WEBVTT 00:00:00.000 --> 00:00:01.931 align:middle line:90% [SQUEAKING] 00:00:01.931 --> 00:00:03.839 align:middle line:90% [RUSTLING] 00:00:03.839 --> 00:00:05.747 align:middle line:90% [CLICKING] 00:00:05.747 --> 00:00:10.517 align:middle line:90% 00:00:10.517 --> 00:00:13.100 align:middle line:84% IAN BALL: So today we're going to talk about Nash equilibrium. 00:00:13.100 --> 00:00:15.280 align:middle line:84% So this is one of the most important concepts 00:00:15.280 --> 00:00:16.880 align:middle line:90% in game theory. 00:00:16.880 --> 00:00:19.040 align:middle line:84% But I want to start by reminding you 00:00:19.040 --> 00:00:25.250 align:middle line:84% a little bit about rationalizability 00:00:25.250 --> 00:00:27.000 align:middle line:84% because I think that's going to provide us 00:00:27.000 --> 00:00:29.260 align:middle line:84% a good foundation for talking about Nash equilibrium, 00:00:29.260 --> 00:00:31.800 align:middle line:84% seeing how these two concepts differ. 00:00:31.800 --> 00:00:34.280 align:middle line:84% So as before and throughout this class, 00:00:34.280 --> 00:00:36.980 align:middle line:84% we're going to be given a strategic form game. 00:00:36.980 --> 00:00:38.900 align:middle line:84% And this notation is going to stay the same. 00:00:38.900 --> 00:00:40.300 align:middle line:84% So we have a strategic form game. 00:00:40.300 --> 00:00:45.020 align:middle line:84% Maybe I'll write strat-form game, 00:00:45.020 --> 00:00:51.280 align:middle line:84% which specifies the set of strategies, the n 00:00:51.280 --> 00:00:53.840 align:middle line:84% players, and then the payoffs that each player gets 00:00:53.840 --> 00:00:58.440 align:middle line:84% as a function of the strategy profiles that are chosen. 00:00:58.440 --> 00:01:00.940 align:middle line:84% And last time, we defined this procedure, 00:01:00.940 --> 00:01:04.260 align:middle line:84% which we called iterated elimination of strictly 00:01:04.260 --> 00:01:05.780 align:middle line:90% dominated strategies. 00:01:05.780 --> 00:01:08.140 align:middle line:84% And this was an iterative algorithm 00:01:08.140 --> 00:01:12.300 align:middle line:84% that gradually narrowed down or pared down the strategies 00:01:12.300 --> 00:01:13.640 align:middle line:90% that the players had. 00:01:13.640 --> 00:01:17.900 align:middle line:84% So we started with, if you remember, the first stage 00:01:17.900 --> 00:01:20.220 align:middle line:84% or stage 0, the initialization stage 00:01:20.220 --> 00:01:22.580 align:middle line:84% was just the set of all strategies, 00:01:22.580 --> 00:01:25.060 align:middle line:90% all strategy profiles. 00:01:25.060 --> 00:01:33.380 align:middle line:84% And then we went to big S, S1, S2, 00:01:33.380 --> 00:01:39.060 align:middle line:84% and we kept going down until maybe we had a generic SK, 00:01:39.060 --> 00:01:43.620 align:middle line:84% and then, at the end, we had something called S infinity. 00:01:43.620 --> 00:01:45.260 align:middle line:84% And it's important to understand-- 00:01:45.260 --> 00:01:48.220 align:middle line:90% maybe I'll write this up here-- 00:01:48.220 --> 00:01:51.300 align:middle line:84% the interpretation of each of these sets. 00:01:51.300 --> 00:01:54.160 align:middle line:84% So below, let me say what is the interpretation? 00:01:54.160 --> 00:01:58.540 align:middle line:90% 00:01:58.540 --> 00:02:00.040 align:middle line:84% The interpretation is that, as we're 00:02:00.040 --> 00:02:03.640 align:middle line:84% willing to make stronger and stronger behavioral assumptions 00:02:03.640 --> 00:02:05.480 align:middle line:84% about the way that the players play, 00:02:05.480 --> 00:02:08.240 align:middle line:84% we're able to make sharper and sharper predictions 00:02:08.240 --> 00:02:11.680 align:middle line:84% about the strategies that we could observe in practice. 00:02:11.680 --> 00:02:13.920 align:middle line:90% So we start with S0. 00:02:13.920 --> 00:02:16.420 align:middle line:90% This imposed no assumptions. 00:02:16.420 --> 00:02:20.520 align:middle line:90% 00:02:20.520 --> 00:02:23.240 align:middle line:84% If we make no assumptions about how people play, then, 00:02:23.240 --> 00:02:24.920 align:middle line:90% of course, anything is possible. 00:02:24.920 --> 00:02:28.040 align:middle line:84% The set of strategy profiles that the players may choose 00:02:28.040 --> 00:02:32.640 align:middle line:84% is unrestricted, but then we move to S1, 00:02:32.640 --> 00:02:37.240 align:middle line:84% where, there, the interpretation is rationality. 00:02:37.240 --> 00:02:39.960 align:middle line:84% What we said is, what if we assume that every player is 00:02:39.960 --> 00:02:43.360 align:middle line:84% rational in the sense that every player plays a best 00:02:43.360 --> 00:02:46.080 align:middle line:84% response to some belief about the opposing player's 00:02:46.080 --> 00:02:47.480 align:middle line:90% strategies? 00:02:47.480 --> 00:02:49.497 align:middle line:84% And if we assume that every player is rational, 00:02:49.497 --> 00:02:51.080 align:middle line:84% then we can narrow down our prediction 00:02:51.080 --> 00:02:53.200 align:middle line:90% in the game to the set S1. 00:02:53.200 --> 00:02:56.620 align:middle line:84% So capital S1 is the set of strategy profiles 00:02:56.620 --> 00:02:59.700 align:middle line:84% that are consistent with every player's rationality. 00:02:59.700 --> 00:03:02.500 align:middle line:84% And just to remember-- maybe I'll put it down here-- 00:03:02.500 --> 00:03:08.020 align:middle line:84% when I write S capital K, this is really a product. 00:03:08.020 --> 00:03:09.100 align:middle line:90% Like this. 00:03:09.100 --> 00:03:12.780 align:middle line:84% So S capital K is a set of strategy profiles 00:03:12.780 --> 00:03:16.620 align:middle line:84% that contain a strategy for player 1, in this set S1 K, 00:03:16.620 --> 00:03:21.260 align:middle line:84% all the way up to a strategy for player n in the set SN K. 00:03:21.260 --> 00:03:23.400 align:middle line:84% And then we could go beyond just rationality. 00:03:23.400 --> 00:03:27.340 align:middle line:84% Remember, we went to S2, and there, we 00:03:27.340 --> 00:03:29.600 align:middle line:84% assumed not only that the players are rational 00:03:29.600 --> 00:03:32.600 align:middle line:84% but that the players know that everyone else is rational. 00:03:32.600 --> 00:03:38.100 align:middle line:84% So we might call this rationality plus knowledge 00:03:38.100 --> 00:03:41.180 align:middle line:90% of rationality. 00:03:41.180 --> 00:03:43.500 align:middle line:84% I'll just write knowledge of R, so rationality 00:03:43.500 --> 00:03:46.522 align:middle line:90% plus knowledge of rationality. 00:03:46.522 --> 00:03:47.980 align:middle line:84% But then, as we keep going down, we 00:03:47.980 --> 00:03:50.020 align:middle line:84% can get to higher- and higher-order knowledge 00:03:50.020 --> 00:03:51.080 align:middle line:90% of rationality. 00:03:51.080 --> 00:03:54.920 align:middle line:84% So when we get to SK, the interpretation of this is SK is 00:03:54.920 --> 00:03:57.200 align:middle line:84% our prediction in the game if we're willing to assume 00:03:57.200 --> 00:04:00.360 align:middle line:84% rationality, knowledge of rationality, 00:04:00.360 --> 00:04:01.890 align:middle line:84% knowledge of others' rationality, 00:04:01.890 --> 00:04:04.140 align:middle line:84% knowledge of others' knowledge of others' rationality, 00:04:04.140 --> 00:04:09.240 align:middle line:84% and so on, all the way down to maybe I'll say rationality plus 00:04:09.240 --> 00:04:11.860 align:middle line:84% "K minus 1 order" knowledge of rationality. 00:04:11.860 --> 00:04:19.200 align:middle line:90% 00:04:19.200 --> 00:04:20.560 align:middle line:90% Why K minus 1? 00:04:20.560 --> 00:04:23.100 align:middle line:84% Well, the first stage, we had rationality 00:04:23.100 --> 00:04:25.760 align:middle line:84% but no knowledge of rationality, so the degree of knowledge 00:04:25.760 --> 00:04:29.200 align:middle line:84% of rationality is always just 1 smaller than our number K. 00:04:29.200 --> 00:04:32.880 align:middle line:84% And when we get up to K, this means the players are rational. 00:04:32.880 --> 00:04:34.380 align:middle line:84% They know everyone else is rational, 00:04:34.380 --> 00:04:36.755 align:middle line:84% they know that everyone else knows that they're rational, 00:04:36.755 --> 00:04:37.420 align:middle line:90% and so on. 00:04:37.420 --> 00:04:39.240 align:middle line:84% And those statements go on until I say 00:04:39.240 --> 00:04:42.553 align:middle line:90% the word knows k minus 1 times. 00:04:42.553 --> 00:04:44.220 align:middle line:84% And then if we want to be really formal, 00:04:44.220 --> 00:04:46.240 align:middle line:84% there's a word for this last set. 00:04:46.240 --> 00:04:48.320 align:middle line:84% So this last set, these are the set 00:04:48.320 --> 00:04:50.220 align:middle line:90% of rationalizable strategies. 00:04:50.220 --> 00:04:55.060 align:middle line:90% 00:04:55.060 --> 00:04:56.540 align:middle line:84% And the interpretation of this is 00:04:56.540 --> 00:04:58.900 align:middle line:84% that these are the strategy profiles that 00:04:58.900 --> 00:05:01.660 align:middle line:84% are consistent with rationality and arbitrarily 00:05:01.660 --> 00:05:03.980 align:middle line:84% high degrees of knowledge of rationality. 00:05:03.980 --> 00:05:05.900 align:middle line:84% So not only does each player everyone else 00:05:05.900 --> 00:05:08.460 align:middle line:84% knows their rational, but I can continue those chains 00:05:08.460 --> 00:05:09.980 align:middle line:90% arbitrarily deep. 00:05:09.980 --> 00:05:11.580 align:middle line:84% And we have a word for that-- that's 00:05:11.580 --> 00:05:14.780 align:middle line:84% called common knowledge of rationality. 00:05:14.780 --> 00:05:27.860 align:middle line:84% So I'll write-- just write R. So rationalizable captures the idea 00:05:27.860 --> 00:05:29.720 align:middle line:84% that not only are the players rational, 00:05:29.720 --> 00:05:32.460 align:middle line:84% but they have common knowledge of rationality in the sense that 00:05:32.460 --> 00:05:35.940 align:middle line:84% they have K-th order knowledge of rationality for any K. 00:05:35.940 --> 00:05:38.160 align:middle line:84% They have first order, second order, third order, 00:05:38.160 --> 00:05:41.700 align:middle line:84% fourth order, 100th order, and so on. 00:05:41.700 --> 00:05:46.580 align:middle line:84% So this is kind of a summary of where we've gone so far. 00:05:46.580 --> 00:05:50.090 align:middle line:84% And I want you to notice how, as we continue to the right here, 00:05:50.090 --> 00:05:52.730 align:middle line:84% we make stronger and stronger behavioral assumptions, 00:05:52.730 --> 00:05:54.890 align:middle line:84% and we get sharper and sharper predictions. 00:05:54.890 --> 00:05:57.450 align:middle line:84% But sometimes, even this set S infinity 00:05:57.450 --> 00:05:59.550 align:middle line:84% is not going to give us a very sharp prediction. 00:05:59.550 --> 00:06:00.990 align:middle line:84% And we want to go a step further. 00:06:00.990 --> 00:06:03.530 align:middle line:84% And that's what Nash equilibrium is going to do today. 00:06:03.530 --> 00:06:06.530 align:middle line:84% So to motivate that, let's start with 00:06:06.530 --> 00:06:10.970 align:middle line:84% or revisit an example that we've done before, which 00:06:10.970 --> 00:06:13.190 align:middle line:90% we call the Boston game, BOS. 00:06:13.190 --> 00:06:14.690 align:middle line:84% And remember, the story of this game 00:06:14.690 --> 00:06:16.930 align:middle line:84% is we had two friends who like each other's company 00:06:16.930 --> 00:06:18.550 align:middle line:84% but have different sports preferences. 00:06:18.550 --> 00:06:21.170 align:middle line:84% So they're choosing whether to go to a Celtics game or a Red 00:06:21.170 --> 00:06:22.070 align:middle line:90% Sox game. 00:06:22.070 --> 00:06:24.550 align:middle line:84% They're both happier if they go together than apart, 00:06:24.550 --> 00:06:27.130 align:middle line:84% but one person prefers the Celtics and one person 00:06:27.130 --> 00:06:28.490 align:middle line:90% prefers the Red Sox. 00:06:28.490 --> 00:06:32.322 align:middle line:84% So if we write out the game like this-- 00:06:32.322 --> 00:06:34.530 align:middle line:84% I think I'll use slightly different numbers this time 00:06:34.530 --> 00:06:37.670 align:middle line:84% to make some of the math later a bit easier. 00:06:37.670 --> 00:06:40.850 align:middle line:90% 00:06:40.850 --> 00:06:42.090 align:middle line:90% So let's go over it. 00:06:42.090 --> 00:06:45.250 align:middle line:84% If we choose differently, that's along this off diagonal, 00:06:45.250 --> 00:06:47.710 align:middle line:84% then we both get a payoff of 0 because we don't 00:06:47.710 --> 00:06:50.710 align:middle line:90% get to be together at the game. 00:06:50.710 --> 00:06:52.690 align:middle line:84% If we both go to the Celtics game, 00:06:52.690 --> 00:06:54.470 align:middle line:84% then we both get a positive payoff. 00:06:54.470 --> 00:06:56.870 align:middle line:84% But it's higher for player 1 because player 1 00:06:56.870 --> 00:06:58.510 align:middle line:90% likes the Celtics more. 00:06:58.510 --> 00:07:01.810 align:middle line:84% And then conversely, if we both go to the Red Sox game, 00:07:01.810 --> 00:07:03.410 align:middle line:90% we both get a positive payout. 00:07:03.410 --> 00:07:06.070 align:middle line:84% But here, it's the second player who prefers the Red Sox 00:07:06.070 --> 00:07:09.150 align:middle line:84% and therefore, gets the higher payoff of 2. 00:07:09.150 --> 00:07:11.950 align:middle line:84% So let's go through our procedure 00:07:11.950 --> 00:07:14.830 align:middle line:84% for calculating best responses here. 00:07:14.830 --> 00:07:17.130 align:middle line:84% If my opponent is going to the Celtics game, 00:07:17.130 --> 00:07:18.922 align:middle line:84% I certainly want to go to the Celtics game. 00:07:18.922 --> 00:07:20.630 align:middle line:84% And if they're going to the Red Sox game, 00:07:20.630 --> 00:07:22.090 align:middle line:84% I want to go to the Red Sox game. 00:07:22.090 --> 00:07:23.590 align:middle line:84% And then we can do it the other way. 00:07:23.590 --> 00:07:25.870 align:middle line:84% Now, I'll put myself in the shoes of player 2. 00:07:25.870 --> 00:07:29.870 align:middle line:84% I'm saying, if player 1 plays C, well, I look at my payoffs. 00:07:29.870 --> 00:07:33.230 align:middle line:84% I prefer C. If player one plays R, I look at my payoffs, 00:07:33.230 --> 00:07:35.230 align:middle line:90% I prefer this. 00:07:35.230 --> 00:07:38.110 align:middle line:84% So now, if we look at that, can anyone 00:07:38.110 --> 00:07:41.390 align:middle line:84% see, what are the set of rationalizable strategies 00:07:41.390 --> 00:07:43.610 align:middle line:84% or strategy profiles in this game? 00:07:43.610 --> 00:07:48.810 align:middle line:84% So maybe we'll separate it out just to be clear. 00:07:48.810 --> 00:07:51.050 align:middle line:84% So what are the set of rationalizable strategies 00:07:51.050 --> 00:07:52.710 align:middle line:90% for player 1 in this game? 00:07:52.710 --> 00:08:02.290 align:middle line:90% 00:08:02.290 --> 00:08:04.250 align:middle line:84% Well, here, it's going to be just C 00:08:04.250 --> 00:08:09.330 align:middle line:84% And R because, here, if my opponent plays C, 00:08:09.330 --> 00:08:13.512 align:middle line:84% I want to play C. If my opponent plays R, I want to play R. 00:08:13.512 --> 00:08:14.970 align:middle line:84% So it turns out everything is going 00:08:14.970 --> 00:08:17.290 align:middle line:84% to be rationalizable in this game. 00:08:17.290 --> 00:08:18.850 align:middle line:90% So we're going to get-- 00:08:18.850 --> 00:08:22.050 align:middle line:84% if we go through the procedure C, R, 00:08:22.050 --> 00:08:26.970 align:middle line:84% and if we go through the procedure, we also get C, R. 00:08:26.970 --> 00:08:29.090 align:middle line:84% And just to make sure we understand the notation, 00:08:29.090 --> 00:08:32.710 align:middle line:84% S infinity is the set of rationalizable profiles. 00:08:32.710 --> 00:08:34.669 align:middle line:84% So this is going to have four elements. 00:08:34.669 --> 00:08:41.770 align:middle line:84% It's going to have CC, CR, RC, and RR. 00:08:41.770 --> 00:08:43.370 align:middle line:84% Sometimes I'll write with parentheses, 00:08:43.370 --> 00:08:45.150 align:middle line:84% but it just gets a little messy here. 00:08:45.150 --> 00:08:47.870 align:middle line:84% So this says there's four strategy profiles. 00:08:47.870 --> 00:08:51.190 align:middle line:84% This is the profile where we both go to the Celtics game. 00:08:51.190 --> 00:08:54.070 align:middle line:84% This is the profile where we both go to the Red Sox game. 00:08:54.070 --> 00:08:56.170 align:middle line:84% And the order here indicates who does what. 00:08:56.170 --> 00:08:59.350 align:middle line:84% So the first player plays C, the second player plays R, 00:08:59.350 --> 00:09:01.630 align:middle line:84% and then, conversely, the first player plays R 00:09:01.630 --> 00:09:03.830 align:middle line:90% and the second player plays C. 00:09:03.830 --> 00:09:07.310 align:middle line:84% So in this case, everything is rationalizable. 00:09:07.310 --> 00:09:11.150 align:middle line:84% So this is a case where rationality and common knowledge 00:09:11.150 --> 00:09:14.710 align:middle line:84% of rationality alone doesn't give us a very good prediction 00:09:14.710 --> 00:09:16.908 align:middle line:84% or doesn't rule out many possibilities. 00:09:16.908 --> 00:09:18.950 align:middle line:84% So we're not able to get a very sharp prediction. 00:09:18.950 --> 00:09:22.430 align:middle line:84% And let's see, so in particular, one rationalizable strategy, 00:09:22.430 --> 00:09:25.750 align:middle line:90% profile, I should say, is this. 00:09:25.750 --> 00:09:29.150 align:middle line:90% So this is rationalizable. 00:09:29.150 --> 00:09:32.110 align:middle line:84% But it may seem like something is going wrong here. 00:09:32.110 --> 00:09:34.607 align:middle line:84% So does this seem like a reasonable prediction 00:09:34.607 --> 00:09:35.690 align:middle line:90% of how people are playing? 00:09:35.690 --> 00:09:39.430 align:middle line:84% Or would you say someone's making a mistake here? 00:09:39.430 --> 00:09:41.670 align:middle line:84% What seems strange about this-- when 00:09:41.670 --> 00:09:45.130 align:middle line:84% player 1 is going to the Red Sox and player 2 is 00:09:45.130 --> 00:09:48.490 align:middle line:90% going to the Celtics game? 00:09:48.490 --> 00:09:48.990 align:middle line:90% Yeah. 00:09:48.990 --> 00:09:51.070 align:middle line:84% AUDIENCE: [INAUDIBLE] since they're both losing. 00:09:51.070 --> 00:09:52.410 align:middle line:90% IAN BALL: They're both losing. 00:09:52.410 --> 00:09:55.310 align:middle line:84% And in a sense, well, I'm going to the Red Sox game, 00:09:55.310 --> 00:09:56.970 align:middle line:84% but if I just changed my behavior 00:09:56.970 --> 00:09:59.083 align:middle line:84% and went to the Celtics game, I'd be better off. 00:09:59.083 --> 00:10:01.750 align:middle line:84% And similarly, the other person who's alone at the Celtics game, 00:10:01.750 --> 00:10:04.042 align:middle line:84% if they move to the Red Sox game, they'd be better off. 00:10:04.042 --> 00:10:07.890 align:middle line:84% So there's a bit of this incongruity 00:10:07.890 --> 00:10:09.490 align:middle line:84% that we say, OK, this is consistent 00:10:09.490 --> 00:10:12.173 align:middle line:84% with rationality and common knowledge of rationality, 00:10:12.173 --> 00:10:14.090 align:middle line:84% but it might feel somehow that the players are 00:10:14.090 --> 00:10:15.490 align:middle line:90% making a mistake. 