WEBVTT 00:00:00.000 --> 00:00:03.368 align:middle line:90% [SQUEAKING] 00:00:03.368 --> 00:00:04.324 align:middle line:90% [RUSTLING] 00:00:04.324 --> 00:00:05.280 align:middle line:90% [CLICKING] 00:00:05.280 --> 00:00:09.347 align:middle line:90% 00:00:09.347 --> 00:00:10.180 align:middle line:90% IAN BALL: OK, great. 00:00:10.180 --> 00:00:13.440 align:middle line:84% So let's now say a little bit about what this course is. 00:00:13.440 --> 00:00:16.400 align:middle line:84% This is a course on game theory or on the economic applications 00:00:16.400 --> 00:00:17.440 align:middle line:90% of game theory. 00:00:17.440 --> 00:00:19.080 align:middle line:84% So I think a natural starting point 00:00:19.080 --> 00:00:21.760 align:middle line:90% is to ask what is game theory. 00:00:21.760 --> 00:00:23.840 align:middle line:84% Well, here's two definitions, both from people 00:00:23.840 --> 00:00:27.320 align:middle line:84% who've won the Nobel Prize for their work in game theory. 00:00:27.320 --> 00:00:28.880 align:middle line:84% The first is by Roger Myerson, who 00:00:28.880 --> 00:00:32.800 align:middle line:84% says that game theory is the study of mathematical models 00:00:32.800 --> 00:00:38.160 align:middle line:84% of conflict and cooperation between intelligent, rational 00:00:38.160 --> 00:00:40.120 align:middle line:90% decision makers. 00:00:40.120 --> 00:00:42.660 align:middle line:84% Robert Aumann, also a Nobel laureate, 00:00:42.660 --> 00:00:45.400 align:middle line:84% gives just a briefer definition, which I think 00:00:45.400 --> 00:00:47.560 align:middle line:90% encapsulates the key ideas. 00:00:47.560 --> 00:00:49.440 align:middle line:84% It's interactive decision theory. 00:00:49.440 --> 00:00:51.970 align:middle line:84% It's about people making decisions 00:00:51.970 --> 00:00:53.220 align:middle line:90% when their decisions interact. 00:00:53.220 --> 00:00:55.500 align:middle line:84% What does it mean for people's decisions to interact? 00:00:55.500 --> 00:00:57.180 align:middle line:84% Well, I think the best way to see it is just an example. 00:00:57.180 --> 00:00:58.847 align:middle line:84% So we're going to play a game as a class 00:00:58.847 --> 00:01:00.440 align:middle line:84% to see what it means to be a hopefully 00:01:00.440 --> 00:01:03.290 align:middle line:84% rational and intelligent interacting decision maker. 00:01:03.290 --> 00:01:05.540 align:middle line:84% And then after that, I'll get into the formal material 00:01:05.540 --> 00:01:06.300 align:middle line:90% on the board. 00:01:06.300 --> 00:01:07.800 align:middle line:84% So to play games in the class, we're 00:01:07.800 --> 00:01:10.365 align:middle line:84% going to use a platform called Moblab. 00:01:10.365 --> 00:01:12.740 align:middle line:84% So let me just describe the game and then I'll hit Begin. 00:01:12.740 --> 00:01:14.900 align:middle line:84% You're each going to be assigned a random number between 1 00:01:14.900 --> 00:01:15.460 align:middle line:90% and 100. 00:01:15.460 --> 00:01:16.680 align:middle line:90% It's going to be an integer. 00:01:16.680 --> 00:01:19.080 align:middle line:84% And then you're each going to make a guess, which is also 00:01:19.080 --> 00:01:20.500 align:middle line:90% a number between 1 and 100. 00:01:20.500 --> 00:01:21.760 align:middle line:90% The winner of the game-- 00:01:21.760 --> 00:01:24.240 align:middle line:84% I'm sorry, there's no real prizes but pride-- 00:01:24.240 --> 00:01:26.800 align:middle line:84% the winner of the game is the person who guesses closest 00:01:26.800 --> 00:01:30.085 align:middle line:84% to 2/3 of the average of everyone's number in the class. 00:01:30.085 --> 00:01:31.960 align:middle line:84% AUDIENCE: The number they guess or the number 00:01:31.960 --> 00:01:33.267 align:middle line:90% they were assigned? 00:01:33.267 --> 00:01:34.600 align:middle line:90% IAN BALL: The number they guess. 00:01:34.600 --> 00:01:35.740 align:middle line:90% So you're assigned a number. 00:01:35.740 --> 00:01:36.240 align:middle line:90% Only you know that. 00:01:36.240 --> 00:01:37.500 align:middle line:90% No one else knows that number. 00:01:37.500 --> 00:01:38.708 align:middle line:90% You're going to make a guess. 00:01:38.708 --> 00:01:41.800 align:middle line:84% And the winning guess is the guess that's closest to 2/3 00:01:41.800 --> 00:01:44.060 align:middle line:84% of the average of everyone in the class's guess. 00:01:44.060 --> 00:01:44.760 align:middle line:90% Yes. 00:01:44.760 --> 00:01:45.720 align:middle line:84% AUDIENCE: Is everyone's number drawn 00:01:45.720 --> 00:01:46.887 align:middle line:90% from a uniform distribution? 00:01:46.887 --> 00:01:47.947 align:middle line:90% IAN BALL: Yes. 00:01:47.947 --> 00:01:48.780 align:middle line:90% Any other questions? 00:01:48.780 --> 00:01:49.080 align:middle line:90% Yes. 00:01:49.080 --> 00:01:49.840 align:middle line:84% AUDIENCE: What's the range again? 00:01:49.840 --> 00:01:50.920 align:middle line:90% IAN BALL: 1 to 100. 00:01:50.920 --> 00:01:51.480 align:middle line:90% Yeah. 00:01:51.480 --> 00:01:52.660 align:middle line:90% Any other questions? 00:01:52.660 --> 00:01:54.700 align:middle line:84% I think these details should be there, 00:01:54.700 --> 00:01:56.938 align:middle line:84% but sometimes they're a little imprecise 00:01:56.938 --> 00:01:57.980 align:middle line:90% in the way they write it. 00:01:57.980 --> 00:01:59.540 align:middle line:90% OK, so let's begin. 00:01:59.540 --> 00:02:04.680 align:middle line:90% 00:02:04.680 --> 00:02:05.760 align:middle line:90% All right. 00:02:05.760 --> 00:02:07.800 align:middle line:90% Here we are. 00:02:07.800 --> 00:02:08.660 align:middle line:90% So let's see. 00:02:08.660 --> 00:02:11.172 align:middle line:84% So here on the horizontal axis are 00:02:11.172 --> 00:02:12.380 align:middle line:90% the guesses that people made. 00:02:12.380 --> 00:02:13.797 align:middle line:84% Again, to clarify, not the numbers 00:02:13.797 --> 00:02:15.820 align:middle line:84% you got, but the guesses that people are making. 00:02:15.820 --> 00:02:18.487 align:middle line:84% And on the vertical axis, we're showing the percentage of people 00:02:18.487 --> 00:02:20.000 align:middle line:90% who made that particular guess. 00:02:20.000 --> 00:02:20.987 align:middle line:90% So we have a few. 00:02:20.987 --> 00:02:22.820 align:middle line:84% It's a bit maybe hard to see with the color, 00:02:22.820 --> 00:02:24.600 align:middle line:84% but we have a few summary statistics here. 00:02:24.600 --> 00:02:29.160 align:middle line:84% We see that the average was 36 roughly. 00:02:29.160 --> 00:02:31.720 align:middle line:84% And that's the dotted purple line here. 00:02:31.720 --> 00:02:35.640 align:middle line:84% And the winning guess was the guess that was closest to 2/3 00:02:35.640 --> 00:02:38.080 align:middle line:90% of 36, so about 24. 00:02:38.080 --> 00:02:40.640 align:middle line:90% And we see that guess is here. 00:02:40.640 --> 00:02:44.160 align:middle line:84% Does anyone want to share their thought process? 00:02:44.160 --> 00:02:46.660 align:middle line:84% Maybe someone who was quite high up here, 00:02:46.660 --> 00:02:48.560 align:middle line:90% what they were thinking. 00:02:48.560 --> 00:02:49.220 align:middle line:90% Or low? 00:02:49.220 --> 00:02:54.543 align:middle line:84% Anyone who's willing to share what they were thinking about? 00:02:54.543 --> 00:02:55.460 align:middle line:90% What about the winner? 00:02:55.460 --> 00:02:56.960 align:middle line:84% They should be able to come forward. 00:02:56.960 --> 00:02:57.650 align:middle line:90% Who won? 00:02:57.650 --> 00:02:58.150 align:middle line:90% Yes. 00:02:58.150 --> 00:02:58.650 align:middle line:90% You won. 00:02:58.650 --> 00:03:00.617 align:middle line:84% OK, tell me, what was your reasoning? 00:03:00.617 --> 00:03:02.450 align:middle line:84% AUDIENCE: I wouldn't replicate the strategy. 00:03:02.450 --> 00:03:03.890 align:middle line:90% I think it's not the best idea. 00:03:03.890 --> 00:03:07.790 align:middle line:84% But what I did was put the mean of a uniform distribution 00:03:07.790 --> 00:03:09.730 align:middle line:84% from 0 to 100 would probably be around 50. 00:03:09.730 --> 00:03:10.478 align:middle line:90% IAN BALL: Yeah. 00:03:10.478 --> 00:03:11.770 align:middle line:90% AUDIENCE: I'd take 2/3 of that. 00:03:11.770 --> 00:03:13.478 align:middle line:84% But I thought probably other people would 00:03:13.478 --> 00:03:15.030 align:middle line:90% be thinking the same thing. 00:03:15.030 --> 00:03:17.570 align:middle line:84% So I downscaled it a bit more by 2/3. 00:03:17.570 --> 00:03:18.590 align:middle line:90% IAN BALL: Great. 00:03:18.590 --> 00:03:19.090 align:middle line:90% Great. 00:03:19.090 --> 00:03:21.330 align:middle line:84% So you're showing the foundation of game theoretic reasoning, 00:03:21.330 --> 00:03:23.010 align:middle line:84% what we talked about, interactive decision making. 00:03:23.010 --> 00:03:24.590 align:middle line:84% You didn't just think about your number. 00:03:24.590 --> 00:03:26.330 align:middle line:84% You had to think about other people's numbers. 00:03:26.330 --> 00:03:28.470 align:middle line:84% And then you had to think about what other people will do given 00:03:28.470 --> 00:03:29.450 align:middle line:90% the numbers that they got. 00:03:29.450 --> 00:03:31.290 align:middle line:84% And you adjusted your strategy accordingly. 00:03:31.290 --> 00:03:33.670 align:middle line:84% And in this case, it worked out pretty well. 00:03:33.670 --> 00:03:37.430 align:middle line:84% Anyone else want to share maybe a bid they're happy with, 00:03:37.430 --> 00:03:39.630 align:middle line:84% a bid they feel they made a mistake on? 00:03:39.630 --> 00:03:42.430 align:middle line:90% Any other thoughts? 00:03:42.430 --> 00:03:43.150 align:middle line:90% Yes. 00:03:43.150 --> 00:03:44.772 align:middle line:90% AUDIENCE: I did 26. 00:03:44.772 --> 00:03:45.480 align:middle line:90% IAN BALL: 26, OK. 00:03:45.480 --> 00:03:46.510 align:middle line:90% So you were pretty close, yeah? 00:03:46.510 --> 00:03:47.050 align:middle line:90% AUDIENCE: Yeah. 00:03:47.050 --> 00:03:48.467 align:middle line:84% I thought maybe something like 30, 00:03:48.467 --> 00:03:50.150 align:middle line:84% and then figured that I was probably 00:03:50.150 --> 00:03:52.650 align:middle line:84% wrong in whatever reason I would have, so I just lowered it. 00:03:52.650 --> 00:03:54.010 align:middle line:90% IAN BALL: OK that's fair. 00:03:54.010 --> 00:03:55.410 align:middle line:84% I mean, maybe they don't want to out themselves, 00:03:55.410 --> 00:03:56.410 align:middle line:90% but I'm kind of curious. 00:03:56.410 --> 00:03:59.170 align:middle line:84% People that guessed very high, any reasoning about that? 00:03:59.170 --> 00:04:00.290 align:middle line:84% No, maybe they don't want to say anything. 00:04:00.290 --> 00:04:02.570 align:middle line:84% OK, let's play it one more time and see what happens. 00:04:02.570 --> 00:04:06.395 align:middle line:90% 00:04:06.395 --> 00:04:09.110 align:middle line:84% Ah, so quite different this time. 00:04:09.110 --> 00:04:15.618 align:middle line:84% So now the average was 21 and the winning choice was 14. 00:04:15.618 --> 00:04:16.410 align:middle line:90% That's interesting. 00:04:16.410 --> 00:04:17.970 align:middle line:90% So this happens every year. 00:04:17.970 --> 00:04:19.670 align:middle line:90% So people are very consistent. 00:04:19.670 --> 00:04:21.850 align:middle line:84% So the distribution has changed a bit. 00:04:21.850 --> 00:04:24.550 align:middle line:84% So some people must have decided to bid less this time. 00:04:24.550 --> 00:04:29.790 align:middle line:84% Anyone who bid less this time, do they want to share why? 00:04:29.790 --> 00:04:31.150 align:middle line:90% Very quiet. 00:04:31.150 --> 00:04:33.478 align:middle line:90% Anyone? 00:04:33.478 --> 00:04:37.510 align:middle line:84% AUDIENCE: It's basically tho idea that you would keep getting 00:04:37.510 --> 00:04:41.990 align:middle line:84% 2/3 and 2/3 and 2/3 of what you previously thought like would be 00:04:41.990 --> 00:04:42.830 align:middle line:90% a good guess. 00:04:42.830 --> 00:04:43.372 align:middle line:90% IAN BALL: OK. 00:04:43.372 --> 00:04:44.455 align:middle line:90% Can you say a little more? 00:04:44.455 --> 00:04:46.330 align:middle line:84% So you thought something was a good guess, 00:04:46.330 --> 00:04:47.705 align:middle line:84% and now you're bidding even less. 00:04:47.705 --> 00:04:48.710 align:middle line:90% And why is that? 00:04:48.710 --> 00:04:50.350 align:middle line:84% AUDIENCE: Like I initially thought 00:04:50.350 --> 00:04:50.810 align:middle line:90% something was a good guess. 00:04:50.810 --> 00:04:53.030 align:middle line:84% And then I was like, OK, but if I think this, then 00:04:53.030 --> 00:04:54.530 align:middle line:84% what if other people think the same? 00:04:54.530 --> 00:04:56.750 align:middle line:84% So I'm going to give 2/3 of that. 00:04:56.750 --> 00:04:58.085 align:middle line:90% But then 2/3 again of that. 00:04:58.085 --> 00:04:59.710 align:middle line:84% And then I'm going to give 2/3 of that. 00:04:59.710 --> 00:05:00.950 align:middle line:90% IAN BALL: Right. 00:05:00.950 --> 00:05:03.390 align:middle line:84% So what we see is that interactive decision making 00:05:03.390 --> 00:05:04.690 align:middle line:90% sometimes creates this regress. 00:05:04.690 --> 00:05:06.810 align:middle line:84% So first, I have to think about my number. 00:05:06.810 --> 00:05:07.950 align:middle line:90% That's certainly relevant. 00:05:07.950 --> 00:05:08.990 align:middle line:84% Then I have to try to think about what 00:05:08.990 --> 00:05:10.312 align:middle line:90% other people are going to do. 00:05:10.312 --> 00:05:12.270 align:middle line:84% So I have to think about what number they have. 00:05:12.270 --> 00:05:14.670 align:middle line:84% But now I have to put myself in the shoes of those other people 00:05:14.670 --> 00:05:15.950 align:middle line:90% and say, what are they bidding? 00:05:15.950 --> 00:05:17.158 align:middle line:90% What are they going to guess? 00:05:17.158 --> 00:05:19.510 align:middle line:84% Well, they're thinking about what I'm doing. 00:05:19.510 --> 00:05:20.970 align:middle line:84% But when they try to think about what I'm doing, 00:05:20.970 --> 00:05:22.810 align:middle line:84% they have to think about what I think about what they're doing. 00:05:22.810 --> 00:05:23.510 align:middle line:84% So I have think about what would they 00:05:23.510 --> 00:05:24.770 align:middle line:84% think about what I think what they think. 00:05:24.770 --> 00:05:26.310 align:middle line:84% And all of a sudden, it gets very, very hard. 00:05:26.310 --> 00:05:28.875 align:middle line:84% And this is exactly what game theory is trying to address. 00:05:28.875 --> 00:05:30.250 align:middle line:84% I think for a while historically, 00:05:30.250 --> 00:05:31.330 align:middle line:84% if you go back to the history of thought, 00:05:31.330 --> 00:05:33.770 align:middle line:84% people thought, it's just an infinite regress. 00:05:33.770 --> 00:05:36.210 align:middle line:84% There's no way we can try to analyze games like this. 00:05:36.210 --> 00:05:37.990 align:middle line:84% And game theory is a formal way to try 00:05:37.990 --> 00:05:40.470 align:middle line:84% to think about this kind of reasoning process. 00:05:40.470 --> 00:05:42.150 align:middle line:90% This also happens every year. 00:05:42.150 --> 00:05:45.390 align:middle line:84% A lot of people shift to 100 after we talk about it. 00:05:45.390 --> 00:05:47.137 align:middle line:90% Why? 00:05:47.137 --> 00:05:48.970 align:middle line:84% I think some people are kind of mischievous. 