WEBVTT 00:00:00.000 --> 00:00:01.932 align:middle line:90% [SQUEAKING] 00:00:01.932 --> 00:00:03.864 align:middle line:90% [RUSTLING] 00:00:03.864 --> 00:00:05.313 align:middle line:90% [CLICKING] 00:00:05.313 --> 00:00:11.900 align:middle line:90% 00:00:11.900 --> 00:00:15.160 align:middle line:84% ANKUR MOITRA: So let me tell you about zero sum games. 00:00:15.160 --> 00:00:18.400 align:middle line:84% Let me start off with the key definition. 00:00:18.400 --> 00:00:19.960 align:middle line:84% And we're going to study what happens 00:00:19.960 --> 00:00:22.080 align:middle line:90% under the simple definition. 00:00:22.080 --> 00:00:24.060 align:middle line:84% So first, let me tell you the abstraction. 00:00:24.060 --> 00:00:27.440 align:middle line:84% And then we'll do some examples to try and make it concrete. 00:00:27.440 --> 00:00:29.920 align:middle line:84% So specifically, we'll be interested in what's called 00:00:29.920 --> 00:00:34.460 align:middle line:90% a two-player zero sum game. 00:00:34.460 --> 00:00:39.760 align:middle line:90% 00:00:39.760 --> 00:00:43.280 align:middle line:84% And a two-player zero sum game has a few components. 00:00:43.280 --> 00:00:47.640 align:middle line:84% So it's described by the following things. 00:00:47.640 --> 00:00:51.600 align:middle line:84% First of all, there is a row player. 00:00:51.600 --> 00:00:54.590 align:middle line:84% A lot of times, we'll call this player Alice. 00:00:54.590 --> 00:00:57.160 align:middle line:90% 00:00:57.160 --> 00:01:03.100 align:middle line:84% And Alice has m different strategies. 00:01:03.100 --> 00:01:05.540 align:middle line:84% These are the moves that she's allowed to make 00:01:05.540 --> 00:01:07.300 align:middle line:90% in the game we're playing. 00:01:07.300 --> 00:01:13.020 align:middle line:84% There's a second player, which is called the column player. 00:01:13.020 --> 00:01:17.060 align:middle line:84% And usually, we call the column player Bob. 00:01:17.060 --> 00:01:20.300 align:middle line:84% These are just the standard conventions. 00:01:20.300 --> 00:01:24.060 align:middle line:84% And Bob is going to have n strategies. 00:01:24.060 --> 00:01:26.460 align:middle line:84% So already, you can see that the game looks 00:01:26.460 --> 00:01:27.960 align:middle line:90% different for the two players. 00:01:27.960 --> 00:01:30.060 align:middle line:84% They have even different action spaces 00:01:30.060 --> 00:01:33.460 align:middle line:84% of what they're allowed to make as a move in the game. 00:01:33.460 --> 00:01:36.380 align:middle line:84% And the crucial thing is the last component, 00:01:36.380 --> 00:01:39.420 align:middle line:84% which tells us what are the outcomes of the game-- who 00:01:39.420 --> 00:01:43.340 align:middle line:84% pays who how much depending on what strategies you play. 00:01:43.340 --> 00:01:46.260 align:middle line:84% So finally, what we have is we have an m 00:01:46.260 --> 00:01:50.770 align:middle line:84% by n matrix, which is called the payoff matrix. 00:01:50.770 --> 00:01:54.340 align:middle line:90% 00:01:54.340 --> 00:01:58.680 align:middle line:84% And we'll call this C. And the way that this matrix C works 00:01:58.680 --> 00:02:06.920 align:middle line:84% is that the entry Cij is the amount that the row player, i.e. 00:02:06.920 --> 00:02:20.080 align:middle line:84% Alice, gets in payment from the column player 00:02:20.080 --> 00:02:28.000 align:middle line:84% if they play i and j respectively. 00:02:28.000 --> 00:02:32.200 align:middle line:84% So notice that the row player, Alice, has m strategies 00:02:32.200 --> 00:02:35.160 align:middle line:84% and there are m rows in this matrix. 00:02:35.160 --> 00:02:37.600 align:middle line:84% The column player, Bob, has n strategies. 00:02:37.600 --> 00:02:39.892 align:middle line:84% And there are n columns in this matrix. 00:02:39.892 --> 00:02:41.600 align:middle line:84% So you should think of this payoff matrix 00:02:41.600 --> 00:02:44.200 align:middle line:84% as being indexed by their strategy set. 00:02:44.200 --> 00:02:46.320 align:middle line:84% Every choice of a strategy for the row player 00:02:46.320 --> 00:02:49.880 align:middle line:84% just selects which row we're interested in. 00:02:49.880 --> 00:02:51.600 align:middle line:84% Every strategy for the column player 00:02:51.600 --> 00:02:54.280 align:middle line:84% just selects which column we're interested in. 00:02:54.280 --> 00:02:58.510 align:middle line:84% And then we look into that entry in the matrix. 00:02:58.510 --> 00:03:02.430 align:middle line:84% And that describes the net payment from Bob to Alice. 00:03:02.430 --> 00:03:04.370 align:middle line:84% That payment could be negative, in which case 00:03:04.370 --> 00:03:06.470 align:middle line:84% the money goes from Alice to Bob. 00:03:06.470 --> 00:03:10.310 align:middle line:84% But in any case, right now, in the case of a zero sum game, 00:03:10.310 --> 00:03:12.430 align:middle line:84% we only have a single payoff matrix 00:03:12.430 --> 00:03:15.390 align:middle line:90% that describes what happens. 00:03:15.390 --> 00:03:17.090 align:middle line:84% And it's called the zero sum game 00:03:17.090 --> 00:03:20.370 align:middle line:84% because, if you add up the wins and losses of Alice and Bob, 00:03:20.370 --> 00:03:21.710 align:middle line:90% it's 0. 00:03:21.710 --> 00:03:23.910 align:middle line:84% So of course, you can consider more general games. 00:03:23.910 --> 00:03:26.310 align:middle line:84% And we'll talk more about that towards the end. 00:03:26.310 --> 00:03:29.270 align:middle line:84% But this will be our basic abstraction. 00:03:29.270 --> 00:03:32.910 align:middle line:84% So let's see some examples to get some intuition for this. 00:03:32.910 --> 00:03:38.190 align:middle line:84% There's a very simple example that can get us started. 00:03:38.190 --> 00:03:41.720 align:middle line:84% This is called the matching pennies game. 00:03:41.720 --> 00:03:45.470 align:middle line:90% 00:03:45.470 --> 00:03:53.230 align:middle line:84% So the way that this works is that Alice and Bob 00:03:53.230 --> 00:04:05.110 align:middle line:84% simultaneously choose either H or T. So in particular, 00:04:05.110 --> 00:04:06.330 align:middle line:90% their strategy space-- 00:04:06.330 --> 00:04:08.350 align:middle line:90% m equals n equals 2. 00:04:08.350 --> 00:04:10.150 align:middle line:84% They only have two choices of actions. 00:04:10.150 --> 00:04:13.250 align:middle line:84% They can choose either heads or tails. 00:04:13.250 --> 00:04:16.649 align:middle line:84% And now, what we have to do is describe what the payoffs are. 00:04:16.649 --> 00:04:21.769 align:middle line:84% So if they match, if these two coins match, 00:04:21.769 --> 00:04:28.330 align:middle line:90% then Alice gets $1 from Bob. 00:04:28.330 --> 00:04:34.730 align:middle line:84% And otherwise, if they don't match, then Alice pays Bob $1 00:04:34.730 --> 00:04:36.250 align:middle line:90% instead. 00:04:36.250 --> 00:04:40.930 align:middle line:84% So this is a very simple example of a zero sum game. 00:04:40.930 --> 00:04:43.330 align:middle line:84% In fact, we can write it in this formalization 00:04:43.330 --> 00:04:45.570 align:middle line:84% that I described right here just by writing down 00:04:45.570 --> 00:04:47.970 align:middle line:90% what the payoff matrix is. 00:04:47.970 --> 00:04:49.530 align:middle line:84% So in this case, the payoff matrix 00:04:49.530 --> 00:04:52.130 align:middle line:90% is just the 2 by 2 matrix. 00:04:52.130 --> 00:04:57.120 align:middle line:84% And in particular, It'll look like this-- 00:04:57.120 --> 00:05:00.750 align:middle line:90% 00:05:00.750 --> 00:05:06.670 align:middle line:84% 1, minus 1, minus 1, 1, where we have some conventions 00:05:06.670 --> 00:05:10.630 align:middle line:84% of what the different rows map to in terms of the strategies. 00:05:10.630 --> 00:05:15.350 align:middle line:84% So the first row maps to the choice of Alice choosing heads. 00:05:15.350 --> 00:05:19.150 align:middle line:84% The second row maps the choice of Alice choosing tails, 00:05:19.150 --> 00:05:22.550 align:middle line:84% and then similarly for Bob, but for the columns. 00:05:22.550 --> 00:05:24.030 align:middle line:84% And so the entries in this matrix 00:05:24.030 --> 00:05:27.190 align:middle line:84% just describe the payoffs of this game. 00:05:27.190 --> 00:05:30.870 align:middle line:84% So let's make sure we have the right intuition for this. 00:05:30.870 --> 00:05:34.190 align:middle line:84% So let's think about the zero sum game. 00:05:34.190 --> 00:05:36.812 align:middle line:84% Which player has an advantage, if any? 00:05:36.812 --> 00:05:38.270 align:middle line:84% So who would you want to be, if you 00:05:38.270 --> 00:05:41.470 align:middle line:84% got to choose to either be the row or the column player? 00:05:41.470 --> 00:05:44.270 align:middle line:90% Or would it not really matter? 00:05:44.270 --> 00:05:47.770 align:middle line:90% Just asking for your intuition. 00:05:47.770 --> 00:05:48.270 align:middle line:90% Yeah? 00:05:48.270 --> 00:05:48.890 align:middle line:90% STUDENT: It doesn't matter. 00:05:48.890 --> 00:05:50.098 align:middle line:90% ANKUR MOITRA: Doesn't matter? 00:05:50.098 --> 00:05:51.150 align:middle line:90% Why? 00:05:51.150 --> 00:05:56.290 align:middle line:84% STUDENT: I mean, it's the same payoff for [INAUDIBLE]. 00:05:56.290 --> 00:05:57.770 align:middle line:84% ANKUR MOITRA: Yeah, that's right. 00:05:57.770 --> 00:06:01.570 align:middle line:84% So this is very intuitive, that neither of these players 00:06:01.570 --> 00:06:02.550 align:middle line:90% has an advantage. 00:06:02.550 --> 00:06:04.810 align:middle line:84% It doesn't matter if you're tasked with playing 00:06:04.810 --> 00:06:07.450 align:middle line:90% the role of Alice or Bob. 00:06:07.450 --> 00:06:11.170 align:middle line:84% But for more complicated games, it's maybe not so obvious 00:06:11.170 --> 00:06:12.890 align:middle line:90% who has an advantage. 00:06:12.890 --> 00:06:14.690 align:middle line:84% And this is where we're going to use ideas 00:06:14.690 --> 00:06:17.570 align:middle line:84% from linear programming to try and understand 00:06:17.570 --> 00:06:20.810 align:middle line:84% who actually has an advantage and in what sense 00:06:20.810 --> 00:06:22.570 align:middle line:90% do they have an advantage. 00:06:22.570 --> 00:06:25.690 align:middle line:84% So maybe for simple games, it's easy to reason about them 00:06:25.690 --> 00:06:27.690 align:middle line:84% just by staring at them and figuring out 00:06:27.690 --> 00:06:29.410 align:middle line:90% who you'd rather be. 00:06:29.410 --> 00:06:33.330 align:middle line:84% But for more complicated games, maybe it's not so simple. 00:06:33.330 --> 00:06:36.850 align:middle line:84% So let's do a more interesting example. 00:06:36.850 --> 00:06:38.610 align:middle line:84% And we'll come back to this example 00:06:38.610 --> 00:06:42.170 align:middle line:84% later on, once we've developed the theory. 00:06:42.170 --> 00:06:46.170 align:middle line:84% So let me just write down the payoff matrix. 00:06:46.170 --> 00:06:51.390 align:middle line:84% So let's say our payoff matrix C is a 3 by 3 matrix. 00:06:51.390 --> 00:06:54.950 align:middle line:84% So each Alice and Bob have three different strategies. 00:06:54.950 --> 00:06:57.390 align:middle line:84% But now, the game won't be symmetric in such 00:06:57.390 --> 00:06:59.390 align:middle line:90% an obvious way. 00:06:59.390 --> 00:07:01.760 align:middle line:84% So the payoffs are going to look like this. 00:07:01.760 --> 00:07:05.910 align:middle line:90% 00:07:05.910 --> 00:07:09.950 align:middle line:84% So Alice can either choose the strategy, let's say, 00:07:09.950 --> 00:07:12.830 align:middle line:90% a1, a2, or a3. 00:07:12.830 --> 00:07:17.550 align:middle line:84% Bob can choose the strategy b1, b2, and b3. 00:07:17.550 --> 00:07:20.390 align:middle line:84% And as before, the entries in this matrix 00:07:20.390 --> 00:07:24.310 align:middle line:84% describe the payoff of one player to another. 00:07:24.310 --> 00:07:26.370 align:middle line:90% So let me get some volunteers. 00:07:26.370 --> 00:07:29.830 align:middle line:90% So who wants to play this game? 00:07:29.830 --> 00:07:32.670 align:middle line:90% I need two volunteers. 00:07:32.670 --> 00:07:34.710 align:middle line:90% Yeah, come on down. 00:07:34.710 --> 00:07:35.440 align:middle line:90% Anyone else? 00:07:35.440 --> 00:07:39.070 align:middle line:90% 00:07:39.070 --> 00:07:40.913 align:middle line:84% No one else wants to be on camera? 00:07:40.913 --> 00:07:42.080 align:middle line:90% I'm going to select someone. 00:07:42.080 --> 00:07:53.740 align:middle line:90% 00:07:53.740 --> 00:07:54.640 align:middle line:90% Don't be shy. 00:07:54.640 --> 00:07:58.100 align:middle line:90% 00:07:58.100 --> 00:08:00.700 align:middle line:84% You guys brought some money with you, right? 00:08:00.700 --> 00:08:01.780 align:middle line:90% No? 00:08:01.780 --> 00:08:03.080 align:middle line:90% OK, that's all right. 00:08:03.080 --> 00:08:05.020 align:middle line:90% I'll spot you. 00:08:05.020 --> 00:08:08.860 align:middle line:84% I have these convenient baggies of monopoly money. 00:08:08.860 --> 00:08:16.440 align:middle line:84% So what I want you to think about is, in this game, 00:08:16.440 --> 00:08:19.020 align:middle line:84% who do you think has the advantage? 00:08:19.020 --> 00:08:20.550 align:middle line:90% So the row or the column player? 00:08:20.550 --> 00:08:23.620 align:middle line:90% 00:08:23.620 --> 00:08:26.220 align:middle line:84% And can we make predictions about, 00:08:26.220 --> 00:08:29.780 align:middle line:84% if they play rationally, what their strategies ought 00:08:29.780 --> 00:08:31.420 align:middle line:90% to converge to? 00:08:31.420 --> 00:08:33.740 align:middle line:90% I think this isn't so obvious. 00:08:33.740 --> 00:08:36.179 align:middle line:90% But let me start with you. 00:08:36.179 --> 00:08:37.929 align:middle line:84% So who do you want to be, Alice or bob? 00:08:37.929 --> 00:08:41.580 align:middle line:90% 00:08:41.580 --> 00:08:43.822 align:middle line:90% STUDENT: I want to be Alice. 00:08:43.822 --> 00:08:45.280 align:middle line:84% ANKUR MOITRA: You want to be Alice? 00:08:45.280 --> 00:08:45.840 align:middle line:90% All right. 00:08:45.840 --> 00:08:49.760 align:middle line:90% 00:08:49.760 --> 00:08:51.413 align:middle line:84% I'm afraid you're stuck with Bob. 00:08:51.413 --> 00:08:53.080 align:middle line:84% All right, so what I'm going to have you 00:08:53.080 --> 00:08:56.960 align:middle line:90% guys do is grab some chalk. 00:08:56.960 --> 00:09:02.420 align:middle line:84% And I'm going to ask you guys to write down 00:09:02.420 --> 00:09:04.360 align:middle line:90% which strategy you pick-- 00:09:04.360 --> 00:09:07.880 align:middle line:90% a1, a2, a3, b1, b2, b3. 