00:10:15.490 --> 00:10:19.230 align:middle line:84% So maybe I'll say this is all a bit informal. 00:10:19.230 --> 00:10:20.430 align:middle line:90% It feels like a mistake. 00:10:20.430 --> 00:10:25.370 align:middle line:90% 00:10:25.370 --> 00:10:26.790 align:middle line:90% And let's try to reconcile this. 00:10:26.790 --> 00:10:29.890 align:middle line:84% Why does this satisfy the definition of rationalizability 00:10:29.890 --> 00:10:31.310 align:middle line:90% but feel like a mistake? 00:10:31.310 --> 00:10:32.685 align:middle line:84% And I think the best way to think 00:10:32.685 --> 00:10:35.730 align:middle line:84% about that is, what is rationalizability really mean? 00:10:35.730 --> 00:10:37.330 align:middle line:84% It really means that a player has 00:10:37.330 --> 00:10:40.430 align:middle line:84% an explanation for why they're doing what they're doing. 00:10:40.430 --> 00:10:42.030 align:middle line:84% So I think rationalizability should 00:10:42.030 --> 00:10:48.730 align:middle line:84% be thought of as it means that a player has an explanation. 00:10:48.730 --> 00:10:51.910 align:middle line:90% 00:10:51.910 --> 00:10:54.310 align:middle line:84% So what is player 1's explanation 00:10:54.310 --> 00:10:56.060 align:middle line:84% for why they're going to the Red Sox game? 00:10:56.060 --> 00:10:58.518 align:middle line:84% They would say, well, it's reasonable, it's rational for me 00:10:58.518 --> 00:10:59.990 align:middle line:84% to go to the Red Sox game because I 00:10:59.990 --> 00:11:04.195 align:middle line:84% believe that my friend is also going to the Red Sox game. 00:11:04.195 --> 00:11:05.570 align:middle line:84% And then we can go a step deeper. 00:11:05.570 --> 00:11:08.430 align:middle line:84% We could say, well, why do you believe that your friend is 00:11:08.430 --> 00:11:09.610 align:middle line:90% going to the Red Sox game? 00:11:09.610 --> 00:11:10.590 align:middle line:84% And you could say, well, I believe 00:11:10.590 --> 00:11:12.330 align:middle line:84% my friend is going to the Red Sox game 00:11:12.330 --> 00:11:14.413 align:middle line:84% because I believe that my friend believes that I'm 00:11:14.413 --> 00:11:15.590 align:middle line:90% going to the Red Sox game. 00:11:15.590 --> 00:11:16.650 align:middle line:90% And then you go a step deeper. 00:11:16.650 --> 00:11:18.750 align:middle line:84% You can say, why do you believe that your friend believes 00:11:18.750 --> 00:11:19.970 align:middle line:84% that you're going to the Red Sox game? 00:11:19.970 --> 00:11:22.270 align:middle line:84% Well, I think my friend believes that I believe 00:11:22.270 --> 00:11:25.130 align:middle line:84% that my friend believes he's going to the Red Sox game, 00:11:25.130 --> 00:11:27.770 align:middle line:84% and that explains his belief, and we can go on and on and on. 00:11:27.770 --> 00:11:30.450 align:middle line:84% And that's precisely what rationalizability captures. 00:11:30.450 --> 00:11:32.550 align:middle line:84% It says you can provide an explanation that 00:11:32.550 --> 00:11:33.678 align:middle line:90% goes arbitrarily deep. 00:11:33.678 --> 00:11:35.470 align:middle line:84% So think of this as the answer to the child 00:11:35.470 --> 00:11:36.877 align:middle line:90% who always says, but why this? 00:11:36.877 --> 00:11:38.210 align:middle line:90% You can always go a step deeper. 00:11:38.210 --> 00:11:39.310 align:middle line:90% Why do they have this belief? 00:11:39.310 --> 00:11:41.685 align:middle line:84% Well, they have this belief because of this other belief. 00:11:41.685 --> 00:11:43.210 align:middle line:90% And that chain never stops. 00:11:43.210 --> 00:11:45.403 align:middle line:84% If it weren't rationalizable, eventually, we'd 00:11:45.403 --> 00:11:46.070 align:middle line:90% reach a problem. 00:11:46.070 --> 00:11:47.890 align:middle line:84% We'd reach a contradiction in that chain. 00:11:47.890 --> 00:11:50.690 align:middle line:84% So rationalizability says, there exists some explanation 00:11:50.690 --> 00:11:51.770 align:middle line:90% for your play. 00:11:51.770 --> 00:11:54.250 align:middle line:84% The problem is your explanation may not be correct. 00:11:54.250 --> 00:11:56.270 align:middle line:84% In theory, it's a good explanation, 00:11:56.270 --> 00:11:58.410 align:middle line:84% but it might not actually comport with the way 00:11:58.410 --> 00:12:00.330 align:middle line:90% that people are playing. 00:12:00.330 --> 00:12:03.230 align:middle line:84% So rationalizability means you have an explanation. 00:12:03.230 --> 00:12:05.490 align:middle line:84% You're behaving optimally, given this explanation 00:12:05.490 --> 00:12:08.050 align:middle line:84% or given these beliefs, but there's no requirement 00:12:08.050 --> 00:12:18.050 align:middle line:84% that the explanation is correct, that your beliefs 00:12:18.050 --> 00:12:19.230 align:middle line:90% or your explanation-- 00:12:19.230 --> 00:12:23.030 align:middle line:84% I'll use these explanations a bit deeper. 00:12:23.030 --> 00:12:28.690 align:middle line:90% 00:12:28.690 --> 00:12:32.370 align:middle line:84% Because if we're down here, my explanation involved my belief 00:12:32.370 --> 00:12:37.010 align:middle line:84% that my friend was also going to the Red Sox game, but in fact, 00:12:37.010 --> 00:12:38.210 align:middle line:90% my friend is not doing that. 00:12:38.210 --> 00:12:39.670 align:middle line:84% My friend is going to the Celtics game, 00:12:39.670 --> 00:12:41.070 align:middle line:90% and that's why I'm doing badly. 00:12:41.070 --> 00:12:43.390 align:middle line:84% So what Nash equilibrium is going to capture 00:12:43.390 --> 00:12:45.430 align:middle line:84% is it's going to go a step beyond simply 00:12:45.430 --> 00:12:47.150 align:middle line:90% providing an explanation. 00:12:47.150 --> 00:12:49.590 align:middle line:84% It's going to require that the explanations we provide 00:12:49.590 --> 00:12:50.730 align:middle line:90% are also correct. 00:12:50.730 --> 00:12:52.790 align:middle line:84% That is, they also accurately describe the way 00:12:52.790 --> 00:12:54.665 align:middle line:84% that our opponents or that our fellow players 00:12:54.665 --> 00:12:56.990 align:middle line:90% are actually playing. 00:12:56.990 --> 00:13:01.670 align:middle line:84% So let's now give the formal definition of this. 00:13:01.670 --> 00:13:04.130 align:middle line:84% So if you learn one thing about game theory, 00:13:04.130 --> 00:13:08.033 align:middle line:84% this is often the definition you see. 00:13:08.033 --> 00:13:09.950 align:middle line:84% In the movie about Nash, they talk about this. 00:13:09.950 --> 00:13:12.450 align:middle line:84% Though, actually, their story is kind of wrong in the movie. 00:13:12.450 --> 00:13:15.050 align:middle line:84% It doesn't make mathematical sense. 00:13:15.050 --> 00:13:16.030 align:middle line:90% But anyway. 00:13:16.030 --> 00:13:18.530 align:middle line:90% OK, so a strategy profile-- 00:13:18.530 --> 00:13:27.887 align:middle line:90% 00:13:27.887 --> 00:13:29.470 align:middle line:84% let me just stop here and say, I think 00:13:29.470 --> 00:13:31.678 align:middle line:84% this is really an important aspect of the definition, 00:13:31.678 --> 00:13:35.090 align:middle line:84% that the thing that a Nash equilibrium will be 00:13:35.090 --> 00:13:36.690 align:middle line:90% is a strategy profile. 00:13:36.690 --> 00:13:38.208 align:middle line:84% And often, a mistake students make 00:13:38.208 --> 00:13:40.250 align:middle line:84% on exams is they're asked to give an equilibrium, 00:13:40.250 --> 00:13:42.517 align:middle line:84% and they often don't give a strategy profile. 00:13:42.517 --> 00:13:44.850 align:middle line:84% They might give the payoffs that the players get if they 00:13:44.850 --> 00:13:46.230 align:middle line:90% use that strategy profile. 00:13:46.230 --> 00:13:48.690 align:middle line:84% But an equilibrium is always a strategy profile 00:13:48.690 --> 00:13:52.022 align:middle line:84% because it's a prediction or recommendation for how 00:13:52.022 --> 00:13:53.230 align:middle line:90% people will play in the game. 00:13:53.230 --> 00:13:54.950 align:middle line:90% So this is the key first part. 00:13:54.950 --> 00:13:57.210 align:middle line:84% It's a profile specifying a strategy 00:13:57.210 --> 00:13:58.810 align:middle line:84% for the first player, all the way 00:13:58.810 --> 00:14:00.970 align:middle line:84% up to a strategy for the n-th player. 00:14:00.970 --> 00:14:04.670 align:middle line:84% So this kind of strategy profile is a Nash equilibrium. 00:14:04.670 --> 00:14:09.850 align:middle line:90% 00:14:09.850 --> 00:14:14.210 align:middle line:84% So this is named after John Nash who won the Nobel 00:14:14.210 --> 00:14:15.030 align:middle line:90% Prize in economics. 00:14:15.030 --> 00:14:16.990 align:middle line:90% He wrote this paper in '48, '49. 00:14:16.990 --> 00:14:19.290 align:middle line:84% He wrote a few papers that introduced this concept. 00:14:19.290 --> 00:14:21.590 align:middle line:84% It's actually interesting to read the original paper. 00:14:21.590 --> 00:14:23.813 align:middle line:84% The original paper is like two paragraphs long, 00:14:23.813 --> 00:14:25.230 align:middle line:84% and he got the Nobel Prize for it. 00:14:25.230 --> 00:14:27.090 align:middle line:84% So if you have good ideas, you don't 00:14:27.090 --> 00:14:28.430 align:middle line:90% need to go on and on and on. 00:14:28.430 --> 00:14:31.010 align:middle line:84% So a strategy profile is a Nash equilibrium 00:14:31.010 --> 00:14:41.310 align:middle line:84% if, for every player i, the following holds. 00:14:41.310 --> 00:14:44.150 align:middle line:84% So I'm going to put myself in the position of player i, 00:14:44.150 --> 00:14:47.670 align:middle line:84% and I'm going to compare the payoff I get from my strategy 00:14:47.670 --> 00:14:49.830 align:middle line:84% to the payoff I could get if I deviated 00:14:49.830 --> 00:14:51.230 align:middle line:90% to a different strategy. 00:14:51.230 --> 00:14:59.310 align:middle line:84% So I'm going to compare ui of si star on the left to ui of si 00:14:59.310 --> 00:15:01.310 align:middle line:90% prime on the right. 00:15:01.310 --> 00:15:04.470 align:middle line:84% So I'm comparing-- I'm player i, and I'm 00:15:04.470 --> 00:15:07.190 align:middle line:84% saying if I choose si star, what I'm 00:15:07.190 --> 00:15:08.950 align:middle line:84% supposed to do in equilibrium, I want 00:15:08.950 --> 00:15:12.150 align:middle line:84% to do better weekly than if I choose some other strategy 00:15:12.150 --> 00:15:13.235 align:middle line:90% si prime. 00:15:13.235 --> 00:15:14.610 align:middle line:84% But I still have to fill this in. 00:15:14.610 --> 00:15:18.150 align:middle line:84% And this is going to say, given the way that the other players 00:15:18.150 --> 00:15:18.870 align:middle line:90% are playing. 00:15:18.870 --> 00:15:22.110 align:middle line:84% So what I'm going to fill in here is s negative i star 00:15:22.110 --> 00:15:24.750 align:middle line:90% and s negative i star. 00:15:24.750 --> 00:15:26.830 align:middle line:84% I can't control how the other people are playing. 00:15:26.830 --> 00:15:28.010 align:middle line:90% So I take that as fixed. 00:15:28.010 --> 00:15:29.030 align:middle line:90% I take that as given. 00:15:29.030 --> 00:15:31.570 align:middle line:84% Given The way that my opponents are playing, 00:15:31.570 --> 00:15:35.130 align:middle line:84% I just do what I can control, I compare the strategy 00:15:35.130 --> 00:15:37.920 align:middle line:84% I'm supposed to choose to any deviation, 00:15:37.920 --> 00:15:40.170 align:middle line:84% and I require that the strategy I'm supposed to choose 00:15:40.170 --> 00:15:52.290 align:middle line:84% is weakly better for all si prime in si. 00:15:52.290 --> 00:15:56.530 align:middle line:84% In particular, si prime could just be si star, 00:15:56.530 --> 00:15:58.770 align:middle line:84% but then this is going to be satisfied automatically 00:15:58.770 --> 00:16:00.603 align:middle line:84% because both sides are going to be the same. 00:16:00.603 --> 00:16:04.130 align:middle line:90% 00:16:04.130 --> 00:16:06.930 align:middle line:84% So let's think of another way to say this. 00:16:06.930 --> 00:16:09.490 align:middle line:84% Another way to say this-- so this is equivalent to this 00:16:09.490 --> 00:16:10.810 align:middle line:90% statement here-- 00:16:10.810 --> 00:16:26.250 align:middle line:84% is that si star is a best response to s negative i star. 00:16:26.250 --> 00:16:28.210 align:middle line:84% So another way of defining a Nash equilibrium 00:16:28.210 --> 00:16:30.250 align:middle line:84% is to say that every player I look at, 00:16:30.250 --> 00:16:33.470 align:middle line:84% if I look at the strategy that player's using, that strategy 00:16:33.470 --> 00:16:36.070 align:middle line:84% better be a best response to the strategies 00:16:36.070 --> 00:16:37.648 align:middle line:84% that the other players are using. 00:16:37.648 --> 00:16:39.190 align:middle line:84% Now, we have to be a bit careful here 00:16:39.190 --> 00:16:41.270 align:middle line:84% because, when we first defined a best response, 00:16:41.270 --> 00:16:44.030 align:middle line:84% we defined a best response to a belief, 00:16:44.030 --> 00:16:46.050 align:middle line:84% not to a strategy of the opponents. 00:16:46.050 --> 00:16:50.115 align:middle line:84% But when I say a best response to this strategy profile, what 00:16:50.115 --> 00:16:51.990 align:middle line:84% I really mean, if I want to be really formal, 00:16:51.990 --> 00:16:54.670 align:middle line:84% is I mean to the belief that assigns probability 00:16:54.670 --> 00:16:59.030 align:middle line:84% 1 to my opponents playing this strategy profile, so to, 00:16:59.030 --> 00:17:03.710 align:middle line:84% more formally, the belief beta negative I 00:17:03.710 --> 00:17:10.310 align:middle line:84% with beta negative i of s negative i star equal 1. 00:17:10.310 --> 00:17:14.650 align:middle line:84% So if my belief is my opponents are certain to play s negative 00:17:14.650 --> 00:17:16.690 align:middle line:84% i star, then the best thing for me to do 00:17:16.690 --> 00:17:20.510 align:middle line:84% or a best thing for me to do is to play si star. 00:17:20.510 --> 00:17:22.950 align:middle line:84% And here, you can immediately see in the definition 00:17:22.950 --> 00:17:25.250 align:middle line:84% the consistency that we're imposing on beliefs. 00:17:25.250 --> 00:17:28.968 align:middle line:84% So I think keeping in mind this interpretation is crucial 00:17:28.968 --> 00:17:30.760 align:middle line:84% because, when you first see the definition, 00:17:30.760 --> 00:17:32.302 align:middle line:84% you might say, where are the beliefs? 00:17:32.302 --> 00:17:35.400 align:middle line:84% Well, because we're requiring consistency of beliefs, 00:17:35.400 --> 00:17:37.780 align:middle line:84% the beliefs are the same as what's actually happening. 00:17:37.780 --> 00:17:40.660 align:middle line:84% So s negative i star is how the other players are actually 00:17:40.660 --> 00:17:41.980 align:middle line:90% playing. 00:17:41.980 --> 00:17:44.343 align:middle line:84% And beta negative i is my correct belief 00:17:44.343 --> 00:17:46.760 align:middle line:84% that the players are playing exactly how they are playing. 00:17:46.760 --> 00:17:49.780 align:middle line:84% So sometimes we can lose track of this distinction between play 00:17:49.780 --> 00:17:51.820 align:middle line:84% and beliefs because our definition 00:17:51.820 --> 00:17:55.020 align:middle line:90% requires that they agree. 00:17:55.020 --> 00:18:01.220 align:middle line:84% Maybe we can give a bit of a less mathematical frame on this. 00:18:01.220 --> 00:18:03.220 align:middle line:84% So another way we might say it is we 00:18:03.220 --> 00:18:15.740 align:middle line:84% might say that no player has a strictly profitable 00:18:15.740 --> 00:18:17.200 align:middle line:90% unilateral deviation. 00:18:17.200 --> 00:18:24.220 align:middle line:90% 00:18:24.220 --> 00:18:26.800 align:middle line:84% So let's break down what this means. 00:18:26.800 --> 00:18:29.680 align:middle line:84% No player-- so let's focus on a single player-- 00:18:29.680 --> 00:18:33.080 align:middle line:84% should have a strictly profitable unilateral deviation. 00:18:33.080 --> 00:18:35.800 align:middle line:84% So a deviation for a player is a strategy 00:18:35.800 --> 00:18:39.640 align:middle line:84% that deviates from or is different from their conjectured 00:18:39.640 --> 00:18:40.700 align:middle line:90% equilibrium strategy. 00:18:40.700 --> 00:18:43.040 align:middle line:84% So we start with si star for player i, 00:18:43.040 --> 00:18:45.880 align:middle line:84% and we contemplate an alternative strategy 00:18:45.880 --> 00:18:47.800 align:middle line:90% for player i. 00:18:47.800 --> 00:18:50.600 align:middle line:84% A unilateral deviation means I'm the only one who's 00:18:50.600 --> 00:18:52.160 align:middle line:90% changing my strategy. 00:18:52.160 --> 00:18:53.960 align:middle line:84% And this is I think the key, maybe 00:18:53.960 --> 00:18:55.960 align:middle line:84% the most important idea that Nash introduced 00:18:55.960 --> 00:18:58.760 align:middle line:84% was that, when I'm reasoning about other people, 00:18:58.760 --> 00:19:00.840 align:middle line:84% I should take their behavior and their strategies 00:19:00.840 --> 00:19:03.760 align:middle line:84% as fixed because I can't directly control what they do. 00:19:03.760 --> 00:19:05.960 align:middle line:84% So when I contemplate deviating, I only 00:19:05.960 --> 00:19:08.040 align:middle line:84% consider a change in my strategy. 00:19:08.040 --> 00:19:10.760 align:middle line:84% And you can see that up here, in this inequality, 00:19:10.760 --> 00:19:12.640 align:middle line:84% that the way that my opponents are playing 00:19:12.640 --> 00:19:15.080 align:middle line:84% is the same on both sides of the inequality. 00:19:15.080 --> 00:19:16.800 align:middle line:84% The only difference between the two sides 00:19:16.800 --> 00:19:19.560 align:middle line:90% is how I'm playing as player i. 00:19:19.560 --> 00:19:23.180 align:middle line:84% And this strategy cannot be-- this deviation cannot be 00:19:23.180 --> 00:19:24.320 align:middle line:90% strictly profitable. 00:19:24.320 --> 00:19:26.487 align:middle line:84% It could be that you have a deviation that gives you 00:19:26.487 --> 00:19:29.160 align:middle line:84% the same payoff as your equilibrium strategy, 00:19:29.160 --> 00:19:31.700 align:middle line:84% but it can't be that you can deviate and do strictly better. 00:19:31.700 --> 00:19:34.060 align:middle line:84% And again, we're using profitability 00:19:34.060 --> 00:19:35.980 align:middle line:84% not in the sense of a monetary payoff 00:19:35.980 --> 00:19:37.440 align:middle line:90% but in the sense of utilities. 