00:05:48.970 --> 00:05:51.345 align:middle line:84% They want to try to shift the results and get it wrong. 00:05:51.345 --> 00:05:53.470 align:middle line:84% So I think this highlights a really important thing 00:05:53.470 --> 00:05:55.512 align:middle line:84% about game theory, that when we study game theory 00:05:55.512 --> 00:05:58.590 align:middle line:84% and when we analyze a game, a really important modeling 00:05:58.590 --> 00:06:02.030 align:middle line:84% choice is writing down what people's preferences actually 00:06:02.030 --> 00:06:03.572 align:middle line:90% are, what people want. 00:06:03.572 --> 00:06:06.030 align:middle line:84% So if you analyze this game assuming that people's goal was 00:06:06.030 --> 00:06:08.470 align:middle line:84% to win, your predictions might be wrong 00:06:08.470 --> 00:06:10.510 align:middle line:84% if some people's goal is not actually to win. 00:06:10.510 --> 00:06:11.968 align:middle line:84% So I think some people's goal might 00:06:11.968 --> 00:06:14.170 align:middle line:84% be to make my prediction look a little silly 00:06:14.170 --> 00:06:16.650 align:middle line:84% or to make their friends laugh or to mess with things. 00:06:16.650 --> 00:06:17.690 align:middle line:90% That's fine. 00:06:17.690 --> 00:06:18.890 align:middle line:90% That's their preference. 00:06:18.890 --> 00:06:20.870 align:middle line:84% But we need to make sure we accurately 00:06:20.870 --> 00:06:23.090 align:middle line:84% capture people's preferences when we model games. 00:06:23.090 --> 00:06:24.590 align:middle line:84% Because if we don't, our predictions 00:06:24.590 --> 00:06:26.030 align:middle line:90% are obviously going to be wrong. 00:06:26.030 --> 00:06:26.990 align:middle line:84% Another thing, though, that we have 00:06:26.990 --> 00:06:29.710 align:middle line:84% to keep in mind is, especially when comparing the results we 00:06:29.710 --> 00:06:32.730 align:middle line:84% get in a lab or in a classroom setting to the real world, 00:06:32.730 --> 00:06:34.225 align:middle line:84% is that the stakes really matter. 00:06:34.225 --> 00:06:36.350 align:middle line:84% So we have a lot of students-- maybe a few students 00:06:36.350 --> 00:06:37.517 align:middle line:90% who are kind of mischievous. 00:06:37.517 --> 00:06:39.897 align:middle line:84% If you won $10 for guessing, well, I 00:06:39.897 --> 00:06:42.230 align:middle line:84% think some of those students wouldn't be so mischievous. 00:06:42.230 --> 00:06:44.330 align:middle line:84% And if you won $100,000 for winning, 00:06:44.330 --> 00:06:46.290 align:middle line:84% I think very few students would be mischievous. 00:06:46.290 --> 00:06:48.830 align:middle line:84% And I think once we get high enough, essentially no students 00:06:48.830 --> 00:06:50.190 align:middle line:90% would be mischievous. 00:06:50.190 --> 00:06:52.372 align:middle line:84% So the stakes and what you actually get for winning 00:06:52.372 --> 00:06:53.830 align:middle line:84% are going to affect the preferences 00:06:53.830 --> 00:06:56.510 align:middle line:90% that people have in this game. 00:06:56.510 --> 00:06:58.090 align:middle line:90% So let's move on. 00:06:58.090 --> 00:07:00.170 align:middle line:84% The plan is to move on to the formal analysis, 00:07:00.170 --> 00:07:03.773 align:middle line:84% unless there are any questions about this game. 00:07:03.773 --> 00:07:06.190 align:middle line:84% I should say this is sometimes called the Keynesian beauty 00:07:06.190 --> 00:07:06.650 align:middle line:90% contest. 00:07:06.650 --> 00:07:08.390 align:middle line:84% So if you want to google it or read about it, 00:07:08.390 --> 00:07:09.410 align:middle line:90% that's what it's often called. 00:07:09.410 --> 00:07:09.910 align:middle line:90% Yes. 00:07:09.910 --> 00:07:12.550 align:middle line:84% AUDIENCE: So the average wasn't the average 00:07:12.550 --> 00:07:16.490 align:middle line:84% of numbers randomly assigned by the game to everyone, 00:07:16.490 --> 00:07:18.250 align:middle line:90% but the average of the choices? 00:07:18.250 --> 00:07:19.250 align:middle line:90% IAN BALL: Exactly right. 00:07:19.250 --> 00:07:20.630 align:middle line:84% So with this many people, the average 00:07:20.630 --> 00:07:22.670 align:middle line:84% of the numbers that people got is almost certain 00:07:22.670 --> 00:07:24.750 align:middle line:90% to be very close to 50. 00:07:24.750 --> 00:07:27.250 align:middle line:84% But this average is the average choice, 00:07:27.250 --> 00:07:29.890 align:middle line:84% so the average guess of people in the class. 00:07:29.890 --> 00:07:32.750 align:middle line:84% And that's why when we move from the first round 00:07:32.750 --> 00:07:35.330 align:middle line:84% to the second round, I didn't see all the realizations, 00:07:35.330 --> 00:07:37.990 align:middle line:84% but it's likely that the average number that people received 00:07:37.990 --> 00:07:39.870 align:middle line:84% was basically the same in the two rounds. 00:07:39.870 --> 00:07:41.700 align:middle line:84% But the average guess changed dramatically 00:07:41.700 --> 00:07:43.950 align:middle line:84% because people behaved differently in the second round 00:07:43.950 --> 00:07:45.310 align:middle line:90% after we had this discussion. 00:07:45.310 --> 00:07:46.685 align:middle line:90% And we could keep doing this. 00:07:46.685 --> 00:07:47.810 align:middle line:90% We have limited class time. 00:07:47.810 --> 00:07:49.370 align:middle line:84% But my guess if we played this 10 times, 00:07:49.370 --> 00:07:51.453 align:middle line:84% we might see people getting very, very close to 0. 00:07:51.453 --> 00:07:55.190 align:middle line:84% And my guess is then some people at 100 to mess with things. 00:07:55.190 --> 00:07:57.370 align:middle line:84% Was there a question here or comment? 00:07:57.370 --> 00:07:58.477 align:middle line:90% Yeah. 00:07:58.477 --> 00:07:59.810 align:middle line:90% AUDIENCE: I didn't get a number. 00:07:59.810 --> 00:08:00.860 align:middle line:90% It was just guest. 00:08:00.860 --> 00:08:03.110 align:middle line:84% I don't know if there was a big number assigned to it. 00:08:03.110 --> 00:08:04.390 align:middle line:90% IAN BALL: Oh, maybe I'm-- 00:08:04.390 --> 00:08:05.010 align:middle line:90% oh yeah, you're right. 00:08:05.010 --> 00:08:05.510 align:middle line:90% Sorry. 00:08:05.510 --> 00:08:06.710 align:middle line:90% There's no number. 00:08:06.710 --> 00:08:07.670 align:middle line:90% Yeah, good point. 00:08:07.670 --> 00:08:09.530 align:middle line:84% I'm getting ahead to the next game we play. 00:08:09.530 --> 00:08:09.810 align:middle line:90% You're right. 00:08:09.810 --> 00:08:09.950 align:middle line:90% Sorry. 00:08:09.950 --> 00:08:11.200 align:middle line:90% You weren't assigned a number. 00:08:11.200 --> 00:08:12.490 align:middle line:90% Yeah, sorry. 00:08:12.490 --> 00:08:14.230 align:middle line:90% That was very confusing. 00:08:14.230 --> 00:08:16.850 align:middle line:84% We play another game where you get a number, and fair enough. 00:08:16.850 --> 00:08:18.058 align:middle line:90% Yeah, you don't get a number. 00:08:18.058 --> 00:08:19.020 align:middle line:90% There's no uniformity. 00:08:19.020 --> 00:08:19.770 align:middle line:90% Yeah, fair enough. 00:08:19.770 --> 00:08:22.470 align:middle line:90% OK, I'm getting ahead of myself. 00:08:22.470 --> 00:08:25.150 align:middle line:90% Thanks for the clarification. 00:08:25.150 --> 00:08:28.530 align:middle line:90% OK, let us get to the board. 00:08:28.530 --> 00:08:50.700 align:middle line:90% 00:08:50.700 --> 00:08:51.240 align:middle line:90% There we go. 00:08:51.240 --> 00:08:53.220 align:middle line:84% So on the problem set, you'll analyze a variant of this game 00:08:53.220 --> 00:08:54.040 align:middle line:90% where you'll get a number. 00:08:54.040 --> 00:08:54.960 align:middle line:90% So that'll be coming. 00:08:54.960 --> 00:09:10.500 align:middle line:90% 00:09:10.500 --> 00:09:11.680 align:middle line:90% Let's open this up. 00:09:11.680 --> 00:09:32.380 align:middle line:90% 00:09:32.380 --> 00:09:35.820 align:middle line:90% OK, so let's get started. 00:09:35.820 --> 00:09:38.340 align:middle line:84% So first let me say a little bit more about what we said 00:09:38.340 --> 00:09:43.900 align:middle line:84% was the definition of game theory 00:09:43.900 --> 00:09:52.020 align:middle line:84% which we said was the study of interacting rational decision 00:09:52.020 --> 00:09:52.680 align:middle line:90% makers. 00:09:52.680 --> 00:10:03.300 align:middle line:90% 00:10:03.300 --> 00:10:06.860 align:middle line:84% So let's break down each of the terms in this definition. 00:10:06.860 --> 00:10:10.180 align:middle line:84% Let's go backwards from easiest to hardest. 00:10:10.180 --> 00:10:12.220 align:middle line:84% So I think the basic thing is that we're 00:10:12.220 --> 00:10:13.600 align:middle line:90% studying decision makers. 00:10:13.600 --> 00:10:15.520 align:middle line:84% We're studying people who make decisions. 00:10:15.520 --> 00:10:17.360 align:middle line:84% And that means people are making choices. 00:10:17.360 --> 00:10:19.020 align:middle line:84% So another good word for decision 00:10:19.020 --> 00:10:20.640 align:middle line:84% that we'll use a lot in the course 00:10:20.640 --> 00:10:23.180 align:middle line:84% is people are making either choices 00:10:23.180 --> 00:10:26.860 align:middle line:84% or we might also call it actions. 00:10:26.860 --> 00:10:29.060 align:middle line:84% And game theory is a very abstract, general 00:10:29.060 --> 00:10:32.860 align:middle line:84% mathematical theory that applies to a huge range of choices 00:10:32.860 --> 00:10:33.820 align:middle line:90% and actions. 00:10:33.820 --> 00:10:35.460 align:middle line:84% So in the game you just played, where 00:10:35.460 --> 00:10:39.900 align:middle line:84% you chose a number to guess, that was an example of a choice. 00:10:39.900 --> 00:10:42.518 align:middle line:84% Whether you go to MIT is another example of a choice or action 00:10:42.518 --> 00:10:44.060 align:middle line:84% that we could analyze in game theory. 00:10:44.060 --> 00:10:49.580 align:middle line:84% What price a firm sets is another example of a choice. 00:10:49.580 --> 00:10:52.500 align:middle line:84% Which country you move to, what job you choose, 00:10:52.500 --> 00:10:54.045 align:middle line:84% how many hours a week you work, these 00:10:54.045 --> 00:10:56.180 align:middle line:84% are all these different questions-- 00:10:56.180 --> 00:10:58.540 align:middle line:84% what platform a political party chooses coming up 00:10:58.540 --> 00:11:00.560 align:middle line:84% to an election, who someone votes for, 00:11:00.560 --> 00:11:03.260 align:middle line:84% these are all actions or choices that people make. 00:11:03.260 --> 00:11:05.580 align:middle line:84% And the generality and abstractness of game theory 00:11:05.580 --> 00:11:08.163 align:middle line:84% is what allows it to be applied to so many different settings. 00:11:08.163 --> 00:11:09.747 align:middle line:84% And throughout the course, we're going 00:11:09.747 --> 00:11:11.940 align:middle line:84% to talk about various applications of game theory, 00:11:11.940 --> 00:11:13.460 align:middle line:84% mainly within economics, but also 00:11:13.460 --> 00:11:16.260 align:middle line:84% to some things beyond economics, like political science 00:11:16.260 --> 00:11:19.380 align:middle line:90% and political bargaining. 00:11:19.380 --> 00:11:20.872 align:middle line:90% What does rational mean? 00:11:20.872 --> 00:11:23.080 align:middle line:84% So this is certainly a sticking point of game theory. 00:11:23.080 --> 00:11:25.120 align:middle line:84% If you've mentioned game theory to someone, 00:11:25.120 --> 00:11:26.500 align:middle line:84% whenever I'm on a plane and people ask me 00:11:26.500 --> 00:11:28.000 align:middle line:84% what I do and I say I'm a game theorist, 00:11:28.000 --> 00:11:29.320 align:middle line:90% I always get the same response. 00:11:29.320 --> 00:11:30.700 align:middle line:90% But aren't people not rational? 00:11:30.700 --> 00:11:33.260 align:middle line:84% You probably don't get asked about game theory on planes 00:11:33.260 --> 00:11:37.260 align:middle line:84% as much as I do, but maybe someday you will. 00:11:37.260 --> 00:11:41.100 align:middle line:84% And I want to be cautious about how we interpret rationality. 00:11:41.100 --> 00:11:43.780 align:middle line:84% In some contexts, this is a very bad assumption. 00:11:43.780 --> 00:11:46.500 align:middle line:90% There's no question about that. 00:11:46.500 --> 00:11:49.060 align:middle line:84% But in other contexts, I think it's a better assumption 00:11:49.060 --> 00:11:51.100 align:middle line:90% than people might think. 00:11:51.100 --> 00:11:53.300 align:middle line:84% And the reason for that is that the way 00:11:53.300 --> 00:11:55.763 align:middle line:84% we use the term rationality within game theory 00:11:55.763 --> 00:11:57.180 align:middle line:84% is a bit different than the way we 00:11:57.180 --> 00:11:59.860 align:middle line:84% use it colloquially in everyday language. 00:11:59.860 --> 00:12:04.360 align:middle line:84% So what you might often hear someone on the street say is, 00:12:04.360 --> 00:12:07.140 align:middle line:84% it's so irrational for this person 00:12:07.140 --> 00:12:10.900 align:middle line:84% to like this political candidate. 00:12:10.900 --> 00:12:12.380 align:middle line:84% In our model of rational choice, we 00:12:12.380 --> 00:12:15.060 align:middle line:84% would never say someone's preferences are irrational. 00:12:15.060 --> 00:12:17.160 align:middle line:84% If that's what they like, that's what they like. 00:12:17.160 --> 00:12:18.660 align:middle line:84% So rationality, I think a better way 00:12:18.660 --> 00:12:21.520 align:middle line:84% to think of it is it's really about consistency. 00:12:21.520 --> 00:12:24.877 align:middle line:90% 00:12:24.877 --> 00:12:26.460 align:middle line:84% The agents in the models that we study 00:12:26.460 --> 00:12:29.180 align:middle line:84% are going to act towards achieving 00:12:29.180 --> 00:12:30.860 align:middle line:90% some consistent objective. 00:12:30.860 --> 00:12:32.880 align:middle line:84% You might disagree with the objective they have. 00:12:32.880 --> 00:12:33.860 align:middle line:84% They might like a different flavor 00:12:33.860 --> 00:12:35.818 align:middle line:84% of ice cream or a different political candidate 00:12:35.818 --> 00:12:38.260 align:middle line:84% or put different weight on income and leisure. 00:12:38.260 --> 00:12:39.322 align:middle line:90% That's fine. 00:12:39.322 --> 00:12:41.280 align:middle line:84% We don't all have to have the same preferences, 00:12:41.280 --> 00:12:44.660 align:middle line:84% but the assumption is that people consistently 00:12:44.660 --> 00:12:48.340 align:middle line:84% act with an eye towards these preferences that they do have. 00:12:48.340 --> 00:12:53.020 align:middle line:84% So sometimes, there's a phrase that we say and I like to say is 00:12:53.020 --> 00:12:54.880 align:middle line:84% that preferences cannot be irrational. 00:12:54.880 --> 00:13:01.820 align:middle line:90% 00:13:01.820 --> 00:13:05.020 align:middle line:84% What can be rational in the context of game theory choices 00:13:05.020 --> 00:13:06.540 align:middle line:90% giving preferences. 00:13:06.540 --> 00:13:09.333 align:middle line:84% So liking chocolate ice cream is not irrational. 