WEBVTT 00:00:00.000 --> 00:00:01.970 align:middle line:90% [SQUEAKING] 00:00:01.970 --> 00:00:03.440 align:middle line:90% [RUSTLING] 00:00:03.440 --> 00:00:04.910 align:middle line:90% [CLICKING] 00:00:04.910 --> 00:00:09.740 align:middle line:90% 00:00:09.740 --> 00:00:12.080 align:middle line:84% IAN BALL: So I wanted to start by playing the beauty 00:00:12.080 --> 00:00:13.160 align:middle line:90% contest again. 00:00:13.160 --> 00:00:14.623 align:middle line:90% Last week-- for two reasons. 00:00:14.623 --> 00:00:16.040 align:middle line:84% Last week, I kind of messed it up, 00:00:16.040 --> 00:00:18.210 align:middle line:84% and I described a mixture of the game. 00:00:18.210 --> 00:00:19.960 align:middle line:84% We actually played a more complicated game 00:00:19.960 --> 00:00:21.560 align:middle line:84% that I play in my other class, so I 00:00:21.560 --> 00:00:23.220 align:middle line:84% want to set the record straight here. 00:00:23.220 --> 00:00:26.200 align:middle line:84% And I also want to see how the results maybe have changed 00:00:26.200 --> 00:00:27.940 align:middle line:90% since the first time we played. 00:00:27.940 --> 00:00:30.640 align:middle line:84% So if you haven't made an account yet, you can log in here 00:00:30.640 --> 00:00:34.480 align:middle line:84% and put in the password or put in the class code, 00:00:34.480 --> 00:00:36.420 align:middle line:90% try to join this game now. 00:00:36.420 --> 00:00:39.480 align:middle line:84% And let me say a little bit about how it works. 00:00:39.480 --> 00:00:42.620 align:middle line:84% So remember, this is the beauty contest. 00:00:42.620 --> 00:00:46.680 align:middle line:84% You're going to guess a number between 1 and 100. 00:00:46.680 --> 00:00:49.200 align:middle line:84% The computer is going to collect everyone's guesses 00:00:49.200 --> 00:00:51.520 align:middle line:84% and compute the average of those guesses. 00:00:51.520 --> 00:00:54.800 align:middle line:84% And the winner is going to be the person whose guess is 00:00:54.800 --> 00:01:01.030 align:middle line:84% closest to 2/3 of the average of the class's guesses. 00:01:01.030 --> 00:01:07.150 align:middle line:84% So let's start, play this, and see what happens. 00:01:07.150 --> 00:01:08.690 align:middle line:90% So did we learn from last week? 00:01:08.690 --> 00:01:09.210 align:middle line:90% Let's see. 00:01:09.210 --> 00:01:12.150 align:middle line:84% So in this case, the average choice was 18. 00:01:12.150 --> 00:01:13.570 align:middle line:90% And the winning choice was 12. 00:01:13.570 --> 00:01:15.612 align:middle line:84% So I think last week, what was it the first time? 00:01:15.612 --> 00:01:18.110 align:middle line:90% Maybe 25 was the winning bid. 00:01:18.110 --> 00:01:21.950 align:middle line:84% So what you see is, again, we're shifting a little bit. 00:01:21.950 --> 00:01:24.190 align:middle line:84% I was actually expecting it might shift a bit more. 00:01:24.190 --> 00:01:26.830 align:middle line:90% No one went to close to 0. 00:01:26.830 --> 00:01:29.070 align:middle line:84% If we really were ultra, ultra rational, 00:01:29.070 --> 00:01:31.575 align:middle line:84% there's an argument that everyone should guess 0. 00:01:31.575 --> 00:01:32.950 align:middle line:84% And we'll see the formal argument 00:01:32.950 --> 00:01:34.290 align:middle line:90% for that later in the course. 00:01:34.290 --> 00:01:36.690 align:middle line:84% But if you're not sure if everyone else is rational, 00:01:36.690 --> 00:01:39.970 align:middle line:84% then maybe this is a pretty reasonable performance. 00:01:39.970 --> 00:01:43.170 align:middle line:84% Again, we had a few people going a little crazy and guessing 100. 00:01:43.170 --> 00:01:46.010 align:middle line:84% I think that's probably not a good idea. 00:01:46.010 --> 00:01:48.090 align:middle line:84% So let's put this down and get started. 00:01:48.090 --> 00:02:04.900 align:middle line:90% 00:02:04.900 --> 00:02:07.420 align:middle line:84% So let me start with just a very brief review of what 00:02:07.420 --> 00:02:11.320 align:middle line:84% we did last class, or I'll say yesterday, 00:02:11.320 --> 00:02:12.440 align:middle line:90% but I guess last class. 00:02:12.440 --> 00:02:15.540 align:middle line:90% 00:02:15.540 --> 00:02:18.753 align:middle line:84% So last class, we talked about individual decision making. 00:02:18.753 --> 00:02:20.420 align:middle line:84% This class, we're going to start talking 00:02:20.420 --> 00:02:22.340 align:middle line:84% about interactive decision making, where 00:02:22.340 --> 00:02:25.080 align:middle line:84% we have multiple players making decisions. 00:02:25.080 --> 00:02:27.140 align:middle line:84% And the best decision for each of them 00:02:27.140 --> 00:02:30.900 align:middle line:84% depends upon the decisions that other players are making. 00:02:30.900 --> 00:02:34.820 align:middle line:84% So just some notation to remember from last class-- 00:02:34.820 --> 00:02:37.000 align:middle line:84% it's really just two pieces you need to remember. 00:02:37.000 --> 00:02:51.700 align:middle line:90% 00:02:51.700 --> 00:02:54.220 align:middle line:84% From last class, we said that there's some set 00:02:54.220 --> 00:02:57.850 align:middle line:90% z of consequences or outcomes. 00:02:57.850 --> 00:03:01.370 align:middle line:84% And we defined a Von Neumann-Morgenstern utility 00:03:01.370 --> 00:03:05.010 align:middle line:84% that assigns some real number to every single consequence 00:03:05.010 --> 00:03:05.790 align:middle line:90% or outcome. 00:03:05.790 --> 00:03:08.410 align:middle line:84% I'll use those two terms interchangeably. 00:03:08.410 --> 00:03:12.290 align:middle line:84% And then we defined capital U to be the expected utility. 00:03:12.290 --> 00:03:19.970 align:middle line:84% So what this does is it assigns to every lottery, every lottery 00:03:19.970 --> 00:03:23.210 align:middle line:84% over consequences, the expected utility 00:03:23.210 --> 00:03:25.670 align:middle line:84% that the agent gets from that lottery. 00:03:25.670 --> 00:03:27.870 align:middle line:84% And remember a lottery would be something like this. 00:03:27.870 --> 00:03:31.650 align:middle line:90% 00:03:31.650 --> 00:03:36.250 align:middle line:84% So if there are m consequences, then a lottery 00:03:36.250 --> 00:03:41.530 align:middle line:84% is a vector p that specifies m different probabilities that 00:03:41.530 --> 00:03:42.650 align:middle line:90% sum to 1. 00:03:42.650 --> 00:03:45.570 align:middle line:84% And the interpretation is that p1 is the probability, 00:03:45.570 --> 00:03:49.130 align:middle line:84% under this lottery, that outcome that the first consequence is 00:03:49.130 --> 00:03:51.770 align:middle line:84% realized, all the way up to pm, which 00:03:51.770 --> 00:03:55.650 align:middle line:84% is the probability that the mth consequence is realized. 00:03:55.650 --> 00:03:57.930 align:middle line:84% And I realized last week, I think 00:03:57.930 --> 00:04:00.290 align:middle line:84% I used p in two different ways, both as a probability 00:04:00.290 --> 00:04:01.270 align:middle line:90% and as a vector. 00:04:01.270 --> 00:04:02.850 align:middle line:90% So here this is a vector. 00:04:02.850 --> 00:04:04.230 align:middle line:90% Maybe I'll put a hat on it here. 00:04:04.230 --> 00:04:06.230 align:middle line:84% But I may not use hats throughout the course. 00:04:06.230 --> 00:04:08.990 align:middle line:84% But keep in mind that p will usually be a vector. 00:04:08.990 --> 00:04:11.630 align:middle line:84% But these little p's are numbers. 00:04:11.630 --> 00:04:15.630 align:middle line:84% These are probabilities inside the vector. 00:04:15.630 --> 00:04:25.410 align:middle line:84% So now today, the plan is to move to interactive decision 00:04:25.410 --> 00:04:25.990 align:middle line:90% making. 00:04:25.990 --> 00:04:31.675 align:middle line:90% 00:04:31.675 --> 00:04:34.050 align:middle line:84% And let me just give an outline of the three topics we're 00:04:34.050 --> 00:04:35.210 align:middle line:90% going to cover. 00:04:35.210 --> 00:04:37.290 align:middle line:84% The first is what we're going to define what 00:04:37.290 --> 00:04:39.070 align:middle line:90% we call an extensive form game. 00:04:39.070 --> 00:04:44.570 align:middle line:90% 00:04:44.570 --> 00:04:46.590 align:middle line:84% This is one way of representing a game. 00:04:46.590 --> 00:04:47.710 align:middle line:90% Well, what is a game? 00:04:47.710 --> 00:04:51.410 align:middle line:84% Well, think of a game as a multi-agent decision problem. 00:04:51.410 --> 00:04:53.880 align:middle line:84% Yesterday, we talked about the decision problem 00:04:53.880 --> 00:04:55.600 align:middle line:90% faced by an individual. 00:04:55.600 --> 00:04:58.280 align:middle line:84% Today, when we think about multiple individuals making 00:04:58.280 --> 00:05:02.680 align:middle line:84% choices that interact, we're going to call that a game. 00:05:02.680 --> 00:05:04.480 align:middle line:84% An extensive form game is going to be 00:05:04.480 --> 00:05:07.360 align:middle line:84% a very detailed representation of the game that 00:05:07.360 --> 00:05:09.920 align:middle line:84% specifies all the details that might possibly 00:05:09.920 --> 00:05:12.760 align:middle line:90% matter about the game. 00:05:12.760 --> 00:05:15.840 align:middle line:84% Then later, we're going to talk about a strategic form 00:05:15.840 --> 00:05:24.560 align:middle line:84% game, which you can think of as a summary of the extensive form 00:05:24.560 --> 00:05:25.160 align:middle line:90% game. 00:05:25.160 --> 00:05:28.000 align:middle line:84% It isolates or identifies the essential elements 00:05:28.000 --> 00:05:30.560 align:middle line:84% of the game for strategic analysis, 00:05:30.560 --> 00:05:34.680 align:middle line:84% and it throws out some of the details that are not explicitly 00:05:34.680 --> 00:05:37.680 align:middle line:84% relevant when we're analyzing the game. 00:05:37.680 --> 00:05:41.200 align:middle line:84% And the key way that we jump from an extensive form game 00:05:41.200 --> 00:05:45.080 align:middle line:84% to a strategic form game is the notion of a strategy. 00:05:45.080 --> 00:05:47.920 align:middle line:84% And I think this is the single most important definition 00:05:47.920 --> 00:05:49.000 align:middle line:90% in the course. 00:05:49.000 --> 00:05:51.280 align:middle line:84% So after we talk about extensive form games, 00:05:51.280 --> 00:05:53.630 align:middle line:84% we'll define what a strategy is, and then we'll 00:05:53.630 --> 00:05:55.270 align:middle line:90% move to strategic form games. 00:05:55.270 --> 00:05:57.590 align:middle line:84% I think most points that people lose 00:05:57.590 --> 00:05:59.510 align:middle line:84% throughout this course on exams are 00:05:59.510 --> 00:06:01.310 align:middle line:84% because they haven't thought carefully 00:06:01.310 --> 00:06:03.270 align:middle line:84% about what a strategy is and what the right definition 00:06:03.270 --> 00:06:04.050 align:middle line:90% of a strategy is. 00:06:04.050 --> 00:06:08.750 align:middle line:84% So I'd really encourage you to focus on this definition. 00:06:08.750 --> 00:06:15.870 align:middle line:84% So let's-- maybe I'll start a new board here, actually. 00:06:15.870 --> 00:06:19.150 align:middle line:84% So I want to start with two examples of extensive form 00:06:19.150 --> 00:06:19.930 align:middle line:90% games. 00:06:19.930 --> 00:06:21.717 align:middle line:84% And then motivated by those examples, 00:06:21.717 --> 00:06:23.050 align:middle line:90% I'll present the general theory. 00:06:23.050 --> 00:06:25.010 align:middle line:84% But I think if I do the general theory first, 00:06:25.010 --> 00:06:27.410 align:middle line:84% it's just a lot of notation, and it's not very concrete. 00:06:27.410 --> 00:06:31.030 align:middle line:84% So let's start with two really simple examples. 00:06:31.030 --> 00:06:33.010 align:middle line:90% The first is an investment game. 00:06:33.010 --> 00:06:38.430 align:middle line:90% 00:06:38.430 --> 00:06:41.230 align:middle line:84% And this will actually only have one strategic player. 00:06:41.230 --> 00:06:43.050 align:middle line:84% Games don't have to have multiple players. 00:06:43.050 --> 00:06:45.170 align:middle line:84% They're usually more interesting when there's multiple players. 00:06:45.170 --> 00:06:47.470 align:middle line:84% But it's allowed for there to just be one player. 00:06:47.470 --> 00:06:50.350 align:middle line:84% So let's think that there is a single player. 00:06:50.350 --> 00:06:52.340 align:middle line:90% We'll call them player 1. 00:06:52.340 --> 00:06:55.070 align:middle line:84% And we're going to represent this game using what's 00:06:55.070 --> 00:06:56.820 align:middle line:84% called a decision tree, which you may have 00:06:56.820 --> 00:06:58.700 align:middle line:90% seen in some other courses. 00:06:58.700 --> 00:07:01.220 align:middle line:84% So this one player is choosing whether to make 00:07:01.220 --> 00:07:04.260 align:middle line:84% a safe investment or a risky investment. 00:07:04.260 --> 00:07:09.080 align:middle line:84% So they have two actions, safe and risky. 00:07:09.080 --> 00:07:11.620 align:middle line:90% 00:07:11.620 --> 00:07:13.140 align:middle line:84% And the safe investment is always 00:07:13.140 --> 00:07:15.060 align:middle line:84% going to pay the same amount of money. 00:07:15.060 --> 00:07:17.500 align:middle line:84% But the risky investment may pay a lot of money, 00:07:17.500 --> 00:07:18.920 align:middle line:90% or it may pay a little money. 00:07:18.920 --> 00:07:20.082 align:middle line:90% They're not sure. 00:07:20.082 --> 00:07:21.540 align:middle line:84% So the way we're going to represent 00:07:21.540 --> 00:07:23.782 align:middle line:84% that is, under the risky investment, 00:07:23.782 --> 00:07:25.740 align:middle line:84% there's actually going to be a move by a player 00:07:25.740 --> 00:07:27.820 align:middle line:90% that we'll call Nature. 00:07:27.820 --> 00:07:30.180 align:middle line:84% Now, Nature is a special kind of player. 00:07:30.180 --> 00:07:33.660 align:middle line:84% They're not a strategic player, like everyone else. 00:07:33.660 --> 00:07:37.380 align:middle line:84% But we often find it helpful to think of them as a player when 00:07:37.380 --> 00:07:38.980 align:middle line:90% we represent this game. 00:07:38.980 --> 00:07:41.820 align:middle line:84% And here, we think of Nature as choosing 00:07:41.820 --> 00:07:44.220 align:middle line:84% what the payoff of this asset is going 00:07:44.220 --> 00:07:46.100 align:middle line:84% to be if they make the risky investment. 00:07:46.100 --> 00:07:52.210 align:middle line:84% And in this example, there's two possibilities. 00:07:52.210 --> 00:07:54.690 align:middle line:90% The payoff is only $3. 00:07:54.690 --> 00:07:58.450 align:middle line:84% The monetary payout is either $3 or $1. 00:07:58.450 --> 00:08:01.130 align:middle line:84% And I have to tell you the probabilities of these two 00:08:01.130 --> 00:08:01.710 align:middle line:90% outcomes. 00:08:01.710 --> 00:08:03.750 align:middle line:84% So whenever Nature chooses something, 00:08:03.750 --> 00:08:07.290 align:middle line:84% Nature chooses that from some distribution, from some lottery. 00:08:07.290 --> 00:08:10.510 align:middle line:84% And I'm going to indicate the probabilities like this, 00:08:10.510 --> 00:08:11.510 align:middle line:90% 1/2 and 1/2. 00:08:11.510 --> 00:08:14.032 align:middle line:90% 00:08:14.032 --> 00:08:15.990 align:middle line:84% Now up here, I don't really need to use Nature, 00:08:15.990 --> 00:08:18.390 align:middle line:84% but maybe I will, just to be really precise. 00:08:18.390 --> 00:08:22.370 align:middle line:84% So here Nature is always going to choose a payout of $2. 00:08:22.370 --> 00:08:25.470 align:middle line:84% So maybe if I really want to be precise, I'll put a 1 here. 00:08:25.470 --> 00:08:26.870 align:middle line:90% And this is a one, not a 7. 00:08:26.870 --> 00:08:30.850 align:middle line:84% Sometimes I get a little crazy with the hat. 00:08:30.850 --> 00:08:33.970 align:middle line:84% So this says, if I choose the safe investment, 00:08:33.970 --> 00:08:36.809 align:middle line:90% I'm certain to get $2. 00:08:36.809 --> 00:08:38.750 align:middle line:84% If I choose the risky investment, 00:08:38.750 --> 00:08:42.850 align:middle line:84% then there's a 1/2 chance that I get $3 and a 1/2 chance that I 00:08:42.850 --> 00:08:44.770 align:middle line:90% get $1. 00:08:44.770 --> 00:08:47.370 align:middle line:84% But I would argue that this is not a complete specification 00:08:47.370 --> 00:08:48.000 align:middle line:90% of the game. 00:08:48.000 --> 00:08:52.300 align:middle line:84% What's missing here, if I want to specify this problem? 00:08:52.300 --> 00:08:56.000 align:middle line:90% What have I not told you? 00:08:56.000 --> 00:08:56.703 align:middle line:90% Yeah? 00:08:56.703 --> 00:08:57.620 align:middle line:90% AUDIENCE: Preferences. 00:08:57.620 --> 00:08:58.828 align:middle line:90% IAN BALL: Preferences, right? 00:08:58.828 --> 00:09:00.400 align:middle line:84% And it's attempting to say, oh, well, 00:09:00.400 --> 00:09:03.300 align:middle line:84% all I want to do is maximize expected monetary amount. 00:09:03.300 --> 00:09:05.200 align:middle line:84% But as we talked about last class, 00:09:05.200 --> 00:09:06.580 align:middle line:90% some people are risk-averse. 00:09:06.580 --> 00:09:07.980 align:middle line:90% Some people are risk-loving. 00:09:07.980 --> 00:09:09.640 align:middle line:90% We have to capture preferences. 00:09:09.640 --> 00:09:11.760 align:middle line:84% So what we need to write down is what my utility 00:09:11.760 --> 00:09:13.840 align:middle line:90% is from each of these outcomes. 00:09:13.840 --> 00:09:17.600 align:middle line:84% So we're going to specify my utility from $2, 00:09:17.600 --> 00:09:22.120 align:middle line:84% my utility from $3, and my utility from $1. 