WEBVTT 00:00:00.000 --> 00:00:04.364 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:04.364 --> 00:00:10.460 align:middle line:90% 00:00:10.460 --> 00:00:12.280 align:middle line:84% IAN BALL: So today we're going to continue 00:00:12.280 --> 00:00:15.240 align:middle line:84% with what we did last week, and we're 00:00:15.240 --> 00:00:18.260 align:middle line:84% going to talk about something called rationalizability. 00:00:18.260 --> 00:00:20.783 align:middle line:90% 00:00:20.783 --> 00:00:22.200 align:middle line:84% And I'll say up front that I think 00:00:22.200 --> 00:00:24.658 align:middle line:84% today is the first class where maybe there'll be a bit more 00:00:24.658 --> 00:00:26.220 align:middle line:90% notation, a bit more math. 00:00:26.220 --> 00:00:28.550 align:middle line:84% So if anything's unclear, if some notation isn't clear, 00:00:28.550 --> 00:00:30.800 align:middle line:84% feel free to ask me questions because there's probably 00:00:30.800 --> 00:00:34.680 align:middle line:84% other students who also find it unclear. 00:00:34.680 --> 00:00:38.360 align:middle line:84% So first, let's remember or recall 00:00:38.360 --> 00:00:41.320 align:middle line:84% how we defined a strategic form game. 00:00:41.320 --> 00:00:43.320 align:middle line:84% So remember, a strategic form game 00:00:43.320 --> 00:00:46.840 align:middle line:84% specified the strategy sets for all of the players, 00:00:46.840 --> 00:00:49.240 align:middle line:84% and also the payoffs for all of the players. 00:00:49.240 --> 00:00:58.280 align:middle line:84% So one way I'll often write it is u from S1 Sn to Rn. 00:00:58.280 --> 00:01:00.500 align:middle line:90% So what does this mean? 00:01:00.500 --> 00:01:04.420 align:middle line:84% This u function assigns to every profile of strategies 00:01:04.420 --> 00:01:06.780 align:middle line:84% some vector of payoffs that specifies 00:01:06.780 --> 00:01:08.480 align:middle line:90% each of the n player's payoffs. 00:01:08.480 --> 00:01:11.340 align:middle line:84% So with this notation, we might think of a strategy profile 00:01:11.340 --> 00:01:15.660 align:middle line:84% S is going to get mapped to a vector u of s, 00:01:15.660 --> 00:01:18.620 align:middle line:84% which is just all the payoffs the players get put together 00:01:18.620 --> 00:01:19.580 align:middle line:90% in a vector. 00:01:19.580 --> 00:01:25.620 align:middle line:84% So we have u1 of s, all the way up to u of s. 00:01:25.620 --> 00:01:28.060 align:middle line:84% This tells us player one's payoff, 00:01:28.060 --> 00:01:30.060 align:middle line:84% if the profile of strategies that's played 00:01:30.060 --> 00:01:32.820 align:middle line:84% is s, all the way up to player N's 00:01:32.820 --> 00:01:37.860 align:middle line:84% payoff, if the profile of strategies that's played is s. 00:01:37.860 --> 00:01:41.060 align:middle line:84% And I'll point out here that we're generally 00:01:41.060 --> 00:01:44.980 align:middle line:84% going to use lowercase u throughout the course. 00:01:44.980 --> 00:01:46.900 align:middle line:84% So in the first couple of lectures 00:01:46.900 --> 00:01:49.940 align:middle line:84% we made this big deal, which I think correctly so, 00:01:49.940 --> 00:01:52.580 align:middle line:84% about the distinction between the von Neumann-Morgenstern 00:01:52.580 --> 00:01:55.700 align:middle line:84% utility, which was defined over consequences, 00:01:55.700 --> 00:01:57.460 align:middle line:90% and then expected utility. 00:01:57.460 --> 00:02:01.230 align:middle line:84% That was capital U. And that was defined over lotteries, 00:02:01.230 --> 00:02:02.690 align:middle line:90% over consequences. 00:02:02.690 --> 00:02:04.490 align:middle line:84% But throughout the course, as we start 00:02:04.490 --> 00:02:06.710 align:middle line:84% talking about beliefs and mixed strategies, 00:02:06.710 --> 00:02:09.229 align:middle line:84% it gets a little cumbersome to think every time, 00:02:09.229 --> 00:02:11.490 align:middle line:84% should this be capital U or lowercase u? 00:02:11.490 --> 00:02:13.610 align:middle line:84% So I'm throughout the course from here on out 00:02:13.610 --> 00:02:16.930 align:middle line:84% going to always use lowercase u, with the understanding that that 00:02:16.930 --> 00:02:19.450 align:middle line:84% can represent expected utility if there's 00:02:19.450 --> 00:02:23.570 align:middle line:84% any beliefs or mixed strategies or moves by nature 00:02:23.570 --> 00:02:24.487 align:middle line:90% in the extensive form. 00:02:24.487 --> 00:02:26.195 align:middle line:84% But as we get to more complicated things, 00:02:26.195 --> 00:02:28.670 align:middle line:84% it's just a little less clear when it's big U and little u. 00:02:28.670 --> 00:02:30.253 align:middle line:84% So I'm just going to stick to little u 00:02:30.253 --> 00:02:33.650 align:middle line:84% as long as we understand the conceptual distinction. 00:02:33.650 --> 00:02:36.450 align:middle line:84% So what we talked about last time is which strategies 00:02:36.450 --> 00:02:37.810 align:middle line:90% were reasonable to play? 00:02:37.810 --> 00:02:42.730 align:middle line:84% So last time, we talked about strategies that were reasonable, 00:02:42.730 --> 00:02:44.970 align:middle line:84% and I put reasonable and unreasonable in quotes 00:02:44.970 --> 00:02:48.430 align:middle line:84% because I don't mean these in really formal terms. 00:02:48.430 --> 00:02:50.590 align:middle line:84% I'm just being a little informal here. 00:02:50.590 --> 00:02:53.250 align:middle line:90% 00:02:53.250 --> 00:02:57.590 align:middle line:84% But let's remember we call this strategy reasonable for a player 00:02:57.590 --> 00:03:01.910 align:middle line:84% if it was a best response to some belief. 00:03:01.910 --> 00:03:05.730 align:middle line:84% So I'll say BR for Best Response to some belief. 00:03:05.730 --> 00:03:09.550 align:middle line:90% 00:03:09.550 --> 00:03:12.030 align:middle line:84% And let's remember what a belief is. 00:03:12.030 --> 00:03:14.590 align:middle line:84% A belief that player i has is a belief 00:03:14.590 --> 00:03:17.910 align:middle line:84% about the strategies chosen by all of the other players. 00:03:17.910 --> 00:03:20.830 align:middle line:84% So whenever I talk about player i's belief, 00:03:20.830 --> 00:03:24.550 align:middle line:84% that's a belief about the partial profile of strategies 00:03:24.550 --> 00:03:27.550 align:middle line:90% played by the other players. 00:03:27.550 --> 00:03:29.070 align:middle line:84% And then we alternatively said we 00:03:29.070 --> 00:03:31.390 align:middle line:84% can think of a strategy as unreasonable 00:03:31.390 --> 00:03:37.990 align:middle line:84% if it is strictly dominated, which we defined last week. 00:03:37.990 --> 00:03:40.270 align:middle line:84% And crucially, it wasn't enough to just think 00:03:40.270 --> 00:03:43.270 align:middle line:84% about strict dominance by another pure strategy. 00:03:43.270 --> 00:03:46.650 align:middle line:84% We had to consider strict dominance by mixed strategies, 00:03:46.650 --> 00:03:49.450 align:middle line:84% so if it's strictly dominated by some mixed strategy. 00:03:49.450 --> 00:03:55.140 align:middle line:90% 00:03:55.140 --> 00:03:57.260 align:middle line:84% And what we showed last week is that these 00:03:57.260 --> 00:03:58.660 align:middle line:90% are two sides of the same coin. 00:03:58.660 --> 00:04:03.460 align:middle line:84% Either a strategy is a best response to some belief, 00:04:03.460 --> 00:04:07.020 align:middle line:84% or it's strictly dominated by some mixed strategy. 00:04:07.020 --> 00:04:09.300 align:middle line:84% But every strategy falls into exactly one 00:04:09.300 --> 00:04:10.660 align:middle line:90% of these two categories. 00:04:10.660 --> 00:04:13.740 align:middle line:84% So you can take a positive view and say, I only 00:04:13.740 --> 00:04:17.420 align:middle line:84% want to play a strategy that's a best response to some belief. 00:04:17.420 --> 00:04:19.519 align:middle line:84% Or you can take the negative view and say, 00:04:19.519 --> 00:04:22.980 align:middle line:84% I don't want to play a strategy that's strictly dominated 00:04:22.980 --> 00:04:23.978 align:middle line:90% by some mixed strategy. 00:04:23.978 --> 00:04:26.020 align:middle line:84% And either approach you take is going to give you 00:04:26.020 --> 00:04:28.460 align:middle line:90% the same set of strategies. 00:04:28.460 --> 00:04:30.888 align:middle line:84% So today, we want to be maybe a little more formal. 00:04:30.888 --> 00:04:33.180 align:middle line:84% I don't really love this terminology, but we'll use it, 00:04:33.180 --> 00:04:36.900 align:middle line:84% and we'll say what it means for a player 00:04:36.900 --> 00:04:39.360 align:middle line:90% to be rational in this context. 00:04:39.360 --> 00:04:43.940 align:middle line:90% We'll say player i is rational-- 00:04:43.940 --> 00:04:46.820 align:middle line:84% well, if player i plays a strategy that's 00:04:46.820 --> 00:04:49.100 align:middle line:84% in this good collection of strategies, 00:04:49.100 --> 00:04:52.360 align:middle line:84% or equivalently, is not in this bad collection of strategies. 00:04:52.360 --> 00:04:58.040 align:middle line:84% So I'll say player i is rational if she plays 00:04:58.040 --> 00:04:59.700 align:middle line:90% a best response to some belief. 00:04:59.700 --> 00:05:09.320 align:middle line:90% 00:05:09.320 --> 00:05:11.800 align:middle line:84% That is, the strategy she chooses 00:05:11.800 --> 00:05:14.960 align:middle line:84% is a best response to some belief about the other player's 00:05:14.960 --> 00:05:17.120 align:middle line:90% strategies. 00:05:17.120 --> 00:05:19.540 align:middle line:84% I want to be a little careful using the term "rational." 00:05:19.540 --> 00:05:22.120 align:middle line:84% This is a word that's used in a lot of different ways 00:05:22.120 --> 00:05:23.580 align:middle line:90% in a lot of different contexts. 00:05:23.580 --> 00:05:25.598 align:middle line:84% But in this specific context of game theory, 00:05:25.598 --> 00:05:27.140 align:middle line:84% this is usually the term that's used. 00:05:27.140 --> 00:05:28.098 align:middle line:90% So I'm going to use it. 00:05:28.098 --> 00:05:30.360 align:middle line:84% But I'll warn you that in other game theory courses 00:05:30.360 --> 00:05:32.100 align:middle line:84% you might take or other things you see, 00:05:32.100 --> 00:05:33.640 align:middle line:84% rationality can be defined to mean 00:05:33.640 --> 00:05:36.840 align:middle line:84% different things because it's an important, difficult concept. 00:05:36.840 --> 00:05:38.960 align:middle line:84% But let's look at this definition. 00:05:38.960 --> 00:05:42.880 align:middle line:84% And it's restrictive in that the player must be playing a best 00:05:42.880 --> 00:05:44.620 align:middle line:90% response to some belief. 00:05:44.620 --> 00:05:46.360 align:middle line:84% So the best response, part of it is 00:05:46.360 --> 00:05:48.240 align:middle line:90% what imposes some restriction. 00:05:48.240 --> 00:05:51.840 align:middle line:84% But notice, there's no restriction on the belief here. 00:05:51.840 --> 00:05:56.980 align:middle line:90% 00:05:56.980 --> 00:05:59.340 align:middle line:84% So the assumption or the restriction 00:05:59.340 --> 00:06:03.780 align:middle line:84% of rationality by itself is a very weak requirement 00:06:03.780 --> 00:06:07.740 align:middle line:84% because it doesn't constrain or discipline the beliefs 00:06:07.740 --> 00:06:11.380 align:middle line:84% that player I forms about the other player's strategies. 00:06:11.380 --> 00:06:14.200 align:middle line:84% So this is purely about player I's rationality, 00:06:14.200 --> 00:06:18.020 align:middle line:84% and it says nothing about the other player's rationality. 00:06:18.020 --> 00:06:22.900 align:middle line:84% But let's think a little bit and let's say what 00:06:22.900 --> 00:06:26.180 align:middle line:84% if not only is player I rational, 00:06:26.180 --> 00:06:31.620 align:middle line:84% but what if player i knows or believes that the other players 00:06:31.620 --> 00:06:32.500 align:middle line:90% are rational too? 00:06:32.500 --> 00:06:52.970 align:middle line:90% 00:06:52.970 --> 00:06:55.668 align:middle line:84% Would this change the way that player i might play, 00:06:55.668 --> 00:06:57.710 align:middle line:84% or would this impose any additional restrictions? 00:06:57.710 --> 00:06:58.460 align:middle line:90% What do you think? 00:06:58.460 --> 00:07:00.890 align:middle line:84% What might change if player i, in addition 00:07:00.890 --> 00:07:03.650 align:middle line:84% to being rational herself, also knows 00:07:03.650 --> 00:07:06.710 align:middle line:84% that the other players she's playing against are rational? 00:07:06.710 --> 00:07:10.930 align:middle line:90% 00:07:10.930 --> 00:07:12.490 align:middle line:90% Yeah. 00:07:12.490 --> 00:07:14.810 align:middle line:84% STUDENT: If you change your belief, 00:07:14.810 --> 00:07:17.530 align:middle line:84% that now she knows what the other players [INAUDIBLE] 00:07:17.530 --> 00:07:19.580 align:middle line:84% in this [INAUDIBLE] strategy [INAUDIBLE]. 00:07:19.580 --> 00:07:20.330 align:middle line:90% IAN BALL: Exactly. 00:07:20.330 --> 00:07:20.830 align:middle line:90% Right. 00:07:20.830 --> 00:07:22.870 align:middle line:84% So-- do you want to say more or-- yeah. 00:07:22.870 --> 00:07:23.370 align:middle line:90% Yeah. 00:07:23.370 --> 00:07:24.170 align:middle line:90% Yeah, great. 00:07:24.170 --> 00:07:26.250 align:middle line:84% So I just didn't want to cut you off. 00:07:26.250 --> 00:07:26.850 align:middle line:90% Exactly. 00:07:26.850 --> 00:07:29.230 align:middle line:84% So before we had no restrictions on beliefs. 00:07:29.230 --> 00:07:30.730 align:middle line:84% We said, all player i is going to do 00:07:30.730 --> 00:07:33.070 align:middle line:84% is they're going to play some best response to a belief, 00:07:33.070 --> 00:07:35.090 align:middle line:84% but we don't know where that belief comes from. 00:07:35.090 --> 00:07:38.410 align:middle line:84% But now if player I knows that the other players are rational, 00:07:38.410 --> 00:07:40.410 align:middle line:84% they shouldn't form a belief that 00:07:40.410 --> 00:07:43.930 align:middle line:84% assigns positive probability to a really weird, bad strategy 00:07:43.930 --> 00:07:46.770 align:middle line:84% that a rational player wouldn't play. 00:07:46.770 --> 00:07:55.190 align:middle line:84% So the idea here is that this is going to restrict or constrain 00:07:55.190 --> 00:07:58.470 align:middle line:84% the beliefs that player i should or reasonably would 00:07:58.470 --> 00:07:59.790 align:middle line:90% form about others-- 00:07:59.790 --> 00:08:04.090 align:middle line:84% constrain beliefs about other strategies. 00:08:04.090 --> 00:08:11.870 align:middle line:90% 00:08:11.870 --> 00:08:12.810 align:middle line:90% And do we stop there? 00:08:12.810 --> 00:08:13.330 align:middle line:90% Are we done? 00:08:13.330 --> 00:08:14.370 align:middle line:90% Or can we keep going? 00:08:14.370 --> 00:08:19.990 align:middle line:90% 00:08:19.990 --> 00:08:20.670 align:middle line:90% Yeah? 00:08:20.670 --> 00:08:23.045 align:middle line:84% STUDENT: I guess the other players from their perspective 00:08:23.045 --> 00:08:26.510 align:middle line:84% would update based on [INAUDIBLE] player 00:08:26.510 --> 00:08:27.883 align:middle line:90% I's [INAUDIBLE]. 00:08:27.883 --> 00:08:28.550 align:middle line:90% IAN BALL: Right. 00:08:28.550 --> 00:08:31.990 align:middle line:84% So you might think, OK, not only is it relevant 00:08:31.990 --> 00:08:34.350 align:middle line:84% if I know that the other players are rational. 