WEBVTT 00:00:00.000 --> 00:00:01.922 align:middle line:90% [SQUEAKING] 00:00:01.922 --> 00:00:03.818 align:middle line:90% [RUSTLING] 00:00:03.818 --> 00:00:08.558 align:middle line:90% [CLICKING] 00:00:08.558 --> 00:00:11.120 align:middle line:84% IAN BALL: So the last topic of the course 00:00:11.120 --> 00:00:13.000 align:middle line:84% that we'll discuss today is going 00:00:13.000 --> 00:00:14.980 align:middle line:84% to be something called common knowledge. 00:00:14.980 --> 00:00:21.440 align:middle line:90% 00:00:21.440 --> 00:00:23.760 align:middle line:84% So throughout the course, we've often said things 00:00:23.760 --> 00:00:28.853 align:middle line:84% like, oh, we've assumed that the game is common knowledge. 00:00:28.853 --> 00:00:31.520 align:middle line:84% But we've never really been very precise about exactly what that 00:00:31.520 --> 00:00:34.120 align:middle line:84% means when we say the game itself is common knowledge. 00:00:34.120 --> 00:00:36.560 align:middle line:84% So today, we're going to introduce formally this concept 00:00:36.560 --> 00:00:37.960 align:middle line:90% of common knowledge. 00:00:37.960 --> 00:00:40.913 align:middle line:84% And this is a topic that has quite a long history 00:00:40.913 --> 00:00:42.580 align:middle line:84% and has been studied across many fields. 00:00:42.580 --> 00:00:44.320 align:middle line:84% So philosophers thought about this issue. 00:00:44.320 --> 00:00:46.277 align:middle line:84% Logicians and mathematicians thought about it. 00:00:46.277 --> 00:00:48.360 align:middle line:84% And then economists have thought about it as well. 00:00:48.360 --> 00:00:53.240 align:middle line:84% So the starting point of our model of common knowledge 00:00:53.240 --> 00:00:56.840 align:middle line:84% is we're going to say there's some set, this is called omega, 00:00:56.840 --> 00:00:58.260 align:middle line:90% of states of the world. 00:00:58.260 --> 00:01:05.069 align:middle line:90% 00:01:05.069 --> 00:01:07.910 align:middle line:84% And there's various interpretations about what 00:01:07.910 --> 00:01:09.310 align:middle line:90% these states of the world are. 00:01:09.310 --> 00:01:13.070 align:middle line:84% You could imagine every state specifying every possible detail 00:01:13.070 --> 00:01:15.750 align:middle line:84% about the world, every arrangement of atoms. 00:01:15.750 --> 00:01:17.530 align:middle line:84% Of course, that's not a very useful model. 00:01:17.530 --> 00:01:19.770 align:middle line:84% So often, when we talk about states of the world, 00:01:19.770 --> 00:01:22.030 align:middle line:84% we talk about the states of the world, the dimensions 00:01:22.030 --> 00:01:24.950 align:middle line:84% or aspects of the world that are relevant to the problem at hand. 00:01:24.950 --> 00:01:26.730 align:middle line:84% And I think this is all a bit abstract. 00:01:26.730 --> 00:01:29.810 align:middle line:84% So as I introduce the model, I want to, on the other side, 00:01:29.810 --> 00:01:33.130 align:middle line:84% introduce some leading examples to make this concrete. 00:01:33.130 --> 00:01:35.590 align:middle line:84% So as a leading example let's imagine 00:01:35.590 --> 00:01:38.968 align:middle line:84% we're learning about the roll of a die. 00:01:38.968 --> 00:01:40.510 align:middle line:84% And therefore, the state of the world 00:01:40.510 --> 00:01:43.990 align:middle line:84% is just going to be these possible values. 00:01:43.990 --> 00:01:46.790 align:middle line:84% Of course, this is a huge simplification, 00:01:46.790 --> 00:01:50.990 align:middle line:84% but I think this is going to be a good way to fix ideas. 00:01:50.990 --> 00:01:52.630 align:middle line:90% So a die is rolled. 00:01:52.630 --> 00:01:55.430 align:middle line:84% It can take any value between 1 and 6. 00:01:55.430 --> 00:01:59.110 align:middle line:84% And it's also helpful to represent these states 00:01:59.110 --> 00:02:00.870 align:middle line:90% kind of geometrically here. 00:02:00.870 --> 00:02:03.055 align:middle line:84% So the exact arrangement doesn't matter. 00:02:03.055 --> 00:02:04.930 align:middle line:84% It just makes it easier to draw the pictures. 00:02:04.930 --> 00:02:07.263 align:middle line:84% So I'm going to think of the states as being like this-- 00:02:07.263 --> 00:02:13.567 align:middle line:90% 1, 2, 3, 4, 5, 6. 00:02:13.567 --> 00:02:15.650 align:middle line:84% And then once we have the set of states the world, 00:02:15.650 --> 00:02:18.550 align:middle line:84% we can then talk about each player's-- it's called 00:02:18.550 --> 00:02:20.290 align:middle line:90% their information partition. 00:02:20.290 --> 00:02:26.830 align:middle line:84% So we're going to say an agent, say agent i, 00:02:26.830 --> 00:02:37.710 align:middle line:90% has information partition Ki. 00:02:37.710 --> 00:02:39.390 align:middle line:84% So the subscript i indicates this 00:02:39.390 --> 00:02:42.793 align:middle line:84% is something-- this represents the information for player i. 00:02:42.793 --> 00:02:44.710 align:middle line:84% And the K comes from the word knowledge, which 00:02:44.710 --> 00:02:45.930 align:middle line:90% we'll talk about later on. 00:02:45.930 --> 00:02:47.710 align:middle line:84% We'll talk about what knowledge is. 00:02:47.710 --> 00:02:49.870 align:middle line:84% Now, partition is just a mathy word 00:02:49.870 --> 00:02:53.170 align:middle line:84% that says we're arranging these different states, 00:02:53.170 --> 00:02:56.075 align:middle line:84% these different dots, and we're splitting them up into groups. 00:02:56.075 --> 00:02:57.200 align:middle line:90% That's what a partition is. 00:02:57.200 --> 00:02:58.660 align:middle line:90% It just puts things into groups. 00:02:58.660 --> 00:03:01.300 align:middle line:84% So in this opening example, let's imagine 00:03:01.300 --> 00:03:02.400 align:middle line:90% the groups are like this. 00:03:02.400 --> 00:03:07.140 align:middle line:90% 00:03:07.140 --> 00:03:10.380 align:middle line:84% So this is an example of one possible information 00:03:10.380 --> 00:03:12.420 align:middle line:90% partition for player i. 00:03:12.420 --> 00:03:18.340 align:middle line:84% We've partitioned the six states into four different groups. 00:03:18.340 --> 00:03:21.500 align:middle line:84% And each of these groups, they're sometimes called a cell. 00:03:21.500 --> 00:03:25.340 align:middle line:84% So we might say that this partition assigns the states 00:03:25.340 --> 00:03:26.880 align:middle line:90% into one of four cells. 00:03:26.880 --> 00:03:29.380 align:middle line:84% The first cell contains states 1 and 2. 00:03:29.380 --> 00:03:31.740 align:middle line:90% This cell contains only state 3. 00:03:31.740 --> 00:03:33.780 align:middle line:90% This cell contains only state 5. 00:03:33.780 --> 00:03:37.220 align:middle line:84% And these cells contain states 4 and 6. 00:03:37.220 --> 00:03:40.420 align:middle line:84% And the word "partition" may remind you when we talked about 00:03:40.420 --> 00:03:43.140 align:middle line:84% extensive form games, we talked about a player's information 00:03:43.140 --> 00:03:43.680 align:middle line:90% partition. 00:03:43.680 --> 00:03:46.080 align:middle line:84% And this is the same concept here. 00:03:46.080 --> 00:03:48.220 align:middle line:84% So what's the interpretation of this? 00:03:48.220 --> 00:03:51.460 align:middle line:84% The interpretation is that a player, player i, 00:03:51.460 --> 00:03:53.200 align:middle line:90% does not learn the actual state. 00:03:53.200 --> 00:03:55.570 align:middle line:84% They don't actually see the roll of the die. 00:03:55.570 --> 00:03:59.515 align:middle line:84% But what they do learn is which cell the state lies in. 00:03:59.515 --> 00:04:00.890 align:middle line:84% So in this case, the player might 00:04:00.890 --> 00:04:03.520 align:middle line:84% learn the state of the world is in this cell. 00:04:03.520 --> 00:04:05.270 align:middle line:84% And if they learn that, then they can say, 00:04:05.270 --> 00:04:08.010 align:middle line:84% well, I know the die was either 1 or 2, 00:04:08.010 --> 00:04:10.690 align:middle line:84% but I can't distinguish between 1 or 2. 00:04:10.690 --> 00:04:13.650 align:middle line:84% And similarly, but in this state, when the player learns 00:04:13.650 --> 00:04:16.029 align:middle line:84% that the state is in this cell of the partition, 00:04:16.029 --> 00:04:18.390 align:middle line:84% they know for certain that the state was 3. 00:04:18.390 --> 00:04:25.730 align:middle line:90% 00:04:25.730 --> 00:04:35.770 align:middle line:84% Now, it's convenient to then introduce or denote Ki of omega 00:04:35.770 --> 00:04:39.587 align:middle line:84% to be the cell of the partition that contains state omega. 00:04:39.587 --> 00:04:41.170 align:middle line:84% This is just a definition that's going 00:04:41.170 --> 00:04:43.650 align:middle line:84% to make it a bit easier to reason about these things. 00:04:43.650 --> 00:04:46.370 align:middle line:84% So this is going to be informally 00:04:46.370 --> 00:04:57.200 align:middle line:84% the cell of the partition containing omega. 00:04:57.200 --> 00:04:59.820 align:middle line:84% So again, I think it's easier to just see an example. 00:04:59.820 --> 00:05:01.760 align:middle line:84% Let's look at our example over here. 00:05:01.760 --> 00:05:10.440 align:middle line:90% What would Ki of, say, 4 be? 00:05:10.440 --> 00:05:14.360 align:middle line:84% Well, Ki of 4 is going to be the cell of the partition that 00:05:14.360 --> 00:05:16.320 align:middle line:90% contains the state 4. 00:05:16.320 --> 00:05:19.240 align:middle line:84% In this case, that's going to be this cell. 00:05:19.240 --> 00:05:21.020 align:middle line:90% And we need to name this cell. 00:05:21.020 --> 00:05:23.760 align:middle line:84% Well, this is the cell that contains states 4 and 6. 00:05:23.760 --> 00:05:27.400 align:middle line:84% So we're going to write here this equals 4, 6. 00:05:27.400 --> 00:05:30.920 align:middle line:84% So what we're saying here is when 00:05:30.920 --> 00:05:33.600 align:middle line:84% the state is 4, the cell that contains 00:05:33.600 --> 00:05:35.905 align:middle line:90% 4 is the cell 4 comma 6. 00:05:35.905 --> 00:05:37.280 align:middle line:84% And then as just one more example 00:05:37.280 --> 00:05:41.720 align:middle line:84% to make sure we're clear, we could say Ki of 3-- 00:05:41.720 --> 00:05:44.560 align:middle line:84% well, the cell containing 3 is just 00:05:44.560 --> 00:05:46.560 align:middle line:90% this cell that only contains 3. 00:05:46.560 --> 00:05:49.000 align:middle line:90% So this is going to be just 3. 00:05:49.000 --> 00:05:53.280 align:middle line:90% 00:05:53.280 --> 00:05:56.020 align:middle line:84% Any questions about just this notation so far? 00:05:56.020 --> 00:05:58.860 align:middle line:90% 00:05:58.860 --> 00:06:04.180 align:middle line:84% So now we can start talking about knowledge. 00:06:04.180 --> 00:06:08.920 align:middle line:90% 00:06:08.920 --> 00:06:11.200 align:middle line:84% So what does it mean to know something? 00:06:11.200 --> 00:06:13.040 align:middle line:84% Well, the player doesn't necessarily 00:06:13.040 --> 00:06:16.360 align:middle line:84% know exactly what the state is, but they 00:06:16.360 --> 00:06:19.520 align:middle line:84% may be able to say that the state satisfies 00:06:19.520 --> 00:06:22.460 align:middle line:84% certain properties, or the state lies in some smaller set. 00:06:22.460 --> 00:06:25.200 align:middle line:84% And that's exactly how we're going to define knowledge. 00:06:25.200 --> 00:06:26.783 align:middle line:84% I guess maybe the most difficult thing 00:06:26.783 --> 00:06:28.908 align:middle line:84% when we first think about knowledge is, what do you 00:06:28.908 --> 00:06:29.960 align:middle line:90% form knowledge about? 00:06:29.960 --> 00:06:32.665 align:middle line:84% When I say I know something, what does that mean? 00:06:32.665 --> 00:06:34.040 align:middle line:84% And the key point is knowledge is 00:06:34.040 --> 00:06:36.510 align:middle line:84% going to be formed about what we call an event. 00:06:36.510 --> 00:06:38.260 align:middle line:84% And we can just think of this intuitively. 