00:09:22.120 --> 00:09:25.080 align:middle line:84% And there's a lot of choices we could make. 00:09:25.080 --> 00:09:35.760 align:middle line:84% But maybe we'll say, 1.5, 2, and 0.9, just as some example. 00:09:35.760 --> 00:09:39.360 align:middle line:84% So it's important to understand that the game specifies 00:09:39.360 --> 00:09:50.710 align:middle line:84% the consequences but also the payoffs from those consequences. 00:09:50.710 --> 00:09:52.590 align:middle line:84% And it can be particularly tempting 00:09:52.590 --> 00:09:54.830 align:middle line:84% when you have monetary payouts, so when 00:09:54.830 --> 00:09:56.950 align:middle line:84% the consequences are money, to get confused 00:09:56.950 --> 00:09:58.170 align:middle line:90% about that distinction. 00:09:58.170 --> 00:09:59.910 align:middle line:84% But there's still a crucial distinction 00:09:59.910 --> 00:10:03.350 align:middle line:84% between the consequence, which here is measured in dollars, 00:10:03.350 --> 00:10:06.590 align:middle line:84% and the utility the player gets from that consequence, which 00:10:06.590 --> 00:10:09.470 align:middle line:90% is measured in utils. 00:10:09.470 --> 00:10:15.270 align:middle line:84% So this here is a Von Neumann-Morgenstern utility, 00:10:15.270 --> 00:10:19.510 align:middle line:90% like we talked about last class. 00:10:19.510 --> 00:10:22.230 align:middle line:90% So that's a very simple game. 00:10:22.230 --> 00:10:25.350 align:middle line:84% There's one strategic player together with Nature. 00:10:25.350 --> 00:10:28.830 align:middle line:84% Let's now do a game that actually has two players. 00:10:28.830 --> 00:10:30.870 align:middle line:84% So let's really get into game theory. 00:10:30.870 --> 00:10:36.830 align:middle line:84% And I'm going to call this the BoS game. 00:10:36.830 --> 00:10:40.590 align:middle line:84% So historically, this game is called The Battle of the Sexes. 00:10:40.590 --> 00:10:43.770 align:middle line:84% I think today I'd like to try to avoid that terminology. 00:10:43.770 --> 00:10:45.060 align:middle line:90% So I'll just call it BoS. 00:10:45.060 --> 00:10:46.280 align:middle line:90% Think of it as Boston. 00:10:46.280 --> 00:10:48.340 align:middle line:84% But if you read old game theory books, 00:10:48.340 --> 00:10:51.580 align:middle line:84% they'll often call this the Battle of the Sexes game. 00:10:51.580 --> 00:10:54.500 align:middle line:84% So what's the idea of this game, or at least the version 00:10:54.500 --> 00:10:56.660 align:middle line:90% we're going to tell? 00:10:56.660 --> 00:10:58.660 align:middle line:84% You have two friends, and they're choosing 00:10:58.660 --> 00:11:00.220 align:middle line:90% between two activities. 00:11:00.220 --> 00:11:02.420 align:middle line:84% So let's say since it's Boston, let's say 00:11:02.420 --> 00:11:06.540 align:middle line:84% going to a Celtics game or going to a Red Sox game. 00:11:06.540 --> 00:11:10.340 align:middle line:84% And the idea is they have to choose which activity to do. 00:11:10.340 --> 00:11:12.300 align:middle line:90% And they're good friends. 00:11:12.300 --> 00:11:13.900 align:middle line:84% So if they go to different activities, 00:11:13.900 --> 00:11:15.000 align:middle line:90% they don't have any fun. 00:11:15.000 --> 00:11:17.640 align:middle line:84% If you go to the Celtics game, and I go to the Red Sox game, 00:11:17.640 --> 00:11:20.140 align:middle line:84% we're each all alone, and we're both sad. 00:11:20.140 --> 00:11:22.480 align:middle line:84% So we'd both like to go to the same activity. 00:11:22.480 --> 00:11:24.460 align:middle line:84% We'd like to be together at that activity. 00:11:24.460 --> 00:11:26.220 align:middle line:84% But one of us would rather that we 00:11:26.220 --> 00:11:28.180 align:middle line:84% be together at the Celtics game because they're 00:11:28.180 --> 00:11:29.740 align:middle line:90% more of a basketball fan. 00:11:29.740 --> 00:11:32.380 align:middle line:84% And the other person would rather that they be together 00:11:32.380 --> 00:11:33.780 align:middle line:90% at a Red Sox game. 00:11:33.780 --> 00:11:36.940 align:middle line:84% So why I think this game is so important, 00:11:36.940 --> 00:11:38.580 align:middle line:84% and why it's gotten a lot of attention, 00:11:38.580 --> 00:11:44.420 align:middle line:90% is it features both conflict-- 00:11:44.420 --> 00:11:46.600 align:middle line:90% maybe conflict and cooperation. 00:11:46.600 --> 00:11:50.380 align:middle line:90% 00:11:50.380 --> 00:11:52.580 align:middle line:84% It has a conflict of interest because there's 00:11:52.580 --> 00:11:55.443 align:middle line:84% one player who'd like that you both go to the Celtics game, 00:11:55.443 --> 00:11:56.860 align:middle line:84% and there's the other player who'd 00:11:56.860 --> 00:11:58.600 align:middle line:84% like you both to go to the Red Sox game. 00:11:58.600 --> 00:12:00.180 align:middle line:84% So you have some conflict of interest 00:12:00.180 --> 00:12:02.340 align:middle line:84% between where you go, but you also-- 00:12:02.340 --> 00:12:04.380 align:middle line:84% and maybe cooperation, or coordination is maybe 00:12:04.380 --> 00:12:05.000 align:middle line:90% a better term. 00:12:05.000 --> 00:12:06.120 align:middle line:90% Let me say coordination. 00:12:06.120 --> 00:12:09.700 align:middle line:90% 00:12:09.700 --> 00:12:11.860 align:middle line:84% But there's also a coordination aspect to it 00:12:11.860 --> 00:12:14.300 align:middle line:84% because you'd like to go to the same activity. 00:12:14.300 --> 00:12:16.380 align:middle line:84% You'd like to be at the same place. 00:12:16.380 --> 00:12:18.860 align:middle line:84% And this combination of conflict and coordination 00:12:18.860 --> 00:12:21.940 align:middle line:84% is present in a lot of situations far 00:12:21.940 --> 00:12:24.740 align:middle line:90% beyond this very simple game. 00:12:24.740 --> 00:12:26.300 align:middle line:90% So let's draw this out. 00:12:26.300 --> 00:12:28.540 align:middle line:84% We're going to say player 1 is first 00:12:28.540 --> 00:12:31.380 align:middle line:84% going to choose whether to go to the Celtics game or the Red Sox 00:12:31.380 --> 00:12:32.060 align:middle line:90% game. 00:12:32.060 --> 00:12:36.300 align:middle line:84% And I'll call this C and R. And let me just 00:12:36.300 --> 00:12:38.140 align:middle line:84% be clear, when I say Celtics game or Red Sox 00:12:38.140 --> 00:12:40.118 align:middle line:84% game, that's not the formal sense of a game 00:12:40.118 --> 00:12:41.160 align:middle line:90% we're talking about here. 00:12:41.160 --> 00:12:43.250 align:middle line:84% That's game as in an athletic game. 00:12:43.250 --> 00:12:46.970 align:middle line:84% So player 1 is choosing between C and R. 00:12:46.970 --> 00:12:58.530 align:middle line:84% And then player 2 is choosing also between C and R. 00:12:58.530 --> 00:13:02.363 align:middle line:84% Now, the outcomes are kind of already immediate from the way 00:13:02.363 --> 00:13:03.030 align:middle line:90% I've written it. 00:13:03.030 --> 00:13:05.050 align:middle line:90% What is the outcome here? 00:13:05.050 --> 00:13:09.450 align:middle line:84% The outcome here is that we watch the Celtics game together. 00:13:09.450 --> 00:13:13.330 align:middle line:84% And the outcome here is that player 1 watches the Red Sox 00:13:13.330 --> 00:13:16.610 align:middle line:84% game by yourself, and player 2 watches the Celtics game 00:13:16.610 --> 00:13:18.050 align:middle line:90% by yourself. 00:13:18.050 --> 00:13:21.090 align:middle line:84% But again, if we want to fully specify the game, 00:13:21.090 --> 00:13:23.210 align:middle line:90% we have to say what payoffs are. 00:13:23.210 --> 00:13:25.130 align:middle line:84% So we're going to write down payoffs here. 00:13:25.130 --> 00:13:28.810 align:middle line:84% And what I'm going to assume is that player 1 is the player who 00:13:28.810 --> 00:13:30.650 align:middle line:90% prefers the Celtics. 00:13:30.650 --> 00:13:32.970 align:middle line:84% So if they both go to the Celtics game, 00:13:32.970 --> 00:13:34.610 align:middle line:90% the payoffs are 4, 3. 00:13:34.610 --> 00:13:36.450 align:middle line:84% Now, notice a key difference here. 00:13:36.450 --> 00:13:38.610 align:middle line:84% With a single-player game, I only 00:13:38.610 --> 00:13:42.520 align:middle line:84% had to specify one utility at each of these consequences. 00:13:42.520 --> 00:13:44.280 align:middle line:84% But now that I have two players, I 00:13:44.280 --> 00:13:48.000 align:middle line:84% have to specify each player's utility from that consequence. 00:13:48.000 --> 00:13:49.720 align:middle line:84% And the convention we're going to use 00:13:49.720 --> 00:13:53.360 align:middle line:84% is that if we have players 1 and 2, the first player's payoff 00:13:53.360 --> 00:13:55.800 align:middle line:84% is going to come first, and the second player's payoff 00:13:55.800 --> 00:13:57.020 align:middle line:90% is going to come second. 00:13:57.020 --> 00:13:59.200 align:middle line:84% So this is how we specify everyone's payoffs 00:13:59.200 --> 00:14:00.500 align:middle line:90% from this consequence. 00:14:00.500 --> 00:14:02.640 align:middle line:84% If we had more players, we'd keep having commas, 00:14:02.640 --> 00:14:05.600 align:middle line:84% and we'd keep adding more numbers. 00:14:05.600 --> 00:14:08.160 align:middle line:84% Now the other one we'll write down is R, R. 00:14:08.160 --> 00:14:10.200 align:middle line:84% So this is, again, where we've coordinated 00:14:10.200 --> 00:14:12.220 align:middle line:84% on going to the same sports event, 00:14:12.220 --> 00:14:14.120 align:middle line:84% but we've coordinated on the Red Sox game. 00:14:14.120 --> 00:14:17.000 align:middle line:84% And player 2 is the one who likes the Red Sox more. 00:14:17.000 --> 00:14:20.260 align:middle line:84% So here, this down here, is going to be 3, 4. 00:14:20.260 --> 00:14:22.800 align:middle line:90% 00:14:22.800 --> 00:14:24.840 align:middle line:84% And then, to keep things simple, we'll 00:14:24.840 --> 00:14:28.080 align:middle line:90% make the other payoff 0, 0. 00:14:28.080 --> 00:14:30.820 align:middle line:84% So the idea is if you go to the game by yourself, you're so sad, 00:14:30.820 --> 00:14:32.060 align:middle line:84% it doesn't even matter who's playing. 00:14:32.060 --> 00:14:33.300 align:middle line:84% You're just sitting by yourself, and you're sad. 00:14:33.300 --> 00:14:34.480 align:middle line:90% You get a payoff of 0. 00:14:34.480 --> 00:14:39.650 align:middle line:90% But here we see the conflict. 00:14:39.650 --> 00:14:40.730 align:middle line:90% So a very simple game-- 00:14:40.730 --> 00:14:45.750 align:middle line:90% 00:14:45.750 --> 00:14:46.490 align:middle line:90% I don't know. 00:14:46.490 --> 00:14:49.130 align:middle line:84% In this game, who would you rather be? 00:14:49.130 --> 00:14:54.470 align:middle line:84% Do you have any intuition about what might happen in this game 00:14:54.470 --> 00:14:57.872 align:middle line:84% or which player you'd like to be? 00:14:57.872 --> 00:14:59.830 align:middle line:84% Who would you rather, be based on the way we've 00:14:59.830 --> 00:15:00.950 align:middle line:90% written the game? 00:15:00.950 --> 00:15:01.850 align:middle line:90% Yeah? 00:15:01.850 --> 00:15:03.147 align:middle line:90% AUDIENCE: Player 1. 00:15:03.147 --> 00:15:04.230 align:middle line:90% IAN BALL: And tell me why. 00:15:04.230 --> 00:15:05.647 align:middle line:84% AUDIENCE: Because they pick first. 00:15:05.647 --> 00:15:07.650 align:middle line:84% IAN BALL: Because they pick first, so exactly. 00:15:07.650 --> 00:15:08.650 align:middle line:90% I'm getting at a crucial thing. 00:15:08.650 --> 00:15:11.270 align:middle line:84% What is implicit in the way I've written this, and what you see 00:15:11.270 --> 00:15:12.812 align:middle line:84% is important for the analysis, is not 00:15:12.812 --> 00:15:15.830 align:middle line:84% just that they pick first, but that player 2 observes the pick 00:15:15.830 --> 00:15:17.230 align:middle line:90% that they make. 00:15:17.230 --> 00:15:19.070 align:middle line:84% So implicitly, the way I've written this 00:15:19.070 --> 00:15:21.788 align:middle line:84% is that player 2 gets to see what player 1 does. 00:15:21.788 --> 00:15:24.330 align:middle line:84% And can you walk me through why if player 2 sees what you do, 00:15:24.330 --> 00:15:27.047 align:middle line:84% why you think player one is in a really strong position? 00:15:27.047 --> 00:15:28.130 align:middle line:90% So what's going to happen? 00:15:28.130 --> 00:15:30.895 align:middle line:84% What's your prediction about what would happen? 00:15:30.895 --> 00:15:32.270 align:middle line:84% AUDIENCE: No matter what player 1 00:15:32.270 --> 00:15:36.083 align:middle line:84% does, player 2 is better off if they follow suit. 00:15:36.083 --> 00:15:36.750 align:middle line:90% IAN BALL: Right. 00:15:36.750 --> 00:15:38.800 align:middle line:90% So in this case, I'm player 1. 00:15:38.800 --> 00:15:39.760 align:middle line:90% I like the Celtics. 00:15:39.760 --> 00:15:41.968 align:middle line:84% What I want to do is I want to go to the Celtics game 00:15:41.968 --> 00:15:43.700 align:middle line:84% and make sure you know I'm going there. 00:15:43.700 --> 00:15:46.452 align:middle line:84% And once player 2 already knows that I'm at the Celtics game, 00:15:46.452 --> 00:15:48.160 align:middle line:84% well, they don't really want to be alone, 00:15:48.160 --> 00:15:50.243 align:middle line:84% so they're going to follow me to the Celtics game. 00:15:50.243 --> 00:15:52.580 align:middle line:84% So in this case, the fact that player 1 moves first, 00:15:52.580 --> 00:15:54.540 align:middle line:84% and that move is observed, gives them 00:15:54.540 --> 00:15:58.220 align:middle line:84% the power to select which game they end up going to. 00:15:58.220 --> 00:16:01.260 align:middle line:84% But what you can see is if player 2 didn't observe 00:16:01.260 --> 00:16:05.260 align:middle line:84% what player one did, then this would be a quite different game. 00:16:05.260 --> 00:16:07.640 align:middle line:84% An alternative game would be we move simultaneously. 00:16:07.640 --> 00:16:09.140 align:middle line:84% It's always a bit harder to motivate 00:16:09.140 --> 00:16:11.120 align:middle line:84% this in the world of instant communication. 00:16:11.120 --> 00:16:12.495 align:middle line:84% But in the old days, people would 00:16:12.495 --> 00:16:14.160 align:middle line:84% have to agree about a place to meet up, 00:16:14.160 --> 00:16:15.700 align:middle line:84% and they might simultaneously choose 00:16:15.700 --> 00:16:17.040 align:middle line:90% where they would go to meet up. 00:16:17.040 --> 00:16:19.100 align:middle line:84% And you might not observe where the other person 00:16:19.100 --> 00:16:20.260 align:middle line:90% had chosen to go. 00:16:20.260 --> 00:16:21.800 align:middle line:90% So we need notation for that. 00:16:21.800 --> 00:16:24.020 align:middle line:84% So if we wanted to say the version of the game 00:16:24.020 --> 00:16:27.440 align:middle line:84% where player 2 does not observe what player 1 did, 00:16:27.440 --> 00:16:31.620 align:middle line:84% we would indicate that using a dotted line here. 00:16:31.620 --> 00:16:32.940 align:middle line:90% So the dotted line-- 00:16:32.940 --> 00:16:34.980 align:middle line:84% maybe I'll write this over here-- 00:16:34.980 --> 00:16:41.230 align:middle line:84% means player 2, P2, does not observe player 1's move. 00:16:41.230 --> 00:16:48.730 align:middle line:90% 00:16:48.730 --> 00:16:50.250 align:middle line:84% So you can think of this essentially 00:16:50.250 --> 00:16:52.250 align:middle line:90% as a simultaneous move game. 00:16:52.250 --> 00:16:55.090 align:middle line:84% Technically, the way we wrote the tree, 00:16:55.090 --> 00:16:59.170 align:middle line:84% player 1 is moving first, and player 2 is moving second. 00:16:59.170 --> 00:17:02.450 align:middle line:84% But because player 2 doesn't get to see what player 1 does, 00:17:02.450 --> 00:17:05.869 align:middle line:84% it's effectively as if they were moving simultaneously. 00:17:05.869 --> 00:17:09.410 align:middle line:90% 00:17:09.410 --> 00:17:14.910 align:middle line:84% Are there any questions on these games here? 00:17:14.910 --> 00:17:18.810 align:middle line:90% 00:17:18.810 --> 00:17:22.569 align:middle line:84% So now what I'd like to do is present the general definition 00:17:22.569 --> 00:17:23.270 align:middle line:90% of a game. 00:17:23.270 --> 00:17:26.430 align:middle line:84% So these are two examples of a game that we want to capture. 00:17:26.430 --> 00:17:28.938 align:middle line:84% And we want to introduce a much more general definition that 00:17:28.938 --> 00:17:30.730 align:middle line:84% can cover all the examples that we're going 00:17:30.730 --> 00:17:33.650 align:middle line:90% to discuss in this course. 00:17:33.650 --> 00:17:35.660 align:middle line:84% So formally-- I'll do it over here. 00:17:35.660 --> 00:17:40.760 align:middle line:90% 00:17:40.760 --> 00:17:42.220 align:middle line:90% An extensive form game-- 00:17:42.220 --> 00:17:46.280 align:middle line:90% 00:17:46.280 --> 00:17:51.160 align:middle line:84% well, first we need a word for this decision tree that I drew. 00:17:51.160 --> 00:17:53.500 align:middle line:84% Formally, this is what's called a rooted tree. 00:17:53.500 --> 00:17:55.740 align:middle line:84% So let me say an extensive form game is a-- 00:17:55.740 --> 00:18:01.320 align:middle line:90% 00:18:01.320 --> 00:18:02.740 align:middle line:90% this is a fancy name. 00:18:02.740 --> 00:18:05.320 align:middle line:84% All it means is it looks like this. 00:18:05.320 --> 00:18:07.600 align:middle line:90% The root is where you start. 00:18:07.600 --> 00:18:08.680 align:middle line:90% This is the root. 00:18:08.680 --> 00:18:11.040 align:middle line:90% And then it comes out like this. 00:18:11.040 --> 00:18:13.180 align:middle line:84% Sometimes people write trees vertically. 