00:08:34.350 --> 00:08:38.190 align:middle line:84% But do I know that the other players know that I'm rational? 00:08:38.190 --> 00:08:40.747 align:middle line:84% Because if the other players know that I'm rational, 00:08:40.747 --> 00:08:42.789 align:middle line:84% that's going to affect the beliefs that they form 00:08:42.789 --> 00:08:45.790 align:middle line:84% about my strategy, and therefore, further 00:08:45.790 --> 00:08:47.890 align:middle line:84% restrict the way they might play, and therefore 00:08:47.890 --> 00:08:50.790 align:middle line:84% further restrict the beliefs that I would form about them. 00:08:50.790 --> 00:08:52.403 align:middle line:84% Now, I think you can see pretty soon 00:08:52.403 --> 00:08:54.070 align:middle line:84% this is going to get pretty complicated, 00:08:54.070 --> 00:08:56.750 align:middle line:84% and just writing it in words is going to become cumbersome. 00:08:56.750 --> 00:08:58.875 align:middle line:84% So now we're going to try to take a formal approach 00:08:58.875 --> 00:09:00.930 align:middle line:84% to this kind of reasoning that captures 00:09:00.930 --> 00:09:05.510 align:middle line:84% what's often called higher order knowledge of rationality. 00:09:05.510 --> 00:09:15.770 align:middle line:90% 00:09:15.770 --> 00:09:17.850 align:middle line:84% So first order knowledge of rationality 00:09:17.850 --> 00:09:20.330 align:middle line:84% means I know other people are rational. 00:09:20.330 --> 00:09:22.890 align:middle line:84% Second order knowledge is I know that they 00:09:22.890 --> 00:09:24.107 align:middle line:90% know that I'm rational. 00:09:24.107 --> 00:09:26.190 align:middle line:84% And third order I know that they know that I know, 00:09:26.190 --> 00:09:27.990 align:middle line:84% and we can see how this goes on and on and on. 00:09:27.990 --> 00:09:30.330 align:middle line:84% And we want to capture these higher degrees of knowledge 00:09:30.330 --> 00:09:32.530 align:middle line:84% about rationality in order to make 00:09:32.530 --> 00:09:35.610 align:middle line:84% more precise or sharper predictions about playing 00:09:35.610 --> 00:09:37.210 align:middle line:90% the game. 00:09:37.210 --> 00:09:39.330 align:middle line:84% So let's go through-- this is a bit abstract. 00:09:39.330 --> 00:09:42.650 align:middle line:84% So let's go through an example to try to convey this idea. 00:09:42.650 --> 00:09:45.660 align:middle line:84% And this is going to be the same example we did last class, 00:09:45.660 --> 00:09:48.400 align:middle line:84% but we're going to add in the payoffs for player two as well. 00:09:48.400 --> 00:09:52.640 align:middle line:84% So we're going to have a game like this. 00:09:52.640 --> 00:09:56.240 align:middle line:84% Remember player one is always the player who chooses rows, 00:09:56.240 --> 00:09:59.200 align:middle line:84% and player two is always the player who chooses columns. 00:09:59.200 --> 00:10:01.560 align:middle line:84% We always put player one's payoffs first. 00:10:01.560 --> 00:10:05.540 align:middle line:84% So we're going to have 2, 0, negative 1, 1. 00:10:05.540 --> 00:10:07.060 align:middle line:90% Let me copy from my notes here. 00:10:07.060 --> 00:10:19.600 align:middle line:90% 00:10:19.600 --> 00:10:24.160 align:middle line:84% So last week, we used these same payoffs for player one, 00:10:24.160 --> 00:10:26.160 align:middle line:84% but we didn't specify any payoffs for player two 00:10:26.160 --> 00:10:27.640 align:middle line:90% when we talked about this game. 00:10:27.640 --> 00:10:30.640 align:middle line:84% And just as a reminder, we represent the game in this nice 00:10:30.640 --> 00:10:33.205 align:middle line:84% normal form, what's sometimes called a bimatrix, 00:10:33.205 --> 00:10:34.580 align:middle line:84% because it's a matrix of numbers, 00:10:34.580 --> 00:10:37.793 align:middle line:84% but each entry has two numbers, so "bit" for two. 00:10:37.793 --> 00:10:39.460 align:middle line:84% But let's remember what's going on here. 00:10:39.460 --> 00:10:42.240 align:middle line:84% S1, the set of strategies for player one, 00:10:42.240 --> 00:10:45.320 align:middle line:84% are just the rows of this bimatrix. 00:10:45.320 --> 00:10:49.820 align:middle line:84% So we have top, middle, and bottom. 00:10:49.820 --> 00:10:55.460 align:middle line:84% And S2 for player two is just left and right. 00:10:55.460 --> 00:10:57.860 align:middle line:84% So let's try to start going through this procedure 00:10:57.860 --> 00:10:59.940 align:middle line:90% of rationality. 00:10:59.940 --> 00:11:01.920 align:middle line:84% So let's first think about, well, 00:11:01.920 --> 00:11:06.060 align:middle line:84% if player one is rational, what strategies might they play? 00:11:06.060 --> 00:11:08.460 align:middle line:84% And a good way of going about this is sometimes helpful 00:11:08.460 --> 00:11:12.460 align:middle line:84% is to say, well, we know that rationality is a bit complicated 00:11:12.460 --> 00:11:14.660 align:middle line:84% because we have to consider all possible beliefs 00:11:14.660 --> 00:11:15.980 align:middle line:90% about the other players. 00:11:15.980 --> 00:11:18.240 align:middle line:84% But let's start with the simpler beliefs, 00:11:18.240 --> 00:11:21.100 align:middle line:84% where we know what strategy the other player is playing. 00:11:21.100 --> 00:11:26.620 align:middle line:84% So if I'm player one, and I know that my opponent is playing L, 00:11:26.620 --> 00:11:28.480 align:middle line:84% then I can just look down and say, well, 00:11:28.480 --> 00:11:32.700 align:middle line:84% I get 2 if I play T. I get 0 if I play M, 00:11:32.700 --> 00:11:34.860 align:middle line:84% and I get negative 1 if I play B. 00:11:34.860 --> 00:11:38.100 align:middle line:84% So the best response for me as player one is to play T. 00:11:38.100 --> 00:11:43.490 align:middle line:84% And we often denote that by underlying the first entry here. 00:11:43.490 --> 00:11:46.330 align:middle line:84% So the fact that we're underlying player one's payoff 00:11:46.330 --> 00:11:49.290 align:middle line:84% means we're thinking about player one's best response 00:11:49.290 --> 00:11:51.570 align:middle line:90% to player two's strategy. 00:11:51.570 --> 00:11:54.370 align:middle line:84% And the fact that it's in the first column tells us 00:11:54.370 --> 00:11:57.610 align:middle line:84% we're thinking about player one's best response to player 00:11:57.610 --> 00:11:59.330 align:middle line:90% two playing L. 00:11:59.330 --> 00:12:01.490 align:middle line:84% Now let's do the same thing if player two 00:12:01.490 --> 00:12:05.210 align:middle line:84% plays R. If player two were to play R, 00:12:05.210 --> 00:12:07.970 align:middle line:84% then player one knows that they get negative 1 if they play 00:12:07.970 --> 00:12:14.050 align:middle line:84% T, 0 if they play M, and 2 if they play B. 2 is the highest 00:12:14.050 --> 00:12:18.250 align:middle line:84% number there, so we're going to underline this. 00:12:18.250 --> 00:12:21.610 align:middle line:84% And we can immediately see two strategies that 00:12:21.610 --> 00:12:24.170 align:middle line:90% are reasonable for player one-- 00:12:24.170 --> 00:12:27.930 align:middle line:84% T and B. Now, the trickier part is, are there any more? 00:12:27.930 --> 00:12:28.950 align:middle line:90% Does anyone remember? 00:12:28.950 --> 00:12:32.788 align:middle line:84% So M here is not a best response to L or R, 00:12:32.788 --> 00:12:34.330 align:middle line:84% but we haven't actually checked if it 00:12:34.330 --> 00:12:36.370 align:middle line:84% could be a best response to a belief where 00:12:36.370 --> 00:12:38.570 align:middle line:84% I think you play L with some probability 00:12:38.570 --> 00:12:40.190 align:middle line:90% and R with some probability. 00:12:40.190 --> 00:12:41.170 align:middle line:90% Does anyone remember? 00:12:41.170 --> 00:12:44.168 align:middle line:84% Is M best response to any belief that player one has? 00:12:44.168 --> 00:12:45.210 align:middle line:90% You're shaking your head. 00:12:45.210 --> 00:12:45.918 align:middle line:90% The answer is no. 00:12:45.918 --> 00:12:46.590 align:middle line:90% And why? 00:12:46.590 --> 00:12:48.690 align:middle line:90% How do we know that? 00:12:48.690 --> 00:12:49.190 align:middle line:90% Yeah? 00:12:49.190 --> 00:12:51.510 align:middle line:84% STUDENT: Because of the theorem that 00:12:51.510 --> 00:12:55.990 align:middle line:84% says either a strategy is a best response to some belief 00:12:55.990 --> 00:12:57.890 align:middle line:90% or is strictly dominated. 00:12:57.890 --> 00:12:58.890 align:middle line:90% IAN BALL: Exactly right. 00:12:58.890 --> 00:13:02.030 align:middle line:84% And you're saying that this is strictly dominated by something? 00:13:02.030 --> 00:13:06.750 align:middle line:90% 00:13:06.750 --> 00:13:07.250 align:middle line:90% It is. 00:13:07.250 --> 00:13:08.850 align:middle line:84% It is going to be strictly dominated by something. 00:13:08.850 --> 00:13:10.308 align:middle line:84% This is what we went over last week 00:13:10.308 --> 00:13:12.230 align:middle line:90% that was a bit of a while ago. 00:13:12.230 --> 00:13:15.950 align:middle line:84% We said that M is strictly dominated by the mixture 00:13:15.950 --> 00:13:19.930 align:middle line:84% of playing T with probability 1/2 and B with probability 1/2. 00:13:19.930 --> 00:13:21.950 align:middle line:84% That's something we checked last week. 00:13:21.950 --> 00:13:26.110 align:middle line:84% So we know that the only, quote unquote, reasonable strategies 00:13:26.110 --> 00:13:29.950 align:middle line:90% for player one are T and B. 00:13:29.950 --> 00:13:34.510 align:middle line:84% So let's maybe think of that step one. 00:13:34.510 --> 00:13:37.190 align:middle line:84% And when I say step one, I'm going to say, let's assume 00:13:37.190 --> 00:13:40.130 align:middle line:84% or suppose that both players are rational. 00:13:40.130 --> 00:13:41.970 align:middle line:84% And just from that assumption alone, 00:13:41.970 --> 00:13:45.170 align:middle line:84% what can we conclude about the strategies that might be played? 00:13:45.170 --> 00:13:49.910 align:middle line:90% So P1 and P2 are both rational. 00:13:49.910 --> 00:13:53.668 align:middle line:90% 00:13:53.668 --> 00:13:55.210 align:middle line:84% Well, if they're both rational, we've 00:13:55.210 --> 00:13:57.335 align:middle line:84% said that that tells us something about what player 00:13:57.335 --> 00:13:58.330 align:middle line:90% one can play. 00:13:58.330 --> 00:14:02.610 align:middle line:84% So I'll write S1 prime here to say, let me collect together 00:14:02.610 --> 00:14:06.130 align:middle line:84% the strategies that player one could play, or might reasonably 00:14:06.130 --> 00:14:09.210 align:middle line:90% play, if player one is rational. 00:14:09.210 --> 00:14:14.290 align:middle line:84% And in this case, we just get T and B. 00:14:14.290 --> 00:14:16.610 align:middle line:84% And now let's try to do the same for player two. 00:14:16.610 --> 00:14:19.450 align:middle line:84% So let's go through the same procedure. 00:14:19.450 --> 00:14:21.370 align:middle line:90% Suppose I'm player two. 00:14:21.370 --> 00:14:26.370 align:middle line:84% Imagine player one is playing T. Then I get 0 if I play left, 00:14:26.370 --> 00:14:27.830 align:middle line:90% and 1 if I play right. 00:14:27.830 --> 00:14:28.910 align:middle line:90% Right is better. 00:14:28.910 --> 00:14:30.150 align:middle line:90% So I'm going to underline it. 00:14:30.150 --> 00:14:33.610 align:middle line:84% And notice here I underlined the second number because here I'm 00:14:33.610 --> 00:14:37.170 align:middle line:84% keeping track of the best response by player two 00:14:37.170 --> 00:14:39.800 align:middle line:84% to the beliefs they have about player one. 00:14:39.800 --> 00:14:41.280 align:middle line:90% And I'll do the same thing here. 00:14:41.280 --> 00:14:44.600 align:middle line:84% If player one plays M, I can get 10 or 0. 00:14:44.600 --> 00:14:48.080 align:middle line:84% 10 is higher, so I'll choose L. And if they play B, 00:14:48.080 --> 00:14:49.760 align:middle line:90% I can get negative 6 or 0. 00:14:49.760 --> 00:14:53.440 align:middle line:90% 0 is higher, so I come here. 00:14:53.440 --> 00:14:55.720 align:middle line:84% And we can immediately see that R 00:14:55.720 --> 00:14:58.480 align:middle line:84% is a best response to some strategy, 00:14:58.480 --> 00:15:01.640 align:middle line:84% and L is a best response to another strategy. 00:15:01.640 --> 00:15:03.920 align:middle line:84% So in this case, we're already done because we already 00:15:03.920 --> 00:15:06.880 align:middle line:84% see that both of these strategies are reasonable. 00:15:06.880 --> 00:15:08.620 align:middle line:90% Each of them is a best response. 00:15:08.620 --> 00:15:11.360 align:middle line:84% In fact, in this case, not only to some belief, but to a belief 00:15:11.360 --> 00:15:15.600 align:middle line:84% that puts all the probability on a single strategy. 00:15:15.600 --> 00:15:22.000 align:middle line:84% So here, we'll say S2 prime equals L and R. 00:15:22.000 --> 00:15:24.120 align:middle line:84% And notice an important point here. 00:15:24.120 --> 00:15:27.040 align:middle line:84% Rationality doesn't always restrict the strategies 00:15:27.040 --> 00:15:28.840 align:middle line:90% that a player might play. 00:15:28.840 --> 00:15:31.600 align:middle line:84% In player one's case, player one's rationality 00:15:31.600 --> 00:15:33.520 align:middle line:90% does throw out one strategy. 00:15:33.520 --> 00:15:35.660 align:middle line:84% It threw out M because M was strictly 00:15:35.660 --> 00:15:37.500 align:middle line:84% dominated by another mixed strategy 00:15:37.500 --> 00:15:39.900 align:middle line:84% and was never a best response to any belief. 00:15:39.900 --> 00:15:42.520 align:middle line:84% But for player two, even if they're rational, 00:15:42.520 --> 00:15:43.558 align:middle line:90% we can't say much more. 00:15:43.558 --> 00:15:45.100 align:middle line:84% We can't say anything else about what 00:15:45.100 --> 00:15:48.100 align:middle line:84% they might play because what's best for them depends 00:15:48.100 --> 00:15:49.660 align:middle line:90% upon what player one does. 00:15:49.660 --> 00:15:52.260 align:middle line:84% And if player one plays T, it's great for them 00:15:52.260 --> 00:15:55.660 align:middle line:84% to play R. And if player one plays M, it's great for them 00:15:55.660 --> 00:15:58.640 align:middle line:84% to play L So we can't rule out any strategies yet. 00:15:58.640 --> 00:16:00.980 align:middle line:90% So that's kind of step one. 00:16:00.980 --> 00:16:06.060 align:middle line:84% Now let's go to step two and impose a bit more-- slightly 00:16:06.060 --> 00:16:07.880 align:middle line:90% strengthen our assumptions. 00:16:07.880 --> 00:16:13.300 align:middle line:84% Let's now say in step two, we'll maintain our assumption 00:16:13.300 --> 00:16:15.620 align:middle line:84% that P1 and P2 are both rational. 00:16:15.620 --> 00:16:18.020 align:middle line:84% But we'll add the further assumption 00:16:18.020 --> 00:16:21.460 align:middle line:84% that each player knows that the other player is rational. 00:16:21.460 --> 00:16:25.020 align:middle line:84% So I'll say "and" to remind you that I'm maintaining 00:16:25.020 --> 00:16:32.340 align:middle line:84% the assumption up here, each knows that the other is 00:16:32.340 --> 00:16:33.040 align:middle line:90% rational. 