00:09:07.880 --> 00:09:10.920 align:middle line:84% Hide your answers from the other person, so they don't see it. 00:09:10.920 --> 00:09:13.480 align:middle line:84% And then we'll reveal them simultaneously and figure out 00:09:13.480 --> 00:09:14.760 align:middle line:90% who's doing better. 00:09:14.760 --> 00:09:16.680 align:middle line:84% So we'll play a few rounds of the game. 00:09:16.680 --> 00:09:20.280 align:middle line:84% So pick a spot on the board and figure out, 00:09:20.280 --> 00:09:26.520 align:middle line:84% based on the payoff matrix, what exactly you want to play. 00:09:26.520 --> 00:09:27.607 align:middle line:90% And we'll see how we do. 00:09:27.607 --> 00:09:28.440 align:middle line:90% STUDENT: [INAUDIBLE] 00:09:28.440 --> 00:09:29.560 align:middle line:84% ANKUR MOITRA: Yeah, why don't you use this one, 00:09:29.560 --> 00:09:30.800 align:middle line:90% so that you guys-- 00:09:30.800 --> 00:09:31.677 align:middle line:90% yeah. 00:09:31.677 --> 00:09:32.760 align:middle line:90% STUDENT: So I just write-- 00:09:32.760 --> 00:09:34.052 align:middle line:90% ANKUR MOITRA: Yeah, yeah, yeah. 00:09:34.052 --> 00:09:37.240 align:middle line:90% 00:09:37.240 --> 00:09:41.140 align:middle line:84% You can ask the audience for help, too, if you want. 00:09:41.140 --> 00:09:41.640 align:middle line:90% STUDENT: Oh. 00:09:41.640 --> 00:09:44.680 align:middle line:90% 00:09:44.680 --> 00:09:46.640 align:middle line:90% STUDENT: [INAUDIBLE] 00:09:46.640 --> 00:09:49.780 align:middle line:84% ANKUR MOITRA: Yeah, maybe cover it up. 00:09:49.780 --> 00:09:52.220 align:middle line:90% Yeah. 00:09:52.220 --> 00:09:52.980 align:middle line:90% OK. 00:09:52.980 --> 00:09:54.700 align:middle line:90% Are you guys ready? 00:09:54.700 --> 00:09:55.480 align:middle line:90% All right. 00:09:55.480 --> 00:09:57.980 align:middle line:90% Reveal your answers. 00:09:57.980 --> 00:10:00.780 align:middle line:90% b1 and a1. 00:10:00.780 --> 00:10:01.400 align:middle line:90% All right. 00:10:01.400 --> 00:10:03.560 align:middle line:90% So, Alice, congratulations. 00:10:03.560 --> 00:10:05.180 align:middle line:90% You win $2. 00:10:05.180 --> 00:10:07.780 align:middle line:90% Can you pay the lady? 00:10:07.780 --> 00:10:08.640 align:middle line:90% STUDENT: Thank you. 00:10:08.640 --> 00:10:11.800 align:middle line:90% 00:10:11.800 --> 00:10:13.800 align:middle line:84% ANKUR MOITRA: All right, let's do another round, 00:10:13.800 --> 00:10:16.008 align:middle line:84% since maybe you guys are just getting the hang of it. 00:10:16.008 --> 00:10:18.260 align:middle line:90% All right, so think hard. 00:10:18.260 --> 00:10:20.440 align:middle line:90% Last time, she played a1. 00:10:20.440 --> 00:10:22.420 align:middle line:84% So maybe you want to change your strategy. 00:10:22.420 --> 00:10:23.000 align:middle line:90% I don't know. 00:10:23.000 --> 00:10:25.700 align:middle line:90% That's up to you. 00:10:25.700 --> 00:10:27.040 align:middle line:90% a1 worked out pretty well. 00:10:27.040 --> 00:10:28.700 align:middle line:84% I don't know if you want to change. 00:10:28.700 --> 00:10:30.810 align:middle line:84% So think hard, and then write down your answer. 00:10:30.810 --> 00:10:50.336 align:middle line:90% 00:10:50.336 --> 00:10:56.440 align:middle line:90% So we've got a1 and b2. 00:10:56.440 --> 00:10:58.120 align:middle line:90% So you owe him $1 now. 00:10:58.120 --> 00:11:01.680 align:middle line:90% 00:11:01.680 --> 00:11:04.000 align:middle line:90% He's learning. 00:11:04.000 --> 00:11:04.980 align:middle line:90% You're still up. 00:11:04.980 --> 00:11:07.800 align:middle line:84% But now, some of your winnings are going away. 00:11:07.800 --> 00:11:09.473 align:middle line:90% Let's do another round. 00:11:09.473 --> 00:11:11.140 align:middle line:84% What do you guys want to play this time? 00:11:11.140 --> 00:11:23.880 align:middle line:90% 00:11:23.880 --> 00:11:26.200 align:middle line:90% OK All right, so let's see. 00:11:26.200 --> 00:11:29.200 align:middle line:90% b3 and a2. 00:11:29.200 --> 00:11:34.950 align:middle line:84% Ooh, you lose three more dollars. 00:11:34.950 --> 00:11:37.760 align:middle line:90% Uh oh. 00:11:37.760 --> 00:11:39.160 align:middle line:90% STUDENT: How much do I have? 00:11:39.160 --> 00:11:41.800 align:middle line:90% ANKUR MOITRA: You have $7 total. 00:11:41.800 --> 00:11:45.710 align:middle line:84% You're going to go bust soon, though, if you keep it up. 00:11:45.710 --> 00:11:47.470 align:middle line:84% All right, let's do another round. 00:11:47.470 --> 00:11:51.350 align:middle line:84% Let's give her a chance to get back on top. 00:11:51.350 --> 00:11:52.790 align:middle line:90% All right. 00:11:52.790 --> 00:11:54.420 align:middle line:84% So write down your next strategy. 00:11:54.420 --> 00:12:02.910 align:middle line:90% 00:12:02.910 --> 00:12:06.430 align:middle line:90% OK, b3 and a1. 00:12:06.430 --> 00:12:07.290 align:middle line:90% All right. 00:12:07.290 --> 00:12:11.100 align:middle line:90% So oh, she just won it all back. 00:12:11.100 --> 00:12:16.270 align:middle line:90% 00:12:16.270 --> 00:12:16.770 align:middle line:90% All right. 00:12:16.770 --> 00:12:18.990 align:middle line:84% So you guys are pretty close to even. 00:12:18.990 --> 00:12:21.410 align:middle line:84% So let me just test your guys' intuition. 00:12:21.410 --> 00:12:23.910 align:middle line:84% You're not necessarily supposed to have intuition about this 00:12:23.910 --> 00:12:24.630 align:middle line:90% yet. 00:12:24.630 --> 00:12:28.950 align:middle line:84% So after seeing them play, who thinks 00:12:28.950 --> 00:12:30.420 align:middle line:90% Alice has the better deal? 00:12:30.420 --> 00:12:34.670 align:middle line:90% 00:12:34.670 --> 00:12:37.630 align:middle line:90% You have to vote for something. 00:12:37.630 --> 00:12:38.810 align:middle line:90% It's OK if you're wrong. 00:12:38.810 --> 00:12:41.270 align:middle line:84% We'll build up intuition as we're going along. 00:12:41.270 --> 00:12:44.410 align:middle line:84% Who thinks Bob has the better deal? 00:12:44.410 --> 00:12:46.870 align:middle line:90% All right, let's try that again. 00:12:46.870 --> 00:12:49.490 align:middle line:84% I got a grand total of like two votes. 00:12:49.490 --> 00:12:52.850 align:middle line:84% Who thinks Alice has the better deal? 00:12:52.850 --> 00:12:53.350 align:middle line:90% All right. 00:12:53.350 --> 00:12:55.250 align:middle line:84% Who thinks Bob has the better deal? 00:12:55.250 --> 00:12:56.850 align:middle line:90% All right, it's a pretty mixed. 00:12:56.850 --> 00:12:58.310 align:middle line:84% All right, so thanks for your help. 00:12:58.310 --> 00:13:02.670 align:middle line:84% And we can figure out the theory of how zero sum games work. 00:13:02.670 --> 00:13:04.930 align:middle line:84% And we're going to be able to answer this question. 00:13:04.930 --> 00:13:07.430 align:middle line:84% But for things like matching pennies, as I told you, 00:13:07.430 --> 00:13:08.470 align:middle line:90% it's very intuitive. 00:13:08.470 --> 00:13:10.330 align:middle line:84% You guys all got the right answer 00:13:10.330 --> 00:13:11.870 align:middle line:90% about who has an advantage. 00:13:11.870 --> 00:13:13.447 align:middle line:90% They're both the same. 00:13:13.447 --> 00:13:15.530 align:middle line:84% But as soon as you move to bigger games-- and this 00:13:15.530 --> 00:13:16.670 align:middle line:90% isn't even much bigger. 00:13:16.670 --> 00:13:19.170 align:middle line:90% This is just a 3 by 3 game. 00:13:19.170 --> 00:13:22.228 align:middle line:84% Already, it's not so obvious who has the advantage. 00:13:22.228 --> 00:13:23.770 align:middle line:84% So one of the things that we're going 00:13:23.770 --> 00:13:26.130 align:middle line:84% to do at the very end of lecture is 00:13:26.130 --> 00:13:29.050 align:middle line:84% we're going to talk about really gigantic games, 00:13:29.050 --> 00:13:33.050 align:middle line:84% games which I could never even write down on the board, 00:13:33.050 --> 00:13:34.730 align:middle line:84% because there are all kinds of games 00:13:34.730 --> 00:13:37.410 align:middle line:84% that show up in things like advertising 00:13:37.410 --> 00:13:39.370 align:middle line:90% for presidential elections. 00:13:39.370 --> 00:13:41.910 align:middle line:84% Maybe you want to figure out how to allocate 00:13:41.910 --> 00:13:44.910 align:middle line:84% your total advertising budget across a whole bunch 00:13:44.910 --> 00:13:48.350 align:middle line:84% of battleground states to figure out how to win the most 00:13:48.350 --> 00:13:49.870 align:middle line:90% electoral votes. 00:13:49.870 --> 00:13:51.590 align:middle line:84% These are exactly the kinds of things 00:13:51.590 --> 00:13:55.195 align:middle line:84% that can and are modeled as gigantic zero sum games. 00:13:55.195 --> 00:13:57.070 align:middle line:84% And we're going to develop the theory for how 00:13:57.070 --> 00:13:58.630 align:middle line:90% to think about these things. 00:13:58.630 --> 00:14:01.150 align:middle line:84% So it's good that we don't have intuition 00:14:01.150 --> 00:14:04.630 align:middle line:90% because we can make progress. 00:14:04.630 --> 00:14:05.130 align:middle line:90% All right. 00:14:05.130 --> 00:14:09.990 align:middle line:84% So let me introduce a little bit more terminology. 00:14:09.990 --> 00:14:13.670 align:middle line:84% Let me tell you what it means to solve a game. 00:14:13.670 --> 00:14:16.230 align:middle line:84% And there will actually be a few different notions 00:14:16.230 --> 00:14:17.570 align:middle line:90% of how to solve games. 00:14:17.570 --> 00:14:20.350 align:middle line:84% And they'll all behave a little bit differently. 00:14:20.350 --> 00:14:23.590 align:middle line:84% So the easiest concept to wrap your head around 00:14:23.590 --> 00:14:27.370 align:middle line:84% is called a pure Nash equilibrium. 00:14:27.370 --> 00:14:33.110 align:middle line:90% 00:14:33.110 --> 00:14:35.870 align:middle line:84% So this is a concept that makes sense 00:14:35.870 --> 00:14:39.270 align:middle line:84% regardless of whether the game is zero sum or not. 00:14:39.270 --> 00:14:44.010 align:middle line:84% For general two-player games, I could have two payoff matrices, 00:14:44.010 --> 00:14:47.850 align:middle line:84% one of which that describes what Alice wins and one of which 00:14:47.850 --> 00:14:49.730 align:middle line:90% describes what Bob wins. 00:14:49.730 --> 00:14:52.170 align:middle line:84% But it's not necessarily a transfer of money from one 00:14:52.170 --> 00:14:53.770 align:middle line:90% to the other. 00:14:53.770 --> 00:14:56.190 align:middle line:84% So a lot of the things we're doing, though, 00:14:56.190 --> 00:15:00.110 align:middle line:84% the only way to think about it as a linear program 00:15:00.110 --> 00:15:02.410 align:middle line:84% will be if we're working in the zero sum game, 00:15:02.410 --> 00:15:05.530 align:middle line:84% even though this concept makes sense more generally. 00:15:05.530 --> 00:15:07.850 align:middle line:90% So what is a Nash equilibrium? 00:15:07.850 --> 00:15:17.970 align:middle line:84% It's a pair of strategies i, j-- so this means 00:15:17.970 --> 00:15:19.750 align:middle line:90% that the row player plays i. 00:15:19.750 --> 00:15:22.050 align:middle line:90% The column player plays j. 00:15:22.050 --> 00:15:23.850 align:middle line:84% And it has the following property, which 00:15:23.850 --> 00:15:25.670 align:middle line:90% I'll just describe informally. 00:15:25.670 --> 00:15:28.850 align:middle line:84% But then we'll make it mathematically precise. 00:15:28.850 --> 00:15:35.290 align:middle line:84% It has the property that no player has any incentive 00:15:35.290 --> 00:15:35.840 align:middle line:90% to deviate. 00:15:35.840 --> 00:15:43.630 align:middle line:90% 00:15:43.630 --> 00:15:47.350 align:middle line:84% How many people have seen the movie A Beautiful Mind? 00:15:47.350 --> 00:15:48.850 align:middle line:90% It butchered this concept. 00:15:48.850 --> 00:15:53.070 align:middle line:84% So please do not get that definition from that movie. 00:15:53.070 --> 00:15:55.790 align:middle line:84% So we'll do the correct definition here. 00:15:55.790 --> 00:15:58.990 align:middle line:84% So what do I mean by incentive to deviate? 00:15:58.990 --> 00:16:02.670 align:middle line:84% Let me turn this into a set of linear inequalities. 00:16:02.670 --> 00:16:05.350 align:middle line:84% That way, we can think about it in a more mathematically 00:16:05.350 --> 00:16:06.950 align:middle line:90% precise way. 00:16:06.950 --> 00:16:11.510 align:middle line:84% What I claim is that the row player, if he's playing strategy 00:16:11.510 --> 00:16:15.070 align:middle line:84% i, and the column player is playing strategy j, 00:16:15.070 --> 00:16:26.830 align:middle line:84% then the row player has no incentive to deviate if and only 00:16:26.830 --> 00:16:35.670 align:middle line:84% if, for all i prime, Ci prime j is at most Cij. 00:16:35.670 --> 00:16:38.740 align:middle line:84% So the way to think about this inequality is 00:16:38.740 --> 00:16:42.620 align:middle line:84% that the column player has already committed to j. 00:16:42.620 --> 00:16:46.660 align:middle line:84% That's what we mean by i, j being a pure Nash equilibrium, 00:16:46.660 --> 00:16:49.780 align:middle line:84% is that row player's decided I'm going to play i. 00:16:49.780 --> 00:16:51.100 align:middle line:90% He's even announced it. 00:16:51.100 --> 00:16:52.620 align:middle line:90% I'm playing i. 00:16:52.620 --> 00:16:55.740 align:middle line:84% The column player has announced I'm playing j2. 00:16:55.740 --> 00:16:59.420 align:middle line:84% And even with that knowledge of what the other player is doing, 00:16:59.420 --> 00:17:01.580 align:middle line:84% they still can't do strictly better 00:17:01.580 --> 00:17:04.339 align:middle line:84% than what their current strategy is. 00:17:04.339 --> 00:17:06.060 align:middle line:84% So if the column player has announced 00:17:06.060 --> 00:17:08.540 align:middle line:84% that they're playing j, the only thing that's relevant 00:17:08.540 --> 00:17:11.660 align:middle line:84% is the j-th column of our payoff matrix. 00:17:11.660 --> 00:17:14.099 align:middle line:84% This would be the payoff if the row player 00:17:14.099 --> 00:17:15.800 align:middle line:90% sticks to his current strategy. 00:17:15.800 --> 00:17:19.300 align:middle line:84% He still plays the strategy he announced, strategy i. 00:17:19.300 --> 00:17:23.000 align:middle line:84% And hypothetically, if instead he changed to another strategy, 00:17:23.000 --> 00:17:25.940 align:middle line:84% i prime, the claim is that he doesn't 00:17:25.940 --> 00:17:32.100 align:middle line:84% get any more money, any positive win for making that deviation. 00:17:32.100 --> 00:17:35.520 align:middle line:84% So he might tie, in terms of changing to another strategy. 