00:19:37.440 --> 00:19:39.580 align:middle line:84% So a profitable deviation is a deviation 00:19:39.580 --> 00:19:42.620 align:middle line:84% that gives me a higher utility, a strictly higher expected 00:19:42.620 --> 00:19:43.320 align:middle line:90% utility. 00:19:43.320 --> 00:19:46.020 align:middle line:90% 00:19:46.020 --> 00:19:49.940 align:middle line:84% So let's go back to this game and see 00:19:49.940 --> 00:19:53.640 align:middle line:84% if we can find the Nash equilibria of this game. 00:19:53.640 --> 00:19:57.940 align:middle line:84% Any guesses on what the Nash equilibria of BOS this game 00:19:57.940 --> 00:19:59.658 align:middle line:84% are going to be based on the definition 00:19:59.658 --> 00:20:01.200 align:middle line:84% or based on the things we've written? 00:20:01.200 --> 00:20:01.700 align:middle line:90% Yeah. 00:20:01.700 --> 00:20:02.962 align:middle line:90% AUDIENCE: Both [INAUDIBLE] 00:20:02.962 --> 00:20:04.420 align:middle line:84% IAN BALL: Right, so these are going 00:20:04.420 --> 00:20:06.860 align:middle line:90% to be the two Nash equilibria. 00:20:06.860 --> 00:20:07.760 align:middle line:90% How can we see this? 00:20:07.760 --> 00:20:11.140 align:middle line:84% Well, let's suppose we're both going to the Celtics game. 00:20:11.140 --> 00:20:14.380 align:middle line:84% Does anyone have a profitable unilateral deviation? 00:20:14.380 --> 00:20:16.540 align:middle line:84% No, because if I were to unilaterally deviate, 00:20:16.540 --> 00:20:19.020 align:middle line:84% we're together at the Celtics game, if I deviate, 00:20:19.020 --> 00:20:20.840 align:middle line:84% I'm by myself with the other game, 00:20:20.840 --> 00:20:22.680 align:middle line:90% so I'm certainly worse off. 00:20:22.680 --> 00:20:25.140 align:middle line:84% And the same is true if we go to the Red Sox game. 00:20:25.140 --> 00:20:26.960 align:middle line:84% If we're both going to the Red Sox game, 00:20:26.960 --> 00:20:30.440 align:middle line:84% if I unilaterally deviate, I'm going to be worse off. 00:20:30.440 --> 00:20:33.880 align:middle line:84% Now, of course, if we're both going to the Red Sox game 00:20:33.880 --> 00:20:37.000 align:middle line:84% and I'm player 1, I would much rather that we both 00:20:37.000 --> 00:20:40.160 align:middle line:84% go to the Celtics game because I'm a Celtics fan. 00:20:40.160 --> 00:20:43.840 align:middle line:84% But moving from here to here is not a unilateral deviation. 00:20:43.840 --> 00:20:46.280 align:middle line:84% All I can control is whether I go to the Red Sox 00:20:46.280 --> 00:20:47.340 align:middle line:90% game or the Celtics game. 00:20:47.340 --> 00:20:50.007 align:middle line:84% Of course, I would love to force you to come to the Celtics game 00:20:50.007 --> 00:20:52.720 align:middle line:84% with me, but that's not something I can do in this game. 00:20:52.720 --> 00:20:54.800 align:middle line:84% So that's the intuitive explanation. 00:20:54.800 --> 00:20:58.080 align:middle line:84% Mathematically, we can see that both numbers are underlined. 00:20:58.080 --> 00:21:01.200 align:middle line:84% And that's actually a great way to compute Nash equilibria 00:21:01.200 --> 00:21:02.480 align:middle line:90% in games like this. 00:21:02.480 --> 00:21:05.160 align:middle line:90% So we go through for player 1. 00:21:05.160 --> 00:21:07.960 align:middle line:84% We underline all of their best responses. 00:21:07.960 --> 00:21:10.120 align:middle line:84% Then we go through player 2 and underline 00:21:10.120 --> 00:21:11.840 align:middle line:90% all of their best responses. 00:21:11.840 --> 00:21:15.620 align:middle line:84% And we look-- so I guess this is an algorithm for finding them. 00:21:15.620 --> 00:21:18.160 align:middle line:90% 00:21:18.160 --> 00:21:23.240 align:middle line:84% So step one is underline best responses. 00:21:23.240 --> 00:21:25.802 align:middle line:90% 00:21:25.802 --> 00:21:27.260 align:middle line:84% And it's crucial that you underline 00:21:27.260 --> 00:21:31.420 align:middle line:84% both players best responses and underline all best responses. 00:21:31.420 --> 00:21:36.640 align:middle line:84% So let me say all best responses for both players. 00:21:36.640 --> 00:21:43.580 align:middle line:90% 00:21:43.580 --> 00:21:48.980 align:middle line:84% And then step 2 is you just check which cells-- 00:21:48.980 --> 00:21:53.820 align:middle line:84% or which boxes in the matrix have both numbers underlined. 00:21:53.820 --> 00:21:57.940 align:middle line:90% So look for cells-- 00:21:57.940 --> 00:21:58.720 align:middle line:90% what is a cell? 00:21:58.720 --> 00:22:01.760 align:middle line:84% Well, a cell in this context is just a strategy profile. 00:22:01.760 --> 00:22:02.760 align:middle line:90% So let's remember that. 00:22:02.760 --> 00:22:17.400 align:middle line:90% 00:22:17.400 --> 00:22:19.700 align:middle line:84% And if you think through this carefully, 00:22:19.700 --> 00:22:22.460 align:middle line:84% you'll see this is just another way of saying the definition. 00:22:22.460 --> 00:22:24.380 align:middle line:84% If both cells are underlined-- well, 00:22:24.380 --> 00:22:26.340 align:middle line:84% if the first player's number is underlined, 00:22:26.340 --> 00:22:28.400 align:middle line:84% that precisely means player 1 is playing 00:22:28.400 --> 00:22:31.160 align:middle line:84% a best response to player 2's strategy. 00:22:31.160 --> 00:22:33.140 align:middle line:84% And if player two's number is underlined, 00:22:33.140 --> 00:22:35.640 align:middle line:84% that means player 2 is playing a best response to player 1's 00:22:35.640 --> 00:22:36.320 align:middle line:90% strategy. 00:22:36.320 --> 00:22:38.640 align:middle line:84% And that's exactly our definition of a Nash 00:22:38.640 --> 00:22:41.160 align:middle line:84% equilibrium, at least in a two-player game. 00:22:41.160 --> 00:22:44.160 align:middle line:84% So this is a good way going through and finding and checking 00:22:44.160 --> 00:22:47.343 align:middle line:90% for Nash equilibrium. 00:22:47.343 --> 00:22:48.760 align:middle line:84% So now I think it's helpful to try 00:22:48.760 --> 00:22:55.040 align:middle line:84% to understand the relationship between some of the solution 00:22:55.040 --> 00:22:59.040 align:middle line:84% concepts and some of the definitions we've given so far. 00:22:59.040 --> 00:23:04.320 align:middle line:84% So so far, we have three, I think, important definitions. 00:23:04.320 --> 00:23:06.340 align:middle line:90% One was rationalizability. 00:23:06.340 --> 00:23:12.120 align:middle line:90% 00:23:12.120 --> 00:23:14.720 align:middle line:84% And we represented the set of rationalizable strategy 00:23:14.720 --> 00:23:15.300 align:middle line:90% profiles. 00:23:15.300 --> 00:23:17.700 align:middle line:90% We called that s infinity. 00:23:17.700 --> 00:23:20.040 align:middle line:84% Then another important notion is Nash Equilibrium. 00:23:20.040 --> 00:23:22.740 align:middle line:90% 00:23:22.740 --> 00:23:25.060 align:middle line:84% And let's call this NE, so let me 00:23:25.060 --> 00:23:29.100 align:middle line:84% think of NE as the set of Nash Equilibrium strategy profiles 00:23:29.100 --> 00:23:32.260 align:middle line:90% in a game. 00:23:32.260 --> 00:23:37.060 align:middle line:84% So just to be clear, this is a subset of S. 00:23:37.060 --> 00:23:38.920 align:middle line:90% What kind of objects are these? 00:23:38.920 --> 00:23:40.600 align:middle line:90% These are strategy profiles. 00:23:40.600 --> 00:23:42.740 align:middle line:84% So rationalizable strategy profiles 00:23:42.740 --> 00:23:45.820 align:middle line:84% live in the set of all strategy profiles. 00:23:45.820 --> 00:23:49.340 align:middle line:84% Then the set of Nash equilibria that's also a subset of big S 00:23:49.340 --> 00:23:52.620 align:middle line:84% because a Nash equilibrium is also a strategy profile. 00:23:52.620 --> 00:23:54.660 align:middle line:84% And then we had another notion, remember, 00:23:54.660 --> 00:23:56.720 align:middle line:84% which was Dominant-Strategy Equilibrium. 00:23:56.720 --> 00:24:01.100 align:middle line:90% 00:24:01.100 --> 00:24:03.140 align:middle line:90% And we'll call that DSE. 00:24:03.140 --> 00:24:08.380 align:middle line:84% And that's also a subset of S. So we have three sets 00:24:08.380 --> 00:24:09.980 align:middle line:90% of strategy profiles here. 00:24:09.980 --> 00:24:12.740 align:middle line:84% Let's remember a dominant-strategy equilibrium, 00:24:12.740 --> 00:24:14.580 align:middle line:84% if you recall from the dominance lecture, 00:24:14.580 --> 00:24:18.440 align:middle line:84% was a strategy profile where each player's strategy is 00:24:18.440 --> 00:24:21.480 align:middle line:90% a weakly dominant strategy. 00:24:21.480 --> 00:24:23.662 align:middle line:90% That was the definition. 00:24:23.662 --> 00:24:25.120 align:middle line:84% So now what we'd like to understand 00:24:25.120 --> 00:24:26.740 align:middle line:84% is the relationship between them. 00:24:26.740 --> 00:24:31.560 align:middle line:90% 00:24:31.560 --> 00:24:33.760 align:middle line:84% If something's a dominant strategy equilibrium, 00:24:33.760 --> 00:24:34.938 align:middle line:90% is it a Nash equilibrium? 00:24:34.938 --> 00:24:36.980 align:middle line:84% If it's rationalizable, is it a Nash equilibrium? 00:24:36.980 --> 00:24:37.840 align:middle line:90% And so on. 00:24:37.840 --> 00:24:39.345 align:middle line:84% I think a good way to get started, 00:24:39.345 --> 00:24:41.220 align:middle line:84% when you're approaching a question like this, 00:24:41.220 --> 00:24:42.760 align:middle line:84% is to work through an example and see if you can 00:24:42.760 --> 00:24:44.280 align:middle line:90% get some inspiration from that. 00:24:44.280 --> 00:24:46.320 align:middle line:84% So let's look at our BOS example. 00:24:46.320 --> 00:24:50.720 align:middle line:84% So here, the set of rationalizable strategies 00:24:50.720 --> 00:24:53.380 align:middle line:90% was all four strategies. 00:24:53.380 --> 00:24:58.520 align:middle line:90% 00:24:58.520 --> 00:25:00.560 align:middle line:84% The set of Nash equilibria in this game 00:25:00.560 --> 00:25:04.540 align:middle line:84% over here with just two strategy profiles right, CC and RR. 00:25:04.540 --> 00:25:11.280 align:middle line:90% 00:25:11.280 --> 00:25:14.000 align:middle line:84% What about the set of dominant-strategy equilibria? 00:25:14.000 --> 00:25:16.980 align:middle line:84% Which profiles were dominant-strategy equilibria 00:25:16.980 --> 00:25:17.800 align:middle line:90% in this game? 00:25:17.800 --> 00:25:23.500 align:middle line:90% 00:25:23.500 --> 00:25:25.420 align:middle line:90% Any thoughts? 00:25:25.420 --> 00:25:28.420 align:middle line:84% Well, let's go through for player 1. 00:25:28.420 --> 00:25:29.080 align:middle line:90% Let's check. 00:25:29.080 --> 00:25:31.400 align:middle line:84% Is C a dominant strategy for player 1? 00:25:31.400 --> 00:25:33.340 align:middle line:84% That is, does C always do better than R 00:25:33.340 --> 00:25:36.140 align:middle line:84% and sometimes do strictly better? 00:25:36.140 --> 00:25:38.740 align:middle line:84% Well, no, because if the other player plays R, 00:25:38.740 --> 00:25:43.020 align:middle line:84% then C does worse than R. C gives me 0, and R gives me 1. 00:25:43.020 --> 00:25:45.620 align:middle line:90% So C does not weakly dominate R. 00:25:45.620 --> 00:25:47.240 align:middle line:84% And now let's ask the other question. 00:25:47.240 --> 00:25:50.020 align:middle line:90% Does R weakly dominate C? 00:25:50.020 --> 00:25:52.805 align:middle line:84% Well, no, because if my opponent goes to the Celtics game, 00:25:52.805 --> 00:25:54.180 align:middle line:84% I'd rather go to the Celtics game 00:25:54.180 --> 00:25:55.780 align:middle line:90% than go to the Red Sox game. 00:25:55.780 --> 00:26:01.082 align:middle line:84% So neither does C dominate R, nor does R dominate C. 00:26:01.082 --> 00:26:02.540 align:middle line:84% And that means that neither of them 00:26:02.540 --> 00:26:05.320 align:middle line:84% is weakly dominant because if it was weakly dominant, 00:26:05.320 --> 00:26:07.460 align:middle line:84% it would have to weakly dominate everything else. 00:26:07.460 --> 00:26:10.200 align:middle line:84% And therefore, we have no dominant strategy equilibria 00:26:10.200 --> 00:26:11.400 align:middle line:90% in this game. 00:26:11.400 --> 00:26:14.837 align:middle line:90% So here, we have none. 00:26:14.837 --> 00:26:16.420 align:middle line:84% Or we might say we have the empty set. 00:26:16.420 --> 00:26:20.600 align:middle line:90% 00:26:20.600 --> 00:26:22.160 align:middle line:84% So at least in this example, we see 00:26:22.160 --> 00:26:24.320 align:middle line:84% that Nash equilibrium is more demanding 00:26:24.320 --> 00:26:25.760 align:middle line:90% than rationalizability. 00:26:25.760 --> 00:26:28.920 align:middle line:84% The Nash equilibria are a subset of the rationalizable strategy 00:26:28.920 --> 00:26:31.800 align:middle line:84% profiles, and dominant-strategy equilibrium 00:26:31.800 --> 00:26:34.395 align:middle line:90% is even more demanding still. 00:26:34.395 --> 00:26:36.020 align:middle line:84% It's a bit weird to say it in this case 00:26:36.020 --> 00:26:38.640 align:middle line:84% because there are none of them, but it's vacuously true 00:26:38.640 --> 00:26:41.200 align:middle line:84% that every dominant strategy equilibrium is a Nash 00:26:41.200 --> 00:26:41.780 align:middle line:90% equilibrium. 00:26:41.780 --> 00:26:42.988 align:middle line:90% They just aren't any of them. 00:26:42.988 --> 00:26:45.860 align:middle line:84% But if there were one, it would be a Nash equilibrium. 00:26:45.860 --> 00:26:47.320 align:middle line:84% So that gives us maybe a conjecture 00:26:47.320 --> 00:26:50.800 align:middle line:84% based on this example about the relationship between them. 00:26:50.800 --> 00:26:53.900 align:middle line:84% And I chose the example, so we wouldn't be led astray. 00:26:53.900 --> 00:26:57.680 align:middle line:84% And indeed, this is a result that, in general, we 00:26:57.680 --> 00:27:03.820 align:middle line:84% have DSE subset of NE subset of S infinity, 00:27:03.820 --> 00:27:06.760 align:middle line:84% and maybe I'll write subset of S just to remind us the space 00:27:06.760 --> 00:27:09.480 align:middle line:90% that we're in over here. 00:27:09.480 --> 00:27:12.780 align:middle line:84% So let's understand what these subset relationships mean. 00:27:12.780 --> 00:27:14.080 align:middle line:90% DSE is a set. 00:27:14.080 --> 00:27:16.700 align:middle line:84% It's the set of all dominant-strategy equilibria 00:27:16.700 --> 00:27:18.900 align:middle line:84% which is a set of strategy profiles. 00:27:18.900 --> 00:27:21.220 align:middle line:84% This is another set, and this is another set. 00:27:21.220 --> 00:27:25.420 align:middle line:84% So this inclusion tells me every dominant-strategy equilibrium 00:27:25.420 --> 00:27:28.340 align:middle line:84% is necessarily a Nash equilibrium. 00:27:28.340 --> 00:27:30.340 align:middle line:84% The converse is not necessarily true. 00:27:30.340 --> 00:27:32.860 align:middle line:84% It could be that something's a Nash equilibrium but not 00:27:32.860 --> 00:27:34.620 align:middle line:90% a dominant-strategy equilibrium. 00:27:34.620 --> 00:27:37.220 align:middle line:84% Can you come up with an example of a Nash equilibrium 00:27:37.220 --> 00:27:39.920 align:middle line:84% that's not a dominant-strategy equilibrium, maybe in this game? 00:27:39.920 --> 00:27:43.460 align:middle line:90% 00:27:43.460 --> 00:27:43.960 align:middle line:90% Yeah. 00:27:43.960 --> 00:27:46.560 align:middle line:84% AUDIENCE: [INAUDIBLE] is a Nash equilibrium, 00:27:46.560 --> 00:27:48.585 align:middle line:84% but it's not a dominant strategy. 00:27:48.585 --> 00:27:49.960 align:middle line:84% IAN BALL: Right, and same with R, 00:27:49.960 --> 00:27:51.220 align:middle line:84% so here, it's easy because there are 00:27:51.220 --> 00:27:52.680 align:middle line:90% no dominant-strategy equilibria. 00:27:52.680 --> 00:27:54.540 align:middle line:84% So both of the Nash equilibria are 00:27:54.540 --> 00:27:56.900 align:middle line:84% examples of Nash equilibria that are not 00:27:56.900 --> 00:27:59.460 align:middle line:90% dominant-strategy equilibria. 00:27:59.460 --> 00:28:01.900 align:middle line:84% And then, similarly, we have, there 00:28:01.900 --> 00:28:04.260 align:middle line:84% are some rationalizable strategy profiles 00:28:04.260 --> 00:28:05.620 align:middle line:90% that are not Nash equilibrium. 00:28:05.620 --> 00:28:10.480 align:middle line:84% In this example, CR and RC are two profiles 00:28:10.480 --> 00:28:15.000 align:middle line:84% that are rationalizable but not Nash equilibrium. 00:28:15.000 --> 00:28:17.300 align:middle line:84% Now, sometimes these sets could agree, 00:28:17.300 --> 00:28:19.300 align:middle line:84% but sometimes this set could be strictly bigger. 00:28:19.300 --> 00:28:22.480 align:middle line:84% But our claim here is just that everything in here is in here, 00:28:22.480 --> 00:28:23.928 align:middle line:84% and everything in here is in here. 00:28:23.928 --> 00:28:26.220 align:middle line:84% And now we'd like to go through the reasoning for that, 00:28:26.220 --> 00:28:28.320 align:middle line:84% but are there any questions on what I mean by this statement 00:28:28.320 --> 00:28:29.780 align:middle line:84% before we go through the argument? 00:28:29.780 --> 00:28:35.720 align:middle line:90% 00:28:35.720 --> 00:28:37.580 align:middle line:84% OK, so let's try to think through it. 00:28:37.580 --> 00:28:41.345 align:middle line:90% So here's the argument. 00:28:41.345 --> 00:28:43.220 align:middle line:84% There's two different things we want to show. 00:28:43.220 --> 00:28:45.820 align:middle line:84% I mean, let's be clear, this is just for context. 00:28:45.820 --> 00:28:47.855 align:middle line:90% This relationship is obvious. 00:28:47.855 --> 00:28:50.480 align:middle line:84% It's just, by definition, every rationalizable strategy profile 00:28:50.480 --> 00:28:51.700 align:middle line:90% is a strategy profile. 00:28:51.700 --> 00:28:53.240 align:middle line:90% So we have two things to check. 00:28:53.240 --> 00:28:54.340 align:middle line:90% So let's do the first one. 00:28:54.340 --> 00:28:59.800 align:middle line:84% The first one is that DSE is a subset of NE. 00:28:59.800 --> 00:29:01.303 align:middle line:84% So that's a mathy way of saying it, 00:29:01.303 --> 00:29:02.720 align:middle line:84% but what we mean is if you give me 00:29:02.720 --> 00:29:05.510 align:middle line:84% a dominant-strategy equilibrium of some game, 00:29:05.510 --> 00:29:08.790 align:middle line:84% I know that it must be a Nash equilibrium of that game. 00:29:08.790 --> 00:29:10.730 align:middle line:84% So let's see how we can show that. 00:29:10.730 --> 00:29:14.150 align:middle line:84% So first, let's start with a dominant-strategy equilibrium. 00:29:14.150 --> 00:29:23.150 align:middle line:84% So suppose S star, which I'll write S1 star up to SN star, 00:29:23.150 --> 00:29:24.610 align:middle line:90% is a DSE. 00:29:24.610 --> 00:29:25.610 align:middle line:90% That's a starting point. 00:29:25.610 --> 00:29:27.710 align:middle line:84% We're just going to take a strategy profile 00:29:27.710 --> 00:29:29.470 align:middle line:84% and make the assumption that it is 00:29:29.470 --> 00:29:32.190 align:middle line:90% a dominant-strategy equilibrium. 