beliefs. 01:10:24.655 --> 01:10:26.530 align:middle line:84% The tricky thing is what I form beliefs about 01:10:26.530 --> 01:10:30.010 align:middle line:90% is t1 to tn and theta. 01:10:30.010 --> 01:10:33.730 align:middle line:84% My belief about t1 to tn by itself isn't that useful, 01:10:33.730 --> 01:10:35.253 align:middle line:84% but when combined with the strategy 01:10:35.253 --> 01:10:36.920 align:middle line:84% the opponents and the other players use, 01:10:36.920 --> 01:10:37.980 align:middle line:90% now it becomes useful. 01:10:37.980 --> 01:10:40.700 align:middle line:84% Just knowing that the other player's mental state is 01:10:40.700 --> 01:10:42.680 align:middle line:90% t2 prime doesn't tell me much. 01:10:42.680 --> 01:10:45.700 align:middle line:84% But in equilibrium, I correctly anticipate 01:10:45.700 --> 01:10:47.820 align:middle line:84% that they're using the strategy s2 star, 01:10:47.820 --> 01:10:50.100 align:middle line:84% so I can anticipate that the action they take 01:10:50.100 --> 01:10:52.560 align:middle line:90% is s2 star of t2 double prime. 01:10:52.560 --> 01:10:55.220 align:middle line:84% And then that gives me information. 01:10:55.220 --> 01:10:56.180 align:middle line:90% Any other questions? 01:10:56.180 --> 01:10:57.640 align:middle line:84% AUDIENCE: Yeah, I think you did just say. 01:10:57.640 --> 01:10:57.900 align:middle line:90% IAN BALL: No, no. 01:10:57.900 --> 01:10:58.620 align:middle line:90% This is good. 01:10:58.620 --> 01:11:03.500 align:middle line:84% AUDIENCE: [INAUDIBLE] another player has, for example, belief 01:11:03.500 --> 01:11:06.980 align:middle line:84% or not belief type t2, then they're going to do this action. 01:11:06.980 --> 01:11:08.603 align:middle line:90% We can't know that for sure. 01:11:08.603 --> 01:11:10.020 align:middle line:84% IAN BALL: So this is getting into, 01:11:10.020 --> 01:11:11.478 align:middle line:84% I guess, the notion of equilibrium. 01:11:11.478 --> 01:11:16.957 align:middle line:90% 01:11:16.957 --> 01:11:19.540 align:middle line:84% I think the short answer is yes if you want to get the problem 01:11:19.540 --> 01:11:21.460 align:middle line:84% right, if you want to be really subtle about what knowledge 01:11:21.460 --> 01:11:22.040 align:middle line:90% means. 01:11:22.040 --> 01:11:25.580 align:middle line:90% 01:11:25.580 --> 01:11:28.500 align:middle line:84% I guess one way of calling this a Bayes Nash equilibrium is 01:11:28.500 --> 01:11:31.300 align:middle line:84% if we all believe that these are the strategy profiles we're 01:11:31.300 --> 01:11:33.950 align:middle line:84% employing, then no one has an incentive 01:11:33.950 --> 01:11:37.870 align:middle line:84% to deviate from the way they're behaving. 01:11:37.870 --> 01:11:40.750 align:middle line:84% So it's capturing the idea of Nash equilibrium 01:11:40.750 --> 01:11:42.430 align:middle line:90% that I'm believing. 01:11:42.430 --> 01:11:44.490 align:middle line:84% So just like in Nash-- in Nash, we say, 01:11:44.490 --> 01:11:47.030 align:middle line:84% I'm believing my opponents are following this strategy. 01:11:47.030 --> 01:11:50.110 align:middle line:84% And given that belief, I'm behaving optimally. 01:11:50.110 --> 01:11:53.110 align:middle line:84% If I believe that my opponents all follow this strategy 01:11:53.110 --> 01:11:55.390 align:middle line:84% profiles, then it's optimal for me 01:11:55.390 --> 01:11:57.355 align:middle line:90% to follow my strategy profile. 01:11:57.355 --> 01:11:58.730 align:middle line:84% And then that works for everyone, 01:11:58.730 --> 01:12:01.030 align:middle line:90% so we have this consistency. 