00:18:13.180 --> 00:18:15.400 align:middle line:84% Sometimes you write them from left to right. 00:18:15.400 --> 00:18:17.200 align:middle line:90% It doesn't really matter. 00:18:17.200 --> 00:18:22.000 align:middle line:84% The formal notion of a tree is that at any point I look at, 00:18:22.000 --> 00:18:25.560 align:middle line:84% there's a unique path back to the root. 00:18:25.560 --> 00:18:27.280 align:middle line:84% So just so we understand what's not 00:18:27.280 --> 00:18:34.200 align:middle line:84% allowed, if I had something like this, 00:18:34.200 --> 00:18:37.800 align:middle line:84% this would not be allowed because here's my root. 00:18:37.800 --> 00:18:40.400 align:middle line:84% If I look at this point, there's two different ways 00:18:40.400 --> 00:18:41.700 align:middle line:90% to get back to the root. 00:18:41.700 --> 00:18:44.060 align:middle line:84% So this is not a graph theory course. 00:18:44.060 --> 00:18:46.060 align:middle line:84% We're not getting into too many technicalities. 00:18:46.060 --> 00:18:48.280 align:middle line:84% But basically, there can't be any loops 00:18:48.280 --> 00:18:50.060 align:middle line:90% or any cycles in the tree. 00:18:50.060 --> 00:18:52.320 align:middle line:84% And that's what we mean by rooted tree. 00:18:52.320 --> 00:18:54.040 align:middle line:84% And the interpretation of the root 00:18:54.040 --> 00:18:55.617 align:middle line:90% is that's where things start. 00:18:55.617 --> 00:18:58.200 align:middle line:84% So remember, implicitly, when we write an extensive form game, 00:18:58.200 --> 00:18:59.700 align:middle line:90% there's some notion of time. 00:18:59.700 --> 00:19:01.780 align:middle line:84% We think of time starting at the root. 00:19:01.780 --> 00:19:03.660 align:middle line:84% And then as we move forward in time, 00:19:03.660 --> 00:19:07.200 align:middle line:90% we move throughout the tree. 00:19:07.200 --> 00:19:09.820 align:middle line:84% Let me erase this, since this is not what we allow. 00:19:09.820 --> 00:19:13.720 align:middle line:90% 00:19:13.720 --> 00:19:15.520 align:middle line:90% I want to make-- 00:19:15.520 --> 00:19:21.680 align:middle line:84% so it's a rooted tree specifying four things. 00:19:21.680 --> 00:19:24.340 align:middle line:84% So I want to first specify each of them kind of informally. 00:19:24.340 --> 00:19:27.500 align:middle line:84% And then I want to talk formally how they appear on the graph. 00:19:27.500 --> 00:19:29.180 align:middle line:84% So there's four things it specifies. 00:19:29.180 --> 00:19:33.750 align:middle line:84% And I like using the acronym PAPI to remember it. 00:19:33.750 --> 00:19:37.927 align:middle line:84% So this is players, actions, payoffs, information. 00:19:37.927 --> 00:19:39.010 align:middle line:90% So let me write that down. 00:19:39.010 --> 00:19:39.630 align:middle line:90% Yes. 00:19:39.630 --> 00:19:41.790 align:middle line:84% AUDIENCE: Are these required to be finite trees? 00:19:41.790 --> 00:19:43.950 align:middle line:90% IAN BALL: Good question. 00:19:43.950 --> 00:19:46.710 align:middle line:84% We're going to do what's often done in an intro game theory 00:19:46.710 --> 00:19:47.210 align:middle line:90% class. 00:19:47.210 --> 00:19:48.830 align:middle line:84% And when we formally define it, we'll 00:19:48.830 --> 00:19:51.333 align:middle line:84% assume things are finite so we don't have issues. 00:19:51.333 --> 00:19:52.750 align:middle line:84% But then we'll go to some examples 00:19:52.750 --> 00:19:53.930 align:middle line:90% where things are infinite. 00:19:53.930 --> 00:19:56.190 align:middle line:84% So basically, what I would say is, 00:19:56.190 --> 00:19:58.888 align:middle line:84% to carefully define infinite games, there are some issues. 00:19:58.888 --> 00:20:00.430 align:middle line:84% So we'll only consider infinite games 00:20:00.430 --> 00:20:01.650 align:middle line:84% where things are very well-behaved. 00:20:01.650 --> 00:20:03.890 align:middle line:84% And we won't formally define exactly what well-behaved is. 00:20:03.890 --> 00:20:06.140 align:middle line:84% It's just if you see an infinite game, trust that it's 00:20:06.140 --> 00:20:08.350 align:middle line:90% going to be well-behaved. 00:20:08.350 --> 00:20:12.110 align:middle line:84% But if we think of the Keynesian beauty contest 00:20:12.110 --> 00:20:13.670 align:middle line:84% that we started with, we required 00:20:13.670 --> 00:20:15.450 align:middle line:84% you to choose a number between 1 and 100. 00:20:15.450 --> 00:20:16.530 align:middle line:90% It had to be an integer. 00:20:16.530 --> 00:20:19.170 align:middle line:84% But what if you could choose a real number between 0 and 100? 00:20:19.170 --> 00:20:21.087 align:middle line:84% Now all of a sudden, we have an infinite game. 00:20:21.087 --> 00:20:23.650 align:middle line:84% And that could be a realistic model or a useful model. 00:20:23.650 --> 00:20:25.790 align:middle line:90% Yeah. 00:20:25.790 --> 00:20:26.370 align:middle line:90% Great. 00:20:26.370 --> 00:20:28.370 align:middle line:90% So P is for players. 00:20:28.370 --> 00:20:33.100 align:middle line:90% 00:20:33.100 --> 00:20:34.660 align:middle line:90% A is for actions. 00:20:34.660 --> 00:20:40.500 align:middle line:90% P is for payoffs. 00:20:40.500 --> 00:20:42.718 align:middle line:90% And I is for information. 00:20:42.718 --> 00:20:44.260 align:middle line:84% So I could define these things, but I 00:20:44.260 --> 00:20:47.260 align:middle line:84% think people should have a pretty intuitive sense of what 00:20:47.260 --> 00:20:47.800 align:middle line:90% these mean. 00:20:47.800 --> 00:20:51.820 align:middle line:84% The players are the agents, the actors, in our game. 00:20:51.820 --> 00:20:55.180 align:middle line:84% The actions are the choices that they make. 00:20:55.180 --> 00:20:57.260 align:middle line:84% So for instance, in this game, player 1 00:20:57.260 --> 00:20:58.820 align:middle line:90% chose between safe and risky. 00:20:58.820 --> 00:21:02.580 align:middle line:84% Over here, player 2 chose between going to the Celtics 00:21:02.580 --> 00:21:05.900 align:middle line:84% game and going to the Red Sox game. 00:21:05.900 --> 00:21:08.440 align:middle line:84% Payoffs are what we define at the end. 00:21:08.440 --> 00:21:09.860 align:middle line:84% This is the utilities that people 00:21:09.860 --> 00:21:13.165 align:middle line:84% get from the consequences of the game. 00:21:13.165 --> 00:21:14.540 align:middle line:84% And then information, we're going 00:21:14.540 --> 00:21:16.880 align:middle line:84% to have to be a bit more careful about defining this, 00:21:16.880 --> 00:21:19.640 align:middle line:84% but it captures what we were initially discussing here, 00:21:19.640 --> 00:21:23.620 align:middle line:84% things like does player 2 know what player 1 chose when 00:21:23.620 --> 00:21:27.140 align:middle line:90% it's player two's turn to move? 00:21:27.140 --> 00:21:30.768 align:middle line:84% And I want to just put a few synonyms up here 00:21:30.768 --> 00:21:32.810 align:middle line:84% because I have a tendency, throughout the course, 00:21:32.810 --> 00:21:34.870 align:middle line:84% to use a few different words to mean the same thing. 00:21:34.870 --> 00:21:36.610 align:middle line:84% And I just want us to be all on the same page about what 00:21:36.610 --> 00:21:37.530 align:middle line:90% those mean. 00:21:37.530 --> 00:21:41.170 align:middle line:84% So I might say action, or I might say moves. 00:21:41.170 --> 00:21:43.290 align:middle line:90% Those mean the same thing. 00:21:43.290 --> 00:21:47.930 align:middle line:84% And I might say payoffs, or I might say utilities. 00:21:47.930 --> 00:21:50.090 align:middle line:90% And those mean the same thing. 00:21:50.090 --> 00:21:54.370 align:middle line:84% One really important thing is payoffs always means utilities. 00:21:54.370 --> 00:21:59.070 align:middle line:84% So even in this game here, here I would say your payoff is 1.5. 00:21:59.070 --> 00:22:01.430 align:middle line:84% What I mean by that is your utility is 1.5. 00:22:01.430 --> 00:22:03.310 align:middle line:84% I'm not talking about the monetary payoff. 00:22:03.310 --> 00:22:04.850 align:middle line:84% So if I don't use the word monetary, 00:22:04.850 --> 00:22:06.510 align:middle line:84% we're always talking about utilities. 00:22:06.510 --> 00:22:07.010 align:middle line:90% Why? 00:22:07.010 --> 00:22:09.730 align:middle line:84% Well, it's really the utilities that matter. 00:22:09.730 --> 00:22:12.230 align:middle line:84% The monetary payoffs are in the background. 00:22:12.230 --> 00:22:15.810 align:middle line:84% But everything really comes down to the utilities. 00:22:15.810 --> 00:22:18.930 align:middle line:84% So players-- I think intuitively we 00:22:18.930 --> 00:22:20.910 align:middle line:84% all know how we need to specify this, 00:22:20.910 --> 00:22:23.650 align:middle line:84% but I just want to be a little bit more formal. 00:22:23.650 --> 00:22:25.450 align:middle line:84% Well, the players, the way we specify it, 00:22:25.450 --> 00:22:29.160 align:middle line:84% is we put labels on the nodes in the tree. 00:22:29.160 --> 00:22:32.480 align:middle line:84% But notice, not every node gets a label. 00:22:32.480 --> 00:22:35.660 align:middle line:84% In this game, these nodes don't get a player label. 00:22:35.660 --> 00:22:37.080 align:middle line:90% They get a utility label. 00:22:37.080 --> 00:22:38.980 align:middle line:84% So really, if we look at this game, 00:22:38.980 --> 00:22:41.640 align:middle line:84% we see there's two different kinds of nodes. 00:22:41.640 --> 00:22:47.345 align:middle line:84% There's the nodes like this, 1, 2, 3 which are decision nodes. 00:22:47.345 --> 00:22:48.720 align:middle line:84% These are nodes in the tree where 00:22:48.720 --> 00:22:51.200 align:middle line:84% people take decisions or choose-- maybe I'll 00:22:51.200 --> 00:22:52.540 align:middle line:90% say actions, moves. 00:22:52.540 --> 00:22:54.700 align:middle line:84% Maybe I'll add one more synonym, decisions. 00:22:54.700 --> 00:22:58.920 align:middle line:90% 00:22:58.920 --> 00:23:01.280 align:middle line:84% And then we have nodes like this, which 00:23:01.280 --> 00:23:02.740 align:middle line:90% are at the end of the game. 00:23:02.740 --> 00:23:05.000 align:middle line:90% And these specify consequences. 00:23:05.000 --> 00:23:08.400 align:middle line:84% So let me say the rooted tree-- let me go a little up here-- 00:23:08.400 --> 00:23:16.000 align:middle line:84% it has nodes in some set H. Why do we call the nodes capital H? 00:23:16.000 --> 00:23:18.200 align:middle line:90% Because H reminds us of history. 00:23:18.200 --> 00:23:20.880 align:middle line:84% And with a tree, every node exactly 00:23:20.880 --> 00:23:22.720 align:middle line:84% tells us the history of play in the game. 00:23:22.720 --> 00:23:24.393 align:middle line:90% Let's look at this. 00:23:24.393 --> 00:23:26.060 align:middle line:84% At his node, well, nothing has happened. 00:23:26.060 --> 00:23:27.710 align:middle line:90% So the history is nothing. 00:23:27.710 --> 00:23:31.590 align:middle line:84% At this node, the history is player 1 chose C. 00:23:31.590 --> 00:23:34.830 align:middle line:84% And at this node the history is player 1 chose C. 00:23:34.830 --> 00:23:38.070 align:middle line:84% And then player two chose R. Notice 00:23:38.070 --> 00:23:43.030 align:middle line:84% that this gets back to the problem with our example here. 00:23:43.030 --> 00:23:47.230 align:middle line:84% If we had two paths that went to the same node, 00:23:47.230 --> 00:23:49.990 align:middle line:84% well now, interpreting this as a history becomes unclear, 00:23:49.990 --> 00:23:52.390 align:middle line:84% because if we're here, well, did we get here 00:23:52.390 --> 00:23:54.190 align:middle line:90% by following that history? 00:23:54.190 --> 00:23:56.703 align:middle line:84% Or did we get here by following that history? 00:23:56.703 --> 00:23:58.870 align:middle line:84% And that shows we've done something wrong in the way 00:23:58.870 --> 00:24:00.090 align:middle line:90% we've specified the game. 00:24:00.090 --> 00:24:03.368 align:middle line:84% We want to uniquely specify the history of actions 00:24:03.368 --> 00:24:04.410 align:middle line:90% at each node in the game. 00:24:04.410 --> 00:24:06.910 align:middle line:90% And that's why we rule that out. 00:24:06.910 --> 00:24:10.750 align:middle line:84% So within the set of nodes H, we'll 00:24:10.750 --> 00:24:15.730 align:middle line:84% say Z, which is a subset of H, these are the terminal nodes. 00:24:15.730 --> 00:24:20.030 align:middle line:90% 00:24:20.030 --> 00:24:23.370 align:middle line:84% And then H minus Z, these are the decision nodes. 00:24:23.370 --> 00:24:26.820 align:middle line:90% 00:24:26.820 --> 00:24:28.368 align:middle line:90% So I could define this formally. 00:24:28.368 --> 00:24:30.160 align:middle line:84% But I think we should know what this means. 00:24:30.160 --> 00:24:34.460 align:middle line:84% Z subset of H means there's some subcollection of nodes 00:24:34.460 --> 00:24:36.220 align:middle line:90% that are the terminal nodes. 00:24:36.220 --> 00:24:39.780 align:middle line:84% In this game here, the set H contains, 00:24:39.780 --> 00:24:43.060 align:middle line:90% well, 1, 2, 3, 4, 5, 6, 7. 00:24:43.060 --> 00:24:46.940 align:middle line:84% So in this game, H would have 7 nodes. 00:24:46.940 --> 00:24:49.300 align:middle line:90% 4 of them are terminal nodes. 00:24:49.300 --> 00:24:51.860 align:middle line:84% So these 4 nodes at the end would 00:24:51.860 --> 00:24:55.120 align:middle line:84% be in the set Z, which is a subset of all the 7. 00:24:55.120 --> 00:24:57.700 align:middle line:84% So we have 7 histories, 7 nodes altogether. 00:24:57.700 --> 00:25:01.580 align:middle line:84% 4 of them are terminal nodes in the set Z. And then you 00:25:01.580 --> 00:25:03.420 align:middle line:84% may not have seen this notation before, 00:25:03.420 --> 00:25:05.780 align:middle line:84% but H set minus Z means everything 00:25:05.780 --> 00:25:08.820 align:middle line:90% that's in H that's not in Z. 00:25:08.820 --> 00:25:13.700 align:middle line:84% So over here in this example, we have 7 nodes altogether. 00:25:13.700 --> 00:25:16.160 align:middle line:90% 4 of them are in Z. What's left? 00:25:16.160 --> 00:25:17.420 align:middle line:90% What's not in Z? 00:25:17.420 --> 00:25:20.580 align:middle line:84% Well, it's these 3 nodes, 1, 2, 3. 00:25:20.580 --> 00:25:22.500 align:middle line:84% And these are called decision nodes 00:25:22.500 --> 00:25:25.610 align:middle line:84% because these are nodes where players make decisions. 00:25:25.610 --> 00:25:28.430 align:middle line:84% Once we get here, there are no decisions to be made. 00:25:28.430 --> 00:25:31.090 align:middle line:90% These are just consequences. 00:25:31.090 --> 00:25:36.010 align:middle line:84% It's not a coincidence that we use the term Z for the terminal 00:25:36.010 --> 00:25:38.770 align:middle line:84% nodes because Z is exactly playing 00:25:38.770 --> 00:25:42.290 align:middle line:84% the role that Z played last week in our individual game. 00:25:42.290 --> 00:25:44.990 align:middle line:84% So at the terminal nodes of our game, 00:25:44.990 --> 00:25:48.570 align:middle line:84% we have some consequence, which lives in some set Z, 00:25:48.570 --> 00:25:51.210 align:middle line:84% and we have Von Neumann-Morgenstern utilities 00:25:51.210 --> 00:25:52.770 align:middle line:90% over those consequences. 00:25:52.770 --> 00:25:56.050 align:middle line:84% Of course, before, we just had one vNM utility. 00:25:56.050 --> 00:25:58.410 align:middle line:84% And now, we have one for every player. 00:25:58.410 --> 00:26:00.210 align:middle line:84% So we'll talk about that in a second. 00:26:00.210 --> 00:26:01.610 align:middle line:90% So what are the players? 00:26:01.610 --> 00:26:06.650 align:middle line:84% We'll usually label them i equals 1 up to n. 00:26:06.650 --> 00:26:09.970 align:middle line:84% So we could have a one player game a two player game. 00:26:09.970 --> 00:26:11.830 align:middle line:90% These are the strategic players. 00:26:11.830 --> 00:26:14.130 align:middle line:84% And then sometimes we also have Nature. 00:26:14.130 --> 00:26:17.868 align:middle line:90% So maybe I'll say and Nature. 00:26:17.868 --> 00:26:19.410 align:middle line:84% So Nature is kind of a special player 00:26:19.410 --> 00:26:21.250 align:middle line:84% that we sometimes also add to the game 00:26:21.250 --> 00:26:23.150 align:middle line:84% if there's uncertainty in the game. 00:26:23.150 --> 00:26:27.650 align:middle line:84% So over here, this game had player 1 as a strategic player, 00:26:27.650 --> 00:26:29.450 align:middle line:90% and it had Nature. 00:26:29.450 --> 00:26:34.890 align:middle line:84% This game had two strategic players player 1 and player 2. 00:26:34.890 --> 00:26:37.570 align:middle line:84% Now, we need the actions, moves, or decisions. 00:26:37.570 --> 00:26:40.090 align:middle line:84% Well, crucially, these are, I'll say, 00:26:40.090 --> 00:26:48.880 align:middle line:84% labeled on the edges of the tree. 00:26:48.880 --> 00:26:50.630 align:middle line:84% And we could be really formal and precise. 00:26:50.630 --> 00:26:53.050 align:middle line:84% But I think it's easier just to see what we mean. 00:26:53.050 --> 00:26:55.330 align:middle line:90% So let's take this history here. 00:26:55.330 --> 00:26:57.490 align:middle line:90% It's player 1's turn to move. 00:26:57.490 --> 00:27:00.730 align:middle line:84% At this node, player 1 has two choices or two actions, 00:27:00.730 --> 00:27:02.410 align:middle line:90% safe and risky. 00:27:02.410 --> 00:27:05.370 align:middle line:84% We represent those two actions with edges coming out 00:27:05.370 --> 00:27:07.530 align:middle line:84% of the node, and we label each edge 00:27:07.530 --> 00:27:10.130 align:middle line:90% with the corresponding action. 