00:13:09.333 --> 00:13:12.000 align:middle line:84% Even if you really like vanilla, if your friend likes chocolate, 00:13:12.000 --> 00:13:12.480 align:middle line:90% that's fine. 00:13:12.480 --> 00:13:13.500 align:middle line:84% But if your friend likes chocolate 00:13:13.500 --> 00:13:15.100 align:middle line:84% and they choose vanilla, we would 00:13:15.100 --> 00:13:17.058 align:middle line:84% call that choice irrational because they're not 00:13:17.058 --> 00:13:19.140 align:middle line:90% maximizing their objective. 00:13:19.140 --> 00:13:23.060 align:middle line:84% Now, consistency-- now I've shifted the goalposts. 00:13:23.060 --> 00:13:24.700 align:middle line:84% Instead of trying to defend rationality 00:13:24.700 --> 00:13:26.340 align:middle line:84% as a game theory assumption, I'm now 00:13:26.340 --> 00:13:28.740 align:middle line:90% trying to mention consistency. 00:13:28.740 --> 00:13:31.520 align:middle line:84% Again, sometimes it applies, sometimes it doesn't. 00:13:31.520 --> 00:13:34.900 align:middle line:84% My guess is your preferences today as 20-year-olds are very 00:13:34.900 --> 00:13:37.570 align:middle line:84% different from the preferences you had 18 years ago. 00:13:37.570 --> 00:13:40.170 align:middle line:84% So if we're looking at a model of people's choices over life, 00:13:40.170 --> 00:13:42.175 align:middle line:84% this probably isn't a very good description, 00:13:42.175 --> 00:13:44.050 align:middle line:84% or at least we might need to enrich the model 00:13:44.050 --> 00:13:44.885 align:middle line:90% to capture that. 00:13:44.885 --> 00:13:47.010 align:middle line:84% But if we're looking at your choices or preferences 00:13:47.010 --> 00:13:51.610 align:middle line:84% today and tomorrow, it probably is a pretty good assumption. 00:13:51.610 --> 00:13:53.003 align:middle line:90% And then interacting. 00:13:53.003 --> 00:13:55.170 align:middle line:84% This is the key thing that distinguishes game theory 00:13:55.170 --> 00:13:57.690 align:middle line:84% from a lot of other decision problems 00:13:57.690 --> 00:13:59.350 align:middle line:90% that we study in economics. 00:13:59.350 --> 00:14:01.530 align:middle line:84% And it's exactly what we saw in this first game. 00:14:01.530 --> 00:14:04.630 align:middle line:84% It's that multiple people are making decisions. 00:14:04.630 --> 00:14:05.790 align:middle line:90% But it's more than that. 00:14:05.790 --> 00:14:08.308 align:middle line:84% There are a lot of situations where we all make decisions, 00:14:08.308 --> 00:14:10.350 align:middle line:84% but there's no interaction between the decisions. 00:14:10.350 --> 00:14:12.392 align:middle line:84% So we don't really have to take into account what 00:14:12.392 --> 00:14:13.625 align:middle line:90% decisions other people make. 00:14:13.625 --> 00:14:15.250 align:middle line:84% What distinguishes game theory and what 00:14:15.250 --> 00:14:17.050 align:middle line:84% I mean by interacting decisions is 00:14:17.050 --> 00:14:19.770 align:middle line:84% that when I'm choosing what decision to make myself, 00:14:19.770 --> 00:14:22.850 align:middle line:84% I have to take into account what other people are going to do. 00:14:22.850 --> 00:14:26.570 align:middle line:84% Because what other people do affects my preferences. 00:14:26.570 --> 00:14:29.410 align:middle line:84% So this is often called-- the situation is often called 00:14:29.410 --> 00:14:31.150 align:middle line:90% strategic interdependence. 00:14:31.150 --> 00:14:38.850 align:middle line:90% 00:14:38.850 --> 00:14:43.130 align:middle line:84% What's best for me depends upon what other people do. 00:14:43.130 --> 00:14:44.810 align:middle line:84% So a classic example where this is used 00:14:44.810 --> 00:14:47.450 align:middle line:84% would be penalty kicks or tennis serves. 00:14:47.450 --> 00:14:49.530 align:middle line:84% Should the shooter shoot left or right? 00:14:49.530 --> 00:14:52.010 align:middle line:84% Well, there's no inherent reason to shoot left or right. 00:14:52.010 --> 00:14:54.210 align:middle line:84% But whether they should shoot left or right 00:14:54.210 --> 00:14:56.550 align:middle line:84% depends upon which way the goalie is going to dive 00:14:56.550 --> 00:14:59.550 align:middle line:84% or which way the tennis serve receiver is planning to play. 00:14:59.550 --> 00:15:01.952 align:middle line:84% The best action for one player depends upon what 00:15:01.952 --> 00:15:02.910 align:middle line:90% other people are doing. 00:15:02.910 --> 00:15:06.090 align:middle line:84% And that's going to be what we study today. 00:15:06.090 --> 00:15:08.010 align:middle line:90% So that's this course. 00:15:08.010 --> 00:15:09.850 align:middle line:90% What about this class today? 00:15:09.850 --> 00:15:13.690 align:middle line:84% We're actually going to drop interacting and just study 00:15:13.690 --> 00:15:15.640 align:middle line:84% a single rational decision maker. 00:15:15.640 --> 00:15:17.390 align:middle line:84% That's going to be our baseline for today. 00:15:17.390 --> 00:15:19.650 align:middle line:84% Because before we can study how rational decision 00:15:19.650 --> 00:15:22.410 align:middle line:84% makers interact, we have to study how rational decision 00:15:22.410 --> 00:15:25.077 align:middle line:90% makers act in isolation. 00:15:25.077 --> 00:15:27.410 align:middle line:84% So today-- and then the rest of the course will be about 00:15:27.410 --> 00:15:29.863 align:middle line:90% interaction-- 00:15:29.863 --> 00:15:31.530 align:middle line:84% today we're going to study what's called 00:15:31.530 --> 00:15:35.410 align:middle line:90% individual decision making. 00:15:35.410 --> 00:15:38.730 align:middle line:84% Of course, up here, we still have individuals. 00:15:38.730 --> 00:15:40.410 align:middle line:84% But the key is that they're interacting 00:15:40.410 --> 00:15:41.270 align:middle line:90% with other individuals. 00:15:41.270 --> 00:15:42.830 align:middle line:84% So maybe individual is not the best term, 00:15:42.830 --> 00:15:43.910 align:middle line:90% but this is what's used. 00:15:43.910 --> 00:15:46.870 align:middle line:84% We're going to study individual decision making. 00:15:46.870 --> 00:15:50.890 align:middle line:90% 00:15:50.890 --> 00:15:52.890 align:middle line:84% I want to make one final comment about the scope 00:15:52.890 --> 00:15:55.570 align:middle line:84% of game theory, which is the question of who's 00:15:55.570 --> 00:15:57.730 align:middle line:90% taking these actions. 00:15:57.730 --> 00:16:00.603 align:middle line:84% We often think of it as an individual making choices. 00:16:00.603 --> 00:16:02.270 align:middle line:84% But it can be applied much more broadly. 00:16:02.270 --> 00:16:05.110 align:middle line:84% So there's a lot of applications of game theory to evolution, 00:16:05.110 --> 00:16:06.970 align:middle line:84% where we think of animals or species 00:16:06.970 --> 00:16:08.830 align:middle line:84% kind of collectively taking actions. 00:16:08.830 --> 00:16:11.730 align:middle line:84% We might also think about firms or organizations, governments 00:16:11.730 --> 00:16:14.510 align:middle line:84% taking actions, committees taking actions. 00:16:14.510 --> 00:16:17.948 align:middle line:84% And I think now with the rise of artificial intelligence, 00:16:17.948 --> 00:16:19.490 align:middle line:84% there's a lot of interest in thinking 00:16:19.490 --> 00:16:21.610 align:middle line:84% of the agents in our models as being 00:16:21.610 --> 00:16:24.343 align:middle line:84% an LLM or some algorithm that's taking an action 00:16:24.343 --> 00:16:25.510 align:middle line:90% and interacting with people. 00:16:25.510 --> 00:16:29.530 align:middle line:84% So one view that is kind of becoming common, 00:16:29.530 --> 00:16:32.002 align:middle line:84% we can debate about it, is that the rationality assumptions 00:16:32.002 --> 00:16:34.210 align:middle line:84% of game theory become more useful and more compelling 00:16:34.210 --> 00:16:35.810 align:middle line:84% in a world of artificial intelligence. 00:16:35.810 --> 00:16:37.602 align:middle line:84% We can debate about that, but that's a view 00:16:37.602 --> 00:16:39.810 align:middle line:90% that some people express. 00:16:39.810 --> 00:16:43.890 align:middle line:84% OK, so here we are, individual decision making. 00:16:43.890 --> 00:16:46.310 align:middle line:84% Let's start with a very simple problem-- 00:16:46.310 --> 00:16:48.930 align:middle line:90% 00:16:48.930 --> 00:16:51.470 align:middle line:84% a simple what I might call decision or choice problem. 00:16:51.470 --> 00:16:54.810 align:middle line:90% 00:16:54.810 --> 00:16:55.790 align:middle line:90% Nothing deep here. 00:16:55.790 --> 00:16:57.582 align:middle line:84% Nothing that interesting about the problem. 00:16:57.582 --> 00:16:59.730 align:middle line:84% But I just want to show how we can set it up. 00:16:59.730 --> 00:17:02.290 align:middle line:84% So let's say that you're choosing between three things. 00:17:02.290 --> 00:17:05.450 align:middle line:84% Let's say you're at a cafe and it's coffee, espresso, and tea. 00:17:05.450 --> 00:17:10.690 align:middle line:90% So we have three choices-- 00:17:10.690 --> 00:17:16.690 align:middle line:84% Coffee which I'll label C, espresso which I'll label E, 00:17:16.690 --> 00:17:19.095 align:middle line:90% and tea, which I'll label T. 00:17:19.095 --> 00:17:20.470 align:middle line:84% And let's not think about prices. 00:17:20.470 --> 00:17:22.030 align:middle line:84% Of course, maybe coffee is cheaper. 00:17:22.030 --> 00:17:23.630 align:middle line:84% But for now, let's not think about prices. 00:17:23.630 --> 00:17:24.470 align:middle line:90% You just get a choice. 00:17:24.470 --> 00:17:25.345 align:middle line:90% Your friend's paying. 00:17:25.345 --> 00:17:28.052 align:middle line:84% You're choosing between these three things. 00:17:28.052 --> 00:17:30.010 align:middle line:84% The first component that we need to think about 00:17:30.010 --> 00:17:32.428 align:middle line:84% in this decision problem is, what are these preferences? 00:17:32.428 --> 00:17:33.970 align:middle line:84% Remember, we said that we're studying 00:17:33.970 --> 00:17:37.730 align:middle line:84% decision makers who consistently maximize some of preference. 00:17:37.730 --> 00:17:40.670 align:middle line:84% So we need to specify this agent's preferences. 00:17:40.670 --> 00:17:44.912 align:middle line:90% 00:17:44.912 --> 00:17:46.370 align:middle line:84% And in this case, their preferences 00:17:46.370 --> 00:17:49.730 align:middle line:84% are going to specify some ordering among these three 00:17:49.730 --> 00:17:50.550 align:middle line:90% choices. 00:17:50.550 --> 00:17:52.850 align:middle line:84% So a great way to represent this in this context 00:17:52.850 --> 00:17:57.150 align:middle line:84% would be C better than E better than T. 00:17:57.150 --> 00:17:59.650 align:middle line:84% So these are probably my preferences. 00:17:59.650 --> 00:18:00.810 align:middle line:90% What does this mean? 00:18:00.810 --> 00:18:04.930 align:middle line:84% It means I prefer drinking coffee to drinking espresso. 00:18:04.930 --> 00:18:07.783 align:middle line:84% I prefer drinking espresso to drinking tea. 00:18:07.783 --> 00:18:09.950 align:middle line:84% And the way I've written this suggests transitivity, 00:18:09.950 --> 00:18:11.970 align:middle line:84% which you will assume that if I prefer coffee 00:18:11.970 --> 00:18:14.330 align:middle line:84% to espresso and espresso to tea, then 00:18:14.330 --> 00:18:16.290 align:middle line:84% I must also prefer coffee to tea. 00:18:16.290 --> 00:18:17.790 align:middle line:84% I won't write that out explicitly, 00:18:17.790 --> 00:18:19.430 align:middle line:90% but that's implied by this. 00:18:19.430 --> 00:18:22.050 align:middle line:84% And there's an implicit transitivity assumption 00:18:22.050 --> 00:18:25.370 align:middle line:90% that we're making. 00:18:25.370 --> 00:18:26.997 align:middle line:84% But I think this actually raises kind 00:18:26.997 --> 00:18:28.580 align:middle line:84% of a difficult philosophical question. 00:18:28.580 --> 00:18:30.600 align:middle line:84% What does it mean to prefer coffee to espresso? 00:18:30.600 --> 00:18:32.360 align:middle line:90% What is a preference? 00:18:32.360 --> 00:18:34.240 align:middle line:90% Philosophers talk about this. 00:18:34.240 --> 00:18:36.682 align:middle line:84% Maybe people who study cognitive sciences go to the lab 00:18:36.682 --> 00:18:38.640 align:middle line:84% and they try to measure people's brain activity 00:18:38.640 --> 00:18:40.240 align:middle line:84% and say, oh, when you drink coffee, 00:18:40.240 --> 00:18:42.427 align:middle line:90% a different neuron activates. 00:18:42.427 --> 00:18:44.760 align:middle line:84% Philosophers will get into these long debates about what 00:18:44.760 --> 00:18:46.040 align:middle line:90% it means to have a preference. 00:18:46.040 --> 00:18:49.280 align:middle line:84% In economics, we take a much more operational view 00:18:49.280 --> 00:18:50.190 align:middle line:90% of preferences. 00:18:50.190 --> 00:18:50.940 align:middle line:90% What does it mean? 00:18:50.940 --> 00:18:52.340 align:middle line:90% And it's almost tautological. 00:18:52.340 --> 00:18:55.000 align:middle line:84% What does it mean that I prefer coffee to espresso? 00:18:55.000 --> 00:18:57.360 align:middle line:84% It means when I'm faced with a choice between coffee 00:18:57.360 --> 00:18:59.067 align:middle line:90% and espresso, I choose coffee. 00:18:59.067 --> 00:18:59.900 align:middle line:90% That's all it means. 00:18:59.900 --> 00:19:03.320 align:middle line:84% So this is the decision based view of preferences that's 00:19:03.320 --> 00:19:04.760 align:middle line:90% predominant in economics. 00:19:04.760 --> 00:19:07.140 align:middle line:84% So let's say what is the meaning of this? 00:19:07.140 --> 00:19:12.640 align:middle line:90% 00:19:12.640 --> 00:19:15.480 align:middle line:84% The meaning is that if I'm given a choice between coffee 00:19:15.480 --> 00:19:21.198 align:middle line:90% and espresso, I choose coffee. 00:19:21.198 --> 00:19:22.740 align:middle line:84% I'm not going to be too precise here. 00:19:22.740 --> 00:19:25.280 align:middle line:84% We might argue about strict preferences 00:19:25.280 --> 00:19:26.420 align:middle line:90% versus weak preferences. 00:19:26.420 --> 00:19:28.460 align:middle line:84% And maybe we can be indifferent between things. 00:19:28.460 --> 00:19:31.440 align:middle line:84% But for now, let's just think about strict preferences. 00:19:31.440 --> 00:19:34.060 align:middle line:84% And preferences are a pairwise object, right? 00:19:34.060 --> 00:19:36.080 align:middle line:84% So a preference says for any two things, 00:19:36.080 --> 00:19:37.700 align:middle line:90% which do I prefer between them? 00:19:37.700 --> 00:19:40.033 align:middle line:84% So one way of thinking about what we mean by preferences 00:19:40.033 --> 00:19:42.080 align:middle line:84% is we're describing what an agent would 00:19:42.080 --> 00:19:46.373 align:middle line:84% choose from any pair or any menu containing two items. 00:19:46.373 --> 00:19:48.040 align:middle line:84% And then from that primitive preference, 00:19:48.040 --> 00:19:52.240 align:middle line:84% we can talk about what they would choose in richer problems. 