01:12:01.030 --> 01:12:04.110 align:middle line:84% But I guess I would say there's a bit of a distinction 01:12:04.110 --> 01:12:07.670 align:middle line:90% between maybe this is a point. 01:12:07.670 --> 01:12:14.710 align:middle line:84% Given ti, I form beliefs about t1 to tn and theta. 01:12:14.710 --> 01:12:17.610 align:middle line:84% These are totally outside the control of people. 01:12:17.610 --> 01:12:19.330 align:middle line:84% This is just something about the world. 01:12:19.330 --> 01:12:20.790 align:middle line:90% I can form those beliefs. 01:12:20.790 --> 01:12:22.350 align:middle line:84% Then the mapping from their types 01:12:22.350 --> 01:12:24.150 align:middle line:84% to the actions they actually take-- that's 01:12:24.150 --> 01:12:26.170 align:middle line:84% something that's under control of a rational decision maker. 01:12:26.170 --> 01:12:27.378 align:middle line:90% And we view that differently. 01:12:27.378 --> 01:12:29.390 align:middle line:84% So generally, I guess this gets to a broader 01:12:29.390 --> 01:12:30.330 align:middle line:90% point of game theory. 01:12:30.330 --> 01:12:33.240 align:middle line:84% We view a fundamental difference between when I form beliefs 01:12:33.240 --> 01:12:38.445 align:middle line:84% about whether a dealt hand is going to be kings or aces 01:12:38.445 --> 01:12:40.820 align:middle line:84% and whether my opponent is going to bet a certain amount, 01:12:40.820 --> 01:12:42.860 align:middle line:84% because one is the choice of a rational player 01:12:42.860 --> 01:12:45.193 align:middle line:84% and the other is just something that's chosen by nature. 01:12:45.193 --> 01:12:47.400 align:middle line:84% So we maintain a distinction there in game theory. 01:12:47.400 --> 01:12:49.663 align:middle line:84% AUDIENCE: You know how another player will 01:12:49.663 --> 01:12:51.080 align:middle line:84% play given a certain hand, but you 01:12:51.080 --> 01:12:54.960 align:middle line:84% don't know whether they actually have that hand. 01:12:54.960 --> 01:12:59.360 align:middle line:84% IAN BALL: Yeah, I would say you have beliefs 01:12:59.360 --> 01:13:01.120 align:middle line:90% about the hand they will have. 01:13:01.120 --> 01:13:03.810 align:middle line:84% And then in equilibrium, the equilibrium 01:13:03.810 --> 01:13:05.560 align:middle line:84% describes consistent beliefs about the way 01:13:05.560 --> 01:13:07.050 align:middle line:90% they're going to play. 01:13:07.050 --> 01:13:08.800 align:middle line:84% But I guess maybe another way of saying it 01:13:08.800 --> 01:13:12.520 align:middle line:90% is if I change the strategies-- 01:13:12.520 --> 01:13:14.640 align:middle line:84% so yeah, this is maybe the key point. 01:13:14.640 --> 01:13:18.100 align:middle line:84% This is independent of strategies. 01:13:18.100 --> 01:13:21.440 align:middle line:84% The probability I assign to the other player having this type 01:13:21.440 --> 01:13:23.585 align:middle line:84% is totally independent of how they play. 01:13:23.585 --> 01:13:25.460 align:middle line:84% This is not a statement about their behavior. 01:13:25.460 --> 01:13:27.320 align:middle line:84% This is just a statement about the world. 01:13:27.320 --> 01:13:29.740 align:middle line:84% Equilibrium is now a statement about behavior. 01:13:29.740 --> 01:13:31.870 align:middle line:84% But there's this fundamental difference there. 01:13:31.870 --> 01:13:34.488 align:middle line:84% But these are subtle things, so these are great questions 01:13:34.488 --> 01:13:35.030 align:middle line:90% to be asking. 