00:06:38.260 --> 00:06:42.760 align:middle line:84% I might say, I know that the die is odd. 00:06:42.760 --> 00:06:45.560 align:middle line:84% And if I say that, I'm making a statement about the event 00:06:45.560 --> 00:06:49.400 align:middle line:84% that the state is either 1, 3, or 5, the possible odd 00:06:49.400 --> 00:06:50.560 align:middle line:90% realizations. 00:06:50.560 --> 00:06:52.650 align:middle line:84% So first, we're going to define an event. 00:06:52.650 --> 00:06:54.333 align:middle line:84% And this notation may be familiar to you 00:06:54.333 --> 00:06:56.250 align:middle line:84% if you've taken a class on probability theory. 00:06:56.250 --> 00:07:01.710 align:middle line:84% An event E is just a subset of the state space omega. 00:07:01.710 --> 00:07:07.970 align:middle line:84% So as an example, e.g., the event E equals 1, 3, 00:07:07.970 --> 00:07:12.210 align:middle line:84% 5, formally, it's a subset of the state space. 00:07:12.210 --> 00:07:13.710 align:middle line:84% But colloquially we might say this 00:07:13.710 --> 00:07:25.930 align:middle line:84% is the event that the die is on, that the state or the die is on. 00:07:25.930 --> 00:07:30.630 align:middle line:90% 00:07:30.630 --> 00:07:33.370 align:middle line:84% So now that we have the first piece, we can talk about 00:07:33.370 --> 00:07:35.590 align:middle line:90% does a player know E? 00:07:35.590 --> 00:07:37.990 align:middle line:84% Do they know that the die is odd? 00:07:37.990 --> 00:07:40.190 align:middle line:84% But we need one more piece of notation 00:07:40.190 --> 00:07:42.230 align:middle line:84% here because, well, whether I know 00:07:42.230 --> 00:07:45.750 align:middle line:84% the die is odd is going to depend on what 00:07:45.750 --> 00:07:47.830 align:middle line:90% the actual value of the die is. 00:07:47.830 --> 00:07:51.100 align:middle line:84% If the die is 6, I'm not going to know that the die is odd. 00:07:51.100 --> 00:07:54.300 align:middle line:84% So our definition is going to describe knowledge 00:07:54.300 --> 00:07:56.540 align:middle line:90% in a particular state. 00:07:56.540 --> 00:08:12.660 align:middle line:84% So we're going to say in state omega, player i knows E. 00:08:12.660 --> 00:08:14.780 align:middle line:84% So notice here that our definition depends 00:08:14.780 --> 00:08:16.740 align:middle line:90% on two or maybe three things. 00:08:16.740 --> 00:08:18.760 align:middle line:84% It depends on what the actual state is. 00:08:18.760 --> 00:08:20.340 align:middle line:84% That's certainly affects knowledge. 00:08:20.340 --> 00:08:22.820 align:middle line:84% It depends on which player we're talking about because some 00:08:22.820 --> 00:08:25.020 align:middle line:84% players know things that other players don't. 00:08:25.020 --> 00:08:28.820 align:middle line:84% And it depends on the fact or the statement or the proposition 00:08:28.820 --> 00:08:31.300 align:middle line:84% that we're actually knowing, this event E. 00:08:31.300 --> 00:08:34.320 align:middle line:84% So we say in state omega player knows E if-- 00:08:34.320 --> 00:08:37.140 align:middle line:90% 00:08:37.140 --> 00:08:40.419 align:middle line:84% well, we're going to just write it in math 00:08:40.419 --> 00:08:51.490 align:middle line:84% and then interpret it, Ki omega is a subset of E. 00:08:51.490 --> 00:08:54.130 align:middle line:90% Let's interpret this. 00:08:54.130 --> 00:08:57.330 align:middle line:84% If the state is omega, well, the player 00:08:57.330 --> 00:08:59.830 align:middle line:84% is informed by their information partition 00:08:59.830 --> 00:09:02.068 align:middle line:84% not necessarily the state is actually omega, 00:09:02.068 --> 00:09:04.610 align:middle line:84% but they're informed of which cell of the partition the state 00:09:04.610 --> 00:09:05.570 align:middle line:90% lies in. 00:09:05.570 --> 00:09:08.270 align:middle line:84% So all the player learns, all the player, quote unquote, 00:09:08.270 --> 00:09:13.770 align:middle line:84% "knows," is that the state lies in this cell of their partition. 00:09:13.770 --> 00:09:16.050 align:middle line:84% And if that cell is a subset of E, 00:09:16.050 --> 00:09:18.770 align:middle line:84% then the player can conclude that the event must 00:09:18.770 --> 00:09:22.350 align:middle line:84% be E. They don't know exactly which state has been realized, 00:09:22.350 --> 00:09:24.430 align:middle line:84% but they know whichever state has been realized, 00:09:24.430 --> 00:09:25.570 align:middle line:90% it must be in here. 00:09:25.570 --> 00:09:27.217 align:middle line:84% And therefore, it must be in here. 00:09:27.217 --> 00:09:29.550 align:middle line:84% And this is exactly how we're going to define knowledge. 00:09:29.550 --> 00:09:33.110 align:middle line:84% So I think it's kind of easier to go through an example again. 00:09:33.110 --> 00:09:37.010 align:middle line:90% 00:09:37.010 --> 00:09:40.330 align:middle line:84% So let's here think of an event, the event 00:09:40.330 --> 00:09:42.890 align:middle line:90% that the die is, at most, 4. 00:09:42.890 --> 00:09:43.430 align:middle line:90% OK. 00:09:43.430 --> 00:09:47.470 align:middle line:84% So that's going to be this event here. 00:09:47.470 --> 00:09:51.570 align:middle line:90% 00:09:51.570 --> 00:09:55.610 align:middle line:84% And we can ask, well, does this player, the player 00:09:55.610 --> 00:09:58.810 align:middle line:84% i whose information partition is represented here, 00:09:58.810 --> 00:10:00.420 align:middle line:90% do they know event E? 00:10:00.420 --> 00:10:02.170 align:middle line:84% Or another way we might say it is, do they 00:10:02.170 --> 00:10:04.590 align:middle line:90% know that event E has occurred? 00:10:04.590 --> 00:10:06.090 align:middle line:84% Well, it's certainly going to depend 00:10:06.090 --> 00:10:09.470 align:middle line:90% on what the die actually was. 00:10:09.470 --> 00:10:14.110 align:middle line:84% So let's ask if the die is 5, does the player know E? 00:10:14.110 --> 00:10:16.890 align:middle line:84% Do they know that the event E has occurred? 00:10:16.890 --> 00:10:20.890 align:middle line:84% No, because this cell is not a subset of 5. 00:10:20.890 --> 00:10:24.010 align:middle line:84% And, in fact, there's only going to be three states-- 00:10:24.010 --> 00:10:26.370 align:middle line:84% this one, this one, and this one-- 00:10:26.370 --> 00:10:28.010 align:middle line:84% where the player knows the event E 00:10:28.010 --> 00:10:32.850 align:middle line:84% So let's write that down over here as an example. 00:10:32.850 --> 00:10:36.050 align:middle line:84% So I guess here I've changed my event E. Let's call it 00:10:36.050 --> 00:10:43.370 align:middle line:90% the event E prime. 00:10:43.370 --> 00:10:48.880 align:middle line:84% So let's consider the event E prime, which is 1, 2, 3, 4. 00:10:48.880 --> 00:10:52.840 align:middle line:84% So this is the event that the die is, at most, 4. 00:10:52.840 --> 00:10:55.560 align:middle line:84% And let's just go through each of our cases. 00:10:55.560 --> 00:10:59.920 align:middle line:84% So if the state is either 1 or 2-- 00:10:59.920 --> 00:11:02.440 align:middle line:90% let's start if the state is 1. 00:11:02.440 --> 00:11:04.840 align:middle line:84% If the state is 1, well, the player 00:11:04.840 --> 00:11:09.560 align:middle line:84% knows that the state lies in this cell 1, 2. 00:11:09.560 --> 00:11:14.240 align:middle line:84% And because the set 1, 2 is a subset of the set E prime, 00:11:14.240 --> 00:11:16.640 align:middle line:84% we conclude by our definition that in state 1 00:11:16.640 --> 00:11:20.880 align:middle line:90% the player knows the event is E. 00:11:20.880 --> 00:11:27.880 align:middle line:84% So we have Ki of 1 is a subset of E prime, which 00:11:27.880 --> 00:11:37.440 align:middle line:84% means player i knows E prime in this state 1. 00:11:37.440 --> 00:11:39.960 align:middle line:84% And we could reason analogously about state 2. 00:11:39.960 --> 00:11:42.320 align:middle line:84% And we could reason analogously about state 3. 00:11:42.320 --> 00:11:43.860 align:middle line:90% Now let's worry about state 4. 00:11:43.860 --> 00:11:47.910 align:middle line:84% I think that's maybe a tricky case. 00:11:47.910 --> 00:11:52.430 align:middle line:84% So state in state 4, the cell of the partition for player i 00:11:52.430 --> 00:11:53.430 align:middle line:90% is the cell 4, 6. 00:11:53.430 --> 00:11:59.710 align:middle line:90% 00:11:59.710 --> 00:12:07.450 align:middle line:84% So in this case, Ki of 4 is not a subset of E prime. 00:12:07.450 --> 00:12:07.950 align:middle line:90% Oops. 00:12:07.950 --> 00:12:13.470 align:middle line:90% 00:12:13.470 --> 00:12:14.270 align:middle line:90% Why? 00:12:14.270 --> 00:12:18.070 align:middle line:84% Well, this cell, 4, 6, is not entirely contained 00:12:18.070 --> 00:12:19.330 align:middle line:90% within the yellow box. 00:12:19.330 --> 00:12:20.770 align:middle line:90% It breaks the yellow box. 00:12:20.770 --> 00:12:24.310 align:middle line:84% And parts of this cell lie outside the yellow box. 00:12:24.310 --> 00:12:37.950 align:middle line:84% So we conclude that in state 4, player i does not know E prime. 00:12:37.950 --> 00:12:40.670 align:middle line:90% Now, notice the difference here. 00:12:40.670 --> 00:12:45.740 align:middle line:84% When the state is 1, the event E prime has occurred, 00:12:45.740 --> 00:12:49.260 align:middle line:84% and the player knows that it has occurred. 00:12:49.260 --> 00:12:53.740 align:middle line:84% In the state 4, event E prime has occurred 00:12:53.740 --> 00:12:56.120 align:middle line:90% because 4 is in E prime. 00:12:56.120 --> 00:12:57.880 align:middle line:84% Yet, the player does not know this. 00:12:57.880 --> 00:12:59.600 align:middle line:84% And then we can have an even easier case. 00:12:59.600 --> 00:13:01.000 align:middle line:90% Let's look at Ki of 5. 00:13:01.000 --> 00:13:05.260 align:middle line:90% 00:13:05.260 --> 00:13:11.260 align:middle line:84% And again, Ki of 5 is not a subset of E prime. 00:13:11.260 --> 00:13:19.060 align:middle line:84% So here, player i does not know E prime. 00:13:19.060 --> 00:13:21.580 align:middle line:84% But notice there's kind of an interesting difference here. 00:13:21.580 --> 00:13:24.040 align:middle line:84% In state 4, E prime has occurred, 00:13:24.040 --> 00:13:26.500 align:middle line:84% but the player i doesn't know it. 00:13:26.500 --> 00:13:28.860 align:middle line:84% In state 5, E prime has not occurred, 00:13:28.860 --> 00:13:30.600 align:middle line:90% and the player doesn't know it. 00:13:30.600 --> 00:13:33.882 align:middle line:90% 00:13:33.882 --> 00:13:35.340 align:middle line:84% And this is consistent with the way 00:13:35.340 --> 00:13:36.940 align:middle line:84% that we drew information partitions 00:13:36.940 --> 00:13:40.740 align:middle line:84% and thought about extensive form games previously. 00:13:40.740 --> 00:13:44.220 align:middle line:84% So I guess the first thing we might ask, well, 00:13:44.220 --> 00:13:46.980 align:middle line:90% we saw three possibilities here. 00:13:46.980 --> 00:13:50.100 align:middle line:90% E prime occurs, and we know it. 00:13:50.100 --> 00:13:52.900 align:middle line:84% E prime occurs, and we don't know it. 00:13:52.900 --> 00:13:55.340 align:middle line:84% And E prime doesn't occur, and we don't know it. 00:13:55.340 --> 00:13:57.980 align:middle line:84% You might ask, is it ever possible that I know as an event 00:13:57.980 --> 00:13:59.742 align:middle line:90% when the event doesn't happen? 00:13:59.742 --> 00:14:01.700 align:middle line:84% And it turns out that's going to be impossible, 00:14:01.700 --> 00:14:04.580 align:middle line:84% and we're going to make that precise next. 00:14:04.580 --> 00:14:09.100 align:middle line:84% So I guess the next definition we need-- 00:14:09.100 --> 00:14:12.420 align:middle line:84% well, we've said what it means for player i 00:14:12.420 --> 00:14:14.700 align:middle line:84% to know an event in a given state. 00:14:14.700 --> 00:14:17.380 align:middle line:84% But now we can talk about well, in which states does 00:14:17.380 --> 00:14:19.460 align:middle line:90% a player know a given event E? 00:14:19.460 --> 00:14:21.740 align:middle line:84% And that itself is another event. 