00:29:32.190 --> 00:29:33.990 align:middle line:84% And we want to now reason through 00:29:33.990 --> 00:29:39.590 align:middle line:84% why this strategy profile must also be a Nash equilibrium. 00:29:39.590 --> 00:29:42.670 align:middle line:84% Well, to show that it's a Nash equilibrium, 00:29:42.670 --> 00:29:46.230 align:middle line:84% we have to check that no player has 00:29:46.230 --> 00:29:48.950 align:middle line:84% a profitable unilateral deviation. 00:29:48.950 --> 00:29:51.950 align:middle line:90% So let's consider a player. 00:29:51.950 --> 00:29:59.670 align:middle line:84% So that means that for, each player i, 00:29:59.670 --> 00:30:02.090 align:middle line:84% let's consider player i, and let's also 00:30:02.090 --> 00:30:04.850 align:middle line:84% consider a potential deviation by player i. 00:30:04.850 --> 00:30:17.250 align:middle line:84% So for each player i and each strategy si prime, 00:30:17.250 --> 00:30:20.810 align:middle line:90% what do we want to show? 00:30:20.810 --> 00:30:23.850 align:middle line:84% We want to show that this deviation is not 00:30:23.850 --> 00:30:26.530 align:middle line:90% profitable for player i. 00:30:26.530 --> 00:30:29.090 align:middle line:84% That is, we want to show that, for player i, 00:30:29.090 --> 00:30:34.272 align:middle line:84% it's better for them to play si star than to play si prime, 00:30:34.272 --> 00:30:36.230 align:middle line:84% given the way that their opponents are playing. 00:30:36.230 --> 00:30:39.140 align:middle line:90% 00:30:39.140 --> 00:30:41.170 align:middle line:90% But that's what we want to show. 00:30:41.170 --> 00:30:44.150 align:middle line:84% All we know is that this is a dominant-strategy equilibrium. 00:30:44.150 --> 00:30:46.788 align:middle line:90% So what's something useful-- 00:30:46.788 --> 00:30:48.330 align:middle line:84% do you have any ideas of how we could 00:30:48.330 --> 00:30:50.930 align:middle line:84% use the fact that this is a dominant-strategy equilibrium 00:30:50.930 --> 00:30:51.830 align:middle line:90% to get us started? 00:30:51.830 --> 00:30:54.890 align:middle line:84% What do we know about si star, given 00:30:54.890 --> 00:30:57.350 align:middle line:84% that s star is a dominant-strategy equilibrium? 00:30:57.350 --> 00:31:02.510 align:middle line:90% 00:31:02.510 --> 00:31:03.430 align:middle line:90% Yeah. 00:31:03.430 --> 00:31:06.510 align:middle line:90% AUDIENCE: si star [INAUDIBLE]? 00:31:06.510 --> 00:31:08.870 align:middle line:84% IAN BALL: Yeah, so let's go through it in turn. 00:31:08.870 --> 00:31:11.090 align:middle line:84% Because it's a dominant-strategy equilibrium, 00:31:11.090 --> 00:31:15.070 align:middle line:84% we know that si star is weakly dominant. 00:31:15.070 --> 00:31:17.110 align:middle line:84% And what does it mean to be weakly dominant? 00:31:17.110 --> 00:31:20.590 align:middle line:84% It means that si star weakly dominates every other strategy 00:31:20.590 --> 00:31:22.270 align:middle line:90% of player i. 00:31:22.270 --> 00:31:25.590 align:middle line:84% But in particular, that means si star weakly 00:31:25.590 --> 00:31:27.270 align:middle line:84% dominates si prime because that's 00:31:27.270 --> 00:31:29.190 align:middle line:90% another strategy of player i. 00:31:29.190 --> 00:31:38.492 align:middle line:84% So we know that si star weakly dominates si prime. 00:31:38.492 --> 00:31:39.450 align:middle line:90% We know more than that. 00:31:39.450 --> 00:31:41.450 align:middle line:84% But here, we're just throwing away information. 00:31:41.450 --> 00:31:43.590 align:middle line:84% If si star weakly dominates everything, 00:31:43.590 --> 00:31:48.270 align:middle line:84% in particular, it weakly dominates si prime. 00:31:48.270 --> 00:31:50.330 align:middle line:84% And now can we reach the conclusion we want? 00:31:50.330 --> 00:31:53.390 align:middle line:90% 00:31:53.390 --> 00:31:56.870 align:middle line:84% Well, yeah, if si star weakly dominates si prime, 00:31:56.870 --> 00:32:01.310 align:middle line:84% that means si star always does weakly better than si prime, 00:32:01.310 --> 00:32:03.730 align:middle line:84% no matter how my opponents are playing, 00:32:03.730 --> 00:32:06.170 align:middle line:84% so if that's true and sometimes strictly better. 00:32:06.170 --> 00:32:08.630 align:middle line:84% So if that's true no matter how my opponents are playing, 00:32:08.630 --> 00:32:10.690 align:middle line:84% it's certainly true in the particular way 00:32:10.690 --> 00:32:12.450 align:middle line:90% my opponents are playing now. 00:32:12.450 --> 00:32:20.130 align:middle line:84% So that tells me that ui of si star si negative-- 00:32:20.130 --> 00:32:23.410 align:middle line:84% s negative i star is greater than or equal to ui 00:32:23.410 --> 00:32:28.090 align:middle line:90% si prime s negative i star. 00:32:28.090 --> 00:32:31.050 align:middle line:84% So because si star weekly dominance si prime, 00:32:31.050 --> 00:32:33.770 align:middle line:84% this inequality would be true whatever I put on both sides 00:32:33.770 --> 00:32:34.270 align:middle line:90% here. 00:32:34.270 --> 00:32:36.890 align:middle line:84% I could put any as long as I put the same thing on both sides. 00:32:36.890 --> 00:32:39.650 align:middle line:84% So in particular, if I put the way my opponents are actually 00:32:39.650 --> 00:32:42.090 align:middle line:84% playing, this is going to be true. 00:32:42.090 --> 00:32:47.770 align:middle line:84% And this precisely means that si star is a best 00:32:47.770 --> 00:32:50.770 align:middle line:90% response to s negative i star. 00:32:50.770 --> 00:33:04.190 align:middle line:84% So I conclude si star is a best response to s negative i star? 00:33:04.190 --> 00:33:06.157 align:middle line:84% Well, I have reason for player i, 00:33:06.157 --> 00:33:07.990 align:middle line:84% but I can go through the exact same argument 00:33:07.990 --> 00:33:09.390 align:middle line:90% for all the other players. 00:33:09.390 --> 00:33:11.870 align:middle line:84% So not only is player i playing a best response 00:33:11.870 --> 00:33:13.470 align:middle line:84% to the way everyone else is playing, 00:33:13.470 --> 00:33:15.670 align:middle line:84% but every single player i is playing a best 00:33:15.670 --> 00:33:17.950 align:middle line:84% response to the way everyone else is playing. 00:33:17.950 --> 00:33:24.368 align:middle line:84% So I'll say, therefore, s star is a Nash equilibrium. 00:33:24.368 --> 00:33:26.410 align:middle line:84% So if you're not so familiar with writing proofs, 00:33:26.410 --> 00:33:28.530 align:middle line:84% this kind of reasoning may be a little tricky. 00:33:28.530 --> 00:33:29.322 align:middle line:90% Let me pause there. 00:33:29.322 --> 00:33:32.377 align:middle line:90% Are any questions on this? 00:33:32.377 --> 00:33:34.710 align:middle line:84% So it's really just about being organized and going step 00:33:34.710 --> 00:33:35.630 align:middle line:90% by step. 00:33:35.630 --> 00:33:37.170 align:middle line:84% None of the steps is too difficult, 00:33:37.170 --> 00:33:38.950 align:middle line:84% but you have to keep in mind where you're 00:33:38.950 --> 00:33:41.830 align:middle line:84% going, where you're starting, and carefully 00:33:41.830 --> 00:33:42.970 align:middle line:90% go through the logic of it. 00:33:42.970 --> 00:33:45.870 align:middle line:90% 00:33:45.870 --> 00:33:47.790 align:middle line:84% But the basic idea is pretty simple. 00:33:47.790 --> 00:33:51.230 align:middle line:84% If what I'm doing is best no matter how my opponents play, 00:33:51.230 --> 00:33:53.190 align:middle line:84% it's certainly best, given the particular way 00:33:53.190 --> 00:33:54.440 align:middle line:90% that my opponents are playing. 00:33:54.440 --> 00:33:57.970 align:middle line:84% And that's the high-level idea of this. 00:33:57.970 --> 00:34:01.690 align:middle line:84% OK, so now let's try to show the second part. 00:34:01.690 --> 00:34:04.890 align:middle line:84% So I have shown that every dominant strategy equilibrium 00:34:04.890 --> 00:34:06.185 align:middle line:90% is a Nash equilibrium. 00:34:06.185 --> 00:34:07.810 align:middle line:84% Notice, I didn't show that there exists 00:34:07.810 --> 00:34:09.350 align:middle line:90% a dominant-strategy equilibrium. 00:34:09.350 --> 00:34:11.188 align:middle line:84% My starting point here was let's take 00:34:11.188 --> 00:34:13.230 align:middle line:84% something that's a dominant-strategy equilibrium. 00:34:13.230 --> 00:34:15.230 align:middle line:84% Maybe there's no dominant strategy equilibrium. 00:34:15.230 --> 00:34:15.949 align:middle line:90% That's OK. 00:34:15.949 --> 00:34:18.210 align:middle line:84% But I'm saying if there is one, it must also 00:34:18.210 --> 00:34:20.570 align:middle line:90% be a Nash equilibrium. 00:34:20.570 --> 00:34:24.770 align:middle line:84% And now let's go see if we can do this next step. 00:34:24.770 --> 00:34:28.330 align:middle line:84% So part 2 is to show that every Nash 00:34:28.330 --> 00:34:32.770 align:middle line:90% equilibrium is rationalizable. 00:34:32.770 --> 00:34:35.090 align:middle line:84% So if I start with something that's Nash equilibrium, 00:34:35.090 --> 00:34:38.650 align:middle line:84% it better be a rationalizable strategy profile. 00:34:38.650 --> 00:34:41.770 align:middle line:84% Now this seems a bit hard because the way 00:34:41.770 --> 00:34:43.712 align:middle line:84% we defined S infinity was really tricky. 00:34:43.712 --> 00:34:45.670 align:middle line:84% We went through this whole iterative procedure. 00:34:45.670 --> 00:34:46.503 align:middle line:90% We had an algorithm. 00:34:46.503 --> 00:34:48.449 align:middle line:84% We had to go through infinitely many steps. 00:34:48.449 --> 00:34:53.670 align:middle line:84% So do you have any ideas about how we might show this? 00:34:53.670 --> 00:34:55.429 align:middle line:84% Any mathematical technique that might be 00:34:55.429 --> 00:34:58.090 align:middle line:90% useful for this kind of proof? 00:34:58.090 --> 00:34:58.590 align:middle line:90% Yeah. 00:34:58.590 --> 00:35:00.657 align:middle line:84% AUDIENCE: Proof by contradiction. 00:35:00.657 --> 00:35:01.490 align:middle line:90% IAN BALL: We could-- 00:35:01.490 --> 00:35:03.070 align:middle line:84% I think a proof by contradiction would work, yeah, 00:35:03.070 --> 00:35:03.970 align:middle line:90% but I saw another hand. 00:35:03.970 --> 00:35:04.230 align:middle line:90% Yeah. 00:35:04.230 --> 00:35:04.730 align:middle line:90% AUDIENCE: Induction. 00:35:04.730 --> 00:35:05.690 align:middle line:84% IAN BALL: Induction is a good idea. 00:35:05.690 --> 00:35:07.150 align:middle line:90% Contradiction would work. 00:35:07.150 --> 00:35:08.210 align:middle line:84% A lot of things, you could convert it. 00:35:08.210 --> 00:35:10.127 align:middle line:84% But I think induction is a really natural idea 00:35:10.127 --> 00:35:13.430 align:middle line:84% because we've defined S infinity inductively. 00:35:13.430 --> 00:35:15.130 align:middle line:90% So we'd like to use induction. 00:35:15.130 --> 00:35:17.172 align:middle line:84% And if you haven't heard of induction, that's OK, 00:35:17.172 --> 00:35:18.172 align:middle line:90% I'll go through it. 00:35:18.172 --> 00:35:19.630 align:middle line:84% OK, but let's first just understand 00:35:19.630 --> 00:35:21.225 align:middle line:90% what we're trying to do. 00:35:21.225 --> 00:35:23.350 align:middle line:84% We're going to follow the same procedure as before. 00:35:23.350 --> 00:35:27.070 align:middle line:90% So suppose s star-- 00:35:27.070 --> 00:35:28.830 align:middle line:90% this is a strategy profile-- 00:35:28.830 --> 00:35:31.170 align:middle line:90% is a Nash equilibrium. 00:35:31.170 --> 00:35:35.590 align:middle line:90% 00:35:35.590 --> 00:35:36.910 align:middle line:90% What do we want to do? 00:35:36.910 --> 00:35:41.870 align:middle line:84% We want to show that s star is rationalizable. 00:35:41.870 --> 00:35:50.090 align:middle line:84% So we want to show that s star is in S infinity. 00:35:50.090 --> 00:35:52.770 align:middle line:90% 00:35:52.770 --> 00:35:55.370 align:middle line:84% But doing that directly is hard because S infinity 00:35:55.370 --> 00:35:58.410 align:middle line:84% is only the result of this iterative procedure. 00:35:58.410 --> 00:36:01.690 align:middle line:84% But remember, from over here, the way 00:36:01.690 --> 00:36:04.690 align:middle line:84% we got S infinity is we just kept making the set smaller 00:36:04.690 --> 00:36:06.290 align:middle line:90% and smaller and smaller. 00:36:06.290 --> 00:36:09.370 align:middle line:84% So one way-- or what's equivalent to showing that s 00:36:09.370 --> 00:36:11.290 align:middle line:90% star is in S infinity-- 00:36:11.290 --> 00:36:17.655 align:middle line:84% so I'll say, or equivalently and easier to show 00:36:17.655 --> 00:36:24.050 align:middle line:90% S star is in Sk for all k. 00:36:24.050 --> 00:36:25.470 align:middle line:90% So let's just say this in words. 00:36:25.470 --> 00:36:27.450 align:middle line:84% We want to show that s star survives 00:36:27.450 --> 00:36:30.028 align:middle line:84% every round of elimination and makes it till the end. 00:36:30.028 --> 00:36:31.570 align:middle line:84% Well, the way we show that is we just 00:36:31.570 --> 00:36:34.253 align:middle line:84% show it makes it to the first step and the second step 00:36:34.253 --> 00:36:36.670 align:middle line:84% and the third step and the fourth step and the fifth step, 00:36:36.670 --> 00:36:38.230 align:middle line:90% and it never gets thrown out. 00:36:38.230 --> 00:36:40.570 align:middle line:84% And therefore, it must survive all the way to the end. 00:36:40.570 --> 00:36:43.970 align:middle line:84% So we want to show that this strategy profile must survive 00:36:43.970 --> 00:36:47.530 align:middle line:90% every round of elimination. 00:36:47.530 --> 00:36:49.870 align:middle line:84% And since we're showing something for every k, 00:36:49.870 --> 00:36:51.210 align:middle line:84% this is where we're going to use induction. 00:36:51.210 --> 00:36:52.335 align:middle line:90% So what do we want to show? 00:36:52.335 --> 00:36:58.190 align:middle line:84% We want to show that, first-- so there's a base case of k 00:36:58.190 --> 00:36:59.070 align:middle line:90% equals 0. 00:36:59.070 --> 00:37:01.750 align:middle line:84% We want to show first that s star is in there 00:37:01.750 --> 00:37:03.293 align:middle line:90% after the zeroth round. 00:37:03.293 --> 00:37:04.710 align:middle line:84% And then we have an inductive step 00:37:04.710 --> 00:37:07.190 align:middle line:84% that says if it's made it to the k-th round, 00:37:07.190 --> 00:37:10.262 align:middle line:84% It's also going to make it to the k plus first round. 00:37:10.262 --> 00:37:11.970 align:middle line:84% And then we can just tie it all together. 00:37:11.970 --> 00:37:14.310 align:middle line:84% If we know it makes it to the zeroth round, 00:37:14.310 --> 00:37:15.850 align:middle line:84% then it must make it to the first. 00:37:15.850 --> 00:37:18.470 align:middle line:84% And if it makes it the first, it must make it to the second 00:37:18.470 --> 00:37:18.970 align:middle line:90% and so on. 00:37:18.970 --> 00:37:21.310 align:middle line:84% And we can just chain our reasoning together. 00:37:21.310 --> 00:37:25.870 align:middle line:84% So for the base case of k equals 0, what we need to show 00:37:25.870 --> 00:37:30.973 align:middle line:90% is that s star is in S0. 00:37:30.973 --> 00:37:32.390 align:middle line:84% And maybe I'll put a question mark 00:37:32.390 --> 00:37:33.770 align:middle line:90% because this is not a claim. 00:37:33.770 --> 00:37:35.610 align:middle line:90% I'm not saying it's true. 00:37:35.610 --> 00:37:36.990 align:middle line:90% It's what we're trying to show. 00:37:36.990 --> 00:37:40.220 align:middle line:90% So is s star in S0? 00:37:40.220 --> 00:37:42.470 align:middle line:84% Well, yeah, because S0 is just the set of all strategy 00:37:42.470 --> 00:37:44.110 align:middle line:90% profiles. 00:37:44.110 --> 00:37:49.170 align:middle line:84% And s star is a strategy profile, so that's pretty easy. 00:37:49.170 --> 00:37:51.250 align:middle line:90% This is just by definition. 00:37:51.250 --> 00:37:54.210 align:middle line:84% The heart of the proof is the inductive step. 00:37:54.210 --> 00:37:55.950 align:middle line:90% So here's the inductive step. 00:37:55.950 --> 00:38:00.730 align:middle line:90% 00:38:00.730 --> 00:38:02.250 align:middle line:84% What we want to show is, in general, 00:38:02.250 --> 00:38:05.370 align:middle line:84% is if s star makes it through the first k rounds, 00:38:05.370 --> 00:38:07.230 align:middle line:84% then it must make it one more round. 00:38:07.230 --> 00:38:09.610 align:middle line:84% It must also make it to the k plus first round. 00:38:09.610 --> 00:38:11.450 align:middle line:84% So what we Want To Show-- and maybe I'll 00:38:11.450 --> 00:38:13.210 align:middle line:84% say WTS because it's always important 00:38:13.210 --> 00:38:15.210 align:middle line:84% it proves to be clear about what you're assuming 00:38:15.210 --> 00:38:16.502 align:middle line:90% and what you're trying to show. 00:38:16.502 --> 00:38:19.730 align:middle line:84% So WTS is just an abbreviation for Want To Show. 00:38:19.730 --> 00:38:26.890 align:middle line:84% What I Want To Show is that if s star is in Sk, 00:38:26.890 --> 00:38:34.290 align:middle line:90% then s star is in Sk plus 1. 00:38:34.290 --> 00:38:37.010 align:middle line:84% So if I haven't thrown it out by the k-th round, 00:38:37.010 --> 00:38:39.350 align:middle line:84% I'm not going to throw it out at the k plus first round. 00:38:39.350 --> 00:38:41.923 align:middle line:90% 00:38:41.923 --> 00:38:43.590 align:middle line:84% I think if we just look at it like this, 00:38:43.590 --> 00:38:45.610 align:middle line:84% it can be a little tricky because, remember, 00:38:45.610 --> 00:38:48.110 align:middle line:84% the way we defined this procedure was we 00:38:48.110 --> 00:38:49.850 align:middle line:90% had to reason player by player. 00:38:49.850 --> 00:38:51.670 align:middle line:84% So let's write that more explicitly. 00:38:51.670 --> 00:38:54.950 align:middle line:84% What does it mean for this strategy profile to be in Sk? 00:38:54.950 --> 00:38:57.870 align:middle line:84% It means the i-th component of that profile, 00:38:57.870 --> 00:39:01.570 align:middle line:84% player i's strategy is in si k for each of the players i. 00:39:01.570 --> 00:39:03.690 align:middle line:84% And I think that's a bit more concrete. 00:39:03.690 --> 00:39:07.110 align:middle line:90% So that means that si star-- 00:39:07.110 --> 00:39:10.510 align:middle line:84% that is, remember, s star is a profile of strategies. 00:39:10.510 --> 00:39:13.670 align:middle line:84% si star is the strategy that player i is actually using. 00:39:13.670 --> 00:39:18.070 align:middle line:90% That must be in si k. 00:39:18.070 --> 00:39:19.610 align:middle line:90% And this is true for all i. 00:39:19.610 --> 00:39:23.990 align:middle line:90% 00:39:23.990 --> 00:39:28.150 align:middle line:84% And now, symmetrically, what we're trying to show 00:39:28.150 --> 00:39:33.970 align:middle line:84% is that si star is in si k plus 1 for all i. 00:39:33.970 --> 00:39:37.670 align:middle line:90% 00:39:37.670 --> 00:39:41.445 align:middle line:84% Now, when we go through proofs and arguments, 00:39:41.445 --> 00:39:43.320 align:middle line:84% we only go through the right way of doing it, 00:39:43.320 --> 00:39:44.737 align:middle line:84% and I think you can sometimes lose 00:39:44.737 --> 00:39:47.580 align:middle line:84% sight of how hard it is because you 00:39:47.580 --> 00:39:49.660 align:middle line:90% don't show all the wrong paths. 