00:19:52.240 --> 00:19:56.880 align:middle line:84% Now, an issue is that these little squiggly inequality signs 00:19:56.880 --> 00:19:58.420 align:middle line:90% are kind of messy to deal with. 00:19:58.420 --> 00:20:00.840 align:middle line:84% So it often becomes more convenient in economics 00:20:00.840 --> 00:20:03.880 align:middle line:84% to use a utility representation of preferences. 00:20:03.880 --> 00:20:05.500 align:middle line:84% So let me go to a new board for this. 00:20:05.500 --> 00:20:07.220 align:middle line:84% And this didn't go all the way up. 00:20:07.220 --> 00:20:08.120 align:middle line:90% Let's see. 00:20:08.120 --> 00:20:08.780 align:middle line:90% There we go. 00:20:08.780 --> 00:20:12.680 align:middle line:90% 00:20:12.680 --> 00:20:14.940 align:middle line:84% And I'll define the word ordinal in a second. 00:20:14.940 --> 00:20:19.560 align:middle line:84% But I'm going to talk about an ordinal utility 00:20:19.560 --> 00:20:21.620 align:middle line:90% representation of preferences. 00:20:21.620 --> 00:20:31.800 align:middle line:90% 00:20:31.800 --> 00:20:32.860 align:middle line:90% Why do we use this? 00:20:32.860 --> 00:20:34.560 align:middle line:90% Well, it's just more convenient. 00:20:34.560 --> 00:20:36.380 align:middle line:84% Especially when we have a lot of things, 00:20:36.380 --> 00:20:38.920 align:middle line:84% specifying all my pairwise preferences between these things 00:20:38.920 --> 00:20:40.280 align:middle line:90% can get kind of cumbersome. 00:20:40.280 --> 00:20:43.880 align:middle line:84% So what we do is we just specify a function called a utility 00:20:43.880 --> 00:20:44.420 align:middle line:90% function. 00:20:44.420 --> 00:20:46.100 align:middle line:84% So let's give just an example here. 00:20:46.100 --> 00:20:53.760 align:middle line:84% So maybe my utility function is u of C equals 5, u of E 00:20:53.760 --> 00:20:59.800 align:middle line:90% equals 4, and u of T equals 1. 00:20:59.800 --> 00:21:02.360 align:middle line:84% This should be one utility representation 00:21:02.360 --> 00:21:03.680 align:middle line:90% of these preferences. 00:21:03.680 --> 00:21:06.600 align:middle line:84% Why do these utilities represent these preferences? 00:21:06.600 --> 00:21:08.760 align:middle line:90% Well, do I prefer C to E? 00:21:08.760 --> 00:21:11.280 align:middle line:84% Yes, because the utility assigned to C 00:21:11.280 --> 00:21:13.880 align:middle line:84% is higher than the utility assigned to E. 00:21:13.880 --> 00:21:15.680 align:middle line:90% Do I prefer E to T? 00:21:15.680 --> 00:21:18.440 align:middle line:84% Well, yes, because the utility assigned to E 00:21:18.440 --> 00:21:21.200 align:middle line:84% is strictly greater than the utility assigned to T. 00:21:21.200 --> 00:21:23.140 align:middle line:84% And if we have a utility function like this, 00:21:23.140 --> 00:21:26.560 align:middle line:84% you can check that our induced preferences are going to satisfy 00:21:26.560 --> 00:21:27.580 align:middle line:90% transitivity as well. 00:21:27.580 --> 00:21:29.420 align:middle line:84% Because if one number is bigger than another 00:21:29.420 --> 00:21:31.587 align:middle line:84% and that other number is bigger than a third number, 00:21:31.587 --> 00:21:35.440 align:middle line:84% then the first number is bigger than the third number. 00:21:35.440 --> 00:21:39.400 align:middle line:84% But we have an issue here that this is only a representation. 00:21:39.400 --> 00:21:41.780 align:middle line:84% So I can also write down an alternative utility function. 00:21:41.780 --> 00:21:44.320 align:middle line:90% Maybe I'll call this u1. 00:21:44.320 --> 00:21:49.300 align:middle line:84% Let me now write down a utility function u2 that's this. 00:21:49.300 --> 00:21:59.692 align:middle line:90% 00:21:59.692 --> 00:22:00.900 align:middle line:90% What about these preferences? 00:22:00.900 --> 00:22:02.442 align:middle line:84% Are these the same preferences or are 00:22:02.442 --> 00:22:06.400 align:middle line:84% these different preferences than the first utility function? 00:22:06.400 --> 00:22:07.760 align:middle line:90% What do people think? 00:22:07.760 --> 00:22:11.285 align:middle line:84% I don't think there's really one right answer. 00:22:11.285 --> 00:22:13.160 align:middle line:84% AUDIENCE: In a marginal sense, it's the same. 00:22:13.160 --> 00:22:14.243 align:middle line:90% IAN BALL: The same, right? 00:22:14.243 --> 00:22:17.860 align:middle line:84% So it is true that if this were my utility representation, 00:22:17.860 --> 00:22:22.440 align:middle line:84% it still says I prefer C to E, I prefer E to T, 00:22:22.440 --> 00:22:26.400 align:middle line:84% and I prefer C to T. So this is an alternative representation 00:22:26.400 --> 00:22:27.760 align:middle line:90% of the same preferences. 00:22:27.760 --> 00:22:31.620 align:middle line:84% And this is what we mean by an ordinal utility representation. 00:22:31.620 --> 00:22:34.640 align:middle line:90% Only the order matters. 00:22:34.640 --> 00:22:35.500 align:middle line:90% Let me say this. 00:22:35.500 --> 00:22:46.360 align:middle line:90% 00:22:46.360 --> 00:22:47.820 align:middle line:84% And because only the order matters, 00:22:47.820 --> 00:22:49.720 align:middle line:90% that has a few implications. 00:22:49.720 --> 00:22:54.280 align:middle line:84% The first is that there's many utility representations 00:22:54.280 --> 00:22:55.540 align:middle line:90% of the same preferences. 00:22:55.540 --> 00:22:58.320 align:middle line:90% 00:22:58.320 --> 00:23:00.220 align:middle line:90% Many ordinal representations. 00:23:00.220 --> 00:23:12.600 align:middle line:90% 00:23:12.600 --> 00:23:14.760 align:middle line:84% And a second I think more subtle point 00:23:14.760 --> 00:23:17.960 align:middle line:84% is that the units and the values in the utility function 00:23:17.960 --> 00:23:18.860 align:middle line:90% don't have meaning. 00:23:18.860 --> 00:23:27.858 align:middle line:90% 00:23:27.858 --> 00:23:29.900 align:middle line:84% In fact, what are the units of utility functions? 00:23:29.900 --> 00:23:31.220 align:middle line:84% They don't really have a unit that makes sense. 00:23:31.220 --> 00:23:33.060 align:middle line:84% Sometimes we call the units utils. 00:23:33.060 --> 00:23:37.440 align:middle line:90% 00:23:37.440 --> 00:23:40.745 align:middle line:84% But from a purely preference standpoint, 00:23:40.745 --> 00:23:42.620 align:middle line:84% you might look at this and you might say, oh, 00:23:42.620 --> 00:23:44.780 align:middle line:84% I like C more in this world than in this world. 00:23:44.780 --> 00:23:47.480 align:middle line:84% But no, the way we think about ordinal representations 00:23:47.480 --> 00:23:50.252 align:middle line:84% is that what matters are your preferences. 00:23:50.252 --> 00:23:51.960 align:middle line:84% And there's a convenient mathematical way 00:23:51.960 --> 00:23:53.722 align:middle line:90% to represent them like this. 00:23:53.722 --> 00:23:55.680 align:middle line:84% And this is another convenient mathematical way 00:23:55.680 --> 00:23:56.740 align:middle line:90% to represent them. 00:23:56.740 --> 00:23:58.660 align:middle line:84% And they represent the same preferences. 00:23:58.660 --> 00:24:00.160 align:middle line:84% And there's no meaning associated 00:24:00.160 --> 00:24:02.740 align:middle line:84% with these utility functions in this ordinal world. 00:24:02.740 --> 00:24:03.240 align:middle line:90% Yes. 00:24:03.240 --> 00:24:05.920 align:middle line:84% AUDIENCE: Would you have to specify a null option, then? 00:24:05.920 --> 00:24:09.760 align:middle line:84% Let's say if the utility is negative 100 for T, 00:24:09.760 --> 00:24:12.520 align:middle line:84% would you have to specify that they could also choose nothing? 00:24:12.520 --> 00:24:14.062 align:middle line:84% IAN BALL: Great, that's a good point. 00:24:14.062 --> 00:24:16.590 align:middle line:84% So when we write a choice problem, it's really crucial-- 00:24:16.590 --> 00:24:19.257 align:middle line:84% and it's implicit in what I did, but I didn't say it explicitly, 00:24:19.257 --> 00:24:22.990 align:middle line:84% so thanks for bringing this up-- that I can make only one choice, 00:24:22.990 --> 00:24:25.343 align:middle line:84% and every possible choice is included. 00:24:25.343 --> 00:24:28.010 align:middle line:84% So if you have four options, if only three of them are included, 00:24:28.010 --> 00:24:30.718 align:middle line:84% that's not going to capture your preference problem-- your choice 00:24:30.718 --> 00:24:31.370 align:middle line:90% problem. 00:24:31.370 --> 00:24:33.402 align:middle line:84% So really, if I were being careful, 00:24:33.402 --> 00:24:35.610 align:middle line:84% probably this is actually a more complicated problem. 00:24:35.610 --> 00:24:38.150 align:middle line:84% And there's a third option, which is not drinking any drink. 00:24:38.150 --> 00:24:40.410 align:middle line:84% And I should really specify the utility here. 00:24:40.410 --> 00:24:42.150 align:middle line:84% So in this simple thing I'm imagining 00:24:42.150 --> 00:24:44.650 align:middle line:84% for some reason you have to make one of these three choices. 00:24:44.650 --> 00:24:45.090 align:middle line:90% But you're right. 00:24:45.090 --> 00:24:46.965 align:middle line:84% In general, when we model a choice problem, 00:24:46.965 --> 00:24:49.590 align:middle line:84% it's important to capture every possible choice the agent could 00:24:49.590 --> 00:24:50.270 align:middle line:90% make. 00:24:50.270 --> 00:24:53.670 align:middle line:84% On the other side, indeed, if it's possible for me 00:24:53.670 --> 00:24:55.390 align:middle line:84% to choose both coffee and espresso, 00:24:55.390 --> 00:24:57.070 align:middle line:84% then that needs to be formally modeled 00:24:57.070 --> 00:25:00.330 align:middle line:84% as another choice, the bundle where I get coffee and espresso, 00:25:00.330 --> 00:25:02.390 align:middle line:84% because that's a separate choice that I can make. 00:25:02.390 --> 00:25:04.150 align:middle line:90% Good point. 00:25:04.150 --> 00:25:06.790 align:middle line:84% OK, so this is ordinal utility representations. 00:25:06.790 --> 00:25:09.070 align:middle line:84% And now I'm going to move to the opposite, 00:25:09.070 --> 00:25:10.170 align:middle line:90% cardinal representations. 00:25:10.170 --> 00:25:12.850 align:middle line:84% And everything I said here is going to go the other way. 00:25:12.850 --> 00:25:14.010 align:middle line:90% So let's be really clear. 00:25:14.010 --> 00:25:15.190 align:middle line:84% Everything I just said is going to be 00:25:15.190 --> 00:25:16.410 align:middle line:90% wrong in a different context. 00:25:16.410 --> 00:25:18.910 align:middle line:90% So let's be very careful. 00:25:18.910 --> 00:25:21.430 align:middle line:90% So this is ordinal. 00:25:21.430 --> 00:25:25.290 align:middle line:84% The issue is that often in game theory, 00:25:25.290 --> 00:25:27.510 align:middle line:84% we face decisions under uncertainty. 00:25:27.510 --> 00:25:29.430 align:middle line:84% And when we face decisions under uncertainty, 00:25:29.430 --> 00:25:32.110 align:middle line:84% we need to model preferences a bit differently. 00:25:32.110 --> 00:25:34.890 align:middle line:84% So let's now move to decisions under uncertainty. 00:25:34.890 --> 00:25:41.830 align:middle line:90% 00:25:41.830 --> 00:25:43.910 align:middle line:84% What is a decision under uncertainty mean? 00:25:43.910 --> 00:25:46.510 align:middle line:84% It means when I make a choice, I don't 00:25:46.510 --> 00:25:48.510 align:middle line:84% know exactly what the consequence of that choice 00:25:48.510 --> 00:25:49.350 align:middle line:90% will be. 00:25:49.350 --> 00:25:51.670 align:middle line:84% When I buy an index fund or buy a stock, 00:25:51.670 --> 00:25:54.630 align:middle line:84% I don't know whether that stock is going to go up or down. 00:25:54.630 --> 00:25:56.350 align:middle line:84% I can't choose-- of course, if I could 00:25:56.350 --> 00:25:58.610 align:middle line:84% choose the stock that made more money, I would choose that one. 00:25:58.610 --> 00:25:59.652 align:middle line:90% But that's not my choice. 00:25:59.652 --> 00:26:02.377 align:middle line:84% I'm choosing one stock that has some distribution over outcomes 00:26:02.377 --> 00:26:04.210 align:middle line:84% that I don't know what the outcomes will be. 00:26:04.210 --> 00:26:05.570 align:middle line:84% And then I might choose another stock. 00:26:05.570 --> 00:26:08.230 align:middle line:84% And again, I don't know exactly what the outcome of that stock 00:26:08.230 --> 00:26:08.890 align:middle line:90% will be. 00:26:08.890 --> 00:26:11.270 align:middle line:84% And that's what we mean by uncertainty. 00:26:11.270 --> 00:26:14.333 align:middle line:84% So let's look at, again, a very, very simple problem, not 00:26:14.333 --> 00:26:15.750 align:middle line:84% economically interesting, but just 00:26:15.750 --> 00:26:18.633 align:middle line:84% to convey the ideas, which is the choice of how to get home. 00:26:18.633 --> 00:26:20.550 align:middle line:84% Let's say you're choosing whether to walk home 00:26:20.550 --> 00:26:21.910 align:middle line:90% or to take the subway. 00:26:21.910 --> 00:26:27.670 align:middle line:84% So you can either walk home or take the subway. 00:26:27.670 --> 00:26:29.763 align:middle line:84% Maybe we'll call that and we'll call this W. 00:26:29.763 --> 00:26:31.930 align:middle line:84% And again, here we're assuming you have to get home. 00:26:31.930 --> 00:26:33.810 align:middle line:84% Maybe there's a third option where you stay at work all day. 00:26:33.810 --> 00:26:35.690 align:middle line:84% But we're not going to model that. 00:26:35.690 --> 00:26:41.528 align:middle line:90% 00:26:41.528 --> 00:26:43.570 align:middle line:84% And now why is this a decision under uncertainty? 00:26:43.570 --> 00:26:45.130 align:middle line:84% Well, you don't know what the weather is going to be like. 00:26:45.130 --> 00:26:46.838 align:middle line:84% You have a long walk home, so the weather 00:26:46.838 --> 00:26:48.190 align:middle line:90% could change as you walk home. 00:26:48.190 --> 00:26:50.607 align:middle line:84% And we're going to assume there's two ways the world could 00:26:50.607 --> 00:26:51.130 align:middle line:90% be again. 00:26:51.130 --> 00:26:52.350 align:middle line:84% Weather it's more complicated, but this 00:26:52.350 --> 00:26:54.010 align:middle line:90% is a simplification of reality. 00:26:54.010 --> 00:26:55.970 align:middle line:84% It could be sunny or it could be rainy. 00:26:55.970 --> 00:26:58.338 align:middle line:90% 00:26:58.338 --> 00:27:00.630 align:middle line:84% Maybe we should really model every possible temperature 00:27:00.630 --> 00:27:02.530 align:middle line:84% and every possible level of precipitation, 00:27:02.530 --> 00:27:04.350 align:middle line:90% but we're not going to do that. 00:27:04.350 --> 00:27:08.410 align:middle line:84% And the way things work is when it's sunny, 00:27:08.410 --> 00:27:11.570 align:middle line:84% you'd rather walk home and when it's rainy, 00:27:11.570 --> 00:27:13.630 align:middle line:90% you'd rather take the subway. 00:27:13.630 --> 00:27:15.830 align:middle line:84% So maybe I'll write this over here. 00:27:15.830 --> 00:27:22.430 align:middle line:90% If sunny, you prefer-- 00:27:22.430 --> 00:27:25.590 align:middle line:84% and I think sometimes I call this the T to avoid the S. 00:27:25.590 --> 00:27:32.470 align:middle line:84% Let's call this the T. Maybe TA, just so we don't see the two 00:27:32.470 --> 00:27:33.670 align:middle line:90% S's. 00:27:33.670 --> 00:27:35.190 align:middle line:90% Sunny, you prefer to walk home. 00:27:35.190 --> 00:27:42.345 align:middle line:84% And if rainy, you prefer to take the T. 00:27:42.345 --> 00:27:43.970 align:middle line:84% So how would you approach this problem? 