00:17:35.520 --> 00:17:38.320 align:middle line:84% But he doesn't do strictly better. 00:17:38.320 --> 00:17:47.960 align:middle line:84% And similarly, the column player has no incentive 00:17:47.960 --> 00:17:52.080 align:middle line:84% to deviate if and only if for all j 00:17:52.080 --> 00:17:58.480 align:middle line:90% prime Cij prime is at least Cij. 00:17:58.480 --> 00:18:02.080 align:middle line:84% Now, just to make sure people are paying attention, 00:18:02.080 --> 00:18:04.040 align:middle line:84% so here, obviously, the deviation 00:18:04.040 --> 00:18:08.160 align:middle line:84% is happening over the choice of i versus i prime. 00:18:08.160 --> 00:18:11.080 align:middle line:84% Here, the deviation is happening over the choice 00:18:11.080 --> 00:18:13.320 align:middle line:90% of j prime versus j. 00:18:13.320 --> 00:18:16.720 align:middle line:84% Why are my inequalities flipped here? 00:18:16.720 --> 00:18:20.440 align:middle line:84% Why is my notion of no incentive to deviate 00:18:20.440 --> 00:18:23.600 align:middle line:84% a less than or equal to constraint here when I change i 00:18:23.600 --> 00:18:24.920 align:middle line:90% to i prime? 00:18:24.920 --> 00:18:27.480 align:middle line:84% And why is it greater than or equal to constraint 00:18:27.480 --> 00:18:29.170 align:middle line:90% when I change j to j prime? 00:18:29.170 --> 00:18:31.680 align:middle line:90% 00:18:31.680 --> 00:18:32.360 align:middle line:90% Yeah? 00:18:32.360 --> 00:18:35.700 align:middle line:84% STUDENT: The payoff is like [INAUDIBLE] player, 00:18:35.700 --> 00:18:37.263 align:middle line:90% so it's the [INAUDIBLE] 00:18:37.263 --> 00:18:38.180 align:middle line:90% ANKUR MOITRA: Exactly. 00:18:38.180 --> 00:18:39.108 align:middle line:90% That's right. 00:18:39.108 --> 00:18:40.900 align:middle line:84% So really, the payoff for the column player 00:18:40.900 --> 00:18:42.160 align:middle line:90% is just negative of this. 00:18:42.160 --> 00:18:45.620 align:middle line:84% So when I multiply through by minus 1, everything switches. 00:18:45.620 --> 00:18:47.560 align:middle line:84% So this is a very natural notion. 00:18:47.560 --> 00:18:50.300 align:middle line:84% This is called a pure Nash equilibrium. 00:18:50.300 --> 00:18:52.980 align:middle line:84% But already, we can see some deficiencies 00:18:52.980 --> 00:18:55.820 align:middle line:90% of pure Nash equilibrium. 00:18:55.820 --> 00:19:00.420 align:middle line:84% So even if we come back to this matching pennies game, 00:19:00.420 --> 00:19:02.920 align:middle line:84% so is there a pure Nash equilibrium? 00:19:02.920 --> 00:19:08.012 align:middle line:90% 00:19:08.012 --> 00:19:08.970 align:middle line:90% What do you guys think? 00:19:08.970 --> 00:19:22.820 align:middle line:90% 00:19:22.820 --> 00:19:28.860 align:middle line:84% I see someone shaking their head in the correct direction No, 00:19:28.860 --> 00:19:29.360 align:middle line:90% there isn't. 00:19:29.360 --> 00:19:30.500 align:middle line:90% Can you tell us why? 00:19:30.500 --> 00:19:32.300 align:middle line:90% Any intuition? 00:19:32.300 --> 00:19:37.360 align:middle line:84% STUDENT: I guess if if I'm the rogue player, 00:19:37.360 --> 00:19:39.360 align:middle line:84% I always want to choose one that will give me 1. 00:19:39.360 --> 00:19:42.892 align:middle line:84% And then if the tails person switches their strategy, 00:19:42.892 --> 00:19:44.100 align:middle line:90% I'll switch to the other one. 00:19:44.100 --> 00:19:45.680 align:middle line:84% ANKUR MOITRA: That's right, that's right. 00:19:45.680 --> 00:19:46.660 align:middle line:90% That's exactly right. 00:19:46.660 --> 00:19:48.560 align:middle line:84% Let me just rephrase it slightly. 00:19:48.560 --> 00:19:51.840 align:middle line:84% A pure Nash equilibrium corresponds 00:19:51.840 --> 00:19:54.280 align:middle line:84% to announcing what your strategies are 00:19:54.280 --> 00:19:57.160 align:middle line:84% publicly and still there being no reason to switch. 00:19:57.160 --> 00:19:59.380 align:middle line:84% So whatever those strategies are, 00:19:59.380 --> 00:20:02.840 align:middle line:84% imagine that Alice announces she's going to play heads, 00:20:02.840 --> 00:20:05.440 align:middle line:84% Bob announces he's going to play heads. 00:20:05.440 --> 00:20:07.580 align:middle line:84% So then who has the incentive to deviate? 00:20:07.580 --> 00:20:08.200 align:middle line:90% Bob. 00:20:08.200 --> 00:20:09.860 align:middle line:84% He doesn't want the pennies to match. 00:20:09.860 --> 00:20:12.600 align:middle line:84% So if he deviates to tails, then he's going to win $1. 00:20:12.600 --> 00:20:14.440 align:middle line:90% And he'll do strictly better. 00:20:14.440 --> 00:20:17.040 align:middle line:84% And that's true for any particular starting 00:20:17.040 --> 00:20:17.860 align:middle line:90% configuration. 00:20:17.860 --> 00:20:21.760 align:middle line:84% There's always one player who has an incentive to deviate. 00:20:21.760 --> 00:20:25.280 align:middle line:84% So unfortunately, there's no pure Nash equilibrium. 00:20:25.280 --> 00:20:28.020 align:middle line:84% Now, even though there wasn't a pure Nash equilibrium, 00:20:28.020 --> 00:20:29.840 align:middle line:84% we can already intuitively see the way 00:20:29.840 --> 00:20:31.540 align:middle line:90% the game would be played. 00:20:31.540 --> 00:20:35.340 align:middle line:84% Maybe Alice and Bob just choose randomly. 00:20:35.340 --> 00:20:38.028 align:middle line:84% And that would be pretty good because in expectation they 00:20:38.028 --> 00:20:38.820 align:middle line:90% would win no money. 00:20:38.820 --> 00:20:41.420 align:middle line:84% And that's probably the best that they can do. 00:20:41.420 --> 00:20:43.940 align:middle line:84% But for this type of game, you might have the intuition 00:20:43.940 --> 00:20:46.100 align:middle line:84% that I cooked this up in such a way 00:20:46.100 --> 00:20:48.860 align:middle line:84% that, again, there's no pure Nash equilibrium. 00:20:48.860 --> 00:20:52.340 align:middle line:84% So when there is a pure Nash, it becomes very obvious how 00:20:52.340 --> 00:20:54.900 align:middle line:90% to play, for the most part. 00:20:54.900 --> 00:20:58.820 align:middle line:84% But when there's no pure Nash equilibrium, all of a sudden, 00:20:58.820 --> 00:21:02.100 align:middle line:84% you have to ask how you should play. 00:21:02.100 --> 00:21:05.380 align:middle line:84% And the key point is that maybe the notion of playing 00:21:05.380 --> 00:21:08.805 align:middle line:84% shouldn't necessarily be a deterministic choice 00:21:08.805 --> 00:21:10.180 align:middle line:84% because the problem is that, when 00:21:10.180 --> 00:21:13.740 align:middle line:84% you make deterministic choices, you can't commit in a way 00:21:13.740 --> 00:21:16.940 align:middle line:84% where you never have an incentive to deviate. 00:21:16.940 --> 00:21:20.000 align:middle line:84% So this takes us to the next notion, 00:21:20.000 --> 00:21:22.500 align:middle line:84% which will be critical for us, which 00:21:22.500 --> 00:21:27.340 align:middle line:84% is a finer grained notion of an equilibrium, which is called 00:21:27.340 --> 00:21:30.060 align:middle line:90% a mixed Nash equilibrium. 00:21:30.060 --> 00:21:33.450 align:middle line:84% Basically, it's one which allows randomization. 00:21:33.450 --> 00:21:34.670 align:middle line:90% So let me define it. 00:21:34.670 --> 00:21:38.450 align:middle line:84% This will be the key thing that we talk about today. 00:21:38.450 --> 00:21:53.130 align:middle line:84% So a mixed Nash equilibrium is a pair. 00:21:53.130 --> 00:21:56.370 align:middle line:84% But now, this pair is going to mean something different. 00:21:56.370 --> 00:22:00.250 align:middle line:90% So it's a pair x, y with-- 00:22:00.250 --> 00:22:02.250 align:middle line:84% and I'll tell you what this means in a second-- 00:22:02.250 --> 00:22:04.970 align:middle line:90% x belonging to delta m-- 00:22:04.970 --> 00:22:06.770 align:middle line:90% I'll tell you what that means-- 00:22:06.770 --> 00:22:12.210 align:middle line:84% y belonging to delta n again with the property 00:22:12.210 --> 00:22:18.130 align:middle line:84% that no player has an incentive to deviate, 00:22:18.130 --> 00:22:20.280 align:middle line:84% has a positive incentive to deviate. 00:22:20.280 --> 00:22:26.730 align:middle line:90% 00:22:26.730 --> 00:22:30.790 align:middle line:84% So I have to tell you what this notation delta sub m means. 00:22:30.790 --> 00:22:40.030 align:middle line:84% And what this means is just a distribution on the set m. 00:22:40.030 --> 00:22:42.390 align:middle line:84% So the way to think about it equivalently 00:22:42.390 --> 00:22:50.990 align:middle line:84% is that x is a vector that has non-negative entries. 00:22:50.990 --> 00:22:55.630 align:middle line:84% And the sum of all of these xi's from i equals 1 to m 00:22:55.630 --> 00:22:57.190 align:middle line:90% is equal to 1. 00:22:57.190 --> 00:22:59.230 align:middle line:90% That's what the constraint is. 00:22:59.230 --> 00:23:01.870 align:middle line:84% So what's going on is that, instead of 00:23:01.870 --> 00:23:06.830 align:middle line:84% the row player choosing one fixed strategy, like strategy i, 00:23:06.830 --> 00:23:10.190 align:middle line:84% he's going to randomize over different strategies. 00:23:10.190 --> 00:23:14.030 align:middle line:84% And the probability that he puts on each different strategy 00:23:14.030 --> 00:23:16.470 align:middle line:84% is given by the associated coordinate 00:23:16.470 --> 00:23:19.270 align:middle line:90% of this corresponding vector x. 00:23:19.270 --> 00:23:22.190 align:middle line:84% So when I write x belonging to delta m, 00:23:22.190 --> 00:23:25.415 align:middle line:84% keep in mind that this is really a set of linear inequalities, 00:23:25.415 --> 00:23:26.790 align:middle line:84% which is great because we've been 00:23:26.790 --> 00:23:28.890 align:middle line:90% talking about linear programs. 00:23:28.890 --> 00:23:32.410 align:middle line:84% It's just the constraint that x is an m dimensional vector, 00:23:32.410 --> 00:23:36.450 align:middle line:84% all of its coordinates are non-negative, and they sum to 1. 00:23:36.450 --> 00:23:39.570 align:middle line:84% Same thing is happening here, except with the fact that it's 00:23:39.570 --> 00:23:44.250 align:middle line:84% a distribution on n, not m, because the column player is 00:23:44.250 --> 00:23:47.430 align:middle line:84% randomizing over the choice of which of the n columns, 00:23:47.430 --> 00:23:49.970 align:middle line:84% because that's the number of strategies. 00:23:49.970 --> 00:23:53.890 align:middle line:84% So this is the same basic idea as a pure Nash equilibrium. 00:23:53.890 --> 00:23:57.210 align:middle line:84% We still have this notion that no player has an incentive 00:23:57.210 --> 00:23:59.050 align:middle line:90% to deviate. 00:23:59.050 --> 00:24:03.010 align:middle line:84% But now, we're augmenting the set of strategies. 00:24:03.010 --> 00:24:08.010 align:middle line:84% So instead of me announcing I'm playing strategy i 00:24:08.010 --> 00:24:10.890 align:middle line:84% and you announcing you're playing strategy j, 00:24:10.890 --> 00:24:12.970 align:middle line:84% I'll instead announce that I'm going 00:24:12.970 --> 00:24:16.090 align:middle line:84% to sample my strategies from the distribution x. 00:24:16.090 --> 00:24:18.610 align:middle line:84% And you will announce that you're 00:24:18.610 --> 00:24:21.890 align:middle line:84% going to sample your strategies from the distribution y. 00:24:21.890 --> 00:24:24.950 align:middle line:84% And we can ask, even after you make that announcement, 00:24:24.950 --> 00:24:28.790 align:middle line:84% if you have any incentive to change up your strategies. 00:24:28.790 --> 00:24:32.270 align:middle line:84% So let's be a bit more precise about what exactly 00:24:32.270 --> 00:24:35.670 align:middle line:84% I mean by no incentive to deviate, 00:24:35.670 --> 00:24:37.830 align:middle line:84% because this will come up later when 00:24:37.830 --> 00:24:40.910 align:middle line:84% we do the proof of a very powerful theorem I'm 00:24:40.910 --> 00:24:43.110 align:middle line:90% going to cover very soon. 00:24:43.110 --> 00:24:45.750 align:middle line:84% So what does it mean for the row player 00:24:45.750 --> 00:24:48.190 align:middle line:84% to have no incentive to deviate now, 00:24:48.190 --> 00:24:52.670 align:middle line:84% when we're talking about randomized strategies? 00:24:52.670 --> 00:24:55.470 align:middle line:84% Well, let's first do the following. 00:24:55.470 --> 00:25:09.350 align:middle line:84% So first of all, the expected payoff to the row player 00:25:09.350 --> 00:25:14.190 align:middle line:84% is the sum over j equals 1 to m-- 00:25:14.190 --> 00:25:21.430 align:middle line:84% the sum-- oh, sorry, i, i equals 1 to m, j equals 1 to n 00:25:21.430 --> 00:25:27.170 align:middle line:90% of Cij xi yj. 00:25:27.170 --> 00:25:29.850 align:middle line:84% And in matrix vector notation, this 00:25:29.850 --> 00:25:36.170 align:middle line:84% is just the quadratic form of x and y on this payoff matrix C. 00:25:36.170 --> 00:25:38.690 align:middle line:84% So all I'm doing is I'm saying, all right, you've 00:25:38.690 --> 00:25:41.410 align:middle line:84% announced that you're playing strategy x 00:25:41.410 --> 00:25:43.610 align:middle line:90% and you're playing strategy y. 00:25:43.610 --> 00:25:45.230 align:middle line:84% The first thing I could care about, 00:25:45.230 --> 00:25:48.410 align:middle line:84% once you've announced that those are your strategies, 00:25:48.410 --> 00:25:52.570 align:middle line:84% is how much does Alice win in expectation. 00:25:52.570 --> 00:25:56.570 align:middle line:84% And this is a very simple probabilistic computation 00:25:56.570 --> 00:26:00.410 align:middle line:84% because all this is is it's the probability that Alice plays 00:26:00.410 --> 00:26:04.010 align:middle line:84% strategy i, because that's given by the i-th coordinate of x, 00:26:04.010 --> 00:26:06.910 align:middle line:84% times the probability that Bob plays strategy j. 00:26:06.910 --> 00:26:09.290 align:middle line:84% They sample their strategies from their distribution 00:26:09.290 --> 00:26:10.810 align:middle line:90% independently. 00:26:10.810 --> 00:26:13.410 align:middle line:84% And then this is whatever the payoff would be. 00:26:13.410 --> 00:26:16.230 align:middle line:84% So this is just computing the expected value, 00:26:16.230 --> 00:26:19.090 align:middle line:84% the expected winnings that Alice has. 00:26:19.090 --> 00:26:22.890 align:middle line:84% And in our case, it works out to a very nice expression in terms 00:26:22.890 --> 00:26:25.880 align:middle line:90% of vectors and matrices. 