00:16:33.040 --> 00:16:40.240 align:middle line:90% 00:16:40.240 --> 00:16:42.840 align:middle line:84% So our numbers are kind of off here. 00:16:42.840 --> 00:16:44.520 align:middle line:84% We could call that step zero, but this 00:16:44.520 --> 00:16:47.940 align:middle line:84% is first order knowledge of rationality here in step two. 00:16:47.940 --> 00:16:54.080 align:middle line:90% 00:16:54.080 --> 00:16:59.840 align:middle line:84% So now let's maybe write S1 prime, double prime, 00:16:59.840 --> 00:17:03.920 align:middle line:84% to be the strategies that player one could play. 00:17:03.920 --> 00:17:06.280 align:middle line:84% If player one is rational and knows that 00:17:06.280 --> 00:17:07.520 align:middle line:90% the other player is rational. 00:17:07.520 --> 00:17:12.720 align:middle line:84% So if player one knows that player two is rational, 00:17:12.720 --> 00:17:14.260 align:middle line:84% what can they conclude from that? 00:17:14.260 --> 00:17:17.000 align:middle line:90% 00:17:17.000 --> 00:17:19.127 align:middle line:84% Well, in this case, actually not much. 00:17:19.127 --> 00:17:20.960 align:middle line:84% They know that instead of the player playing 00:17:20.960 --> 00:17:23.537 align:middle line:84% something in here, they're going to play something in here. 00:17:23.537 --> 00:17:24.579 align:middle line:90% Well, those are the same. 00:17:24.579 --> 00:17:27.800 align:middle line:84% So actually in this case, player one's knowledge of player two's 00:17:27.800 --> 00:17:30.080 align:middle line:84% rationality doesn't change things. 00:17:30.080 --> 00:17:32.590 align:middle line:84% And we're actually going to get the same answer down here. 00:17:32.590 --> 00:17:35.870 align:middle line:84% We're going to get T and B. Because in this case, 00:17:35.870 --> 00:17:39.070 align:middle line:84% my knowledge of player two's rationality doesn't actually 00:17:39.070 --> 00:17:41.678 align:middle line:84% restrict the beliefs that I form. 00:17:41.678 --> 00:17:42.970 align:middle line:90% But let's do the second player. 00:17:42.970 --> 00:17:45.630 align:middle line:84% I think this will be a bit clearer. 00:17:45.630 --> 00:17:47.430 align:middle line:90% What about player two? 00:17:47.430 --> 00:17:51.950 align:middle line:84% So now that player two knows that player one is rational, 00:17:51.950 --> 00:17:54.830 align:middle line:84% how does that change player two's beliefs? 00:17:54.830 --> 00:17:55.655 align:middle line:90% Yeah? 00:17:55.655 --> 00:17:57.030 align:middle line:84% STUDENT: [INAUDIBLE] because they 00:17:57.030 --> 00:18:00.430 align:middle line:84% know that player one won't play M, so it's so not rational. 00:18:00.430 --> 00:18:01.170 align:middle line:90% IAN BALL: Great. 00:18:01.170 --> 00:18:03.050 align:middle line:90% So let's go through that slowly. 00:18:03.050 --> 00:18:08.050 align:middle line:84% So if I'm player two, if I know that player one is rational, 00:18:08.050 --> 00:18:10.690 align:middle line:84% I can reason through the way player one would behave. 00:18:10.690 --> 00:18:13.010 align:middle line:84% And I can say if player one is rational, 00:18:13.010 --> 00:18:15.430 align:middle line:84% they're only going to play T or B. Those are the only two 00:18:15.430 --> 00:18:17.088 align:middle line:90% strategies they could play. 00:18:17.088 --> 00:18:18.630 align:middle line:84% And then I'm going to look up and I'm 00:18:18.630 --> 00:18:22.550 align:middle line:84% going to say, OK, if player one plays T, then I'm 00:18:22.550 --> 00:18:27.030 align:middle line:84% better off playing R. And if player one plays B, 00:18:27.030 --> 00:18:29.230 align:middle line:84% then I'm also better off playing R. 00:18:29.230 --> 00:18:31.330 align:middle line:84% Now we have to be a little careful here 00:18:31.330 --> 00:18:33.450 align:middle line:84% because I might also think, well, 00:18:33.450 --> 00:18:36.250 align:middle line:84% maybe they'll play T with some probability and B 00:18:36.250 --> 00:18:37.510 align:middle line:90% with some probability. 00:18:37.510 --> 00:18:39.290 align:middle line:84% But it turns out even if they played those 00:18:39.290 --> 00:18:42.370 align:middle line:84% with some probability, you'd still be better off playing R. 00:18:42.370 --> 00:18:46.370 align:middle line:84% And the reason is that, well, R is strictly better than L 00:18:46.370 --> 00:18:49.330 align:middle line:84% given T, and strictly better than L given B. 00:18:49.330 --> 00:18:51.570 align:middle line:84% So it's not too hard to check that if they 00:18:51.570 --> 00:18:55.650 align:middle line:84% formed a belief about those two things, R would still be better. 00:18:55.650 --> 00:19:03.450 align:middle line:84% So now we go down to R. So what changed when we moved 00:19:03.450 --> 00:19:05.290 align:middle line:90% from step one to step two? 00:19:05.290 --> 00:19:07.370 align:middle line:84% Notice the set of strategies that we 00:19:07.370 --> 00:19:09.490 align:middle line:84% think player two might play, or could play, 00:19:09.490 --> 00:19:12.910 align:middle line:84% has gotten smaller, not because player two is rational. 00:19:12.910 --> 00:19:15.050 align:middle line:84% We already assumed player two is rational. 00:19:15.050 --> 00:19:17.290 align:middle line:84% But what further restricted our prediction, 00:19:17.290 --> 00:19:19.570 align:middle line:84% or sharpened our prediction, was our assumption 00:19:19.570 --> 00:19:22.570 align:middle line:84% that player two not only was rational herself, 00:19:22.570 --> 00:19:25.590 align:middle line:84% but knew that player one was also rational. 00:19:25.590 --> 00:19:27.970 align:middle line:84% And by thinking through how her opponent being rational 00:19:27.970 --> 00:19:31.200 align:middle line:84% might play, player two was able to restrict things. 00:19:31.200 --> 00:19:33.220 align:middle line:84% So now let's see if we can go one step further. 00:19:33.220 --> 00:19:38.400 align:middle line:90% 00:19:38.400 --> 00:19:40.140 align:middle line:84% So now it gets a bit cumbersome to write, 00:19:40.140 --> 00:19:43.880 align:middle line:84% but maybe I'll say step three is what? 00:19:43.880 --> 00:19:47.920 align:middle line:84% It's that, and each player knows that each other player 00:19:47.920 --> 00:19:49.820 align:middle line:84% knows that the other player is rational. 00:19:49.820 --> 00:19:52.400 align:middle line:84% This is a bit trickier to say, but-- 00:19:52.400 --> 00:20:03.920 align:middle line:84% and knows that the other knows that they are rational. 00:20:03.920 --> 00:20:06.420 align:middle line:90% It's a bit sloppy, but-- 00:20:06.420 --> 00:20:11.240 align:middle line:90% 00:20:11.240 --> 00:20:14.160 align:middle line:84% So if we read through this from player one's perspective, 00:20:14.160 --> 00:20:18.080 align:middle line:84% player one knows that the other player, player two 00:20:18.080 --> 00:20:21.000 align:middle line:84% knows that they, player one, are rational. 00:20:21.000 --> 00:20:25.040 align:middle line:84% And similarly, player two knows that player one knows 00:20:25.040 --> 00:20:27.080 align:middle line:90% that player two is rational. 00:20:27.080 --> 00:20:29.160 align:middle line:84% So we might-- if we want to write this in words, 00:20:29.160 --> 00:20:31.860 align:middle line:84% it's like one knows that two knows that one is rational, 00:20:31.860 --> 00:20:34.560 align:middle line:84% and two knows that one knows that two is rational. 00:20:34.560 --> 00:20:36.760 align:middle line:84% It's two different statements here, 00:20:36.760 --> 00:20:38.520 align:middle line:84% and the arrows indicate knowledge. 00:20:38.520 --> 00:20:41.500 align:middle line:84% This is that two knows that one is rational, 00:20:41.500 --> 00:20:44.300 align:middle line:84% and this is that one knows that two knows that one 00:20:44.300 --> 00:20:46.780 align:middle line:90% is rational, and so on. 00:20:46.780 --> 00:20:51.180 align:middle line:84% And maybe now we'll call this S1 triple prime 00:20:51.180 --> 00:20:54.000 align:middle line:90% and S2 triple prime. 00:20:54.000 --> 00:20:57.220 align:middle line:90% 00:20:57.220 --> 00:20:59.895 align:middle line:84% Well, S2 triple prime is pretty easy because player two, 00:20:59.895 --> 00:21:02.020 align:middle line:84% we've already narrowed it down to the only strategy 00:21:02.020 --> 00:21:02.760 align:middle line:90% they have left. 00:21:02.760 --> 00:21:07.460 align:middle line:84% So that's just still going to be R. That's not going to change. 00:21:07.460 --> 00:21:11.140 align:middle line:90% But what about player one? 00:21:11.140 --> 00:21:12.740 align:middle line:90% What is player one-- 00:21:12.740 --> 00:21:14.580 align:middle line:84% what is reasonable for player one 00:21:14.580 --> 00:21:19.000 align:middle line:84% to play given the second order knowledge of rationality? 00:21:19.000 --> 00:21:22.380 align:middle line:90% 00:21:22.380 --> 00:21:23.940 align:middle line:90% Thoughts? 00:21:23.940 --> 00:21:25.300 align:middle line:90% Yeah? 00:21:25.300 --> 00:21:26.200 align:middle line:90% B, right? 00:21:26.200 --> 00:21:31.400 align:middle line:84% Because now if player one knows that player two must play R, 00:21:31.400 --> 00:21:34.200 align:middle line:84% then they look up, and they say, well, if player two plays R, 00:21:34.200 --> 00:21:35.960 align:middle line:84% I've already actually underlined it. 00:21:35.960 --> 00:21:38.640 align:middle line:84% B is my best response to R. So if I 00:21:38.640 --> 00:21:43.080 align:middle line:84% know that player two is playing R, then my best response is B. 00:21:43.080 --> 00:21:45.100 align:middle line:84% And notice here the structure of this 00:21:45.100 --> 00:21:46.440 align:middle line:90% when I go through the steps. 00:21:46.440 --> 00:21:51.000 align:middle line:84% We write S1 triple prime and S2 triple prime next to each other. 00:21:51.000 --> 00:21:54.840 align:middle line:84% But what defines S1 triple prime is actually my knowledge 00:21:54.840 --> 00:21:59.160 align:middle line:84% that player two is playing a strategy in S2 double prime. 00:21:59.160 --> 00:22:02.880 align:middle line:84% And similarly, S2 triple prime is driven by player two's 00:22:02.880 --> 00:22:05.480 align:middle line:84% beliefs about what player one is going 00:22:05.480 --> 00:22:06.780 align:middle line:90% to play from S1 double prime. 00:22:06.780 --> 00:22:09.360 align:middle line:84% So we're always kind of looking back to the step before. 00:22:09.360 --> 00:22:13.240 align:middle line:84% The step before pinned down how my opponent would play. 00:22:13.240 --> 00:22:16.040 align:middle line:84% And now adding one other layer of knowledge 00:22:16.040 --> 00:22:18.740 align:middle line:84% tells me that I know my opponent will play that way, 00:22:18.740 --> 00:22:20.680 align:middle line:84% and that may allow me to further restrict 00:22:20.680 --> 00:22:23.640 align:middle line:84% the strategy that I use, this crisscross pattern as we 00:22:23.640 --> 00:22:25.310 align:middle line:90% go down. 00:22:25.310 --> 00:22:27.070 align:middle line:90% And now we're done. 00:22:27.070 --> 00:22:30.410 align:middle line:84% In this case, there's no strategies left to deal with, 00:22:30.410 --> 00:22:31.630 align:middle line:90% and we can check. 00:22:31.630 --> 00:22:33.830 align:middle line:84% Not a coincidence here that we have 00:22:33.830 --> 00:22:36.910 align:middle line:84% one strategy left at the end, one strategy profile 00:22:36.910 --> 00:22:38.950 align:middle line:90% left at the end, BR. 00:22:38.950 --> 00:22:43.830 align:middle line:84% And notice that both be that both numbers in that cell 00:22:43.830 --> 00:22:48.150 align:middle line:84% are underlined, which reminds us that B is a best response to R, 00:22:48.150 --> 00:22:51.630 align:middle line:84% and R is a best response to B, which makes sense 00:22:51.630 --> 00:22:54.870 align:middle line:84% and will hold a bit more generally. 00:22:54.870 --> 00:23:00.888 align:middle line:84% So notice here that here, this approach, or we 00:23:00.888 --> 00:23:03.430 align:middle line:84% might say rationalizability-- or we haven't defined that yet. 00:23:03.430 --> 00:23:13.350 align:middle line:84% But let's say here, we get a unique prediction. 00:23:13.350 --> 00:23:18.910 align:middle line:90% 00:23:18.910 --> 00:23:21.790 align:middle line:84% When we apply this algorithm, eventually we got to a point 00:23:21.790 --> 00:23:24.970 align:middle line:84% where each player only had one strategy left, 00:23:24.970 --> 00:23:26.690 align:middle line:90% but that doesn't always hold. 00:23:26.690 --> 00:23:29.290 align:middle line:84% So let's emphasize this is not always the case, 00:23:29.290 --> 00:23:30.310 align:middle line:90% as we'll see later. 00:23:30.310 --> 00:23:34.850 align:middle line:90% 00:23:34.850 --> 00:23:37.270 align:middle line:84% But when this is true, we have a name for it. 00:23:37.270 --> 00:23:40.330 align:middle line:84% So when we have a game where we can go through this procedure, 00:23:40.330 --> 00:23:43.590 align:middle line:84% and we get a unique prediction for each player's strategy, 00:23:43.590 --> 00:23:45.690 align:middle line:84% we call that game dominance solvable. 00:23:45.690 --> 00:23:48.150 align:middle line:84% So when this does happen, it's not guaranteed. 00:23:48.150 --> 00:23:53.830 align:middle line:84% But when it does happen, we say the game is dominance solvable. 00:23:53.830 --> 00:23:56.650 align:middle line:90% 00:23:56.650 --> 00:23:57.550 align:middle line:90% Underline this. 00:23:57.550 --> 00:24:02.210 align:middle line:90% 00:24:02.210 --> 00:24:04.210 align:middle line:84% Why do we call it dominance solvable? 00:24:04.210 --> 00:24:06.950 align:middle line:84% Well, simply by reasoning about dominance-- remember, 00:24:06.950 --> 00:24:09.190 align:middle line:84% we often reasoned about best responses, 00:24:09.190 --> 00:24:11.410 align:middle line:84% but that's equivalent to reasoning about dominance 00:24:11.410 --> 00:24:13.010 align:middle line:90% by our theorem from last week. 00:24:13.010 --> 00:24:14.690 align:middle line:84% So all we really needed to use is 00:24:14.690 --> 00:24:18.158 align:middle line:84% that players don't play strictly dominated strategies. 00:24:18.158 --> 00:24:20.450 align:middle line:84% They know their opponents don't play strictly dominated 00:24:20.450 --> 00:24:21.075 align:middle line:90% strategies. 00:24:21.075 --> 00:24:23.200 align:middle line:84% They know their opponents know that they don't play 00:24:23.200 --> 00:24:24.700 align:middle line:90% strictly dominated strategies. 00:24:24.700 --> 00:24:25.960 align:middle line:84% They know their opponents know that they 00:24:25.960 --> 00:24:27.560 align:middle line:84% know that their opponents know that they don't play strictly 00:24:27.560 --> 00:24:29.060 align:middle line:90% dominated strategies and so on. 00:24:29.060 --> 00:24:30.820 align:middle line:84% And simply by reasoning about dominance, 00:24:30.820 --> 00:24:32.920 align:middle line:84% we were able to, quote unquote, "solve the game," 00:24:32.920 --> 00:24:39.440 align:middle line:84% in the sense of identifying a unique prediction in the game. 00:24:39.440 --> 00:24:41.340 align:middle line:90% Any questions on this so far? 00:24:41.340 --> 00:24:45.760 align:middle line:90% 00:24:45.760 --> 00:24:47.660 align:middle line:84% So now we'd like to formalize this. 00:24:47.660 --> 00:24:49.180 align:middle line:84% We went through it in an example. 00:24:49.180 --> 00:24:53.120 align:middle line:84% But let's introduce the formal algorithm that is going 00:24:53.120 --> 00:24:54.840 align:middle line:90% to go through this procedure. 00:24:54.840 --> 00:24:57.700 align:middle line:84% So this algorithm is called rationalizability. 00:24:57.700 --> 00:25:00.320 align:middle line:90% 00:25:00.320 --> 00:25:03.440 align:middle line:90% Or it has two names. 