00:14:21.740 --> 00:14:35.240 align:middle line:84% So now the event that player i knows E is given by-- 00:14:35.240 --> 00:14:39.340 align:middle line:90% 00:14:39.340 --> 00:14:43.850 align:middle line:84% well, we're going to call it Ki of E. 00:14:43.850 --> 00:14:47.210 align:middle line:84% So this is going to be the set of all states in which player 00:14:47.210 --> 00:14:50.690 align:middle line:84% i knows that event E has occurred. 00:14:50.690 --> 00:14:53.970 align:middle line:84% So let's write this down mathematically. 00:14:53.970 --> 00:14:56.970 align:middle line:90% It's the state omega. 00:14:56.970 --> 00:15:02.410 align:middle line:84% First, maybe we'll write it in words, that at omega player 00:15:02.410 --> 00:15:09.530 align:middle line:84% i knows E. So this is itself an event because it's 00:15:09.530 --> 00:15:11.050 align:middle line:90% a collection of states. 00:15:11.050 --> 00:15:12.930 align:middle line:84% And it's the collection of all states, 00:15:12.930 --> 00:15:17.330 align:middle line:84% such in that state player i knows that event E has occurred. 00:15:17.330 --> 00:15:20.730 align:middle line:84% And now we can just plug in our definition 00:15:20.730 --> 00:15:24.690 align:middle line:84% from here to make this more precise. 00:15:24.690 --> 00:15:31.270 align:middle line:84% And we can say, this is the set of omega omega, such that-- 00:15:31.270 --> 00:15:37.010 align:middle line:90% 00:15:37.010 --> 00:15:40.480 align:middle line:84% so all I've done is I've replaced this informal statement 00:15:40.480 --> 00:15:44.360 align:middle line:84% that player i knows E to a more precise statement of what 00:15:44.360 --> 00:15:47.660 align:middle line:84% that knowledge means, according to this definition over here. 00:15:47.660 --> 00:15:53.000 align:middle line:90% 00:15:53.000 --> 00:15:56.600 align:middle line:84% And now we can try to formalize this property that a player can 00:15:56.600 --> 00:16:00.680 align:middle line:84% only know something if it's true, that if in some state 00:16:00.680 --> 00:16:03.720 align:middle line:84% I know an event E holds, then it must be the case that the event 00:16:03.720 --> 00:16:04.780 align:middle line:90% E actually holds. 00:16:04.780 --> 00:16:06.400 align:middle line:84% And let's state that mathematically and then prove 00:16:06.400 --> 00:16:06.900 align:middle line:90% it. 00:16:06.900 --> 00:16:12.760 align:middle line:90% 00:16:12.760 --> 00:16:21.680 align:middle line:84% So here's our maybe first very basic result. Ki of E 00:16:21.680 --> 00:16:30.400 align:middle line:84% is a subset of E. So let's understand what this means. 00:16:30.400 --> 00:16:33.200 align:middle line:84% This says in any state of the world in which player 00:16:33.200 --> 00:16:37.110 align:middle line:84% i knows that E has occurred, it must be the case 00:16:37.110 --> 00:16:38.990 align:middle line:84% that that state is actually in E. 00:16:38.990 --> 00:16:41.650 align:middle line:84% And therefore, the event E has in fact occurred. 00:16:41.650 --> 00:16:45.190 align:middle line:90% 00:16:45.190 --> 00:16:48.097 align:middle line:84% And the proof of this is actually-- 00:16:48.097 --> 00:16:49.930 align:middle line:84% I mean, I think conceptually it may be hard, 00:16:49.930 --> 00:16:53.990 align:middle line:84% but mathematically pretty straightforward. 00:16:53.990 --> 00:16:59.790 align:middle line:84% We want to argue that this set is a subset of this set. 00:16:59.790 --> 00:17:02.230 align:middle line:84% So we're going to take an arbitrary thing in this set 00:17:02.230 --> 00:17:04.030 align:middle line:84% and argue that it must be in this set. 00:17:04.030 --> 00:17:13.510 align:middle line:84% So let's choose some state omega that's in Ki of E. 00:17:13.510 --> 00:17:16.990 align:middle line:84% And now let's just apply our definition. 00:17:16.990 --> 00:17:22.510 align:middle line:84% So what does it mean that state omega is in Ki of E? 00:17:22.510 --> 00:17:25.650 align:middle line:84% If we apply our definition, what can we conclude about omega? 00:17:25.650 --> 00:17:29.428 align:middle line:90% 00:17:29.428 --> 00:17:30.970 align:middle line:84% We just want to go to our definition. 00:17:30.970 --> 00:17:31.530 align:middle line:90% Yeah, Amy? 00:17:31.530 --> 00:17:33.470 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:17:33.470 --> 00:17:34.190 align:middle line:90% IAN BALL: Right. 00:17:34.190 --> 00:17:36.110 align:middle line:84% So this is just by definition, right? 00:17:36.110 --> 00:17:38.090 align:middle line:84% If we choose omega in here, well, 00:17:38.090 --> 00:17:43.727 align:middle line:84% by definition, that means Ki of omega is a subset of E. 00:17:43.727 --> 00:17:44.810 align:middle line:90% We haven't done much here. 00:17:44.810 --> 00:17:46.130 align:middle line:90% We've just used our definition. 00:17:46.130 --> 00:17:49.697 align:middle line:90% 00:17:49.697 --> 00:17:51.530 align:middle line:84% And then I would argue we're basically done. 00:17:51.530 --> 00:17:53.030 align:middle line:84% What's the last thing we need to see 00:17:53.030 --> 00:17:56.910 align:middle line:84% when you want to argue that, in fact, omega is in E? 00:17:56.910 --> 00:17:58.010 align:middle line:90% So what do we need to do? 00:17:58.010 --> 00:18:01.930 align:middle line:90% 00:18:01.930 --> 00:18:02.430 align:middle line:90% Yeah? 00:18:02.430 --> 00:18:03.847 align:middle line:84% AUDIENCE: Omega [INAUDIBLE] of Ki. 00:18:03.847 --> 00:18:04.472 align:middle line:90% IAN BALL: Yeah. 00:18:04.472 --> 00:18:05.770 align:middle line:90% So we're basically done. 00:18:05.770 --> 00:18:09.870 align:middle line:84% Ki omega is the cell of the partition that contains omega. 00:18:09.870 --> 00:18:12.030 align:middle line:84% So, in particular, omega is in here. 00:18:12.030 --> 00:18:15.870 align:middle line:84% So what that tells us is that omega 00:18:15.870 --> 00:18:20.870 align:middle line:84% is in Ki of omega, which is a subset of E. 00:18:20.870 --> 00:18:27.217 align:middle line:90% And therefore, omega is in E. 00:18:27.217 --> 00:18:28.550 align:middle line:90% Maybe that's just a mathier way. 00:18:28.550 --> 00:18:30.670 align:middle line:90% We can look at it over here. 00:18:30.670 --> 00:18:33.710 align:middle line:84% If I'm in some state, and the cell containing 00:18:33.710 --> 00:18:37.060 align:middle line:84% that state is entirely contained in this yellow box, 00:18:37.060 --> 00:18:39.160 align:middle line:84% then the state must be in the yellow box. 00:18:39.160 --> 00:18:41.620 align:middle line:90% That's all we're saying. 00:18:41.620 --> 00:18:44.098 align:middle line:84% And this is kind of a fundamental property 00:18:44.098 --> 00:18:44.640 align:middle line:90% of knowledge. 00:18:44.640 --> 00:18:46.980 align:middle line:84% I should maybe point out an alternative approach 00:18:46.980 --> 00:18:51.060 align:middle line:84% to modeling knowledge is not to work with a primitive partition, 00:18:51.060 --> 00:18:55.380 align:middle line:84% but to start with this function that tells us which events do 00:18:55.380 --> 00:18:57.340 align:middle line:84% we know, and then imposing axioms 00:18:57.340 --> 00:18:58.632 align:middle line:90% or properties on this function. 00:18:58.632 --> 00:19:00.840 align:middle line:84% And if you impose enough properties on this function, 00:19:00.840 --> 00:19:03.220 align:middle line:84% you can show that it must be derived from some partition 00:19:03.220 --> 00:19:03.840 align:middle line:90% in this way. 00:19:03.840 --> 00:19:08.340 align:middle line:84% But we're not going to go through this entire exercise. 00:19:08.340 --> 00:19:11.140 align:middle line:84% So now we've just talked about things 00:19:11.140 --> 00:19:12.940 align:middle line:90% that a single player knows. 00:19:12.940 --> 00:19:14.940 align:middle line:84% Now we're going to talk about multiple players 00:19:14.940 --> 00:19:16.380 align:middle line:90% at the same time. 00:19:16.380 --> 00:19:25.560 align:middle line:84% So let's come back over here, do a really simple example. 00:19:25.560 --> 00:19:38.730 align:middle line:90% 00:19:38.730 --> 00:19:42.130 align:middle line:84% So now let's consider we have two players. 00:19:42.130 --> 00:19:46.450 align:middle line:84% We have players one and two, and each player 00:19:46.450 --> 00:19:48.330 align:middle line:84% may know different things about the state. 00:19:48.330 --> 00:19:49.788 align:middle line:84% And therefore, each player is going 00:19:49.788 --> 00:19:51.990 align:middle line:84% to have their own information partition. 00:19:51.990 --> 00:19:56.330 align:middle line:84% So player one is going to have their information partition K1, 00:19:56.330 --> 00:20:00.350 align:middle line:84% and player two is going to have their information partition K2. 00:20:00.350 --> 00:20:06.370 align:middle line:90% 00:20:06.370 --> 00:20:09.150 align:middle line:84% And let's represent the state as we did before. 00:20:09.150 --> 00:20:13.850 align:middle line:90% 00:20:13.850 --> 00:20:19.515 align:middle line:84% So this is state 1, 2, 3, 4, 5, 6. 00:20:19.515 --> 00:20:21.390 align:middle line:84% And I think it's easier now if we use colors. 00:20:21.390 --> 00:20:30.410 align:middle line:84% So let's put player one's information partition in blue. 00:20:30.410 --> 00:20:37.000 align:middle line:84% And let's say that player one's information partition 00:20:37.000 --> 00:20:37.900 align:middle line:90% looks like this. 00:20:37.900 --> 00:20:42.560 align:middle line:90% 00:20:42.560 --> 00:20:44.680 align:middle line:84% And now let's say that player two's information 00:20:44.680 --> 00:20:47.580 align:middle line:90% partition looks like this. 00:20:47.580 --> 00:20:51.600 align:middle line:90% 00:20:51.600 --> 00:20:53.840 align:middle line:84% So intuitively, what does player one learn? 00:20:53.840 --> 00:20:57.480 align:middle line:84% Player one learns either that the die is 1 or 2, 00:20:57.480 --> 00:21:01.800 align:middle line:84% or that the die is 3 or 4, or that the die is 5 or 6. 00:21:01.800 --> 00:21:03.180 align:middle line:90% What does player two learn? 00:21:03.180 --> 00:21:06.920 align:middle line:84% They learn either that the die is odd or that the die is even. 00:21:06.920 --> 00:21:09.240 align:middle line:84% So now we can talk about the different information 00:21:09.240 --> 00:21:12.080 align:middle line:90% that the players have. 00:21:12.080 --> 00:21:18.260 align:middle line:84% And now let's draw an event here. 00:21:18.260 --> 00:21:23.080 align:middle line:90% 00:21:23.080 --> 00:21:29.360 align:middle line:84% So let's let E be the event 1, 2, 3, 4, 5. 00:21:29.360 --> 00:21:35.600 align:middle line:90% 00:21:35.600 --> 00:21:39.160 align:middle line:84% So now maybe first, let's ask, what is the-- 00:21:39.160 --> 00:21:44.800 align:middle line:84% let's describe the event K1 of E. 00:21:44.800 --> 00:21:48.000 align:middle line:84% So this is going to be in which states does player one know 00:21:48.000 --> 00:21:50.200 align:middle line:90% that event E has occurred? 00:21:50.200 --> 00:21:52.080 align:middle line:90% Any ideas here? 00:21:52.080 --> 00:21:54.320 align:middle line:84% What is this going to be Yeah, Amy? 00:21:54.320 --> 00:21:56.135 align:middle line:90% AUDIENCE: 1, 2, 3, 4. 00:21:56.135 --> 00:21:56.760 align:middle line:90% IAN BALL: Yeah. 00:21:56.760 --> 00:21:58.660 align:middle line:84% And actually, we can just write them all together. 00:21:58.660 --> 00:21:58.760 align:middle line:90% Yeah. 00:21:58.760 --> 00:21:59.740 align:middle line:90% So 1, 2, 3, 4. 00:21:59.740 --> 00:22:01.772 align:middle line:90% In any of these states-- 00:22:01.772 --> 00:22:02.980 align:middle line:90% so that's an important point. 00:22:02.980 --> 00:22:05.293 align:middle line:84% The object here, it's not a collection of cells. 00:22:05.293 --> 00:22:06.460 align:middle line:90% It's a collection of states. 00:22:06.460 --> 00:22:10.540 align:middle line:84% So it is the states in this cell and the states in this cell. 00:22:10.540 --> 00:22:14.240 align:middle line:90% So altogether, we get 1, 2, 3 4. 