00:27:43.970 --> 00:27:46.470 align:middle line:84% And this is a very simple problem, 00:27:46.470 --> 00:27:48.843 align:middle line:84% a simple choice people make all the time. 00:27:48.843 --> 00:27:50.010 align:middle line:90% How would you approach this? 00:27:50.010 --> 00:27:53.670 align:middle line:90% 00:27:53.670 --> 00:27:57.670 align:middle line:84% What do you do when you-- do you just randomly go home or do 00:27:57.670 --> 00:28:00.630 align:middle line:90% you think a little bit? 00:28:00.630 --> 00:28:01.370 align:middle line:90% What's relevant? 00:28:01.370 --> 00:28:03.910 align:middle line:84% What do you need to know to approach this problem? 00:28:03.910 --> 00:28:04.630 align:middle line:90% Yeah. 00:28:04.630 --> 00:28:05.610 align:middle line:84% AUDIENCE: You check the weather app. 00:28:05.610 --> 00:28:07.110 align:middle line:84% IAN BALL: You check the weather app. 00:28:07.110 --> 00:28:08.277 align:middle line:90% And what does that tell you? 00:28:08.277 --> 00:28:09.652 align:middle line:84% AUDIENCE: If it's sunny or rainy. 00:28:09.652 --> 00:28:10.230 align:middle line:90% IAN BALL: OK. 00:28:10.230 --> 00:28:12.328 align:middle line:84% And it might tell you-- and really, it 00:28:12.328 --> 00:28:14.370 align:middle line:84% might give you some-- or do you want to go ahead? 00:28:14.370 --> 00:28:14.630 align:middle line:90% Yeah. 00:28:14.630 --> 00:28:15.850 align:middle line:84% AUDIENCE: It'd be how likely it is. 00:28:15.850 --> 00:28:16.110 align:middle line:90% IAN BALL: Yeah. 00:28:16.110 --> 00:28:17.730 align:middle line:84% It's going to tell you-- it might say sun or rain, 00:28:17.730 --> 00:28:19.610 align:middle line:84% but also it might give you a likelihood of sun or rain. 00:28:19.610 --> 00:28:21.032 align:middle line:84% And oftentimes, you're not certain 00:28:21.032 --> 00:28:22.490 align:middle line:84% if it's going to be sunny or rainy. 00:28:22.490 --> 00:28:23.573 align:middle line:90% You have some uncertainty. 00:28:23.573 --> 00:28:25.450 align:middle line:84% That's the key thing that we're having here. 00:28:25.450 --> 00:28:28.088 align:middle line:84% So this problem, if you knew the weather, would be very easy. 00:28:28.088 --> 00:28:30.130 align:middle line:84% If you knew for certain it was going to be sunny, 00:28:30.130 --> 00:28:31.310 align:middle line:90% we know you'd walk home. 00:28:31.310 --> 00:28:32.890 align:middle line:84% And if you knew for certain it was going to rain, 00:28:32.890 --> 00:28:33.670 align:middle line:90% you'd take the T. 00:28:33.670 --> 00:28:35.418 align:middle line:84% The issue is that you have uncertainty. 00:28:35.418 --> 00:28:37.710 align:middle line:84% And indeed, when you check the weather app, what you do 00:28:37.710 --> 00:28:38.930 align:middle line:90% is you form beliefs. 00:28:38.930 --> 00:28:41.550 align:middle line:84% And this is going to be kind of a foundational idea 00:28:41.550 --> 00:28:53.810 align:middle line:84% in this course, that in the face of uncertainty, what do you do? 00:28:53.810 --> 00:28:54.770 align:middle line:90% You form beliefs. 00:28:54.770 --> 00:28:57.350 align:middle line:90% 00:28:57.350 --> 00:29:00.070 align:middle line:84% So in game theory, we take a fundamentally Bayesian view 00:29:00.070 --> 00:29:00.890 align:middle line:90% of the world. 00:29:00.890 --> 00:29:02.290 align:middle line:84% When we have uncertainty, we say, 00:29:02.290 --> 00:29:04.930 align:middle line:84% well, we don't know whether it's going to be sunny or rainy. 00:29:04.930 --> 00:29:06.597 align:middle line:84% But we might say something like, there's 00:29:06.597 --> 00:29:08.260 align:middle line:90% a 20% chance that it's sunny. 00:29:08.260 --> 00:29:09.600 align:middle line:90% So we'll assign beliefs. 00:29:09.600 --> 00:29:12.300 align:middle line:84% And in this case, let's represent our belief 00:29:12.300 --> 00:29:15.180 align:middle line:84% by the probability p that it's sunny. 00:29:15.180 --> 00:29:18.500 align:middle line:84% And then correspondingly, we have the probability 1 minus 00:29:18.500 --> 00:29:21.100 align:middle line:90% p that it's raining. 00:29:21.100 --> 00:29:23.220 align:middle line:90% So these are my beliefs. 00:29:23.220 --> 00:29:23.760 align:middle line:90% OK? 00:29:23.760 --> 00:29:27.180 align:middle line:84% And now I'm going to fill in some utilities here. 00:29:27.180 --> 00:29:35.740 align:middle line:90% I'm going to say 7, 2, 5, 5. 00:29:35.740 --> 00:29:38.080 align:middle line:84% So this is a representation of my preferences. 00:29:38.080 --> 00:29:41.180 align:middle line:84% And indeed, when it's sunny, I prefer walking to taking the T. 00:29:41.180 --> 00:29:46.620 align:middle line:84% And when it's rainy, I prefer taking the T to walking. 00:29:46.620 --> 00:29:48.820 align:middle line:84% So now we've written things down. 00:29:48.820 --> 00:29:49.680 align:middle line:90% Now I'd ask again. 00:29:49.680 --> 00:29:52.220 align:middle line:84% Now we've specified our beliefs, which would you choose? 00:29:52.220 --> 00:29:55.066 align:middle line:84% How would you approach this choice? 00:29:55.066 --> 00:29:56.683 align:middle line:84% AUDIENCE: Find an expected utility. 00:29:56.683 --> 00:29:58.100 align:middle line:84% IAN BALL: So in this course, we're 00:29:58.100 --> 00:29:59.560 align:middle line:84% going to assume expected utility. 00:29:59.560 --> 00:30:01.840 align:middle line:84% But I want to point out that that's not so obvious. 00:30:01.840 --> 00:30:03.760 align:middle line:84% One thing you might say you might say, look, 00:30:03.760 --> 00:30:06.460 align:middle line:90% I'm an optimistic person. 00:30:06.460 --> 00:30:10.180 align:middle line:84% If I walk, the best thing that can happen is I get 7. 00:30:10.180 --> 00:30:12.820 align:middle line:84% If I take the T, the best thing that can happen is I get 5. 00:30:12.820 --> 00:30:14.120 align:middle line:90% 7 is higher than 5. 00:30:14.120 --> 00:30:15.020 align:middle line:90% I'm going to walk. 00:30:15.020 --> 00:30:16.493 align:middle line:90% I'm an optimistic person. 00:30:16.493 --> 00:30:18.160 align:middle line:84% You might say, I'm a pessimistic person. 00:30:18.160 --> 00:30:21.060 align:middle line:84% I'm really afraid of the worst-case outcome. 00:30:21.060 --> 00:30:23.340 align:middle line:90% So when I walk, I could get 2. 00:30:23.340 --> 00:30:25.560 align:middle line:84% When I take the T, the worst I can get is 5. 00:30:25.560 --> 00:30:27.763 align:middle line:84% That's better, so I'm going to take the T. 00:30:27.763 --> 00:30:29.180 align:middle line:84% I think the view in game theory is 00:30:29.180 --> 00:30:31.540 align:middle line:84% that optimism and pessimism are better 00:30:31.540 --> 00:30:33.740 align:middle line:90% reflected in your beliefs. 00:30:33.740 --> 00:30:36.980 align:middle line:84% And we're generally going to assume in this course expected 00:30:36.980 --> 00:30:37.640 align:middle line:90% utility. 00:30:37.640 --> 00:30:42.660 align:middle line:90% 00:30:42.660 --> 00:30:44.900 align:middle line:84% But I want to point out that this is an assumption. 00:30:44.900 --> 00:30:46.760 align:middle line:90% There's an axiomatic foundation. 00:30:46.760 --> 00:30:49.220 align:middle line:84% So decision theorists think a lot about the question, 00:30:49.220 --> 00:30:51.340 align:middle line:84% why should we use expected utility? 00:30:51.340 --> 00:30:53.860 align:middle line:84% Or more concretely, if we assume that agents 00:30:53.860 --> 00:30:56.753 align:middle line:84% use expected utility, what assumption are we making? 00:30:56.753 --> 00:30:58.420 align:middle line:84% What does that mean about their behavior 00:30:58.420 --> 00:30:59.753 align:middle line:90% when we assume expected utility? 00:30:59.753 --> 00:31:01.300 align:middle line:84% How restrictive is that assumption? 00:31:01.300 --> 00:31:03.283 align:middle line:84% And there's an appendix in the lecture notes 00:31:03.283 --> 00:31:04.700 align:middle line:84% that's posted online where you can 00:31:04.700 --> 00:31:07.840 align:middle line:84% look at this axiomatic approach to expected utility theory. 00:31:07.840 --> 00:31:09.340 align:middle line:84% But in this course, we're just going 00:31:09.340 --> 00:31:12.400 align:middle line:84% to take it as given that people always use expected utility. 00:31:12.400 --> 00:31:13.400 align:middle line:90% So what does that mean? 00:31:13.400 --> 00:31:16.020 align:middle line:90% It's two steps. 00:31:16.020 --> 00:31:17.520 align:middle line:90% First, they form beliefs. 00:31:17.520 --> 00:31:27.123 align:middle line:90% 00:31:27.123 --> 00:31:28.540 align:middle line:84% Now, how reasonable is it to think 00:31:28.540 --> 00:31:30.260 align:middle line:84% that people form beliefs about the thing 00:31:30.260 --> 00:31:31.440 align:middle line:90% they're uncertain about? 00:31:31.440 --> 00:31:33.420 align:middle line:84% Well, I think it depends on the context. 00:31:33.420 --> 00:31:35.557 align:middle line:84% In this problem, I think it's very reasonable. 00:31:35.557 --> 00:31:36.640 align:middle line:90% You open your weather app. 00:31:36.640 --> 00:31:39.040 align:middle line:84% It gives you some probability of rain or sun. 00:31:39.040 --> 00:31:40.680 align:middle line:84% You might think, oh, I don't really trust the weather app. 00:31:40.680 --> 00:31:42.200 align:middle line:84% It always overestimates or underestimates. 00:31:42.200 --> 00:31:42.908 align:middle line:90% You might adjust. 00:31:42.908 --> 00:31:44.300 align:middle line:90% But you'll form your belief. 00:31:44.300 --> 00:31:49.140 align:middle line:84% But if I said, form your belief about what 00:31:49.140 --> 00:31:51.540 align:middle line:84% is the probability that in 15 years 00:31:51.540 --> 00:31:54.460 align:middle line:84% your income will be between these two numbers. 00:31:54.460 --> 00:31:56.402 align:middle line:84% That's pretty hard to form beliefs about. 00:31:56.402 --> 00:31:57.860 align:middle line:84% So I think this assumption that you 00:31:57.860 --> 00:32:00.020 align:middle line:84% start by forming beliefs about what you don't know 00:32:00.020 --> 00:32:03.120 align:middle line:84% is a very realistic assumption in very simple problems. 00:32:03.120 --> 00:32:05.500 align:middle line:84% But as we get to very, very rich problems and complicated 00:32:05.500 --> 00:32:08.300 align:middle line:84% problems, you might worry that this is not 00:32:08.300 --> 00:32:09.680 align:middle line:90% a very well-founded assumption. 00:32:09.680 --> 00:32:11.600 align:middle line:84% When we get to complicated choices, 00:32:11.600 --> 00:32:15.180 align:middle line:84% maybe your choice about what job to take today 00:32:15.180 --> 00:32:16.780 align:middle line:84% depend on your belief about how likely 00:32:16.780 --> 00:32:20.380 align:middle line:84% it is that you have a grandchild in 40 years who's short on money 00:32:20.380 --> 00:32:23.840 align:middle line:84% and needs the money that you saved based on your job today. 00:32:23.840 --> 00:32:24.800 align:middle line:90% Or let's go deeper. 00:32:24.800 --> 00:32:26.740 align:middle line:84% What's the probability that in 10 generations, 00:32:26.740 --> 00:32:31.020 align:middle line:84% your 10th degree grandchild is going to be short on money? 00:32:31.020 --> 00:32:32.980 align:middle line:84% Wow, that's really hard to form beliefs about. 00:32:32.980 --> 00:32:35.480 align:middle line:84% So in that context, maybe this is an unrealistic assumption, 00:32:35.480 --> 00:32:36.500 align:middle line:84% but it's going to be the assumption we 00:32:36.500 --> 00:32:37.200 align:middle line:90% make in this class. 00:32:37.200 --> 00:32:39.450 align:middle line:84% And we're going to apply this to models where we think 00:32:39.450 --> 00:32:40.740 align:middle line:90% it's a reasonable assumption. 00:32:40.740 --> 00:32:42.940 align:middle line:84% And then the next thing that you do, 00:32:42.940 --> 00:32:44.440 align:middle line:84% given that you formed your beliefs, 00:32:44.440 --> 00:32:46.880 align:middle line:84% is you compute your expected utility from each choice. 00:32:46.880 --> 00:32:55.980 align:middle line:90% 00:32:55.980 --> 00:32:57.620 align:middle line:84% But now we have a bit of an issue. 00:32:57.620 --> 00:33:00.980 align:middle line:84% Because I set up here when I talked about ordinal utility 00:33:00.980 --> 00:33:04.338 align:middle line:84% representations that the numbers didn't matter. 00:33:04.338 --> 00:33:06.380 align:middle line:84% But now all of a sudden, the numbers will matter. 00:33:06.380 --> 00:33:09.620 align:middle line:84% Because when I compute the expectation, my expected utility 00:33:09.620 --> 00:33:12.940 align:middle line:84% from walking, if 7 were 700, that's 00:33:12.940 --> 00:33:15.500 align:middle line:84% going to change my expected utility. 00:33:15.500 --> 00:33:19.140 align:middle line:84% So we have to really crucially understand that in this model, 00:33:19.140 --> 00:33:20.880 align:middle line:90% these utilities are different. 00:33:20.880 --> 00:33:23.300 align:middle line:84% These are no longer ordinal utilities. 00:33:23.300 --> 00:33:28.420 align:middle line:84% So in the decision making under uncertainty-- 00:33:28.420 --> 00:33:43.220 align:middle line:84% so in this context, utility is cardinal as opposed to ordinal. 00:33:43.220 --> 00:33:46.000 align:middle line:84% Ordinal means only the order matters. 00:33:46.000 --> 00:33:48.240 align:middle line:84% That's where the word ordinal comes from. 00:33:48.240 --> 00:33:52.020 align:middle line:84% Cardinal means the size or the actual numbers matter. 00:33:52.020 --> 00:33:55.300 align:middle line:84% And the way we distinguish this is we 00:33:55.300 --> 00:33:57.180 align:middle line:84% say that this utility function here 00:33:57.180 --> 00:33:59.830 align:middle line:84% is a Von Neumann-Morgenstern utility function. 00:33:59.830 --> 00:34:12.929 align:middle line:90% 00:34:12.929 --> 00:34:14.750 align:middle line:84% And basically, throughout this course, 00:34:14.750 --> 00:34:17.130 align:middle line:84% we're going to focus on this case of uncertainty. 00:34:17.130 --> 00:34:20.090 align:middle line:84% And essentially, all the utility functions we work with 00:34:20.090 --> 00:34:23.810 align:middle line:84% are going to be cardinal utility functions or Von Neumann utility 00:34:23.810 --> 00:34:26.630 align:middle line:84% functions rather than ordinal utility functions. 00:34:26.630 --> 00:34:29.290 align:middle line:84% And just to understand the landscape, 00:34:29.290 --> 00:34:33.190 align:middle line:84% up here, we didn't make any expected utility assumptions. 00:34:33.190 --> 00:34:34.889 align:middle line:84% This was a very abstract approach 00:34:34.889 --> 00:34:36.650 align:middle line:90% to any preference problem. 00:34:36.650 --> 00:34:38.370 align:middle line:84% When we're in the world of uncertainty 00:34:38.370 --> 00:34:41.889 align:middle line:84% and we restrict attention to expected utility preferences, 00:34:41.889 --> 00:34:43.550 align:middle line:84% in that more restricted environment, 00:34:43.550 --> 00:34:45.050 align:middle line:84% we're always going to work with what 00:34:45.050 --> 00:34:46.889 align:middle line:84% are called Von Neumann-Morgenstern utility 00:34:46.889 --> 00:34:51.610 align:middle line:84% functions, where the numerical value actually matters. 