00:26:25.880 --> 00:26:29.480 align:middle line:84% But now, we can talk about what it means for the row player 00:26:29.480 --> 00:26:32.360 align:middle line:90% to have no incentive to deviate. 00:26:32.360 --> 00:26:33.960 align:middle line:84% So in this case, when the row player 00:26:33.960 --> 00:26:37.120 align:middle line:84% is playing a randomized strategy, what I claim 00:26:37.120 --> 00:26:48.200 align:middle line:84% is that the row player has no incentive to deviate if and only 00:26:48.200 --> 00:26:56.280 align:middle line:84% if for all i prime the sum from j equals 1 to n of C i 00:26:56.280 --> 00:27:03.720 align:middle line:84% prime j yj is at most x transpose Cy. 00:27:03.720 --> 00:27:05.453 align:middle line:84% So what's going on with this expression? 00:27:05.453 --> 00:27:07.120 align:middle line:84% There's a reason I'm taking a little bit 00:27:07.120 --> 00:27:09.086 align:middle line:90% to build up towards it. 00:27:09.086 --> 00:27:12.320 align:middle line:84% You should see that already it Bears some similarity 00:27:12.320 --> 00:27:15.160 align:middle line:84% with what's going on here, where the inequality is 00:27:15.160 --> 00:27:17.280 align:middle line:84% less than or equal to, but I'm allowing 00:27:17.280 --> 00:27:19.320 align:middle line:84% some variation because I'm changing 00:27:19.320 --> 00:27:23.660 align:middle line:84% my strategy from my strategy x that 00:27:23.660 --> 00:27:26.300 align:middle line:84% has this payoff on the right-hand side 00:27:26.300 --> 00:27:31.060 align:middle line:84% to instead being the strategy that just selects i prime. 00:27:31.060 --> 00:27:33.340 align:middle line:84% So this is really calculating what 00:27:33.340 --> 00:27:38.500 align:middle line:84% would happen if I play according to x against y versus 00:27:38.500 --> 00:27:41.160 align:middle line:84% I decide I'm going to just play strategy. 00:27:41.160 --> 00:27:45.180 align:middle line:90% i prime against the strategy yj. 00:27:45.180 --> 00:27:48.020 align:middle line:84% So what's going on right here in this expression 00:27:48.020 --> 00:27:50.640 align:middle line:84% is it's the same expected value computation. 00:27:50.640 --> 00:27:53.140 align:middle line:84% Bob is still playing according to y. 00:27:53.140 --> 00:27:56.980 align:middle line:84% And this is the probability that Bob selects strategy j. 00:27:56.980 --> 00:28:00.200 align:middle line:84% But Alice is now deviating to the strategy i prime. 00:28:00.200 --> 00:28:02.900 align:middle line:84% So she always selects strategy i prime. 00:28:02.900 --> 00:28:05.380 align:middle line:90% And this is her expected payoff. 00:28:05.380 --> 00:28:07.700 align:middle line:84% And I want that this expected payoff 00:28:07.700 --> 00:28:11.220 align:middle line:84% is less than or equal to what I would get if I just stuck 00:28:11.220 --> 00:28:13.820 align:middle line:90% to my pre-announced strategy. 00:28:13.820 --> 00:28:16.580 align:middle line:84% So even though you can interpret this 00:28:16.580 --> 00:28:18.917 align:middle line:84% as a system of linear inequalities, 00:28:18.917 --> 00:28:20.500 align:middle line:84% it's important to be able to translate 00:28:20.500 --> 00:28:23.880 align:middle line:84% this back into words to understand what it's modeling. 00:28:23.880 --> 00:28:25.090 align:middle line:90% So does this make sense? 00:28:25.090 --> 00:28:27.640 align:middle line:90% 00:28:27.640 --> 00:28:29.280 align:middle line:90% Yeah? 00:28:29.280 --> 00:28:31.120 align:middle line:90% All right. 00:28:31.120 --> 00:28:33.100 align:middle line:84% So let's talk about the column player. 00:28:33.100 --> 00:28:40.320 align:middle line:84% And then I'm going to ask you a question, a somewhat 00:28:40.320 --> 00:28:42.680 align:middle line:90% tricky question. 00:28:42.680 --> 00:28:47.320 align:middle line:90% That'll be good food for later. 00:28:47.320 --> 00:28:56.000 align:middle line:84% So similarly, the column player has no incentive 00:28:56.000 --> 00:29:07.640 align:middle line:84% to deviate if and only if for all i prime the sum from j 00:29:07.640 --> 00:29:11.200 align:middle line:90% equals 1 to n C-- 00:29:11.200 --> 00:29:17.680 align:middle line:84% oh, sorry, sum for all j prime, the sum from I equals 1 to m 00:29:17.680 --> 00:29:25.580 align:middle line:84% of Cij prime xi is greater than or equal to the payoff. 00:29:25.580 --> 00:29:28.860 align:middle line:90% So this works the same way. 00:29:28.860 --> 00:29:32.060 align:middle line:84% Bob is just deviating to a single strategy j prime. 00:29:32.060 --> 00:29:34.260 align:middle line:84% The inequality gets flipped because the payments are 00:29:34.260 --> 00:29:36.700 align:middle line:90% going to Alice, not to Bob. 00:29:36.700 --> 00:29:38.035 align:middle line:90% But this is what it means. 00:29:38.035 --> 00:29:38.660 align:middle line:90% At least, This. 00:29:38.660 --> 00:29:43.100 align:middle line:84% Is how to turn this definition of a mixed Nash equilibrium 00:29:43.100 --> 00:29:46.900 align:middle line:84% into a system of linear inequalities. 00:29:46.900 --> 00:29:49.290 align:middle line:84% Now, let me ask a diagnostic question. 00:29:49.290 --> 00:29:51.890 align:middle line:90% 00:29:51.890 --> 00:30:00.420 align:middle line:84% So the way that I've defined it, these inequalities, 00:30:00.420 --> 00:30:06.860 align:middle line:84% they capture deviating from a randomized strategy, 00:30:06.860 --> 00:30:09.770 align:middle line:90% but to a pure strategy. 00:30:09.770 --> 00:30:15.100 align:middle line:90% 00:30:15.100 --> 00:30:21.022 align:middle line:84% Why not allow more general deviations? 00:30:21.022 --> 00:30:30.515 align:middle line:90% 00:30:30.515 --> 00:30:31.890 align:middle line:84% So the way to think about this is 00:30:31.890 --> 00:30:36.850 align:middle line:84% when I defined the notion of incentive to deviate right here, 00:30:36.850 --> 00:30:38.658 align:middle line:84% you started off with a strategy i. 00:30:38.658 --> 00:30:40.450 align:middle line:84% And you considered what would happen if you 00:30:40.450 --> 00:30:42.850 align:middle line:90% deviate to a strategy i prime. 00:30:42.850 --> 00:30:46.370 align:middle line:84% So you started off with a deterministic strategy i. 00:30:46.370 --> 00:30:52.730 align:middle line:84% And you deviated to another deterministic strategy i prime. 00:30:52.730 --> 00:30:55.530 align:middle line:84% Here, I define the notion of a mixed Nash 00:30:55.530 --> 00:30:57.690 align:middle line:84% equilibrium in terms of the players having 00:30:57.690 --> 00:31:00.210 align:middle line:90% randomized strategies. 00:31:00.210 --> 00:31:03.070 align:middle line:84% Alice is announcing that she plays according to x. 00:31:03.070 --> 00:31:04.030 align:middle line:90% Bob is announcing. 00:31:04.030 --> 00:31:06.090 align:middle line:90% He plays according to y. 00:31:06.090 --> 00:31:09.050 align:middle line:84% But my notion of deviation was just 00:31:09.050 --> 00:31:11.610 align:middle line:84% Alice deviating from her randomized strategy 00:31:11.610 --> 00:31:16.790 align:middle line:84% x to a deterministic strategy i prime. 00:31:16.790 --> 00:31:19.430 align:middle line:84% Bob's notion of deviation was deviating 00:31:19.430 --> 00:31:23.270 align:middle line:84% from a randomized strategy y to a deterministic strategy 00:31:23.270 --> 00:31:24.630 align:middle line:90% j prime. 00:31:24.630 --> 00:31:27.190 align:middle line:84% So why didn't I consider deviations 00:31:27.190 --> 00:31:29.050 align:middle line:90% to randomized strategies? 00:31:29.050 --> 00:31:32.510 align:middle line:90% 00:31:32.510 --> 00:31:34.630 align:middle line:90% This is an important concept. 00:31:34.630 --> 00:31:36.830 align:middle line:84% And we'll see it a few times in lecture. 00:31:36.830 --> 00:31:38.290 align:middle line:90% Anyone have any intuition? 00:31:38.290 --> 00:31:52.830 align:middle line:90% 00:31:52.830 --> 00:31:54.020 align:middle line:90% I claim it's the same. 00:31:54.020 --> 00:31:56.670 align:middle line:90% 00:31:56.670 --> 00:32:01.510 align:middle line:84% So imagine that Alice had a deviation 00:32:01.510 --> 00:32:03.510 align:middle line:84% of some other randomized strategy x 00:32:03.510 --> 00:32:06.590 align:middle line:90% prime that did better. 00:32:06.590 --> 00:32:08.750 align:middle line:84% What I claim is the only way that she 00:32:08.750 --> 00:32:12.030 align:middle line:84% has a deviation to an x prime that does better 00:32:12.030 --> 00:32:14.870 align:middle line:84% is if there's some fixed deterministic strategy that 00:32:14.870 --> 00:32:18.570 align:middle line:84% does better too, because how well she does when 00:32:18.570 --> 00:32:23.730 align:middle line:84% I take x and replace it with x prime is just an average of how 00:32:23.730 --> 00:32:27.490 align:middle line:84% well she does for each of the different row strategies 00:32:27.490 --> 00:32:30.090 align:middle line:90% she can take. 00:32:30.090 --> 00:32:31.856 align:middle line:84% So this is a very important concept. 00:32:31.856 --> 00:32:35.810 align:middle line:90% 00:32:35.810 --> 00:32:39.930 align:middle line:84% And what I'm claiming here, purely game theoretically, 00:32:39.930 --> 00:32:44.850 align:middle line:84% is that if Bob has pre-announced that his strategy is y 00:32:44.850 --> 00:32:48.610 align:middle line:84% and he's committed to that, then I don't have an incentive 00:32:48.610 --> 00:32:50.010 align:middle line:90% to deviate to-- 00:32:50.010 --> 00:32:52.570 align:middle line:84% I have an incentive to deviate to some randomized 00:32:52.570 --> 00:32:56.690 align:middle line:84% x prime if and only if there's a fixed deterministic strategy i 00:32:56.690 --> 00:32:59.970 align:middle line:84% prime that I have an incentive to deviate from. 00:32:59.970 --> 00:33:01.790 align:middle line:90% So it's not totally obvious. 00:33:01.790 --> 00:33:03.450 align:middle line:84% But hopefully, it'll become clearer 00:33:03.450 --> 00:33:05.010 align:middle line:90% over the course of the lecture. 00:33:05.010 --> 00:33:09.370 align:middle line:84% That's really a statement that has to do with weak duality. 00:33:09.370 --> 00:33:11.410 align:middle line:84% So when we talked about linear programming, 00:33:11.410 --> 00:33:13.950 align:middle line:84% we talked about weak and strong duality. 00:33:13.950 --> 00:33:17.450 align:middle line:84% This is really, as we'll see, the crux of what's going on. 00:33:17.450 --> 00:33:19.990 align:middle line:84% And there's a view to think about what 00:33:19.990 --> 00:33:24.270 align:middle line:84% weak duality is that happens purely through zero sum games. 00:33:24.270 --> 00:33:28.190 align:middle line:84% So in fact, the starting point of understanding linear programs 00:33:28.190 --> 00:33:30.710 align:middle line:90% was actually zero sum games. 00:33:30.710 --> 00:33:33.410 align:middle line:84% So it happened with von Neumann's famous work. 00:33:33.410 --> 00:33:36.750 align:middle line:84% And it was, of course, famously augmented 00:33:36.750 --> 00:33:40.390 align:middle line:84% to not necessarily zero sum games by Nash. 00:33:40.390 --> 00:33:42.110 align:middle line:84% But this is really how people first 00:33:42.110 --> 00:33:45.670 align:middle line:84% thought about the concepts of weak and strong duality. 00:33:45.670 --> 00:33:50.150 align:middle line:84% So now, I get to tell you about probably my favorite theorem 00:33:50.150 --> 00:33:50.780 align:middle line:90% in class. 00:33:50.780 --> 00:33:57.030 align:middle line:90% 00:33:57.030 --> 00:33:58.060 align:middle line:90% Let me write it up here. 00:33:58.060 --> 00:34:00.990 align:middle line:90% 00:34:00.990 --> 00:34:05.870 align:middle line:84% This is an amazing result, which we will prove today. 00:34:05.870 --> 00:34:09.429 align:middle line:84% But the depth of this result, I think, 00:34:09.429 --> 00:34:12.350 align:middle line:84% you appreciate more and more over time 00:34:12.350 --> 00:34:16.929 align:middle line:84% because of how many amazing consequences it has. 00:34:16.929 --> 00:34:21.690 align:middle line:84% So this is something called the min max theorem. 00:34:21.690 --> 00:34:24.759 align:middle line:84% And it was proven by von Neumann. 00:34:24.759 --> 00:34:27.770 align:middle line:90% 00:34:27.770 --> 00:34:30.710 align:middle line:84% Let me just write it down as a mathematical statement. 00:34:30.710 --> 00:34:33.449 align:middle line:84% There is some value lambda star, which 00:34:33.449 --> 00:34:41.690 align:middle line:84% is equal to the max over all x belonging to delta m of the min 00:34:41.690 --> 00:34:48.530 align:middle line:84% over all y belonging to delta n of x transpose Cy. 00:34:48.530 --> 00:34:50.370 align:middle line:84% But I claim that this is actually 00:34:50.370 --> 00:34:56.050 align:middle line:84% the same thing as switching the max and the min-- 00:34:56.050 --> 00:35:03.690 align:middle line:84% so the min over all y in delta n the max over all x in delta 00:35:03.690 --> 00:35:08.930 align:middle line:84% m of the same thing, x transpose Cy. 00:35:08.930 --> 00:35:11.900 align:middle line:84% So this is my favorite theorem in the entire class 00:35:11.900 --> 00:35:14.680 align:middle line:90% we're going to cover. 00:35:14.680 --> 00:35:17.200 align:middle line:84% So what this statement is even saying 00:35:17.200 --> 00:35:19.700 align:middle line:84% and why it's so deep and powerful, 00:35:19.700 --> 00:35:21.580 align:middle line:84% this takes a little bit to digest. 00:35:21.580 --> 00:35:24.280 align:middle line:84% So let's go through that right now. 00:35:24.280 --> 00:35:27.280 align:middle line:84% So right now, all I said-- it's called von Neumann's min max 00:35:27.280 --> 00:35:28.160 align:middle line:90% theorem-- 00:35:28.160 --> 00:35:31.920 align:middle line:84% is that these two things are the same, interchanging 00:35:31.920 --> 00:35:32.960 align:middle line:90% the max and the min. 00:35:32.960 --> 00:35:35.560 align:middle line:90% 00:35:35.560 --> 00:35:38.800 align:middle line:84% And I'm going to call these two expressions 1 and 2. 00:35:38.800 --> 00:35:44.160 align:middle line:84% So let's interpret this expression game theoretically, 00:35:44.160 --> 00:35:46.640 align:middle line:84% in terms of what the semantics are for these two 00:35:46.640 --> 00:35:48.060 align:middle line:90% different expressions. 00:35:48.060 --> 00:36:03.520 align:middle line:90% 00:36:03.520 --> 00:36:05.390 align:middle line:90% So what does this result imply? 00:36:05.390 --> 00:36:09.520 align:middle line:90% 00:36:09.520 --> 00:36:12.940 align:middle line:84% Well, what I claim is that it's really telling us 00:36:12.940 --> 00:36:15.340 align:middle line:84% that two different ways to play the game-- actually, 00:36:15.340 --> 00:36:18.240 align:middle line:84% three different ways are all equivalent. 00:36:18.240 --> 00:36:20.060 align:middle line:90% It doesn't matter. 