00:22:14.240 --> 00:22:18.020 align:middle line:84% And then we can ask, what about player two? 00:22:18.020 --> 00:22:21.520 align:middle line:90% 00:22:21.520 --> 00:22:23.440 align:middle line:84% Well, similarly player two, it's only 00:22:23.440 --> 00:22:26.840 align:middle line:84% going to be 1, 3, and 5 because these are the only states that 00:22:26.840 --> 00:22:29.550 align:middle line:84% are in a cell that's contained in the yellow box. 00:22:29.550 --> 00:22:32.250 align:middle line:90% So we get 1, 3, 5. 00:22:32.250 --> 00:22:34.950 align:middle line:90% 00:22:34.950 --> 00:22:39.270 align:middle line:84% But now we can ask another question, which is, what 00:22:39.270 --> 00:22:43.730 align:middle line:90% is K1 of E intersect K2 of E? 00:22:43.730 --> 00:22:46.830 align:middle line:90% 00:22:46.830 --> 00:22:48.830 align:middle line:84% So let's try to interpret this set. 00:22:48.830 --> 00:22:52.870 align:middle line:84% This is the set of states where player one knows E, 00:22:52.870 --> 00:22:57.370 align:middle line:84% and player two knows E. And if we just read it off here, 00:22:57.370 --> 00:22:59.810 align:middle line:84% we can see these are the states 1 and 3. 00:22:59.810 --> 00:23:09.390 align:middle line:90% 00:23:09.390 --> 00:23:15.510 align:middle line:84% So we can say in state 3, player one knows E has occurred, 00:23:15.510 --> 00:23:18.110 align:middle line:84% and player two knows E has occurred. 00:23:18.110 --> 00:23:22.870 align:middle line:84% But now we can start asking maybe a more confusing question. 00:23:22.870 --> 00:23:25.910 align:middle line:84% Does player one know that both players 00:23:25.910 --> 00:23:28.435 align:middle line:90% know that event E has occurred? 00:23:28.435 --> 00:23:30.560 align:middle line:84% So let's try to think through what that would mean. 00:23:30.560 --> 00:23:33.380 align:middle line:90% So we want to ask-- 00:23:33.380 --> 00:23:42.180 align:middle line:84% we want to look at the set K1 of K1 of E intersect K2 of E. 00:23:42.180 --> 00:23:44.740 align:middle line:90% So this means I'm player one. 00:23:44.740 --> 00:23:48.300 align:middle line:84% In which states do I know that I know event E, 00:23:48.300 --> 00:23:50.145 align:middle line:84% and my opponent, player two, knows events E? 00:23:50.145 --> 00:23:52.520 align:middle line:84% So now we can start thinking about interactive knowledge. 00:23:52.520 --> 00:23:54.980 align:middle line:84% What do I know about what other players know? 00:23:54.980 --> 00:23:57.800 align:middle line:84% And it's kind of a mouthful to say it, 00:23:57.800 --> 00:24:00.740 align:middle line:84% but mathematically it's not too hard. 00:24:00.740 --> 00:24:02.380 align:middle line:90% We've just said what this is. 00:24:02.380 --> 00:24:04.000 align:middle line:90% This is 1, 3. 00:24:04.000 --> 00:24:07.860 align:middle line:90% 00:24:07.860 --> 00:24:09.980 align:middle line:90% So what is K1 of 1, 3? 00:24:09.980 --> 00:24:20.260 align:middle line:90% 00:24:20.260 --> 00:24:20.760 align:middle line:90% Yeah? 00:24:20.760 --> 00:24:21.620 align:middle line:84% AUDIENCE: Would it be the empty set? 00:24:21.620 --> 00:24:22.995 align:middle line:84% IAN BALL: It's just the empty set 00:24:22.995 --> 00:24:25.810 align:middle line:84% because there are no states in which player 00:24:25.810 --> 00:24:30.810 align:middle line:84% one's cell of his partition is a subset of 1, 3. 00:24:30.810 --> 00:24:32.650 align:middle line:90% So this is one possibility. 00:24:32.650 --> 00:24:34.210 align:middle line:90% It can be the case-- 00:24:34.210 --> 00:24:36.950 align:middle line:84% so we can say informally, what does this mean? 00:24:36.950 --> 00:24:41.310 align:middle line:84% It means there are states of the world where both players 1 and 2 00:24:41.310 --> 00:24:43.050 align:middle line:90% know that the event is E. 00:24:43.050 --> 00:24:46.410 align:middle line:84% But there is no state of the world where player one knows 00:24:46.410 --> 00:24:52.290 align:middle line:84% that both players know that the event is E. 00:24:52.290 --> 00:24:56.890 align:middle line:84% And in fact, this motivates one more definition. 00:24:56.890 --> 00:25:01.470 align:middle line:84% Let's define-- and this is what we might call mutual knowledge. 00:25:01.470 --> 00:25:07.010 align:middle line:90% 00:25:07.010 --> 00:25:10.122 align:middle line:84% So it gets a bit messy to keep talking about, do 00:25:10.122 --> 00:25:12.330 align:middle line:84% I know that player one knows something and player two 00:25:12.330 --> 00:25:14.372 align:middle line:84% knows something and player three knows something. 00:25:14.372 --> 00:25:16.570 align:middle line:84% So let's just introduce notation for the event 00:25:16.570 --> 00:25:19.810 align:middle line:84% that everyone knows that some event has occurred. 00:25:19.810 --> 00:25:42.450 align:middle line:84% So we'll say in state omega, event E is mutual knowledge if-- 00:25:42.450 --> 00:25:49.970 align:middle line:84% well, Ki omega is a subset of E for all i, 00:25:49.970 --> 00:25:53.330 align:middle line:84% for all players i that we're talking about. 00:25:53.330 --> 00:25:55.650 align:middle line:84% Maybe we'll say i equals 1 through n, because we often 00:25:55.650 --> 00:25:56.670 align:middle line:90% work with n players. 00:25:56.670 --> 00:26:04.370 align:middle line:90% 00:26:04.370 --> 00:26:06.710 align:middle line:84% And then once we have this definition, 00:26:06.710 --> 00:26:08.470 align:middle line:90% we can just naturally define-- 00:26:08.470 --> 00:26:15.770 align:middle line:90% 00:26:15.770 --> 00:26:18.410 align:middle line:84% well, here I've said, in a given state 00:26:18.410 --> 00:26:22.370 align:middle line:84% an event is mutual knowledge if everyone knows it. 00:26:22.370 --> 00:26:24.320 align:middle line:84% But now we can ask about the event 00:26:24.320 --> 00:26:26.920 align:middle line:84% that everyone knows something, or the event 00:26:26.920 --> 00:26:28.500 align:middle line:84% that this event is mutual knowledge. 00:26:28.500 --> 00:26:30.417 align:middle line:84% I know it's getting to be a bit of a mouthful, 00:26:30.417 --> 00:26:32.560 align:middle line:84% but we can give another definition, 00:26:32.560 --> 00:26:38.420 align:middle line:84% and we'll just say K of E is equal to the intersection-- 00:26:38.420 --> 00:26:40.160 align:middle line:90% or maybe I'll write it this way. 00:26:40.160 --> 00:26:48.720 align:middle line:90% K1 of E intersect Kn of E. 00:26:48.720 --> 00:26:55.960 align:middle line:84% So this is precisely the set of states 00:26:55.960 --> 00:26:58.640 align:middle line:84% where player one knows event E. Player 00:26:58.640 --> 00:27:03.560 align:middle line:84% two knows event E. And everyone else knows event E. 00:27:03.560 --> 00:27:12.000 align:middle line:84% So if we want to rewrite this equation star over here using 00:27:12.000 --> 00:27:19.760 align:middle line:84% our new notation, we could write that is the event K1 of K of E. 00:27:19.760 --> 00:27:23.270 align:middle line:84% So in the notation, E is an event. 00:27:23.270 --> 00:27:28.030 align:middle line:84% K of E is the event that E is mutual knowledge. 00:27:28.030 --> 00:27:31.190 align:middle line:84% It's the event that everyone knows E is true. 00:27:31.190 --> 00:27:34.390 align:middle line:84% And K1 of K of E is the event that player one 00:27:34.390 --> 00:27:38.070 align:middle line:84% knows that everyone knows that the event E is true. 00:27:38.070 --> 00:27:40.950 align:middle line:84% So what this highlights is that mutual knowledge sounds pretty 00:27:40.950 --> 00:27:44.350 align:middle line:84% strong, but it's actually not quite as strong 00:27:44.350 --> 00:27:45.550 align:middle line:90% as we might think. 00:27:45.550 --> 00:27:48.590 align:middle line:84% Because it may be that we all know something, but there's 00:27:48.590 --> 00:27:51.230 align:middle line:84% a player who doesn't know that we all know something. 00:27:51.230 --> 00:27:54.230 align:middle line:84% And that suggests that we might need to go a step deeper. 00:27:54.230 --> 00:27:58.110 align:middle line:84% So we could then look at, what is the interpretation k 00:27:58.110 --> 00:28:02.710 align:middle line:84% squared of E is equal to K of K of E? 00:28:02.710 --> 00:28:06.930 align:middle line:84% So in words, what would you describe this as? 00:28:06.930 --> 00:28:09.430 align:middle line:90% This is the event that what? 00:28:09.430 --> 00:28:13.030 align:middle line:90% K of K of E. Yeah? 00:28:13.030 --> 00:28:15.678 align:middle line:84% AUDIENCE: Everyone knows that everyone else knows. 00:28:15.678 --> 00:28:16.970 align:middle line:90% IAN BALL: Knows that E is true. 00:28:16.970 --> 00:28:19.350 align:middle line:90% So we might call-- 00:28:19.350 --> 00:28:21.820 align:middle line:90% we call this mutual knowledge. 00:28:21.820 --> 00:28:24.100 align:middle line:84% It's kind of first-order mutual knowledge. 00:28:24.100 --> 00:28:26.438 align:middle line:84% This we might call second-order mutual knowledge. 00:28:26.438 --> 00:28:27.480 align:middle line:90% Everyone knows something. 00:28:27.480 --> 00:28:31.860 align:middle line:84% And on top of that, everyone knows that everyone knows it. 00:28:31.860 --> 00:28:41.740 align:middle line:84% So this is maybe second-order mutual knowledge, 00:28:41.740 --> 00:28:43.000 align:middle line:90% but that's still too weak. 00:28:43.000 --> 00:28:44.580 align:middle line:84% Because it could be that everyone knows something, 00:28:44.580 --> 00:28:46.400 align:middle line:84% and everyone knows that everyone knows something. 00:28:46.400 --> 00:28:48.020 align:middle line:84% But maybe people don't know that everyone 00:28:48.020 --> 00:28:49.228 align:middle line:90% knows that everyone knows it. 00:28:49.228 --> 00:28:50.540 align:middle line:90% So have to go a step deeper. 00:28:50.540 --> 00:28:57.100 align:middle line:84% And we finally get to maybe what I'll say common knowledge, CK 00:28:57.100 --> 00:29:01.253 align:middle line:84% of E. Now we can formally define this thing that we've 00:29:01.253 --> 00:29:03.420 align:middle line:84% been informally talking about throughout the course. 00:29:03.420 --> 00:29:13.740 align:middle line:84% CK of E is K of E intersect K squared of E Intersect K cubed 00:29:13.740 --> 00:29:17.500 align:middle line:90% of E, and it goes on forever. 00:29:17.500 --> 00:29:20.660 align:middle line:84% So now we have a formal definition of common knowledge. 00:29:20.660 --> 00:29:23.260 align:middle line:90% Event E is common knowledge. 00:29:23.260 --> 00:29:26.020 align:middle line:84% Or in other words, the set of states in which event 00:29:26.020 --> 00:29:28.140 align:middle line:84% E is common knowledge are precisely 00:29:28.140 --> 00:29:30.140 align:middle line:84% the states in which everyone knows 00:29:30.140 --> 00:29:34.380 align:middle line:84% E. Everyone knows that everyone knows E. Everyone knows 00:29:34.380 --> 00:29:36.380 align:middle line:84% that everyone knows that everyone knows E. 00:29:36.380 --> 00:29:39.300 align:middle line:84% And this goes on no matter how long the length of this chain 00:29:39.300 --> 00:29:39.800 align:middle line:90% is. 00:29:39.800 --> 00:29:42.060 align:middle line:84% And this is a very, very demanding property 00:29:42.060 --> 00:29:44.380 align:middle line:84% that we've implicitly been kind of relying on 00:29:44.380 --> 00:29:45.400 align:middle line:90% throughout the course. 00:29:45.400 --> 00:29:47.030 align:middle line:84% Now, I think you might think, well, why 00:29:47.030 --> 00:29:48.280 align:middle line:90% are we going through all this? 00:29:48.280 --> 00:29:49.380 align:middle line:90% This is kind of-- 00:29:49.380 --> 00:29:51.100 align:middle line:90% we're just playing math games. 00:29:51.100 --> 00:29:52.780 align:middle line:84% Does this really have any content? 00:29:52.780 --> 00:29:56.120 align:middle line:84% So I now, for the next part of the class, 00:29:56.120 --> 00:29:57.900 align:middle line:84% want to go over a famous riddle that 00:29:57.900 --> 00:30:01.100 align:middle line:84% illustrates how different common knowledge of something 00:30:01.100 --> 00:30:02.802 align:middle line:84% can be from just mutual knowledge, 00:30:02.802 --> 00:30:04.760 align:middle line:84% or from first- or second-order mutual knowledge 00:30:04.760 --> 00:30:08.240 align:middle line:84% So this is kind of a classic riddle that we'll go over. 00:30:08.240 --> 00:30:31.210 align:middle line:90% 00:30:31.210 --> 00:30:34.128 align:middle line:90% So this is sometimes called-- 00:30:34.128 --> 00:30:34.670 align:middle line:90% I don't know. 00:30:34.670 --> 00:30:36.570 align:middle line:84% It has many names-- the sage and the children. 