00:34:51.610 --> 00:34:54.010 align:middle line:84% So let's see how we would approach this problem 00:34:54.010 --> 00:34:56.210 align:middle line:90% in this VNM context. 00:34:56.210 --> 00:34:58.330 align:middle line:90% So we have two choices. 00:34:58.330 --> 00:35:03.432 align:middle line:84% We have walking and taking the T. 00:35:03.432 --> 00:35:04.890 align:middle line:84% And what I'm generally going to use 00:35:04.890 --> 00:35:07.930 align:middle line:84% is this Von Neumann-Morgenstern utility function u. 00:35:07.930 --> 00:35:11.330 align:middle line:90% This is a small u. 00:35:11.330 --> 00:35:13.670 align:middle line:84% But then when I compute expected utilities, 00:35:13.670 --> 00:35:15.750 align:middle line:84% your expected utility is going to be a big U. 00:35:15.750 --> 00:35:19.530 align:middle line:84% So I'm going to now compute here U of walking 00:35:19.530 --> 00:35:22.690 align:middle line:84% and U of taken the T. And this is a big U. 00:35:22.690 --> 00:35:25.190 align:middle line:84% And of course with handwriting, it's not obvious what it is. 00:35:25.190 --> 00:35:27.315 align:middle line:84% So if you're ever unsure, you can ask me. 00:35:27.315 --> 00:35:28.690 align:middle line:84% But let's make sure we understand 00:35:28.690 --> 00:35:32.050 align:middle line:84% that these functions are defined for different objects. 00:35:32.050 --> 00:35:35.370 align:middle line:84% My cardinal utility tells me what 00:35:35.370 --> 00:35:38.130 align:middle line:84% is my utility from walking when it's sunny? 00:35:38.130 --> 00:35:40.310 align:middle line:84% What is my utility from walking when it's rainy? 00:35:40.310 --> 00:35:42.450 align:middle line:84% And it gives me these four numbers. 00:35:42.450 --> 00:35:46.930 align:middle line:84% My big U, my expected utility, isn't assigned 00:35:46.930 --> 00:35:49.850 align:middle line:90% to a weather-choice pair. 00:35:49.850 --> 00:35:52.130 align:middle line:84% It's assigned just to the choice. 00:35:52.130 --> 00:35:55.570 align:middle line:84% So for my big U, I can say, what is my expected utility 00:35:55.570 --> 00:35:57.250 align:middle line:90% from walking? 00:35:57.250 --> 00:35:59.170 align:middle line:84% My little you tells me my utility 00:35:59.170 --> 00:36:01.450 align:middle line:84% from walking in each state of the world. 00:36:01.450 --> 00:36:02.750 align:middle line:90% So let's just compute this. 00:36:02.750 --> 00:36:04.378 align:middle line:90% It's going to be quite easy. 00:36:04.378 --> 00:36:05.670 align:middle line:90% The math here is not the point. 00:36:05.670 --> 00:36:07.850 align:middle line:90% It's about conceptually. 00:36:07.850 --> 00:36:10.970 align:middle line:84% So if I walk, well, with probability p I get 7 00:36:10.970 --> 00:36:13.170 align:middle line:84% and with probability 1 minus p I get 2. 00:36:13.170 --> 00:36:21.330 align:middle line:84% So my expected utility is p times 7 plus 1 minus p times 2. 00:36:21.330 --> 00:36:26.410 align:middle line:84% And if I take the T, well, it's p times 5 00:36:26.410 --> 00:36:30.627 align:middle line:90% plus 1 minus p times 5. 00:36:30.627 --> 00:36:31.710 align:middle line:90% Well, this is easy, right? 00:36:31.710 --> 00:36:33.990 align:middle line:84% The expectation of this is just going to be 5. 00:36:33.990 --> 00:36:35.170 align:middle line:90% It's 5 either way. 00:36:35.170 --> 00:36:38.210 align:middle line:90% So this is 5. 00:36:38.210 --> 00:36:41.970 align:middle line:90% And then here what do I get? 00:36:41.970 --> 00:36:42.690 align:middle line:90% I get 2. 00:36:42.690 --> 00:36:45.090 align:middle line:90% Minus 2p, so plus 5p. 00:36:45.090 --> 00:36:48.850 align:middle line:90% 00:36:48.850 --> 00:36:50.690 align:middle line:90% So which do I prefer? 00:36:50.690 --> 00:36:52.570 align:middle line:90% Well, it depends on p, right? 00:36:52.570 --> 00:36:56.130 align:middle line:84% If p is really, really small-- if p gets close to 0, 00:36:56.130 --> 00:36:57.810 align:middle line:90% then this becomes 2. 00:36:57.810 --> 00:36:59.370 align:middle line:90% And I prefer 5 to 2. 00:36:59.370 --> 00:37:01.290 align:middle line:90% So I prefer taking the T. 00:37:01.290 --> 00:37:02.203 align:middle line:90% That makes sense. 00:37:02.203 --> 00:37:04.370 align:middle line:84% If I look outside and it looks really, really rainy, 00:37:04.370 --> 00:37:07.930 align:middle line:84% I'm going to take the T. If p is really, really high-- well, 00:37:07.930 --> 00:37:09.070 align:middle line:90% let's say p is 1. 00:37:09.070 --> 00:37:10.650 align:middle line:90% I know it's going to be sunny. 00:37:10.650 --> 00:37:13.930 align:middle line:90% Then if I walk, I get 7. 00:37:13.930 --> 00:37:15.430 align:middle line:90% Yes, if I walk, I get 7. 00:37:15.430 --> 00:37:17.330 align:middle line:90% If I take the T, I get 5. 00:37:17.330 --> 00:37:19.170 align:middle line:90% 7 is more than 5. 00:37:19.170 --> 00:37:22.370 align:middle line:90% So therefore, I prefer to walk. 00:37:22.370 --> 00:37:24.070 align:middle line:84% When will I walk rather than take the T? 00:37:24.070 --> 00:37:25.910 align:middle line:84% Well, I have to compare these two numbers. 00:37:25.910 --> 00:37:29.510 align:middle line:84% So when is 2 plus 5p greater than or equal to 5? 00:37:29.510 --> 00:37:30.630 align:middle line:90% Let's ask that question. 00:37:30.630 --> 00:37:37.010 align:middle line:90% 00:37:37.010 --> 00:37:40.410 align:middle line:84% Well, it's if 5p is greater than or equal to 3. 00:37:40.410 --> 00:37:46.370 align:middle line:84% So if p is greater than or equal to 3/5, right? 00:37:46.370 --> 00:37:46.870 align:middle line:90% All right? 00:37:46.870 --> 00:37:48.410 align:middle line:90% Yeah. 00:37:48.410 --> 00:37:50.990 align:middle line:84% I'm sure someone will check the math. 00:37:50.990 --> 00:37:54.330 align:middle line:90% 00:37:54.330 --> 00:37:56.090 align:middle line:90% Yeah. 00:37:56.090 --> 00:37:59.010 align:middle line:84% AUDIENCE: Why is it greater than 3/5? 00:37:59.010 --> 00:38:00.230 align:middle line:90% IAN BALL: Ah, good point. 00:38:00.230 --> 00:38:04.090 align:middle line:84% So when p is exactly 3/5, I'm indifferent between the two. 00:38:04.090 --> 00:38:05.710 align:middle line:90% So I need some word for that. 00:38:05.710 --> 00:38:10.210 align:middle line:84% So I guess I would say I weakly prefer walking to taking the T 00:38:10.210 --> 00:38:12.250 align:middle line:84% if p is greater than or equal to 3/5. 00:38:12.250 --> 00:38:17.010 align:middle line:84% And I strictly prefer walking to taking the T if p is strictly 00:38:17.010 --> 00:38:18.730 align:middle line:90% greater than 3/5. 00:38:18.730 --> 00:38:20.490 align:middle line:84% AUDIENCE: But wouldn't that make sense 00:38:20.490 --> 00:38:25.690 align:middle line:84% that the equal part would be for the T, not for walking? 00:38:25.690 --> 00:38:27.290 align:middle line:90% IAN BALL: Sorry, say it again. 00:38:27.290 --> 00:38:31.010 align:middle line:84% AUDIENCE: So since it's greater or equal, when 00:38:31.010 --> 00:38:32.380 align:middle line:90% this happens, we walk, right? 00:38:32.380 --> 00:38:33.130 align:middle line:90% IAN BALL: Exactly. 00:38:33.130 --> 00:38:33.850 align:middle line:90% Yeah. 00:38:33.850 --> 00:38:37.810 align:middle line:84% AUDIENCE: So why are we putting equal in walking? 00:38:37.810 --> 00:38:40.770 align:middle line:84% Is there like-- do we say like when it's equal, 00:38:40.770 --> 00:38:41.510 align:middle line:90% it's indifferent? 00:38:41.510 --> 00:38:43.232 align:middle line:90% Do we decide which one? 00:38:43.232 --> 00:38:44.190 align:middle line:90% IAN BALL: You're right. 00:38:44.190 --> 00:38:45.690 align:middle line:84% So there's a bit of ambiguity there. 00:38:45.690 --> 00:38:48.330 align:middle line:84% So maybe I'll put-- if this makes it-- let's say this, yeah. 00:38:48.330 --> 00:38:51.030 align:middle line:84% I would say, let's focus on the case where p is not 3/5. 00:38:51.030 --> 00:38:52.050 align:middle line:90% Then it's pretty clear. 00:38:52.050 --> 00:38:54.110 align:middle line:84% When p is exactly 3/5, you're right, 00:38:54.110 --> 00:38:56.530 align:middle line:84% we wouldn't have an inherent preference for walking over 00:38:56.530 --> 00:38:59.370 align:middle line:84% taking the T. In that case, we would be indifferent between 00:38:59.370 --> 00:39:00.510 align:middle line:90% walking and taking the T. 00:39:00.510 --> 00:39:03.157 align:middle line:84% So maybe I'll make a note up here and sometimes use 00:39:03.157 --> 00:39:03.990 align:middle line:90% a squiggle for that. 00:39:03.990 --> 00:39:08.570 align:middle line:84% So we might say W squiggle T, I'm indifferent between walking 00:39:08.570 --> 00:39:17.290 align:middle line:84% and taking the T, exactly if p equals 3/5. 00:39:17.290 --> 00:39:19.108 align:middle line:84% And in this case, it's a strict preference. 00:39:19.108 --> 00:39:20.650 align:middle line:84% So here what this says, which I think 00:39:20.650 --> 00:39:22.810 align:middle line:84% is an intuitive model of decision making, 00:39:22.810 --> 00:39:25.970 align:middle line:84% that I'm going to choose to walk home when I check my weather app 00:39:25.970 --> 00:39:28.610 align:middle line:84% and I think it's sufficiently likely that it's sunny. 00:39:28.610 --> 00:39:29.910 align:middle line:90% How sufficiently likely? 00:39:29.910 --> 00:39:32.670 align:middle line:84% Well, in this case, it's exactly 60% chance of sun 00:39:32.670 --> 00:39:35.330 align:middle line:84% that makes it worth it for me to walk home. 00:39:35.330 --> 00:39:37.690 align:middle line:84% Now, where did we get this number from? 00:39:37.690 --> 00:39:41.402 align:middle line:84% It depended on the numerical values over here. 00:39:41.402 --> 00:39:43.110 align:middle line:84% If these numerical values were different, 00:39:43.110 --> 00:39:45.750 align:middle line:84% if I made this 700, well, then I'm basically always going 00:39:45.750 --> 00:39:48.000 align:middle line:84% to walk home unless I'm absolutely certain, basically, 00:39:48.000 --> 00:39:49.040 align:middle line:90% it's going to rain. 00:39:49.040 --> 00:39:53.360 align:middle line:84% So that again shows us that the cardinal values of these utility 00:39:53.360 --> 00:39:56.440 align:middle line:84% functions are going to affect the decisions that we make. 00:39:56.440 --> 00:39:57.700 align:middle line:90% Any other question on this? 00:39:57.700 --> 00:39:58.412 align:middle line:90% Yeah. 00:39:58.412 --> 00:40:00.120 align:middle line:84% AUDIENCE: So in the case when it's equal, 00:40:00.120 --> 00:40:03.700 align:middle line:84% is there-- like, now we use expected utility. 00:40:03.700 --> 00:40:07.000 align:middle line:84% So you can see which expected utility is bigger. 00:40:07.000 --> 00:40:10.240 align:middle line:84% But in the case it's equal, is there a rule we should follow? 00:40:10.240 --> 00:40:12.960 align:middle line:84% Or based on the numbers we have in the grid-- 00:40:12.960 --> 00:40:15.173 align:middle line:84% IAN BALL: No, I would say the right way 00:40:15.173 --> 00:40:16.340 align:middle line:90% to think about preferences-- 00:40:16.340 --> 00:40:18.940 align:middle line:84% I was being a little loose over there to make it quicker. 00:40:18.940 --> 00:40:21.333 align:middle line:84% But I'd say really, there's three possibilities 00:40:21.333 --> 00:40:22.500 align:middle line:90% when you compare two things. 00:40:22.500 --> 00:40:24.208 align:middle line:84% You can strictly prefer one to the other. 00:40:24.208 --> 00:40:26.737 align:middle line:84% You can strictly prefer the other to the first one. 00:40:26.737 --> 00:40:27.820 align:middle line:90% Or you can be indifferent. 00:40:27.820 --> 00:40:29.278 align:middle line:84% There's really three possibilities. 00:40:29.278 --> 00:40:31.043 align:middle line:84% And when I compare my expected utilities, 00:40:31.043 --> 00:40:32.460 align:middle line:84% there's again three possibilities. 00:40:32.460 --> 00:40:34.780 align:middle line:84% Either one is strictly higher, the other is strictly higher, 00:40:34.780 --> 00:40:36.100 align:middle line:90% or we're exactly indifferent. 00:40:36.100 --> 00:40:38.650 align:middle line:84% And in general, when people are indifferent, 00:40:38.650 --> 00:40:40.900 align:middle line:84% it's harder to make predictions about what they'll do. 00:40:40.900 --> 00:40:41.700 align:middle line:90% They might randomize. 00:40:41.700 --> 00:40:42.380 align:middle line:90% They might choose one. 00:40:42.380 --> 00:40:43.100 align:middle line:90% They might choose the other. 00:40:43.100 --> 00:40:44.600 align:middle line:84% We're not going to really take a strong stance on what 00:40:44.600 --> 00:40:45.743 align:middle line:90% people do in that case. 00:40:45.743 --> 00:40:46.660 align:middle line:90% Thanks for clarifying. 00:40:46.660 --> 00:40:47.160 align:middle line:90% Yeah. 00:40:47.160 --> 00:40:50.320 align:middle line:90% Any other questions? 00:40:50.320 --> 00:40:50.900 align:middle line:90% All right. 00:40:50.900 --> 00:40:52.600 align:middle line:84% So now let's be a bit more formal. 00:40:52.600 --> 00:40:57.100 align:middle line:84% This was a very simple example, but let's formalize this a bit. 00:40:57.100 --> 00:40:58.460 align:middle line:90% I guess I have one more board. 00:40:58.460 --> 00:41:07.000 align:middle line:90% 00:41:07.000 --> 00:41:09.400 align:middle line:90% So maybe I'll say VNM. 00:41:09.400 --> 00:41:11.980 align:middle line:84% This is Von Neumann-Morgenstern that I said before. 00:41:11.980 --> 00:41:14.100 align:middle line:84% And I'll call this the VNM setting. 00:41:14.100 --> 00:41:17.423 align:middle line:90% 00:41:17.423 --> 00:41:18.840 align:middle line:84% So generally, this is the approach 00:41:18.840 --> 00:41:20.220 align:middle line:84% we're going to take in this course. 00:41:20.220 --> 00:41:22.262 align:middle line:84% We're going to start with a really simple example 00:41:22.262 --> 00:41:23.400 align:middle line:90% to motivate the theory. 00:41:23.400 --> 00:41:25.905 align:middle line:84% We're then going to present a more general abstract model. 00:41:25.905 --> 00:41:28.280 align:middle line:84% And then we're going to go back to more concrete and more 00:41:28.280 --> 00:41:30.020 align:middle line:84% interesting applications that are more substantive. 00:41:30.020 --> 00:41:31.580 align:middle line:84% So that's the three steps we'll take. 00:41:31.580 --> 00:41:32.840 align:middle line:90% So what's the setting here? 00:41:32.840 --> 00:41:36.600 align:middle line:84% Well, there's going to be a set Z, which we'll say is the-- 00:41:36.600 --> 00:41:41.480 align:middle line:84% and I'll call this Z1 through Zm. 00:41:41.480 --> 00:41:46.980 align:middle line:84% And this is the set of maybe outcomes or consequences. 00:41:46.980 --> 00:41:49.513 align:middle line:90% 00:41:49.513 --> 00:41:51.680 align:middle line:84% And we have to be really careful about how we define 00:41:51.680 --> 00:41:53.080 align:middle line:90% outcomes and consequences. 00:41:53.080 --> 00:41:55.440 align:middle line:84% So in this game that we described, 00:41:55.440 --> 00:41:57.880 align:middle line:84% there are actually four outcomes-- 00:41:57.880 --> 00:42:00.880 align:middle line:84% I walk home in the sun, I walk home in the rain, 00:42:00.880 --> 00:42:03.520 align:middle line:84% I take the T in the sun, and I take the T in the rain. 00:42:03.520 --> 00:42:06.240 align:middle line:84% It's important to understand the outcome has to capture 00:42:06.240 --> 00:42:07.420 align:middle line:90% everything about my choice. 