00:27:10.130 --> 00:27:12.650 align:middle line:84% Similarly, over here, when it's player 2's 00:27:12.650 --> 00:27:14.810 align:middle line:84% is turn to move at this node, player 2 00:27:14.810 --> 00:27:18.990 align:middle line:84% has two choices, C and R. We represent those with edges. 00:27:18.990 --> 00:27:21.760 align:middle line:84% And we label them according to the actions that they are. 00:27:21.760 --> 00:27:24.330 align:middle line:84% And if you have a math background, 00:27:24.330 --> 00:27:26.580 align:middle line:84% you may think, oh, these labels are kind of arbitrary. 00:27:26.580 --> 00:27:29.000 align:middle line:84% It turns out the labels are quite fundamental and something 00:27:29.000 --> 00:27:29.580 align:middle line:90% we'll do later. 00:27:29.580 --> 00:27:30.913 align:middle line:90% So the labels are not arbitrary. 00:27:30.913 --> 00:27:34.200 align:middle line:90% I'll make that point now. 00:27:34.200 --> 00:27:37.220 align:middle line:84% Payoffs or utilities-- well, based on what I've said, 00:27:37.220 --> 00:27:40.080 align:middle line:84% can anyone guess how we're going to specify the payoffs, 00:27:40.080 --> 00:27:44.400 align:middle line:90% utilities in this game? 00:27:44.400 --> 00:27:45.180 align:middle line:90% Any thoughts? 00:27:45.180 --> 00:27:49.640 align:middle line:90% 00:27:49.640 --> 00:27:50.240 align:middle line:90% Shy? 00:27:50.240 --> 00:27:51.240 align:middle line:90% Yeah? 00:27:51.240 --> 00:27:52.767 align:middle line:90% AUDIENCE: A set-- 00:27:52.767 --> 00:27:53.600 align:middle line:90% IAN BALL: Say again? 00:27:53.600 --> 00:27:55.760 align:middle line:84% AUDIENCE: A set of utilities associated with [INAUDIBLE]? 00:27:55.760 --> 00:27:56.360 align:middle line:90% IAN BALL: Exactly. 00:27:56.360 --> 00:27:58.277 align:middle line:84% And the way we're going to write that is we're 00:27:58.277 --> 00:28:00.600 align:middle line:84% going to have n utility functions, one for each player. 00:28:00.600 --> 00:28:08.880 align:middle line:84% So we're going to have utility function ui from Z to R 00:28:08.880 --> 00:28:12.240 align:middle line:90% for i equals 1 through n. 00:28:12.240 --> 00:28:13.340 align:middle line:90% What does that mean? 00:28:13.340 --> 00:28:15.360 align:middle line:84% It says for each player, i, we're 00:28:15.360 --> 00:28:19.070 align:middle line:84% going to specify that player i's Von Neumann-Morgenstern utility 00:28:19.070 --> 00:28:21.550 align:middle line:84% function over all of the outcomes. 00:28:21.550 --> 00:28:24.710 align:middle line:84% So this is going to assign a real number in R 00:28:24.710 --> 00:28:27.350 align:middle line:84% to every single outcome of the game. 00:28:27.350 --> 00:28:29.750 align:middle line:84% Now, notice we've grouped things a little differently. 00:28:29.750 --> 00:28:31.870 align:middle line:84% Here I'm grouping things by player. 00:28:31.870 --> 00:28:35.670 align:middle line:84% So for player i, this function says, how much utility 00:28:35.670 --> 00:28:39.670 align:middle line:84% does that player i get from every single consequence? 00:28:39.670 --> 00:28:41.550 align:middle line:84% Well, the way we wrote it on the tree, 00:28:41.550 --> 00:28:43.830 align:middle line:84% we actually grouped it by consequences. 00:28:43.830 --> 00:28:48.510 align:middle line:84% So associated with this node is u1 of that node, comma, 00:28:48.510 --> 00:28:49.790 align:middle line:90% u2 of that node. 00:28:49.790 --> 00:28:55.670 align:middle line:84% So if you want to connect it to the definition, 00:28:55.670 --> 00:29:00.510 align:middle line:84% all these numbers here would correspond to the function u1. 00:29:00.510 --> 00:29:04.470 align:middle line:84% And all these numbers here would correspond to the function u2. 00:29:04.470 --> 00:29:09.390 align:middle line:84% So technically what I put here is u1 of this outcome, comma, 00:29:09.390 --> 00:29:14.550 align:middle line:84% u2 of this outcome or consequence, same thing. 00:29:14.550 --> 00:29:16.250 align:middle line:84% And these are, I should emphasize, 00:29:16.250 --> 00:29:19.418 align:middle line:84% Von Neumann-Morgenstern utilities, 00:29:19.418 --> 00:29:21.460 align:middle line:84% just like we said last week, so that when players 00:29:21.460 --> 00:29:25.460 align:middle line:84% face uncertainty, they evaluate lotteries over consequences 00:29:25.460 --> 00:29:27.040 align:middle line:90% using expected utility. 00:29:27.040 --> 00:29:31.060 align:middle line:90% 00:29:31.060 --> 00:29:31.740 align:middle line:90% Great. 00:29:31.740 --> 00:29:33.700 align:middle line:84% And then the last thing is information 00:29:33.700 --> 00:29:36.220 align:middle line:84% is going to be represented with something called information 00:29:36.220 --> 00:29:36.980 align:middle line:90% sets. 00:29:36.980 --> 00:29:40.460 align:middle line:84% And that's going to take a new board to talk about. 00:29:40.460 --> 00:29:42.460 align:middle line:84% So let me pause here and see, are any questions 00:29:42.460 --> 00:29:44.040 align:middle line:90% on what I've done so far? 00:29:44.040 --> 00:29:55.865 align:middle line:90% 00:29:55.865 --> 00:29:57.440 align:middle line:84% It's a little tight at the bottom. 00:29:57.440 --> 00:30:01.104 align:middle line:90% 00:30:01.104 --> 00:30:05.100 align:middle line:84% So now, let's talk about information sets. 00:30:05.100 --> 00:30:10.020 align:middle line:90% So for each player i-- 00:30:10.020 --> 00:30:11.680 align:middle line:84% so let me just say it in words first. 00:30:11.680 --> 00:30:13.420 align:middle line:90% We want to look at the game. 00:30:13.420 --> 00:30:16.570 align:middle line:84% And we want to look at all the nodes where 00:30:16.570 --> 00:30:19.553 align:middle line:90% player i is called to move. 00:30:19.553 --> 00:30:20.470 align:middle line:90% And this is important. 00:30:20.470 --> 00:30:23.530 align:middle line:84% So up here, when I said players i equals 1 through n, 00:30:23.530 --> 00:30:26.050 align:middle line:84% part of naming those players is also 00:30:26.050 --> 00:30:29.650 align:middle line:84% assigning a player to every single node in the tree. 00:30:29.650 --> 00:30:33.750 align:middle line:84% And that means that if I fix a given player i, I can ask, 00:30:33.750 --> 00:30:35.490 align:middle line:84% what are the nodes in the tree where 00:30:35.490 --> 00:30:37.730 align:middle line:90% that player i is called to act? 00:30:37.730 --> 00:30:40.530 align:middle line:84% So which of the decision nodes is player 00:30:40.530 --> 00:30:42.330 align:middle line:90% i the one making the decision? 00:30:42.330 --> 00:30:46.150 align:middle line:84% So I'll say for each player i, the decision nodes of player i-- 00:30:46.150 --> 00:30:55.530 align:middle line:90% 00:30:55.530 --> 00:30:59.043 align:middle line:84% and they'd all be connected, but just to give an illustration, 00:30:59.043 --> 00:31:00.710 align:middle line:84% maybe the nodes are something like this. 00:31:00.710 --> 00:31:02.085 align:middle line:84% There'd be lines in between them, 00:31:02.085 --> 00:31:04.170 align:middle line:84% but let's just focus on the nodes for now. 00:31:04.170 --> 00:31:06.970 align:middle line:84% The decision nodes of player i are partitioned 00:31:06.970 --> 00:31:08.010 align:middle line:90% into information sets. 00:31:08.010 --> 00:31:09.390 align:middle line:84% And I'll explain what that means. 00:31:09.390 --> 00:31:22.480 align:middle line:90% 00:31:22.480 --> 00:31:25.160 align:middle line:84% A partition is a fancy math word to say, 00:31:25.160 --> 00:31:26.908 align:middle line:90% we have a group of things. 00:31:26.908 --> 00:31:28.200 align:middle line:90% We have a collection of things. 00:31:28.200 --> 00:31:30.440 align:middle line:84% And we're going to put them into groups. 00:31:30.440 --> 00:31:32.680 align:middle line:84% We're just going to break them up into groups. 00:31:32.680 --> 00:31:37.480 align:middle line:84% So for instance, one way of partitioning this would be that. 00:31:37.480 --> 00:31:38.380 align:middle line:90% That's a partition. 00:31:38.380 --> 00:31:39.560 align:middle line:90% I have five things. 00:31:39.560 --> 00:31:42.320 align:middle line:84% I put two things together in this group, these two things 00:31:42.320 --> 00:31:45.105 align:middle line:84% together in this group, and one thing by itself in this group. 00:31:45.105 --> 00:31:46.980 align:middle line:84% So the rules of partition are what you think. 00:31:46.980 --> 00:31:48.640 align:middle line:90% Nothing could be in two groups. 00:31:48.640 --> 00:31:51.740 align:middle line:84% And nothing can be not in any group. 00:31:51.740 --> 00:31:53.540 align:middle line:84% So everything's in exactly one group. 00:31:53.540 --> 00:31:56.180 align:middle line:84% And that group could be it by itself, like we have here. 00:31:56.180 --> 00:31:56.820 align:middle line:90% That's allowed. 00:31:56.820 --> 00:31:59.120 align:middle line:84% But think of you have a pile of candy on the ground, 00:31:59.120 --> 00:32:03.160 align:middle line:84% and you just push it into piles on the ground. 00:32:03.160 --> 00:32:05.680 align:middle line:90% What is the interpretation? 00:32:05.680 --> 00:32:09.760 align:middle line:84% The interpretation is that player i can only 00:32:09.760 --> 00:32:14.070 align:middle line:84% tell which group they're in, which information 00:32:14.070 --> 00:32:16.150 align:middle line:84% set they're in, but not which node they're 00:32:16.150 --> 00:32:17.870 align:middle line:90% in within that information set. 00:32:17.870 --> 00:32:18.890 align:middle line:90% So let me say that. 00:32:18.890 --> 00:32:21.590 align:middle line:90% 00:32:21.590 --> 00:32:30.870 align:middle line:84% So player I observes the information set. 00:32:30.870 --> 00:32:33.190 align:middle line:84% So in this case, we have three information sets, maybe 00:32:33.190 --> 00:32:36.270 align:middle line:90% i1, i2, and i3. 00:32:36.270 --> 00:32:38.150 align:middle line:84% Whenever the player moves, they know which 00:32:38.150 --> 00:32:39.670 align:middle line:90% information set they're in. 00:32:39.670 --> 00:32:42.310 align:middle line:84% But they don't know which node inside the information 00:32:42.310 --> 00:32:43.210 align:middle line:90% set they're in. 00:32:43.210 --> 00:33:00.670 align:middle line:90% 00:33:00.670 --> 00:33:03.412 align:middle line:84% Now, of course, if a node is by itself, 00:33:03.412 --> 00:33:04.870 align:middle line:84% then they effectively observe which 00:33:04.870 --> 00:33:06.510 align:middle line:84% node they're at because if I know 00:33:06.510 --> 00:33:08.630 align:middle line:84% I'm in this information set, then I certainly 00:33:08.630 --> 00:33:10.270 align:middle line:90% know which node I'm at. 00:33:10.270 --> 00:33:12.990 align:middle line:84% But if I know only that I'm at this information set, 00:33:12.990 --> 00:33:17.830 align:middle line:84% I'm not sure whether I'm at this node or I'm at this node. 00:33:17.830 --> 00:33:21.690 align:middle line:84% Now here, that's exactly what we were capturing on this graph. 00:33:21.690 --> 00:33:23.350 align:middle line:84% But sometimes it's cleaner, instead 00:33:23.350 --> 00:33:26.070 align:middle line:84% of drawing a circle around the two nodes, 00:33:26.070 --> 00:33:27.910 align:middle line:84% just to put a dotted line between them. 00:33:27.910 --> 00:33:31.670 align:middle line:84% So what we did here is exactly representing this information 00:33:31.670 --> 00:33:32.530 align:middle line:90% partition. 00:33:32.530 --> 00:33:34.730 align:middle line:84% But we just did it in a slightly more compact way. 00:33:34.730 --> 00:33:38.130 align:middle line:84% So often, an alternative thing-- maybe I'll give an example. 00:33:38.130 --> 00:33:40.630 align:middle line:84% If you had three nodes, or specifically 00:33:40.630 --> 00:33:43.630 align:middle line:84% if you have two nodes, you might draw a dotted line between them 00:33:43.630 --> 00:33:44.530 align:middle line:90% instead of a circle. 00:33:44.530 --> 00:33:45.988 align:middle line:84% And that just means the same thing. 00:33:45.988 --> 00:33:48.368 align:middle line:84% But it means things get a little less messy. 00:33:48.368 --> 00:33:50.410 align:middle line:84% And I should say, this is not an information set. 00:33:50.410 --> 00:33:52.690 align:middle line:84% This is just me indicating something. 00:33:52.690 --> 00:33:56.290 align:middle line:90% So pretend that's not there. 00:33:56.290 --> 00:33:57.970 align:middle line:84% And let's go to the interpretation. 00:33:57.970 --> 00:34:00.262 align:middle line:84% So what we said, when these are in the same information 00:34:00.262 --> 00:34:03.790 align:middle line:84% set, when it's player 2's turn to move, what do they know? 00:34:03.790 --> 00:34:06.590 align:middle line:84% They know that they're either at this history, 00:34:06.590 --> 00:34:08.110 align:middle line:90% or they're at this history. 00:34:08.110 --> 00:34:09.750 align:middle line:84% So let's say that, in words, what they 00:34:09.750 --> 00:34:12.699 align:middle line:84% know is player 1 has chosen something. 00:34:12.699 --> 00:34:15.219 align:middle line:84% But they don't know which thing player 1 has chosen. 00:34:15.219 --> 00:34:18.659 align:middle line:84% Either player 1 chose to go to the Celtics game 00:34:18.659 --> 00:34:21.080 align:middle line:84% or player 1 chose to go to the Red Sox game. 00:34:21.080 --> 00:34:23.040 align:middle line:84% They know player 1 chose one of those things, 00:34:23.040 --> 00:34:25.420 align:middle line:90% but they don't know which one. 00:34:25.420 --> 00:34:28.100 align:middle line:84% If we erase this information set, 00:34:28.100 --> 00:34:31.659 align:middle line:84% now these two nodes are in separate information sets. 00:34:31.659 --> 00:34:33.760 align:middle line:84% And if they're in separate information sets, 00:34:33.760 --> 00:34:37.300 align:middle line:84% player 2 does observe the choice that player 1 makes because they 00:34:37.300 --> 00:34:40.940 align:middle line:84% know either they're at this singleton information set that 00:34:40.940 --> 00:34:43.076 align:middle line:84% only contains this node or they're 00:34:43.076 --> 00:34:44.659 align:middle line:84% at this singleton information set that 00:34:44.659 --> 00:34:45.840 align:middle line:90% only contains this node. 00:34:45.840 --> 00:34:48.440 align:middle line:84% So they know exactly what choice player 1 made. 00:34:48.440 --> 00:34:50.060 align:middle line:84% And this is the formal way that we're 00:34:50.060 --> 00:34:52.179 align:middle line:90% going to model observability. 00:34:52.179 --> 00:34:53.060 align:middle line:90% Yes. 00:34:53.060 --> 00:34:55.380 align:middle line:84% AUDIENCE: For simultaneous decision making, 00:34:55.380 --> 00:34:57.560 align:middle line:84% do we prefer to use the information segmentation, 00:34:57.560 --> 00:34:59.700 align:middle line:84% or do we ever take a decision node 00:34:59.700 --> 00:35:02.230 align:middle line:84% and have two players kind of belonging to that decision? 00:35:02.230 --> 00:35:04.480 align:middle line:84% IAN BALL: So we're always going to do the first thing. 00:35:04.480 --> 00:35:07.640 align:middle line:84% So we every node always has exactly one player. 00:35:07.640 --> 00:35:09.113 align:middle line:84% If we want to capture simultaneity, 00:35:09.113 --> 00:35:11.030 align:middle line:84% We always capture it through information sets. 00:35:11.030 --> 00:35:11.790 align:middle line:90% It's a great question. 00:35:11.790 --> 00:35:14.090 align:middle line:84% So nodes are never going to have two players associated 00:35:14.090 --> 00:35:15.215 align:middle line:90% with them, always just one. 00:35:15.215 --> 00:35:15.910 align:middle line:90% Good question. 00:35:15.910 --> 00:35:18.610 align:middle line:90% 00:35:18.610 --> 00:35:24.370 align:middle line:84% So one final comment is if a player can observe everything, 00:35:24.370 --> 00:35:27.570 align:middle line:84% then what partition does that correspond to, 00:35:27.570 --> 00:35:30.810 align:middle line:84% if I observe everything, I have no uncertainty about who's 00:35:30.810 --> 00:35:33.570 align:middle line:90% played before me. 00:35:33.570 --> 00:35:35.690 align:middle line:84% Well, then just everything's in its own group. 00:35:35.690 --> 00:35:37.590 align:middle line:84% So that would be I have a circle over this. 00:35:37.590 --> 00:35:39.290 align:middle line:90% I a circle over this. 00:35:39.290 --> 00:35:41.530 align:middle line:84% And in that case, we sometimes call these-- maybe 00:35:41.530 --> 00:35:45.530 align:middle line:84% I'll give a word for that, a singleton information set. 00:35:45.530 --> 00:35:48.530 align:middle line:84% A singleton is just a set with one node inside it. 00:35:48.530 --> 00:35:50.950 align:middle line:84% So if I'm ever at a singleton information set, 00:35:50.950 --> 00:35:53.250 align:middle line:84% I know exactly which node I'm at. 00:35:53.250 --> 00:35:55.770 align:middle line:84% And if a player always knows exactly where they are, 00:35:55.770 --> 00:35:58.690 align:middle line:84% then their partition consists only of singletons. 00:35:58.690 --> 00:36:01.710 align:middle line:84% None of their information sets contains more than one node. 00:36:01.710 --> 00:36:05.370 align:middle line:90% 00:36:05.370 --> 00:36:07.280 align:middle line:90% Any questions here? 00:36:07.280 --> 00:36:07.937 align:middle line:90% Yes? 00:36:07.937 --> 00:36:09.520 align:middle line:84% AUDIENCE: What about other information 00:36:09.520 --> 00:36:13.480 align:middle line:84% that might be relevant for decisions, such as the payoffs? 00:36:13.480 --> 00:36:14.640 align:middle line:90% IAN BALL: Good question. 00:36:14.640 --> 00:36:18.720 align:middle line:84% So here what we've assumed is that the payoffs 00:36:18.720 --> 00:36:20.360 align:middle line:90% of each consequence are known. 00:36:20.360 --> 00:36:23.940 align:middle line:84% If there's some uncertainty about what my payoff, payout is, 00:36:23.940 --> 00:36:27.320 align:middle line:84% then we need to model that as different consequences. 