00:30:36.570 --> 00:30:37.112 align:middle line:90% I don't know. 00:30:37.112 --> 00:30:39.710 align:middle line:84% Or the teacher-- there's so many stories we could use. 00:30:39.710 --> 00:30:41.510 align:middle line:84% Let's use the teacher and the students. 00:30:41.510 --> 00:30:47.570 align:middle line:90% 00:30:47.570 --> 00:30:49.390 align:middle line:84% And here's the-- the story, of course, 00:30:49.390 --> 00:30:51.428 align:middle line:84% it's kind of a bit of a silly puzzle, 00:30:51.428 --> 00:30:53.470 align:middle line:84% but I think it illustrates the ideas really well. 00:30:53.470 --> 00:30:54.790 align:middle line:90% So we have three students. 00:30:54.790 --> 00:30:57.850 align:middle line:90% 00:30:57.850 --> 00:30:59.790 align:middle line:90% Call them 1, 2, and 3. 00:30:59.790 --> 00:31:02.370 align:middle line:90% 00:31:02.370 --> 00:31:05.460 align:middle line:84% And there's various versions of it. 00:31:05.460 --> 00:31:06.710 align:middle line:90% We'll use the version of hats. 00:31:06.710 --> 00:31:08.150 align:middle line:84% So each student is wearing a hat. 00:31:08.150 --> 00:31:13.300 align:middle line:90% 00:31:13.300 --> 00:31:16.220 align:middle line:84% The student's hat is white or red. 00:31:16.220 --> 00:31:18.960 align:middle line:90% 00:31:18.960 --> 00:31:19.900 align:middle line:90% Why do we use hats? 00:31:19.900 --> 00:31:21.260 align:middle line:84% Well, the point is, when you're wearing a hat, 00:31:21.260 --> 00:31:23.480 align:middle line:84% you can see other people's hats, but not your own. 00:31:23.480 --> 00:31:25.740 align:middle line:84% So you can observe the color of everyone else's hat, 00:31:25.740 --> 00:31:29.120 align:middle line:84% but you can't observe the color of your own hat. 00:31:29.120 --> 00:31:41.000 align:middle line:84% And so we'll say that each student observes 00:31:41.000 --> 00:31:45.180 align:middle line:84% the other's hats, but not her own. 00:31:45.180 --> 00:32:05.280 align:middle line:90% 00:32:05.280 --> 00:32:09.180 align:middle line:84% So let's suppose that all the hats are actually red. 00:32:09.180 --> 00:32:16.150 align:middle line:90% 00:32:16.150 --> 00:32:17.730 align:middle line:90% We should put it in red. 00:32:17.730 --> 00:32:18.970 align:middle line:90% So all hats are red. 00:32:18.970 --> 00:32:21.110 align:middle line:90% This is an assumption. 00:32:21.110 --> 00:32:23.710 align:middle line:84% It's really a statement about what the true state omega is. 00:32:23.710 --> 00:32:27.190 align:middle line:84% So the true state is that all the hats are red. 00:32:27.190 --> 00:32:30.810 align:middle line:84% And then the teacher is going to ask all the students, 00:32:30.810 --> 00:32:32.290 align:middle line:84% do you know what color your hat is? 00:32:32.290 --> 00:32:39.790 align:middle line:90% 00:32:39.790 --> 00:32:41.628 align:middle line:84% Now, for this riddle to work out, 00:32:41.628 --> 00:32:44.170 align:middle line:84% it must be the case that we're assuming everyone is truthful. 00:32:44.170 --> 00:32:46.110 align:middle line:90% No one's lying to the teacher. 00:32:46.110 --> 00:32:49.230 align:middle line:84% So the teacher asks, do you know your hat color? 00:32:49.230 --> 00:32:52.750 align:middle line:84% And we should be a little precise 00:32:52.750 --> 00:32:54.790 align:middle line:84% when it means know your hat color means either 00:32:54.790 --> 00:32:58.310 align:middle line:84% you know hat is red, or you know your hat is white. 00:32:58.310 --> 00:33:01.630 align:middle line:84% That's what it means to know your hat color. 00:33:01.630 --> 00:33:04.190 align:middle line:84% What would the student's answer be if they're truthful? 00:33:04.190 --> 00:33:05.450 align:middle line:90% They look around. 00:33:05.450 --> 00:33:07.430 align:middle line:90% Do they know their hat color? 00:33:07.430 --> 00:33:07.955 align:middle line:90% No, right? 00:33:07.955 --> 00:33:09.330 align:middle line:84% They can see other people's hats, 00:33:09.330 --> 00:33:10.550 align:middle line:90% but they can't see their own. 00:33:10.550 --> 00:33:11.650 align:middle line:90% They know others hat colors. 00:33:11.650 --> 00:33:12.450 align:middle line:90% They don't know their own. 00:33:12.450 --> 00:33:13.610 align:middle line:90% So do your own hat color? 00:33:13.610 --> 00:33:16.190 align:middle line:90% The answer is no. 00:33:16.190 --> 00:33:18.450 align:middle line:84% Then the teacher makes an announcement. 00:33:18.450 --> 00:33:22.590 align:middle line:84% The teacher says-- so the teacher announces 00:33:22.590 --> 00:33:24.010 align:middle line:90% at least one hat is red. 00:33:24.010 --> 00:33:29.143 align:middle line:90% 00:33:29.143 --> 00:33:30.810 align:middle line:84% Have people heard of this riddle before? 00:33:30.810 --> 00:33:31.330 align:middle line:90% I'm curious. 00:33:31.330 --> 00:33:31.830 align:middle line:90% Anyone? 00:33:31.830 --> 00:33:33.130 align:middle line:90% One person, two. 00:33:33.130 --> 00:33:34.530 align:middle line:90% OK, so not too many. 00:33:34.530 --> 00:33:36.895 align:middle line:84% So the teacher says at least one hat is red. 00:33:36.895 --> 00:33:39.270 align:middle line:84% And now she's going to go through and ask these questions 00:33:39.270 --> 00:33:41.230 align:middle line:84% again to the students one by one. 00:33:41.230 --> 00:33:44.410 align:middle line:84% So she's going to go to student 1, S1, and say, 00:33:44.410 --> 00:33:46.030 align:middle line:90% do you know your hat color? 00:33:46.030 --> 00:33:48.590 align:middle line:84% And still, I mean, knowing that at least one hat is 00:33:48.590 --> 00:33:51.070 align:middle line:84% red doesn't give that student any information 00:33:51.070 --> 00:33:51.970 align:middle line:90% about their own hat. 00:33:51.970 --> 00:33:54.070 align:middle line:84% So that student is going to say no. 00:33:54.070 --> 00:33:56.185 align:middle line:84% And then the teacher goes to student 2 and says, 00:33:56.185 --> 00:33:57.310 align:middle line:90% do you know your hat color? 00:33:57.310 --> 00:33:59.470 align:middle line:90% And the student says, no. 00:33:59.470 --> 00:34:02.330 align:middle line:84% And crucially, when the teacher is asking these students, 00:34:02.330 --> 00:34:04.350 align:middle line:84% the other students get to observe the answers. 00:34:04.350 --> 00:34:06.100 align:middle line:84% So these are public questioning, and we're 00:34:06.100 --> 00:34:07.630 align:middle line:90% assuming everyone's truthful. 00:34:07.630 --> 00:34:09.630 align:middle line:84% And then she goes to student three, 00:34:09.630 --> 00:34:10.969 align:middle line:90% and that student says yes. 00:34:10.969 --> 00:34:13.739 align:middle line:90% Now I know my hat color. 00:34:13.739 --> 00:34:15.300 align:middle line:90% Now, I think people-- 00:34:15.300 --> 00:34:17.300 align:middle line:84% this, I think, was first developed in the 1950s. 00:34:17.300 --> 00:34:19.258 align:middle line:84% These mathematicians were really shocked by it. 00:34:19.258 --> 00:34:20.719 align:middle line:90% So why is this kind of shocking? 00:34:20.719 --> 00:34:22.940 align:middle line:90% I hope people are shocked. 00:34:22.940 --> 00:34:25.500 align:middle line:84% It's shocking-- well, it doesn't feel 00:34:25.500 --> 00:34:28.340 align:middle line:84% like the teacher's announcement should have made any difference. 00:34:28.340 --> 00:34:30.620 align:middle line:84% The teacher announced something that everyone knew. 00:34:30.620 --> 00:34:33.179 align:middle line:84% The teacher said at least one hat his red. 00:34:33.179 --> 00:34:35.100 align:middle line:84% Well, if everyone's had his red, any student 00:34:35.100 --> 00:34:37.017 align:middle line:84% can look around and, in fact, see that there's 00:34:37.017 --> 00:34:38.420 align:middle line:90% at least two red hats. 00:34:38.420 --> 00:34:42.300 align:middle line:84% So why would the announcement that at least one hat is red 00:34:42.300 --> 00:34:44.699 align:middle line:84% change the dynamics that come after? 00:34:44.699 --> 00:34:45.360 align:middle line:90% Any thoughts? 00:34:45.360 --> 00:34:47.842 align:middle line:84% The teacher has said something that everyone knows. 00:34:47.842 --> 00:34:49.800 align:middle line:84% So why does this announcement make a difference 00:34:49.800 --> 00:34:52.580 align:middle line:84% using the language-- maybe using the language we've said before, 00:34:52.580 --> 00:34:54.739 align:middle line:84% how could saying something, announcing something 00:34:54.739 --> 00:34:57.040 align:middle line:84% that everyone already knows make a difference? 00:34:57.040 --> 00:34:58.460 align:middle line:90% Yeah? 00:34:58.460 --> 00:35:00.910 align:middle line:84% AUDIENCE: If player one-- or student one 00:35:00.910 --> 00:35:03.740 align:middle line:84% and student two, both sides, one hat is red. 00:35:03.740 --> 00:35:06.820 align:middle line:84% And both, one that they don't share [INAUDIBLE]. 00:35:06.820 --> 00:35:09.780 align:middle line:90% 00:35:09.780 --> 00:35:10.900 align:middle line:90% IAN BALL: I didn't follow. 00:35:10.900 --> 00:35:13.858 align:middle line:84% AUDIENCE: [INAUDIBLE] based on one student [INAUDIBLE]. 00:35:13.858 --> 00:35:15.650 align:middle line:84% IAN BALL: So student three is learning from 00:35:15.650 --> 00:35:17.310 align:middle line:84% based on what student one and student said. 00:35:17.310 --> 00:35:18.190 align:middle line:90% But let's be clear. 00:35:18.190 --> 00:35:20.370 align:middle line:84% If the teacher had not made this announcement, 00:35:20.370 --> 00:35:22.703 align:middle line:84% and the teacher kept saying, do you know your hat color? 00:35:22.703 --> 00:35:25.390 align:middle line:84% Everyone would say no, and we would just go around forever, 00:35:25.390 --> 00:35:26.070 align:middle line:90% actually. 00:35:26.070 --> 00:35:29.090 align:middle line:84% So the answers to the questions are fundamentally changed 00:35:29.090 --> 00:35:31.270 align:middle line:84% by this announcement that at least one hat is red. 00:35:31.270 --> 00:35:33.270 align:middle line:84% So why does this announcement make a difference? 00:35:33.270 --> 00:35:35.970 align:middle line:84% What is changed by this announcement using maybe a hint 00:35:35.970 --> 00:35:37.190 align:middle line:84% about what we've talked about over here. 00:35:37.190 --> 00:35:37.690 align:middle line:90% Yeah? 00:35:37.690 --> 00:35:39.107 align:middle line:84% AUDIENCE: So like everyone already 00:35:39.107 --> 00:35:40.590 align:middle line:84% knew that at least one hat is red, 00:35:40.590 --> 00:35:42.490 align:middle line:84% but not everyone knew that everyone knew it. 00:35:42.490 --> 00:35:42.730 align:middle line:90% IAN BALL: Yeah. 00:35:42.730 --> 00:35:44.150 align:middle line:84% In fact, everyone knew that everyone knew it. 00:35:44.150 --> 00:35:45.410 align:middle line:84% But it's not the case that everyone knew that everyone 00:35:45.410 --> 00:35:46.535 align:middle line:90% knew that everyone knew it. 00:35:46.535 --> 00:35:48.650 align:middle line:84% So why did this announcement make a difference? 00:35:48.650 --> 00:35:51.782 align:middle line:84% It took something that was mutual knowledge-- in fact, 00:35:51.782 --> 00:35:53.990 align:middle line:84% if we're careful, it's second-order mutual knowledge, 00:35:53.990 --> 00:35:55.890 align:middle line:84% but it was not common knowledge, and it 00:35:55.890 --> 00:35:57.450 align:middle line:90% made that common knowledge. 00:35:57.450 --> 00:35:59.530 align:middle line:84% So announcements about things that everyone know 00:35:59.530 --> 00:36:00.555 align:middle line:90% can change things. 00:36:00.555 --> 00:36:01.930 align:middle line:84% If that thing that everyone knows 00:36:01.930 --> 00:36:04.610 align:middle line:84% is only mutual knowledge and not common knowledge. 