00:39:49.660 --> 00:39:51.560 align:middle line:84% And there's an easy mistake people make here. 00:39:51.560 --> 00:39:53.100 align:middle line:84% What's really tempting at this point 00:39:53.100 --> 00:39:55.740 align:middle line:84% is to say, OK, let me reason player by player. 00:39:55.740 --> 00:39:57.740 align:middle line:84% So I want to look at player i, and I 00:39:57.740 --> 00:40:01.180 align:middle line:84% want to say if player i's strategy has survived up 00:40:01.180 --> 00:40:04.260 align:middle line:84% to round k, then player i strategies should survive up 00:40:04.260 --> 00:40:07.020 align:middle line:90% to round k plus 1. 00:40:07.020 --> 00:40:09.260 align:middle line:84% That's not going to get you very far. 00:40:09.260 --> 00:40:12.980 align:middle line:84% And the reason is that which strategies for player i 00:40:12.980 --> 00:40:15.260 align:middle line:84% survive up to round k plus 1 depend 00:40:15.260 --> 00:40:17.220 align:middle line:84% on which strategies of player i's opponents 00:40:17.220 --> 00:40:20.433 align:middle line:84% have survived up to round k because, when I'm choosing, 00:40:20.433 --> 00:40:22.100 align:middle line:84% when we go through the algorithm, what's 00:40:22.100 --> 00:40:23.980 align:middle line:90% the best response for player i? 00:40:23.980 --> 00:40:27.540 align:middle line:84% It depends on what strategies of my opponents 00:40:27.540 --> 00:40:28.860 align:middle line:90% have survived so far. 00:40:28.860 --> 00:40:31.043 align:middle line:84% So we really have to reason simultaneously 00:40:31.043 --> 00:40:31.960 align:middle line:90% about all the players. 00:40:31.960 --> 00:40:33.710 align:middle line:84% We can't just reason through it separately 00:40:33.710 --> 00:40:36.500 align:middle line:84% because which strategies survive for player i depend 00:40:36.500 --> 00:40:40.060 align:middle line:84% on which strategies of the other players have survived so far. 00:40:40.060 --> 00:40:43.980 align:middle line:84% So let's try to go through this argument. 00:40:43.980 --> 00:40:54.880 align:middle line:90% 00:40:54.880 --> 00:40:57.160 align:middle line:90% OK, so here's the argument. 00:40:57.160 --> 00:40:59.020 align:middle line:90% So let's consider player i. 00:40:59.020 --> 00:41:05.120 align:middle line:90% 00:41:05.120 --> 00:41:08.200 align:middle line:84% So we know that si star is in si k. 00:41:08.200 --> 00:41:10.920 align:middle line:84% But I've argued that that's actually not so useful for us. 00:41:10.920 --> 00:41:14.920 align:middle line:84% What's more important from the perspective of player i? 00:41:14.920 --> 00:41:16.960 align:middle line:84% What do I care about as I'm player i? 00:41:16.960 --> 00:41:19.920 align:middle line:84% I care about what strategies my opponents might play. 00:41:19.920 --> 00:41:26.240 align:middle line:84% So the important thing is that we know that s negative i 00:41:26.240 --> 00:41:30.600 align:middle line:90% star is in s negative i k. 00:41:30.600 --> 00:41:34.480 align:middle line:84% This is the key part of our-- so our inductive assumption 00:41:34.480 --> 00:41:38.680 align:middle line:84% is that everyone's strategy has survived so far? 00:41:38.680 --> 00:41:41.140 align:middle line:84% But the relevant aspect of that assumption 00:41:41.140 --> 00:41:43.220 align:middle line:84% is the fact that all of my opponent strategies 00:41:43.220 --> 00:41:44.780 align:middle line:84% have survived so far because that's 00:41:44.780 --> 00:41:46.860 align:middle line:84% what's going to determine what strategies 00:41:46.860 --> 00:41:48.500 align:middle line:90% are best responses for me. 00:41:48.500 --> 00:41:49.640 align:middle line:90% So this is what we know. 00:41:49.640 --> 00:41:52.380 align:middle line:90% 00:41:52.380 --> 00:42:05.520 align:middle line:84% And what we want to show is that si star is in si k plus 1. 00:42:05.520 --> 00:42:07.872 align:middle line:90% So this is what we know. 00:42:07.872 --> 00:42:09.080 align:middle line:90% This is what we want to show. 00:42:09.080 --> 00:42:11.663 align:middle line:84% I think this is often a good way to try to get through proofs. 00:42:11.663 --> 00:42:13.288 align:middle line:84% So I think the first thing we should do 00:42:13.288 --> 00:42:14.800 align:middle line:84% is just write down some definitions. 00:42:14.800 --> 00:42:19.260 align:middle line:90% What is si k plus 1? 00:42:19.260 --> 00:42:21.520 align:middle line:84% What is the definition of si k plus 1? 00:42:21.520 --> 00:42:29.260 align:middle line:90% 00:42:29.260 --> 00:42:30.860 align:middle line:90% Yeah, over here. 00:42:30.860 --> 00:42:34.980 align:middle line:90% In the Boston sweatshirt, yeah. 00:42:34.980 --> 00:42:38.040 align:middle line:90% AUDIENCE: It's like all-- 00:42:38.040 --> 00:42:45.000 align:middle line:84% it's the set of strategies that are a best response to something 00:42:45.000 --> 00:42:46.860 align:middle line:90% in si k. 00:42:46.860 --> 00:42:47.860 align:middle line:90% IAN BALL: Exactly right. 00:42:47.860 --> 00:42:49.060 align:middle line:90% And we have a notation for that. 00:42:49.060 --> 00:42:50.518 align:middle line:84% That's exactly what it is in words. 00:42:50.518 --> 00:42:53.080 align:middle line:84% And in math-- we'll just convert that into math, 00:42:53.080 --> 00:42:56.600 align:middle line:90% is s negative i k. 00:42:56.600 --> 00:43:01.280 align:middle line:84% So which strategies survive up to the k plus first round? 00:43:01.280 --> 00:43:03.840 align:middle line:84% It's all the strategies for player i 00:43:03.840 --> 00:43:07.120 align:middle line:84% that are best responses for player i 00:43:07.120 --> 00:43:11.880 align:middle line:84% to some belief over the strategies in s negative i k? 00:43:11.880 --> 00:43:14.200 align:middle line:84% So these are the strategies I consider it possible 00:43:14.200 --> 00:43:16.020 align:middle line:90% that my opponents might play. 00:43:16.020 --> 00:43:20.520 align:middle line:84% And I want to think about all best responses to that. 00:43:20.520 --> 00:43:22.720 align:middle line:84% So now we just have to put things together. 00:43:22.720 --> 00:43:27.120 align:middle line:84% We know that s negative i star is in s negative i k. 00:43:27.120 --> 00:43:30.800 align:middle line:84% We want to show that si star is a best response to something 00:43:30.800 --> 00:43:32.920 align:middle line:90% in s negative i k. 00:43:32.920 --> 00:43:35.420 align:middle line:90% What's the key step? 00:43:35.420 --> 00:43:36.320 align:middle line:90% What's the leap here? 00:43:36.320 --> 00:43:38.320 align:middle line:90% So why is this true? 00:43:38.320 --> 00:43:44.140 align:middle line:90% 00:43:44.140 --> 00:43:44.860 align:middle line:90% Yeah. 00:43:44.860 --> 00:43:47.900 align:middle line:84% AUDIENCE: So si star is by definition a best response 00:43:47.900 --> 00:43:50.340 align:middle line:84% to s negative i star, and that's what we just 00:43:50.340 --> 00:43:52.398 align:middle line:84% showed because their game best [INAUDIBLE]. 00:43:52.398 --> 00:43:53.440 align:middle line:90% IAN BALL: Exactly, right. 00:43:53.440 --> 00:43:55.260 align:middle line:84% So by the definition of Nash equilibrium-- 00:43:55.260 --> 00:43:56.420 align:middle line:90% let me explain this. 00:43:56.420 --> 00:43:58.580 align:middle line:84% By the definition of Nash equilibrium, 00:43:58.580 --> 00:44:02.380 align:middle line:84% we know that si star is a best response to s negative i star. 00:44:02.380 --> 00:44:10.180 align:middle line:90% 00:44:10.180 --> 00:44:12.760 align:middle line:84% And remember, this is a small s, this is a big S, 00:44:12.760 --> 00:44:14.180 align:middle line:90% this is a small s. 00:44:14.180 --> 00:44:16.740 align:middle line:84% So we know that si star is a best 00:44:16.740 --> 00:44:19.660 align:middle line:90% response to s negative i star. 00:44:19.660 --> 00:44:21.940 align:middle line:84% And we know by our inductive hypothesis 00:44:21.940 --> 00:44:26.020 align:middle line:84% that s star is a strategy in s negative i k. 00:44:26.020 --> 00:44:28.500 align:middle line:84% So it just follows by definition that, therefore, 00:44:28.500 --> 00:44:31.540 align:middle line:84% s negative i star is a best response to some belief 00:44:31.540 --> 00:44:33.120 align:middle line:90% about things in s negative i k. 00:44:33.120 --> 00:44:34.720 align:middle line:84% Specifically, it's a best response 00:44:34.720 --> 00:44:38.360 align:middle line:84% to the belief that puts all the probability on this strategy. 00:44:38.360 --> 00:44:39.680 align:middle line:90% So we're done. 00:44:39.680 --> 00:44:44.400 align:middle line:84% We've shown that si star is in si k plus 1. 00:44:44.400 --> 00:44:47.140 align:middle line:84% And if we reason exactly the same way for all the players, 00:44:47.140 --> 00:44:49.520 align:middle line:84% we'll show that this holds for every single player i, 00:44:49.520 --> 00:44:51.340 align:middle line:84% and now we've completed our inductive step. 00:44:51.340 --> 00:44:56.640 align:middle line:90% 00:44:56.640 --> 00:44:59.420 align:middle line:84% So again, I think each individual step is not too hard. 00:44:59.420 --> 00:45:01.680 align:middle line:84% But piecing it together and thinking through this 00:45:01.680 --> 00:45:02.460 align:middle line:90% is a bit tricky. 00:45:02.460 --> 00:45:05.380 align:middle line:84% So again, are there any questions on this argument? 00:45:05.380 --> 00:45:11.000 align:middle line:90% 00:45:11.000 --> 00:45:14.660 align:middle line:84% So now let's go a little farther. 00:45:14.660 --> 00:45:16.260 align:middle line:90% Let's go to another example. 00:45:16.260 --> 00:45:31.237 align:middle line:90% 00:45:31.237 --> 00:45:32.820 align:middle line:84% This is another classic game, and this 00:45:32.820 --> 00:45:37.060 align:middle line:90% is called hide-and-seek. 00:45:37.060 --> 00:45:39.080 align:middle line:84% So think of this as the childhood game. 00:45:39.080 --> 00:45:40.663 align:middle line:84% Someone hides, the other person seeks, 00:45:40.663 --> 00:45:42.372 align:middle line:84% and the hider doesn't want to get caught, 00:45:42.372 --> 00:45:44.660 align:middle line:84% and the seeker does want to catch the other person. 00:45:44.660 --> 00:45:46.715 align:middle line:84% Of course, there's more substantive examples. 00:45:46.715 --> 00:45:48.840 align:middle line:84% You might think of someone cheating on their taxes, 00:45:48.840 --> 00:45:50.548 align:middle line:84% trying to hide, and then the IRS seeking, 00:45:50.548 --> 00:45:53.173 align:middle line:84% trying to figure out which line of their taxes they cheated on. 00:45:53.173 --> 00:45:55.200 align:middle line:84% So I think it has broader implications, 00:45:55.200 --> 00:45:57.120 align:middle line:84% but we'll keep with the hide and seek story. 00:45:57.120 --> 00:45:58.940 align:middle line:84% So let's say player 1 is the hider, 00:45:58.940 --> 00:46:01.260 align:middle line:84% and there's two places they can hide. 00:46:01.260 --> 00:46:03.380 align:middle line:84% And player 2 is the seeker, and there's two places 00:46:03.380 --> 00:46:05.700 align:middle line:90% they can look for the person. 00:46:05.700 --> 00:46:10.980 align:middle line:84% So the seeker is going to win if they look where the person hid. 00:46:10.980 --> 00:46:12.940 align:middle line:84% So that means we're going to have-- 00:46:12.940 --> 00:46:15.300 align:middle line:84% give myself a little more space-- 00:46:15.300 --> 00:46:18.340 align:middle line:90% negative 1, 1, negative 1, 1. 00:46:18.340 --> 00:46:22.260 align:middle line:84% Because here, person 1 hid in spot A, 00:46:22.260 --> 00:46:24.500 align:middle line:84% person 2 looked for them in spot A. 00:46:24.500 --> 00:46:26.540 align:middle line:84% That means person 1 gets caught, so they 00:46:26.540 --> 00:46:28.540 align:middle line:84% get a payoff of negative 1, and the other player 00:46:28.540 --> 00:46:30.080 align:middle line:90% gets a payoff of 1. 00:46:30.080 --> 00:46:34.560 align:middle line:84% And the same is true if they both hide and seek in spot B. 00:46:34.560 --> 00:46:37.840 align:middle line:84% But then, over here, we're going to have the opposite. 00:46:37.840 --> 00:46:41.680 align:middle line:84% So here, if i, as player 1, hide in spot B and player 2 00:46:41.680 --> 00:46:44.000 align:middle line:84% looks for me in A, then they're not going to find me. 00:46:44.000 --> 00:46:45.618 align:middle line:84% And therefore, I get a payoff of 1 00:46:45.618 --> 00:46:47.660 align:middle line:84% and the other player gets a payoff of negative 1. 00:46:47.660 --> 00:46:50.280 align:middle line:84% You could also think as the hider as the penalty taker 00:46:50.280 --> 00:46:52.000 align:middle line:90% and the seeker as the goalie. 00:46:52.000 --> 00:46:54.083 align:middle line:84% There's a lot of different stories you could tell. 00:46:54.083 --> 00:46:56.152 align:middle line:90% 00:46:56.152 --> 00:46:58.360 align:middle line:84% So notice a difference between this game and the game 00:46:58.360 --> 00:46:59.440 align:middle line:90% we discussed before. 00:46:59.440 --> 00:47:03.300 align:middle line:84% So this is a game of pure conflict of interest. 00:47:03.300 --> 00:47:10.240 align:middle line:90% 00:47:10.240 --> 00:47:12.502 align:middle line:84% In the Boston game, there were both. 00:47:12.502 --> 00:47:13.960 align:middle line:84% There was some conflict of interest 00:47:13.960 --> 00:47:16.085 align:middle line:84% because one person liked the Red Sox and one person 00:47:16.085 --> 00:47:16.980 align:middle line:90% liked the Celtics. 00:47:16.980 --> 00:47:18.440 align:middle line:84% But there was also some common interest 00:47:18.440 --> 00:47:19.920 align:middle line:84% because they both liked each other's company, 00:47:19.920 --> 00:47:21.560 align:middle line:90% and they wanted to be together. 00:47:21.560 --> 00:47:23.600 align:middle line:84% Here, it's purely a conflict of interest. 00:47:23.600 --> 00:47:25.180 align:middle line:90% I want us as player 1. 00:47:25.180 --> 00:47:28.060 align:middle line:84% I want us to hide in different places. 00:47:28.060 --> 00:47:29.560 align:middle line:90% So maybe I'll say that. 00:47:29.560 --> 00:47:33.100 align:middle line:90% So what does player 1 want? 00:47:33.100 --> 00:47:36.420 align:middle line:84% They want them to look and hide in different places, maybe 00:47:36.420 --> 00:47:37.560 align:middle line:90% different locations. 00:47:37.560 --> 00:47:40.260 align:middle line:90% 00:47:40.260 --> 00:47:43.600 align:middle line:84% And player 2 wants them to be in the same location. 00:47:43.600 --> 00:47:47.385 align:middle line:90% 00:47:47.385 --> 00:47:49.760 align:middle line:84% Of course, the part is you can only choose your location. 00:47:49.760 --> 00:47:53.940 align:middle line:84% If you chose the other person's location, then it would be easy. 00:47:53.940 --> 00:47:55.780 align:middle line:84% So as a first step for this, let's go 00:47:55.780 --> 00:47:59.980 align:middle line:84% through our standard procedure where we compute player 1's best 00:47:59.980 --> 00:48:04.060 align:middle line:84% responses to player 2 and player 2's best responses to player 1. 00:48:04.060 --> 00:48:05.680 align:middle line:90% So let's do it slowly. 00:48:05.680 --> 00:48:11.000 align:middle line:84% If I'm player 1, if player 2 guesses A, then I want to hide. 00:48:11.000 --> 00:48:15.340 align:middle line:84% So I want to do B. And if player 2 guesses B, 00:48:15.340 --> 00:48:17.560 align:middle line:84% I want to hide and be in a different place. 00:48:17.560 --> 00:48:20.340 align:middle line:90% So I would like to be A. 00:48:20.340 --> 00:48:21.900 align:middle line:84% Now let's look at it the other way. 00:48:21.900 --> 00:48:25.960 align:middle line:84% Now let's look at the perspective of player 2. 00:48:25.960 --> 00:48:30.600 align:middle line:84% If player 1 plays A, then player 2 does better if they also go A 00:48:30.600 --> 00:48:32.280 align:middle line:84% because they want to find the person. 00:48:32.280 --> 00:48:39.060 align:middle line:84% And if player 1 goes B, then player two wants to go B. 00:48:39.060 --> 00:48:41.960 align:middle line:84% So we've computed the best responses. 00:48:41.960 --> 00:48:45.080 align:middle line:84% We can look, and if we follow our algorithm, 00:48:45.080 --> 00:48:47.680 align:middle line:84% where are the Nash equilibria in this game? 00:48:47.680 --> 00:48:51.260 align:middle line:84% We want to look for a cell where both numbers are underlined. 00:48:51.260 --> 00:48:54.040 align:middle line:84% But here, we have a problem because none of the cells 00:48:54.040 --> 00:48:55.800 align:middle line:90% are underlined. 00:48:55.800 --> 00:48:59.458 align:middle line:84% So what this tells us is that this game has no-- 00:48:59.458 --> 00:49:02.000 align:middle line:84% and this will be important in the second-- pure-strategy Nash 00:49:02.000 --> 00:49:02.780 align:middle line:90% equilibrium. 00:49:02.780 --> 00:49:07.040 align:middle line:90% 00:49:07.040 --> 00:49:09.100 align:middle line:84% And we can think through it reasonably right. 00:49:09.100 --> 00:49:12.320 align:middle line:90% 00:49:12.320 --> 00:49:14.720 align:middle line:84% In order for us to have a Nash equilibrium, 00:49:14.720 --> 00:49:17.560 align:middle line:84% neither player can have a profitable deviation. 00:49:17.560 --> 00:49:20.680 align:middle line:84% But if we're in the same place, then someone certainly 00:49:20.680 --> 00:49:22.640 align:middle line:90% has a profitable deviation. 00:49:22.640 --> 00:49:26.320 align:middle line:84% The hider would rather deviate if we're in the same place. 00:49:26.320 --> 00:49:27.695 align:middle line:84% And if we're in different places, 00:49:27.695 --> 00:49:29.487 align:middle line:84% then, again, someone would like to deviate, 00:49:29.487 --> 00:49:31.780 align:middle line:84% the seeker would like to deviate and change places 00:49:31.780 --> 00:49:34.180 align:middle line:84% so that they find the person who's hiding. 00:49:34.180 --> 00:49:36.400 align:middle line:84% So because of this immense conflict of interest, 00:49:36.400 --> 00:49:38.600 align:middle line:84% we can't have a pure-strategy Nash equilibrium 00:49:38.600 --> 00:49:41.083 align:middle line:84% because someone's losing, and whoever's losing 00:49:41.083 --> 00:49:43.500 align:middle line:84% would rather change their behavior so that they could win. 00:49:43.500 --> 00:49:45.820 align:middle line:84% This is a zero-sum game in a formal sense 00:49:45.820 --> 00:49:47.673 align:middle line:84% that we'll talk about more later. 00:49:47.673 --> 00:49:49.340 align:middle line:84% But we'd still like to make a prediction 00:49:49.340 --> 00:49:50.660 align:middle line:90% about play in this game. 00:49:50.660 --> 00:49:52.558 align:middle line:84% If we just use rationalizability, 00:49:52.558 --> 00:49:54.100 align:middle line:84% well, anything could happen because I 00:49:54.100 --> 00:49:56.240 align:middle line:84% can hide an A if I think the other person's going to B, 00:49:56.240 --> 00:49:57.660 align:middle line:84% and I can hide in B If I think they're going to A, 00:49:57.660 --> 00:49:59.540 align:middle line:90% and we can explain everything. 