00:36:27.320 --> 00:36:29.200 align:middle line:84% So just like we did here, basically, 00:36:29.200 --> 00:36:30.980 align:middle line:84% when I chose the risky investment, 00:36:30.980 --> 00:36:33.620 align:middle line:84% we didn't assign a utility directly to the investment. 00:36:33.620 --> 00:36:36.040 align:middle line:84% What we said is, well, we don't know how the investment 00:36:36.040 --> 00:36:37.060 align:middle line:90% is going to turn out. 00:36:37.060 --> 00:36:38.580 align:middle line:84% And we said, if I get different utilities, 00:36:38.580 --> 00:36:39.960 align:middle line:84% then those are really different outcomes 00:36:39.960 --> 00:36:41.173 align:middle line:90% or different consequences. 00:36:41.173 --> 00:36:42.340 align:middle line:90% So we need to split them up. 00:36:42.340 --> 00:36:45.280 align:middle line:84% So I would say, consequences always 00:36:45.280 --> 00:36:47.520 align:middle line:84% pin down and specify utilities exactly. 00:36:47.520 --> 00:36:50.208 align:middle line:84% If you feel like you're not sure what your utility is, 00:36:50.208 --> 00:36:52.000 align:middle line:84% then you should split those consequences up 00:36:52.000 --> 00:36:53.600 align:middle line:90% and represent them differently. 00:36:53.600 --> 00:36:56.080 align:middle line:84% AUDIENCE: What about in relation to other players? 00:36:56.080 --> 00:36:57.760 align:middle line:90% IAN BALL: Same thing. 00:36:57.760 --> 00:37:00.200 align:middle line:84% So if there's some consequence, if I'm not 00:37:00.200 --> 00:37:02.900 align:middle line:84% sure what the other player's payoff is, 00:37:02.900 --> 00:37:05.400 align:middle line:84% then we should represent that as two different consequences, 00:37:05.400 --> 00:37:07.317 align:middle line:84% where the other player gets different payoffs. 00:37:07.317 --> 00:37:09.930 align:middle line:84% Maybe I'll get the same payoff at those two consequences, 00:37:09.930 --> 00:37:11.930 align:middle line:84% but the other player will get different payoffs. 00:37:11.930 --> 00:37:14.910 align:middle line:90% 00:37:14.910 --> 00:37:17.410 align:middle line:84% And we'll see examples of that throughout the course. 00:37:17.410 --> 00:37:20.110 align:middle line:84% So I think this framework is actually much more general 00:37:20.110 --> 00:37:21.492 align:middle line:90% than it may at first seem. 00:37:21.492 --> 00:37:22.950 align:middle line:84% A lot of these assumptions-- oh, we 00:37:22.950 --> 00:37:24.367 align:middle line:84% know exactly what our payoffs are. 00:37:24.367 --> 00:37:27.910 align:middle line:84% We know all these things-- can capture much richer environments 00:37:27.910 --> 00:37:29.950 align:middle line:84% with the appropriate interpretation of what 00:37:29.950 --> 00:37:32.210 align:middle line:84% the information sets are, what Nature is, 00:37:32.210 --> 00:37:34.610 align:middle line:90% and what the consequences are. 00:37:34.610 --> 00:37:35.110 align:middle line:90% Yeah. 00:37:35.110 --> 00:37:38.030 align:middle line:84% And was there a question here or no? 00:37:38.030 --> 00:37:40.070 align:middle line:90% Maybe not. 00:37:40.070 --> 00:37:42.830 align:middle line:84% Well, I think there's one problem with this definition. 00:37:42.830 --> 00:37:45.362 align:middle line:84% And so we need a little more structure here. 00:37:45.362 --> 00:37:48.070 align:middle line:84% The problem with this definition is what if I had two information 00:37:48.070 --> 00:37:52.270 align:middle line:84% sets or two nodes-- let's take this-- 00:37:52.270 --> 00:37:54.212 align:middle line:84% that are in the same information set. 00:37:54.212 --> 00:37:56.670 align:middle line:84% But the actions that I'm allowed to take at those two nodes 00:37:56.670 --> 00:37:58.790 align:middle line:90% are different. 00:37:58.790 --> 00:38:00.990 align:middle line:84% Now we have a problem because we've implicitly 00:38:00.990 --> 00:38:02.827 align:middle line:84% assumed that I know what my actions are. 00:38:02.827 --> 00:38:04.410 align:middle line:84% If I didn't know what my actions were, 00:38:04.410 --> 00:38:06.120 align:middle line:84% then there'd be no way I could even take the action. 00:38:06.120 --> 00:38:06.953 align:middle line:90% So we're in trouble. 00:38:06.953 --> 00:38:09.740 align:middle line:84% So we've assumed I know what actions I can choose. 00:38:09.740 --> 00:38:13.227 align:middle line:84% But if we had two nodes where I had different feasible actions, 00:38:13.227 --> 00:38:15.560 align:middle line:84% then I would definitely know that I'm at different nodes 00:38:15.560 --> 00:38:17.477 align:middle line:84% because I could tell that at one of the nodes, 00:38:17.477 --> 00:38:19.620 align:middle line:84% I had a different action available. 00:38:19.620 --> 00:38:22.820 align:middle line:84% So we need to impose a restriction on the way we define 00:38:22.820 --> 00:38:25.740 align:middle line:84% information sets so that property doesn't-- that never 00:38:25.740 --> 00:38:26.400 align:middle line:90% occurs. 00:38:26.400 --> 00:38:30.500 align:middle line:90% 00:38:30.500 --> 00:38:34.600 align:middle line:84% So conditions on information sets-- 00:38:34.600 --> 00:38:40.380 align:middle line:90% 00:38:40.380 --> 00:38:42.722 align:middle line:84% and there's really two main conditions that we need. 00:38:42.722 --> 00:38:45.180 align:middle line:84% And the first main condition is capturing what I just said, 00:38:45.180 --> 00:38:50.740 align:middle line:90% that the player must have-- 00:38:50.740 --> 00:39:02.260 align:middle line:84% so I'll say player i has same feasible actions at all nodes 00:39:02.260 --> 00:39:04.030 align:middle line:90% in the same information set. 00:39:04.030 --> 00:39:15.810 align:middle line:90% 00:39:15.810 --> 00:39:18.330 align:middle line:84% So just to make it a little more concrete, 00:39:18.330 --> 00:39:21.890 align:middle line:84% if we want to put this in the same information set, 00:39:21.890 --> 00:39:25.450 align:middle line:84% this is OK here because player 2 is choosing between C and R 00:39:25.450 --> 00:39:27.770 align:middle line:84% at this node, and they're choosing between C and R 00:39:27.770 --> 00:39:28.810 align:middle line:90% at this node. 00:39:28.810 --> 00:39:32.050 align:middle line:84% But if I tried to add something else-- let's say, H, 00:39:32.050 --> 00:39:33.210 align:middle line:90% I stay home-- 00:39:33.210 --> 00:39:35.210 align:middle line:84% now we have a problem because when 00:39:35.210 --> 00:39:37.010 align:middle line:84% the player gets to this node, and he 00:39:37.010 --> 00:39:39.210 align:middle line:84% knows he's able to stay home, he knows 00:39:39.210 --> 00:39:40.770 align:middle line:90% that he must be at this node. 00:39:40.770 --> 00:39:43.110 align:middle line:84% And if he sees that staying home is not feasible, 00:39:43.110 --> 00:39:44.490 align:middle line:84% he knows he must be at this node, 00:39:44.490 --> 00:39:47.210 align:middle line:84% and therefore, those can't be in the same information set. 00:39:47.210 --> 00:39:49.530 align:middle line:90% So let me erase this. 00:39:49.530 --> 00:39:52.330 align:middle line:90% But that's all that this says. 00:39:52.330 --> 00:39:55.770 align:middle line:84% Notice if all of a player's information sets 00:39:55.770 --> 00:39:59.690 align:middle line:84% are singletons, well, then this is automatically satisfied. 00:39:59.690 --> 00:40:02.660 align:middle line:84% If I only have one node in my information set, then of course 00:40:02.660 --> 00:40:04.805 align:middle line:84% I have the same actions feasible at every node 00:40:04.805 --> 00:40:06.180 align:middle line:84% because there's only one of them. 00:40:06.180 --> 00:40:09.040 align:middle line:84% So this is what we might say vacuously true, automatically 00:40:09.040 --> 00:40:11.620 align:middle line:84% true with a singleton information set. 00:40:11.620 --> 00:40:13.560 align:middle line:84% But it has bite if my information 00:40:13.560 --> 00:40:16.318 align:middle line:90% set has multiple nodes. 00:40:16.318 --> 00:40:18.360 align:middle line:84% And then there's one more restriction that we put 00:40:18.360 --> 00:40:23.580 align:middle line:84% on and we say that the root-- remember this is the initial 00:40:23.580 --> 00:40:24.080 align:middle line:90% node-- 00:40:24.080 --> 00:40:27.005 align:middle line:90% 00:40:27.005 --> 00:40:29.140 align:middle line:84% is in a singleton information set. 00:40:29.140 --> 00:40:35.960 align:middle line:90% 00:40:35.960 --> 00:40:38.450 align:middle line:84% The idea would be if we don't know where the game starts, 00:40:38.450 --> 00:40:40.200 align:middle line:84% and we don't know where we are, then there 00:40:40.200 --> 00:40:42.560 align:middle line:84% must be some prior move that we didn't observe, 00:40:42.560 --> 00:40:44.960 align:middle line:84% and we should model that explicitly. 00:40:44.960 --> 00:40:47.440 align:middle line:84% So if you don't know where you start, then often the way 00:40:47.440 --> 00:40:50.040 align:middle line:84% we capture that is we say that Nature is initially 00:40:50.040 --> 00:40:52.500 align:middle line:84% making a move that determines where we start. 00:40:52.500 --> 00:40:56.060 align:middle line:84% But then Nature has a node that starts the game. 00:40:56.060 --> 00:40:58.240 align:middle line:84% So the root or the initial node must 00:40:58.240 --> 00:40:59.960 align:middle line:90% be in its own information set. 00:40:59.960 --> 00:41:03.420 align:middle line:84% And you see that that was satisfied here and here. 00:41:03.420 --> 00:41:08.480 align:middle line:90% 00:41:08.480 --> 00:41:14.136 align:middle line:90% A few final points-- 00:41:14.136 --> 00:41:18.320 align:middle line:90% let me check the time. 00:41:18.320 --> 00:41:22.440 align:middle line:84% So I've been a little sloppy in a few things in that I've 00:41:22.440 --> 00:41:24.050 align:middle line:84% been talking a lot about players, 00:41:24.050 --> 00:41:26.300 align:middle line:84% but Nature is really different than the other players. 00:41:26.300 --> 00:41:28.960 align:middle line:84% So we need to specify some different rules for Nature. 00:41:28.960 --> 00:41:31.260 align:middle line:84% So let me say, Nature is different. 00:41:31.260 --> 00:41:37.440 align:middle line:90% 00:41:37.440 --> 00:41:38.980 align:middle line:84% If you just look up at these trees, 00:41:38.980 --> 00:41:42.240 align:middle line:84% can you see any differences when we 00:41:42.240 --> 00:41:44.912 align:middle line:84% label a node to be Nature, rather than a strategic player, 00:41:44.912 --> 00:41:46.620 align:middle line:84% what's different when it's Nature's turn? 00:41:46.620 --> 00:41:47.320 align:middle line:90% Yeah? 00:41:47.320 --> 00:41:50.380 align:middle line:84% AUDIENCE: Nature works on probability. 00:41:50.380 --> 00:41:51.380 align:middle line:90% IAN BALL: Exactly right. 00:41:51.380 --> 00:41:54.680 align:middle line:84% So whereas a strategic player chooses an action at an edge, 00:41:54.680 --> 00:41:57.880 align:middle line:84% Nature just chooses probabilities. 00:41:57.880 --> 00:42:10.430 align:middle line:84% So edges have probabilities, not actions. 00:42:10.430 --> 00:42:12.370 align:middle line:84% And that's because Nature is not strategic. 00:42:12.370 --> 00:42:14.470 align:middle line:84% These just probabilities are given to us. 00:42:14.470 --> 00:42:17.590 align:middle line:84% With strategic players, we can't assume that a player is always 00:42:17.590 --> 00:42:20.055 align:middle line:84% going to go to the Celtics game with probability 0.2. 00:42:20.055 --> 00:42:21.430 align:middle line:84% That's the whole point what we're 00:42:21.430 --> 00:42:23.490 align:middle line:84% trying to figure out, whereas Nature, 00:42:23.490 --> 00:42:26.870 align:middle line:84% we do think of it like a lottery, like a roulette wheel. 00:42:26.870 --> 00:42:29.410 align:middle line:84% These probabilities are just fixed as part of the game. 00:42:29.410 --> 00:42:31.630 align:middle line:84% And all that nature does is it mechanically 00:42:31.630 --> 00:42:34.670 align:middle line:84% draws an edge according to those probabilities. 00:42:34.670 --> 00:42:37.530 align:middle line:84% Any other things you notice that are different about Nature? 00:42:37.530 --> 00:42:40.630 align:middle line:90% 00:42:40.630 --> 00:42:42.910 align:middle line:84% Well, one thing, Nature doesn't have payoffs. 00:42:42.910 --> 00:42:45.170 align:middle line:84% When we labeled payoffs at the end of the game, 00:42:45.170 --> 00:42:47.950 align:middle line:84% we only specified payoffs for the strategic players 00:42:47.950 --> 00:42:49.730 align:middle line:84% in the game, players 1 through n. 00:42:49.730 --> 00:42:51.850 align:middle line:90% We didn't label it for Nature. 00:42:51.850 --> 00:42:53.450 align:middle line:90% So Nature has no payoffs. 00:42:53.450 --> 00:42:58.480 align:middle line:90% 00:42:58.480 --> 00:42:59.980 align:middle line:84% And then the final thing is remember 00:42:59.980 --> 00:43:04.140 align:middle line:84% when I talked about information sets, I said, for each player i. 00:43:04.140 --> 00:43:07.060 align:middle line:84% What I meant by that is each strategic player i. 00:43:07.060 --> 00:43:09.220 align:middle line:84% Nature also doesn't have information sets. 00:43:09.220 --> 00:43:11.180 align:middle line:84% Wherever Nature is, they just mechanically 00:43:11.180 --> 00:43:12.640 align:middle line:90% draw their probabilities. 00:43:12.640 --> 00:43:15.663 align:middle line:84% There's no sense of what Nature knows about other moves. 00:43:15.663 --> 00:43:17.080 align:middle line:84% We don't need to worry about that. 00:43:17.080 --> 00:43:18.500 align:middle line:90% So we have no information sets. 00:43:18.500 --> 00:43:21.425 align:middle line:90% 00:43:21.425 --> 00:43:24.100 align:middle line:84% Or an equivalent way of saying it is every info set 00:43:24.100 --> 00:43:24.960 align:middle line:90% is a singleton. 00:43:24.960 --> 00:43:32.420 align:middle line:90% 00:43:32.420 --> 00:43:35.940 align:middle line:84% I guess this is convenient because I said the root must 00:43:35.940 --> 00:43:37.780 align:middle line:90% be a singleton information set. 00:43:37.780 --> 00:43:40.113 align:middle line:84% Well, if the root is a move by Nature, 00:43:40.113 --> 00:43:42.780 align:middle line:84% technically, I should say Nature has a Singleton information set 00:43:42.780 --> 00:43:45.240 align:middle line:84% if I want to be consistent with this definition. 00:43:45.240 --> 00:43:47.040 align:middle line:84% But that's nothing about interpretation. 00:43:47.040 --> 00:43:49.940 align:middle line:84% It's just we allow Nature to start the game. 00:43:49.940 --> 00:43:52.100 align:middle line:90% And then I guess, we often-- 00:43:52.100 --> 00:43:53.740 align:middle line:84% it can sometimes get a bit cumbersome 00:43:53.740 --> 00:43:55.240 align:middle line:90% to keep writing Nature. 00:43:55.240 --> 00:44:00.030 align:middle line:84% So often we just draw a circle to indicate Nature's nodes. 00:44:00.030 --> 00:44:06.410 align:middle line:90% 00:44:06.410 --> 00:44:09.490 align:middle line:84% So for strategic players, we have a filled-in circle. 00:44:09.490 --> 00:44:13.110 align:middle line:84% And for Nature, we have a hollow or empty circle. 00:44:13.110 --> 00:44:17.090 align:middle line:90% 00:44:17.090 --> 00:44:17.590 align:middle line:90% Great. 00:44:17.590 --> 00:44:20.470 align:middle line:84% So now that we've talked about extensive form games, 00:44:20.470 --> 00:44:22.090 align:middle line:84% I'd like to move on to strategies. 00:44:22.090 --> 00:44:24.665 align:middle line:84% Unless there's any questions on extensive form games, 00:44:24.665 --> 00:44:25.790 align:middle line:90% we're going to move on now. 00:44:25.790 --> 00:44:36.776 align:middle line:90% 00:44:36.776 --> 00:44:39.190 align:middle line:84% And maybe I'll say this is kind of implicit, 00:44:39.190 --> 00:44:41.970 align:middle line:84% but Nature also has no strategies, 00:44:41.970 --> 00:44:43.390 align:middle line:90% which we're about to define. 00:44:43.390 --> 00:44:47.250 align:middle line:90% 00:44:47.250 --> 00:44:48.630 align:middle line:90% So what is a strategy? 00:44:48.630 --> 00:44:53.010 align:middle line:90% 00:44:53.010 --> 00:44:56.200 align:middle line:84% Well, I want you to put yourself in the shoes of someone 00:44:56.200 --> 00:44:59.760 align:middle line:90% approaching this game. 00:44:59.760 --> 00:45:01.760 align:middle line:84% Before the game has started, you're 00:45:01.760 --> 00:45:04.720 align:middle line:84% contemplating how you're going to play in this game. 00:45:04.720 --> 00:45:06.620 align:middle line:84% And the strategy is the answer to that. 00:45:06.620 --> 00:45:13.700 align:middle line:84% The strategy is a complete contingent plan. 00:45:13.700 --> 00:45:20.440 align:middle line:90% 00:45:20.440 --> 00:45:22.520 align:middle line:90% So it's a plan of action. 00:45:22.520 --> 00:45:24.020 align:middle line:90% You analyze it at the beginning. 00:45:24.020 --> 00:45:26.380 align:middle line:84% You're an outsider, or you're one of the players, 00:45:26.380 --> 00:45:29.580 align:middle line:84% but you're kind of evaluating and assessing the game. 00:45:29.580 --> 00:45:31.080 align:middle line:84% And you look at all the places where 00:45:31.080 --> 00:45:34.100 align:middle line:84% you might be called upon to move, and you plan ahead. 00:45:34.100 --> 00:45:36.200 align:middle line:84% You make a plan for what actions you're 00:45:36.200 --> 00:45:40.000 align:middle line:90% going to take at every node. 00:45:40.000 --> 00:45:41.540 align:middle line:90% It's a complete plan. 00:45:41.540 --> 00:45:43.520 align:middle line:84% You have to specify what you're going 00:45:43.520 --> 00:45:47.240 align:middle line:84% to do at every possible contingency that arises. 