00:36:04.610 --> 00:36:07.090 align:middle line:84% So the crucial thing about this announcement 00:36:07.090 --> 00:36:14.240 align:middle line:84% is it makes an event go from mutual knowledge, 00:36:14.240 --> 00:36:18.680 align:middle line:84% everyone knows it, to common knowledge. 00:36:18.680 --> 00:36:23.600 align:middle line:90% 00:36:23.600 --> 00:36:26.000 align:middle line:84% And I think this really highlights 00:36:26.000 --> 00:36:28.620 align:middle line:84% how fundamentally different these two concepts are. 00:36:28.620 --> 00:36:31.460 align:middle line:84% So let's try to work through and see how we actually get this. 00:36:31.460 --> 00:36:33.180 align:middle line:90% And then we'll conclude. 00:36:33.180 --> 00:36:46.440 align:middle line:90% 00:36:46.440 --> 00:36:48.700 align:middle line:84% So first, we have to model the state space. 00:36:48.700 --> 00:36:50.760 align:middle line:84% So each of the three students has a hat, 00:36:50.760 --> 00:36:53.640 align:middle line:84% and the hat can either be white or red. 00:36:53.640 --> 00:36:55.420 align:middle line:84% I find it easier to use binary here. 00:36:55.420 --> 00:36:57.240 align:middle line:84% So let's say we're going to say white is 00:36:57.240 --> 00:37:01.520 align:middle line:84% going to correspond to 0, and red is going to correspond to 1. 00:37:01.520 --> 00:37:04.000 align:middle line:90% So what is a state omega? 00:37:04.000 --> 00:37:09.320 align:middle line:84% Well, a state omega is going to be something like 010. 00:37:09.320 --> 00:37:12.480 align:middle line:84% This would mean player one's hat is white, 00:37:12.480 --> 00:37:15.380 align:middle line:84% player two's hat is red, and player three's hat is white. 00:37:15.380 --> 00:37:17.520 align:middle line:84% So I'm going to represent the state 00:37:17.520 --> 00:37:20.840 align:middle line:84% by 3 binary digits, each one indicating 00:37:20.840 --> 00:37:23.560 align:middle line:84% whether that corresponding student's hat is red. 00:37:23.560 --> 00:37:27.060 align:middle line:84% So I'll just note here, this is student two's hat color. 00:37:27.060 --> 00:37:31.360 align:middle line:90% 00:37:31.360 --> 00:37:34.240 align:middle line:84% Now, this means that we have eight states. 00:37:34.240 --> 00:37:37.680 align:middle line:84% And it turns out to be quite useful to represent the states 00:37:37.680 --> 00:37:39.640 align:middle line:90% as the corners of a cube. 00:37:39.640 --> 00:37:43.660 align:middle line:84% So let me think about different axes here. 00:37:43.660 --> 00:37:46.480 align:middle line:90% 00:37:46.480 --> 00:37:53.560 align:middle line:84% So we have the first axis, the second axis, and the third axis. 00:37:53.560 --> 00:37:57.020 align:middle line:84% This will kind of test my drawing abilities here. 00:37:57.020 --> 00:38:02.880 align:middle line:90% 00:38:02.880 --> 00:38:06.590 align:middle line:90% I think-- how do I do this? 00:38:06.590 --> 00:38:20.110 align:middle line:84% So we need- have I drawn that right? 00:38:20.110 --> 00:38:23.550 align:middle line:90% It keeps-- kind of jumps at you. 00:38:23.550 --> 00:38:25.390 align:middle line:90% Let me see how I did it here. 00:38:25.390 --> 00:38:30.602 align:middle line:84% OK Yeah, let me draw this a bit differently. 00:38:30.602 --> 00:38:32.310 align:middle line:84% I don't know if it makes more sense to me 00:38:32.310 --> 00:38:36.270 align:middle line:84% or it will make more sense to other people, too. 00:38:36.270 --> 00:38:39.910 align:middle line:90% I think I should-- 00:38:39.910 --> 00:38:41.950 align:middle line:90% yeah. 00:38:41.950 --> 00:38:43.130 align:middle line:90% I'm struggling here. 00:38:43.130 --> 00:38:47.030 align:middle line:90% 00:38:47.030 --> 00:38:49.390 align:middle line:84% I should have done better in drawing class here. 00:38:49.390 --> 00:38:51.630 align:middle line:90% Let's see. 00:38:51.630 --> 00:38:54.630 align:middle line:84% It makes sense in my paper, but then I can't draw it here. 00:38:54.630 --> 00:38:56.970 align:middle line:90% OK, I think it should be-- 00:38:56.970 --> 00:38:57.470 align:middle line:90% yeah. 00:38:57.470 --> 00:39:01.910 align:middle line:90% 00:39:01.910 --> 00:39:04.110 align:middle line:90% Yeah. 00:39:04.110 --> 00:39:05.820 align:middle line:90% There we go. 00:39:05.820 --> 00:39:08.220 align:middle line:90% That looks better. 00:39:08.220 --> 00:39:16.380 align:middle line:84% So we have-- so this is state 111. 00:39:16.380 --> 00:39:21.500 align:middle line:90% This is state 000. 00:39:21.500 --> 00:39:28.180 align:middle line:84% And as we move on this axis, we increase component 1 by one. 00:39:28.180 --> 00:39:33.800 align:middle line:84% As we move along this axis, we increase component 2 by one. 00:39:33.800 --> 00:39:38.340 align:middle line:84% And as we move along this axis, we increase component 3 by one. 00:39:38.340 --> 00:39:42.100 align:middle line:90% So my picture-- 00:39:42.100 --> 00:39:45.020 align:middle line:90% I think-- there we go. 00:39:45.020 --> 00:39:49.140 align:middle line:84% So now let's think about the information partitions. 00:39:49.140 --> 00:39:54.900 align:middle line:84% So what is player one's information partition? 00:39:54.900 --> 00:39:57.940 align:middle line:84% Well, player one's partition has four cells. 00:39:57.940 --> 00:40:00.300 align:middle line:84% The player one observes the colors 00:40:00.300 --> 00:40:02.100 align:middle line:84% of the other two students' hats, but they 00:40:02.100 --> 00:40:03.610 align:middle line:90% don't observe their own hat. 00:40:03.610 --> 00:40:05.650 align:middle line:84% So we can think of each cell of the partition 00:40:05.650 --> 00:40:07.490 align:middle line:90% as being a horizontal line. 00:40:07.490 --> 00:40:14.606 align:middle line:90% 00:40:14.606 --> 00:40:17.270 align:middle line:84% And it may get a bit messy when I keep drawing everyone else's. 00:40:17.270 --> 00:40:22.130 align:middle line:84% Player two's is going to be like this. 00:40:22.130 --> 00:40:25.490 align:middle line:84% And then player three's partition 00:40:25.490 --> 00:40:27.890 align:middle line:90% is going to be like this. 00:40:27.890 --> 00:40:30.210 align:middle line:90% It's going to be [INAUDIBLE]. 00:40:30.210 --> 00:40:31.790 align:middle line:84% So let's understand what that means. 00:40:31.790 --> 00:40:36.010 align:middle line:90% 00:40:36.010 --> 00:40:39.170 align:middle line:84% In this cell of the partition, player one 00:40:39.170 --> 00:40:41.890 align:middle line:84% sees that the other two player's hats are white, 00:40:41.890 --> 00:40:44.150 align:middle line:84% but player one doesn't know her own hat color. 00:40:44.150 --> 00:40:47.930 align:middle line:84% She knows her hat can either be white or red. 00:40:47.930 --> 00:40:51.090 align:middle line:90% Let's go up to this cell. 00:40:51.090 --> 00:40:55.090 align:middle line:90% Well, this point here is 001. 00:40:55.090 --> 00:40:58.170 align:middle line:84% So here, player one sees that player two's hat is white 00:40:58.170 --> 00:41:00.450 align:middle line:84% and player three's hat is red, but she 00:41:00.450 --> 00:41:02.010 align:middle line:90% doesn't know her own hat color. 00:41:02.010 --> 00:41:07.570 align:middle line:84% And that's why these two are going to be like this. 00:41:07.570 --> 00:41:13.670 align:middle line:84% So this is-- any questions about this information structure? 00:41:13.670 --> 00:41:14.170 align:middle line:90% Great. 00:41:14.170 --> 00:41:17.490 align:middle line:84% So now let me try to formalize my claim. 00:41:17.490 --> 00:41:20.650 align:middle line:84% My claim here was that the event, that at least one hat is 00:41:20.650 --> 00:41:23.990 align:middle line:84% read, is mutual knowledge but not common knowledge. 00:41:23.990 --> 00:41:27.170 align:middle line:90% So let's try to check that. 00:41:27.170 --> 00:41:28.310 align:middle line:90% So let's write this down. 00:41:28.310 --> 00:41:32.130 align:middle line:84% This event E-- maybe I'll write E greater than or equal to 1. 00:41:32.130 --> 00:41:35.510 align:middle line:84% This is the event that at least one student's hat is red. 00:41:35.510 --> 00:41:41.970 align:middle line:90% 00:41:41.970 --> 00:41:43.710 align:middle line:90% I'll write this as a triple. 00:41:43.710 --> 00:41:46.530 align:middle line:90% 00:41:46.530 --> 00:41:49.530 align:middle line:84% So the event that at least one student's hat is red 00:41:49.530 --> 00:41:54.330 align:middle line:84% is the event that omega 1 plus omega 2 plus omega 3 00:41:54.330 --> 00:41:58.010 align:middle line:90% is greater than or equal to 1. 00:41:58.010 --> 00:42:00.050 align:middle line:84% In fact, if we draw it on the graph, 00:42:00.050 --> 00:42:03.133 align:middle line:84% this is the event that's everything except this point. 00:42:03.133 --> 00:42:05.800 align:middle line:84% So it's the event containing all seven points of the cube, which 00:42:05.800 --> 00:42:07.640 align:middle line:84% I've quite messily drawn, but hopefully it 00:42:07.640 --> 00:42:12.640 align:middle line:84% looks-- you can see it now, except this point here. 00:42:12.640 --> 00:42:15.940 align:middle line:84% So now we're going to have to do a bit of calculation. 00:42:15.940 --> 00:42:17.940 align:middle line:90% So let's come over here. 00:42:17.940 --> 00:42:28.280 align:middle line:90% 00:42:28.280 --> 00:42:35.200 align:middle line:84% So now let's ask, what is K1 of E greater than 1? 00:42:35.200 --> 00:42:38.320 align:middle line:84% So this is, in which states does player one 00:42:38.320 --> 00:42:41.920 align:middle line:84% know that at least one player's hat is red? 00:42:41.920 --> 00:42:43.000 align:middle line:90% I want to be clear here. 00:42:43.000 --> 00:42:45.240 align:middle line:84% Over here, I was talking about the true state 00:42:45.240 --> 00:42:47.160 align:middle line:90% being red, red, red. 00:42:47.160 --> 00:42:49.387 align:middle line:84% But for these statements about knowledge, 00:42:49.387 --> 00:42:51.220 align:middle line:84% they don't depend on what the true state is. 00:42:51.220 --> 00:42:52.800 align:middle line:84% These are just statements about in which state 00:42:52.800 --> 00:42:53.500 align:middle line:90% something is known. 00:42:53.500 --> 00:42:55.167 align:middle line:84% They don't depend on the realized state. 00:42:55.167 --> 00:42:57.840 align:middle line:90% So what is this going to be? 00:42:57.840 --> 00:43:02.710 align:middle line:84% It's omega and omega such that what? 00:43:02.710 --> 00:43:03.230 align:middle line:90% Yeah? 00:43:03.230 --> 00:43:05.880 align:middle line:84% AUDIENCE: Omega 2 plus omega 3 is greater than omega 4. 00:43:05.880 --> 00:43:06.630 align:middle line:90% IAN BALL: Exactly. 00:43:06.630 --> 00:43:08.430 align:middle line:84% So I only know at least someone's hat 00:43:08.430 --> 00:43:10.790 align:middle line:90% is red if I see a red hat. 00:43:10.790 --> 00:43:12.010 align:middle line:90% And I'm player one. 00:43:12.010 --> 00:43:13.390 align:middle line:90% I see hats two and three. 00:43:13.390 --> 00:43:14.290 align:middle line:90% I look at them. 00:43:14.290 --> 00:43:16.070 align:middle line:84% If either of those is red, then I 00:43:16.070 --> 00:43:17.890 align:middle line:84% know at least one person's hat is red. 00:43:17.890 --> 00:43:21.990 align:middle line:84% So this is the event that W2 plus W3 00:43:21.990 --> 00:43:24.910 align:middle line:90% is greater than or equal to 1. 00:43:24.910 --> 00:43:30.010 align:middle line:84% And symmetrically, for player 2 it's the same thing, 00:43:30.010 --> 00:43:34.230 align:middle line:84% but it's W1 plus W3 is greater than or equal to 1. 00:43:34.230 --> 00:43:36.190 align:middle line:84% Because if I'm player two, I can only 00:43:36.190 --> 00:43:39.230 align:middle line:84% observe the hats of player one and player three. 00:43:39.230 --> 00:43:40.730 align:middle line:90% And then finally, K3. 00:43:40.730 --> 00:43:49.670 align:middle line:90% 00:43:49.670 --> 00:43:53.550 align:middle line:84% So now that we've done these calculations, 00:43:53.550 --> 00:43:58.400 align:middle line:84% we can say, well, what is K of E, which, remember, 00:43:58.400 --> 00:44:17.780 align:middle line:84% is the intersection K1 of E intersect K2 of E. Well, 00:44:17.780 --> 00:44:22.900 align:middle line:90% W2 plus W3 has to be at least 1. 