00:49:59.540 --> 00:50:01.700 align:middle line:84% So I don't know, how would you play this game? 00:50:01.700 --> 00:50:04.860 align:middle line:84% If you were taking penalty kicks in the World Cup, 00:50:04.860 --> 00:50:08.400 align:middle line:84% how would you-- what strategy would you use? 00:50:08.400 --> 00:50:08.900 align:middle line:90% Yeah. 00:50:08.900 --> 00:50:09.680 align:middle line:90% AUDIENCE: Mixed-strategy. 00:50:09.680 --> 00:50:11.513 align:middle line:84% IAN BALL: You'd use a mixed strategy, right? 00:50:11.513 --> 00:50:12.320 align:middle line:90% So you'd randomize. 00:50:12.320 --> 00:50:15.120 align:middle line:84% And in this case, how do you think you would randomize? 00:50:15.120 --> 00:50:17.900 align:middle line:90% 00:50:17.900 --> 00:50:21.043 align:middle line:84% Would you play A 99% of the time? 00:50:21.043 --> 00:50:21.960 align:middle line:90% AUDIENCE: 1/2 and 1/2. 00:50:21.960 --> 00:50:24.543 align:middle line:84% IAN BALL: Probably 1/2 and 1/2 because you don't want yourself 00:50:24.543 --> 00:50:25.520 align:middle line:90% to be predictable. 00:50:25.520 --> 00:50:27.820 align:middle line:84% If you look at the data on World Cup penalty takers, 00:50:27.820 --> 00:50:29.153 align:middle line:90% that's a pretty good prediction. 00:50:29.153 --> 00:50:31.128 align:middle line:84% I mean, sometimes right-footed players 00:50:31.128 --> 00:50:32.920 align:middle line:84% have a bit of a left advantage, and there's 00:50:32.920 --> 00:50:33.860 align:middle line:90% all these subtleties. 00:50:33.860 --> 00:50:37.240 align:middle line:84% But roughly, people mix 50/50 because if I develop 00:50:37.240 --> 00:50:39.320 align:middle line:84% a reputation, if everyone knows that-- 00:50:39.320 --> 00:50:41.777 align:middle line:84% if you think about playing hide and seek with a young kid, 00:50:41.777 --> 00:50:43.860 align:middle line:84% sometimes they just always hide in the same place. 00:50:43.860 --> 00:50:45.520 align:middle line:90% It's pretty easy to beat them. 00:50:45.520 --> 00:50:47.820 align:middle line:84% And even if you don't always hide in the same place, 00:50:47.820 --> 00:50:49.740 align:middle line:84% if you hid in the same place 95% of the time, 00:50:49.740 --> 00:50:51.380 align:middle line:84% you'd also be pretty easy to find. 00:50:51.380 --> 00:50:53.902 align:middle line:84% So it seems pretty intuitive that you'd want to do 1/2, 1/2. 00:50:53.902 --> 00:50:54.860 align:middle line:90% So that's a good guess. 00:50:54.860 --> 00:50:55.780 align:middle line:90% And that's right. 00:50:55.780 --> 00:50:59.480 align:middle line:84% So it turns out it will have a mixed-strategy Nash equilibrium. 00:50:59.480 --> 00:51:06.638 align:middle line:84% But 1/2, 1/2, meaning player 1 randomizes 1/2, 00:51:06.638 --> 00:51:08.680 align:middle line:84% 1/2 between A and B, let me write this a bit more 00:51:08.680 --> 00:51:09.920 align:middle line:90% explicitly. 00:51:09.920 --> 00:51:16.080 align:middle line:84% 1/2 A plus 1/2 B comma 1/2 A plus 1/2 B. 00:51:16.080 --> 00:51:21.210 align:middle line:84% This means player 1 mixes 50/50 and player 2 mixes 50/50. 00:51:21.210 --> 00:51:24.310 align:middle line:84% This is going to be a Nash equilibrium in mixed strategies. 00:51:24.310 --> 00:51:26.310 align:middle line:90% So I'd say this is-- 00:51:26.310 --> 00:51:28.850 align:middle line:84% maybe I'll use the acronym Mixed-Strategy. 00:51:28.850 --> 00:51:32.703 align:middle line:84% NE, MSNE, mixed-strategy Nash equilibrium. 00:51:32.703 --> 00:51:34.370 align:middle line:84% But I haven't even defined what that is. 00:51:34.370 --> 00:51:35.947 align:middle line:90% So let's now define that. 00:51:35.947 --> 00:51:37.530 align:middle line:84% I think it's exactly what you'd think. 00:51:37.530 --> 00:52:01.870 align:middle line:90% 00:52:01.870 --> 00:52:06.050 align:middle line:84% So definition, a mixed-strategy profile-- 00:52:06.050 --> 00:52:12.350 align:middle line:90% 00:52:12.350 --> 00:52:14.070 align:middle line:84% and now we'll use sigma instead of s 00:52:14.070 --> 00:52:15.650 align:middle line:90% to remind us that this is mixed. 00:52:15.650 --> 00:52:20.330 align:middle line:84% So sigma star equals sigma 1 star, 00:52:20.330 --> 00:52:26.310 align:middle line:84% sigma n star is a Nash equilibrium-- 00:52:26.310 --> 00:52:31.570 align:middle line:90% 00:52:31.570 --> 00:52:42.690 align:middle line:84% and we'll give exactly the same definition, for each player i, 00:52:42.690 --> 00:52:43.710 align:middle line:90% what happens? 00:52:43.710 --> 00:52:51.530 align:middle line:84% Well, I'm comparing now what I'm supposed to do as player i 00:52:51.530 --> 00:52:57.890 align:middle line:84% to a deviation, which I'll maybe call sigma i prime. 00:52:57.890 --> 00:53:00.770 align:middle line:84% So this is the mixed-strategy I'm actually playing. 00:53:00.770 --> 00:53:03.410 align:middle line:84% This is the mixed-strategy I could deviate to. 00:53:03.410 --> 00:53:06.490 align:middle line:84% And I want this to be optimal given the way my opponents are 00:53:06.490 --> 00:53:07.030 align:middle line:90% playing. 00:53:07.030 --> 00:53:16.170 align:middle line:90% 00:53:16.170 --> 00:53:17.870 align:middle line:90% So this is some new notation. 00:53:17.870 --> 00:53:21.270 align:middle line:84% We've been using capital SI to denote all the pure strategies 00:53:21.270 --> 00:53:22.350 align:middle line:90% of player i. 00:53:22.350 --> 00:53:25.532 align:middle line:84% It's often convenient to use, this, it's the summation symbol, 00:53:25.532 --> 00:53:26.990 align:middle line:84% and this is capital Sigma, and this 00:53:26.990 --> 00:53:29.130 align:middle line:84% is the set of all mixed strategies by player i. 00:53:29.130 --> 00:53:35.070 align:middle line:90% 00:53:35.070 --> 00:53:37.510 align:middle line:84% So we use lowercase s for pure strategy, 00:53:37.510 --> 00:53:40.510 align:middle line:84% uppercase S for a set of pure strategies, 00:53:40.510 --> 00:53:43.150 align:middle line:84% lowercase sigma for a mixed strategy, 00:53:43.150 --> 00:53:47.710 align:middle line:84% and uppercase Sigma for a set of mixed strategies. 00:53:47.710 --> 00:53:52.070 align:middle line:84% So it's the same idea, given the way my opponents are playing, 00:53:52.070 --> 00:53:53.850 align:middle line:90% my strategy is a best response. 00:53:53.850 --> 00:53:55.950 align:middle line:84% My mixed strategy does weakly better 00:53:55.950 --> 00:53:58.390 align:middle line:84% than any other mixed strategy I could use. 00:53:58.390 --> 00:53:59.970 align:middle line:84% But if you look at this definition, 00:53:59.970 --> 00:54:03.190 align:middle line:84% it's pretty messy because there's so many different mixed 00:54:03.190 --> 00:54:05.570 align:middle line:90% strategies I could deviate to. 00:54:05.570 --> 00:54:07.810 align:middle line:84% Even in a game with just two strategies, 00:54:07.810 --> 00:54:11.070 align:middle line:84% I could deviate to any probability between 0 and 1. 00:54:11.070 --> 00:54:12.853 align:middle line:84% That's a lot of deviations to deal with. 00:54:12.853 --> 00:54:14.270 align:middle line:84% But fortunately, we don't actually 00:54:14.270 --> 00:54:15.570 align:middle line:90% have to worry about that. 00:54:15.570 --> 00:54:19.590 align:middle line:84% So it turns out, even if I were to check every mixed strategy, 00:54:19.590 --> 00:54:22.370 align:middle line:84% it's actually enough just to check my pure strategy 00:54:22.370 --> 00:54:23.690 align:middle line:90% deviations. 00:54:23.690 --> 00:54:27.450 align:middle line:84% So it turns out this is the same as-- 00:54:27.450 --> 00:54:34.410 align:middle line:84% the same thing, but with si prime for all si prime. 00:54:34.410 --> 00:54:36.290 align:middle line:84% So when I use the two dashes, I'm 00:54:36.290 --> 00:54:38.783 align:middle line:84% just saying everything else in the definition is the same. 00:54:38.783 --> 00:54:40.450 align:middle line:84% But instead of saying, it must do better 00:54:40.450 --> 00:54:42.370 align:middle line:84% than any mixed strategy, I'm saying 00:54:42.370 --> 00:54:45.970 align:middle line:84% it must do weakly better than any pure strategy. 00:54:45.970 --> 00:54:48.030 align:middle line:84% And we could give a formal math proof, 00:54:48.030 --> 00:54:50.530 align:middle line:84% but let's just think of the intuition. 00:54:50.530 --> 00:54:53.970 align:middle line:84% If I'd rather do what I'm doing than deviate 00:54:53.970 --> 00:54:56.370 align:middle line:84% to any pure strategy, then by choosing 00:54:56.370 --> 00:54:58.790 align:middle line:84% to flip a coin and mix over those pure strategies, 00:54:58.790 --> 00:55:01.210 align:middle line:84% I can't do strictly better because if I mix over 00:55:01.210 --> 00:55:03.330 align:middle line:84% those pure strategies, my payoff is just 00:55:03.330 --> 00:55:05.610 align:middle line:84% going to be an average of my payoff 00:55:05.610 --> 00:55:07.530 align:middle line:90% from all the pure deviations. 00:55:07.530 --> 00:55:08.970 align:middle line:84% So if each of the pure deviations 00:55:08.970 --> 00:55:11.690 align:middle line:84% makes me weakly worse off, then a mixture of them 00:55:11.690 --> 00:55:13.330 align:middle line:84% must also make me weakly worse off. 00:55:13.330 --> 00:55:14.348 align:middle line:90% That's the intuition. 00:55:14.348 --> 00:55:16.390 align:middle line:84% You could write down a formal mathematical proof. 00:55:16.390 --> 00:55:19.710 align:middle line:84% But the point is, it's really important here 00:55:19.710 --> 00:55:22.130 align:middle line:84% that what I am doing can be mixed. 00:55:22.130 --> 00:55:24.010 align:middle line:84% So I want people to really understand this. 00:55:24.010 --> 00:55:24.810 align:middle line:90% This is crucial. 00:55:24.810 --> 00:55:27.510 align:middle line:90% 00:55:27.510 --> 00:55:30.430 align:middle line:84% It's crucial that we're allowing my equilibrium strategy 00:55:30.430 --> 00:55:32.350 align:middle line:90% to be mixed and not pure. 00:55:32.350 --> 00:55:34.930 align:middle line:84% But when I contemplate the deviations, 00:55:34.930 --> 00:55:37.330 align:middle line:84% it's enough to only look at pure strategies. 00:55:37.330 --> 00:55:39.850 align:middle line:84% And that's a key difference here. 00:55:39.850 --> 00:55:42.612 align:middle line:84% So indeed, how can we tell that mixtures matter? 00:55:42.612 --> 00:55:44.070 align:middle line:84% Well, already in this game here, we 00:55:44.070 --> 00:55:47.090 align:middle line:84% saw that it didn't have a pure-strategy Nash equilibrium. 00:55:47.090 --> 00:55:49.290 align:middle line:84% But it does have a mixed-strategy Nash equilibrium. 00:55:49.290 --> 00:55:52.510 align:middle line:84% So the fact that we allow the equilibrium strategy to be mixed 00:55:52.510 --> 00:55:53.830 align:middle line:90% is essential. 00:55:53.830 --> 00:55:55.950 align:middle line:84% But when you're contemplating deviations, 00:55:55.950 --> 00:55:59.070 align:middle line:84% you only have to worry about pure strategies. 00:55:59.070 --> 00:56:00.670 align:middle line:84% And let's take that randomization 00:56:00.670 --> 00:56:01.570 align:middle line:90% idea little further. 00:56:01.570 --> 00:56:03.930 align:middle line:84% Why would I ever want to play a mixed strategy? 00:56:03.930 --> 00:56:07.070 align:middle line:84% I mean, why am I willing to flip a coin and sometimes go to A 00:56:07.070 --> 00:56:08.425 align:middle line:90% and sometimes go to B? 00:56:08.425 --> 00:56:11.050 align:middle line:84% I mean, if A is better, wouldn't I just always want to go to A? 00:56:11.050 --> 00:56:13.470 align:middle line:84% And if B is better, wouldn't I just always want to go to B? 00:56:13.470 --> 00:56:15.012 align:middle line:84% Why would I ever want to flip a coin? 00:56:15.012 --> 00:56:17.090 align:middle line:84% Or why would I ever be willing to flip a coin 00:56:17.090 --> 00:56:21.570 align:middle line:84% and play A some of the time and B some of the time? 00:56:21.570 --> 00:56:22.590 align:middle line:90% What must be true? 00:56:22.590 --> 00:56:27.410 align:middle line:90% 00:56:27.410 --> 00:56:28.218 align:middle line:90% If I'm-- yeah. 00:56:28.218 --> 00:56:30.010 align:middle line:84% AUDIENCE: A is better in some circumstances 00:56:30.010 --> 00:56:31.352 align:middle line:90% while B is better [INAUDIBLE]. 00:56:31.352 --> 00:56:32.810 align:middle line:84% IAN BALL: OK, that's true, but what 00:56:32.810 --> 00:56:35.393 align:middle line:84% about given my belief, given the way my opponents are playing, 00:56:35.393 --> 00:56:36.230 align:middle line:90% what must be true? 00:56:36.230 --> 00:56:39.370 align:middle line:90% 00:56:39.370 --> 00:56:41.730 align:middle line:84% If I strictly preferred A to B, given 00:56:41.730 --> 00:56:43.250 align:middle line:84% the way my opponents were playing, 00:56:43.250 --> 00:56:45.490 align:middle line:84% would I want to mix between A and B? 00:56:45.490 --> 00:56:46.010 align:middle line:90% No. 00:56:46.010 --> 00:56:46.550 align:middle line:90% Yeah. 00:56:46.550 --> 00:56:48.133 align:middle line:84% AUDIENCE: Does your opponent also have 00:56:48.133 --> 00:56:49.730 align:middle line:90% to be playing a mixed strategy? 00:56:49.730 --> 00:56:52.830 align:middle line:84% IAN BALL: So that's related, but maybe not quite where we are. 00:56:52.830 --> 00:56:54.750 align:middle line:84% So if I'm willing to mix between A and B, 00:56:54.750 --> 00:56:57.250 align:middle line:84% it must be that I'm indifferent between A and B. 00:56:57.250 --> 00:56:58.430 align:middle line:90% That's the key thing. 00:56:58.430 --> 00:57:00.890 align:middle line:84% If I strictly preferred A to B, I 00:57:00.890 --> 00:57:03.410 align:middle line:84% would never play B with positive probability at all. 00:57:03.410 --> 00:57:05.690 align:middle line:84% And if I strictly preferred B, I would never 00:57:05.690 --> 00:57:07.470 align:middle line:84% play A with positive probability? 00:57:07.470 --> 00:57:09.590 align:middle line:84% So I'm only willing to flip a coin 00:57:09.590 --> 00:57:12.983 align:middle line:84% and sometimes play A and sometimes play B if I'm exactly 00:57:12.983 --> 00:57:14.150 align:middle line:90% indifferent between A and B. 00:57:14.150 --> 00:57:15.250 align:middle line:90% Now this gets to your point. 00:57:15.250 --> 00:57:17.050 align:middle line:84% Why will I be indifferent between A and B? 00:57:17.050 --> 00:57:21.150 align:middle line:84% It'll be precisely because my opponent is mixing. 00:57:21.150 --> 00:57:23.490 align:middle line:90% So this is a key observation. 00:57:23.490 --> 00:57:28.470 align:middle line:90% 00:57:28.470 --> 00:57:30.170 align:middle line:90% Maybe I'll start a new board. 00:57:30.170 --> 00:57:38.150 align:middle line:90% 00:57:38.150 --> 00:57:40.330 align:middle line:90% So this is the key observation. 00:57:40.330 --> 00:57:47.630 align:middle line:90% 00:57:47.630 --> 00:57:51.790 align:middle line:90% So mixing is optimal. 00:57:51.790 --> 00:57:54.190 align:middle line:84% Only if you're indifferent between the things 00:57:54.190 --> 00:57:55.090 align:middle line:90% you're mixing over. 00:57:55.090 --> 00:58:03.990 align:middle line:90% 00:58:03.990 --> 00:58:07.080 align:middle line:84% And in fact, we can say something even stronger. 00:58:07.080 --> 00:58:08.830 align:middle line:84% It's not just enough that I'm indifferent, 00:58:08.830 --> 00:58:11.530 align:middle line:84% but I'm only willing to mix over pure strategies that 00:58:11.530 --> 00:58:13.610 align:middle line:90% are best responses. 00:58:13.610 --> 00:58:15.690 align:middle line:90% So it must be-- 00:58:15.690 --> 00:58:28.030 align:middle line:84% and in fact, it's only optimal to mix over pure best responses. 00:58:28.030 --> 00:58:35.192 align:middle line:90% 00:58:35.192 --> 00:58:36.150 align:middle line:90% So what does that mean? 00:58:36.150 --> 00:58:38.010 align:middle line:90% Well, it's just a silly example. 00:58:38.010 --> 00:58:41.210 align:middle line:84% What if I had A, B, and C and given 00:58:41.210 --> 00:58:44.250 align:middle line:84% the way my opponents were playing, A gave me 1, 00:58:44.250 --> 00:58:47.410 align:middle line:90% B gave me 1, and C gave me 2. 00:58:47.410 --> 00:58:49.080 align:middle line:90% Do I want to mix over A and B? 00:58:49.080 --> 00:58:51.330 align:middle line:84% Well, no, even though I'm indifferent between A and B, 00:58:51.330 --> 00:58:52.830 align:middle line:84% that's necessary, but it's not sufficient. 00:58:52.830 --> 00:58:54.580 align:middle line:84% I certainly don't want to mix over A and B 00:58:54.580 --> 00:58:55.810 align:middle line:90% because C is strictly better. 00:58:55.810 --> 00:58:57.590 align:middle line:84% So if I'm willing to mix, I must be indifferent. 00:58:57.590 --> 00:58:58.530 align:middle line:84% But sometimes people get confused, 00:58:58.530 --> 00:59:00.590 align:middle line:84% and they get it backwards and they say, if I'm indifferent, 00:59:00.590 --> 00:59:01.752 align:middle line:90% I must be willing to mix. 00:59:01.752 --> 00:59:03.710 align:middle line:84% But no, it's not just enough to be indifferent. 00:59:03.710 --> 00:59:06.030 align:middle line:84% It also has to be optimal what you're mixing over. 00:59:06.030 --> 00:59:08.090 align:middle line:84% And that's precisely what I'm saying here. 00:59:08.090 --> 00:59:10.690 align:middle line:84% It's only optimal to mix over pure best responses. 00:59:10.690 --> 00:59:14.270 align:middle line:84% So if I added D at 2, now, indeed, I'd 00:59:14.270 --> 00:59:20.110 align:middle line:84% be willing to mix between C and D. 00:59:20.110 --> 00:59:22.790 align:middle line:90% So now let's try to check-- 00:59:22.790 --> 00:59:25.070 align:middle line:90% going back to this game-- 00:59:25.070 --> 00:59:29.870 align:middle line:84% that 1/2, 1/2 is actually a Nash equilibrium. 00:59:29.870 --> 00:59:31.730 align:middle line:84% So I think the best way to do it is to say, 00:59:31.730 --> 00:59:34.790 align:middle line:84% well, let's take the perspective of player 2. 00:59:34.790 --> 00:59:38.470 align:middle line:84% And let's suppose that player 1 mixes 1/2, 1/2. 00:59:38.470 --> 00:59:41.525 align:middle line:84% What payoff does player 2 get from playing A and playing B? 00:59:41.525 --> 00:59:44.150 align:middle line:84% So what I'm going to imagine is I'm going to add a new strategy 00:59:44.150 --> 00:59:50.670 align:middle line:84% here, 1/2 A, 1/2 B. So I'm adding a new row to this matrix 00:59:50.670 --> 00:59:54.150 align:middle line:84% that's going to represent the payoffs if player 1 chooses this 00:59:54.150 --> 00:59:55.610 align:middle line:90% mixed strategy. 00:59:55.610 --> 00:59:57.110 align:middle line:84% So in here, what I want to write is, 00:59:57.110 --> 00:59:59.550 align:middle line:84% what are the expected payoffs if player 1 00:59:59.550 --> 01:00:02.970 align:middle line:84% uses this mixed strategy and player 2 plays B? 01:00:02.970 --> 01:00:04.450 align:middle line:90% Sorry, plays A. 01:00:04.450 --> 01:00:06.590 align:middle line:84% And then, over here, what am I going to write? 