00:36:20.060 --> 00:36:25.020 align:middle line:84% So expression 1, where, I have a max and then a min-- 00:36:25.020 --> 00:36:27.020 align:middle line:84% I claim that what's going on with the way 00:36:27.020 --> 00:36:28.700 align:middle line:84% that I've ordered the max and the min 00:36:28.700 --> 00:36:32.100 align:middle line:84% is that it's a certain semantics for how the game is played, 00:36:32.100 --> 00:36:36.100 align:middle line:84% which sounds like it's unfair to the column player 00:36:36.100 --> 00:36:46.540 align:middle line:84% because what's going on is that the column player chooses 00:36:46.540 --> 00:37:03.580 align:middle line:84% a distribution on strategies, announces it, 00:37:03.580 --> 00:37:06.250 align:middle line:84% and the row player best response. 00:37:06.250 --> 00:37:16.280 align:middle line:90% 00:37:16.280 --> 00:37:17.670 align:middle line:90% Did I get these backwards? 00:37:17.670 --> 00:37:18.170 align:middle line:90% One second. 00:37:18.170 --> 00:37:26.120 align:middle line:90% 00:37:26.120 --> 00:37:28.020 align:middle line:90% Yeah, I think it's backwards. 00:37:28.020 --> 00:37:31.480 align:middle line:90% All right, let's do 2. 00:37:31.480 --> 00:37:34.120 align:middle line:84% So what's going on in expression 2 00:37:34.120 --> 00:37:38.680 align:middle line:84% is that first we have this outer min. 00:37:38.680 --> 00:37:41.440 align:middle line:90% So first, this-- oh, wait. 00:37:41.440 --> 00:37:42.480 align:middle line:90% Let's see. 00:37:42.480 --> 00:37:44.220 align:middle line:84% Nope, I had it right the first time. 00:37:44.220 --> 00:37:45.060 align:middle line:90% My mistake. 00:37:45.060 --> 00:37:46.548 align:middle line:90% All right. 00:37:46.548 --> 00:37:48.340 align:middle line:84% It doesn't matter where they're equivalent, 00:37:48.340 --> 00:37:51.535 align:middle line:84% but let's just make sure we get the parsing correct. 00:37:51.535 --> 00:37:53.820 align:middle line:84% So imagine that I take that expression. 00:37:53.820 --> 00:37:58.000 align:middle line:84% I cover up the max first in my expression 1. 00:37:58.000 --> 00:38:00.600 align:middle line:84% So what's going on is that the column player 00:38:00.600 --> 00:38:02.920 align:middle line:90% is playing this min strategy. 00:38:02.920 --> 00:38:08.580 align:middle line:84% So he's trying to find this strategy y that's 00:38:08.580 --> 00:38:11.220 align:middle line:90% a distribution on columns. 00:38:11.220 --> 00:38:16.820 align:middle line:84% But now, once he's chosen what that strategy is, 00:38:16.820 --> 00:38:19.420 align:middle line:84% once he's chosen what that strategy is, 00:38:19.420 --> 00:38:25.100 align:middle line:84% then the max player is allowed to choose the best response. 00:38:25.100 --> 00:38:28.940 align:middle line:84% So even after the column player is fixed and committed 00:38:28.940 --> 00:38:31.660 align:middle line:84% to playing y, then the row player 00:38:31.660 --> 00:38:35.088 align:middle line:84% chooses the strategy x that maximizes it. 00:38:35.088 --> 00:38:37.380 align:middle line:84% I'm going to get this slightly-- yeah, let me just make 00:38:37.380 --> 00:38:38.750 align:middle line:90% sure I got these things right. 00:38:38.750 --> 00:38:45.120 align:middle line:90% 00:38:45.120 --> 00:38:45.620 align:middle line:90% Yeah. 00:38:45.620 --> 00:38:50.100 align:middle line:90% 00:38:50.100 --> 00:38:51.140 align:middle line:90% All right. 00:38:51.140 --> 00:38:55.740 align:middle line:84% I might have gotten it backwards, but that's OK. 00:38:55.740 --> 00:38:58.360 align:middle line:84% And for strategy 2, it'll be the same, 00:38:58.360 --> 00:39:00.490 align:middle line:90% but the row player goes first. 00:39:00.490 --> 00:39:07.160 align:middle line:90% 00:39:07.160 --> 00:39:08.960 align:middle line:84% So the way to think about it is that, 00:39:08.960 --> 00:39:11.800 align:middle line:84% if the way that I described the semantics of the game 00:39:11.800 --> 00:39:14.840 align:middle line:84% to begin with started with one of the players 00:39:14.840 --> 00:39:17.240 align:middle line:84% having to pre-commit to their strategy, 00:39:17.240 --> 00:39:20.200 align:middle line:84% a priori that would have sounded like I really 00:39:20.200 --> 00:39:25.160 align:middle line:84% changed who had an advantage because now it 00:39:25.160 --> 00:39:29.625 align:middle line:84% has to be that Alice is going first or Bob is going first. 00:39:29.625 --> 00:39:31.000 align:middle line:84% That person has to commit to what 00:39:31.000 --> 00:39:32.833 align:middle line:90% their randomized strategy is. 00:39:32.833 --> 00:39:35.000 align:middle line:84% See, the way to think about it is that, when we just 00:39:35.000 --> 00:39:38.160 align:middle line:84% had the notion of a pure Nash equilibrium, 00:39:38.160 --> 00:39:40.560 align:middle line:84% then all of a sudden changing the order that the two 00:39:40.560 --> 00:39:44.240 align:middle line:84% players play completely changes the meaning of the game. 00:39:44.240 --> 00:39:47.400 align:middle line:84% So think about the matching pennies example. 00:39:47.400 --> 00:39:49.440 align:middle line:84% If Alice goes first, you definitely 00:39:49.440 --> 00:39:52.160 align:middle line:84% want to be Bob because, whatever Alice chooses, 00:39:52.160 --> 00:39:54.040 align:middle line:84% you can choose the penny to not match. 00:39:54.040 --> 00:39:56.320 align:middle line:90% And then you always win $1. 00:39:56.320 --> 00:39:59.640 align:middle line:84% So when you're committing to deterministic strategies, 00:39:59.640 --> 00:40:03.080 align:middle line:84% there's a huge advantage to who commits first to their strategy 00:40:03.080 --> 00:40:05.350 align:middle line:90% and who gets to adapt. 00:40:05.350 --> 00:40:07.630 align:middle line:84% But what we're saying here is that it actually 00:40:07.630 --> 00:40:11.830 align:middle line:84% doesn't matter who goes first, that the best that you can 00:40:11.830 --> 00:40:14.150 align:middle line:84% do in either of these two scenarios, 00:40:14.150 --> 00:40:16.370 align:middle line:84% where Alice pre-commits to a distribution, 00:40:16.370 --> 00:40:20.470 align:middle line:84% then Bob goes or vice versa, is actually the same value. 00:40:20.470 --> 00:40:23.670 align:middle line:90% It's the value lambda star. 00:40:23.670 --> 00:40:27.350 align:middle line:84% So in fact, what this min max theorem is really telling 00:40:27.350 --> 00:40:30.410 align:middle line:84% us is that these two ways of playing the game, 00:40:30.410 --> 00:40:32.630 align:middle line:84% and even the way where we simultaneously 00:40:32.630 --> 00:40:35.190 align:middle line:84% choose what our distribution on strategies are, 00:40:35.190 --> 00:40:38.190 align:middle line:90% are all completely equivalent. 00:40:38.190 --> 00:40:41.450 align:middle line:84% So this may not yet seem like a very powerful result. 00:40:41.450 --> 00:40:45.110 align:middle line:84% But I'll give you some more intuition later. 00:40:45.110 --> 00:40:51.590 align:middle line:84% But for right now, let's prove one very powerful thing. 00:40:51.590 --> 00:40:53.770 align:middle line:84% So when I talked about a pure Nash equilibrium, 00:40:53.770 --> 00:40:56.450 align:middle line:84% that was a very natural solution concept. 00:40:56.450 --> 00:40:58.630 align:middle line:84% But unfortunately, we had simple games, 00:40:58.630 --> 00:41:01.070 align:middle line:84% like the matching pennies example, that 00:41:01.070 --> 00:41:04.610 align:middle line:90% had no pure Nash equilibrium. 00:41:04.610 --> 00:41:08.370 align:middle line:84% Now, a mixed Nash equilibrium is more promising 00:41:08.370 --> 00:41:11.450 align:middle line:84% because we've augmented the strategy space so that Alice 00:41:11.450 --> 00:41:13.810 align:middle line:90% and Bob choose distributions. 00:41:13.810 --> 00:41:17.090 align:middle line:84% What I claim, the corollary of this theorem 00:41:17.090 --> 00:41:21.570 align:middle line:84% is that every zero sum game has a mixed Nash equilibrium. 00:41:21.570 --> 00:41:23.673 align:middle line:84% So that's the first thing we're going to prove. 00:41:23.673 --> 00:41:25.090 align:middle line:84% And then we'll come back and we'll 00:41:25.090 --> 00:41:28.810 align:middle line:84% prove the min max theorem based on our understanding of LP 00:41:28.810 --> 00:41:30.850 align:middle line:90% duality. 00:41:30.850 --> 00:41:31.350 align:middle line:90% All right. 00:41:31.350 --> 00:41:33.530 align:middle line:90% So let me state that corollary. 00:41:33.530 --> 00:41:35.270 align:middle line:90% And then let's prove it. 00:41:35.270 --> 00:41:51.130 align:middle line:90% 00:41:51.130 --> 00:42:02.150 align:middle line:84% I claim every two player zero sum game has a mixed Nash 00:42:02.150 --> 00:42:02.930 align:middle line:90% equilibrium. 00:42:02.930 --> 00:42:16.870 align:middle line:90% 00:42:16.870 --> 00:42:19.310 align:middle line:84% So the main question is, where am I 00:42:19.310 --> 00:42:23.150 align:middle line:84% going to get my strategies that are purportedly mixed Nash 00:42:23.150 --> 00:42:23.970 align:middle line:90% equilibrium? 00:42:23.970 --> 00:42:28.190 align:middle line:90% 00:42:28.190 --> 00:42:29.860 align:middle line:90% This part's not so obvious. 00:42:29.860 --> 00:42:32.110 align:middle line:84% But it's really going to come from the two expressions 00:42:32.110 --> 00:42:33.790 align:middle line:90% right here. 00:42:33.790 --> 00:42:38.230 align:middle line:84% So here's how the proof is going to work. 00:42:38.230 --> 00:42:42.990 align:middle line:84% We're going to let lambda star equal 00:42:42.990 --> 00:42:49.510 align:middle line:84% the max min, which, of course, is equal to the min max. 00:42:49.510 --> 00:42:52.030 align:middle line:84% And now, the main question is, where 00:42:52.030 --> 00:42:54.710 align:middle line:90% are my strategy is going to be? 00:42:54.710 --> 00:43:01.310 align:middle line:84% So what I'm going to do is I'm going to let x star and y star-- 00:43:01.310 --> 00:43:04.850 align:middle line:84% these are my purported mixed Nash equilibrium. 00:43:04.850 --> 00:43:18.330 align:middle line:84% They'll be the optimal strategies In 2 and 1, 00:43:18.330 --> 00:43:20.250 align:middle line:90% respectively. 00:43:20.250 --> 00:43:23.816 align:middle line:84% So the way to think about it is in these different scenarios. 00:43:23.816 --> 00:43:28.850 align:middle line:84% So x star is the answer when the row 00:43:28.850 --> 00:43:31.490 align:middle line:90% player is forced to go first. 00:43:31.490 --> 00:43:33.850 align:middle line:84% y star is the answer when the column 00:43:33.850 --> 00:43:37.090 align:middle line:90% player is forced to go first. 00:43:37.090 --> 00:43:40.490 align:middle line:84% And what I'm going to prove is that these pair of strategies 00:43:40.490 --> 00:43:45.210 align:middle line:84% really are a mixed Nash equilibrium. 00:43:45.210 --> 00:43:47.540 align:middle line:84% So now, let's put it all together. 00:43:47.540 --> 00:43:59.310 align:middle line:90% 00:43:59.310 --> 00:44:02.310 align:middle line:84% So we can take x star transpose Cy. 00:44:02.310 --> 00:44:12.990 align:middle line:90% 00:44:12.990 --> 00:44:17.190 align:middle line:84% And I claim we have the following inequality. 00:44:17.190 --> 00:44:20.910 align:middle line:84% So what does this inequality mean in words? 00:44:20.910 --> 00:44:24.150 align:middle line:84% So we know that lambda star is the payoff when 00:44:24.150 --> 00:44:29.322 align:middle line:84% these two players play x star and y star respectively. 00:44:29.322 --> 00:44:32.870 align:middle line:84% And so what's going on here is that when the row player 00:44:32.870 --> 00:44:38.510 align:middle line:84% pre-commits to x star, I claim that, no matter what the column 00:44:38.510 --> 00:44:42.630 align:middle line:84% player does for his strategy, that the payoff is 00:44:42.630 --> 00:44:44.280 align:middle line:90% at least lambda star. 00:44:44.280 --> 00:44:47.110 align:middle line:90% 00:44:47.110 --> 00:44:51.590 align:middle line:84% That's what it means for this formulation in 2 00:44:51.590 --> 00:44:56.630 align:middle line:84% to have the value lambda star, is that I can choose my x star 00:44:56.630 --> 00:44:57.700 align:middle line:90% strategy. 00:44:57.700 --> 00:45:02.380 align:middle line:84% And no matter what happens after I pre-commit to that strategy, 00:45:02.380 --> 00:45:06.220 align:middle line:84% I get the value at least lambda star, no matter 00:45:06.220 --> 00:45:07.860 align:middle line:90% what that strategy is. 00:45:07.860 --> 00:45:12.380 align:middle line:84% So this is literally just translating the statement 00:45:12.380 --> 00:45:15.740 align:middle line:84% of how we're playing the game in scenario 2 00:45:15.740 --> 00:45:19.140 align:middle line:84% into a system of linear inequalities, the same way 00:45:19.140 --> 00:45:21.526 align:middle line:90% that we've been doing thus far. 00:45:21.526 --> 00:45:24.700 align:middle line:84% I'm going to call this set of inequalities A 00:45:24.700 --> 00:45:27.500 align:middle line:84% because I'm going to have to put it together. 00:45:27.500 --> 00:45:30.540 align:middle line:84% And similarly, we can do the same thing the other direction. 00:45:30.540 --> 00:45:35.540 align:middle line:90% 00:45:35.540 --> 00:45:42.540 align:middle line:84% x transpose Cy star is at most lambda star. 00:45:42.540 --> 00:45:47.580 align:middle line:84% And this holds for all x that belongs to this distribution 00:45:47.580 --> 00:45:49.955 align:middle line:90% on m things. 00:45:49.955 --> 00:45:54.460 align:middle line:84% And this holds because, when the column player has 00:45:54.460 --> 00:45:57.320 align:middle line:84% to go first, this is what it means 00:45:57.320 --> 00:45:59.960 align:middle line:84% to have a good strategy in scenario 1, 00:45:59.960 --> 00:46:02.920 align:middle line:84% is that the column player chooses this distribution y 00:46:02.920 --> 00:46:03.760 align:middle line:90% star. 00:46:03.760 --> 00:46:06.040 align:middle line:84% And no matter what the row player does, 00:46:06.040 --> 00:46:10.240 align:middle line:84% we still get the value at most lambda squared-- 00:46:10.240 --> 00:46:10.880 align:middle line:90% lambda star. 00:46:10.880 --> 00:46:14.120 align:middle line:84% We're losing at most lambda star dollars. 