00:25:03.440 --> 00:25:09.440 align:middle line:84% So it's either called rationalizability, 00:25:09.440 --> 00:25:11.440 align:middle line:84% and rationalize ability is really 00:25:11.440 --> 00:25:15.052 align:middle line:84% a statement about the strategies that survive this algorithm. 00:25:15.052 --> 00:25:16.760 align:middle line:84% So in this example, we would say strategy 00:25:16.760 --> 00:25:20.140 align:middle line:84% B is rationalizable for player B. 00:25:20.140 --> 00:25:23.920 align:middle line:84% And strategy R is rationalizable for player two. 00:25:23.920 --> 00:25:26.500 align:middle line:84% Sorry-- player one and player two. 00:25:26.500 --> 00:25:28.163 align:middle line:84% But the procedure we use is often 00:25:28.163 --> 00:25:29.580 align:middle line:84% goes by a different name, which is 00:25:29.580 --> 00:25:34.920 align:middle line:84% the iterated elimination of strictly dominated strategies. 00:25:34.920 --> 00:25:51.060 align:middle line:90% 00:25:51.060 --> 00:25:56.500 align:middle line:84% And sometimes this goes by the acronym iterated elimination 00:25:56.500 --> 00:25:58.900 align:middle line:84% of strictly dominated strategies because it's 00:25:58.900 --> 00:26:00.660 align:middle line:90% a bit of a mouthful. 00:26:00.660 --> 00:26:02.060 align:middle line:90% What's the idea? 00:26:02.060 --> 00:26:05.660 align:middle line:84% At each step, we implicitly threw out the strategies 00:26:05.660 --> 00:26:07.200 align:middle line:90% that were strictly dominated. 00:26:07.200 --> 00:26:09.720 align:middle line:84% We eliminated the strategies that were strictly dominated. 00:26:09.720 --> 00:26:12.443 align:middle line:84% And we did this recursively, or iteratively. 00:26:12.443 --> 00:26:14.860 align:middle line:84% And that's why we call it iterated elimination of strictly 00:26:14.860 --> 00:26:16.860 align:middle line:90% dominated strategies. 00:26:16.860 --> 00:26:18.480 align:middle line:84% To formally define this procedure, 00:26:18.480 --> 00:26:19.660 align:middle line:90% we need some notation. 00:26:19.660 --> 00:26:21.700 align:middle line:84% It was easy in this example to work through it. 00:26:21.700 --> 00:26:23.400 align:middle line:84% But to do it more generally we're going 00:26:23.400 --> 00:26:25.360 align:middle line:90% to need some general notation. 00:26:25.360 --> 00:26:29.000 align:middle line:84% So we really need two pieces of notation to start. 00:26:29.000 --> 00:26:33.000 align:middle line:90% So let's consider player i. 00:26:33.000 --> 00:26:41.800 align:middle line:84% We're given player i and some collection of strategy profiles 00:26:41.800 --> 00:26:43.000 align:middle line:90% of the other players. 00:26:43.000 --> 00:26:45.120 align:middle line:84% So we're given player i, and I'll 00:26:45.120 --> 00:26:53.440 align:middle line:84% say S negative i prime, which is a subset of S negative i. 00:26:53.440 --> 00:26:57.780 align:middle line:84% So remember, S negative i is the collection of all profiles, 00:26:57.780 --> 00:27:00.320 align:middle line:84% or we might say partial profiles, of strategies 00:27:00.320 --> 00:27:02.200 align:middle line:90% of my opponents. 00:27:02.200 --> 00:27:05.800 align:middle line:84% And S negative i prime is just an arbitrary subset 00:27:05.800 --> 00:27:08.420 align:middle line:84% of those strategies, of those strategy profiles. 00:27:08.420 --> 00:27:10.360 align:middle line:84% So it's just some collection of strategies 00:27:10.360 --> 00:27:12.920 align:middle line:90% that my opponents might play. 00:27:12.920 --> 00:27:15.400 align:middle line:84% And now we want to define two things. 00:27:15.400 --> 00:27:22.470 align:middle line:84% The first thing is BRi of S negative i prime. 00:27:22.470 --> 00:27:25.310 align:middle line:84% So what this is going to be is the set 00:27:25.310 --> 00:27:28.950 align:middle line:84% of strategies for player i that are 00:27:28.950 --> 00:27:33.990 align:middle line:84% best responses to some belief about my opponent's strategies 00:27:33.990 --> 00:27:37.110 align:middle line:84% that only puts positive probability on strategies 00:27:37.110 --> 00:27:37.808 align:middle line:90% in here. 00:27:37.808 --> 00:27:38.850 align:middle line:90% So let's go through this. 00:27:38.850 --> 00:27:42.190 align:middle line:84% Remember, we said we have to think through what beliefs are 00:27:42.190 --> 00:27:43.910 align:middle line:84% reasonable, and we only want to keep 00:27:43.910 --> 00:27:46.470 align:middle line:84% track of strategies that are best responses 00:27:46.470 --> 00:27:48.190 align:middle line:90% to certain reasonable beliefs. 00:27:48.190 --> 00:27:50.447 align:middle line:84% So let's define this a bit more formally. 00:27:50.447 --> 00:27:52.030 align:middle line:84% First of all, the key thing about this 00:27:52.030 --> 00:27:58.670 align:middle line:84% is these are strategies for player I. So here, as I said, 00:27:58.670 --> 00:28:01.390 align:middle line:84% I'm going to use some notation that maybe we 00:28:01.390 --> 00:28:03.030 align:middle line:90% haven't done before. 00:28:03.030 --> 00:28:05.310 align:middle line:90% I'm defining a set here. 00:28:05.310 --> 00:28:07.630 align:middle line:84% And the way I'm building up the set 00:28:07.630 --> 00:28:11.030 align:middle line:84% using set builder notation is I'm going to say this set 00:28:11.030 --> 00:28:13.830 align:middle line:84% contains every strategy little si 00:28:13.830 --> 00:28:18.490 align:middle line:84% in the set big Si that satisfies some additional property. 00:28:18.490 --> 00:28:20.910 align:middle line:84% So this is kind of the universe of things I'm considering. 00:28:20.910 --> 00:28:23.690 align:middle line:84% And then I'm going to impose some constraints over here that 00:28:23.690 --> 00:28:28.770 align:middle line:84% narrow down which elements of this set I actually allow for. 00:28:28.770 --> 00:28:31.650 align:middle line:84% So what does it mean for Si to be in here? 00:28:31.650 --> 00:28:34.810 align:middle line:84% Well, a strategy Si is in this set 00:28:34.810 --> 00:28:47.250 align:middle line:84% if Si is a best response to some belief, beta negative i. 00:28:47.250 --> 00:28:48.890 align:middle line:84% Now if I just stop there, we'd be 00:28:48.890 --> 00:28:50.150 align:middle line:90% back to what we did last week. 00:28:50.150 --> 00:28:51.692 align:middle line:84% We would just be thinking about being 00:28:51.692 --> 00:28:53.970 align:middle line:90% a best response to some belief. 00:28:53.970 --> 00:28:57.450 align:middle line:84% But now I want to impose the restriction on this belief. 00:28:57.450 --> 00:29:00.730 align:middle line:84% I want to say it's a best response to a belief that only 00:29:00.730 --> 00:29:04.370 align:middle line:84% puts positive probability on strategies in this set, not 00:29:04.370 --> 00:29:05.730 align:middle line:90% any strategy. 00:29:05.730 --> 00:29:10.690 align:middle line:84% So on some belief beta negative i is satisfying. 00:29:10.690 --> 00:29:12.230 align:middle line:84% Well, what's a way of saying this? 00:29:12.230 --> 00:29:15.630 align:middle line:84% Well, this belief assigns some probability 00:29:15.630 --> 00:29:18.590 align:middle line:84% to the opponent's strategies being in this set. 00:29:18.590 --> 00:29:22.190 align:middle line:84% And I'm going to require that that probability is 1. 00:29:22.190 --> 00:29:28.210 align:middle line:84% So beta negative i of S negative i prime equals 1. 00:29:28.210 --> 00:29:32.070 align:middle line:90% 00:29:32.070 --> 00:29:34.550 align:middle line:84% So it's not enough for a strategy 00:29:34.550 --> 00:29:36.710 align:middle line:84% to be a best response to some belief. 00:29:36.710 --> 00:29:40.470 align:middle line:84% It has to be a best response to a particular kind of belief, 00:29:40.470 --> 00:29:44.710 align:middle line:84% a belief that maybe puts-- it could put probabilities 00:29:44.710 --> 00:29:46.530 align:middle line:84% on different strategies in this set, 00:29:46.530 --> 00:29:49.110 align:middle line:84% but it can't put any probability on strategies 00:29:49.110 --> 00:29:50.630 align:middle line:90% that are outside this set. 00:29:50.630 --> 00:29:52.790 align:middle line:84% So the interpretation of this is if I 00:29:52.790 --> 00:29:55.750 align:middle line:84% know that my opponents are only going to play strategies 00:29:55.750 --> 00:30:00.510 align:middle line:84% in this set, what strategies for me are reasonable? 00:30:00.510 --> 00:30:02.310 align:middle line:84% And it's exactly the strategies that 00:30:02.310 --> 00:30:04.230 align:middle line:84% are best responses to a belief that 00:30:04.230 --> 00:30:06.770 align:middle line:84% doesn't put any probability outside that set, 00:30:06.770 --> 00:30:09.350 align:middle line:84% or equivalently, puts all the probability 00:30:09.350 --> 00:30:11.610 align:middle line:90% on the strategies in this set. 00:30:11.610 --> 00:30:15.700 align:middle line:90% 00:30:15.700 --> 00:30:19.540 align:middle line:84% Now, we said from last week that there 00:30:19.540 --> 00:30:21.120 align:middle line:84% are two ways of thinking about this. 00:30:21.120 --> 00:30:24.300 align:middle line:84% We can think about what I should play and what I shouldn't play. 00:30:24.300 --> 00:30:27.220 align:middle line:84% So now we want to look at the other side of the coin, which 00:30:27.220 --> 00:30:29.980 align:middle line:84% is, what are the strategies that are strictly dominated 00:30:29.980 --> 00:30:32.380 align:middle line:90% by some mixed strategy? 00:30:32.380 --> 00:30:33.520 align:middle line:90% So let's go here. 00:30:33.520 --> 00:30:39.540 align:middle line:90% 00:30:39.540 --> 00:30:45.020 align:middle line:84% So what this stands for is strictly 00:30:45.020 --> 00:30:46.720 align:middle line:90% conditionally dominated. 00:30:46.720 --> 00:30:54.620 align:middle line:90% 00:30:54.620 --> 00:30:57.020 align:middle line:84% So last week we talked about what 00:30:57.020 --> 00:31:00.140 align:middle line:84% it means for a strategy to be strictly dominated 00:31:00.140 --> 00:31:01.500 align:middle line:90% by another strategy. 00:31:01.500 --> 00:31:03.940 align:middle line:84% And it means some other strategy does better however 00:31:03.940 --> 00:31:05.860 align:middle line:90% my opponent's play. 00:31:05.860 --> 00:31:08.465 align:middle line:84% But now we want to be a bit more permissive. 00:31:08.465 --> 00:31:10.340 align:middle line:84% Again, we're going to have the interpretation 00:31:10.340 --> 00:31:14.600 align:middle line:84% that I know that my opponent's strategies lie in here. 00:31:14.600 --> 00:31:18.075 align:middle line:84% So if I know that my opponent's strategies are in here, 00:31:18.075 --> 00:31:20.700 align:middle line:84% for dominance, I'm not going to require a strategy to do better 00:31:20.700 --> 00:31:22.520 align:middle line:90% however my opponent's play. 00:31:22.520 --> 00:31:25.700 align:middle line:84% It's going to be enough if that strategy does better however 00:31:25.700 --> 00:31:28.360 align:middle line:84% my opponent's play in this restricted set. 00:31:28.360 --> 00:31:30.500 align:middle line:84% And that's what the conditional dominance means. 00:31:30.500 --> 00:31:32.760 align:middle line:84% It means conditional on my opponent's playing 00:31:32.760 --> 00:31:35.680 align:middle line:90% a strategy in this set. 00:31:35.680 --> 00:31:38.880 align:middle line:84% Conditional dominance means one strategy always does strictly 00:31:38.880 --> 00:31:40.380 align:middle line:90% better than the other strategy. 00:31:40.380 --> 00:31:43.560 align:middle line:84% So let me write it in words, and then we'll be a bit more formal. 00:31:43.560 --> 00:31:49.600 align:middle line:84% So again, this is a collection of strategies si and big Si, 00:31:49.600 --> 00:31:58.540 align:middle line:84% such that Si is strictly conditionally dominated. 00:31:58.540 --> 00:32:07.440 align:middle line:90% 00:32:07.440 --> 00:32:09.395 align:middle line:84% Now, if I say conditionally dominated, 00:32:09.395 --> 00:32:10.770 align:middle line:84% strictly conditionally dominated, 00:32:10.770 --> 00:32:12.730 align:middle line:84% I have to say what it's conditional on. 00:32:12.730 --> 00:32:15.710 align:middle line:84% So strictly conditionally dominated 00:32:15.710 --> 00:32:20.910 align:middle line:84% I'll say given S negative i prime. 00:32:20.910 --> 00:32:21.970 align:middle line:90% So that's my condition. 00:32:21.970 --> 00:32:24.270 align:middle line:84% The condition is I know my opponents are going 00:32:24.270 --> 00:32:26.630 align:middle line:84% to play something in S negative i prime, 00:32:26.630 --> 00:32:29.110 align:middle line:84% and I'm interested in conditional dominance given 00:32:29.110 --> 00:32:30.463 align:middle line:90% that condition. 00:32:30.463 --> 00:32:32.130 align:middle line:84% So it's strictly conditionally dominated 00:32:32.130 --> 00:32:34.270 align:middle line:90% given S negative i by what? 00:32:34.270 --> 00:32:38.930 align:middle line:90% By some mixed strategy. 00:32:38.930 --> 00:32:41.910 align:middle line:90% 00:32:41.910 --> 00:32:43.630 align:middle line:84% And I think it's often tricky to get-- 00:32:43.630 --> 00:32:45.350 align:middle line:90% it's easy to get confused here. 00:32:45.350 --> 00:32:48.590 align:middle line:84% Is it dominated by a mixed strategy of player i's or player 00:32:48.590 --> 00:32:49.333 align:middle line:90% i's opponents? 00:32:49.333 --> 00:32:50.250 align:middle line:90% What should come here? 00:32:50.250 --> 00:32:57.730 align:middle line:90% 00:32:57.730 --> 00:32:58.230 align:middle line:90% Yeah? 00:32:58.230 --> 00:32:58.670 align:middle line:90% STUDENT: Player i. 00:32:58.670 --> 00:32:59.910 align:middle line:90% IAN BALL: Player i, right? 00:32:59.910 --> 00:33:02.550 align:middle line:84% So it's always player i's strategies 00:33:02.550 --> 00:33:04.850 align:middle line:84% that dominate player I's other strategies. 00:33:04.850 --> 00:33:06.850 align:middle line:84% The beliefs are about the opponent's strategies, 00:33:06.850 --> 00:33:08.410 align:middle line:84% but the dominance relation is always 00:33:08.410 --> 00:33:10.350 align:middle line:84% between strategies by the same player, 00:33:10.350 --> 00:33:13.030 align:middle line:84% because the player is only comparing their own strategies. 00:33:13.030 --> 00:33:16.270 align:middle line:84% So by some mixed strategy sigma negative i. 00:33:16.270 --> 00:33:21.890 align:middle line:90% 00:33:21.890 --> 00:33:23.490 align:middle line:84% So here's the definition, or here's 00:33:23.490 --> 00:33:24.760 align:middle line:90% the definition of this set. 00:33:24.760 --> 00:33:27.010 align:middle line:84% But I've of cheated a bit because I've used a new word 00:33:27.010 --> 00:33:28.490 align:middle line:90% that I haven't yet defined. 00:33:28.490 --> 00:33:30.110 align:middle line:84% So now I need to define that new word. 00:33:30.110 --> 00:33:33.630 align:middle line:84% What does it mean for Si to be strictly conditionally dominated 00:33:33.630 --> 00:33:37.090 align:middle line:84% given S negative i prime by some mixed strategy, sigma i? 00:33:37.090 --> 00:33:39.010 align:middle line:84% Well, we already said it intuitively. 00:33:39.010 --> 00:33:42.530 align:middle line:84% It means that whatever strategy profile my opponents 00:33:42.530 --> 00:33:46.810 align:middle line:84% play in this set, sigma i is strictly better for me as player 00:33:46.810 --> 00:33:49.650 align:middle line:90% than strategy Si. 00:33:49.650 --> 00:33:52.370 align:middle line:84% And now we just need to translate this intuitive idea 00:33:52.370 --> 00:33:54.210 align:middle line:90% into math. 00:33:54.210 --> 00:34:01.730 align:middle line:84% So I'm comparing, as player i, what I get from sigma i 00:34:01.730 --> 00:34:05.010 align:middle line:90% to what I get from Si. 00:34:05.010 --> 00:34:07.970 align:middle line:84% And again here, we're using our lowercase u's 00:34:07.970 --> 00:34:09.550 align:middle line:84% even though what I'm plugging in here 00:34:09.550 --> 00:34:11.810 align:middle line:84% is not a pure strategy but a mixed strategy. 