01:00:06.590 --> 01:00:09.530 align:middle line:84% I'm going to write the payoffs-- if player 1 plays this mixed 01:00:09.530 --> 01:00:14.810 align:middle line:84% strategy and player 2 plays B. And it turns out, this-- 01:00:14.810 --> 01:00:18.010 align:middle line:90% well, let's go through it. 01:00:18.010 --> 01:00:22.170 align:middle line:84% If player 2 plays and player 1 plays A with probability 1/2 01:00:22.170 --> 01:00:26.850 align:middle line:84% and B with probability 1/2, then what is the payoff of-- 01:00:26.850 --> 01:00:29.050 align:middle line:84% what are the payoffs going to be for the players? 01:00:29.050 --> 01:00:33.090 align:middle line:84% Well, player 1 is going to get negative 1 with probability 1/2 01:00:33.090 --> 01:00:37.330 align:middle line:84% and 1 with probability 1/2, which just gives them 0. 01:00:37.330 --> 01:00:40.490 align:middle line:84% And player 2 is also going to get 1 with probability 1/2 01:00:40.490 --> 01:00:42.190 align:middle line:84% and negative 1 with probability 1/2. 01:00:42.190 --> 01:00:44.378 align:middle line:84% So we just get 0, and if we go to this side, 01:00:44.378 --> 01:00:46.170 align:middle line:84% we're actually just going to get 0's again. 01:00:46.170 --> 01:00:49.370 align:middle line:84% And mathematically, the way we form this row of the matrix 01:00:49.370 --> 01:00:52.770 align:middle line:84% is we just take this row, multiply it by half, 01:00:52.770 --> 01:00:55.990 align:middle line:84% take this row, multiply it by half, and add them together. 01:00:55.990 --> 01:00:59.890 align:middle line:84% And when we do that, we exactly get 0, 0, 0, 0. 01:00:59.890 --> 01:01:02.830 align:middle line:84% So in particular, the key thing about this 01:01:02.830 --> 01:01:06.070 align:middle line:84% is that these numbers are the same. 01:01:06.070 --> 01:01:13.230 align:middle line:84% So what this tells me is if player 1 plays 1/2 A plus 1/2 B, 01:01:13.230 --> 01:01:17.710 align:middle line:84% then player 2 gets a payoff of 0 if they play A and a payoff of 0 01:01:17.710 --> 01:01:21.510 align:middle line:84% if they play B. And that means that player 2 is indifferent 01:01:21.510 --> 01:01:24.390 align:middle line:84% between A and B, and in fact, A and B are both pure best 01:01:24.390 --> 01:01:25.350 align:middle line:90% responses. 01:01:25.350 --> 01:01:27.190 align:middle line:84% So if I'm indifferent between A and B, 01:01:27.190 --> 01:01:30.790 align:middle line:84% then I'm also willing to mix over A and B. And therefore, 01:01:30.790 --> 01:01:34.870 align:middle line:84% it's a best response for player 2 to mix 1/2, 1/2. 01:01:34.870 --> 01:01:39.190 align:middle line:84% So technically, what I've shown is if player 1 is mixing 1/2, 01:01:39.190 --> 01:01:42.888 align:middle line:84% 1/2, it's a best response for player 2 to mix 1/2, 1/2. 01:01:42.888 --> 01:01:44.930 align:middle line:84% So technically, if I want to finish the argument, 01:01:44.930 --> 01:01:46.710 align:middle line:84% I need to show the other direction, 01:01:46.710 --> 01:01:49.910 align:middle line:84% that if player 2 mixes 1/2, 1/2, then it's a best response 01:01:49.910 --> 01:01:52.390 align:middle line:84% for player 1 to mix 1/2, 1/2, but it's going to be entirely 01:01:52.390 --> 01:01:52.930 align:middle line:90% symmetric. 01:01:52.930 --> 01:01:55.290 align:middle line:84% I could add another column here and go 0, 0, 01:01:55.290 --> 01:01:58.745 align:middle line:90% but we can check that-- 01:01:58.745 --> 01:01:59.870 align:middle line:90% I'll just put a check mark. 01:01:59.870 --> 01:02:00.690 align:middle line:90% We checked it. 01:02:00.690 --> 01:02:01.550 align:middle line:90% It's an equilibrium. 01:02:01.550 --> 01:02:03.870 align:middle line:84% Don't do that on your homework, but that's all we need. 01:02:03.870 --> 01:02:08.530 align:middle line:90% 01:02:08.530 --> 01:02:11.930 align:middle line:84% Now, let's return to the Boston game. 01:02:11.930 --> 01:02:13.370 align:middle line:90% I don't know if we have it. 01:02:13.370 --> 01:02:14.992 align:middle line:90% I may have to write it again. 01:02:14.992 --> 01:02:15.950 align:middle line:90% You can write it again. 01:02:15.950 --> 01:02:17.490 align:middle line:90% That's OK. 01:02:17.490 --> 01:02:19.710 align:middle line:90% So now let's go to the BOS game. 01:02:19.710 --> 01:02:21.290 align:middle line:90% This is a bit trickier. 01:02:21.290 --> 01:02:29.750 align:middle line:84% And we have Celtics, Red Sox, Celtics, Red Sox 2, 1, 1, 2, 0, 01:02:29.750 --> 01:02:32.210 align:middle line:90% 0, 0, 0. 01:02:32.210 --> 01:02:35.290 align:middle line:84% So this game, we already found some pure equilibria. 01:02:35.290 --> 01:02:36.450 align:middle line:90% We found two of them. 01:02:36.450 --> 01:02:39.130 align:middle line:84% We said there's one equilibrium where we both play C, 01:02:39.130 --> 01:02:41.810 align:middle line:84% and there's one equilibrium where we both play R. 01:02:41.810 --> 01:02:44.990 align:middle line:84% But it turns out, there's also a mixed-strategy equilibrium. 01:02:44.990 --> 01:02:50.730 align:middle line:84% So here, what we'd like to do is find a mixed-strategy Nash 01:02:50.730 --> 01:02:52.390 align:middle line:90% equilibrium of this game. 01:02:52.390 --> 01:02:55.290 align:middle line:90% 01:02:55.290 --> 01:02:58.720 align:middle line:84% Now, of course, we could say the pure-strategy Nash equilibria 01:02:58.720 --> 01:03:00.920 align:middle line:84% we already found are technically mixed-strategy Nash 01:03:00.920 --> 01:03:04.420 align:middle line:84% equilibria where people's mixing probabilities are just 1 and 0. 01:03:04.420 --> 01:03:07.280 align:middle line:84% But what I mean is something that's actually mixed. 01:03:07.280 --> 01:03:10.340 align:middle line:84% So what I really mean is a mixed-strategy Nash equilibrium 01:03:10.340 --> 01:03:12.660 align:middle line:84% that's not a pure-strategy Nash equilibrium. 01:03:12.660 --> 01:03:15.720 align:middle line:90% 01:03:15.720 --> 01:03:18.617 align:middle line:84% OK, well, I think the first step is let's 01:03:18.617 --> 01:03:19.700 align:middle line:90% just write some notations. 01:03:19.700 --> 01:03:20.980 align:middle line:84% So we're going to look for an equilibrium. 01:03:20.980 --> 01:03:22.480 align:middle line:84% We're going to look for some mixing. 01:03:22.480 --> 01:03:26.160 align:middle line:84% So let's look for an equilibrium of the form-- well, 01:03:26.160 --> 01:03:31.120 align:middle line:84% player 1 is going to put, say, probability P1 on C. 01:03:31.120 --> 01:03:35.360 align:middle line:84% And let's say player 2 is going to put probability P2 on C. 01:03:35.360 --> 01:03:38.840 align:middle line:84% So our goal is to find a Nash equilibrium of the form player 01:03:38.840 --> 01:03:44.440 align:middle line:84% 1 plays P1 times C plus 1 minus P1 times R. 01:03:44.440 --> 01:03:47.120 align:middle line:84% So what this means is, with probability P1, 01:03:47.120 --> 01:03:49.480 align:middle line:84% player 1 goes to the Celtics game, 01:03:49.480 --> 01:03:52.640 align:middle line:84% and with probability 1 minus P1, player 1 01:03:52.640 --> 01:03:54.520 align:middle line:90% goes to the Red Sox game. 01:03:54.520 --> 01:03:55.920 align:middle line:84% And then, similarly for player 2, 01:03:55.920 --> 01:03:57.920 align:middle line:84% but we have potentially a different number here. 01:03:57.920 --> 01:04:06.386 align:middle line:90% 01:04:06.386 --> 01:04:09.700 align:middle line:84% And we want to find an equilibrium of this form 01:04:09.700 --> 01:04:12.820 align:middle line:84% where maybe what we'd like is we want it to actually involve 01:04:12.820 --> 01:04:13.320 align:middle line:90% mixing. 01:04:13.320 --> 01:04:15.740 align:middle line:90% So let's say P1-- 01:04:15.740 --> 01:04:19.500 align:middle line:84% 0 is less than P1 is less than 1, and 0 is less than P2 01:04:19.500 --> 01:04:20.360 align:middle line:90% is less than 1. 01:04:20.360 --> 01:04:21.000 align:middle line:90% This is what we want. 01:04:21.000 --> 01:04:23.292 align:middle line:84% We want something where each player is actually mixing. 01:04:23.292 --> 01:04:27.535 align:middle line:84% They're not just playing one strategy with probability 1. 01:04:27.535 --> 01:04:29.160 align:middle line:84% So I guess the trick is to think, well, 01:04:29.160 --> 01:04:30.660 align:middle line:90% what must be true here? 01:04:30.660 --> 01:04:36.460 align:middle line:84% So if we want this to be a mixed-strategy Nash equilibrium, 01:04:36.460 --> 01:04:41.220 align:middle line:84% it must be that player 1 is willing to mix between C and R, 01:04:41.220 --> 01:04:42.910 align:middle line:84% given the way that player 2 is playing. 01:04:42.910 --> 01:04:45.160 align:middle line:84% And based on what we said before, what does that mean? 01:04:45.160 --> 01:04:49.240 align:middle line:84% What must be true if player 1 is willing to mix between C and R? 01:04:49.240 --> 01:04:52.120 align:middle line:90% 01:04:52.120 --> 01:04:52.620 align:middle line:90% Yeah. 01:04:52.620 --> 01:04:56.000 align:middle line:84% AUDIENCE: Player 1 [INAUDIBLE] C, R [INAUDIBLE]. 01:04:56.000 --> 01:04:58.480 align:middle line:84% IAN BALL: Exactly, and then the other direction. 01:04:58.480 --> 01:05:00.600 align:middle line:84% Player 2 must also be indifferent between C 01:05:00.600 --> 01:05:02.705 align:middle line:90% and R, given player 1 strategy. 01:05:02.705 --> 01:05:04.080 align:middle line:84% So the way that we as the analyst 01:05:04.080 --> 01:05:06.280 align:middle line:84% often solve for an equilibrium like this is we 01:05:06.280 --> 01:05:07.920 align:middle line:84% look for mixing probabilities that 01:05:07.920 --> 01:05:09.140 align:middle line:90% make the players indifferent. 01:05:09.140 --> 01:05:12.200 align:middle line:84% So let's look at this, there's two steps. 01:05:12.200 --> 01:05:20.540 align:middle line:84% So the mixture P1 must make player 2 indifferent. 01:05:20.540 --> 01:05:31.500 align:middle line:90% 01:05:31.500 --> 01:05:32.500 align:middle line:90% And this is a key point. 01:05:32.500 --> 01:05:35.280 align:middle line:84% You might think I made a typo here, but I didn't actually. 01:05:35.280 --> 01:05:37.760 align:middle line:84% It turns out that the probability with which player 01:05:37.760 --> 01:05:42.880 align:middle line:84% 1 mixes is what determines whether player 2 is indifferent 01:05:42.880 --> 01:05:47.320 align:middle line:84% because player 1's mixing probabilities in equilibrium 01:05:47.320 --> 01:05:49.800 align:middle line:84% determine the beliefs that player 2 has. 01:05:49.800 --> 01:05:52.140 align:middle line:84% So whether player 2 wants to play 01:05:52.140 --> 01:05:56.140 align:middle line:84% C or R depends on their beliefs about player 1 01:05:56.140 --> 01:05:57.820 align:middle line:84% and the frequency with which player 01:05:57.820 --> 01:06:00.260 align:middle line:84% 1 plays C or R, so it's, actually, player 1's 01:06:00.260 --> 01:06:03.740 align:middle line:84% mixing probability that is responsible for determining 01:06:03.740 --> 01:06:05.500 align:middle line:90% whether player 2 is indifferent. 01:06:05.500 --> 01:06:07.820 align:middle line:84% And conversely, it's player 2's mixing 01:06:07.820 --> 01:06:11.540 align:middle line:84% probability that determines whether player 1 is indifferent. 01:06:11.540 --> 01:06:14.140 align:middle line:84% So when you try to solve for these equilibrium, 01:06:14.140 --> 01:06:15.695 align:middle line:90% it all looks nice. 01:06:15.695 --> 01:06:17.820 align:middle line:84% We have two indifference conditions and two numbers 01:06:17.820 --> 01:06:18.680 align:middle line:90% to solve for. 01:06:18.680 --> 01:06:21.260 align:middle line:84% Mathematically, it's pretty easy, but the thing to remember 01:06:21.260 --> 01:06:24.780 align:middle line:84% is that player 2's indifference condition is actually, 01:06:24.780 --> 01:06:26.860 align:middle line:84% in this case, going to be determined by P1, 01:06:26.860 --> 01:06:29.180 align:middle line:84% and player 1's going to be determined by P2. 01:06:29.180 --> 01:06:31.160 align:middle line:84% So player 2's indifferent condition, 01:06:31.160 --> 01:06:35.460 align:middle line:84% we need to make sure that player 2 gets the same payoff from C 01:06:35.460 --> 01:06:40.180 align:middle line:84% as from R. So let's look at player 2. 01:06:40.180 --> 01:06:46.340 align:middle line:84% If player 2 plays C, well, they get a payoff of 1 01:06:46.340 --> 01:06:50.800 align:middle line:84% if player 1 plays C and a payoff of 0 if player 1 plays R. 01:06:50.800 --> 01:06:54.440 align:middle line:84% So what is their expected payoff? 01:06:54.440 --> 01:06:57.240 align:middle line:84% Their expected payoff from playing C 01:06:57.240 --> 01:07:03.640 align:middle line:84% is going to be P1 times 1 plus 1 minus P1 times 0. 01:07:03.640 --> 01:07:05.560 align:middle line:84% But the 0 disappears, and we just 01:07:05.560 --> 01:07:08.320 align:middle line:90% get P1 in the left-hand side. 01:07:08.320 --> 01:07:13.800 align:middle line:84% So here, P1 is the utility, the expected utility 01:07:13.800 --> 01:07:16.140 align:middle line:90% for player 2 from playing C. 01:07:16.140 --> 01:07:19.200 align:middle line:84% So maybe I'll put a C here and an R here 01:07:19.200 --> 01:07:24.280 align:middle line:84% just to keep track of what we're comparing. 01:07:24.280 --> 01:07:25.960 align:middle line:90% And this makes sense. 01:07:25.960 --> 01:07:28.640 align:middle line:84% If I'm player 2, and I go to the Celtics game, 01:07:28.640 --> 01:07:30.820 align:middle line:84% I know that if my opponent goes to the Red Sox game, 01:07:30.820 --> 01:07:32.160 align:middle line:90% I get a payoff of 0. 01:07:32.160 --> 01:07:34.080 align:middle line:84% So my only utility comes from if they also 01:07:34.080 --> 01:07:37.100 align:middle line:84% go to the Celtics game, that happens with probability P1. 01:07:37.100 --> 01:07:39.160 align:middle line:84% And if it does happen, I get a payoff of 1 01:07:39.160 --> 01:07:40.827 align:middle line:84% because I like the Red Sox more-- 01:07:40.827 --> 01:07:42.660 align:middle line:84% I only get a payoff of one from the Celtics. 01:07:42.660 --> 01:07:43.743 align:middle line:90% What about the other case? 01:07:43.743 --> 01:07:45.040 align:middle line:90% If we have the R-- 01:07:45.040 --> 01:07:47.460 align:middle line:84% if I play r, what is my payoff going to be? 01:07:47.460 --> 01:07:50.540 align:middle line:90% 01:07:50.540 --> 01:07:51.233 align:middle line:90% Yeah. 01:07:51.233 --> 01:07:52.900 align:middle line:84% AUDIENCE: [INAUDIBLE] 2 times 1 minus R. 01:07:52.900 --> 01:07:54.000 align:middle line:90% IAN BALL: Exactly, right. 01:07:54.000 --> 01:07:57.580 align:middle line:90% 01:07:57.580 --> 01:08:01.220 align:middle line:84% So here, again, I get no payoff if they go the other way. 01:08:01.220 --> 01:08:04.580 align:middle line:84% 1 minus P1 is the probability that my friend also 01:08:04.580 --> 01:08:05.662 align:middle line:90% goes to the Red Sox game. 01:08:05.662 --> 01:08:08.120 align:middle line:84% But now there's a 2 here because I really like the Red Sox. 01:08:08.120 --> 01:08:09.870 align:middle line:84% Or if they go to the Red Sox game with me, 01:08:09.870 --> 01:08:12.540 align:middle line:84% I get a payoff of 2 rather than 1. 01:08:12.540 --> 01:08:14.660 align:middle line:84% And now let's do the same thing for player 1. 01:08:14.660 --> 01:08:15.820 align:middle line:90% I'm player 1. 01:08:15.820 --> 01:08:19.500 align:middle line:84% If I play C, given that my opponent is playing 01:08:19.500 --> 01:08:21.939 align:middle line:84% C with probability P2, what is my payoff 01:08:21.939 --> 01:08:23.120 align:middle line:90% going to be as player 1? 01:08:23.120 --> 01:08:26.700 align:middle line:90% 01:08:26.700 --> 01:08:28.859 align:middle line:90% Well, it's going to be 2. 01:08:28.859 --> 01:08:33.779 align:middle line:84% I get 2 with probability P2 and 0 with probability 1 minus P2, 01:08:33.779 --> 01:08:35.439 align:middle line:90% so I just get 2 P2. 01:08:35.439 --> 01:08:38.859 align:middle line:90% 01:08:38.859 --> 01:08:43.620 align:middle line:84% And then if I play R, again, 0 with probability P2, 1 01:08:43.620 --> 01:08:47.460 align:middle line:84% with probability 1 minus P2, so I get 1 minus P2. 01:08:47.460 --> 01:08:54.160 align:middle line:90% 01:08:54.160 --> 01:08:58.279 align:middle line:84% And we can just do some algebra here. 01:08:58.279 --> 01:09:05.960 align:middle line:84% And if we pull this over, we're going to get P2 equals 1/3. 01:09:05.960 --> 01:09:08.020 align:middle line:90% And then P1 equals 2/3. 01:09:08.020 --> 01:09:13.788 align:middle line:90% 01:09:13.788 --> 01:09:15.580 align:middle line:84% So I want to caution here against, I think, 01:09:15.580 --> 01:09:17.660 align:middle line:84% a common misinterpretation of this. 01:09:17.660 --> 01:09:22.040 align:middle line:90% 01:09:22.040 --> 01:09:24.880 align:middle line:84% What a lot of people say is they say, OK, this makes sense. 01:09:24.880 --> 01:09:28.300 align:middle line:84% Player 1 goes to the Celtics game more often. 01:09:28.300 --> 01:09:31.520 align:middle line:84% P1 is 2/3, more than 1/2, because player 1 likes 01:09:31.520 --> 01:09:34.000 align:middle line:90% the Celtics. 01:09:34.000 --> 01:09:38.120 align:middle line:84% And player 2 goes to the Red Sox game more. 01:09:38.120 --> 01:09:40.819 align:middle line:84% Player 2 goes to Celtics game only with probability 1/3, 01:09:40.819 --> 01:09:43.319 align:middle line:84% which means they go to the Red Sox game with probability 2/3 01:09:43.319 --> 01:09:47.340 align:middle line:84% because player 2 likes the Red Sox more. 01:09:47.340 --> 01:09:49.892 align:middle line:90% Is that right? 01:09:49.892 --> 01:09:51.600 align:middle line:84% Do you see a problem with that reasoning? 01:09:51.600 --> 01:09:52.359 align:middle line:90% I don't know, what do you think? 01:09:52.359 --> 01:09:53.526 align:middle line:90% Does that seem right to you? 01:09:53.526 --> 01:10:00.060 align:middle line:90% 01:10:00.060 --> 01:10:02.340 align:middle line:84% So it turns out that the reason, if we really 01:10:02.340 --> 01:10:04.500 align:middle line:84% look through the equilibrium, the reason 01:10:04.500 --> 01:10:08.620 align:middle line:84% that player 1 goes to the Celtics a lot 01:10:08.620 --> 01:10:11.140 align:middle line:84% is not because they like the Celtics more 01:10:11.140 --> 01:10:14.300 align:middle line:84% but because their opponent likes the Celtics less. 01:10:14.300 --> 01:10:16.860 align:middle line:84% Player 1's mixing probabilities have 01:10:16.860 --> 01:10:19.380 align:middle line:90% to make player 2 indifferent. 01:10:19.380 --> 01:10:22.580 align:middle line:84% And because player 2 doesn't like the Celtics very much, 01:10:22.580 --> 01:10:25.220 align:middle line:84% they're only willing to go to the Celtics game 01:10:25.220 --> 01:10:28.180 align:middle line:84% if they believe that player 1 is very often going to the Celtics 01:10:28.180 --> 01:10:29.340 align:middle line:90% game. 01:10:29.340 --> 01:10:31.040 align:middle line:84% And let's do the reverse reasoning. 