01:13:35.030 --> 01:13:35.530 align:middle line:90% Yes. 01:13:35.530 --> 01:13:37.910 align:middle line:90% AUDIENCE: So that just means-- 01:13:37.910 --> 01:13:40.750 align:middle line:84% so here, this is just over other people's beliefs. 01:13:40.750 --> 01:13:42.850 align:middle line:84% But then here, it's like what you actually 01:13:42.850 --> 01:13:45.760 align:middle line:84% think of what other people are thinking. 01:13:45.760 --> 01:13:46.510 align:middle line:90% IAN BALL: Exactly. 01:13:46.510 --> 01:13:50.730 align:middle line:84% So by definition of my mental state, 01:13:50.730 --> 01:13:52.690 align:middle line:84% my mental state defines the beliefs 01:13:52.690 --> 01:13:54.930 align:middle line:84% I have about other people's mental states. 01:13:54.930 --> 01:13:58.130 align:middle line:84% But that alone doesn't allow me to form beliefs 01:13:58.130 --> 01:14:00.255 align:middle line:84% about other people's actions, because in general, I 01:14:00.255 --> 01:14:02.172 align:middle line:84% don't know the mapping from their mental state 01:14:02.172 --> 01:14:03.010 align:middle line:90% to their actions. 01:14:03.010 --> 01:14:08.170 align:middle line:84% We've defined equilibrium to be a profile of strategies that 01:14:08.170 --> 01:14:10.490 align:middle line:84% allows me to link my beliefs about their mental states 01:14:10.490 --> 01:14:12.550 align:middle line:84% to my beliefs about their actions, 01:14:12.550 --> 01:14:14.810 align:middle line:84% and we apply the classical idea of Nash equilibrium 01:14:14.810 --> 01:14:17.890 align:middle line:84% that says that this method of behavior 01:14:17.890 --> 01:14:19.970 align:middle line:84% is stable and consistent, in the sense 01:14:19.970 --> 01:14:22.450 align:middle line:84% that if everyone correctly believes that everyone's playing 01:14:22.450 --> 01:14:24.450 align:middle line:84% this way, then no one's going to want to deviate 01:14:24.450 --> 01:14:27.290 align:middle line:90% from this method of playing. 01:14:27.290 --> 01:14:29.100 align:middle line:84% AUDIENCE: So this is based on what 01:14:29.100 --> 01:14:32.060 align:middle line:84% you think other people are going to play, 01:14:32.060 --> 01:14:34.340 align:middle line:84% but you already based that on what 01:14:34.340 --> 01:14:35.840 align:middle line:90% you think their beliefs will be. 01:14:35.840 --> 01:14:39.340 align:middle line:90% 01:14:39.340 --> 01:14:40.860 align:middle line:90% IAN BALL: Yeah. 01:14:40.860 --> 01:14:49.980 align:middle line:84% I would say-- yeah, I think the short answer is this calculation 01:14:49.980 --> 01:14:51.280 align:middle line:90% is not an equilibrium object. 01:14:51.280 --> 01:14:53.980 align:middle line:84% This is a calculation that's a property of the game itself. 01:14:53.980 --> 01:14:56.820 align:middle line:84% This calculation is a property of the strategies 01:14:56.820 --> 01:14:58.355 align:middle line:90% and of the equilibrium concept. 01:14:58.355 --> 01:14:59.480 align:middle line:90% Maybe I'll just stop there. 01:14:59.480 --> 01:15:02.420 align:middle line:84% And let me say one more thing that maybe is useful, 01:15:02.420 --> 01:15:06.860 align:middle line:84% to see how we can tie this in to our notion of Nash equilibrium. 01:15:06.860 --> 01:15:24.380 align:middle line:90% 01:15:24.380 --> 01:15:26.750 align:middle line:84% Actually, one thing I should have said before-- 01:15:26.750 --> 01:15:30.988 align:middle line:84% when we're computing these conditional expectations, 01:15:30.988 --> 01:15:33.030 align:middle line:84% you get into some weird issues when you condition 01:15:33.030 --> 01:15:34.290 align:middle line:90% on 0 probability events. 