00:44:22.900 --> 00:44:25.860 align:middle line:90% W1 plus W3 has to be at least 1. 00:44:25.860 --> 00:44:29.380 align:middle line:84% And W1 plus W2 has to be at least 1. 00:44:29.380 --> 00:44:31.180 align:middle line:84% It turns out this is exactly going 00:44:31.180 --> 00:44:36.140 align:middle line:84% to be the event that at least two hats are red. 00:44:36.140 --> 00:44:37.440 align:middle line:90% Let's see how we can see this. 00:44:37.440 --> 00:44:48.100 align:middle line:90% 00:44:48.100 --> 00:44:50.980 align:middle line:84% I think it's kind of easier maybe to reason in words. 00:44:50.980 --> 00:44:55.060 align:middle line:84% If only one person's hat is red, then the person with the red hat 00:44:55.060 --> 00:44:56.740 align:middle line:90% sees two white hats. 00:44:56.740 --> 00:44:59.820 align:middle line:84% So they're not certain that at least one person's hat is red. 00:44:59.820 --> 00:45:01.720 align:middle line:84% But if at least two hats are red, 00:45:01.720 --> 00:45:03.700 align:middle line:84% then everyone must see at least one red hat, 00:45:03.700 --> 00:45:06.020 align:middle line:84% and therefore, know that at least one hat is red. 00:45:06.020 --> 00:45:09.080 align:middle line:84% So formally, if we maintain this notation here, 00:45:09.080 --> 00:45:14.360 align:middle line:84% this is E greater than or equal to 2. 00:45:14.360 --> 00:45:17.260 align:middle line:90% 00:45:17.260 --> 00:45:20.740 align:middle line:84% So let's see if we can continue this pattern. 00:45:20.740 --> 00:45:26.460 align:middle line:84% So we've said that K of E greater than or equal to 1 00:45:26.460 --> 00:45:29.780 align:middle line:90% is E greater than or equal to 2. 00:45:29.780 --> 00:45:33.620 align:middle line:84% The event that everyone knows at least one hat is red 00:45:33.620 --> 00:45:36.300 align:middle line:84% is exactly the event that at least two hats are red. 00:45:36.300 --> 00:45:38.740 align:middle line:84% You need one more red hat for people 00:45:38.740 --> 00:45:41.060 align:middle line:84% to know that at least one hat is red. 00:45:41.060 --> 00:45:46.540 align:middle line:84% And now let's ask, what is K squared of E greater than 00:45:46.540 --> 00:45:48.540 align:middle line:90% or equal to 1? 00:45:48.540 --> 00:45:52.620 align:middle line:84% This is the event that everyone knows that everyone knows 00:45:52.620 --> 00:45:54.460 align:middle line:90% that at least one hat is red. 00:45:54.460 --> 00:45:59.730 align:middle line:84% Well, from what we said over here, this is K of E 00:45:59.730 --> 00:46:02.530 align:middle line:90% greater than or equal to 2. 00:46:02.530 --> 00:46:06.650 align:middle line:84% This means everyone knows that at least two hats are red. 00:46:06.650 --> 00:46:07.390 align:middle line:90% And any guesses? 00:46:07.390 --> 00:46:09.410 align:middle line:84% Can we just reason through in which 00:46:09.410 --> 00:46:12.050 align:middle line:84% states does everyone know that at least two of the three hats 00:46:12.050 --> 00:46:12.710 align:middle line:90% are red? 00:46:12.710 --> 00:46:17.050 align:middle line:90% 00:46:17.050 --> 00:46:17.550 align:middle line:90% Yeah? 00:46:17.550 --> 00:46:18.690 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:46:18.690 --> 00:46:20.970 align:middle line:84% IAN BALL: All three hats are red. 00:46:20.970 --> 00:46:24.550 align:middle line:90% So this is going to be E3. 00:46:24.550 --> 00:46:27.410 align:middle line:90% 00:46:27.410 --> 00:46:33.130 align:middle line:84% Because if anyone's hat was white, 00:46:33.130 --> 00:46:36.690 align:middle line:84% then a person seeing that white hat 00:46:36.690 --> 00:46:38.345 align:middle line:84% would only be seeing one red hat, 00:46:38.345 --> 00:46:39.970 align:middle line:84% and they wouldn't be certain that there 00:46:39.970 --> 00:46:42.650 align:middle line:90% are at least two red hats. 00:46:42.650 --> 00:46:45.250 align:middle line:90% So now, final step. 00:46:45.250 --> 00:46:49.730 align:middle line:84% What is K3 of E greater than or equal to 1? 00:46:49.730 --> 00:46:52.790 align:middle line:90% This is K of E3. 00:46:52.790 --> 00:47:02.640 align:middle line:90% 00:47:02.640 --> 00:47:06.800 align:middle line:84% So in which states does everyone know 00:47:06.800 --> 00:47:08.900 align:middle line:90% that there are three red hats? 00:47:08.900 --> 00:47:09.400 align:middle line:90% Yeah? 00:47:09.400 --> 00:47:12.783 align:middle line:84% AUDIENCE: [INAUDIBLE] red hats [INAUDIBLE]. 00:47:12.783 --> 00:47:13.700 align:middle line:90% IAN BALL: Let's think. 00:47:13.700 --> 00:47:14.997 align:middle line:90% So do we know-- 00:47:14.997 --> 00:47:16.580 align:middle line:84% let's say everyone has three red hats. 00:47:16.580 --> 00:47:18.820 align:middle line:84% Do we all know that everyone has three red hats? 00:47:18.820 --> 00:47:19.320 align:middle line:90% No. 00:47:19.320 --> 00:47:21.800 align:middle line:84% So actually, this is the empty set. 00:47:21.800 --> 00:47:24.040 align:middle line:84% Because even if everyone has three red hats, 00:47:24.040 --> 00:47:25.300 align:middle line:90% we don't all know that. 00:47:25.300 --> 00:47:28.020 align:middle line:84% Because I only know the hats of the other two people. 00:47:28.020 --> 00:47:32.680 align:middle line:90% 00:47:32.680 --> 00:47:43.440 align:middle line:84% So here, we can formalize our claim in state omega equals 111. 00:47:43.440 --> 00:47:46.280 align:middle line:84% Well, the event at least one person's hat 00:47:46.280 --> 00:47:49.640 align:middle line:90% is red is mutual knowledge. 00:47:49.640 --> 00:47:51.720 align:middle line:84% It's second-order mutual knowledge, 00:47:51.720 --> 00:47:55.310 align:middle line:84% but it's not third-order mutual knowledge. 00:47:55.310 --> 00:48:00.990 align:middle line:84% So in this state, the event E1 is maybe 00:48:00.990 --> 00:48:04.830 align:middle line:84% I'll say second, but not third-order mutual knowledge. 00:48:04.830 --> 00:48:17.830 align:middle line:90% 00:48:17.830 --> 00:48:20.630 align:middle line:84% And I think the reason this puzzle is so confusing to people 00:48:20.630 --> 00:48:22.390 align:middle line:84% is that we have pretty good intuition 00:48:22.390 --> 00:48:23.972 align:middle line:90% about mutual knowledge. 00:48:23.972 --> 00:48:25.430 align:middle line:84% We have pretty weak intuition about 00:48:25.430 --> 00:48:26.650 align:middle line:90% second-order mutual knowledge. 00:48:26.650 --> 00:48:29.010 align:middle line:84% And once we get to third- or fourth-order mutual knowledge, 00:48:29.010 --> 00:48:30.885 align:middle line:84% it's really hard for us to reason through it. 00:48:30.885 --> 00:48:33.190 align:middle line:84% It feels a lot like common knowledge. 00:48:33.190 --> 00:48:35.270 align:middle line:84% So in this game, because this event 00:48:35.270 --> 00:48:37.010 align:middle line:84% is second-order mutual knowledge, 00:48:37.010 --> 00:48:39.170 align:middle line:84% it feels a lot like common knowledge. 00:48:39.170 --> 00:48:41.950 align:middle line:84% But in fact, it's not actually third-order mutual knowledge, 00:48:41.950 --> 00:48:44.503 align:middle line:84% and therefore, certainly is not common knowledge. 00:48:44.503 --> 00:48:47.170 align:middle line:84% And that's why this announcement can make such a big difference. 00:48:47.170 --> 00:48:49.930 align:middle line:84% Let me pause to see if there's any questions on this, 00:48:49.930 --> 00:48:53.710 align:middle line:84% and then we'll see if we can actually resolve the riddle. 00:48:53.710 --> 00:48:54.830 align:middle line:90% Questions? 00:48:54.830 --> 00:48:56.110 align:middle line:90% Yeah? 00:48:56.110 --> 00:48:58.090 align:middle line:84% AUDIENCE: Like, I guess if we were to continue, 00:48:58.090 --> 00:49:00.790 align:middle line:84% does that mean like the other students would also start to 00:49:00.790 --> 00:49:03.150 align:middle line:90% [INAUDIBLE]? 00:49:03.150 --> 00:49:05.410 align:middle line:84% IAN BALL: You're saying if we ask more questions here? 00:49:05.410 --> 00:49:05.910 align:middle line:90% AUDIENCE: Yeah. 00:49:05.910 --> 00:49:06.070 align:middle line:90% [INAUDIBLE] 00:49:06.070 --> 00:49:07.370 align:middle line:84% IAN BALL: If we ask a fourth question. 00:49:07.370 --> 00:49:07.910 align:middle line:84% AUDIENCE: If we just keep going around. 00:49:07.910 --> 00:49:09.450 align:middle line:84% IAN BALL: Yeah, it turns out it would actually cycle. 00:49:09.450 --> 00:49:10.890 align:middle line:84% Then it gets even more confusing, 00:49:10.890 --> 00:49:13.030 align:middle line:84% and it's going to depend on what the state is. 00:49:13.030 --> 00:49:14.530 align:middle line:84% But you can actually have situations 00:49:14.530 --> 00:49:19.767 align:middle line:84% where it would cycle where you might think, 00:49:19.767 --> 00:49:21.850 align:middle line:84% oh, we're just learning more and more information. 00:49:21.850 --> 00:49:24.225 align:middle line:84% But yeah you can actually have this kind of weird cycling 00:49:24.225 --> 00:49:26.830 align:middle line:84% pattern where the subsequent answers now change, 00:49:26.830 --> 00:49:28.550 align:middle line:90% and we keep going on forever. 00:49:28.550 --> 00:49:29.590 align:middle line:90% Yeah. 00:49:29.590 --> 00:49:31.150 align:middle line:90% Any other questions? 00:49:31.150 --> 00:49:31.710 align:middle line:90% Yes? 00:49:31.710 --> 00:49:32.668 align:middle line:90% AUDIENCE: Do you mean-- 00:49:32.668 --> 00:49:35.710 align:middle line:84% [INAUDIBLE] answers [INAUDIBLE] that should be-- yes. 00:49:35.710 --> 00:49:38.310 align:middle line:84% [INAUDIBLE] students actually don't know their hat color, 00:49:38.310 --> 00:49:41.470 align:middle line:90% or a student [INAUDIBLE]. 00:49:41.470 --> 00:49:45.130 align:middle line:90% IAN BALL: So you're right. 00:49:45.130 --> 00:49:50.460 align:middle line:84% So if I know my hat color, I can never subsequently forget 00:49:50.460 --> 00:49:51.220 align:middle line:90% my hat color. 00:49:51.220 --> 00:49:51.720 align:middle line:90% That's true. 00:49:51.720 --> 00:49:53.400 align:middle line:84% So what am I thinking in this example? 00:49:53.400 --> 00:49:54.900 align:middle line:84% Let me-- I should double-check this. 00:49:54.900 --> 00:50:02.960 align:middle line:90% 00:50:02.960 --> 00:50:03.460 align:middle line:90% Yeah. 00:50:03.460 --> 00:50:05.960 align:middle line:84% So more information can ever cause you to not know something 00:50:05.960 --> 00:50:07.380 align:middle line:90% that you previously knew. 00:50:07.380 --> 00:50:09.880 align:middle line:84% But there is a similar riddle where these answers can cycle. 00:50:09.880 --> 00:50:12.445 align:middle line:84% So they must be asking about do you know something about-- 00:50:12.445 --> 00:50:14.820 align:middle line:84% it must be something about something other people knowing 00:50:14.820 --> 00:50:15.440 align:middle line:90% something about you. 00:50:15.440 --> 00:50:16.200 align:middle line:90% Let me follow up with that. 00:50:16.200 --> 00:50:17.260 align:middle line:90% Yeah, let me double-check that. 00:50:17.260 --> 00:50:19.593 align:middle line:84% You're right, that more information-- adding information 00:50:19.593 --> 00:50:26.540 align:middle line:84% can never cause you to forget something you once knew. 00:50:26.540 --> 00:50:29.300 align:middle line:84% So now let's see if we can actually resolve the puzzle 00:50:29.300 --> 00:50:31.480 align:middle line:90% and see what's happening here. 00:50:31.480 --> 00:50:34.420 align:middle line:90% 00:50:34.420 --> 00:50:39.180 align:middle line:84% So I think it's best to go to the picture 00:50:39.180 --> 00:50:40.800 align:middle line:84% here and go through the questions. 00:50:40.800 --> 00:50:47.180 align:middle line:90% 00:50:47.180 --> 00:50:48.730 align:middle line:90% So here we are. 00:50:48.730 --> 00:50:52.970 align:middle line:84% The teacher, I guess, first makes an announcement. 00:50:52.970 --> 00:50:57.590 align:middle line:90% I'll go here-- so these stages. 00:50:57.590 --> 00:51:02.053 align:middle line:90% 00:51:02.053 --> 00:51:03.470 align:middle line:84% The teacher makes an announcement. 