00:34:11.810 --> 00:34:14.429 align:middle line:84% But remember what we said at the beginning, 00:34:14.429 --> 00:34:16.350 align:middle line:84% the interpretation is that this is just 00:34:16.350 --> 00:34:20.510 align:middle line:84% my expected utility over the lotteries induced 00:34:20.510 --> 00:34:22.670 align:middle line:90% by this mixed strategy. 00:34:22.670 --> 00:34:25.909 align:middle line:84% So we're comparing sigma i to Si. 00:34:25.909 --> 00:34:28.909 align:middle line:84% We want Si to be dominated by sigma i. 00:34:28.909 --> 00:34:31.389 align:middle line:84% So we know the inequality has to go this way. 00:34:31.389 --> 00:34:34.772 align:middle line:84% And it's strict dominance, so it has to be a strict inequality. 00:34:34.772 --> 00:34:36.230 align:middle line:84% But we're not going to require this 00:34:36.230 --> 00:34:38.370 align:middle line:84% for every strategy of my opponents. 00:34:38.370 --> 00:34:40.670 align:middle line:84% We're only going to require it for the strategies that 00:34:40.670 --> 00:34:41.989 align:middle line:90% satisfy the condition. 00:34:41.989 --> 00:34:46.130 align:middle line:84% So this is going to be S negative i 00:34:46.130 --> 00:34:52.830 align:middle line:84% S negative i, but for which S negative i for all S negative i 00:34:52.830 --> 00:34:57.090 align:middle line:84% that are elements of S negative i prime. 00:34:57.090 --> 00:35:02.790 align:middle line:90% 00:35:02.790 --> 00:35:04.860 align:middle line:84% Now if there's two players this is pretty easy. 00:35:04.860 --> 00:35:07.698 align:middle line:84% S negative i is just a strategy of my opponent. 00:35:07.698 --> 00:35:09.240 align:middle line:84% But if there's more than two players, 00:35:09.240 --> 00:35:11.460 align:middle line:84% remember, say there's three players, 00:35:11.460 --> 00:35:14.800 align:middle line:84% then S negative i-- let's say i is 1. 00:35:14.800 --> 00:35:18.300 align:middle line:84% S negative i specifies what strategy player two chooses 00:35:18.300 --> 00:35:21.620 align:middle line:84% and what strategy player three chooses. 00:35:21.620 --> 00:35:24.060 align:middle line:84% And in this case, S negative i prime 00:35:24.060 --> 00:35:29.260 align:middle line:84% is a collection of strategy pairs chosen by players 00:35:29.260 --> 00:35:30.660 align:middle line:90% two and player three. 00:35:30.660 --> 00:35:32.280 align:middle line:84% They're still choosing separately, 00:35:32.280 --> 00:35:33.780 align:middle line:84% but we're just putting them together 00:35:33.780 --> 00:35:37.300 align:middle line:90% mathematically into a vector. 00:35:37.300 --> 00:35:40.760 align:middle line:90% Are there any questions on this? 00:35:40.760 --> 00:35:44.140 align:middle line:84% I think this is maybe the heaviest definition we've done. 00:35:44.140 --> 00:35:45.080 align:middle line:90% Any questions? 00:35:45.080 --> 00:35:47.780 align:middle line:90% 00:35:47.780 --> 00:35:50.820 align:middle line:90% OK, great. 00:35:50.820 --> 00:35:55.100 align:middle line:84% So now let's try to-- maybe I'll move to a new board. 00:35:55.100 --> 00:36:00.060 align:middle line:84% Let's try to recall what our theorem said 00:36:00.060 --> 00:36:02.600 align:middle line:90% last week in this new notation. 00:36:02.600 --> 00:36:12.080 align:middle line:90% 00:36:12.080 --> 00:36:13.780 align:middle line:84% So often when we introduce new notation, 00:36:13.780 --> 00:36:15.720 align:middle line:84% it becomes a lot easier to express ideas 00:36:15.720 --> 00:36:17.560 align:middle line:90% that we've already expressed. 00:36:17.560 --> 00:36:27.000 align:middle line:84% So let's recall the theorem from last week. 00:36:27.000 --> 00:36:29.920 align:middle line:84% Well, what we said is that a strategy is either 00:36:29.920 --> 00:36:32.140 align:middle line:84% a best response or it's strictly dominated. 00:36:32.140 --> 00:36:33.080 align:middle line:90% It's not both. 00:36:33.080 --> 00:36:35.000 align:middle line:84% But remember, last week we didn't 00:36:35.000 --> 00:36:37.780 align:middle line:84% worry about strict conditional dominance. 00:36:37.780 --> 00:36:39.920 align:middle line:84% We didn't worry about restricting beliefs. 00:36:39.920 --> 00:36:43.400 align:middle line:84% So it was a special case where S negative i prime was just 00:36:43.400 --> 00:36:44.240 align:middle line:90% S negative i. 00:36:44.240 --> 00:36:46.440 align:middle line:84% There wasn't any restriction on these things. 00:36:46.440 --> 00:36:51.520 align:middle line:84% So our theorem from last week basically said two things. 00:36:51.520 --> 00:37:02.020 align:middle line:84% It said Si equals BRi of S negative i union SCD 00:37:02.020 --> 00:37:05.540 align:middle line:90% i of S negative i. 00:37:05.540 --> 00:37:09.040 align:middle line:84% Let's try to go through this in words. 00:37:09.040 --> 00:37:11.940 align:middle line:84% And first, let's think about what the union means. 00:37:11.940 --> 00:37:14.700 align:middle line:90% This is a set of strategies. 00:37:14.700 --> 00:37:16.500 align:middle line:90% This is a set of strategies. 00:37:16.500 --> 00:37:18.220 align:middle line:84% When I take the union, I'm thinking 00:37:18.220 --> 00:37:19.927 align:middle line:84% of the larger set that puts together 00:37:19.927 --> 00:37:21.760 align:middle line:84% things that are in either of those two sets. 00:37:21.760 --> 00:37:23.552 align:middle line:84% So it's like I have two bags of strategies, 00:37:23.552 --> 00:37:25.580 align:middle line:84% and I pour them together into one big bag that 00:37:25.580 --> 00:37:27.700 align:middle line:90% contains all of the strategies. 00:37:27.700 --> 00:37:34.860 align:middle line:84% So in words this says, every strategy in Si 00:37:34.860 --> 00:37:39.180 align:middle line:84% is either a best response to some belief. 00:37:39.180 --> 00:37:42.060 align:middle line:84% Notice, here I'm not-- there's no prime here. 00:37:42.060 --> 00:37:45.180 align:middle line:84% So this is a best response to an unrestricted belief. 00:37:45.180 --> 00:37:46.740 align:middle line:84% It would only have a restriction if I 00:37:46.740 --> 00:37:50.900 align:middle line:84% chose as S prime a smaller set of strategies. 00:37:50.900 --> 00:37:55.980 align:middle line:84% So another way of saying this is if we go to this definition, 00:37:55.980 --> 00:37:59.890 align:middle line:84% well, if S negative i prime is just S negative i, then the fact 00:37:59.890 --> 00:38:02.490 align:middle line:84% that beta negative i of S negative i equals 1 00:38:02.490 --> 00:38:03.890 align:middle line:90% is not a restriction at all. 00:38:03.890 --> 00:38:06.450 align:middle line:84% Because every belief assigns probability 1 00:38:06.450 --> 00:38:09.330 align:middle line:90% to the set of all strategies. 00:38:09.330 --> 00:38:14.490 align:middle line:84% So every strategy is either a best response to some belief, 00:38:14.490 --> 00:38:16.510 align:middle line:90% or it's strictly dominated. 00:38:16.510 --> 00:38:20.090 align:middle line:84% And again, this means strictly conditionally dominated. 00:38:20.090 --> 00:38:22.370 align:middle line:84% But strictly conditionally dominated 00:38:22.370 --> 00:38:24.730 align:middle line:84% conditional on any possible strategy 00:38:24.730 --> 00:38:26.710 align:middle line:84% just reduces to strict dominance. 00:38:26.710 --> 00:38:28.570 align:middle line:84% So strict dominance is a special case 00:38:28.570 --> 00:38:30.330 align:middle line:90% of strict conditional dominance. 00:38:30.330 --> 00:38:32.190 align:middle line:84% But this is only one part of the theorem. 00:38:32.190 --> 00:38:33.730 align:middle line:90% What else did our theorem say? 00:38:33.730 --> 00:38:35.570 align:middle line:84% I mean, this says that every strategy 00:38:35.570 --> 00:38:37.865 align:middle line:90% is in one of these two sets. 00:38:37.865 --> 00:38:39.490 align:middle line:84% But it said one more-- there's one more 00:38:39.490 --> 00:38:41.450 align:middle line:90% thing we know about these sets. 00:38:41.450 --> 00:38:42.090 align:middle line:90% Yeah? 00:38:42.090 --> 00:38:43.970 align:middle line:84% STUDENT: Each strategy is exactly one of the sets. 00:38:43.970 --> 00:38:44.630 align:middle line:90% IAN BALL: Exactly one. 00:38:44.630 --> 00:38:45.430 align:middle line:90% So what's missing here? 00:38:45.430 --> 00:38:47.013 align:middle line:84% What haven't we-- what does this allow 00:38:47.013 --> 00:38:48.945 align:middle line:90% for that we want to rule out? 00:38:48.945 --> 00:38:51.070 align:middle line:84% STUDENT: The last four do not have either function. 00:38:51.070 --> 00:38:53.550 align:middle line:84% IAN BALL: But what we know is that that can't happen. 00:38:53.550 --> 00:38:57.110 align:middle line:84% So what we also know is that the intersection is empty. 00:38:57.110 --> 00:39:00.850 align:middle line:84% That is, no strategy is in both of these sets. 00:39:00.850 --> 00:39:05.470 align:middle line:84% So this is our "at least one" statement. 00:39:05.470 --> 00:39:06.710 align:middle line:90% And now we're going to-- 00:39:06.710 --> 00:39:11.470 align:middle line:84% just using the notation we said last week, 00:39:11.470 --> 00:39:14.110 align:middle line:84% this statement says that every strategy is in at least one 00:39:14.110 --> 00:39:15.450 align:middle line:90% of these two sets. 00:39:15.450 --> 00:39:16.950 align:middle line:84% But our second part of our statement 00:39:16.950 --> 00:39:20.190 align:middle line:84% was that every strategy was in at most one. 00:39:20.190 --> 00:39:22.070 align:middle line:84% What does it mean to be in at most one? 00:39:22.070 --> 00:39:24.590 align:middle line:90% It means you're not in both. 00:39:24.590 --> 00:39:31.670 align:middle line:84% So what that means is BRi of S negative i intersection SCD of S 00:39:31.670 --> 00:39:34.550 align:middle line:90% negative i is empty. 00:39:34.550 --> 00:39:37.430 align:middle line:84% And the math notation for that, I'll just write this. 00:39:37.430 --> 00:39:40.670 align:middle line:84% So again, this is a collection of strategies. 00:39:40.670 --> 00:39:42.410 align:middle line:84% This is a collection of strategies. 00:39:42.410 --> 00:39:43.830 align:middle line:90% I should have an i here. 00:39:43.830 --> 00:39:45.470 align:middle line:84% And all this notation says in math 00:39:45.470 --> 00:39:48.950 align:middle line:84% is that no strategy is in both of these sets. 00:39:48.950 --> 00:39:51.050 align:middle line:84% The intersection of these two sets is empty. 00:39:51.050 --> 00:40:03.214 align:middle line:90% 00:40:03.214 --> 00:40:06.660 align:middle line:84% It turns out, though, that this holds more generally. 00:40:06.660 --> 00:40:11.780 align:middle line:84% So last week we had the special case with S negative i, 00:40:11.780 --> 00:40:14.900 align:middle line:84% where we didn't put any restrictions on these beliefs. 00:40:14.900 --> 00:40:20.100 align:middle line:84% It turns out this holds more generally if we put primes here. 00:40:20.100 --> 00:40:23.560 align:middle line:90% So this holds for any subset. 00:40:23.560 --> 00:40:42.540 align:middle line:90% 00:40:42.540 --> 00:40:46.020 align:middle line:90% So let's work through this. 00:40:46.020 --> 00:40:48.240 align:middle line:90% We started with S negative i. 00:40:48.240 --> 00:40:50.440 align:middle line:84% Now we're using a smaller set of strategies. 00:40:50.440 --> 00:40:53.640 align:middle line:84% So let's try to understand what happens to these two sets. 00:40:53.640 --> 00:40:58.840 align:middle line:90% 00:40:58.840 --> 00:41:03.200 align:middle line:84% If S negative i prime gets smaller, 00:41:03.200 --> 00:41:05.000 align:middle line:84% so I'm considering fewer strategies 00:41:05.000 --> 00:41:09.440 align:middle line:84% that my opponents might play, what happens to these two sets? 00:41:09.440 --> 00:41:13.120 align:middle line:84% So this best response set-- does this smaller or bigger 00:41:13.120 --> 00:41:15.925 align:middle line:84% as S negative i prime gets smaller? 00:41:15.925 --> 00:41:16.800 align:middle line:90% STUDENT: [INAUDIBLE]. 00:41:16.800 --> 00:41:18.222 align:middle line:90% IAN BALL: Say again. 00:41:18.222 --> 00:41:19.180 align:middle line:90% I think someone had it. 00:41:19.180 --> 00:41:20.360 align:middle line:90% Yeah, smaller, right? 00:41:20.360 --> 00:41:21.820 align:middle line:90% So this gets smaller. 00:41:21.820 --> 00:41:22.380 align:middle line:90% Why? 00:41:22.380 --> 00:41:26.420 align:middle line:84% Well, I'm not allowed as many beliefs right. 00:41:26.420 --> 00:41:28.720 align:middle line:84% This is the set of all strategies that are best 00:41:28.720 --> 00:41:30.800 align:middle line:90% response to certain beliefs. 00:41:30.800 --> 00:41:33.320 align:middle line:84% If I shrink the set S negative i prime, 00:41:33.320 --> 00:41:35.680 align:middle line:84% it says I'm narrowing down the strategies 00:41:35.680 --> 00:41:38.320 align:middle line:84% that my opponents might play, which narrows down 00:41:38.320 --> 00:41:40.298 align:middle line:90% the beliefs I might hold. 00:41:40.298 --> 00:41:41.840 align:middle line:84% So how many strategies do I have that 00:41:41.840 --> 00:41:44.440 align:middle line:84% are best responses to this smaller, narrower set 00:41:44.440 --> 00:41:45.260 align:middle line:90% of beliefs? 00:41:45.260 --> 00:41:46.460 align:middle line:90% Fewer strategies. 00:41:46.460 --> 00:41:49.720 align:middle line:84% So this set is going to get smaller. 00:41:49.720 --> 00:41:50.920 align:middle line:90% What about this set? 00:41:50.920 --> 00:41:52.380 align:middle line:84% As S negative i prime gets smaller, 00:41:52.380 --> 00:41:54.740 align:middle line:84% what's going to happen to this set? 00:41:54.740 --> 00:41:56.132 align:middle line:90% It's got to get bigger. 00:41:56.132 --> 00:41:57.840 align:middle line:84% Well, one way-- I'll answer in a second-- 00:41:57.840 --> 00:42:01.353 align:middle line:84% One way to see this is well, the union is staying the same size. 00:42:01.353 --> 00:42:03.520 align:middle line:84% So if one gets smaller, the other has to get bigger. 00:42:03.520 --> 00:42:06.740 align:middle line:84% But let's think through it mathematically. 00:42:06.740 --> 00:42:10.500 align:middle line:84% A strategy is dominated by another strategy 00:42:10.500 --> 00:42:12.980 align:middle line:84% if that other strategy is just better, no matter what 00:42:12.980 --> 00:42:14.820 align:middle line:90% happens in here. 00:42:14.820 --> 00:42:18.860 align:middle line:84% It gets easier to be dominated if there are fewer strategies 00:42:18.860 --> 00:42:20.580 align:middle line:84% that my opponents could be playing. 00:42:20.580 --> 00:42:22.780 align:middle line:84% So it's easier to be conditionally dominated 00:42:22.780 --> 00:42:25.020 align:middle line:90% if the condition is stronger. 00:42:25.020 --> 00:42:29.140 align:middle line:84% One way of thinking about it is just counting the inequalities. 00:42:29.140 --> 00:42:31.740 align:middle line:84% If S negative i prime is a really big set, 00:42:31.740 --> 00:42:33.620 align:middle line:84% we have a lot of inequalities here. 00:42:33.620 --> 00:42:35.900 align:middle line:84% These inequalities have to hold for every element 00:42:35.900 --> 00:42:37.340 align:middle line:90% of this big set. 00:42:37.340 --> 00:42:39.400 align:middle line:84% If Si prime becomes a smaller set, 00:42:39.400 --> 00:42:40.940 align:middle line:90% there are fewer things to check. 