00:45:47.240 --> 00:45:52.040 align:middle line:84% So I'm very often asked in section and office hours, 00:45:52.040 --> 00:45:55.230 align:middle line:84% do I really have to specify this contingency here? 00:45:55.230 --> 00:45:56.730 align:middle line:84% I've never been asked that question, 00:45:56.730 --> 00:45:57.855 align:middle line:90% and the answer has been no. 00:45:57.855 --> 00:45:58.862 align:middle line:90% It's always been yes. 00:45:58.862 --> 00:46:01.070 align:middle line:84% So if you're unsure, do I really need to specify this 00:46:01.070 --> 00:46:02.150 align:middle line:90% as part of a strategy? 00:46:02.150 --> 00:46:04.150 align:middle line:84% Unless you're different than every other student 00:46:04.150 --> 00:46:06.608 align:middle line:84% I've had before, the answer is yes, you have to specify it. 00:46:06.608 --> 00:46:09.227 align:middle line:90% So when in doubt, specify it. 00:46:09.227 --> 00:46:11.310 align:middle line:84% It turns out that's going to make your life-- that 00:46:11.310 --> 00:46:12.530 align:middle line:90% kind of makes your life easier. 00:46:12.530 --> 00:46:15.010 align:middle line:84% But I think we'll see that as we go throughout the course. 00:46:15.010 --> 00:46:18.190 align:middle line:84% But I'll just say that maybe my answer is yes, you 00:46:18.190 --> 00:46:24.950 align:middle line:84% have to include it, everywhere, even when 00:46:24.950 --> 00:46:28.110 align:middle line:90% you think you don't need to. 00:46:28.110 --> 00:46:30.335 align:middle line:84% And contingent is capturing the fact 00:46:30.335 --> 00:46:31.710 align:middle line:84% that you're going to learn things 00:46:31.710 --> 00:46:33.550 align:middle line:90% over the course of the game. 00:46:33.550 --> 00:46:36.430 align:middle line:84% So if, for instance, these two nodes 00:46:36.430 --> 00:46:38.310 align:middle line:84% are on different information sets, 00:46:38.310 --> 00:46:41.790 align:middle line:84% then player 2 is going to learn what player 1 does. 00:46:41.790 --> 00:46:43.510 align:middle line:84% And therefore, their plan is implicitly 00:46:43.510 --> 00:46:44.890 align:middle line:90% going to be contingent plan. 00:46:44.890 --> 00:46:47.650 align:middle line:84% It's not going to say, I'm going to go to the Celtics game, 00:46:47.650 --> 00:46:50.030 align:middle line:84% or I'm going to go to the Red Sox game. 00:46:50.030 --> 00:46:53.280 align:middle line:84% It might say something like, I'm going to go to the Celtics game 00:46:53.280 --> 00:46:55.580 align:middle line:84% if my friend goes to the Celtics game, 00:46:55.580 --> 00:46:57.240 align:middle line:84% and I'm going to go to the Red Sox game 00:46:57.240 --> 00:46:59.080 align:middle line:84% if my friend goes to the Red Sox game. 00:46:59.080 --> 00:47:03.340 align:middle line:84% And that's what's capturing the contingency in a strategy. 00:47:03.340 --> 00:47:05.040 align:middle line:90% So that's the intuitive idea. 00:47:05.040 --> 00:47:06.160 align:middle line:90% What is it formally? 00:47:06.160 --> 00:47:09.500 align:middle line:90% 00:47:09.500 --> 00:47:13.800 align:middle line:84% Formally, it specifies an action at every information set. 00:47:13.800 --> 00:47:25.940 align:middle line:90% 00:47:25.940 --> 00:47:31.380 align:middle line:84% So often a good way to write a strategy is 00:47:31.380 --> 00:47:34.300 align:middle line:84% to put out some spaces for each of the information 00:47:34.300 --> 00:47:35.660 align:middle line:90% sets a player has. 00:47:35.660 --> 00:47:39.220 align:middle line:84% So let's say a player has four information sets. 00:47:39.220 --> 00:47:42.140 align:middle line:84% Maybe this is information set 1, information 00:47:42.140 --> 00:47:47.580 align:middle line:84% set 2, information set 3, and information set 4. 00:47:47.580 --> 00:47:51.100 align:middle line:84% And the way you specify a strategy is to specify an action 00:47:51.100 --> 00:47:52.900 align:middle line:90% here-- maybe I'll call it A1-- 00:47:52.900 --> 00:47:56.120 align:middle line:84% an action here, an action here, an action here. 00:47:56.120 --> 00:47:58.900 align:middle line:90% 00:47:58.900 --> 00:48:00.988 align:middle line:84% So how many information sets I have 00:48:00.988 --> 00:48:02.780 align:middle line:84% are going to be determined by the structure 00:48:02.780 --> 00:48:04.380 align:middle line:90% of the extensive form game. 00:48:04.380 --> 00:48:07.020 align:middle line:84% But once I have the extensive form game written down, 00:48:07.020 --> 00:48:09.660 align:middle line:84% I can list out all the information 00:48:09.660 --> 00:48:11.500 align:middle line:90% sets for a given player. 00:48:11.500 --> 00:48:13.500 align:middle line:84% And then I can specify a strategy 00:48:13.500 --> 00:48:16.460 align:middle line:84% by specifying what action the player will take at each 00:48:16.460 --> 00:48:18.037 align:middle line:90% of these information sets. 00:48:18.037 --> 00:48:19.620 align:middle line:84% So the interpretation of this strategy 00:48:19.620 --> 00:48:23.420 align:middle line:84% would be if I find myself in information set 1, 00:48:23.420 --> 00:48:25.380 align:middle line:90% I will take action A1. 00:48:25.380 --> 00:48:28.060 align:middle line:84% If I find myself at information set I2, 00:48:28.060 --> 00:48:30.500 align:middle line:84% I will take action A2, and so on. 00:48:30.500 --> 00:48:34.060 align:middle line:84% And of course, the action that I take at information 00:48:34.060 --> 00:48:38.440 align:middle line:84% set I1 must be a feasible action at that information set. 00:48:38.440 --> 00:48:43.900 align:middle line:84% If it was infeasible, then it wouldn't make sense. 00:48:43.900 --> 00:48:46.020 align:middle line:90% But here's a question. 00:48:46.020 --> 00:48:50.410 align:middle line:84% I said this action must be feasible at an information set. 00:48:50.410 --> 00:48:52.570 align:middle line:84% But an information set has many nodes. 00:48:52.570 --> 00:48:53.790 align:middle line:90% So how does that make sense? 00:48:53.790 --> 00:48:58.370 align:middle line:84% Isn't feasibility a property of a node, not an information set? 00:48:58.370 --> 00:48:58.925 align:middle line:90% Yeah? 00:48:58.925 --> 00:49:01.050 align:middle line:84% AUDIENCE: One of the properties of information sets 00:49:01.050 --> 00:49:03.510 align:middle line:84% was that every node had to have the same feasible actions. 00:49:03.510 --> 00:49:04.260 align:middle line:90% IAN BALL: Exactly. 00:49:04.260 --> 00:49:06.210 align:middle line:84% So because every node in information set 00:49:06.210 --> 00:49:09.770 align:middle line:84% has the same feasible actions, I can think of the feasible 00:49:09.770 --> 00:49:12.050 align:middle line:84% actions at an information set because it's 00:49:12.050 --> 00:49:14.870 align:middle line:84% the same whichever node in that information set I look at. 00:49:14.870 --> 00:49:16.690 align:middle line:84% And therefore, it's meaningful to say, 00:49:16.690 --> 00:49:18.650 align:middle line:84% I'm going to take action A1, which 00:49:18.650 --> 00:49:20.790 align:middle line:84% is feasible at this information set. 00:49:20.790 --> 00:49:23.130 align:middle line:84% It just means it's feasible at every single node 00:49:23.130 --> 00:49:25.730 align:middle line:84% in the information set that I might be at. 00:49:25.730 --> 00:49:27.270 align:middle line:84% And again, this is a contingency. 00:49:27.270 --> 00:49:28.850 align:middle line:90% We often say, if it-- 00:49:28.850 --> 00:49:29.510 align:middle line:90% we'll do this. 00:49:29.510 --> 00:49:31.007 align:middle line:90% But if it rains, we'll do this. 00:49:31.007 --> 00:49:32.590 align:middle line:84% That's exactly what this is capturing. 00:49:32.590 --> 00:49:33.230 align:middle line:90% We'll do this. 00:49:33.230 --> 00:49:34.522 align:middle line:90% But if it rains, we'll do this. 00:49:34.522 --> 00:49:35.830 align:middle line:90% And if it snows, we'll do this. 00:49:35.830 --> 00:49:38.490 align:middle line:84% And if it's sunny, we'll do this. 00:49:38.490 --> 00:49:44.570 align:middle line:90% So let's now take-- 00:49:44.570 --> 00:49:45.910 align:middle line:90% let's go over here. 00:49:45.910 --> 00:49:55.800 align:middle line:90% 00:49:55.800 --> 00:49:58.200 align:middle line:84% So we've defined an extensive form game. 00:49:58.200 --> 00:49:59.560 align:middle line:90% We've defined a strategy. 00:49:59.560 --> 00:50:02.660 align:middle line:84% I now want to describe what a strategic form game is. 00:50:02.660 --> 00:50:03.900 align:middle line:90% That's actually really easy. 00:50:03.900 --> 00:50:05.280 align:middle line:84% But then what's harder is showing 00:50:05.280 --> 00:50:07.880 align:middle line:84% how we move from an extensive form game 00:50:07.880 --> 00:50:10.280 align:middle line:84% to an equivalent strategic form game. 00:50:10.280 --> 00:50:13.220 align:middle line:84% So let's first talk about a strategic form game. 00:50:13.220 --> 00:50:21.680 align:middle line:90% 00:50:21.680 --> 00:50:23.640 align:middle line:84% And the idea is, well, look, we wrote down 00:50:23.640 --> 00:50:26.320 align:middle line:84% this really complicated extensive form game. 00:50:26.320 --> 00:50:29.160 align:middle line:84% But once we understand the concept of a strategy, 00:50:29.160 --> 00:50:31.360 align:middle line:84% we can really simplify this game a lot. 00:50:31.360 --> 00:50:32.880 align:middle line:90% What happens in this game? 00:50:32.880 --> 00:50:34.840 align:middle line:90% Each player chooses a strategy. 00:50:34.840 --> 00:50:36.760 align:middle line:84% They choose a complete contingent plan. 00:50:36.760 --> 00:50:39.140 align:middle line:84% And then based on those complete contingent plans, 00:50:39.140 --> 00:50:42.600 align:middle line:84% we can calculate what's going to happen in the game. 00:50:42.600 --> 00:50:44.600 align:middle line:84% So the strategic form representation 00:50:44.600 --> 00:50:47.510 align:middle line:84% is a way of reducing the game simply 00:50:47.510 --> 00:50:49.030 align:middle line:90% to its strategic components. 00:50:49.030 --> 00:50:50.350 align:middle line:90% So what do we specify? 00:50:50.350 --> 00:50:52.650 align:middle line:84% We specify the players as before. 00:50:52.650 --> 00:50:56.110 align:middle line:90% 00:50:56.110 --> 00:50:57.770 align:middle line:90% We specify the strategies. 00:50:57.770 --> 00:51:00.990 align:middle line:90% 00:51:00.990 --> 00:51:03.130 align:middle line:84% Well, well call them-- we'll have sets. 00:51:03.130 --> 00:51:07.150 align:middle line:90% So we have S1 up to Sn. 00:51:07.150 --> 00:51:11.070 align:middle line:84% Big S1 is the set of strategies for player 1. 00:51:11.070 --> 00:51:13.430 align:middle line:84% So every element, every member of this set, 00:51:13.430 --> 00:51:15.710 align:middle line:84% is a particular strategy for player 1. 00:51:15.710 --> 00:51:18.430 align:middle line:84% We list all those strategies together and put them together 00:51:18.430 --> 00:51:20.670 align:middle line:90% in a set called Big S1. 00:51:20.670 --> 00:51:24.470 align:middle line:84% And we might use notation like little s1 00:51:24.470 --> 00:51:26.830 align:middle line:84% is a particular strategy that lives 00:51:26.830 --> 00:51:28.810 align:middle line:90% inside this set of strategies. 00:51:28.810 --> 00:51:32.670 align:middle line:90% 00:51:32.670 --> 00:51:35.350 align:middle line:84% And then it's often going to be helpful to have notation 00:51:35.350 --> 00:51:39.110 align:middle line:84% for profiles of strategies because if we want to understand 00:51:39.110 --> 00:51:40.830 align:middle line:84% what's going to happen in a game, 00:51:40.830 --> 00:51:43.590 align:middle line:84% it's not enough just to know player 1's strategy. 00:51:43.590 --> 00:51:45.480 align:middle line:84% We need to know, what is player 1 doing? 00:51:45.480 --> 00:51:46.700 align:middle line:90% What is their strategy? 00:51:46.700 --> 00:51:48.300 align:middle line:90% What is player 2's strategy? 00:51:48.300 --> 00:51:50.860 align:middle line:84% All the way up to what is player n's strategy? 00:51:50.860 --> 00:51:59.500 align:middle line:84% So we're going to define a set, capital S is equal to S1 times 00:51:59.500 --> 00:52:00.360 align:middle line:90% up to Sn. 00:52:00.360 --> 00:52:01.380 align:middle line:90% So what is this? 00:52:01.380 --> 00:52:06.460 align:middle line:84% This is the set of all tuples or pairs, triples, however 00:52:06.460 --> 00:52:08.100 align:middle line:90% many we have, of strategies. 00:52:08.100 --> 00:52:09.940 align:middle line:84% So as an example, an element of this 00:52:09.940 --> 00:52:17.020 align:middle line:84% might be something like S equals S1 up to Sn. 00:52:17.020 --> 00:52:18.720 align:middle line:90% So this is the element symbol. 00:52:18.720 --> 00:52:23.360 align:middle line:84% So what I'm saying is S here is a strategy profile. 00:52:23.360 --> 00:52:27.060 align:middle line:90% 00:52:27.060 --> 00:52:29.360 align:middle line:84% So this notation will come up throughout the year. 00:52:29.360 --> 00:52:32.100 align:middle line:84% So I think it's really important to make sure we understand it. 00:52:32.100 --> 00:52:34.900 align:middle line:84% Little s is a profile of strategies. 00:52:34.900 --> 00:52:37.500 align:middle line:84% It specifies a strategy for player 1, 00:52:37.500 --> 00:52:39.580 align:middle line:84% a strategy for player 2, all the way up 00:52:39.580 --> 00:52:41.860 align:middle line:90% to a strategy for player n. 00:52:41.860 --> 00:52:47.170 align:middle line:84% So sn is the strategy by player n in the particular strategy 00:52:47.170 --> 00:52:49.210 align:middle line:90% profile s. 00:52:49.210 --> 00:52:52.730 align:middle line:84% This strategy profile s lives in this set big S 00:52:52.730 --> 00:52:56.410 align:middle line:84% This is the collection of all strategy profiles in the game. 00:52:56.410 --> 00:52:56.950 align:middle line:90% Yeah? 00:52:56.950 --> 00:52:58.450 align:middle line:84% AUDIENCE: So when we say that, we're 00:52:58.450 --> 00:53:02.130 align:middle line:84% saying it's every combination of every strategy any player could 00:53:02.130 --> 00:53:02.630 align:middle line:90% think of. 00:53:02.630 --> 00:53:03.730 align:middle line:90% IAN BALL: Exactly right, yeah. 00:53:03.730 --> 00:53:05.397 align:middle line:84% So that's what this product notation is. 00:53:05.397 --> 00:53:07.350 align:middle line:84% We choose one strategy for player 1. 00:53:07.350 --> 00:53:09.850 align:middle line:84% We choose one strategy for player 2, all the way up to n. 00:53:09.850 --> 00:53:12.170 align:middle line:84% And any collection of those choices 00:53:12.170 --> 00:53:14.630 align:middle line:84% is going to give us the strategy profile. 00:53:14.630 --> 00:53:16.930 align:middle line:90% Yeah. 00:53:16.930 --> 00:53:19.470 align:middle line:84% And then the last thing is just payoffs. 00:53:19.470 --> 00:53:25.410 align:middle line:90% 00:53:25.410 --> 00:53:32.850 align:middle line:84% And payoffs, we need a function ui from S to R for i 00:53:32.850 --> 00:53:33.870 align:middle line:90% equals 1 to n. 00:53:33.870 --> 00:53:37.930 align:middle line:90% 00:53:37.930 --> 00:53:40.050 align:middle line:84% Now, some people are tempted to put an i here 00:53:40.050 --> 00:53:41.640 align:middle line:90% because they see an i here. 00:53:41.640 --> 00:53:42.640 align:middle line:90% But that would be wrong. 00:53:42.640 --> 00:53:44.480 align:middle line:84% If we put an eye here, that would 00:53:44.480 --> 00:53:48.583 align:middle line:84% mean that player i's payoff only depends on player i's strategy. 00:53:48.583 --> 00:53:50.000 align:middle line:84% If that were the case, we wouldn't 00:53:50.000 --> 00:53:51.400 align:middle line:90% need to do game theory at all. 00:53:51.400 --> 00:53:53.327 align:middle line:84% That's just an individual decision problem. 00:53:53.327 --> 00:53:55.160 align:middle line:84% What makes it interesting is that the payoff 00:53:55.160 --> 00:53:58.980 align:middle line:84% that I get as player i depends not only on what I do, 00:53:58.980 --> 00:54:00.720 align:middle line:84% but also on what everyone else does. 00:54:00.720 --> 00:54:02.800 align:middle line:84% And that's why there's no i here. 00:54:02.800 --> 00:54:05.000 align:middle line:84% And the payoff of player i is a function 00:54:05.000 --> 00:54:06.500 align:middle line:90% of the strategy profile. 00:54:06.500 --> 00:54:08.720 align:middle line:84% It says, for instance, if there's two players, 00:54:08.720 --> 00:54:13.040 align:middle line:84% if player 1 chooses strategy S1 and player 2 00:54:13.040 --> 00:54:15.200 align:middle line:84% chooses strategy S2, what is going 00:54:15.200 --> 00:54:18.405 align:middle line:84% to be the payoff for player i from that combination 00:54:18.405 --> 00:54:19.780 align:middle line:84% of strategies, or in other words, 00:54:19.780 --> 00:54:21.180 align:middle line:90% from that strategy profile? 00:54:21.180 --> 00:54:25.560 align:middle line:90% 00:54:25.560 --> 00:54:27.900 align:middle line:84% So strategic form, it's really simple, 00:54:27.900 --> 00:54:30.600 align:middle line:84% but you have to make sure you write down the set of strategies 00:54:30.600 --> 00:54:33.600 align:middle line:84% correctly, and you have to make sure you write down the payoffs 00:54:33.600 --> 00:54:34.300 align:middle line:90% correctly. 00:54:34.300 --> 00:54:35.800 align:middle line:84% So what we're now going to describe 00:54:35.800 --> 00:54:38.640 align:middle line:84% is how we can move when we're given an extensive form 00:54:38.640 --> 00:54:41.970 align:middle line:84% game, how we can reduce it to a strategic form game, 00:54:41.970 --> 00:54:45.070 align:middle line:84% reduce it to its strategic form representation. 