00:51:03.470 --> 00:51:08.350 align:middle line:84% Basically, they say, the event is E greater than or equal to 1. 00:51:08.350 --> 00:51:13.010 align:middle line:84% The teacher says at least one person's hat is red. 00:51:13.010 --> 00:51:15.930 align:middle line:84% But crucially, this announcement makes this event 00:51:15.930 --> 00:51:17.370 align:middle line:90% common knowledge. 00:51:17.370 --> 00:51:25.930 align:middle line:90% 00:51:25.930 --> 00:51:27.615 align:middle line:84% And what it means is now everyone 00:51:27.615 --> 00:51:29.490 align:middle line:84% knows that everyone knows that everyone knows 00:51:29.490 --> 00:51:31.610 align:middle line:90% that we're not in this state. 00:51:31.610 --> 00:51:33.670 align:middle line:84% So let me put an x over this state. 00:51:33.670 --> 00:51:38.330 align:middle line:90% 00:51:38.330 --> 00:51:43.210 align:middle line:84% So now we're going to go to player one, 00:51:43.210 --> 00:51:48.467 align:middle line:84% and we're going to ask player one, do you know your hat color? 00:51:48.467 --> 00:51:50.300 align:middle line:84% That is, do you know that your hat is white? 00:51:50.300 --> 00:51:51.800 align:middle line:84% Or do you know that you had his red? 00:51:51.800 --> 00:51:52.960 align:middle line:90% Or are you not sure? 00:51:52.960 --> 00:51:56.840 align:middle line:84% And let's go through each of player one's information sets. 00:51:56.840 --> 00:52:00.540 align:middle line:84% So if the state of the world is in this information set, 00:52:00.540 --> 00:52:02.920 align:middle line:84% does player one know their hat color? 00:52:02.920 --> 00:52:03.468 align:middle line:90% No. 00:52:03.468 --> 00:52:05.760 align:middle line:84% Because player one thinks either they're in this state, 00:52:05.760 --> 00:52:08.682 align:middle line:84% or they're in this state, and they don't know their hat color. 00:52:08.682 --> 00:52:10.140 align:middle line:84% If they're in this information set, 00:52:10.140 --> 00:52:12.520 align:middle line:84% player one also doesn't know their hat color, and same 00:52:12.520 --> 00:52:16.080 align:middle line:90% with this information set. 00:52:16.080 --> 00:52:20.600 align:middle line:84% But if player one is in this information set, well, now, 00:52:20.600 --> 00:52:24.277 align:middle line:84% because we've announced that the state is not 000. 00:52:24.277 --> 00:52:26.360 align:middle line:84% Basically, what's happened is this information set 00:52:26.360 --> 00:52:27.380 align:middle line:90% has split into two. 00:52:27.380 --> 00:52:30.000 align:middle line:90% 00:52:30.000 --> 00:52:31.300 align:middle line:90% The partition has changed. 00:52:31.300 --> 00:52:33.500 align:middle line:84% And in fact, if we're really formal about it, 00:52:33.500 --> 00:52:39.800 align:middle line:84% that's true for this one and also for this one. 00:52:39.800 --> 00:52:41.520 align:middle line:84% It gets messy to keep writing this. 00:52:41.520 --> 00:52:46.920 align:middle line:84% But the announcement that the state is not 000 splits up any 00:52:46.920 --> 00:52:49.880 align:middle line:84% of the information sets, any of the cells that includes this 00:52:49.880 --> 00:52:51.300 align:middle line:90% state 000. 00:52:51.300 --> 00:52:55.880 align:middle line:84% So it splits up this one, this one, and this one. 00:52:55.880 --> 00:52:58.920 align:middle line:84% So now when we ask player one, do you 00:52:58.920 --> 00:53:01.320 align:middle line:84% know your hat color, and player one says 00:53:01.320 --> 00:53:06.640 align:middle line:84% no, what do all the other players now 00:53:06.640 --> 00:53:07.620 align:middle line:90% learn about the state? 00:53:07.620 --> 00:53:11.580 align:middle line:84% What is the one state that they can rule out? 00:53:11.580 --> 00:53:12.080 align:middle line:90% Yeah? 00:53:12.080 --> 00:53:13.277 align:middle line:90% AUDIENCE: The bottom one. 00:53:13.277 --> 00:53:14.360 align:middle line:90% IAN BALL: This one, right? 00:53:14.360 --> 00:53:16.920 align:middle line:84% Because if the state were here, player one 00:53:16.920 --> 00:53:22.520 align:middle line:84% would know for certain that his hat was red. 00:53:22.520 --> 00:53:26.640 align:middle line:84% So now the players-- we break this up. 00:53:26.640 --> 00:53:28.345 align:middle line:84% And I want to be really clear here, 00:53:28.345 --> 00:53:29.720 align:middle line:84% what these questions are doing is 00:53:29.720 --> 00:53:31.560 align:middle line:84% they're changing the information partitions 00:53:31.560 --> 00:53:34.200 align:middle line:90% that the players have. 00:53:34.200 --> 00:53:37.480 align:middle line:84% Again, this one is going to split up, 00:53:37.480 --> 00:53:39.500 align:middle line:84% and this one is going to split up. 00:53:39.500 --> 00:53:42.440 align:middle line:90% 00:53:42.440 --> 00:53:47.630 align:middle line:84% Because now all the players learn that we're not here. 00:53:47.630 --> 00:53:49.950 align:middle line:90% Now let's go to player two. 00:53:49.950 --> 00:53:53.550 align:middle line:84% So next, player two is asked, do your hat color? 00:53:53.550 --> 00:53:55.670 align:middle line:90% Well, let's look. 00:53:55.670 --> 00:53:59.110 align:middle line:84% If the state is in this information 00:53:59.110 --> 00:54:01.550 align:middle line:84% set or this cell for player two, then player two 00:54:01.550 --> 00:54:03.710 align:middle line:90% doesn't know their hat color. 00:54:03.710 --> 00:54:06.070 align:middle line:84% If it's in this cell, then player two also 00:54:06.070 --> 00:54:07.870 align:middle line:90% doesn't know their hat color. 00:54:07.870 --> 00:54:10.530 align:middle line:84% But if it's in either of these four points, 00:54:10.530 --> 00:54:12.090 align:middle line:84% the player does know their hat color. 00:54:12.090 --> 00:54:14.750 align:middle line:84% Because now these four points are 00:54:14.750 --> 00:54:17.990 align:middle line:84% in distinct singleton information sets for player two. 00:54:17.990 --> 00:54:21.190 align:middle line:84% Because previously, these two were in the same information set 00:54:21.190 --> 00:54:24.710 align:middle line:84% in the same cell, but that was split by the first announcement. 00:54:24.710 --> 00:54:27.090 align:middle line:84% And then these two were in the same cell, 00:54:27.090 --> 00:54:30.910 align:middle line:84% but that was split by player one's answer. 00:54:30.910 --> 00:54:36.470 align:middle line:84% So when player two says no, I don't know my hat color, 00:54:36.470 --> 00:54:40.410 align:middle line:84% that tells us that the state is not here or here. 00:54:40.410 --> 00:54:44.460 align:middle line:90% 00:54:44.460 --> 00:54:48.500 align:middle line:84% So now everyone knows that the state 00:54:48.500 --> 00:54:52.600 align:middle line:84% must be up in this higher horizontal plane. 00:54:52.600 --> 00:54:56.620 align:middle line:90% 00:54:56.620 --> 00:55:01.500 align:middle line:84% Player one knows we're either in this information 00:55:01.500 --> 00:55:03.500 align:middle line:84% set or this information set, and they actually 00:55:03.500 --> 00:55:04.940 align:middle line:90% know which one they're in. 00:55:04.940 --> 00:55:06.920 align:middle line:90% Player two sees this. 00:55:06.920 --> 00:55:10.020 align:middle line:84% But player one now, player three actually 00:55:10.020 --> 00:55:12.040 align:middle line:90% now perfectly knows the state. 00:55:12.040 --> 00:55:12.540 align:middle line:90% Why? 00:55:12.540 --> 00:55:15.780 align:middle line:84% Because if the state is here, then player three knows it. 00:55:15.780 --> 00:55:17.847 align:middle line:84% If the state is here, player three knows it. 00:55:17.847 --> 00:55:19.680 align:middle line:84% If the state is here, player three knows it. 00:55:19.680 --> 00:55:21.680 align:middle line:84% And if the state is here, player three knows it. 00:55:21.680 --> 00:55:24.460 align:middle line:84% So now player three, when the state is actually 111, 00:55:24.460 --> 00:55:27.840 align:middle line:84% is able to conclude that the state must be here. 00:55:27.840 --> 00:55:29.560 align:middle line:84% And therefore, the answer is yes. 00:55:29.560 --> 00:55:34.540 align:middle line:90% 00:55:34.540 --> 00:55:36.240 align:middle line:84% I think this can be a little confusing. 00:55:36.240 --> 00:55:36.840 align:middle line:90% So yeah? 00:55:36.840 --> 00:55:38.900 align:middle line:84% AUDIENCE: Please repeat what you said about 00:55:38.900 --> 00:55:41.770 align:middle line:84% after player's two answer, how that allows 00:55:41.770 --> 00:55:44.490 align:middle line:90% player one to know a state? 00:55:44.490 --> 00:55:45.610 align:middle line:90% IAN BALL: Yes. 00:55:45.610 --> 00:55:49.510 align:middle line:90% So before player two answers-- 00:55:49.510 --> 00:55:51.350 align:middle line:84% I think maybe another way of looking at it. 00:55:51.350 --> 00:55:54.828 align:middle line:90% Let's focus on player three. 00:55:54.828 --> 00:55:56.370 align:middle line:84% So maybe another way of looking at it 00:55:56.370 --> 00:55:58.210 align:middle line:84% without breaking information sets 00:55:58.210 --> 00:56:03.090 align:middle line:84% is player three's information set is this one. 00:56:03.090 --> 00:56:05.690 align:middle line:84% Because the state is 111, we're saying 00:56:05.690 --> 00:56:09.250 align:middle line:84% suppose the realized state is that everyone's hat is red. 00:56:09.250 --> 00:56:14.490 align:middle line:84% So that means that player three knows either my hat is red 00:56:14.490 --> 00:56:19.410 align:middle line:84% and the other two player's hats are red, or my hat is white, 00:56:19.410 --> 00:56:21.970 align:middle line:84% and the other two player's hats are red. 00:56:21.970 --> 00:56:22.830 align:middle line:90% Put this up here. 00:56:22.830 --> 00:56:25.570 align:middle line:90% 00:56:25.570 --> 00:56:27.122 align:middle line:84% These are the only two possibilities 00:56:27.122 --> 00:56:29.330 align:middle line:84% that player three entertains because they can observe 00:56:29.330 --> 00:56:31.970 align:middle line:90% the other two player's hats. 00:56:31.970 --> 00:56:36.410 align:middle line:84% Now player three now says, well, wait a second. 00:56:36.410 --> 00:56:37.990 align:middle line:90% Suppose the state were this. 00:56:37.990 --> 00:56:40.890 align:middle line:90% 00:56:40.890 --> 00:56:45.790 align:middle line:84% If the state were 110, and that means that my hat is white, 00:56:45.790 --> 00:56:49.050 align:middle line:84% but the other two player's hats are red, 00:56:49.050 --> 00:56:52.930 align:middle line:84% then let me put myself in the shoes of player two. 00:56:52.930 --> 00:56:55.410 align:middle line:84% So if this were the state, then player 00:56:55.410 --> 00:56:59.890 align:middle line:84% two's information set would be this. 00:56:59.890 --> 00:57:05.330 align:middle line:84% But player two has learned from player one's answer 00:57:05.330 --> 00:57:07.690 align:middle line:90% that this can't be the state. 00:57:07.690 --> 00:57:09.970 align:middle line:84% So if this were the state, then player two 00:57:09.970 --> 00:57:12.910 align:middle line:84% would know for certain that this were the state. 00:57:12.910 --> 00:57:15.890 align:middle line:84% And therefore, player two would have said yes to this question. 00:57:15.890 --> 00:57:18.698 align:middle line:84% So it must be the case because player two says no, 00:57:18.698 --> 00:57:19.990 align:middle line:90% that we can't be in this state. 00:57:19.990 --> 00:57:21.670 align:middle line:84% And therefore, we must be in this state. 00:57:21.670 --> 00:57:22.750 align:middle line:84% And knowing that we're in this state, 00:57:22.750 --> 00:57:23.990 align:middle line:90% I know that my hat is red. 00:57:23.990 --> 00:57:27.290 align:middle line:90% 00:57:27.290 --> 00:57:27.790 align:middle line:90% Great. 00:57:27.790 --> 00:57:30.050 align:middle line:84% So I think I want to end a bit early since it's the last class. 00:57:30.050 --> 00:57:30.630 align:middle line:90% I'll stop there. 00:57:30.630 --> 00:57:31.130 align:middle line:90% Thanks. 00:57:31.130 --> 00:57:33.980 align:middle line:90% [APPLAUSE] 00:57:33.980 --> 00:57:39.000 align:middle line:90%