00:46:14.120 --> 00:46:18.320 align:middle line:84% So I'll call this system of linear inequalities B. 00:46:18.320 --> 00:46:24.280 align:middle line:84% And together, what these two things 00:46:24.280 --> 00:46:30.280 align:middle line:84% imply is that they imply that x star transpose 00:46:30.280 --> 00:46:36.670 align:middle line:90% Cy star is equal to lambda star. 00:46:36.670 --> 00:46:38.720 align:middle line:84% Just by putting these two constraints 00:46:38.720 --> 00:46:41.920 align:middle line:84% that we have over all y and over all x together, 00:46:41.920 --> 00:46:45.080 align:middle line:84% we get exactly what we wanted, that the payoff really 00:46:45.080 --> 00:46:48.280 align:middle line:90% is lambda star. 00:46:48.280 --> 00:46:51.180 align:middle line:84% And so what this means, when we put it all together, 00:46:51.180 --> 00:47:10.540 align:middle line:84% is that A plus B plus C mean that the expected win loss is 00:47:10.540 --> 00:47:15.330 align:middle line:90% lambda star and no worse. 00:47:15.330 --> 00:47:19.580 align:middle line:90% 00:47:19.580 --> 00:47:21.860 align:middle line:84% So no one can actually do better. 00:47:21.860 --> 00:47:25.060 align:middle line:84% So really, just the statement of this min max 00:47:25.060 --> 00:47:29.540 align:middle line:84% equals max min is really just the translation of the statement 00:47:29.540 --> 00:47:32.500 align:middle line:84% that no one has an incentive to deviate, 00:47:32.500 --> 00:47:35.460 align:middle line:84% as long as we're careful about how we get those min max 00:47:35.460 --> 00:47:36.630 align:middle line:90% strategies. 00:47:36.630 --> 00:47:38.460 align:middle line:84% And the way we get the min max strategy 00:47:38.460 --> 00:47:40.260 align:middle line:90% is we piece it together. 00:47:40.260 --> 00:47:42.442 align:middle line:84% One of the strategies comes from this scenario 00:47:42.442 --> 00:47:43.900 align:middle line:84% of how we play the game, where it's 00:47:43.900 --> 00:47:45.940 align:middle line:90% unfair to the column player. 00:47:45.940 --> 00:47:47.580 align:middle line:84% And one comes from this scenario, 00:47:47.580 --> 00:47:49.900 align:middle line:84% where it's unfair to the row player 00:47:49.900 --> 00:47:52.180 align:middle line:90% instead who has to go first. 00:47:52.180 --> 00:47:54.160 align:middle line:90% So are there any questions? 00:47:54.160 --> 00:47:55.820 align:middle line:84% That's the first main punch line, 00:47:55.820 --> 00:48:00.280 align:middle line:84% is that all zero sum games with two players 00:48:00.280 --> 00:48:03.440 align:middle line:84% are guaranteed to have a mixed Nash equilibrium. 00:48:03.440 --> 00:48:06.960 align:middle line:84% And that follows just from von Neumann's min max theorem, 00:48:06.960 --> 00:48:08.960 align:middle line:90% which we're now going to prove. 00:48:08.960 --> 00:48:12.560 align:middle line:90% So any questions? 00:48:12.560 --> 00:48:14.120 align:middle line:90% Good? 00:48:14.120 --> 00:48:15.320 align:middle line:90% All right. 00:48:15.320 --> 00:48:17.190 align:middle line:84% So now, let's prove the min max theorem. 00:48:17.190 --> 00:48:34.600 align:middle line:90% 00:48:34.600 --> 00:48:37.400 align:middle line:84% The first thing I'm going to do, before I prove the min max 00:48:37.400 --> 00:48:39.560 align:middle line:84% theorem, is I'm going to tell you 00:48:39.560 --> 00:48:43.160 align:middle line:84% a lemma that makes precise something that I asserted 00:48:43.160 --> 00:48:47.640 align:middle line:84% earlier, that the notion of deviation-- 00:48:47.640 --> 00:48:49.540 align:middle line:84% you can, without loss of generality, 00:48:49.540 --> 00:48:52.310 align:middle line:90% just deviate to pure strategies. 00:48:52.310 --> 00:48:54.230 align:middle line:90% Let's make this precise. 00:48:54.230 --> 00:48:55.330 align:middle line:90% And then we'll prove it. 00:48:55.330 --> 00:48:57.690 align:middle line:84% And we'll see how it's really weak duality, 00:48:57.690 --> 00:48:59.020 align:middle line:90% types of arguments and action. 00:48:59.020 --> 00:49:02.990 align:middle line:90% 00:49:02.990 --> 00:49:05.070 align:middle line:84% So here's my key helper lemma we're 00:49:05.070 --> 00:49:07.750 align:middle line:84% going to use to prove the min max theorem. 00:49:07.750 --> 00:49:11.750 align:middle line:90% I claim that the constraint-- 00:49:11.750 --> 00:49:17.270 align:middle line:84% or rather, the system of constraints x 00:49:17.270 --> 00:49:21.390 align:middle line:84% transpose C xy is at least lambda. 00:49:21.390 --> 00:49:25.550 align:middle line:84% And this is true for all y in delta n. 00:49:25.550 --> 00:49:28.670 align:middle line:84% Let me call this the constraint system 3. 00:49:28.670 --> 00:49:40.070 align:middle line:84% I claim it is equivalent to the constraint x transpose 00:49:40.070 --> 00:49:44.110 align:middle line:90% C Ej is at least lambda. 00:49:44.110 --> 00:49:48.750 align:middle line:84% And this is true for all j belonging to n. 00:49:48.750 --> 00:49:52.530 align:middle line:84% And I'm going to call this constraint system 4. 00:49:52.530 --> 00:49:57.650 align:middle line:84% So this just makes precise the thing which I asserted earlier. 00:49:57.650 --> 00:50:01.530 align:middle line:84% I'm just saying that, when we have a system of constraints, 00:50:01.530 --> 00:50:04.010 align:middle line:84% this is an extraordinarily complicated system 00:50:04.010 --> 00:50:06.570 align:middle line:84% of constraints because there are infinitely 00:50:06.570 --> 00:50:10.750 align:middle line:84% many constraints, one for every choice of y 00:50:10.750 --> 00:50:13.450 align:middle line:84% that's a distribution on n things. 00:50:13.450 --> 00:50:16.290 align:middle line:84% And I want the constraint that, no matter what 00:50:16.290 --> 00:50:21.450 align:middle line:84% you choose for y, my payoff when I play x is at least lambda. 00:50:21.450 --> 00:50:23.330 align:middle line:84% That's what this constraint is saying. 00:50:23.330 --> 00:50:25.850 align:middle line:84% No matter what randomized strategy you try, 00:50:25.850 --> 00:50:27.490 align:middle line:90% I get at least lambda. 00:50:27.490 --> 00:50:30.050 align:middle line:84% I claim that this system of constraints that's 00:50:30.050 --> 00:50:35.370 align:middle line:84% captured by this equation 3 is equivalent to this much simpler 00:50:35.370 --> 00:50:38.430 align:middle line:84% system of constraints that, when I play x, 00:50:38.430 --> 00:50:42.290 align:middle line:84% I just have to consider strategies for you that 00:50:42.290 --> 00:50:45.050 align:middle line:90% are deterministic strategies. 00:50:45.050 --> 00:50:48.770 align:middle line:84% So here, Ej is the vector that has 1, 00:50:48.770 --> 00:50:53.190 align:middle line:84% 1 in the j-th coordinate and the rest of it is 0's. 00:50:53.190 --> 00:50:55.110 align:middle line:84% So what this corresponds to is instead 00:50:55.110 --> 00:50:57.250 align:middle line:84% of y being a general distribution, 00:50:57.250 --> 00:51:00.910 align:middle line:84% it's now a fixed choice of playing strategy j. 00:51:00.910 --> 00:51:03.230 align:middle line:84% And I just need to ensure that against every one 00:51:03.230 --> 00:51:07.110 align:middle line:84% of your individual strategies I get at least lambda. 00:51:07.110 --> 00:51:10.550 align:middle line:84% So this is the crux of what I was saying earlier, that I only 00:51:10.550 --> 00:51:13.230 align:middle line:84% need to consider deviations which 00:51:13.230 --> 00:51:15.470 align:middle line:90% are deterministic strategies. 00:51:15.470 --> 00:51:17.590 align:middle line:84% It's really the statement that the system 00:51:17.590 --> 00:51:19.430 align:middle line:84% of linear inequalities are actually 00:51:19.430 --> 00:51:20.770 align:middle line:90% equivalent to each other. 00:51:20.770 --> 00:51:24.070 align:middle line:84% You can derive one from the other. 00:51:24.070 --> 00:51:26.110 align:middle line:90% So let's prove this fact. 00:51:26.110 --> 00:51:27.280 align:middle line:90% The proof is very simple. 00:51:27.280 --> 00:51:31.710 align:middle line:90% 00:51:31.710 --> 00:51:35.750 align:middle line:84% So one direction is obvious because, whenever 00:51:35.750 --> 00:51:39.510 align:middle line:84% I satisfy the constraints in 3, I better satisfy 00:51:39.510 --> 00:51:44.470 align:middle line:84% the constraints in 4 because this is just a subset of them. 00:51:44.470 --> 00:51:48.153 align:middle line:84% It's a very particular set of distributions I look at. 00:51:48.153 --> 00:51:49.570 align:middle line:84% But the more interesting direction 00:51:49.570 --> 00:51:52.650 align:middle line:84% is when I satisfy all the constraints in 4, 00:51:52.650 --> 00:51:57.850 align:middle line:84% how exactly can I prove that I satisfy the constraints in 3? 00:51:57.850 --> 00:52:03.970 align:middle line:84% So here, the key point is that if I satisfy all the constraints 00:52:03.970 --> 00:52:08.570 align:middle line:84% in 4 and I want to prove I satisfy the constraints in 3, 00:52:08.570 --> 00:52:13.170 align:middle line:84% what I'm going to do is I can take an arbitrary constraint 00:52:13.170 --> 00:52:15.810 align:middle line:90% y that belongs to the set 3. 00:52:15.810 --> 00:52:18.410 align:middle line:84% And what I can do is I can take this vector. 00:52:18.410 --> 00:52:21.630 align:middle line:84% And I can break it up into the sum of simpler vectors. 00:52:21.630 --> 00:52:25.010 align:middle line:84% So it's the same thing as the sum from j equals 1 to n 00:52:25.010 --> 00:52:27.810 align:middle line:90% of yj Ej. 00:52:27.810 --> 00:52:30.690 align:middle line:84% Remember, Ej is just this placeholder that has a 1 00:52:30.690 --> 00:52:34.530 align:middle line:84% in the j-th coordinate and has 0's everywhere else. 00:52:34.530 --> 00:52:36.550 align:middle line:84% So this is just a simple equality, 00:52:36.550 --> 00:52:39.770 align:middle line:84% breaking up this vector y into each one 00:52:39.770 --> 00:52:43.650 align:middle line:84% of the pieces for each one of its coordinates. 00:52:43.650 --> 00:52:45.330 align:middle line:84% And now, I'm in business because, when 00:52:45.330 --> 00:52:49.270 align:middle line:84% I consider an arbitrary constraint of the form 00:52:49.270 --> 00:52:54.070 align:middle line:84% x transpose Cy is at least lambda, 00:52:54.070 --> 00:53:04.870 align:middle line:84% I claim that I can derive it as follows. 00:53:04.870 --> 00:53:15.263 align:middle line:84% I can take the sum from J equals 1 to n of yj x transpose C Ej. 00:53:15.263 --> 00:53:18.190 align:middle line:90% 00:53:18.190 --> 00:53:21.990 align:middle line:84% Let me just put the inequality inside. 00:53:21.990 --> 00:53:24.910 align:middle line:84% So the way to think about it is that if I take this vector y 00:53:24.910 --> 00:53:28.270 align:middle line:84% and apply this substitution and use linearity, 00:53:28.270 --> 00:53:31.910 align:middle line:84% I can just pull out all the yj's and the summation. 00:53:31.910 --> 00:53:35.990 align:middle line:84% And now, what's going on is that this inequality right here 00:53:35.990 --> 00:53:39.610 align:middle line:84% is just a weighted linear combination, 00:53:39.610 --> 00:53:42.870 align:middle line:84% non-negative linear combination, of the inequalities that 00:53:42.870 --> 00:53:44.510 align:middle line:90% arise in 3. 00:53:44.510 --> 00:53:46.260 align:middle line:84% So each one of these inequalities 00:53:46.260 --> 00:53:49.900 align:middle line:84% shows up in 3 because it corresponds to an Ej. 00:53:49.900 --> 00:53:52.100 align:middle line:84% And this tells me that I can actually 00:53:52.100 --> 00:53:55.540 align:middle line:84% go from satisfying these equations in 3, 00:53:55.540 --> 00:53:58.500 align:middle line:84% add them up in order to derive any constraint I 00:53:58.500 --> 00:54:02.180 align:middle line:84% want that arises in the constraint system 3. 00:54:02.180 --> 00:54:07.860 align:middle line:84% So this should remind you of weak duality 00:54:07.860 --> 00:54:11.500 align:middle line:84% because what's going on when you write the dual to an LP 00:54:11.500 --> 00:54:15.100 align:middle line:84% is you're really asking, when I have a maximization problem, 00:54:15.100 --> 00:54:17.900 align:middle line:84% how can I set up a companion LP that's 00:54:17.900 --> 00:54:20.380 align:middle line:84% a minimization problem that derives 00:54:20.380 --> 00:54:23.940 align:middle line:84% an upper bound on the largest value of the max I can get? 00:54:23.940 --> 00:54:26.020 align:middle line:84% The way that you form the dual is really 00:54:26.020 --> 00:54:29.020 align:middle line:84% by taking the inequalities in your original system 00:54:29.020 --> 00:54:32.060 align:middle line:84% and adding them up with the appropriate variables. 00:54:32.060 --> 00:54:33.820 align:middle line:84% So your dual is just asking what's 00:54:33.820 --> 00:54:37.300 align:middle line:84% the choice of how to add up those inequalities to derive 00:54:37.300 --> 00:54:38.900 align:middle line:90% the best constraint. 00:54:38.900 --> 00:54:40.803 align:middle line:84% That's exactly how you prove weak duality. 00:54:40.803 --> 00:54:42.220 align:middle line:84% And it's really the crux of what's 00:54:42.220 --> 00:54:46.680 align:middle line:84% going on from a game theoretic standpoint here, too. 00:54:46.680 --> 00:54:48.480 align:middle line:90% So are there any questions? 00:54:48.480 --> 00:54:50.880 align:middle line:90% Make sense? 00:54:50.880 --> 00:54:55.880 align:middle line:84% Once we have this lemma, now let's prove the min max theorem. 00:54:55.880 --> 00:54:58.440 align:middle line:84% And because we already have all of the tools 00:54:58.440 --> 00:55:01.840 align:middle line:84% of linear programming duality, we're 00:55:01.840 --> 00:55:04.270 align:middle line:84% just going to prove it by appealing to LP duality. 00:55:04.270 --> 00:55:07.720 align:middle line:90% 00:55:07.720 --> 00:55:10.650 align:middle line:84% So let's prove the min max theorem using this helper lemma. 00:55:10.650 --> 00:55:38.703 align:middle line:90% 00:55:38.703 --> 00:55:40.120 align:middle line:84% So what we're going to do is we're 00:55:40.120 --> 00:55:48.810 align:middle line:84% going to write down an LP for playing in scenario 1. 00:55:48.810 --> 00:55:51.900 align:middle line:90% 00:55:51.900 --> 00:55:54.620 align:middle line:84% So the proof strategy is very simple for this. 00:55:54.620 --> 00:55:56.260 align:middle line:84% What we're going to do is I told you 00:55:56.260 --> 00:55:57.780 align:middle line:84% that the min max theorem is really 00:55:57.780 --> 00:56:00.780 align:middle line:84% about saying that two different scenarios for playing the game 00:56:00.780 --> 00:56:02.260 align:middle line:90% are equivalent. 00:56:02.260 --> 00:56:04.140 align:middle line:84% We're going to write down an LP that 00:56:04.140 --> 00:56:06.980 align:middle line:84% captures each of these scenarios, that capture scenario 00:56:06.980 --> 00:56:10.900 align:middle line:90% 1 and that captures scenario 2. 00:56:10.900 --> 00:56:12.340 align:middle line:84% And what we're going to observe is 00:56:12.340 --> 00:56:16.180 align:middle line:84% that those two LPs are actually going to be dual to each other. 