01:15:34.290 --> 01:15:35.748 align:middle line:84% So what I've of assumed throughout, 01:15:35.748 --> 01:15:36.873 align:middle line:90% and maybe I'll say it now-- 01:15:36.873 --> 01:15:38.110 align:middle line:90% I should have said it before. 01:15:38.110 --> 01:15:41.610 align:middle line:84% Assume that this distribution, P, is strictly positive. 01:15:41.610 --> 01:15:46.710 align:middle line:90% 01:15:46.710 --> 01:15:48.270 align:middle line:84% So it assigns positive probability 01:15:48.270 --> 01:15:56.550 align:middle line:84% to everything, every combination of states and types. 01:15:56.550 --> 01:15:57.078 align:middle line:90% Say again? 01:15:57.078 --> 01:15:58.870 align:middle line:84% AUDIENCE: The simple case in the beginning. 01:15:58.870 --> 01:15:59.530 align:middle line:90% IAN BALL: Good, OK. 01:15:59.530 --> 01:16:01.488 align:middle line:84% So I was going to try to avoid that simplicity. 01:16:01.488 --> 01:16:02.570 align:middle line:90% But no, no. 01:16:02.570 --> 01:16:03.330 align:middle line:90% So let me say no. 01:16:03.330 --> 01:16:05.830 align:middle line:84% So when I say strictly positive, really what I mean is-- 01:16:05.830 --> 01:16:07.050 align:middle line:90% you're exactly right. 01:16:07.050 --> 01:16:10.330 align:middle line:90% 01:16:10.330 --> 01:16:10.830 align:middle line:90% Yeah. 01:16:10.830 --> 01:16:13.745 align:middle line:90% 01:16:13.745 --> 01:16:14.370 align:middle line:90% I should do it. 01:16:14.370 --> 01:16:16.245 align:middle line:84% I was going to skip a step, but you're right. 01:16:16.245 --> 01:16:17.290 align:middle line:90% It's good not to. 01:16:17.290 --> 01:16:21.760 align:middle line:90% 01:16:21.760 --> 01:16:23.010 align:middle line:90% So this is really what I mean. 01:16:23.010 --> 01:16:25.540 align:middle line:84% It's not that every combination has positive probability, 01:16:25.540 --> 01:16:27.923 align:middle line:84% but every type of every player gets positive probability. 01:16:27.923 --> 01:16:29.840 align:middle line:84% And the reason I didn't want to write this out 01:16:29.840 --> 01:16:30.900 align:middle line:90% is what does this mean? 01:16:30.900 --> 01:16:32.720 align:middle line:84% I have to take a summation in order to compute this. 01:16:32.720 --> 01:16:34.637 align:middle line:84% This is a summation over all the other things. 01:16:34.637 --> 01:16:36.480 align:middle line:84% So it's true that there are some 0s here, 01:16:36.480 --> 01:16:40.040 align:middle line:84% but the probability that player 2 has type 0 is 1. 01:16:40.040 --> 01:16:42.562 align:middle line:84% The probability that player 1 has type H is 1/2, 01:16:42.562 --> 01:16:45.020 align:middle line:84% and the probability that the other player is type E is 1/2. 01:16:45.020 --> 01:16:47.020 align:middle line:84% So that's really what I mean by positive probability. 01:16:47.020 --> 01:16:48.400 align:middle line:84% And that's actually about loss, because if something 01:16:48.400 --> 01:16:49.720 align:middle line:84% has 0 probability, I can just throw it out 01:16:49.720 --> 01:16:51.363 align:middle line:84% and not include it in the type space. 01:16:51.363 --> 01:16:52.780 align:middle line:84% So thanks for catching me on that. 01:16:52.780 --> 01:16:55.020 align:middle line:84% I tried to skip a step, but you kept me honest. 01:16:55.020 --> 01:16:56.120 align:middle line:90% Good. 01:16:56.120 --> 01:16:58.500 align:middle line:84% And you'll see that's because when I'm conditioning, 01:16:58.500 --> 01:17:00.957 align:middle line:84% I'm conditioning on an event about my type alone. 