00:54:45.070 --> 00:54:50.130 align:middle line:84% So let me go to a new board over here. 00:54:50.130 --> 00:54:50.930 align:middle line:90% Do we have one? 00:54:50.930 --> 00:55:12.430 align:middle line:90% 00:55:12.430 --> 00:55:14.570 align:middle line:84% So what's our procedure for going from-- 00:55:14.570 --> 00:55:26.150 align:middle line:90% 00:55:26.150 --> 00:55:27.610 align:middle line:90% so it's not always the case. 00:55:27.610 --> 00:55:29.710 align:middle line:84% But often when we start by considering 00:55:29.710 --> 00:55:33.390 align:middle line:84% some strategic situation, we write down all the details 00:55:33.390 --> 00:55:35.950 align:middle line:84% in the extensive form to make sure we specify everything 00:55:35.950 --> 00:55:37.270 align:middle line:90% exactly right. 00:55:37.270 --> 00:55:39.530 align:middle line:84% And then when it's time to analyze the game, 00:55:39.530 --> 00:55:42.750 align:middle line:84% it's often more convenient to reduce this extensive form 00:55:42.750 --> 00:55:45.470 align:middle line:84% to this more compact and convenient strategic form. 00:55:45.470 --> 00:55:47.730 align:middle line:84% But if you try to jump right to the strategic form, 00:55:47.730 --> 00:55:48.610 align:middle line:90% you might make a mistake. 00:55:48.610 --> 00:55:50.870 align:middle line:84% So it's good to be really careful about how you set up 00:55:50.870 --> 00:55:53.710 align:middle line:84% the extensive form and then convert things 00:55:53.710 --> 00:55:54.738 align:middle line:90% to the strategic form. 00:55:54.738 --> 00:55:55.530 align:middle line:90% So how do we do it? 00:55:55.530 --> 00:56:03.430 align:middle line:84% Well, there's two steps or two hard steps. 00:56:03.430 --> 00:56:06.072 align:middle line:84% One step is just look at who the players are. 00:56:06.072 --> 00:56:07.030 align:middle line:90% But that's pretty easy. 00:56:07.030 --> 00:56:08.770 align:middle line:84% Once we have an extensive form game, 00:56:08.770 --> 00:56:10.270 align:middle line:84% we see who all the players are, and the players 00:56:10.270 --> 00:56:11.312 align:middle line:90% are going to be the same. 00:56:11.312 --> 00:56:12.350 align:middle line:90% So that's easy. 00:56:12.350 --> 00:56:16.190 align:middle line:84% The hard thing, or the two hard steps, 00:56:16.190 --> 00:56:20.890 align:middle line:84% are that we have to define the strategy sets. 00:56:20.890 --> 00:56:26.230 align:middle line:90% 00:56:26.230 --> 00:56:27.768 align:middle line:84% So we'll do an example in a second. 00:56:27.768 --> 00:56:29.310 align:middle line:84% But a strategy set would be something 00:56:29.310 --> 00:56:32.213 align:middle line:84% like a collection of all vectors that look like this. 00:56:32.213 --> 00:56:34.630 align:middle line:84% That would be an example of what the strategy set will be. 00:56:34.630 --> 00:56:36.430 align:middle line:84% But to define it correctly, you have 00:56:36.430 --> 00:56:39.245 align:middle line:84% to make sure you specify all the information sets correctly 00:56:39.245 --> 00:56:40.620 align:middle line:84% and all the possible actions that 00:56:40.620 --> 00:56:43.260 align:middle line:84% can be chosen at those information sets. 00:56:43.260 --> 00:56:54.780 align:middle line:84% And then the second step is to compute the payoff functions 00:56:54.780 --> 00:57:02.640 align:middle line:84% ui from S to R. That is, once I've defined the strategy sets, 00:57:02.640 --> 00:57:06.540 align:middle line:84% I still have to say, if you give me a strategy profile, 00:57:06.540 --> 00:57:09.140 align:middle line:84% specifying a complete contingent plan for each 00:57:09.140 --> 00:57:11.500 align:middle line:84% of the players, what is the payoff 00:57:11.500 --> 00:57:14.740 align:middle line:84% that player i is going to get from that profile? 00:57:14.740 --> 00:57:16.020 align:middle line:90% So why is this hard? 00:57:16.020 --> 00:57:18.620 align:middle line:84% It's hard because, from the extensive form, 00:57:18.620 --> 00:57:22.460 align:middle line:84% what we know is the payoff each player gets from a consequence, 00:57:22.460 --> 00:57:24.540 align:middle line:84% from a terminal node in the game. 00:57:24.540 --> 00:57:26.620 align:middle line:84% And now we have to compute what payoff 00:57:26.620 --> 00:57:28.900 align:middle line:84% they get from a strategy profile, which 00:57:28.900 --> 00:57:31.300 align:middle line:84% is not the same thing as a consequence. 00:57:31.300 --> 00:57:32.640 align:middle line:90% So how do we do that? 00:57:32.640 --> 00:57:35.610 align:middle line:84% Let me give you a maybe somewhat cryptic formula. 00:57:35.610 --> 00:57:37.870 align:middle line:84% And then we'll do an example to see how we do it. 00:57:37.870 --> 00:57:42.010 align:middle line:84% But the key is that a strategy profile 00:57:42.010 --> 00:57:51.810 align:middle line:84% s, if we focus on a particular strategy profile 00:57:51.810 --> 00:57:56.170 align:middle line:84% s, that is going to induce in the extensive form game 00:57:56.170 --> 00:57:59.170 align:middle line:90% a lottery over terminal nodes. 00:57:59.170 --> 00:58:02.850 align:middle line:84% It's going to induce a lottery over consequences. 00:58:02.850 --> 00:58:07.010 align:middle line:90% So maybe I'll write z of s. 00:58:07.010 --> 00:58:08.690 align:middle line:90% And I'll show how that happens. 00:58:08.690 --> 00:58:13.530 align:middle line:84% But a strategy profile induces a lottery over consequences. 00:58:13.530 --> 00:58:16.270 align:middle line:84% And once we can compute that lottery over consequences, 00:58:16.270 --> 00:58:19.410 align:middle line:84% we can compute each player's expected utility 00:58:19.410 --> 00:58:21.990 align:middle line:84% from that lottery over consequences, because remember, 00:58:21.990 --> 00:58:24.970 align:middle line:84% we've already defined utilities over consequences. 00:58:24.970 --> 00:58:27.050 align:middle line:84% And that expected utility is going 00:58:27.050 --> 00:58:30.290 align:middle line:84% to be the new definition of the utility function 00:58:30.290 --> 00:58:32.272 align:middle line:84% that we assign to this strategy profile. 00:58:32.272 --> 00:58:33.230 align:middle line:90% So we'll do an example. 00:58:33.230 --> 00:58:35.680 align:middle line:84% But the step is a strategy profile 00:58:35.680 --> 00:58:38.720 align:middle line:84% determines a lottery over consequences, which allows us 00:58:38.720 --> 00:58:40.500 align:middle line:90% to compute expected utilities. 00:58:40.500 --> 00:58:43.040 align:middle line:84% And those are going to be the payoffs in our strategic form 00:58:43.040 --> 00:58:43.600 align:middle line:90% game. 00:58:43.600 --> 00:58:45.660 align:middle line:90% So let's do an example. 00:58:45.660 --> 00:58:48.720 align:middle line:90% 00:58:48.720 --> 00:58:52.480 align:middle line:90% Let me go here. 00:58:52.480 --> 00:58:54.260 align:middle line:90% And we'll end with this example. 00:58:54.260 --> 00:58:59.440 align:middle line:90% 00:58:59.440 --> 00:59:02.320 align:middle line:84% So we're going to think of the Boston game we did before. 00:59:02.320 --> 00:59:03.840 align:middle line:84% But it's a little artificial just 00:59:03.840 --> 00:59:06.640 align:middle line:84% to illustrate ideas that Nature is 00:59:06.640 --> 00:59:09.480 align:middle line:84% going to flip a coin to determine the order in which we 00:59:09.480 --> 00:59:11.240 align:middle line:90% move in this game. 00:59:11.240 --> 00:59:17.120 align:middle line:84% So first, Nature is going to flip a coin, say 1/2, 1/2, 00:59:17.120 --> 00:59:18.740 align:middle line:90% to give myself some more space. 00:59:18.740 --> 00:59:23.680 align:middle line:90% 00:59:23.680 --> 00:59:28.440 align:middle line:84% And if the coin is, say, heads, then player 1 00:59:28.440 --> 00:59:31.880 align:middle line:84% is going to move first, and their move 00:59:31.880 --> 00:59:33.870 align:middle line:90% is not going to be observed. 00:59:33.870 --> 00:59:35.830 align:middle line:90% So how do we represent this? 00:59:35.830 --> 00:59:37.670 align:middle line:90% Player 1 moves first. 00:59:37.670 --> 00:59:41.430 align:middle line:90% They choose Celtics or Red Sox. 00:59:41.430 --> 00:59:43.550 align:middle line:90% Their move is not observed. 00:59:43.550 --> 00:59:46.830 align:middle line:90% So that means player 2-- 00:59:46.830 --> 00:59:49.050 align:middle line:90% this is player 2. 00:59:49.050 --> 00:59:51.950 align:middle line:84% And we put a dotted line here to indicate player 2 does not 00:59:51.950 --> 00:59:54.150 align:middle line:90% observe the move by player 1. 00:59:54.150 --> 00:59:59.602 align:middle line:84% And then player 2 chooses Celtics or Red Sox. 00:59:59.602 --> 01:00:01.310 align:middle line:84% And you can see, it's going to get pretty 01:00:01.310 --> 01:00:03.358 align:middle line:84% messy to write all the payoffs, so for now, I'm 01:00:03.358 --> 01:00:04.650 align:middle line:90% not going to write the payoffs. 01:00:04.650 --> 01:00:08.190 align:middle line:84% But the payoffs will be the same as in the other game. 01:00:08.190 --> 01:00:12.310 align:middle line:84% And if we do it the other way, let's say if nature flips tails, 01:00:12.310 --> 01:00:15.190 align:middle line:84% then player 2 is going to move first. 01:00:15.190 --> 01:00:19.110 align:middle line:84% But then their move is going to be observed by player 1 01:00:19.110 --> 01:00:21.670 align:middle line:90% So now we have player 2 here. 01:00:21.670 --> 01:00:27.430 align:middle line:84% They're going to choose between Celtics and Red Sox. 01:00:27.430 --> 01:00:30.790 align:middle line:84% Now it's player 1's turn to move. 01:00:30.790 --> 01:00:34.980 align:middle line:84% We don't draw a dotted line here because player 1 observes 01:00:34.980 --> 01:00:36.200 align:middle line:90% what player 2 does. 01:00:36.200 --> 01:00:39.140 align:middle line:90% 01:00:39.140 --> 01:00:42.500 align:middle line:90% And then we have CR, CR. 01:00:42.500 --> 01:00:44.548 align:middle line:84% So let's make sure we understand the game. 01:00:44.548 --> 01:00:45.840 align:middle line:90% Nature is going to flip a coin. 01:00:45.840 --> 01:00:46.900 align:middle line:90% These are the rules. 01:00:46.900 --> 01:00:50.220 align:middle line:84% If the coin is heads, player 1 decides first, 01:00:50.220 --> 01:00:52.060 align:middle line:84% then player 2 decides without having 01:00:52.060 --> 01:00:54.580 align:middle line:90% observed what player 1 does. 01:00:54.580 --> 01:00:58.080 align:middle line:84% If Nature flips tails, then player 2 goes first, 01:00:58.080 --> 01:01:01.400 align:middle line:84% and then player 1 goes after observing what player 2 did. 01:01:01.400 --> 01:01:02.580 align:middle line:90% Yes? 01:01:02.580 --> 01:01:04.700 align:middle line:84% AUDIENCE: So here then, does player 2 01:01:04.700 --> 01:01:07.593 align:middle line:90% know if they're going first? 01:01:07.593 --> 01:01:08.260 align:middle line:90% IAN BALL: Great. 01:01:08.260 --> 01:01:10.100 align:middle line:84% So let's look at the information sets. 01:01:10.100 --> 01:01:15.020 align:middle line:84% So how many information sets does player 2 have in this game? 01:01:15.020 --> 01:01:17.220 align:middle line:90% I'll open it up to everyone. 01:01:17.220 --> 01:01:17.720 align:middle line:90% 2. 01:01:17.720 --> 01:01:22.000 align:middle line:84% We have information set here and an information set here. 01:01:22.000 --> 01:01:23.920 align:middle line:84% Remember, this is actually an information set. 01:01:23.920 --> 01:01:25.040 align:middle line:90% I just haven't drawn it. 01:01:25.040 --> 01:01:27.420 align:middle line:84% So once we write out all the information sets explicitly, 01:01:27.420 --> 01:01:29.320 align:middle line:84% we can always answer questions like this. 01:01:29.320 --> 01:01:32.090 align:middle line:84% So when I'm here, well, I know that I'm either at one 01:01:32.090 --> 01:01:32.990 align:middle line:90% of these two nodes. 01:01:32.990 --> 01:01:35.770 align:middle line:84% I know that I'm not here, which means I know 01:01:35.770 --> 01:01:37.230 align:middle line:90% the choice that Nature made. 01:01:37.230 --> 01:01:38.650 align:middle line:90% I know they flipped heads. 01:01:38.650 --> 01:01:42.685 align:middle line:84% But I don't know whether player one chose C or R. 01:01:42.685 --> 01:01:44.310 align:middle line:84% AUDIENCE: But you could, theoretically, 01:01:44.310 --> 01:01:46.410 align:middle line:84% even though they're not at the same line? 01:01:46.410 --> 01:01:48.570 align:middle line:84% IAN BALL: A different game would be I put all three 01:01:48.570 --> 01:01:50.150 align:middle line:84% of these in the same information set. 01:01:50.150 --> 01:01:51.910 align:middle line:84% That would be a different game that I could write down, 01:01:51.910 --> 01:01:53.270 align:middle line:84% but that's not the game I've written down here. 01:01:53.270 --> 01:01:53.770 align:middle line:90% Exactly. 01:01:53.770 --> 01:01:55.610 align:middle line:90% Yeah. 01:01:55.610 --> 01:01:58.430 align:middle line:90% Any other questions on this? 01:01:58.430 --> 01:02:01.670 align:middle line:84% So now, the hardest step is figuring out strategies. 01:02:01.670 --> 01:02:04.970 align:middle line:90% 01:02:04.970 --> 01:02:06.770 align:middle line:84% So I think, especially when you're just 01:02:06.770 --> 01:02:09.610 align:middle line:84% getting used to this, whenever you're asked to write down 01:02:09.610 --> 01:02:11.590 align:middle line:84% strategies, the first question should be, 01:02:11.590 --> 01:02:14.290 align:middle line:84% how many information sets does each player have? 01:02:14.290 --> 01:02:16.270 align:middle line:84% Because often when people get strategies wrong, 01:02:16.270 --> 01:02:18.562 align:middle line:84% it's because they've gotten the information sets wrong. 01:02:18.562 --> 01:02:22.770 align:middle line:84% So let's look at player 1 and player 2. 01:02:22.770 --> 01:02:26.590 align:middle line:84% So first, how many information sets does player 1 have? 01:02:26.590 --> 01:02:32.290 align:middle line:90% 01:02:32.290 --> 01:02:32.810 align:middle line:90% Yeah? 01:02:32.810 --> 01:02:33.170 align:middle line:90% AUDIENCE: Three. 01:02:33.170 --> 01:02:33.837 align:middle line:90% IAN BALL: Three. 01:02:33.837 --> 01:02:37.045 align:middle line:84% Well, they have three nodes, 1, 2, and 3. 01:02:37.045 --> 01:02:39.170 align:middle line:84% And remember what I said is I'm not going to draw-- 01:02:39.170 --> 01:02:40.150 align:middle line:90% or I hope I said this. 01:02:40.150 --> 01:02:42.170 align:middle line:84% I'm not going to draw the singleton ones because it's just 01:02:42.170 --> 01:02:43.030 align:middle line:90% a little messy. 01:02:43.030 --> 01:02:46.768 align:middle line:90% So technically, it's like this. 01:02:46.768 --> 01:02:48.310 align:middle line:84% But I'm just not going to draw those. 01:02:48.310 --> 01:02:50.190 align:middle line:84% So if you don't see the nodes connected, 01:02:50.190 --> 01:02:53.090 align:middle line:84% that means each node is in its own information set. 01:02:53.090 --> 01:02:55.230 align:middle line:84% So player 1 has three information sets. 01:02:55.230 --> 01:02:56.890 align:middle line:84% And each of these information sets 01:02:56.890 --> 01:02:58.190 align:middle line:90% is what we call a singleton. 01:02:58.190 --> 01:02:59.890 align:middle line:90% It has a single node in it. 01:02:59.890 --> 01:03:04.370 align:middle line:90% So player 1 has three info sets. 01:03:04.370 --> 01:03:05.770 align:middle line:90% And what about player 2? 01:03:05.770 --> 01:03:10.210 align:middle line:84% How many information sets does player 2 have? 01:03:10.210 --> 01:03:11.450 align:middle line:90% Two, right? 01:03:11.450 --> 01:03:14.390 align:middle line:84% Information set 1 and information set 2. 01:03:14.390 --> 01:03:20.250 align:middle line:90% 01:03:20.250 --> 01:03:23.230 align:middle line:84% So why is it good to ask how many information sets are first? 01:03:23.230 --> 01:03:25.570 align:middle line:84% Because you can think of a strategy as a vector. 01:03:25.570 --> 01:03:27.640 align:middle line:84% And the number of places in the vector 01:03:27.640 --> 01:03:29.720 align:middle line:84% is going to be the number of information sets. 01:03:29.720 --> 01:03:32.700 align:middle line:84% So each information set is going to have a slot in this vector. 01:03:32.700 --> 01:03:34.920 align:middle line:84% And then we're going to fill in those slots. 01:03:34.920 --> 01:03:38.140 align:middle line:84% So let's think of strategies for player 1. 01:03:38.140 --> 01:03:39.840 align:middle line:90% We're going to have three slots. 01:03:39.840 --> 01:03:43.280 align:middle line:84% And strategies for player 2, we're going to have two slots. 01:03:43.280 --> 01:03:44.000 align:middle line:90% Why slots? 01:03:44.000 --> 01:03:47.000 align:middle line:84% Because a strategy specifies something for each information 01:03:47.000 --> 01:03:47.563 align:middle line:90% set. 01:03:47.563 --> 01:03:49.480 align:middle line:84% So we're just going to specify a slot wherever 01:03:49.480 --> 01:03:51.000 align:middle line:90% we need to put something. 01:03:51.000 --> 01:03:53.760 align:middle line:90% 01:03:53.760 --> 01:03:57.600 align:middle line:84% Now, the next thing is it's kind of helpful often 01:03:57.600 --> 01:03:59.000 align:middle line:90% to number the information sets. 01:03:59.000 --> 01:04:01.660 align:middle line:84% On a problem set, you could either explicitly number them, 01:04:01.660 --> 01:04:04.080 align:middle line:84% or maybe you could indicate how you're going to number it 01:04:04.080 --> 01:04:05.500 align:middle line:90% so that the TA understands. 01:04:05.500 --> 01:04:09.960 align:middle line:84% But it's important that you're clear about which of these slots 01:04:09.960 --> 01:04:11.960 align:middle line:84% each information set corresponds to. 01:04:11.960 --> 01:04:21.085 align:middle line:84% So maybe I'll label player 1's information sets I, 2I, and 3I. 01:04:21.085 --> 01:04:22.960 align:middle line:84% I didn't want to use 1, 2, and 3 because that 01:04:22.960 --> 01:04:25.200 align:middle line:90% is confusing with players. 