01:10:31.040 --> 01:10:34.060 align:middle line:84% Why does player two go to the Red Sox game a lot? 01:10:34.060 --> 01:10:36.860 align:middle line:84% It's true that player 2 likes the Red Sox a lot, 01:10:36.860 --> 01:10:38.788 align:middle line:84% but the real reason for this equilibrium 01:10:38.788 --> 01:10:40.580 align:middle line:84% is that they have to go to the Red Sox game 01:10:40.580 --> 01:10:44.120 align:middle line:84% a lot to make player 1 willing to go to the Red Sox game 01:10:44.120 --> 01:10:45.842 align:middle line:84% because player 1 prefers the Celtics, 01:10:45.842 --> 01:10:47.800 align:middle line:84% so they're only going to go to the Red Sox game 01:10:47.800 --> 01:10:49.800 align:middle line:84% if they think it's more likely than not 01:10:49.800 --> 01:10:51.920 align:middle line:84% that player 2 goes to the Red Sox game. 01:10:51.920 --> 01:10:54.480 align:middle line:84% So I think that's just a key point 01:10:54.480 --> 01:11:00.415 align:middle line:84% here about what's driving the structure of this equilibrium. 01:11:00.415 --> 01:11:02.040 align:middle line:84% And I think that should make you wonder 01:11:02.040 --> 01:11:05.480 align:middle line:84% a bit because I shouldn't care about making 01:11:05.480 --> 01:11:07.000 align:middle line:90% my opponent indifferent. 01:11:07.000 --> 01:11:15.092 align:middle line:84% So I think an issue, maybe a question, 01:11:15.092 --> 01:11:22.020 align:middle line:84% is if I'm player 1 here, given the way my opponent is playing, 01:11:22.020 --> 01:11:24.600 align:middle line:84% I am indifferent between C and R. 01:11:24.600 --> 01:11:27.480 align:middle line:84% So why should I mix with exactly the right probability that 01:11:27.480 --> 01:11:29.960 align:middle line:90% makes my opponent indifferent? 01:11:29.960 --> 01:11:32.520 align:middle line:84% It doesn't really seem like, if I'm indifferent, I mean, 01:11:32.520 --> 01:11:35.140 align:middle line:84% why go through the difficulty of doing all the randomization? 01:11:35.140 --> 01:11:37.540 align:middle line:84% And I think that's a question we'll just leave for now. 01:11:37.540 --> 01:11:39.500 align:middle line:84% We'll talk a bit more throughout the course about this, 01:11:39.500 --> 01:11:41.292 align:middle line:84% but I think it's something to keep in mind, 01:11:41.292 --> 01:11:44.360 align:middle line:84% that it's true, I'm willing to mix, but why do I actually mix? 01:11:44.360 --> 01:11:49.660 align:middle line:90% 01:11:49.660 --> 01:11:51.818 align:middle line:90% We do see it in practice. 01:11:51.818 --> 01:11:52.360 align:middle line:90% I don't know. 01:11:52.360 --> 01:11:53.120 align:middle line:90% I will open it up. 01:11:53.120 --> 01:11:54.078 align:middle line:90% Are there any thoughts? 01:11:54.078 --> 01:11:57.462 align:middle line:84% I mean, why do you think poker players actually do mix? 01:11:57.462 --> 01:11:59.420 align:middle line:84% They're indifferent, they should be indifferent 01:11:59.420 --> 01:12:01.120 align:middle line:84% if everyone else is mixing just right. 01:12:01.120 --> 01:12:02.400 align:middle line:90% Why do they mix? 01:12:02.400 --> 01:12:06.660 align:middle line:90% 01:12:06.660 --> 01:12:08.780 align:middle line:84% Why would you mix if you took penalties? 01:12:08.780 --> 01:12:10.400 align:middle line:84% What would go wrong if you didn't mix? 01:12:10.400 --> 01:12:13.900 align:middle line:90% 01:12:13.900 --> 01:12:15.293 align:middle line:90% So yeah. 01:12:15.293 --> 01:12:17.460 align:middle line:84% AUDIENCE: They would know what you were going to do, 01:12:17.460 --> 01:12:19.800 align:middle line:84% and they would update their strategy based on that. 01:12:19.800 --> 01:12:21.300 align:middle line:84% IAN BALL: Right, so often, the story 01:12:21.300 --> 01:12:23.217 align:middle line:84% we tell about mixed strategies is often really 01:12:23.217 --> 01:12:25.837 align:middle line:84% a story about a dynamic process where the game is repeated. 01:12:25.837 --> 01:12:28.420 align:middle line:84% And for this reason, we might think that a mixed-strategy Nash 01:12:28.420 --> 01:12:31.580 align:middle line:84% equilibrium is a more compelling prediction if the game is 01:12:31.580 --> 01:12:32.980 align:middle line:90% repeated over time. 01:12:32.980 --> 01:12:34.860 align:middle line:84% You don't want to have a reputation 01:12:34.860 --> 01:12:37.500 align:middle line:84% as someone who always shoots left because then your opponents 01:12:37.500 --> 01:12:38.543 align:middle line:90% would update on that. 01:12:38.543 --> 01:12:40.460 align:middle line:84% Now, then we have to really think about, well, 01:12:40.460 --> 01:12:42.580 align:middle line:84% what does it mean to develop a reputation for something? 01:12:42.580 --> 01:12:44.240 align:middle line:84% And that takes us down a whole new path 01:12:44.240 --> 01:12:46.490 align:middle line:84% that people have tried to formalize and think through. 01:12:46.490 --> 01:12:49.160 align:middle line:84% It's not clear exactly what it means. 01:12:49.160 --> 01:12:52.160 align:middle line:90% So that's one story. 01:12:52.160 --> 01:12:53.800 align:middle line:84% I guess, another story I would say 01:12:53.800 --> 01:12:58.280 align:middle line:84% is sometimes mixing is not literal mixing, 01:12:58.280 --> 01:13:01.080 align:middle line:84% but what mixing means is you're making a decision based 01:13:01.080 --> 01:13:05.120 align:middle line:84% on something that the other person isn't aware of. 01:13:05.120 --> 01:13:07.120 align:middle line:90% So it may be that-- 01:13:07.120 --> 01:13:10.560 align:middle line:90% 01:13:10.560 --> 01:13:12.580 align:middle line:84% let's take, I don't know, how someone dresses. 01:13:12.580 --> 01:13:13.680 align:middle line:84% You may think, oh, if someone just 01:13:13.680 --> 01:13:15.840 align:middle line:84% randomizes the way they dress, but really, they 01:13:15.840 --> 01:13:17.600 align:middle line:84% wear certain colors on certain weather 01:13:17.600 --> 01:13:19.500 align:middle line:84% because that makes them feel happy. 01:13:19.500 --> 01:13:21.800 align:middle line:84% But to you, not knowing their careful algorithm 01:13:21.800 --> 01:13:22.760 align:middle line:84% for how they choose their clothes 01:13:22.760 --> 01:13:25.280 align:middle line:84% as a function of the weather, it just appears random to you. 01:13:25.280 --> 01:13:26.960 align:middle line:84% Or similarly, maybe a penalty kicker, 01:13:26.960 --> 01:13:29.868 align:middle line:84% when they come to the spot, feels a direction. 01:13:29.868 --> 01:13:31.660 align:middle line:84% One of their feet feels a little different, 01:13:31.660 --> 01:13:33.160 align:middle line:84% they just feel a direction to go to, 01:13:33.160 --> 01:13:36.080 align:middle line:84% and then they make a deterministic choice about which 01:13:36.080 --> 01:13:38.740 align:middle line:84% direction to shoot, but because you, as the opponent, 01:13:38.740 --> 01:13:41.160 align:middle line:84% can't see that signal that they're basing their shot on, 01:13:41.160 --> 01:13:42.380 align:middle line:90% it appears random to you. 01:13:42.380 --> 01:13:44.640 align:middle line:84% So sometimes we literally observe randomization, 01:13:44.640 --> 01:13:47.460 align:middle line:84% and sometimes behaviors appear random because they're 01:13:47.460 --> 01:13:49.900 align:middle line:84% based upon things that we can't actually observe. 01:13:49.900 --> 01:13:52.660 align:middle line:84% So just one thing to keep in mind. 01:13:52.660 --> 01:13:57.580 align:middle line:90% And I think that's good. 01:13:57.580 --> 01:14:01.300 align:middle line:90% Let me finish with one last-- 01:14:01.300 --> 01:14:02.320 align:middle line:90% two last things. 01:14:02.320 --> 01:14:05.260 align:middle line:90% 01:14:05.260 --> 01:14:10.340 align:middle line:84% So notice that, in the game that we looked at here, 01:14:10.340 --> 01:14:14.140 align:middle line:84% this game did not have a pure-strategy Nash equilibrium. 01:14:14.140 --> 01:14:16.940 align:middle line:84% But when we moved to mixed-strategy Nash equilibrium, 01:14:16.940 --> 01:14:19.158 align:middle line:84% it did have a mixed strategy Nash equilibrium. 01:14:19.158 --> 01:14:20.700 align:middle line:84% And the question is, could we come up 01:14:20.700 --> 01:14:23.300 align:middle line:84% with some really weird game where there's no equilibrium. 01:14:23.300 --> 01:14:25.383 align:middle line:84% That maybe wouldn't be so very good for our theory 01:14:25.383 --> 01:14:27.550 align:middle line:84% because then we wouldn't have a prediction about how 01:14:27.550 --> 01:14:28.060 align:middle line:90% people play. 01:14:28.060 --> 01:14:30.260 align:middle line:84% And fortunately, it turns out-- and this is really 01:14:30.260 --> 01:14:35.730 align:middle line:84% Nash's key result in 1948-- that that's never going to happen. 01:14:35.730 --> 01:14:40.970 align:middle line:84% Every game, within reason, has a mixed strategy Nash equilibrium. 01:14:40.970 --> 01:14:47.230 align:middle line:84% So more precisely, let me say every finite strategic-form 01:14:47.230 --> 01:14:47.730 align:middle line:90% game-- 01:14:47.730 --> 01:14:52.650 align:middle line:90% 01:14:52.650 --> 01:14:55.150 align:middle line:84% would be a little more precise, I'll say-- has at least one. 01:14:55.150 --> 01:15:12.250 align:middle line:90% 01:15:12.250 --> 01:15:13.210 align:middle line:90% So at least one. 01:15:13.210 --> 01:15:14.440 align:middle line:90% It might have only one. 01:15:14.440 --> 01:15:15.190 align:middle line:90% It might have two. 01:15:15.190 --> 01:15:16.023 align:middle line:90% It might have three. 01:15:16.023 --> 01:15:18.650 align:middle line:84% It may have infinitely many, but it has at least one. 01:15:18.650 --> 01:15:21.530 align:middle line:84% And when I say mixed strategy, that includes pure strategy 01:15:21.530 --> 01:15:22.550 align:middle line:90% as a special case. 01:15:22.550 --> 01:15:25.137 align:middle line:84% So I might say mixed or pure, but remember, 01:15:25.137 --> 01:15:26.970 align:middle line:84% we're going to think of a pure-strategy Nash 01:15:26.970 --> 01:15:31.290 align:middle line:84% equilibrium as a special case of a mixed-strategy Nash 01:15:31.290 --> 01:15:33.670 align:middle line:84% equilibrium where the players mix 01:15:33.670 --> 01:15:36.150 align:middle line:90% with degenerate probabilities. 01:15:36.150 --> 01:15:38.750 align:middle line:84% Now, this is not something we're going to prove in the course. 01:15:38.750 --> 01:15:41.630 align:middle line:84% This is actually quite a deep result. So remember, 01:15:41.630 --> 01:15:43.870 align:middle line:84% a couple of weeks ago, last week, we 01:15:43.870 --> 01:15:46.450 align:middle line:84% talked about this dominance/best response duality, 01:15:46.450 --> 01:15:47.730 align:middle line:90% and I also didn't prove that. 01:15:47.730 --> 01:15:49.010 align:middle line:90% That is somewhat deep. 01:15:49.010 --> 01:15:49.850 align:middle line:90% This is much deeper. 01:15:49.850 --> 01:15:52.510 align:middle line:84% So this is maybe the mathematically deepest result 01:15:52.510 --> 01:15:53.510 align:middle line:90% in the course. 01:15:53.510 --> 01:15:56.370 align:middle line:84% And it uses something called a fixed-point theorem. 01:15:56.370 --> 01:16:04.693 align:middle line:90% 01:16:04.693 --> 01:16:06.110 align:middle line:84% And if you want to google it, it's 01:16:06.110 --> 01:16:09.090 align:middle line:84% either sometimes attributed to Brouwer or Kakutani. 01:16:09.090 --> 01:16:11.870 align:middle line:90% 01:16:11.870 --> 01:16:15.250 align:middle line:84% And these are, actually, quite deep mathematical results. 01:16:15.250 --> 01:16:17.310 align:middle line:84% So one of the mathematical results 01:16:17.310 --> 01:16:19.730 align:middle line:84% was actually proven, I think, in 1944. 01:16:19.730 --> 01:16:22.170 align:middle line:84% And then Nash proved this result in 1948. 01:16:22.170 --> 01:16:24.390 align:middle line:84% So it was really using cutting-edge math 01:16:24.390 --> 01:16:26.550 align:middle line:84% to prove this new result. Maybe not so cutting edge 01:16:26.550 --> 01:16:30.110 align:middle line:84% anymore in 1944, but at the time, the gap between the math 01:16:30.110 --> 01:16:33.730 align:middle line:84% frontier and the econ frontier was actually very, very close. 01:16:33.730 --> 01:16:36.410 align:middle line:84% If you want to see the proof, you 01:16:36.410 --> 01:16:38.030 align:middle line:90% can look in the lecture notes. 01:16:38.030 --> 01:16:41.610 align:middle line:84% The lecture notes give a proof building upon these theorems. 01:16:41.610 --> 01:16:43.690 align:middle line:84% The real depth is proving these theorems, which 01:16:43.690 --> 01:16:44.610 align:middle line:90% we're not going to talk about. 01:16:44.610 --> 01:16:45.690 align:middle line:84% We're not going to talk about this proof 01:16:45.690 --> 01:16:47.910 align:middle line:84% either, but just want you to keep that in mind. 01:16:47.910 --> 01:16:49.990 align:middle line:84% And before we go, we have three more minutes. 01:16:49.990 --> 01:16:52.810 align:middle line:84% So let me just conclude with a few interpretations of Nash 01:16:52.810 --> 01:16:55.770 align:middle line:90% equilibrium. 01:16:55.770 --> 01:16:58.650 align:middle line:84% I think we talk about Nash equilibrium 01:16:58.650 --> 01:17:00.690 align:middle line:84% as the crowning achievement of game theory. 01:17:00.690 --> 01:17:03.050 align:middle line:84% But over the last seven years, people 01:17:03.050 --> 01:17:05.210 align:middle line:84% have always been searching for justifications 01:17:05.210 --> 01:17:06.230 align:middle line:90% of Nash equilibrium. 01:17:06.230 --> 01:17:08.850 align:middle line:84% And I would actually argue that Nash equilibrium does not 01:17:08.850 --> 01:17:11.430 align:middle line:84% have as compelling of a justification as, say, 01:17:11.430 --> 01:17:12.462 align:middle line:90% rationalizability does. 01:17:12.462 --> 01:17:14.170 align:middle line:84% Of course, it makes stronger assumptions, 01:17:14.170 --> 01:17:16.503 align:middle line:84% and it gets sharper predictions, but whether we actually 01:17:16.503 --> 01:17:18.750 align:middle line:84% find it compelling, I think, is not totally obvious. 01:17:18.750 --> 01:17:21.430 align:middle line:84% Again, on exams, if I ask you to solve for a Nash equilibrium, 01:17:21.430 --> 01:17:21.950 align:middle line:90% you have to solve it. 01:17:21.950 --> 01:17:24.283 align:middle line:84% You can't just say, I don't believe in Nash equilibrium, 01:17:24.283 --> 01:17:27.130 align:middle line:84% but I do think it's helpful to think about how compelling this 01:17:27.130 --> 01:17:27.630 align:middle line:90% is. 01:17:27.630 --> 01:17:31.110 align:middle line:84% So I want to present a few interpretations of this. 01:17:31.110 --> 01:17:36.870 align:middle line:84% So one interpretation is about a stable convention. 01:17:36.870 --> 01:17:39.850 align:middle line:84% So the idea of Nash equilibrium is really about stability. 01:17:39.850 --> 01:17:41.910 align:middle line:84% It's that given the way people are behaving, 01:17:41.910 --> 01:17:43.430 align:middle line:90% no one wants to deviate. 01:17:43.430 --> 01:17:46.310 align:middle line:84% So a great, I think, example of this is the side of the road 01:17:46.310 --> 01:17:47.600 align:middle line:90% that we drive on. 01:17:47.600 --> 01:17:49.350 align:middle line:84% Why do we all drive on the right in the US 01:17:49.350 --> 01:17:50.725 align:middle line:84% and drive on the left in England? 01:17:50.725 --> 01:17:53.410 align:middle line:84% You might say, oh, you get a ticket if you drove on the left, 01:17:53.410 --> 01:17:54.990 align:middle line:84% but that's not really the reason. 01:17:54.990 --> 01:17:56.850 align:middle line:84% It's really because you're afraid you'd get hit by a car. 01:17:56.850 --> 01:17:58.950 align:middle line:84% So if everyone drives on the right, it's best for you 01:17:58.950 --> 01:17:59.970 align:middle line:90% to drive on the right. 01:17:59.970 --> 01:18:01.510 align:middle line:84% If everyone drives on the left, it's best for you 01:18:01.510 --> 01:18:02.410 align:middle line:90% to drive on the left. 01:18:02.410 --> 01:18:04.910 align:middle line:84% These are both stable conventions, 01:18:04.910 --> 01:18:06.700 align:middle line:84% and we don't really need to ticket people 01:18:06.700 --> 01:18:08.450 align:middle line:84% for driving on the wrong side of the road. 01:18:08.450 --> 01:18:10.825 align:middle line:84% I mean, we do, but that's not really what motivates them, 01:18:10.825 --> 01:18:13.550 align:middle line:84% I think, to drive on the correct side of the road. 01:18:13.550 --> 01:18:16.610 align:middle line:84% Another is you can think of it as a self-enforcing agreement. 01:18:16.610 --> 01:18:21.430 align:middle line:90% 01:18:21.430 --> 01:18:23.710 align:middle line:84% Again, self-enforcing because no one 01:18:23.710 --> 01:18:25.770 align:middle line:84% wants to unilaterally deviate from the agreement. 01:18:25.770 --> 01:18:27.312 align:middle line:84% And I think this explains the success 01:18:27.312 --> 01:18:29.470 align:middle line:84% of a lot of multilateral international agreements. 01:18:29.470 --> 01:18:31.730 align:middle line:84% When countries sign treaties, there's 01:18:31.730 --> 01:18:34.590 align:middle line:84% no court that can adjudicate a treaty between countries, 01:18:34.590 --> 01:18:37.290 align:middle line:90% or generally, there's not. 01:18:37.290 --> 01:18:39.240 align:middle line:84% So why do people follow the treaty? 01:18:39.240 --> 01:18:41.490 align:middle line:84% Well, often when treaties are set up in the right way, 01:18:41.490 --> 01:18:43.790 align:middle line:84% people don't have an incentive to unilaterally deviate. 01:18:43.790 --> 01:18:45.050 align:middle line:84% And even though there's not some court that 01:18:45.050 --> 01:18:47.170 align:middle line:84% would throw a country in prison, they often 01:18:47.170 --> 01:18:49.890 align:middle line:84% are still motivated to follow these agreements. 01:18:49.890 --> 01:18:54.650 align:middle line:84% And then another interpretation is as a steady state. 01:18:54.650 --> 01:18:56.970 align:middle line:84% And this gets to a discussion of penalty kicks. 01:18:56.970 --> 01:18:58.790 align:middle line:84% If you were to always shoot right, 01:18:58.790 --> 01:19:00.370 align:middle line:84% the goalies would start always guessing that you 01:19:00.370 --> 01:19:01.270 align:middle line:90% were going to shoot right. 01:19:01.270 --> 01:19:03.870 align:middle line:84% So then you might start shooting left, and then, over time, 01:19:03.870 --> 01:19:06.370 align:middle line:84% we think that you might converge to the steady state. 01:19:06.370 --> 01:19:09.730 align:middle line:84% Whether there's actually a real justification 01:19:09.730 --> 01:19:12.110 align:middle line:84% for dynamics converging is actually a bit more subtle. 01:19:12.110 --> 01:19:15.475 align:middle line:84% But the point is, if you are in a Nash equilibrium, it's steady, 01:19:15.475 --> 01:19:17.350 align:middle line:84% and there's no reason to move away from that. 01:19:17.350 --> 01:19:19.030 align:middle line:84% And that's the final interpretation, 01:19:19.030 --> 01:19:20.070 align:middle line:90% and I'll leave it there. 01:19:20.070 --> 01:19:22.360 align:middle line:90% So I'll see everyone next week. 01:19:22.360 --> 01:19:26.000 align:middle line:90%