00:42:40.940 --> 00:42:43.020 align:middle line:84% It's easier to be dominated conditional 00:42:43.020 --> 00:42:44.640 align:middle line:90% on a smaller set of strategies. 00:42:44.640 --> 00:42:47.940 align:middle line:84% And therefore, the set of things that are dominated gets larger. 00:42:47.940 --> 00:42:49.320 align:middle line:90% That's what we see here. 00:42:49.320 --> 00:42:55.090 align:middle line:90% 00:42:55.090 --> 00:42:58.290 align:middle line:84% And in the extreme case-- maybe one thing to think through. 00:42:58.290 --> 00:43:03.730 align:middle line:84% In the extreme case, what if this just has a single element? 00:43:03.730 --> 00:43:05.350 align:middle line:90% So special case. 00:43:05.350 --> 00:43:22.810 align:middle line:90% 00:43:22.810 --> 00:43:25.930 align:middle line:84% So what if S negative i prime contains a single strategy 00:43:25.930 --> 00:43:27.830 align:middle line:90% profile S negative i prime. 00:43:27.830 --> 00:43:30.690 align:middle line:84% So this is big S negative i prime. 00:43:30.690 --> 00:43:34.090 align:middle line:84% This is little s negative i prime. 00:43:34.090 --> 00:43:35.510 align:middle line:90% Then what happens to this set? 00:43:35.510 --> 00:43:37.670 align:middle line:84% What is this set going to be in words? 00:43:37.670 --> 00:43:42.170 align:middle line:90% 00:43:42.170 --> 00:43:43.530 align:middle line:90% Yeah? 00:43:43.530 --> 00:43:45.917 align:middle line:84% STUDENT: Maybe [INAUDIBLE] be dominant. 00:43:45.917 --> 00:43:46.750 align:middle line:90% IAN BALL: Not quite. 00:43:46.750 --> 00:43:48.125 align:middle line:84% So it's actually just going to be 00:43:48.125 --> 00:43:51.105 align:middle line:84% the strategies that are best responses to this strategy. 00:43:51.105 --> 00:43:53.230 align:middle line:84% So remember, before we got into all this fancy sets 00:43:53.230 --> 00:43:56.430 align:middle line:84% of strategies, we just defined what it is for one strategy 00:43:56.430 --> 00:43:59.310 align:middle line:84% to be a best response to another strategy. 00:43:59.310 --> 00:44:02.310 align:middle line:84% So if this set just has a single element in it, 00:44:02.310 --> 00:44:05.110 align:middle line:84% it just says this is the collection of strategies 00:44:05.110 --> 00:44:08.310 align:middle line:84% that are best responses to this particular strategy 00:44:08.310 --> 00:44:10.407 align:middle line:90% profile of my opponent. 00:44:10.407 --> 00:44:11.490 align:middle line:90% But then this gets tricky. 00:44:11.490 --> 00:44:13.750 align:middle line:84% Then that means this is everything that's not a best 00:44:13.750 --> 00:44:16.087 align:middle line:90% response to this strategy. 00:44:16.087 --> 00:44:17.170 align:middle line:90% So how does that work out? 00:44:17.170 --> 00:44:19.890 align:middle line:84% Why is everything else strictly dominated then? 00:44:19.890 --> 00:44:20.990 align:middle line:90% That seems a bit tricky. 00:44:20.990 --> 00:44:24.390 align:middle line:84% That seems like we have a lot of strategies in there. 00:44:24.390 --> 00:44:28.350 align:middle line:84% Remember, strict conditional dominance given one strategy 00:44:28.350 --> 00:44:29.990 align:middle line:90% is really simple. 00:44:29.990 --> 00:44:34.070 align:middle line:84% All it means to be in this set is that some other strategy 00:44:34.070 --> 00:44:38.110 align:middle line:84% does strictly better given this strategy of my opponents. 00:44:38.110 --> 00:44:41.030 align:middle line:84% But that's exactly the same as saying this strategy is not 00:44:41.030 --> 00:44:41.697 align:middle line:90% a best response. 00:44:41.697 --> 00:44:43.613 align:middle line:84% If there's only one strategy of my opponent's, 00:44:43.613 --> 00:44:45.990 align:middle line:84% either a strategy is a best response to my opponent's 00:44:45.990 --> 00:44:49.260 align:middle line:84% strategy or it's not, meaning another strategy 00:44:49.260 --> 00:44:52.013 align:middle line:84% does strictly better given that strategy of my opponents. 00:44:52.013 --> 00:44:54.180 align:middle line:84% And that's exactly what strict conditional dominance 00:44:54.180 --> 00:44:55.423 align:middle line:90% means in this special case. 00:44:55.423 --> 00:44:56.840 align:middle line:84% That may have been a little quick. 00:44:56.840 --> 00:44:57.780 align:middle line:90% Sorry. 00:44:57.780 --> 00:44:58.840 align:middle line:90% Any questions here? 00:44:58.840 --> 00:44:59.500 align:middle line:90% Yes? 00:44:59.500 --> 00:45:04.540 align:middle line:84% STUDENT: So is it always strict that every time we 00:45:04.540 --> 00:45:08.020 align:middle line:90% basically make Si-- 00:45:08.020 --> 00:45:13.500 align:middle line:84% when we shrink the S negative i prime, that the Ri S negative i 00:45:13.500 --> 00:45:16.020 align:middle line:90% prime is going to shrink? 00:45:16.020 --> 00:45:17.888 align:middle line:84% Or could it sometimes stay the same? 00:45:17.888 --> 00:45:19.680 align:middle line:84% IAN BALL: It could sometimes stay the same. 00:45:19.680 --> 00:45:23.720 align:middle line:84% And, I mean, one way to see that is-- 00:45:23.720 --> 00:45:25.520 align:middle line:84% well, it's going to end eventually. 00:45:25.520 --> 00:45:27.080 align:middle line:84% So eventually, we're not going to be able to take things out. 00:45:27.080 --> 00:45:28.340 align:middle line:90% That's one way of seeing it. 00:45:28.340 --> 00:45:28.840 align:middle line:90% But you're right. 00:45:28.840 --> 00:45:30.423 align:middle line:84% We'll go through an example like this. 00:45:30.423 --> 00:45:36.820 align:middle line:84% So it's not always the case that at each step things will shrink. 00:45:36.820 --> 00:45:39.673 align:middle line:84% But once nothing is thrown out, once everything 00:45:39.673 --> 00:45:41.340 align:middle line:84% stays the same for one round, then we're 00:45:41.340 --> 00:45:42.760 align:middle line:84% basically done with the algorithm. 00:45:42.760 --> 00:45:44.120 align:middle line:84% But we'll talk about that in a second. 00:45:44.120 --> 00:45:44.700 align:middle line:90% Yeah. 00:45:44.700 --> 00:45:45.200 align:middle line:90% OK. 00:45:45.200 --> 00:45:49.320 align:middle line:84% Any more questions about these relations here? 00:45:49.320 --> 00:45:51.520 align:middle line:84% So now that we have our notation, 00:45:51.520 --> 00:45:53.668 align:middle line:84% we can formally define our algorithm. 00:45:53.668 --> 00:45:54.460 align:middle line:90% It took us a while. 00:45:54.460 --> 00:45:56.260 align:middle line:90% We started by motivating it. 00:45:56.260 --> 00:45:57.350 align:middle line:90% We gave these definitions. 00:45:57.350 --> 00:45:59.600 align:middle line:84% But with these definitions, we can write the algorithm 00:45:59.600 --> 00:46:00.860 align:middle line:90% in a really short clean way. 00:46:00.860 --> 00:46:02.500 align:middle line:90% So let's go over this. 00:46:02.500 --> 00:46:22.340 align:middle line:90% 00:46:22.340 --> 00:46:23.980 align:middle line:90% So here's our algorithm. 00:46:23.980 --> 00:46:27.480 align:middle line:90% 00:46:27.480 --> 00:46:30.160 align:middle line:84% Maybe I'll write it kind in a more computer sciencey way. 00:46:30.160 --> 00:46:36.160 align:middle line:84% So what's our input to the algorithm? 00:46:36.160 --> 00:46:39.160 align:middle line:84% The input is just a strategic form game. 00:46:39.160 --> 00:46:40.740 align:middle line:84% And I'll be a little more precise-- 00:46:40.740 --> 00:46:42.940 align:middle line:84% let's say a finite strategic form game. 00:46:42.940 --> 00:46:55.124 align:middle line:90% 00:46:55.124 --> 00:46:57.620 align:middle line:84% And if I just write it like we did at the beginning, 00:46:57.620 --> 00:46:59.780 align:middle line:90% it's just a function u. 00:46:59.780 --> 00:47:03.020 align:middle line:90% And maybe I'll write S to Rn. 00:47:03.020 --> 00:47:04.620 align:middle line:84% And in a sense, all of game theory 00:47:04.620 --> 00:47:07.460 align:middle line:84% is just studying functions like this. 00:47:07.460 --> 00:47:09.780 align:middle line:84% It's a function that says for every strategy 00:47:09.780 --> 00:47:13.580 align:middle line:84% profile the players choose, what are all the players payoffs? 00:47:13.580 --> 00:47:15.700 align:middle line:84% And this is a vector that tells me 00:47:15.700 --> 00:47:18.260 align:middle line:84% player one's payoff, player two's payoff, all the way up 00:47:18.260 --> 00:47:19.460 align:middle line:90% to player n's payoff. 00:47:19.460 --> 00:47:21.337 align:middle line:84% And you can see this simple function gives me 00:47:21.337 --> 00:47:22.420 align:middle line:90% everything I need to know. 00:47:22.420 --> 00:47:26.500 align:middle line:84% Because the n up here tells me how many players there are. 00:47:26.500 --> 00:47:30.480 align:middle line:84% The S here tells me all the strategy profiles, 00:47:30.480 --> 00:47:33.940 align:middle line:84% and therefore, all the strategy sets for each of the n players. 00:47:33.940 --> 00:47:36.560 align:middle line:84% And then the function itself tells me all the payoffs. 00:47:36.560 --> 00:47:39.112 align:middle line:84% So that's really all I need to know. 00:47:39.112 --> 00:47:40.820 align:middle line:84% Maybe I'll put a star by finite, and I'll 00:47:40.820 --> 00:47:44.970 align:middle line:84% say, what we often do throughout the course is when 00:47:44.970 --> 00:47:47.610 align:middle line:84% we talk about formal results, we assume 00:47:47.610 --> 00:47:50.850 align:middle line:84% things are finite to make sure nothing bad happens. 00:47:50.850 --> 00:47:52.970 align:middle line:84% The reason we do that is when you move away 00:47:52.970 --> 00:47:55.690 align:middle line:84% from the finite case, there are some weird cases 00:47:55.690 --> 00:47:57.130 align:middle line:90% that can go wrong. 00:47:57.130 --> 00:47:59.735 align:middle line:84% But then we'll discuss a lot of examples that aren't finite 00:47:59.735 --> 00:48:02.110 align:middle line:84% but where things are well behaved and nothing goes wrong. 00:48:02.110 --> 00:48:05.310 align:middle line:84% So we will apply these ideas to the infinite case, 00:48:05.310 --> 00:48:07.452 align:middle line:84% but we just don't want to make any formal claims 00:48:07.452 --> 00:48:09.410 align:middle line:84% about the infinite case because you can come up 00:48:09.410 --> 00:48:11.730 align:middle line:84% with some really weird infinite discontinuous games 00:48:11.730 --> 00:48:12.910 align:middle line:90% where bad things happen. 00:48:12.910 --> 00:48:17.010 align:middle line:84% But we will-- if on an exam I say, 00:48:17.010 --> 00:48:19.272 align:middle line:84% compute the rationalizable strategies in this game, 00:48:19.272 --> 00:48:21.230 align:middle line:84% you can't just say, oh, it's not a finite game. 00:48:21.230 --> 00:48:22.330 align:middle line:90% I'm not going to do it. 00:48:22.330 --> 00:48:23.580 align:middle line:90% You can't get out that easily. 00:48:23.580 --> 00:48:26.110 align:middle line:84% So we are going to apply these ideas to infinite games. 00:48:26.110 --> 00:48:29.050 align:middle line:84% But I'll formally define it here for the finite case. 00:48:29.050 --> 00:48:33.790 align:middle line:84% So we're going to define things recursively. 00:48:33.790 --> 00:48:36.990 align:middle line:84% So let's start with our-- maybe I'll call it the base case. 00:48:36.990 --> 00:48:39.272 align:middle line:90% 00:48:39.272 --> 00:48:41.730 align:middle line:84% And the idea is we just want to go through the algorithm we 00:48:41.730 --> 00:48:42.470 align:middle line:90% did before. 00:48:42.470 --> 00:48:45.070 align:middle line:84% We want to start with all the strategies. 00:48:45.070 --> 00:48:47.190 align:middle line:84% And at each point, we want to throw out strategies 00:48:47.190 --> 00:48:48.830 align:middle line:84% that we don't think players will play. 00:48:48.830 --> 00:48:51.070 align:middle line:84% And we want to whittle down the set of strategies 00:48:51.070 --> 00:48:54.030 align:middle line:84% until we get to the good ones at the end. 00:48:54.030 --> 00:48:55.170 align:middle line:90% So the base case-- 00:48:55.170 --> 00:48:57.870 align:middle line:90% 00:48:57.870 --> 00:49:00.310 align:middle line:90% we'll index things by k. 00:49:00.310 --> 00:49:02.710 align:middle line:84% So I is always going to index players. 00:49:02.710 --> 00:49:06.470 align:middle line:84% k will indicate the steps in this algorithm. 00:49:06.470 --> 00:49:10.470 align:middle line:84% So with k equals 0, well, we just say 00:49:10.470 --> 00:49:17.155 align:middle line:84% Si0 equals Si for all I equals 1 through n. 00:49:17.155 --> 00:49:18.530 align:middle line:84% So we haven't done anything here. 00:49:18.530 --> 00:49:22.150 align:middle line:84% We've just said at step zero, what strategies are left? 00:49:22.150 --> 00:49:24.350 align:middle line:84% Well, the strategies that we started with. 00:49:24.350 --> 00:49:26.273 align:middle line:84% And that's true for each of the n players. 00:49:26.273 --> 00:49:27.190 align:middle line:90% So we've done nothing. 00:49:27.190 --> 00:49:31.170 align:middle line:84% So far, we've just formally gotten the algorithm going. 00:49:31.170 --> 00:49:32.410 align:middle line:90% We've initialized things. 00:49:32.410 --> 00:49:34.950 align:middle line:90% 00:49:34.950 --> 00:49:36.770 align:middle line:84% And now let's look at the inductive step. 00:49:36.770 --> 00:49:41.572 align:middle line:90% 00:49:41.572 --> 00:49:44.130 align:middle line:84% And if you haven't taken an algorithms course 00:49:44.130 --> 00:49:48.170 align:middle line:84% or seen proofs by induction or seen recursive definitions, 00:49:48.170 --> 00:49:53.370 align:middle line:84% it just means after we start with k0, 00:49:53.370 --> 00:49:56.265 align:middle line:84% well, next we're going to go to step one. 00:49:56.265 --> 00:49:57.890 align:middle line:84% And then we're going to go to step two. 00:49:57.890 --> 00:49:59.470 align:middle line:84% And then we're going to go to step three. 00:49:59.470 --> 00:50:01.762 align:middle line:84% But it takes a long time to write out all those things. 00:50:01.762 --> 00:50:03.910 align:middle line:84% So we're going to reason at an arbitrary step. 00:50:03.910 --> 00:50:05.410 align:middle line:84% We're going to say, let's say we've 00:50:05.410 --> 00:50:07.610 align:middle line:90% gone through steps up to k. 00:50:07.610 --> 00:50:10.295 align:middle line:84% How are we going to perform step k plus 1? 00:50:10.295 --> 00:50:12.170 align:middle line:84% And if you know how to go from k to k plus 1, 00:50:12.170 --> 00:50:14.650 align:middle line:90% well, we already know zero. 00:50:14.650 --> 00:50:17.410 align:middle line:84% So then we can go from 0 to 1, and then 1 to 2, 00:50:17.410 --> 00:50:19.530 align:middle line:90% and then 2 to 3, and so on. 00:50:19.530 --> 00:50:21.170 align:middle line:90% So the inductive step-- 00:50:21.170 --> 00:50:22.990 align:middle line:90% well, what's our starting point? 00:50:22.990 --> 00:50:27.090 align:middle line:84% We're imagining that we've already defined these sets up 00:50:27.090 --> 00:50:28.250 align:middle line:90% to stage k. 00:50:28.250 --> 00:50:32.010 align:middle line:90% So what we have is we're given-- 00:50:32.010 --> 00:50:33.610 align:middle line:90% maybe I'll write it out-- 00:50:33.610 --> 00:50:40.360 align:middle line:90% S1k all the way up to Snk. 00:50:40.360 --> 00:50:42.400 align:middle line:84% So these are the strategies for player 1 00:50:42.400 --> 00:50:46.193 align:middle line:84% that have survived the rounds up to round k. 00:50:46.193 --> 00:50:48.360 align:middle line:84% I guess we're starting at 0, so it's really k plus 1 00:50:48.360 --> 00:50:50.520 align:middle line:84% and them, but that rounds up to k. 00:50:50.520 --> 00:50:53.880 align:middle line:84% And Snk is the set of strategies for player n 00:50:53.880 --> 00:50:56.040 align:middle line:84% that have survived so far up to round k. 00:50:56.040 --> 00:50:59.240 align:middle line:84% And what we want to do is define which strategies are going 00:50:59.240 --> 00:51:03.720 align:middle line:90% to be left at round k plus 1. 