01:04:25.200 --> 01:04:31.830 align:middle line:84% And then for player 2, I guess I'll do I and 2I. 01:04:31.830 --> 01:04:46.110 align:middle line:90% 01:04:46.110 --> 01:04:47.990 align:middle line:90% So now, let's fill this in. 01:04:47.990 --> 01:04:49.450 align:middle line:90% We can just give an example. 01:04:49.450 --> 01:04:51.670 align:middle line:84% So we have three information sets. 01:04:51.670 --> 01:04:53.830 align:middle line:84% To specify a strategy, we just have 01:04:53.830 --> 01:04:58.230 align:middle line:84% to put Celtics or Red Sox here, Celtics or Red Sox here, 01:04:58.230 --> 01:05:00.332 align:middle line:90% and Celtics or Red Sox here. 01:05:00.332 --> 01:05:01.790 align:middle line:84% So how many strategies are we going 01:05:01.790 --> 01:05:03.610 align:middle line:90% to have in total for player 1? 01:05:03.610 --> 01:05:07.070 align:middle line:90% 01:05:07.070 --> 01:05:08.410 align:middle line:90% Eight strategies. 01:05:08.410 --> 01:05:12.670 align:middle line:90% 01:05:12.670 --> 01:05:14.310 align:middle line:90% And what about for player 2? 01:05:14.310 --> 01:05:15.450 align:middle line:90% How many strategies? 01:05:15.450 --> 01:05:22.230 align:middle line:90% 01:05:22.230 --> 01:05:23.470 align:middle line:90% Four, right? 01:05:23.470 --> 01:05:28.105 align:middle line:84% So in general, it's a bit more complicated 01:05:28.105 --> 01:05:29.980 align:middle line:84% if you have more than two moves, but whenever 01:05:29.980 --> 01:05:32.100 align:middle line:84% you have two moves at every information set, 01:05:32.100 --> 01:05:34.500 align:middle line:84% the reason we get 8 is this is 2 to the 3. 01:05:34.500 --> 01:05:37.460 align:middle line:84% And the reason we get 4 is this is 2 to the 2 01:05:37.460 --> 01:05:39.840 align:middle line:84% because we have two choices here. 01:05:39.840 --> 01:05:42.213 align:middle line:84% For each of those two choices, we have two choices here. 01:05:42.213 --> 01:05:44.380 align:middle line:84% And for each of those two choices, we have two more. 01:05:44.380 --> 01:05:46.440 align:middle line:90% So you can-- it's like a tree. 01:05:46.440 --> 01:05:47.660 align:middle line:90% It keeps doubling each time. 01:05:47.660 --> 01:05:51.620 align:middle line:84% And eventually, you'll get eight. 01:05:51.620 --> 01:05:52.880 align:middle line:90% So that's a lot to write down. 01:05:52.880 --> 01:05:54.680 align:middle line:84% If we really want to specify this game, 01:05:54.680 --> 01:05:57.627 align:middle line:84% we have to specify utilities for every strategy profile. 01:05:57.627 --> 01:05:58.960 align:middle line:90% So we're not going to do it now. 01:05:58.960 --> 01:06:00.700 align:middle line:84% But let's just understand, if there's 01:06:00.700 --> 01:06:04.180 align:middle line:84% eight strategies for player 1 and four strategies for player 01:06:04.180 --> 01:06:09.392 align:middle line:84% 2, then all together, we have 32 strategy profiles, 01:06:09.392 --> 01:06:11.600 align:middle line:84% because for each of the eight strategies of player 1, 01:06:11.600 --> 01:06:14.300 align:middle line:84% we can combine it with the strategy for player 2. 01:06:14.300 --> 01:06:16.100 align:middle line:84% And then for each of those profiles, 01:06:16.100 --> 01:06:19.500 align:middle line:84% we have to specify two numbers, the payoff for player 1 01:06:19.500 --> 01:06:20.840 align:middle line:90% and the payoff for player 2. 01:06:20.840 --> 01:06:22.330 align:middle line:84% So altogether, we need to specify 01:06:22.330 --> 01:06:25.410 align:middle line:84% 64 numbers in this really, really small game. 01:06:25.410 --> 01:06:29.795 align:middle line:84% And I think this is illustrating why games like chess, no one's 01:06:29.795 --> 01:06:31.170 align:middle line:84% been able to even-- we can't even 01:06:31.170 --> 01:06:32.670 align:middle line:84% deal with a number of strategies in chess. 01:06:32.670 --> 01:06:34.930 align:middle line:84% It's many, many times larger than the number of atoms 01:06:34.930 --> 01:06:36.888 align:middle line:84% in the universe, like the atoms in the universe 01:06:36.888 --> 01:06:40.810 align:middle line:84% to the atoms of the universe, crazy, crazy numbers. 01:06:40.810 --> 01:06:44.270 align:middle line:84% And the reason for that is, well, you can see here, 01:06:44.270 --> 01:06:47.470 align:middle line:84% even in the simplest possible game, it's combinatorial. 01:06:47.470 --> 01:06:49.010 align:middle line:90% It's exponential. 01:06:49.010 --> 01:06:52.770 align:middle line:84% So let's just do one example to see how 01:06:52.770 --> 01:06:54.570 align:middle line:90% we can calculate these payoffs. 01:06:54.570 --> 01:07:02.050 align:middle line:84% So let's say player 1 is going to choose-- 01:07:02.050 --> 01:07:04.670 align:middle line:84% let's say-- it doesn't really matter. 01:07:04.670 --> 01:07:09.050 align:middle line:90% I'll just choose it arbitrarily. 01:07:09.050 --> 01:07:09.990 align:middle line:90% Let's do that. 01:07:09.990 --> 01:07:13.770 align:middle line:84% And let's say player 2 is choosing RC. 01:07:13.770 --> 01:07:16.530 align:middle line:90% 01:07:16.530 --> 01:07:18.290 align:middle line:84% So what I'd like to do is I'm going 01:07:18.290 --> 01:07:24.600 align:middle line:84% to take this strategy by player 1 and this strategy by player 2, 01:07:24.600 --> 01:07:27.160 align:middle line:84% and I want to compute what the payoffs are 01:07:27.160 --> 01:07:28.480 align:middle line:90% going to be for each player. 01:07:28.480 --> 01:07:31.200 align:middle line:84% So I'm basically filling in two of those 64 numbers 01:07:31.200 --> 01:07:32.720 align:middle line:84% that I have to fill in if I wanted 01:07:32.720 --> 01:07:34.520 align:middle line:90% to fully specify the game. 01:07:34.520 --> 01:07:40.640 align:middle line:90% So what's often helpful is-- 01:07:40.640 --> 01:07:42.480 align:middle line:90% let me start with player 1. 01:07:42.480 --> 01:07:45.680 align:middle line:84% And I'm going to go to each of player 1's information sets. 01:07:45.680 --> 01:07:47.760 align:middle line:84% And I'm going to highlight the edge 01:07:47.760 --> 01:07:50.520 align:middle line:84% that the player would choose at that information set 01:07:50.520 --> 01:07:52.080 align:middle line:90% under this strategy. 01:07:52.080 --> 01:07:54.400 align:middle line:90% So information set 1 is here. 01:07:54.400 --> 01:07:57.920 align:middle line:84% If player 1 finds himself at this contingency, 01:07:57.920 --> 01:08:01.720 align:middle line:84% this strategy specifies that that player chooses C. 01:08:01.720 --> 01:08:05.000 align:middle line:84% So I'm going to highlight this here. 01:08:05.000 --> 01:08:07.480 align:middle line:84% If player 1 finds himself at this contingency, 01:08:07.480 --> 01:08:12.960 align:middle line:84% he's going to choose R. And at this contingency, 01:08:12.960 --> 01:08:14.380 align:middle line:90% he's going to choose R as well. 01:08:14.380 --> 01:08:17.880 align:middle line:90% 01:08:17.880 --> 01:08:19.080 align:middle line:90% Now let's go to player 2. 01:08:19.080 --> 01:08:23.510 align:middle line:84% At this contingency-- this is information set 1-- 01:08:23.510 --> 01:08:29.470 align:middle line:84% player 2 chooses R. So I'll fill this in with R here. 01:08:29.470 --> 01:08:30.750 align:middle line:90% And now, this one is trickier. 01:08:30.750 --> 01:08:33.510 align:middle line:84% At this information set, she chooses 01:08:33.510 --> 01:08:36.250 align:middle line:90% C. But now what do I fill in? 01:08:36.250 --> 01:08:39.090 align:middle line:84% Well what I have to do is I have to fill in C at both nodes. 01:08:39.090 --> 01:08:42.310 align:middle line:90% 01:08:42.310 --> 01:08:42.830 align:middle line:90% Why? 01:08:42.830 --> 01:08:45.029 align:middle line:84% Well, when player 2 gets here, player 2 01:08:45.029 --> 01:08:47.189 align:middle line:84% can't observe which node she's at. 01:08:47.189 --> 01:08:50.370 align:middle line:84% She just always does C because that's what the strategy says. 01:08:50.370 --> 01:08:52.490 align:middle line:84% So if it turned out she were at this node, 01:08:52.490 --> 01:08:54.109 align:middle line:90% this is what she would choose. 01:08:54.109 --> 01:08:56.529 align:middle line:84% If she were at this node, this is what she would choose. 01:08:56.529 --> 01:08:58.029 align:middle line:84% She doesn't know which one she's at. 01:08:58.029 --> 01:09:01.590 align:middle line:84% She just plays C at that information set. 01:09:01.590 --> 01:09:06.109 align:middle line:84% So now that we've filled this in, what we want to do 01:09:06.109 --> 01:09:07.870 align:middle line:84% is kind of trace through the graph 01:09:07.870 --> 01:09:10.310 align:middle line:84% to try to compute what the induced lottery 01:09:10.310 --> 01:09:12.710 align:middle line:84% over consequences is going to be. 01:09:12.710 --> 01:09:15.510 align:middle line:90% So we start here. 01:09:15.510 --> 01:09:19.380 align:middle line:84% If Nature chooses heads, which happens with probability 1/2, 01:09:19.380 --> 01:09:21.100 align:middle line:90% we go here. 01:09:21.100 --> 01:09:25.660 align:middle line:84% Then player 1 chooses C. And then player 2 chooses C. 01:09:25.660 --> 01:09:32.220 align:middle line:84% So we're going to get this outcome with probability 1/2. 01:09:32.220 --> 01:09:36.340 align:middle line:84% If Nature chooses tails, then player 2 chooses R. 01:09:36.340 --> 01:09:39.620 align:middle line:84% And then player 1 chooses R. So we're going to get this outcome 01:09:39.620 --> 01:09:40.580 align:middle line:90% with probability 1/2. 01:09:40.580 --> 01:09:43.740 align:middle line:90% 01:09:43.740 --> 01:09:48.819 align:middle line:84% So remember over here, I said a strategy profile for the players 01:09:48.819 --> 01:09:52.779 align:middle line:84% induces a lottery over consequences. 01:09:52.779 --> 01:09:54.780 align:middle line:84% That's exactly what happened over here. 01:09:54.780 --> 01:09:57.020 align:middle line:90% We computed a strategy profile. 01:09:57.020 --> 01:09:59.260 align:middle line:84% This is the particular strategy profile. 01:09:59.260 --> 01:10:01.100 align:middle line:90% What is the lottery it induces? 01:10:01.100 --> 01:10:04.260 align:middle line:84% The induced lottery is we get to this history with probability 01:10:04.260 --> 01:10:08.180 align:middle line:84% 1/2, which means we both go to the Celtics game after nature 01:10:08.180 --> 01:10:10.020 align:middle line:90% flipped the coin heads. 01:10:10.020 --> 01:10:13.980 align:middle line:84% And we get to this consequence also with probability 1/2. 01:10:13.980 --> 01:10:15.080 align:middle line:90% What is this consequence? 01:10:15.080 --> 01:10:16.460 align:middle line:90% Well, it's this full history. 01:10:16.460 --> 01:10:18.120 align:middle line:90% Nature chooses tales. 01:10:18.120 --> 01:10:19.960 align:middle line:90% Then player 2 chooses Red Sox. 01:10:19.960 --> 01:10:22.460 align:middle line:84% And then player 1 chooses Red Sox. 01:10:22.460 --> 01:10:26.420 align:middle line:84% And now if we go back, we can compute the expected payoffs 01:10:26.420 --> 01:10:30.460 align:middle line:84% from this strategy profile by looking at 1/2 times 01:10:30.460 --> 01:10:34.920 align:middle line:84% the utilities here plus 1/2 times the utilities here. 01:10:34.920 --> 01:10:38.140 align:middle line:84% And that's going to give each player's expected payoff 01:10:38.140 --> 01:10:39.740 align:middle line:90% from this strategy profile. 01:10:39.740 --> 01:10:40.940 align:middle line:90% That gives us two numbers. 01:10:40.940 --> 01:10:42.140 align:middle line:90% And we only have 62 to go. 01:10:42.140 --> 01:10:44.140 align:middle line:84% Obviously, I'm not going to ask you to do things 01:10:44.140 --> 01:10:45.223 align:middle line:90% like that on problem sets. 01:10:45.223 --> 01:10:49.020 align:middle line:84% But that's, in principle, how it would work. 01:10:49.020 --> 01:10:51.460 align:middle line:90% Any questions on this? 01:10:51.460 --> 01:10:51.960 align:middle line:90% Yes? 01:10:51.960 --> 01:10:53.377 align:middle line:84% AUDIENCE: Can I ask again, why are 01:10:53.377 --> 01:10:56.260 align:middle line:84% we introducing the concept of the Nature of flipping a coin? 01:10:56.260 --> 01:11:00.260 align:middle line:84% Is this a correct transition of the game, in the extensive form 01:11:00.260 --> 01:11:02.400 align:middle line:90% game, we were looking at before? 01:11:02.400 --> 01:11:05.220 align:middle line:84% IAN BALL: So I would say, this is a new extensive form game. 01:11:05.220 --> 01:11:07.490 align:middle line:84% So I defined this new extensive form game 01:11:07.490 --> 01:11:09.240 align:middle line:84% to try to illustrate some of the concepts. 01:11:09.240 --> 01:11:11.115 align:middle line:84% So this is not the same as the extensive form 01:11:11.115 --> 01:11:13.220 align:middle line:90% game we did before. 01:11:13.220 --> 01:11:15.650 align:middle line:84% The reason it's important to introduce moves of Nature 01:11:15.650 --> 01:11:19.450 align:middle line:84% is that a lot of games in reality involve randomness. 01:11:19.450 --> 01:11:22.033 align:middle line:84% So the way that we're going to capture randomness more broadly 01:11:22.033 --> 01:11:23.283 align:middle line:90% is going to be through nature. 01:11:23.283 --> 01:11:25.830 align:middle line:84% This is kind of a silly example, with nature flipping a coin. 01:11:25.830 --> 01:11:27.690 align:middle line:84% But in reality, it might be I set 01:11:27.690 --> 01:11:31.120 align:middle line:84% my price, the price at my gas station to be this. 01:11:31.120 --> 01:11:32.870 align:middle line:84% And then there's a war in the Middle East. 01:11:32.870 --> 01:11:34.287 align:middle line:84% And then the price of oil changes. 01:11:34.287 --> 01:11:35.750 align:middle line:90% And that's a choice of Nature. 01:11:35.750 --> 01:11:38.170 align:middle line:84% So it's going to be more substantive things. 01:11:38.170 --> 01:11:40.712 align:middle line:84% I set my price, and then there's an antitrust case 01:11:40.712 --> 01:11:43.170 align:middle line:84% against my competitor, things like that I can't anticipate, 01:11:43.170 --> 01:11:44.570 align:middle line:84% and then I form beliefs over, are 01:11:44.570 --> 01:11:46.430 align:middle line:84% going to be represented by moves of Nature. 01:11:46.430 --> 01:11:49.290 align:middle line:84% But I want to be clear that this extensive form game is not 01:11:49.290 --> 01:11:52.110 align:middle line:84% the same as this extensive form game. 01:11:52.110 --> 01:11:55.410 align:middle line:84% These are two different games that illustrate different ideas. 01:11:55.410 --> 01:11:56.770 align:middle line:90% Any other questions? 01:11:56.770 --> 01:11:57.348 align:middle line:90% Yeah? 01:11:57.348 --> 01:11:59.890 align:middle line:84% AUDIENCE: If you wanted to make it so that the players didn't 01:11:59.890 --> 01:12:03.595 align:middle line:84% know who was choosing first, would you just 01:12:03.595 --> 01:12:04.970 align:middle line:84% put all the nodes for each player 01:12:04.970 --> 01:12:06.510 align:middle line:84% into one information set, basically? 01:12:06.510 --> 01:12:07.770 align:middle line:84% IAN BALL: Yeah, you could do that. 01:12:07.770 --> 01:12:09.330 align:middle line:84% And that would be legal in this game 01:12:09.330 --> 01:12:12.370 align:middle line:84% because we just have to check, are two properties satisfied? 01:12:12.370 --> 01:12:14.240 align:middle line:84% Is there a singleton that starts things? 01:12:14.240 --> 01:12:15.040 align:middle line:90% Yes. 01:12:15.040 --> 01:12:19.680 align:middle line:84% And does the player have the same set of feasible actions 01:12:19.680 --> 01:12:21.460 align:middle line:84% at every node in the information set? 01:12:21.460 --> 01:12:22.720 align:middle line:90% And that would still be true. 01:12:22.720 --> 01:12:25.160 align:middle line:84% I will warn you that there are some weird things you 01:12:25.160 --> 01:12:28.160 align:middle line:84% can do with information sets, what we-- so we've 01:12:28.160 --> 01:12:29.673 align:middle line:90% imposed a few conditions. 01:12:29.673 --> 01:12:31.340 align:middle line:84% There's a few things we haven't imposed. 01:12:31.340 --> 01:12:33.680 align:middle line:84% So a natural assumption is that players don't forget 01:12:33.680 --> 01:12:35.193 align:middle line:90% what they previously knew. 01:12:35.193 --> 01:12:36.860 align:middle line:84% This is something called perfect recall. 01:12:36.860 --> 01:12:39.120 align:middle line:84% You might find it strange if a player takes a move 01:12:39.120 --> 01:12:43.080 align:middle line:84% and then later forgets what move she previously took. 01:12:43.080 --> 01:12:44.980 align:middle line:84% We haven't ruled that out explicitly. 01:12:44.980 --> 01:12:46.987 align:middle line:84% But every game we're going to study, maybe 01:12:46.987 --> 01:12:48.820 align:middle line:84% with the exception of the last day of class, 01:12:48.820 --> 01:12:50.260 align:middle line:84% will have this property of perfect recall. 01:12:50.260 --> 01:12:51.740 align:middle line:84% And without it, you can have some weird things. 01:12:51.740 --> 01:12:54.360 align:middle line:84% On the other hand, is perfect recall really a good assumption? 01:12:54.360 --> 01:12:55.760 align:middle line:84% You probably don't remember a lot of things 01:12:55.760 --> 01:12:57.010 align:middle line:90% you did when you were a child. 01:12:57.010 --> 01:12:59.160 align:middle line:84% So maybe in some cases, imperfect recall 01:12:59.160 --> 01:13:00.180 align:middle line:90% is a good model. 01:13:00.180 --> 01:13:04.880 align:middle line:84% But that's something that is up to the modeler in this course. 01:13:04.880 --> 01:13:06.600 align:middle line:90% Any other questions? 01:13:06.600 --> 01:13:10.150 align:middle line:84% So let me stop there, and I'll see everyone on Thursday. 01:13:10.150 --> 01:13:13.000 align:middle line:90%