00:42:07.420 --> 00:42:08.940 align:middle line:90% It's not directly my choice. 00:42:08.940 --> 00:42:10.700 align:middle line:84% I'm not directly choosing outcomes. 00:42:10.700 --> 00:42:13.640 align:middle line:84% I'm choosing effectively lotteries over outcomes. 00:42:13.640 --> 00:42:15.640 align:middle line:84% And the outcome has to capture everything that's 00:42:15.640 --> 00:42:17.880 align:middle line:90% relevant in the situation. 00:42:17.880 --> 00:42:20.920 align:middle line:84% So we have this set of outcomes or consequences. 00:42:20.920 --> 00:42:23.960 align:middle line:84% And then we have the Von Neumann-Morgenstern utility 00:42:23.960 --> 00:42:26.680 align:middle line:90% function little u. 00:42:26.680 --> 00:42:28.800 align:middle line:84% Now, just as we saw over here, little u 00:42:28.800 --> 00:42:31.400 align:middle line:84% assigned a number to each of our four outcomes. 00:42:31.400 --> 00:42:36.680 align:middle line:84% So U is going to be a function from Z to R. 00:42:36.680 --> 00:42:38.680 align:middle line:84% So I'm going to use this function notation a lot 00:42:38.680 --> 00:42:39.640 align:middle line:90% in the course. 00:42:39.640 --> 00:42:41.760 align:middle line:84% When I put a letter and then a colon, 00:42:41.760 --> 00:42:43.520 align:middle line:84% that means this is a function that 00:42:43.520 --> 00:42:46.340 align:middle line:84% assigns to every item in this set, 00:42:46.340 --> 00:42:49.240 align:middle line:84% every object in this set, something in this set. 00:42:49.240 --> 00:42:52.932 align:middle line:84% So in this case, it assigns to every outcome Z is-- 00:42:52.932 --> 00:42:54.640 align:middle line:84% I'll write it like this underneath-- what 00:42:54.640 --> 00:42:59.160 align:middle line:84% we'll call u of Z. And u of Z is the cardinal Von 00:42:59.160 --> 00:43:04.920 align:middle line:84% Neumann-Morgenstern utility I get from the outcome Z. 00:43:04.920 --> 00:43:09.800 align:middle line:84% Now, given the set of outcomes and my Von Neumann-Morgenstern 00:43:09.800 --> 00:43:14.320 align:middle line:84% utility, we can consider my expected utility. 00:43:14.320 --> 00:43:15.500 align:middle line:90% Let me write it this. 00:43:15.500 --> 00:43:20.480 align:middle line:90% 00:43:20.480 --> 00:43:22.710 align:middle line:84% So my expected utility is the big U. 00:43:22.710 --> 00:43:24.460 align:middle line:84% And what is this going to be defined over? 00:43:24.460 --> 00:43:27.043 align:middle line:84% It's really crucial that this is defined over a different set. 00:43:27.043 --> 00:43:29.220 align:middle line:84% So what do I assign expected utilities to? 00:43:29.220 --> 00:43:32.220 align:middle line:90% 00:43:32.220 --> 00:43:32.720 align:middle line:90% Yeah. 00:43:32.720 --> 00:43:34.960 align:middle line:84% AUDIENCE: Would it be a set of actions? 00:43:34.960 --> 00:43:37.720 align:middle line:84% IAN BALL: So in this context, it was a set of actions. 00:43:37.720 --> 00:43:39.520 align:middle line:84% But in this abstract model-- good question. 00:43:39.520 --> 00:43:40.102 align:middle line:90% Yeah. 00:43:40.102 --> 00:43:41.560 align:middle line:84% AUDIENCE: Like the state you're in. 00:43:41.560 --> 00:43:44.780 align:middle line:84% IAN BALL: So I would say-- may be a bit of a vague question. 00:43:44.780 --> 00:43:48.330 align:middle line:84% I would say the set of lotteries over outcomes. 00:43:48.330 --> 00:43:49.080 align:middle line:90% So let's be clear. 00:43:49.080 --> 00:43:51.140 align:middle line:84% Here, you're right, we were choosing actions. 00:43:51.140 --> 00:43:53.360 align:middle line:84% We chose whether to walk or take the MBTA. 00:43:53.360 --> 00:43:56.200 align:middle line:84% But the way we think about those actions in this abstract setting 00:43:56.200 --> 00:43:58.300 align:middle line:84% is that each one corresponds to a lottery. 00:43:58.300 --> 00:44:01.160 align:middle line:84% Walking is the lottery where with probability p, 00:44:01.160 --> 00:44:04.440 align:middle line:84% I walk in the sun, and with probability 1 minus p, 00:44:04.440 --> 00:44:06.120 align:middle line:90% I walk in the rain. 00:44:06.120 --> 00:44:08.880 align:middle line:84% And similarly, the MBTA is a different lottery 00:44:08.880 --> 00:44:11.100 align:middle line:84% where with probability p, I get this outcome, 00:44:11.100 --> 00:44:13.540 align:middle line:84% and with probability 1 minus p, I get this outcome. 00:44:13.540 --> 00:44:15.480 align:middle line:84% So we need some notation for lotteries. 00:44:15.480 --> 00:44:19.830 align:middle line:84% I'm going to use throughout the course delta of Z. 00:44:19.830 --> 00:44:21.580 align:middle line:84% I don't want the notation to scare people. 00:44:21.580 --> 00:44:22.520 align:middle line:84% It's just going to make it easier 00:44:22.520 --> 00:44:23.980 align:middle line:90% for us to talk about things. 00:44:23.980 --> 00:44:25.740 align:middle line:90% So Z is our set of outcomes. 00:44:25.740 --> 00:44:29.880 align:middle line:84% Delta of Z is the set of lotteries over these outcomes. 00:44:29.880 --> 00:44:34.080 align:middle line:84% So in this context where we only had two outcomes, a lottery just 00:44:34.080 --> 00:44:37.260 align:middle line:84% specified two numbers, p and 1 minus p. 00:44:37.260 --> 00:44:39.497 align:middle line:84% And because those numbers sum to 1, 00:44:39.497 --> 00:44:41.330 align:middle line:84% it was really only one number that mattered. 00:44:41.330 --> 00:44:44.350 align:middle line:84% Once we knew p, we were able to calculate 1 minus p. 00:44:44.350 --> 00:44:46.790 align:middle line:84% But in general, we're looking at a situation 00:44:46.790 --> 00:44:49.350 align:middle line:84% where there are m outcomes instead of just two. 00:44:49.350 --> 00:44:50.530 align:middle line:90% So what is a lottery? 00:44:50.530 --> 00:44:53.110 align:middle line:90% 00:44:53.110 --> 00:44:57.950 align:middle line:84% So it contains what I'll call lotteries. 00:44:57.950 --> 00:45:00.870 align:middle line:84% And again, I should be clear the use of lottery colloquially 00:45:00.870 --> 00:45:01.955 align:middle line:90% is different from in econ. 00:45:01.955 --> 00:45:04.330 align:middle line:84% We're not literally talking about buying lottery tickets. 00:45:04.330 --> 00:45:05.790 align:middle line:84% We're using this more abstractly. 00:45:05.790 --> 00:45:10.910 align:middle line:84% It's a vector p that specifies the probability of each 00:45:10.910 --> 00:45:13.030 align:middle line:90% of these outcomes. 00:45:13.030 --> 00:45:16.350 align:middle line:90% So p1 to pm. 00:45:16.350 --> 00:45:18.910 align:middle line:84% The lottery says with probability p1, 00:45:18.910 --> 00:45:20.990 align:middle line:90% you get consequence Z1. 00:45:20.990 --> 00:45:24.510 align:middle line:84% With probability pm, you get consequence Zm. 00:45:24.510 --> 00:45:25.570 align:middle line:90% These are probabilities. 00:45:25.570 --> 00:45:27.170 align:middle line:84% So what has to be true about them? 00:45:27.170 --> 00:45:28.450 align:middle line:90% They have to be non-negative. 00:45:28.450 --> 00:45:30.070 align:middle line:84% You can't have negative probabilities. 00:45:30.070 --> 00:45:33.310 align:middle line:90% And they have to sum to 1. 00:45:33.310 --> 00:45:41.470 align:middle line:84% So say p1 up to pm are greater than or equal to 0. 00:45:41.470 --> 00:45:45.770 align:middle line:90% And p1 plus pm equals 1. 00:45:45.770 --> 00:45:50.190 align:middle line:90% 00:45:50.190 --> 00:45:53.750 align:middle line:84% So delta of Z is the set of all of these probability vectors 00:45:53.750 --> 00:45:58.510 align:middle line:84% or lotteries that satisfy these properties. 00:45:58.510 --> 00:46:00.310 align:middle line:84% And now our expected utility function 00:46:00.310 --> 00:46:07.830 align:middle line:84% u is a function from delta Z to R. 00:46:07.830 --> 00:46:10.630 align:middle line:84% It says, for any lottery, what is my expected 00:46:10.630 --> 00:46:12.630 align:middle line:90% utility from that lottery? 00:46:12.630 --> 00:46:14.870 align:middle line:84% So again, using the notation above, 00:46:14.870 --> 00:46:19.710 align:middle line:84% we would call a lottery p, and that would go to u of p. 00:46:19.710 --> 00:46:21.370 align:middle line:84% So this distinction is crucial, right? 00:46:21.370 --> 00:46:23.810 align:middle line:84% A VNM utility says for each outcome, 00:46:23.810 --> 00:46:26.070 align:middle line:84% what is my utility from that outcome? 00:46:26.070 --> 00:46:30.550 align:middle line:84% And expected utility big U says for each lottery over outcomes, 00:46:30.550 --> 00:46:33.750 align:middle line:84% what is my expected utility from that lottery? 00:46:33.750 --> 00:46:34.693 align:middle line:90% Yes, question? 00:46:34.693 --> 00:46:36.110 align:middle line:84% AUDIENCE: Just to clarify, when we 00:46:36.110 --> 00:46:39.390 align:middle line:84% say lotteries, that's what we're modeling your decision as 00:46:39.390 --> 00:46:41.530 align:middle line:84% is opting into a specific lottery. 00:46:41.530 --> 00:46:44.350 align:middle line:84% IAN BALL: So in the face of uncertainty, 00:46:44.350 --> 00:46:47.270 align:middle line:84% we're going to imagine that each choice that you make 00:46:47.270 --> 00:46:49.717 align:middle line:84% is going to induce some lottery of outcomes. 00:46:49.717 --> 00:46:52.050 align:middle line:84% And we're going to represent the choice by that lottery. 00:46:52.050 --> 00:46:54.738 align:middle line:84% This is an important point, because it could be that-- 00:46:54.738 --> 00:46:56.530 align:middle line:84% yeah, there's an embedded assumption there. 00:46:56.530 --> 00:46:57.870 align:middle line:84% But I'll just say yeah, that's what we're assuming. 00:46:57.870 --> 00:46:58.550 align:middle line:90% Yes. 00:46:58.550 --> 00:47:01.127 align:middle line:84% AUDIENCE: Can you just say what u would be in this case? 00:47:01.127 --> 00:47:01.710 align:middle line:90% IAN BALL: Yes. 00:47:01.710 --> 00:47:03.270 align:middle line:84% So let me write the definition and then I'll 00:47:03.270 --> 00:47:03.930 align:middle line:90% give you an example. 00:47:03.930 --> 00:47:04.430 align:middle line:90% Exactly. 00:47:04.430 --> 00:47:05.550 align:middle line:90% AUDIENCE: The little one. 00:47:05.550 --> 00:47:06.175 align:middle line:90% IAN BALL: Yeah. 00:47:06.175 --> 00:47:07.270 align:middle line:90% Yeah, exactly. 00:47:07.270 --> 00:47:09.130 align:middle line:84% So let's first give the general definition. 00:47:09.130 --> 00:47:10.338 align:middle line:90% Then I'll go to this example. 00:47:10.338 --> 00:47:12.910 align:middle line:90% So u of p equals what? 00:47:12.910 --> 00:47:16.590 align:middle line:84% Well, I'm just going to sum over all the possible outcomes. 00:47:16.590 --> 00:47:18.128 align:middle line:90% There are m outcomes. 00:47:18.128 --> 00:47:19.670 align:middle line:84% So I'm going to take-- and let me not 00:47:19.670 --> 00:47:21.190 align:middle line:84% use summation notation, because I think that makes 00:47:21.190 --> 00:47:22.482 align:middle line:90% it just seem harder than it is. 00:47:22.482 --> 00:47:25.850 align:middle line:90% So let me write p1 u of Z1. 00:47:25.850 --> 00:47:34.230 align:middle line:90% 00:47:34.230 --> 00:47:38.550 align:middle line:84% So I'm saying with probability p1, I get outcome Z1. 00:47:38.550 --> 00:47:39.970 align:middle line:90% And this is my utility. 00:47:39.970 --> 00:47:42.230 align:middle line:84% And I sum it all the way up until I get to pn. 00:47:42.230 --> 00:47:44.190 align:middle line:84% This is the probability of outcome n. 00:47:44.190 --> 00:47:46.030 align:middle line:90% And I get utility this. 00:47:46.030 --> 00:47:48.270 align:middle line:84% So if we wanted to really formally write 00:47:48.270 --> 00:47:50.790 align:middle line:90% this in the example over here-- 00:47:50.790 --> 00:47:58.230 align:middle line:84% so in the example, walking corresponded 00:47:58.230 --> 00:47:59.522 align:middle line:90% to a particular lottery, right? 00:47:59.522 --> 00:48:01.480 align:middle line:84% Now, if I really want to write out the example, 00:48:01.480 --> 00:48:03.450 align:middle line:84% I need to write out all four of these things. 00:48:03.450 --> 00:48:11.150 align:middle line:84% So maybe I'll call these Z1, Z2, Z3, Z4. 00:48:11.150 --> 00:48:11.650 align:middle line:90% OK? 00:48:11.650 --> 00:48:14.750 align:middle line:84% These are the four consequences of my actions. 00:48:14.750 --> 00:48:17.570 align:middle line:90% Now, what is walking here? 00:48:17.570 --> 00:48:19.470 align:middle line:84% We need to specify four numbers-- 00:48:19.470 --> 00:48:23.550 align:middle line:90% p1, p2, p3, p4. 00:48:23.550 --> 00:48:26.630 align:middle line:84% What are these four numbers in this example? 00:48:26.630 --> 00:48:31.510 align:middle line:84% Well, if I walk, I either get this box with probability p 00:48:31.510 --> 00:48:34.390 align:middle line:84% or this box with probability 1 minus p. 00:48:34.390 --> 00:48:37.870 align:middle line:84% So of these four numbers, two of them are 0. 00:48:37.870 --> 00:48:40.930 align:middle line:90% p3 and p4 are both 0. 00:48:40.930 --> 00:48:44.050 align:middle line:90% 00:48:44.050 --> 00:48:44.550 align:middle line:90% Why? 00:48:44.550 --> 00:48:48.107 align:middle line:84% Because it's impossible for me to get 00:48:48.107 --> 00:48:49.190 align:middle line:90% one of those consequences. 00:48:49.190 --> 00:48:50.565 align:middle line:84% If I choose to walk home, there's 00:48:50.565 --> 00:48:52.210 align:middle line:90% no way-- oh, I got unlucky. 00:48:52.210 --> 00:48:55.510 align:middle line:84% I'm now on the T. We're not imagining that can happen. 00:48:55.510 --> 00:48:58.030 align:middle line:84% What can happen is you can either walk home when it gets 00:48:58.030 --> 00:48:59.870 align:middle line:84% wet or you walk home and you don't. 00:48:59.870 --> 00:49:02.530 align:middle line:90% So we have Z1 is going to be p. 00:49:02.530 --> 00:49:05.910 align:middle line:90% 00:49:05.910 --> 00:49:07.830 align:middle line:90% 1 minus p here. 00:49:07.830 --> 00:49:13.030 align:middle line:84% And if we compute it, what is u of this particular p 00:49:13.030 --> 00:49:14.070 align:middle line:90% going to be? 00:49:14.070 --> 00:49:21.830 align:middle line:84% Well, we go through and we get pu of Z1 plus 1 minus pu 00:49:21.830 --> 00:49:23.968 align:middle line:90% of Z2 plus 0 plus 0. 00:49:23.968 --> 00:49:25.510 align:middle line:84% I'm not going to include the last two 00:49:25.510 --> 00:49:27.350 align:middle line:90% terms because they disappear. 00:49:27.350 --> 00:49:34.780 align:middle line:84% And this is going to be exactly what we said before, p times 7 00:49:34.780 --> 00:49:38.357 align:middle line:90% plus 1 minus p times 2. 00:49:38.357 --> 00:49:39.440 align:middle line:90% Which we already computed. 00:49:39.440 --> 00:49:40.440 align:middle line:90% Any questions on that? 00:49:40.440 --> 00:49:43.820 align:middle line:90% 00:49:43.820 --> 00:49:44.780 align:middle line:90% Yes. 00:49:44.780 --> 00:49:47.072 align:middle line:84% AUDIENCE: So when we're actually getting the value that 00:49:47.072 --> 00:49:49.740 align:middle line:84% goes from p to a real number, there's 00:49:49.740 --> 00:49:53.020 align:middle line:84% already an implicit choice, like here implicitly we're walking 00:49:53.020 --> 00:49:55.195 align:middle line:84% and we're not calculating this particular T. 00:49:55.195 --> 00:49:55.820 align:middle line:90% IAN BALL: Yeah. 00:49:55.820 --> 00:49:59.813 align:middle line:84% So what I would say is in principle, you can evaluate-- 00:49:59.813 --> 00:50:02.480 align:middle line:84% and this is the key distinction between preferences and choices. 