00:56:16.180 --> 00:56:19.700 align:middle line:84% So that's the way that the proof of the min max theorem goes. 00:56:19.700 --> 00:56:23.780 align:middle line:84% So I'll suggestively write p for primal. 00:56:23.780 --> 00:56:29.530 align:middle line:84% So we want to maximize lambda such that-- 00:56:29.530 --> 00:56:34.820 align:middle line:90% 00:56:34.820 --> 00:56:42.200 align:middle line:84% one second-- ah, this would actually be scenario 2. 00:56:42.200 --> 00:56:44.080 align:middle line:90% So let's do scenario 2. 00:56:44.080 --> 00:56:45.420 align:middle line:90% Sorry for getting it backwards. 00:56:45.420 --> 00:56:46.240 align:middle line:90% All right. 00:56:46.240 --> 00:56:49.600 align:middle line:84% So for scenario 2, we want to maximize lambda subject 00:56:49.600 --> 00:56:54.160 align:middle line:84% to the constraint that x transpose 00:56:54.160 --> 00:56:59.440 align:middle line:90% C Ej is at least lambda. 00:56:59.440 --> 00:57:04.600 align:middle line:84% This should be true for all j in my strategy set n. 00:57:04.600 --> 00:57:09.920 align:middle line:84% I want that the sum of the xi's from i equals 1 to m 00:57:09.920 --> 00:57:18.560 align:middle line:84% is equal to 1 and all of the xi's are non-negative for all i. 00:57:18.560 --> 00:57:20.000 align:middle line:84% So really, what's going on is I'm 00:57:20.000 --> 00:57:23.360 align:middle line:84% trying to maximize this value lambda that's 00:57:23.360 --> 00:57:25.760 align:middle line:90% a lower bound on how well I do. 00:57:25.760 --> 00:57:29.760 align:middle line:84% But part of the action is really the choice of what the x is. 00:57:29.760 --> 00:57:32.160 align:middle line:84% So I want to find a strategy x, which 00:57:32.160 --> 00:57:35.840 align:middle line:84% is a distribution on the strategies, with the property 00:57:35.840 --> 00:57:40.440 align:middle line:84% that no matter what the other player plays in response-- 00:57:40.440 --> 00:57:41.940 align:middle line:90% I've chosen x. 00:57:41.940 --> 00:57:44.440 align:middle line:84% No matter what the other player plays in response, 00:57:44.440 --> 00:57:47.060 align:middle line:84% for every strategy j, I will guarantee you 00:57:47.060 --> 00:57:50.260 align:middle line:84% that, in average, I win at least lambda. 00:57:50.260 --> 00:57:53.300 align:middle line:84% And now, I don't just want to find an arbitrary x that 00:57:53.300 --> 00:57:55.060 align:middle line:90% satisfies this condition. 00:57:55.060 --> 00:57:58.300 align:middle line:84% But I want to make lambda as large as possible. 00:57:58.300 --> 00:58:02.460 align:middle line:84% So I want to maximize my safe value so that there exists 00:58:02.460 --> 00:58:04.460 align:middle line:84% a strategy x so that, no matter what 00:58:04.460 --> 00:58:08.300 align:middle line:84% you do, I get at least this value lambda in expectation. 00:58:08.300 --> 00:58:11.460 align:middle line:84% And this is the same thing as scenario 2 00:58:11.460 --> 00:58:14.060 align:middle line:84% because, once the row player has committed 00:58:14.060 --> 00:58:18.340 align:middle line:84% to playing this strategy x, then what would the column player do? 00:58:18.340 --> 00:58:21.320 align:middle line:84% They would choose the j that minimizes this quantity, 00:58:21.320 --> 00:58:23.740 align:middle line:84% so they lose the least amount of money, 00:58:23.740 --> 00:58:25.620 align:middle line:84% which is the same thing as asking that it's 00:58:25.620 --> 00:58:28.500 align:middle line:90% for always at least lambda. 00:58:28.500 --> 00:58:30.060 align:middle line:90% So that's my primal. 00:58:30.060 --> 00:58:39.400 align:middle line:84% And now, I can do the same thing in scenario 1. 00:58:39.400 --> 00:58:42.650 align:middle line:90% 00:58:42.650 --> 00:58:46.050 align:middle line:84% So I'll suggestively write d for dual here. 00:58:46.050 --> 00:58:50.130 align:middle line:84% And here, what I would do is I would minimize over all lambda. 00:58:50.130 --> 00:58:55.770 align:middle line:84% I would want the constraint that Ei transpose Cy is 00:58:55.770 --> 00:59:01.970 align:middle line:90% at most lambda for all i in m. 00:59:01.970 --> 00:59:03.650 align:middle line:84% And of course, I want the constraints 00:59:03.650 --> 00:59:06.810 align:middle line:84% that my vector y is a distribution-- 00:59:06.810 --> 00:59:11.450 align:middle line:84% so the sum from j equals 1 to n of yj equals 1. 00:59:11.450 --> 00:59:17.450 align:middle line:84% All the yj's are non-negative on each of the coordinates. 00:59:17.450 --> 00:59:20.690 align:middle line:84% And so here, it's the same type of thing, but in reverse. 00:59:20.690 --> 00:59:23.290 align:middle line:90% My column player is going first. 00:59:23.290 --> 00:59:25.690 align:middle line:84% My column player is choosing a strategy so that 00:59:25.690 --> 00:59:28.210 align:middle line:84% no matter what the row player does, 00:59:28.210 --> 00:59:30.510 align:middle line:84% you can guarantee that an expectation 00:59:30.510 --> 00:59:33.850 align:middle line:90% I lose at most lambda dollars. 00:59:33.850 --> 00:59:37.110 align:middle line:84% And I want my lambda, my upper bound, on how much 00:59:37.110 --> 00:59:40.320 align:middle line:84% I lose in expectation, to be as small as possible. 00:59:40.320 --> 00:59:45.070 align:middle line:84% It could even be negative if I win dollars in expectation. 00:59:45.070 --> 00:59:49.550 align:middle line:90% So these are the two LPs. 00:59:49.550 --> 00:59:51.670 align:middle line:84% And the proof of the min max theorem 00:59:51.670 --> 00:59:57.430 align:middle line:84% now is very simple because what you can do 00:59:57.430 --> 01:00:10.570 align:middle line:84% is you can check that P and D are duals to each other. 01:00:10.570 --> 01:00:16.350 align:middle line:90% 01:00:16.350 --> 01:00:21.340 align:middle line:90% And moreover, both are feasible. 01:00:21.340 --> 01:00:26.610 align:middle line:90% 01:00:26.610 --> 01:00:27.110 align:middle line:90% Why? 01:00:27.110 --> 01:00:30.210 align:middle line:84% Because I can just choose any x that's a distribution, 01:00:30.210 --> 01:00:32.190 align:middle line:90% any y that's a distribution. 01:00:32.190 --> 01:00:35.970 align:middle line:84% That'll be some finite value for lambda 01:00:35.970 --> 01:00:37.970 align:middle line:90% in both of these two scenarios. 01:00:37.970 --> 01:00:40.170 align:middle line:90% So feasibility is not a problem. 01:00:40.170 --> 01:00:42.610 align:middle line:84% And once you have that both are feasible, 01:00:42.610 --> 01:00:45.220 align:middle line:84% you know that strong duality holds. 01:00:45.220 --> 01:00:53.810 align:middle line:90% 01:00:53.810 --> 01:01:02.580 align:middle line:84% So we know that their optimal values are equal. 01:01:02.580 --> 01:01:05.130 align:middle line:90% 01:01:05.130 --> 01:01:08.290 align:middle line:84% So this gives us von Neumann's min max theorem. 01:01:08.290 --> 01:01:16.370 align:middle line:84% p is equal to max min, which is equal to d, which 01:01:16.370 --> 01:01:18.700 align:middle line:90% is the same thing as min max. 01:01:18.700 --> 01:01:22.370 align:middle line:90% 01:01:22.370 --> 01:01:29.970 align:middle line:84% So this is also not just a cool application of LP duality, 01:01:29.970 --> 01:01:33.470 align:middle line:84% but it really gives you a better way to think about LP duality. 01:01:33.470 --> 01:01:35.590 align:middle line:90% | what is weak duality? 01:01:35.590 --> 01:01:39.350 align:middle line:84% Well, one way to think about what that concept means 01:01:39.350 --> 01:01:42.470 align:middle line:84% is, when I'm in scenario p and I'm 01:01:42.470 --> 01:01:45.470 align:middle line:84% playing according to this scenario 2, 01:01:45.470 --> 01:01:50.510 align:middle line:84% one way I can prove for you that you can't get a very large value 01:01:50.510 --> 01:01:55.310 align:middle line:84% lambda is if I just have to exhibit a good strategy 01:01:55.310 --> 01:01:57.190 align:middle line:90% for the other player. 01:01:57.190 --> 01:02:00.710 align:middle line:84% Once I have a feasible solution to this linear program, 01:02:00.710 --> 01:02:03.510 align:middle line:84% that tells me how the column player has to play. 01:02:03.510 --> 01:02:07.070 align:middle line:84% And that tells me that my row player can't do all that well. 01:02:07.070 --> 01:02:10.630 align:middle line:84% So these two things act as certificates of each other. 01:02:10.630 --> 01:02:13.090 align:middle line:84% Whenever I have a feasible solution to one side, 01:02:13.090 --> 01:02:15.270 align:middle line:84% then it implies something about the other side 01:02:15.270 --> 01:02:17.630 align:middle line:90% not being too good. 01:02:17.630 --> 01:02:20.950 align:middle line:84% And that's really the crux of what weak duality is all about. 01:02:20.950 --> 01:02:23.650 align:middle line:84% And strong duality just tells us that, actually, 01:02:23.650 --> 01:02:27.430 align:middle line:84% the best certificate we can get for p from d 01:02:27.430 --> 01:02:32.150 align:middle line:84% is actually equal to the optimal value that we can obtain. 01:02:32.150 --> 01:02:32.930 align:middle line:90% All right. 01:02:32.930 --> 01:02:34.990 align:middle line:90% So any questions? 01:02:34.990 --> 01:02:36.530 align:middle line:90% Make sense? 01:02:36.530 --> 01:02:37.490 align:middle line:90% All right. 01:02:37.490 --> 01:02:39.310 align:middle line:84% So we covered a lot of ground already. 01:02:39.310 --> 01:02:41.110 align:middle line:84% So we talked about zero sum games. 01:02:41.110 --> 01:02:42.810 align:middle line:84% We talked about pure Nash equilibrium 01:02:42.810 --> 01:02:45.050 align:middle line:84% and how sometimes they don't exist. 01:02:45.050 --> 01:02:49.130 align:middle line:84% We extended our solution concept to mixed Nash equilibrium. 01:02:49.130 --> 01:02:51.890 align:middle line:84% And once we had mixed Nash equilibrium, then 01:02:51.890 --> 01:02:54.490 align:middle line:84% there were a lot of basic questions like, 01:02:54.490 --> 01:02:56.610 align:middle line:90% do they always exist? 01:02:56.610 --> 01:03:00.650 align:middle line:84% We proved that they do exist by courtesy of the von Neumann min 01:03:00.650 --> 01:03:04.890 align:middle line:84% max theorem that told us where x star and y star came from. 01:03:04.890 --> 01:03:07.170 align:middle line:84% And then we proved the min max theorem just 01:03:07.170 --> 01:03:10.920 align:middle line:90% as a consequence of LP duality. 01:03:10.920 --> 01:03:12.170 align:middle line:90% So one of the amazing things-- 01:03:12.170 --> 01:03:16.490 align:middle line:84% I told you that this is one of my favorite theorems. 01:03:16.490 --> 01:03:19.570 align:middle line:84% So actually, first let me tell you the-- 01:03:19.570 --> 01:03:23.210 align:middle line:84% let's go back to the example we had earlier, where 01:03:23.210 --> 01:03:26.770 align:middle line:90% I had you guys play that game. 01:03:26.770 --> 01:03:31.620 align:middle line:84% So we can ask the same questions that I asked back then. 01:03:31.620 --> 01:03:39.240 align:middle line:90% 01:03:39.240 --> 01:03:41.320 align:middle line:84% So when I wrote down that C matrix-- let's 01:03:41.320 --> 01:03:42.730 align:middle line:84% just write it down one more time. 01:03:42.730 --> 01:03:45.360 align:middle line:90% 01:03:45.360 --> 01:03:50.360 align:middle line:84% So I had this payoff matrix C that was 3 by 3-- 01:03:50.360 --> 01:04:00.010 align:middle line:84% 2, minus 1, 3, 2, 2, minus 3, minus 1, 2, minus 3. 01:04:00.010 --> 01:04:03.880 align:middle line:90% 01:04:03.880 --> 01:04:06.360 align:middle line:84% So we can ask the same types of questions. 01:04:06.360 --> 01:04:08.530 align:middle line:90% Who's in the better position? 01:04:08.530 --> 01:04:15.080 align:middle line:90% 01:04:15.080 --> 01:04:17.800 align:middle line:84% Maybe you still wouldn't know the answer. 01:04:17.800 --> 01:04:21.100 align:middle line:84% But how could I find out who's in the better position? 01:04:21.100 --> 01:04:23.320 align:middle line:84% So how would we answer this question now, 01:04:23.320 --> 01:04:25.100 align:middle line:84% now that we have all this technology? 01:04:25.100 --> 01:04:31.780 align:middle line:90% 01:04:31.780 --> 01:04:33.030 align:middle line:90% What should I try and compute? 01:04:33.030 --> 01:04:38.380 align:middle line:90% 01:04:38.380 --> 01:04:39.000 align:middle line:90% Yeah? 01:04:39.000 --> 01:04:41.125 align:middle line:84% STUDENT: I guess the optimal solution [INAUDIBLE]. 01:04:41.125 --> 01:04:42.500 align:middle line:84% ANKUR MOITRA: Yeah, that's right. 01:04:42.500 --> 01:04:46.620 align:middle line:84% So one thing I can do is I can compute the game value, lambda 01:04:46.620 --> 01:04:47.120 align:middle line:90% star. 01:04:47.120 --> 01:04:54.820 align:middle line:90% 01:04:54.820 --> 01:04:59.020 align:middle line:84% Lambda star, which is called the game value. 01:04:59.020 --> 01:05:01.000 align:middle line:84% Remember, the game value captures, 01:05:01.000 --> 01:05:03.820 align:middle line:84% if both players played optimally, 01:05:03.820 --> 01:05:07.020 align:middle line:84% who would win how much in expectation? 01:05:07.020 --> 01:05:09.020 align:middle line:84% So one way I can answer this question of who's 01:05:09.020 --> 01:05:12.100 align:middle line:84% in the better position is I can compute the game value, lambda 01:05:12.100 --> 01:05:13.100 align:middle line:90% star. 01:05:13.100 --> 01:05:16.540 align:middle line:84% Now, in simple examples like the matching pennies, 01:05:16.540 --> 01:05:20.940 align:middle line:84% computing the game value is easy because it's 0. 01:05:20.940 --> 01:05:22.480 align:middle line:84% We know what the best strategy is. 01:05:22.480 --> 01:05:23.780 align:middle line:90% Every player plays randomly. 01:05:23.780 --> 01:05:27.260 align:middle line:84% And then everyone wins nothing in expectation. 01:05:27.260 --> 01:05:29.160 align:middle line:84% But now, for more complicated games, 01:05:29.160 --> 01:05:31.880 align:middle line:84% where it's not so obvious who you would want to be, 01:05:31.880 --> 01:05:35.320 align:middle line:84% I can just feed this to my linear programming solver. 01:05:35.320 --> 01:05:39.000 align:middle line:84% And I can compute whatever the lambda star is. 01:05:39.000 --> 01:05:45.120 align:middle line:84% So in fact, the thing you can show is that lambda star is 01:05:45.120 --> 01:05:48.890 align:middle line:90% actually 1/3, 1/3 of $1. 01:05:48.890 --> 01:05:51.680 align:middle line:90% 01:05:51.680 --> 01:05:53.800 align:middle line:90% So what this means is that-- 01:05:53.800 --> 01:05:57.720 align:middle line:84% so who should I be, the row or the column player? 01:05:57.720 --> 01:05:58.660 align:middle line:90% The row player. 