01:17:00.957 --> 01:17:02.540 align:middle line:84% I'm not conditioning on richer things, 01:17:02.540 --> 01:17:07.320 align:middle line:84% and that's why I only need that to be positive probability. 01:17:07.320 --> 01:17:09.620 align:middle line:84% So if we want to understand what's going on here, 01:17:09.620 --> 01:17:14.480 align:middle line:84% one thing we could do is we could define a new utility 01:17:14.480 --> 01:17:16.460 align:middle line:84% function directly over strategies. 01:17:16.460 --> 01:17:18.320 align:middle line:84% So we could try to reduce this game 01:17:18.320 --> 01:17:20.260 align:middle line:84% and express it purely in strategic form. 01:17:20.260 --> 01:17:22.060 align:middle line:90% So we could say-- 01:17:22.060 --> 01:17:27.060 align:middle line:84% I'm going to write capital ui of s1 up to sN. 01:17:27.060 --> 01:17:28.940 align:middle line:84% And this is going to say, what is my 01:17:28.940 --> 01:17:34.260 align:middle line:84% ex ante expected utility if we all follow these strategies? 01:17:34.260 --> 01:17:36.300 align:middle line:84% And now I'm going to take an expectation not 01:17:36.300 --> 01:17:40.540 align:middle line:84% conditional on ti-- just an expectation before we 01:17:40.540 --> 01:17:42.140 align:middle line:90% learn any of our types. 01:17:42.140 --> 01:17:54.820 align:middle line:84% And it's going to be ui of s1 t1 and tN theta, 01:17:54.820 --> 01:17:56.660 align:middle line:84% where here, the expectation is just 01:17:56.660 --> 01:17:59.140 align:middle line:84% with respect to this probability distribution, P, 01:17:59.140 --> 01:18:01.820 align:middle line:84% here, because the probability distribution, P, specifies 01:18:01.820 --> 01:18:05.038 align:middle line:84% the joint distribution of t1 up to tN and theta, 01:18:05.038 --> 01:18:07.080 align:middle line:84% so it makes sense to talk about this expectation. 01:18:07.080 --> 01:18:08.660 align:middle line:84% It's a bit of a weird object to think about 01:18:08.660 --> 01:18:10.500 align:middle line:84% because this is this weird calculation that you 01:18:10.500 --> 01:18:12.000 align:middle line:84% make before you're born and you know 01:18:12.000 --> 01:18:13.280 align:middle line:90% whether you like blue or red. 01:18:13.280 --> 01:18:15.620 align:middle line:84% It's a weird thing, but we can define this. 01:18:15.620 --> 01:18:18.780 align:middle line:84% So now once we've defined this, we have a classic strategic form 01:18:18.780 --> 01:18:19.320 align:middle line:90% game. 01:18:19.320 --> 01:18:22.070 align:middle line:84% We can say what is the expected utility of each player, 01:18:22.070 --> 01:18:25.870 align:middle line:84% i, as a function of the strategies that are used? 01:18:25.870 --> 01:18:28.690 align:middle line:90% And now we have a theorem-- 01:18:28.690 --> 01:18:30.550 align:middle line:90% it's a pretty basic theorem. 01:18:30.550 --> 01:18:45.470 align:middle line:84% But we say that a strategy profile, s star, equals s1 star 01:18:45.470 --> 01:18:53.790 align:middle line:84% to sN star, is a Bayes Nash equilibrium of the Bayesian game 01:18:53.790 --> 01:19:02.430 align:middle line:84% if and only if s star is a Nash equilibrium 01:19:02.430 --> 01:19:04.610 align:middle line:84% of the strategic form game defined here. 01:19:04.610 --> 01:19:26.080 align:middle line:90% 01:19:26.080 --> 01:19:28.000 align:middle line:84% And I think this is a good homework exercise 01:19:28.000 --> 01:19:32.220 align:middle line:84% to try to go through and verify this. 01:19:32.220 --> 01:19:33.100 align:middle line:90% Yes. 01:19:33.100 --> 