00:50:02.480 --> 00:50:04.060 align:middle line:84% In principle, you could say, if I 00:50:04.060 --> 00:50:06.980 align:middle line:84% were offered a lottery between these outcomes, this lottery 00:50:06.980 --> 00:50:09.180 align:middle line:84% versus this lottery, what would I prefer? 00:50:09.180 --> 00:50:10.440 align:middle line:90% Those are your preferences. 00:50:10.440 --> 00:50:13.100 align:middle line:84% But in reality, you're given a limited menu of lotteries 00:50:13.100 --> 00:50:14.520 align:middle line:84% that you can actually choose from. 00:50:14.520 --> 00:50:16.200 align:middle line:84% And you're going to make a choice between those lotteries. 00:50:16.200 --> 00:50:19.220 align:middle line:84% So I could say in my head, what would I prefer between a stock 00:50:19.220 --> 00:50:22.460 align:middle line:84% that always pays a return of 1,000% and a stock that always 00:50:22.460 --> 00:50:23.835 align:middle line:84% pays a return of negative 1,000%? 00:50:23.835 --> 00:50:25.002 align:middle line:90% I know which one I'd prefer. 00:50:25.002 --> 00:50:26.380 align:middle line:84% And I can compute my preferences. 00:50:26.380 --> 00:50:27.900 align:middle line:84% But there may not actually be a company 00:50:27.900 --> 00:50:29.180 align:middle line:84% listed on the stock market that is 00:50:29.180 --> 00:50:30.320 align:middle line:90% going to give me that lottery. 00:50:30.320 --> 00:50:32.237 align:middle line:84% In reality, I'm going to face a limited choice 00:50:32.237 --> 00:50:34.000 align:middle line:90% set, a limited menu of options. 00:50:34.000 --> 00:50:35.920 align:middle line:84% Each of those options will induce a lottery, 00:50:35.920 --> 00:50:37.740 align:middle line:84% and I'll make a choice among this smaller 00:50:37.740 --> 00:50:38.920 align:middle line:90% collection of lotteries. 00:50:38.920 --> 00:50:41.740 align:middle line:84% In this case, that collection had two lotteries, the walking 00:50:41.740 --> 00:50:43.480 align:middle line:84% lottery and the taking the T lottery. 00:50:43.480 --> 00:50:45.480 align:middle line:84% AUDIENCE: When you're calculating this function, 00:50:45.480 --> 00:50:48.646 align:middle line:90% you've already chosen a lottery. 00:50:48.646 --> 00:50:50.740 align:middle line:84% Or you calculate this function based off of that. 00:50:50.740 --> 00:50:51.920 align:middle line:84% IAN BALL: I would say it's a function, 00:50:51.920 --> 00:50:53.960 align:middle line:84% so it's defined for every single lottery. 00:50:53.960 --> 00:50:56.860 align:middle line:84% I can say for every single lottery what my expected utility 00:50:56.860 --> 00:50:59.100 align:middle line:84% is, even for lotteries that I may not have the option 00:50:59.100 --> 00:51:01.740 align:middle line:90% to choose in a given situation. 00:51:01.740 --> 00:51:02.970 align:middle line:90% Yes. 00:51:02.970 --> 00:51:05.540 align:middle line:84% AUDIENCE: So even though p is determined on our beliefs, 00:51:05.540 --> 00:51:07.660 align:middle line:84% when it's a function of big U, do we just 00:51:07.660 --> 00:51:11.580 align:middle line:84% take it as exogenous and optimize based off of the things 00:51:11.580 --> 00:51:12.520 align:middle line:90% that we can control? 00:51:12.520 --> 00:51:13.520 align:middle line:90% IAN BALL: Good question. 00:51:13.520 --> 00:51:14.820 align:middle line:84% So this is going to be a big difference 00:51:14.820 --> 00:51:17.480 align:middle line:84% between individual decision making and interactive decision 00:51:17.480 --> 00:51:17.980 align:middle line:90% making. 00:51:17.980 --> 00:51:20.420 align:middle line:84% In individual decision making, all of this uncertainty 00:51:20.420 --> 00:51:21.720 align:middle line:90% is just going to be exogenous. 00:51:21.720 --> 00:51:23.200 align:middle line:84% And that's the model I've written down. 00:51:23.200 --> 00:51:25.242 align:middle line:84% But we'll see when we get to interactive decision 00:51:25.242 --> 00:51:27.580 align:middle line:84% making, well, how do I form beliefs about the choices 00:51:27.580 --> 00:51:28.600 align:middle line:90% that other people make? 00:51:28.600 --> 00:51:29.900 align:middle line:84% And then we're going to have to impose more 00:51:29.900 --> 00:51:31.025 align:middle line:90% structure on these beliefs. 00:51:31.025 --> 00:51:33.608 align:middle line:84% But for the individual context, we think of them as exogenous. 00:51:33.608 --> 00:51:35.442 align:middle line:84% Through introspection, you form your beliefs 00:51:35.442 --> 00:51:37.560 align:middle line:84% about what nature does, not what other players do. 00:51:37.560 --> 00:51:38.060 align:middle line:90% Yep. 00:51:38.060 --> 00:51:39.260 align:middle line:90% Thank you. 00:51:39.260 --> 00:51:42.660 align:middle line:84% OK, so one final thing in the last five minutes. 00:51:42.660 --> 00:51:46.420 align:middle line:84% There's a special case, which is of a lot of importance. 00:51:46.420 --> 00:51:54.140 align:middle line:84% And I'll say the key special case 00:51:54.140 --> 00:51:56.160 align:middle line:84% is going to be what we call money lotteries. 00:51:56.160 --> 00:52:00.560 align:middle line:90% 00:52:00.560 --> 00:52:02.060 align:middle line:84% So these are more like the lotteries 00:52:02.060 --> 00:52:04.580 align:middle line:90% that you hear about in reality. 00:52:04.580 --> 00:52:06.560 align:middle line:84% And why do I call it a special case? 00:52:06.560 --> 00:52:09.580 align:middle line:84% Well, it's the special case where Z, the set of outcomes, 00:52:09.580 --> 00:52:12.580 align:middle line:84% is just a set of monetary rewards. 00:52:12.580 --> 00:52:17.700 align:middle line:84% So I'll say Z is equal to R. And this means the real number line. 00:52:17.700 --> 00:52:19.162 align:middle line:90% Now, before I said Z was finite. 00:52:19.162 --> 00:52:20.620 align:middle line:84% Here I'm going to be a little loose 00:52:20.620 --> 00:52:22.340 align:middle line:90% and allow Z to be infinite. 00:52:22.340 --> 00:52:25.680 align:middle line:84% So the outcome here is I get $1, I get $10, 00:52:25.680 --> 00:52:26.680 align:middle line:90% I get a billion dollars. 00:52:26.680 --> 00:52:27.762 align:middle line:90% These are all outcomes. 00:52:27.762 --> 00:52:28.720 align:middle line:90% So what is the lottery? 00:52:28.720 --> 00:52:31.660 align:middle line:90% Let's consider two lotteries. 00:52:31.660 --> 00:52:35.520 align:middle line:90% One lottery maybe pays $10. 00:52:35.520 --> 00:52:55.780 align:middle line:90% 00:52:55.780 --> 00:52:57.700 align:middle line:84% So let's now consider two lotteries. 00:52:57.700 --> 00:52:59.672 align:middle line:84% Now here, with infinitely many things, 00:52:59.672 --> 00:53:00.880 align:middle line:90% it could be more complicated. 00:53:00.880 --> 00:53:02.140 align:middle line:84% But I'm going to look at lotteries 00:53:02.140 --> 00:53:04.340 align:middle line:84% where there's a small number of things that could actually 00:53:04.340 --> 00:53:05.000 align:middle line:90% happen. 00:53:05.000 --> 00:53:06.508 align:middle line:84% So under this lottery, there's only 00:53:06.508 --> 00:53:07.800 align:middle line:90% two outcomes that could happen. 00:53:07.800 --> 00:53:10.060 align:middle line:84% Every other outcome has probability 0. 00:53:10.060 --> 00:53:15.460 align:middle line:84% Under this lottery, with probability 0.99, I get $10. 00:53:15.460 --> 00:53:18.040 align:middle line:84% And with probability 0.01, I get nothing. 00:53:18.040 --> 00:53:19.340 align:middle line:90% I get 0. 00:53:19.340 --> 00:53:24.380 align:middle line:84% With this lottery, I get $1,000 with probability 0.01 and I get 00:53:24.380 --> 00:53:26.620 align:middle line:90% 0 with probability 0.99. 00:53:26.620 --> 00:53:27.840 align:middle line:90% So two different lotteries. 00:53:27.840 --> 00:53:29.240 align:middle line:90% They fit within our framework. 00:53:29.240 --> 00:53:31.920 align:middle line:84% And we can compute our expected utility of these lotteries. 00:53:31.920 --> 00:53:32.420 align:middle line:90% How? 00:53:32.420 --> 00:53:37.620 align:middle line:84% Well, to do that, we need our VNM utility function, which 00:53:37.620 --> 00:53:41.140 align:middle line:84% is a function, remember from Z to R. But in this case, 00:53:41.140 --> 00:53:47.940 align:middle line:84% Z is just R. But let's understand, this is money 00:53:47.940 --> 00:53:49.400 align:middle line:90% and this is utility. 00:53:49.400 --> 00:53:53.100 align:middle line:90% 00:53:53.100 --> 00:53:56.040 align:middle line:84% So which of these lotteries would you prefer? 00:53:56.040 --> 00:53:57.040 align:middle line:90% There's no right answer. 00:53:57.040 --> 00:53:59.660 align:middle line:84% Again, preferences are not irrational. 00:53:59.660 --> 00:54:02.600 align:middle line:84% But what's your preference between these? 00:54:02.600 --> 00:54:06.840 align:middle line:90% 00:54:06.840 --> 00:54:07.340 align:middle line:90% Yeah. 00:54:07.340 --> 00:54:08.380 align:middle line:90% AUDIENCE: I like the first one. 00:54:08.380 --> 00:54:09.060 align:middle line:84% IAN BALL: You like the first one? 00:54:09.060 --> 00:54:09.220 align:middle line:90% OK. 00:54:09.220 --> 00:54:10.400 align:middle line:90% I think most people would. 00:54:10.400 --> 00:54:12.980 align:middle line:90% And why is that? 00:54:12.980 --> 00:54:19.660 align:middle line:84% AUDIENCE: Because I don't know, it seems like 1,000 divided 00:54:19.660 --> 00:54:22.360 align:middle line:84% by whatever seems to be about 10, 00:54:22.360 --> 00:54:25.100 align:middle line:84% and then it feels more certain because it's 0.99. 00:54:25.100 --> 00:54:25.900 align:middle line:90% But I don't know. 00:54:25.900 --> 00:54:26.200 align:middle line:90% IAN BALL: Right. 00:54:26.200 --> 00:54:27.280 align:middle line:84% So that's a great way of thinking about it. 00:54:27.280 --> 00:54:28.738 align:middle line:84% So the first thing you might say is 00:54:28.738 --> 00:54:31.100 align:middle line:84% let's compare the expected values of these lotteries. 00:54:31.100 --> 00:54:34.020 align:middle line:84% The expected value of this is exactly $10. 00:54:34.020 --> 00:54:37.860 align:middle line:84% 100th of 1,000 has expected value $10. 00:54:37.860 --> 00:54:39.720 align:middle line:84% What's the expected value of this? 00:54:39.720 --> 00:54:42.060 align:middle line:90% It's $9.90. 00:54:42.060 --> 00:54:45.260 align:middle line:84% So this has a lower monetary expected value. 00:54:45.260 --> 00:54:48.060 align:middle line:84% But despite it having a lower monetary expected value, 00:54:48.060 --> 00:54:50.940 align:middle line:84% it might have a higher expected utility. 00:54:50.940 --> 00:54:54.680 align:middle line:84% And many people would choose this lottery over this lottery 00:54:54.680 --> 00:54:57.580 align:middle line:84% exactly because of what you said, it's safer. 00:54:57.580 --> 00:55:01.060 align:middle line:84% So what's key is through our Von Neumann-Morgenstern utility 00:55:01.060 --> 00:55:04.217 align:middle line:84% function u, when we evaluate lotteries, 00:55:04.217 --> 00:55:06.300 align:middle line:84% we don't just compare the expected monetary amount 00:55:06.300 --> 00:55:08.740 align:middle line:84% of the lotteries, we compare the expected utility 00:55:08.740 --> 00:55:11.700 align:middle line:90% between the lotteries. 00:55:11.700 --> 00:55:14.420 align:middle line:84% And what you're describing is what 00:55:14.420 --> 00:55:18.810 align:middle line:84% a lot of people feature in this context is, I'll say often, 00:55:18.810 --> 00:55:23.090 align:middle line:90% u is a concave function. 00:55:23.090 --> 00:55:27.330 align:middle line:84% So if I were to draw u for a lot of people, 00:55:27.330 --> 00:55:29.930 align:middle line:90% their u is something like this. 00:55:29.930 --> 00:55:32.090 align:middle line:84% And when u is concave, it means people 00:55:32.090 --> 00:55:34.790 align:middle line:90% are what we call risk averse. 00:55:34.790 --> 00:55:37.970 align:middle line:90% 00:55:37.970 --> 00:55:40.610 align:middle line:84% So in this case, you exhibited risk aversion. 00:55:40.610 --> 00:55:44.210 align:middle line:84% You preferred this lottery to this lottery, 00:55:44.210 --> 00:55:48.090 align:middle line:84% even though it had a smaller expected value. 00:55:48.090 --> 00:55:51.410 align:middle line:84% And if we want to formally define 00:55:51.410 --> 00:55:55.690 align:middle line:84% what does risk aversion mean, a risk averse person 00:55:55.690 --> 00:55:59.970 align:middle line:84% is someone who always prefers a certainty 00:55:59.970 --> 00:56:03.155 align:middle line:84% at the expected value of a lottery to the lottery itself. 00:56:03.155 --> 00:56:05.530 align:middle line:84% So let me just write this mathematically and then explain 00:56:05.530 --> 00:56:06.030 align:middle line:90% it. 00:56:06.030 --> 00:56:12.090 align:middle line:90% 00:56:12.090 --> 00:56:16.350 align:middle line:84% Someone is risk averse if deciding between a lottery p-- 00:56:16.350 --> 00:56:17.910 align:middle line:90% let's take this as an example. 00:56:17.910 --> 00:56:19.410 align:middle line:90% This is one lottery p. 00:56:19.410 --> 00:56:22.450 align:middle line:84% And another lottery that pays the expected value of p 00:56:22.450 --> 00:56:23.690 align:middle line:90% with certainty. 00:56:23.690 --> 00:56:26.747 align:middle line:84% In this case, that would be exactly $10. 00:56:26.747 --> 00:56:28.330 align:middle line:84% Someone who's risk averse would always 00:56:28.330 --> 00:56:31.290 align:middle line:84% prefer getting $10 for sure to getting 00:56:31.290 --> 00:56:34.670 align:middle line:84% a lottery whose expected monetary amount was $10. 00:56:34.670 --> 00:56:36.610 align:middle line:84% And mathematically, that's captured 00:56:36.610 --> 00:56:40.052 align:middle line:84% by the concavity of the utility function here, u. 00:56:40.052 --> 00:56:41.510 align:middle line:84% For this to be a strict preference, 00:56:41.510 --> 00:56:43.090 align:middle line:90% we need p to be non-degenerate. 00:56:43.090 --> 00:56:45.630 align:middle line:84% 1p would be you get something with probability 1. 00:56:45.630 --> 00:56:47.550 align:middle line:84% So as long as it's non-degenerate, 00:56:47.550 --> 00:56:49.870 align:middle line:84% this is the meaning of risk aversion. 00:56:49.870 --> 00:56:51.830 align:middle line:84% And this kind of answers a puzzle, I mean, 00:56:51.830 --> 00:56:53.790 align:middle line:84% going back to the 1700s where people said, 00:56:53.790 --> 00:56:55.808 align:middle line:84% why would you choose this lottery over this one? 00:56:55.808 --> 00:56:57.350 align:middle line:84% This one has a higher expected value. 00:56:57.350 --> 00:56:59.570 align:middle line:84% And the answer is, well, your utility function 00:56:59.570 --> 00:57:01.390 align:middle line:90% is not the same as money. 00:57:01.390 --> 00:57:03.310 align:middle line:84% You don't maximize expected money, 00:57:03.310 --> 00:57:04.910 align:middle line:90% you maximize expected utility. 00:57:04.910 --> 00:57:08.370 align:middle line:84% And some people's utilities can be concave like this. 00:57:08.370 --> 00:57:12.120 align:middle line:84% So let me stop there and I will see everyone next week. 00:57:12.120 --> 00:57:16.000 align:middle line:90%