00:51:03.720 --> 00:51:05.880 align:middle line:90% So what do we need to do? 00:51:05.880 --> 00:51:15.800 align:middle line:84% We're going to define maybe here, say, for each i, Sik 00:51:15.800 --> 00:51:16.582 align:middle line:90% plus 1. 00:51:16.582 --> 00:51:18.040 align:middle line:84% So before I write down the formula, 00:51:18.040 --> 00:51:19.720 align:middle line:84% let's just understand what we have. 00:51:19.720 --> 00:51:22.560 align:middle line:90% We have S1k through Snk. 00:51:22.560 --> 00:51:28.800 align:middle line:84% We want to define Sik plus 1 for every player I. So Sik plus 1 00:51:28.800 --> 00:51:31.360 align:middle line:84% is the set of strategies for player i 00:51:31.360 --> 00:51:34.060 align:middle line:84% that survive up to round k plus 1. 00:51:34.060 --> 00:51:36.920 align:middle line:90% 00:51:36.920 --> 00:51:39.580 align:middle line:84% Any ideas based on the notation we've done 00:51:39.580 --> 00:51:42.680 align:middle line:90% given so far what could be here? 00:51:42.680 --> 00:51:45.780 align:middle line:84% I mean, intuitively, we want to keep in all the strategies 00:51:45.780 --> 00:51:49.460 align:middle line:84% that are best responses to some belief about our opponents 00:51:49.460 --> 00:51:52.940 align:middle line:84% that's consistent with what we've calculated so far. 00:51:52.940 --> 00:51:55.820 align:middle line:84% So what we know at stage k is that it's only 00:51:55.820 --> 00:51:58.460 align:middle line:84% reasonable for player one to play something in here, 00:51:58.460 --> 00:52:00.940 align:middle line:84% all the way up for player n to play something in here. 00:52:00.940 --> 00:52:03.460 align:middle line:84% So given that knowledge that player i has, 00:52:03.460 --> 00:52:05.540 align:middle line:84% what is reasonable for player i to play 00:52:05.540 --> 00:52:08.740 align:middle line:90% at this round of the algorithm? 00:52:08.740 --> 00:52:11.820 align:middle line:90% Any thoughts? 00:52:11.820 --> 00:52:12.720 align:middle line:90% Yeah? 00:52:12.720 --> 00:52:18.780 align:middle line:84% STUDENT: [INAUDIBLE] i of S minus the other-- 00:52:18.780 --> 00:52:19.780 align:middle line:90% IAN BALL: Minus i, yeah. 00:52:19.780 --> 00:52:21.740 align:middle line:90% STUDENT: To the k. 00:52:21.740 --> 00:52:22.780 align:middle line:90% IAN BALL: Exactly right. 00:52:22.780 --> 00:52:29.860 align:middle line:84% So this is going to be BRi of S negative i to the k. 00:52:29.860 --> 00:52:32.120 align:middle line:84% Now let me first define what this is. 00:52:32.120 --> 00:52:33.508 align:middle line:84% So I think we know this notation. 00:52:33.508 --> 00:52:34.300 align:middle line:90% But let's be clear. 00:52:34.300 --> 00:52:42.050 align:middle line:84% This is something like S1k all the way up to Si minus 1k times 00:52:42.050 --> 00:52:46.270 align:middle line:90% Si plus 1k Snk. 00:52:46.270 --> 00:52:48.290 align:middle line:84% I think that makes it seem fancier than it is. 00:52:48.290 --> 00:52:53.410 align:middle line:84% All it means is I know the strategy set 00:52:53.410 --> 00:52:55.510 align:middle line:84% at the k-th round for each of my opponents, 00:52:55.510 --> 00:52:57.350 align:middle line:84% and I just want to put all those together, 00:52:57.350 --> 00:52:59.050 align:middle line:90% making sure to skip myself. 00:52:59.050 --> 00:53:01.450 align:middle line:84% Because I'm player i, I don't need to form beliefs 00:53:01.450 --> 00:53:02.930 align:middle line:90% about my own strategies. 00:53:02.930 --> 00:53:05.370 align:middle line:84% So this is-- you can think of it as-- 00:53:05.370 --> 00:53:09.170 align:middle line:84% or it exactly is the set of strategy profiles 00:53:09.170 --> 00:53:11.930 align:middle line:84% of my opponent, such that each player's strategy 00:53:11.930 --> 00:53:15.570 align:middle line:84% in that profile lies in the corresponding strategy set 00:53:15.570 --> 00:53:17.190 align:middle line:84% that we got at the end of round k. 00:53:17.190 --> 00:53:20.170 align:middle line:90% 00:53:20.170 --> 00:53:22.690 align:middle line:84% And more mathematically, from our definition here, 00:53:22.690 --> 00:53:25.010 align:middle line:84% these are the set of strategies for player i 00:53:25.010 --> 00:53:27.890 align:middle line:84% that are best responses to some belief player 00:53:27.890 --> 00:53:31.010 align:middle line:84% i could hold that only puts positive probability 00:53:31.010 --> 00:53:34.437 align:middle line:84% on strategies that are in this special restricted set. 00:53:34.437 --> 00:53:36.270 align:middle line:84% Now, there's another definition we could use 00:53:36.270 --> 00:53:37.650 align:middle line:90% using our theorem over here. 00:53:37.650 --> 00:53:39.710 align:middle line:84% How else could we define this step that 00:53:39.710 --> 00:53:41.010 align:middle line:90% would give us the same answer? 00:53:41.010 --> 00:53:43.850 align:middle line:90% 00:53:43.850 --> 00:53:44.350 align:middle line:90% Yeah? 00:53:44.350 --> 00:53:48.230 align:middle line:84% STUDENT: We could define the set as basically Si, 00:53:48.230 --> 00:53:55.670 align:middle line:84% and you take out the [INAUDIBLE] SCD i S negative i to the k. 00:53:55.670 --> 00:53:56.670 align:middle line:90% IAN BALL: Exactly. 00:53:56.670 --> 00:53:59.490 align:middle line:84% So just like we always said, there's two ways we can do it. 00:53:59.490 --> 00:54:02.950 align:middle line:84% We can keep in what's good, or we can throw out what's bad. 00:54:02.950 --> 00:54:07.990 align:middle line:84% So we can also define it as everything in Si 00:54:07.990 --> 00:54:10.230 align:middle line:84% except-- so when I use this set minus thing, 00:54:10.230 --> 00:54:11.970 align:middle line:84% I want to say the set of things-- 00:54:11.970 --> 00:54:14.950 align:middle line:84% so everything that's in here, but not in here. 00:54:14.950 --> 00:54:20.470 align:middle line:84% And that's SCD i of S negative ik. 00:54:20.470 --> 00:54:23.430 align:middle line:90% 00:54:23.430 --> 00:54:25.230 align:middle line:84% And our theorem over here exactly 00:54:25.230 --> 00:54:28.390 align:middle line:84% tells us that those two are the same. 00:54:28.390 --> 00:54:32.070 align:middle line:84% So you could always use this definition. 00:54:32.070 --> 00:54:33.990 align:middle line:84% You could always use this definition. 00:54:33.990 --> 00:54:36.170 align:middle line:84% It's not going to make a difference. 00:54:36.170 --> 00:54:39.370 align:middle line:84% This definition is maybe more consistent with the name 00:54:39.370 --> 00:54:42.170 align:middle line:84% "iterated elimination of strictly dominated strategies." 00:54:42.170 --> 00:54:44.090 align:middle line:84% That's literally what we're doing here. 00:54:44.090 --> 00:54:47.130 align:middle line:84% But we could also say iterated retention 00:54:47.130 --> 00:54:48.498 align:middle line:90% of best response strategies. 00:54:48.498 --> 00:54:50.290 align:middle line:84% And that's basically what we're doing here. 00:54:50.290 --> 00:54:51.457 align:middle line:90% Though, I just made that up. 00:54:51.457 --> 00:54:52.970 align:middle line:90% That's not a name that's used. 00:54:52.970 --> 00:54:54.350 align:middle line:84% But that's one thing we could do. 00:54:54.350 --> 00:54:57.550 align:middle line:90% 00:54:57.550 --> 00:54:58.050 align:middle line:90% Great. 00:54:58.050 --> 00:55:01.490 align:middle line:84% So we're not done yet because all I've 00:55:01.490 --> 00:55:04.050 align:middle line:84% told you is how to go through each step. 00:55:04.050 --> 00:55:05.170 align:middle line:90% Step-- oh, Yes 00:55:05.170 --> 00:55:08.330 align:middle line:84% STUDENT: Why isn't the S sub i S sub i to the k? 00:55:08.330 --> 00:55:11.490 align:middle line:90% 00:55:11.490 --> 00:55:12.890 align:middle line:90% IAN BALL: Here? 00:55:12.890 --> 00:55:14.450 align:middle line:90% That's a good point. 00:55:14.450 --> 00:55:17.530 align:middle line:84% It turns out you could do that, and you get the same answer. 00:55:17.530 --> 00:55:21.848 align:middle line:84% So in the infinite case, you have to be a little careful. 00:55:21.848 --> 00:55:23.390 align:middle line:84% But in the finite case, you're right. 00:55:23.390 --> 00:55:27.130 align:middle line:84% If you put Si to the k here, you would get the same answer. 00:55:27.130 --> 00:55:32.820 align:middle line:84% I think conceptually, I like to think-- 00:55:32.820 --> 00:55:35.240 align:middle line:84% I think this is conceptually more-- 00:55:35.240 --> 00:55:37.598 align:middle line:84% better captures it, that here I am. 00:55:37.598 --> 00:55:38.140 align:middle line:90% I'm player i. 00:55:38.140 --> 00:55:40.680 align:middle line:84% All I know at this stage is that this 00:55:40.680 --> 00:55:42.200 align:middle line:84% is what the players could do, and I 00:55:42.200 --> 00:55:44.580 align:middle line:84% want to throw out things that are strictly dominated. 00:55:44.580 --> 00:55:47.600 align:middle line:84% So it doesn't-- there's no reason for me to really think 00:55:47.600 --> 00:55:48.860 align:middle line:90% back to what I did before. 00:55:48.860 --> 00:55:49.960 align:middle line:84% So I think conceptually maybe this 00:55:49.960 --> 00:55:51.700 align:middle line:84% is more in the spirit of the algorithm. 00:55:51.700 --> 00:55:53.800 align:middle line:84% But you're exactly right, that mathematically this 00:55:53.800 --> 00:55:56.520 align:middle line:90% is going to be a subset of Sik. 00:55:56.520 --> 00:55:59.420 align:middle line:84% And if I put Sik here, and often when we do the algorithm, 00:55:59.420 --> 00:56:02.720 align:middle line:84% that's kind of what we implicitly do in our heads. 00:56:02.720 --> 00:56:06.087 align:middle line:84% So you could put the k here, and it wouldn't-- in the finite 00:56:06.087 --> 00:56:07.420 align:middle line:90% case, it wouldn't change things. 00:56:07.420 --> 00:56:08.980 align:middle line:84% You have to be a bit careful in the infinite case. 00:56:08.980 --> 00:56:09.480 align:middle line:90% Yeah. 00:56:09.480 --> 00:56:15.960 align:middle line:90% 00:56:15.960 --> 00:56:16.780 align:middle line:90% Great. 00:56:16.780 --> 00:56:20.832 align:middle line:90% So we've defined these steps. 00:56:20.832 --> 00:56:22.540 align:middle line:84% But in general, this could go on forever. 00:56:22.540 --> 00:56:24.248 align:middle line:84% So if we wanted to be really formal, then 00:56:24.248 --> 00:56:27.742 align:middle line:84% what is the output of the algorithm? 00:56:27.742 --> 00:56:29.200 align:middle line:84% Well the output of the algorithm is 00:56:29.200 --> 00:56:31.700 align:middle line:84% we think of which strategies remain after we 00:56:31.700 --> 00:56:33.540 align:middle line:90% do this infinitely many times. 00:56:33.540 --> 00:56:35.520 align:middle line:84% It's a little weird to think about. 00:56:35.520 --> 00:56:39.460 align:middle line:84% But the output technically we might call Si infinity. 00:56:39.460 --> 00:56:41.660 align:middle line:84% And this is the set of strategies that 00:56:41.660 --> 00:56:43.130 align:middle line:90% survive every single round. 00:56:43.130 --> 00:56:44.880 align:middle line:84% So if we want to be really mathy about it, 00:56:44.880 --> 00:56:52.820 align:middle line:84% we would say this is Si0, intersect Si1, intersect Si2, 00:56:52.820 --> 00:56:55.180 align:middle line:90% intersect, and dot, dot, dot. 00:56:55.180 --> 00:56:58.220 align:middle line:84% And we could write that as the intersection from k 00:56:58.220 --> 00:57:02.500 align:middle line:90% equals 0 to infinity of Sik. 00:57:02.500 --> 00:57:04.080 align:middle line:84% So formally what does the set mean? 00:57:04.080 --> 00:57:06.420 align:middle line:84% It means I only include a strategy 00:57:06.420 --> 00:57:10.900 align:middle line:84% in this set if it survives round zero, it survives round one, 00:57:10.900 --> 00:57:13.340 align:middle line:84% it survives round two, it survives round three, 00:57:13.340 --> 00:57:15.460 align:middle line:90% and on forever. 00:57:15.460 --> 00:57:24.180 align:middle line:84% And we call this the set of rationalizable strategies 00:57:24.180 --> 00:57:25.080 align:middle line:90% for player i. 00:57:25.080 --> 00:57:31.677 align:middle line:90% 00:57:31.677 --> 00:57:33.260 align:middle line:84% So if you wanted to be really precise, 00:57:33.260 --> 00:57:36.280 align:middle line:84% we might say that iterated elimination of strictly 00:57:36.280 --> 00:57:39.800 align:middle line:84% dominated strategies is an algorithm or a procedure, 00:57:39.800 --> 00:57:41.760 align:middle line:84% and that the output of that procedure 00:57:41.760 --> 00:57:43.580 align:middle line:84% is the set of rationalizable strategies. 00:57:43.580 --> 00:57:47.120 align:middle line:84% You might say rationalizabilty is a property of a strategy. 00:57:47.120 --> 00:57:50.573 align:middle line:84% And IESDS is a procedure that we go through 00:57:50.573 --> 00:57:52.240 align:middle line:84% in order to identify the strategies that 00:57:52.240 --> 00:57:53.400 align:middle line:90% have that property. 00:57:53.400 --> 00:57:55.440 align:middle line:84% But on exams, we're not going to be too picky 00:57:55.440 --> 00:57:57.900 align:middle line:84% about exactly this distinction between the algorithm 00:57:57.900 --> 00:58:00.190 align:middle line:90% and the output of the algorithm. 00:58:00.190 --> 00:58:01.440 align:middle line:90% But you might say, well, wait. 00:58:01.440 --> 00:58:03.080 align:middle line:90% We have no hope of doing this. 00:58:03.080 --> 00:58:04.980 align:middle line:84% We have to go through infinitely many steps. 00:58:04.980 --> 00:58:06.600 align:middle line:90% That's a mess. 00:58:06.600 --> 00:58:12.240 align:middle line:84% So fortunately, when we're in a finite game, 00:58:12.240 --> 00:58:15.840 align:middle line:84% the algorithm is always going to end after finitely many steps. 00:58:15.840 --> 00:58:17.560 align:middle line:90% And what do I mean by that? 00:58:17.560 --> 00:58:24.260 align:middle line:84% So I'll say fortunately, this ends after finitely many steps. 00:58:24.260 --> 00:58:32.910 align:middle line:90% 00:58:32.910 --> 00:58:34.050 align:middle line:90% Now, what does ending mean? 00:58:34.050 --> 00:58:37.150 align:middle line:84% It doesn't mean I only have one strategy left for each player. 00:58:37.150 --> 00:58:40.615 align:middle line:84% That happened in our example, but that doesn't always happen. 00:58:40.615 --> 00:58:42.990 align:middle line:84% But what ending means, kind of going back to the question 00:58:42.990 --> 00:58:45.430 align:middle line:84% earlier, is I get to a point where I can't throw out 00:58:45.430 --> 00:58:46.950 align:middle line:90% any more strategies. 00:58:46.950 --> 00:58:49.370 align:middle line:84% And if I don't throw any more out at one round, 00:58:49.370 --> 00:58:50.670 align:middle line:84% then I'm not going to be able to throw any more out 00:58:50.670 --> 00:58:51.650 align:middle line:90% at the next round. 00:58:51.650 --> 00:58:53.650 align:middle line:84% And basically, the algorithm is just going to become static. 00:58:53.650 --> 00:58:55.567 align:middle line:84% There's going to be nothing else that happens. 