01:19:35.292 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:19:35.292 --> 01:19:37.500 align:middle line:84% IAN BALL: All of these are implicitly defined over p. 01:19:37.500 --> 01:19:38.890 align:middle line:90% Maybe I'll put p here. 01:19:38.890 --> 01:19:40.640 align:middle line:84% But I guess the point I was trying to make 01:19:40.640 --> 01:19:44.280 align:middle line:84% is that this is a conditional expectation using 01:19:44.280 --> 01:19:47.360 align:middle line:84% conditional probabilities given ti. 01:19:47.360 --> 01:19:50.720 align:middle line:84% This is an unconditional expectation 01:19:50.720 --> 01:19:53.880 align:middle line:84% using unconditional probabilities over P. 01:19:53.880 --> 01:19:55.680 align:middle line:84% That's the only point I wanted to make. 01:19:55.680 --> 01:19:57.800 align:middle line:84% AUDIENCE: Based on the p in Bayes' rule. 01:19:57.800 --> 01:20:00.200 align:middle line:90% IAN BALL: It's the same p, yeah. 01:20:00.200 --> 01:20:03.280 align:middle line:84% Sorry, this is the P based on Bayes' rule. 01:20:03.280 --> 01:20:04.840 align:middle line:90% P is a distribution. 01:20:04.840 --> 01:20:08.183 align:middle line:84% Using P, I can compute conditional probabilities here. 01:20:08.183 --> 01:20:09.600 align:middle line:84% Here, I don't even have to compute 01:20:09.600 --> 01:20:11.560 align:middle line:84% conditional probabilities because P is already 01:20:11.560 --> 01:20:15.380 align:middle line:84% a distribution over t1 up to tN and theta. 01:20:15.380 --> 01:20:22.810 align:middle line:90% 01:20:22.810 --> 01:20:26.370 align:middle line:84% And maybe just two words about the intuition for this. 01:20:26.370 --> 01:20:31.810 align:middle line:84% What this says is whatever type I have, I'm behaving optimally. 01:20:31.810 --> 01:20:34.130 align:middle line:84% This is saying, let's imagine I'm in the position 01:20:34.130 --> 01:20:37.770 align:middle line:90% before I learn my type. 01:20:37.770 --> 01:20:40.770 align:middle line:84% it's saying that my complete contingent plan is 01:20:40.770 --> 01:20:43.930 align:middle line:84% optimal in expectation, but my complete contingent plan 01:20:43.930 --> 01:20:46.870 align:middle line:84% is only optimal if, at every realization of my type, 01:20:46.870 --> 01:20:48.010 align:middle line:90% it's optimal. 01:20:48.010 --> 01:20:50.690 align:middle line:84% And that's the connection, that if I anticipate 01:20:50.690 --> 01:20:53.110 align:middle line:84% that I'm going to make a mistake at some type realization, 01:20:53.110 --> 01:20:56.050 align:middle line:84% then there must be a mistake in my complete contingent plan. 01:20:56.050 --> 01:20:57.130 align:middle line:84% And that's the connection here, but I 01:20:57.130 --> 01:20:58.672 align:middle line:84% think it's good to maybe write it out 01:20:58.672 --> 01:20:59.910 align:middle line:90% in algebra as an exercise. 01:20:59.910 --> 01:21:00.510 align:middle line:90% But let me stop there. 01:21:00.510 --> 01:21:02.177 align:middle line:84% I'm happy to stick around for questions, 01:21:02.177 --> 01:21:03.350 align:middle line:90% but let me conclude there. 01:21:03.350 --> 01:21:05.670 align:middle line:90% And I know this is abstract. 01:21:05.670 --> 01:21:07.990 align:middle line:84% It's natural for this not to make so much sense 01:21:07.990 --> 01:21:10.407 align:middle line:84% the first time, but I think doing problem sets will really 01:21:10.407 --> 01:21:11.070 align:middle line:90% reinforce this. 01:21:11.070 --> 01:21:12.620 align:middle line:90% Thanks. 01:21:12.620 --> 01:21:15.000 align:middle line:90%