01:05:58.660 --> 01:06:01.280 align:middle line:84% So if we'd played with real money, 01:06:01.280 --> 01:06:03.000 align:middle line:84% then you actually would have won money 01:06:03.000 --> 01:06:06.240 align:middle line:84% if you played according to the min max optimal strategy. 01:06:06.240 --> 01:06:09.640 align:middle line:84% Maybe sometimes you get unlucky and you lose some money. 01:06:09.640 --> 01:06:13.680 align:middle line:84% But it's not actually this game of whack-a-mole 01:06:13.680 --> 01:06:16.280 align:middle line:84% where one player plays a good strategy and the other player 01:06:16.280 --> 01:06:19.240 align:middle line:84% switches to try and get a better strategy 01:06:19.240 --> 01:06:21.360 align:middle line:90% and do better in response. 01:06:21.360 --> 01:06:23.960 align:middle line:84% In the beginning, you lost money. 01:06:23.960 --> 01:06:26.000 align:middle line:90% And then you quickly came back. 01:06:26.000 --> 01:06:28.980 align:middle line:84% But we know what this game of whack-a-mole converges 01:06:28.980 --> 01:06:31.860 align:middle line:84% to, at least if both players play rationally, 01:06:31.860 --> 01:06:35.100 align:middle line:84% because both players should really just play 01:06:35.100 --> 01:06:37.700 align:middle line:90% their mixed Nash equilibrium. 01:06:37.700 --> 01:06:42.540 align:middle line:84% And you can check that the x star that does this-- 01:06:42.540 --> 01:06:43.650 align:middle line:90% let me just write it down. 01:06:43.650 --> 01:06:48.140 align:middle line:90% 01:06:48.140 --> 01:06:51.820 align:middle line:84% So you can actually solve this just by hand 01:06:51.820 --> 01:06:55.620 align:middle line:84% or plugging it into an LP solver. 01:06:55.620 --> 01:07:03.420 align:middle line:84% And what you'll end up getting is that x star is equal to 5/9, 01:07:03.420 --> 01:07:06.540 align:middle line:90% 0, 4/9. 01:07:06.540 --> 01:07:14.460 align:middle line:84% And y star is equal to 0, 2/3, and 1/3. 01:07:14.460 --> 01:07:22.620 align:middle line:84% And we can check that x star transpose C is equal to this 01:07:22.620 --> 01:07:27.970 align:middle line:90% vector 2/3, 1/3, 1/3. 01:07:27.970 --> 01:07:31.410 align:middle line:84% So that's my proof that the game value is at least 1/3, 01:07:31.410 --> 01:07:34.570 align:middle line:84% is when I play that randomized strategy when I am the row 01:07:34.570 --> 01:07:38.650 align:middle line:84% player, well, this vector right here just describes whatever 01:07:38.650 --> 01:07:43.970 align:middle line:84% the column player responds to, what would be my expected value. 01:07:43.970 --> 01:07:47.850 align:middle line:84% The column player should never play this strategy. 01:07:47.850 --> 01:07:51.330 align:middle line:84% And they'll only play among these two strategies. 01:07:51.330 --> 01:07:56.820 align:middle line:84% And I've guaranteed myself that I'll win at least 1/3 of $1. 01:07:56.820 --> 01:08:01.730 align:middle line:84% So did our two volunteers play optimally? 01:08:01.730 --> 01:08:03.270 align:middle line:90% So you can think about it. 01:08:03.270 --> 01:08:06.982 align:middle line:84% What this tells us is that, in the optimal strategy, 01:08:06.982 --> 01:08:08.690 align:middle line:84% you should actually never play, if you're 01:08:08.690 --> 01:08:11.650 align:middle line:90% the row player, strategy 2. 01:08:11.650 --> 01:08:13.113 align:middle line:90% But that happened. 01:08:13.113 --> 01:08:14.530 align:middle line:84% If you were the column player, you 01:08:14.530 --> 01:08:16.022 align:middle line:90% should never play strategy 1. 01:08:16.022 --> 01:08:17.189 align:middle line:90% I don't mean to pick on you. 01:08:17.189 --> 01:08:17.990 align:middle line:90% It's a hard game. 01:08:17.990 --> 01:08:21.130 align:middle line:84% That's why I picked it, was because it's not obvious. 01:08:21.130 --> 01:08:23.569 align:middle line:84% And this is really what's powerful about this solution 01:08:23.569 --> 01:08:27.149 align:middle line:84% concept, is we took something that we had no idea how to play 01:08:27.149 --> 01:08:29.170 align:middle line:84% and we just figured out how to play it. 01:08:29.170 --> 01:08:31.830 align:middle line:84% We figured out how to play it optimally. 01:08:31.830 --> 01:08:34.930 align:middle line:84% Now, in fact, let me ask you an even tougher question 01:08:34.930 --> 01:08:36.510 align:middle line:84% because really, my goal is not just 01:08:36.510 --> 01:08:38.890 align:middle line:84% to teach you about zero sum games, 01:08:38.890 --> 01:08:42.270 align:middle line:84% but to give it to you as a vantage point for how 01:08:42.270 --> 01:08:45.630 align:middle line:84% to think about the key concepts from linear programming 01:08:45.630 --> 01:08:50.350 align:middle line:84% in a more intuitive way, at least in a way I like to. 01:08:50.350 --> 01:08:54.950 align:middle line:84% So one thing you'll notice, I take x star and I take x star 01:08:54.950 --> 01:09:03.510 align:middle line:84% transpose times C. And I get this vector 2/3, 1/3, 1/3. 01:09:03.510 --> 01:09:07.069 align:middle line:84% Now, only these things are tight with the game value. 01:09:07.069 --> 01:09:08.670 align:middle line:84% These are the only things that match 01:09:08.670 --> 01:09:12.590 align:middle line:90% what I claim my lambda star is. 01:09:12.590 --> 01:09:15.149 align:middle line:84% What concept from linear programming theory 01:09:15.149 --> 01:09:19.750 align:middle line:84% tells me why I must have a 0 here? 01:09:19.750 --> 01:09:22.710 align:middle line:90% This is a very tricky question. 01:09:22.710 --> 01:09:23.590 align:middle line:90% We already covered. 01:09:23.590 --> 01:09:24.970 align:middle line:90% Weak duality. 01:09:24.970 --> 01:09:27.609 align:middle line:84% Weak duality in the context of game theory 01:09:27.609 --> 01:09:31.130 align:middle line:84% was how these two different scenarios for playing the game-- 01:09:31.130 --> 01:09:32.770 align:middle line:84% one which seems to advantage the row 01:09:32.770 --> 01:09:35.649 align:middle line:84% player, one which seems to advantage the column player-- 01:09:35.649 --> 01:09:37.689 align:middle line:84% figuring out how to play one of those scenarios 01:09:37.689 --> 01:09:40.490 align:middle line:84% bounds what's possible in the other scenario. 01:09:40.490 --> 01:09:42.370 align:middle line:90% That was weak duality. 01:09:42.370 --> 01:09:47.090 align:middle line:84% I claim that when I take x star transpose times C, and x star 01:09:47.090 --> 01:09:51.050 align:middle line:84% and y star are my purported optimal solutions, 01:09:51.050 --> 01:09:55.490 align:middle line:84% and I look at where I have tight inequalities 01:09:55.490 --> 01:09:57.370 align:middle line:84% that these values for the payoffs 01:09:57.370 --> 01:10:01.170 align:middle line:84% are actually equal to lambda star, the game value, 01:10:01.170 --> 01:10:07.450 align:middle line:84% that anything that's not tight has to have a matching zero. 01:10:07.450 --> 01:10:08.670 align:middle line:90% What concept? 01:10:08.670 --> 01:10:10.410 align:middle line:90% I hope Peter covered this. 01:10:10.410 --> 01:10:11.130 align:middle line:90% Yeah? 01:10:11.130 --> 01:10:13.090 align:middle line:90% STUDENT: Is it basic? 01:10:13.090 --> 01:10:15.650 align:middle line:84% ANKUR MOITRA: It's related, very close. 01:10:15.650 --> 01:10:18.890 align:middle line:84% What about complementary slackness? 01:10:18.890 --> 01:10:21.230 align:middle line:84% So this is complementary slackness 01:10:21.230 --> 01:10:26.990 align:middle line:84% because whenever I have an inequality that's not tight, 01:10:26.990 --> 01:10:29.530 align:middle line:84% like this one right here-- this is not a tight inequality. 01:10:29.530 --> 01:10:32.190 align:middle line:84% It's not equal to the lambda star-- 01:10:32.190 --> 01:10:36.270 align:middle line:84% I better have the corresponding variable is equal to 0. 01:10:36.270 --> 01:10:39.150 align:middle line:84% The only way that I can have strong duality hold 01:10:39.150 --> 01:10:41.470 align:middle line:90% is as long as that's true. 01:10:41.470 --> 01:10:47.610 align:middle line:84% And if instead I took C times y star and I wrote it out, 01:10:47.610 --> 01:10:51.790 align:middle line:90% I would get 1/3, 1/3, 1/3. 01:10:51.790 --> 01:10:53.650 align:middle line:90% So I don't get any constraints. 01:10:53.650 --> 01:10:59.390 align:middle line:84% But if instead I had some value was not equal to 1/3, 01:10:59.390 --> 01:11:03.910 align:middle line:84% then the corresponding thing in x had better be equal to 0, too. 01:11:03.910 --> 01:11:06.830 align:middle line:84% See, the way to think about complementary slackness is that 01:11:06.830 --> 01:11:09.950 align:middle line:84% if this is my purported optimal solution, 01:11:09.950 --> 01:11:13.350 align:middle line:84% if x and y star are my purported optimal pair, 01:11:13.350 --> 01:11:18.270 align:middle line:84% then as soon as y star has some positive mass on this outcome, 01:11:18.270 --> 01:11:22.170 align:middle line:84% then I actually am winning more in expectation than 1/3. 01:11:22.170 --> 01:11:25.290 align:middle line:84% So my game value cannot be the correct answer. 01:11:25.290 --> 01:11:28.370 align:middle line:84% So really, also you can think about complementary slackness 01:11:28.370 --> 01:11:32.410 align:middle line:84% as this constraint about the compatibility of their two 01:11:32.410 --> 01:11:38.130 align:middle line:84% strategies and the associated vectors of expected payoff. 01:11:38.130 --> 01:11:39.850 align:middle line:90% So let me briefly mention-- 01:11:39.850 --> 01:11:43.690 align:middle line:84% this is way beyond the scope of this class. 01:11:43.690 --> 01:11:48.250 align:middle line:84% But one of the reasons this is my favorite result, at least 01:11:48.250 --> 01:11:51.690 align:middle line:84% in the things I teach in undergraduate classes, 01:11:51.690 --> 01:11:56.650 align:middle line:84% is actually my PhD thesis started in some way from this. 01:11:56.650 --> 01:11:59.930 align:middle line:84% So I had a result that I wanted to show 01:11:59.930 --> 01:12:02.950 align:middle line:84% in graph theory, which I'm not going to explain what it is. 01:12:02.950 --> 01:12:06.810 align:middle line:84% But it's a very complicated and surprising result. 01:12:06.810 --> 01:12:11.250 align:middle line:84% So I remember I went into my advisor's office in Akamai. 01:12:11.250 --> 01:12:14.130 align:middle line:84% And then I told him I proved this amazing theorem 01:12:14.130 --> 01:12:18.290 align:middle line:84% about all kinds of graphs, about a structural theorem about them. 01:12:18.290 --> 01:12:21.720 align:middle line:84% And he said, there's no way you proved that. 01:12:21.720 --> 01:12:23.600 align:middle line:84% That doesn't even sound true to me. 01:12:23.600 --> 01:12:26.360 align:middle line:84% So he started writing some concrete examples. 01:12:26.360 --> 01:12:29.520 align:middle line:84% What would your sparsifier look like for this graph? 01:12:29.520 --> 01:12:30.800 align:middle line:90% I couldn't do it. 01:12:30.800 --> 01:12:32.100 align:middle line:90% He gave me another example. 01:12:32.100 --> 01:12:33.020 align:middle line:90% I couldn't do it. 01:12:33.020 --> 01:12:34.158 align:middle line:90% He gave me another example. 01:12:34.158 --> 01:12:35.700 align:middle line:84% I think he started to feel bad for me 01:12:35.700 --> 01:12:37.617 align:middle line:84% because he seemed to get the hint that maybe I 01:12:37.617 --> 01:12:39.280 align:middle line:90% hadn't actually proven it. 01:12:39.280 --> 01:12:41.040 align:middle line:84% And he said, how could you prove this 01:12:41.040 --> 01:12:45.360 align:middle line:84% if you can't construct any of them on the graphs I give you? 01:12:45.360 --> 01:12:47.060 align:middle line:84% And the way that I proved it was, 01:12:47.060 --> 01:12:49.880 align:middle line:84% I actually appealed to a gigantic zero sum 01:12:49.880 --> 01:12:53.780 align:middle line:84% game, where I was playing one of the players, the row player, 01:12:53.780 --> 01:12:55.600 align:middle line:84% and he was playing the column player. 01:12:55.600 --> 01:12:58.920 align:middle line:84% So I didn't have to prove that such things existed by actually 01:12:58.920 --> 01:13:01.000 align:middle line:90% constructing them myself. 01:13:01.000 --> 01:13:05.320 align:middle line:84% I just had to show that any strategy he had for disproving 01:13:05.320 --> 01:13:08.800 align:middle line:90% their existence didn't work. 01:13:08.800 --> 01:13:10.760 align:middle line:84% And that gives you a very indirect way 01:13:10.760 --> 01:13:13.520 align:middle line:84% to prove existence of pretty exotic objects, things 01:13:13.520 --> 01:13:16.440 align:middle line:84% which are hard to wrap your head around otherwise. 01:13:16.440 --> 01:13:18.700 align:middle line:84% So even though you might not see this until 01:13:18.700 --> 01:13:21.540 align:middle line:84% much later if you see it, these types of things 01:13:21.540 --> 01:13:23.780 align:middle line:84% are really under the hood of a huge amount 01:13:23.780 --> 01:13:25.740 align:middle line:90% of combinatorial optimization. 01:13:25.740 --> 01:13:27.580 align:middle line:84% It's not just about zero sum games 01:13:27.580 --> 01:13:30.140 align:middle line:84% I can write down because sometimes it's 01:13:30.140 --> 01:13:33.140 align:middle line:84% about truly gigantic zero sum games that are just 01:13:33.140 --> 01:13:35.140 align:middle line:84% exponentially large or impossibly 01:13:35.140 --> 01:13:37.340 align:middle line:90% large to wrap your head around. 01:13:37.340 --> 01:13:40.660 align:middle line:84% And instead of this whole idea about playing the game 01:13:40.660 --> 01:13:42.940 align:middle line:84% in two different scenarios, instead 01:13:42.940 --> 01:13:45.660 align:middle line:84% of me having to prove to him that they existed, 01:13:45.660 --> 01:13:48.780 align:middle line:84% I just had to prove that he couldn't prove me wrong. 01:13:48.780 --> 01:13:51.700 align:middle line:84% This gives you a great way to prove all kinds of exotic things 01:13:51.700 --> 01:13:54.020 align:middle line:84% that otherwise sound counterintuitive 01:13:54.020 --> 01:13:56.060 align:middle line:90% and like they should be false. 01:13:56.060 --> 01:13:59.020 align:middle line:84% So what's really amazing is when you take a lot of results across 01:13:59.020 --> 01:14:02.223 align:middle line:84% theoretical computer science that sound false to you and you 01:14:02.223 --> 01:14:04.140 align:middle line:84% don't have the intuition-- it sounds like they 01:14:04.140 --> 01:14:05.317 align:middle line:90% shouldn't exist-- 01:14:05.317 --> 01:14:07.900 align:middle line:84% it turns out that, when you pull on the thread, a lot of times 01:14:07.900 --> 01:14:11.330 align:middle line:84% this is what's under the hood, many times. 01:14:11.330 --> 01:14:17.000 align:middle line:90%