00:58:55.567 --> 00:58:58.730 align:middle line:84% So formally, what it means is for some big K-- 00:58:58.730 --> 00:59:01.270 align:middle line:90% 00:59:01.270 --> 00:59:06.550 align:middle line:84% so after big K steps, subsequent steps don't change anything. 00:59:06.550 --> 00:59:12.830 align:middle line:84% So I'll say for some big K, SiK-- 00:59:12.830 --> 00:59:14.470 align:middle line:90% that's a big K-- 00:59:14.470 --> 00:59:23.770 align:middle line:84% equals SiK plus 1 equals SiK plus 2, and so on for all i. 00:59:23.770 --> 00:59:27.250 align:middle line:90% 00:59:27.250 --> 00:59:29.490 align:middle line:84% So what that says is once I get to stage 00:59:29.490 --> 00:59:34.630 align:middle line:84% K, if I were to apply another step of the algorithm, 00:59:34.630 --> 00:59:36.072 align:middle line:90% nothing would change. 00:59:36.072 --> 00:59:38.530 align:middle line:84% And if I were to apply one more step, nothing would change. 00:59:38.530 --> 00:59:40.490 align:middle line:84% How do you know you've gotten to that point? 00:59:40.490 --> 00:59:42.730 align:middle line:84% Well, as long as you get to a step 00:59:42.730 --> 00:59:46.970 align:middle line:84% where nothing is thrown out, you're done. 00:59:46.970 --> 00:59:48.370 align:middle line:90% But let's be really careful. 00:59:48.370 --> 00:59:50.890 align:middle line:84% It's not enough for there to be one player where 00:59:50.890 --> 00:59:52.370 align:middle line:90% you throw nothing out. 00:59:52.370 --> 00:59:54.630 align:middle line:84% If you throw strategies out for any player, 00:59:54.630 --> 00:59:55.970 align:middle line:90% you might have to keep going. 00:59:55.970 --> 00:59:59.130 align:middle line:84% So the algorithm is only done when every single player's 00:59:59.130 --> 01:00:03.130 align:middle line:84% strategy set stays the same from one round to the next. 01:00:03.130 --> 01:00:09.490 align:middle line:84% You can even see in our example here, when we went from step one 01:00:09.490 --> 01:00:11.650 align:middle line:84% to step two, player two's strategy 01:00:11.650 --> 01:00:14.930 align:middle line:84% set-- sorry-- player one's strategy set did not change. 01:00:14.930 --> 01:00:17.730 align:middle line:84% But the algorithm wasn't done because player two's strategy 01:00:17.730 --> 01:00:18.870 align:middle line:90% set did change. 01:00:18.870 --> 01:00:20.890 align:middle line:84% And for that reason, we had to do one more step. 01:00:20.890 --> 01:00:22.730 align:middle line:84% So it's only when everyone's strategy set 01:00:22.730 --> 01:00:24.260 align:middle line:90% is fixed that you're done. 01:00:24.260 --> 01:00:27.840 align:middle line:90% 01:00:27.840 --> 01:00:31.620 align:middle line:84% So keep going until nothing changes, and then you're done. 01:00:31.620 --> 01:00:33.740 align:middle line:84% And what's left are the rationalizable strategies. 01:00:33.740 --> 01:00:35.940 align:middle line:84% Now, let me give a tip, I think, for exams, 01:00:35.940 --> 01:00:39.680 align:middle line:84% for problem sets when you're actually doing this on your own. 01:00:39.680 --> 01:00:42.020 align:middle line:84% This algorithm has a-- or before I do the tip, 01:00:42.020 --> 01:00:44.302 align:middle line:84% let me say, how do we know that it 01:00:44.302 --> 01:00:46.760 align:middle line:84% has to end if we want to be a little mathy and think, wait. 01:00:46.760 --> 01:00:48.020 align:middle line:90% How can we be sure it ends? 01:00:48.020 --> 01:00:50.640 align:middle line:90% Why can't it go on forever? 01:00:50.640 --> 01:00:51.340 align:middle line:90% Anyone know? 01:00:51.340 --> 01:00:55.720 align:middle line:90% 01:00:55.720 --> 01:00:58.400 align:middle line:84% Well, there's only finitely many strategies to begin with. 01:00:58.400 --> 01:01:00.580 align:middle line:84% And at each round, we just throw strategies out. 01:01:00.580 --> 01:01:01.560 align:middle line:90% We never get bigger. 01:01:01.560 --> 01:01:03.080 align:middle line:84% So if we have finitely many things, 01:01:03.080 --> 01:01:04.580 align:middle line:84% and we keep throwing something out, 01:01:04.580 --> 01:01:05.997 align:middle line:84% we can't throw things out forever. 01:01:05.997 --> 01:01:07.620 align:middle line:84% Eventually, we can't get any smaller. 01:01:07.620 --> 01:01:08.120 align:middle line:90% Yes? 01:01:08.120 --> 01:01:11.040 align:middle line:84% STUDENT: Does that mean you can never swap strategies? 01:01:11.040 --> 01:01:14.690 align:middle line:84% Like, [INAUDIBLE] like, throw one out [INAUDIBLE]. 01:01:14.690 --> 01:01:15.440 align:middle line:90% IAN BALL: Exactly. 01:01:15.440 --> 01:01:16.260 align:middle line:90% That will never happen. 01:01:16.260 --> 01:01:16.400 align:middle line:90% Yeah. 01:01:16.400 --> 01:01:18.520 align:middle line:84% So at each step, these sets are getting smaller 01:01:18.520 --> 01:01:19.538 align:middle line:90% and smaller and smaller. 01:01:19.538 --> 01:01:21.080 align:middle line:84% And this exactly goes to the property 01:01:21.080 --> 01:01:24.700 align:middle line:84% we were describing about how the set of best responses 01:01:24.700 --> 01:01:27.740 align:middle line:84% gets smaller as the set of opponent strategies 01:01:27.740 --> 01:01:28.540 align:middle line:90% gets smaller. 01:01:28.540 --> 01:01:30.980 align:middle line:84% So at each round, everyone else's strategy set 01:01:30.980 --> 01:01:31.853 align:middle line:90% gets smaller. 01:01:31.853 --> 01:01:33.520 align:middle line:84% But that means the set of best responses 01:01:33.520 --> 01:01:35.740 align:middle line:84% I have gets smaller, which means the set of best responses 01:01:35.740 --> 01:01:36.800 align:middle line:90% they have gets smaller. 01:01:36.800 --> 01:01:38.883 align:middle line:84% And at each stage, we're just throwing things out. 01:01:38.883 --> 01:01:40.140 align:middle line:90% Another way to see-- 01:01:40.140 --> 01:01:40.640 align:middle line:90% well, yeah. 01:01:40.640 --> 01:01:43.100 align:middle line:84% I guess it's not immediately obvious from this. 01:01:43.100 --> 01:01:45.740 align:middle line:84% But again, this set gets smaller, 01:01:45.740 --> 01:01:46.960 align:middle line:90% and this set gets bigger. 01:01:46.960 --> 01:01:52.083 align:middle line:84% We throw out more things, and we keep fewer things. 01:01:52.083 --> 01:01:54.500 align:middle line:84% If we didn't have that property, then it could go forever. 01:01:54.500 --> 01:01:56.020 align:middle line:84% We'd worry that we might get in cycles where 01:01:56.020 --> 01:01:58.580 align:middle line:84% we take a strategy out, put it back in, take it out, put it 01:01:58.580 --> 01:01:59.080 align:middle line:90% back in. 01:01:59.080 --> 01:01:59.830 align:middle line:90% That can't happen. 01:01:59.830 --> 01:02:00.640 align:middle line:90% There's no cycles. 01:02:00.640 --> 01:02:02.960 align:middle line:84% We just get smaller and smaller until we stop. 01:02:02.960 --> 01:02:03.460 align:middle line:90% Yes? 01:02:03.460 --> 01:02:05.520 align:middle line:84% STUDENT: So when you say it's a finite strategic form game, 01:02:05.520 --> 01:02:06.780 align:middle line:84% does that mean the number of players 01:02:06.780 --> 01:02:09.120 align:middle line:84% is finite, as is the number of strategies for a player? 01:02:09.120 --> 01:02:10.340 align:middle line:90% IAN BALL: Yes. 01:02:10.340 --> 01:02:11.320 align:middle line:90% So n is finite. 01:02:11.320 --> 01:02:13.737 align:middle line:84% I mean, we're never really going to think of n infinite. 01:02:13.737 --> 01:02:15.320 align:middle line:84% But technically, it means n is finite. 01:02:15.320 --> 01:02:16.903 align:middle line:84% But the real substantive content of it 01:02:16.903 --> 01:02:18.480 align:middle line:84% is that the Si's are also finite. 01:02:18.480 --> 01:02:21.140 align:middle line:90% Yeah. 01:02:21.140 --> 01:02:23.620 align:middle line:90% So here's my tip for doing this. 01:02:23.620 --> 01:02:25.120 align:middle line:84% Because of this property that things 01:02:25.120 --> 01:02:31.160 align:middle line:84% get smaller at every stage, if you throw something out 01:02:31.160 --> 01:02:33.098 align:middle line:84% that you're not supposed to, you're done. 01:02:33.098 --> 01:02:34.140 align:middle line:90% You're really in trouble. 01:02:34.140 --> 01:02:34.980 align:middle line:90% You've messed up. 01:02:34.980 --> 01:02:36.400 align:middle line:90% There's no coming back from it. 01:02:36.400 --> 01:02:37.640 align:middle line:84% But if you don't throw something out 01:02:37.640 --> 01:02:39.057 align:middle line:84% that you're supposed to throw out, 01:02:39.057 --> 01:02:40.902 align:middle line:84% you can always throw it out next time. 01:02:40.902 --> 01:02:42.360 align:middle line:84% So it's always safer to keep things 01:02:42.360 --> 01:02:44.903 align:middle line:84% in because once you throw it out, it can't come back. 01:02:44.903 --> 01:02:46.320 align:middle line:84% But if you keep it in, well, maybe 01:02:46.320 --> 01:02:47.440 align:middle line:84% it just stays in a little too long, 01:02:47.440 --> 01:02:49.080 align:middle line:84% and you'll throw it out next round. 01:02:49.080 --> 01:02:52.520 align:middle line:84% So it's better to err on the side of inclusion. 01:02:52.520 --> 01:02:55.280 align:middle line:84% And for that reason, my suggestion 01:02:55.280 --> 01:02:57.160 align:middle line:84% would be when you're going through it, 01:02:57.160 --> 01:03:00.160 align:middle line:90% think about dominance. 01:03:00.160 --> 01:03:06.500 align:middle line:84% So I would say use SCD during the algorithm. 01:03:06.500 --> 01:03:09.440 align:middle line:90% 01:03:09.440 --> 01:03:10.340 align:middle line:90% Why? 01:03:10.340 --> 01:03:12.960 align:middle line:84% Well, if you're looking for strategies that are strictly 01:03:12.960 --> 01:03:16.640 align:middle line:84% dominated and you miss one, that means 01:03:16.640 --> 01:03:18.950 align:middle line:84% you forget to throw something out, 01:03:18.950 --> 01:03:20.990 align:middle line:84% and you can throw it out next time. 01:03:20.990 --> 01:03:23.550 align:middle line:84% If you look for strategies that are best responses 01:03:23.550 --> 01:03:25.750 align:middle line:84% and you miss one, then you're going 01:03:25.750 --> 01:03:29.430 align:middle line:84% to throw that strategy out, and you can never recover from that. 01:03:29.430 --> 01:03:31.470 align:middle line:84% Now, the only problem, though, is at the end, 01:03:31.470 --> 01:03:33.590 align:middle line:84% if I'm using this definition, there's 01:03:33.590 --> 01:03:37.430 align:middle line:84% a risk that I might have kept in too many strategies at the end. 01:03:37.430 --> 01:03:39.450 align:middle line:84% So my advice would be at the last stage, 01:03:39.450 --> 01:03:42.550 align:middle line:84% once you think you're done, then use this definition, 01:03:42.550 --> 01:03:44.430 align:middle line:84% and check that everything that's left 01:03:44.430 --> 01:03:47.870 align:middle line:84% is a best response to some belief about your opponents. 01:03:47.870 --> 01:03:53.410 align:middle line:84% So use SCD during the algorithm, and then use BR at the end. 01:03:53.410 --> 01:03:55.950 align:middle line:90% 01:03:55.950 --> 01:03:59.430 align:middle line:84% And this is a safe way to make sure you never throw anything 01:03:59.430 --> 01:04:01.397 align:middle line:84% out you shouldn't throughout the algorithm. 01:04:01.397 --> 01:04:03.230 align:middle line:84% But then at the very end, check that there's 01:04:03.230 --> 01:04:05.910 align:middle line:84% nothing left that you should have thrown out. 01:04:05.910 --> 01:04:08.350 align:middle line:84% So that would be my advice for how you go about it if you 01:04:08.350 --> 01:04:09.130 align:middle line:90% want to be safe. 01:04:09.130 --> 01:04:10.710 align:middle line:84% Maybe it's a bit slower, but I think 01:04:10.710 --> 01:04:13.670 align:middle line:90% that's a safe way to do it. 01:04:13.670 --> 01:04:14.930 align:middle line:90% Any questions on that? 01:04:14.930 --> 01:04:18.230 align:middle line:90% 01:04:18.230 --> 01:04:18.730 align:middle line:90% Great. 01:04:18.730 --> 01:04:21.070 align:middle line:90% So let's now do a few examples. 01:04:21.070 --> 01:04:35.210 align:middle line:90% 01:04:35.210 --> 01:04:40.370 align:middle line:84% So another simple example would be the prisoner's dilemma-- 01:04:40.370 --> 01:04:45.590 align:middle line:84% so cooperate defect, cooperate defect. 01:04:45.590 --> 01:04:52.970 align:middle line:90% 01:04:52.970 --> 01:04:56.010 align:middle line:84% I think last time we may have used the phrases m and f, mum 01:04:56.010 --> 01:04:56.633 align:middle line:90% and fink. 01:04:56.633 --> 01:04:58.050 align:middle line:84% But in the prisoner's dilemma it's 01:04:58.050 --> 01:05:01.850 align:middle line:84% also common to use cooperate, defect, cooperate, defect. 01:05:01.850 --> 01:05:05.450 align:middle line:84% Remember, cooperate means cooperate 01:05:05.450 --> 01:05:07.550 align:middle line:84% with the other prisoner, not with the authorities. 01:05:07.550 --> 01:05:08.870 align:middle line:90% That can be a little confusing. 01:05:08.870 --> 01:05:11.410 align:middle line:84% So cooperation is a property of what the players in the game 01:05:11.410 --> 01:05:12.250 align:middle line:90% are doing. 01:05:12.250 --> 01:05:14.770 align:middle line:84% So it doesn't mean testify against the other prisoner. 01:05:14.770 --> 01:05:16.920 align:middle line:84% It means cooperate with the other prisoner 01:05:16.920 --> 01:05:18.480 align:middle line:90% by not testifying. 01:05:18.480 --> 01:05:20.240 align:middle line:84% So here in the prisoner's dilemma, 01:05:20.240 --> 01:05:22.220 align:middle line:84% we have-- let's go through the algorithm. 01:05:22.220 --> 01:05:26.280 align:middle line:90% 01:05:26.280 --> 01:05:30.260 align:middle line:90% Well, first, what is S10? 01:05:30.260 --> 01:05:33.200 align:middle line:90% 01:05:33.200 --> 01:05:35.840 align:middle line:84% Well, this is just the set of strategies for player one. 01:05:35.840 --> 01:05:36.940 align:middle line:90% We already see what it is. 01:05:36.940 --> 01:05:38.300 align:middle line:90% It's C, D. So that's easy. 01:05:38.300 --> 01:05:42.440 align:middle line:90% 01:05:42.440 --> 01:05:47.800 align:middle line:90% And then S20 also equals C, D. 01:05:47.800 --> 01:05:51.980 align:middle line:84% So now let's go-- so this is step k equals 0. 01:05:51.980 --> 01:05:52.900 align:middle line:90% Here's k here. 01:05:52.900 --> 01:05:55.560 align:middle line:90% 01:05:55.560 --> 01:05:57.240 align:middle line:90% Now let's go to step one. 01:05:57.240 --> 01:05:59.100 align:middle line:90% So in step one, what happens? 01:05:59.100 --> 01:06:03.320 align:middle line:84% Let's follow our tip, and let's only throw out 01:06:03.320 --> 01:06:05.480 align:middle line:90% strictly dominated strategies. 01:06:05.480 --> 01:06:06.820 align:middle line:90% So what do we throw out here? 01:06:06.820 --> 01:06:13.740 align:middle line:90% 01:06:13.740 --> 01:06:19.500 align:middle line:84% Well, cooperate is strictly dominated by defect. 01:06:19.500 --> 01:06:24.540 align:middle line:84% If my opponent cooperates, I do strictly better by defecting. 01:06:24.540 --> 01:06:27.940 align:middle line:84% And if my opponent defects, I also do strictly better 01:06:27.940 --> 01:06:29.260 align:middle line:90% by defecting. 01:06:29.260 --> 01:06:33.580 align:middle line:84% So D strictly dominates C for each player. 01:06:33.580 --> 01:06:35.580 align:middle line:90% That means I throw out D. 01:06:35.580 --> 01:06:36.680 align:middle line:90% And what do I keep in? 01:06:36.680 --> 01:06:37.620 align:middle line:90% Well, C. Sorry. 01:06:37.620 --> 01:06:41.400 align:middle line:84% I throw out C, and I keep in D. Let me say that again. 01:06:41.400 --> 01:06:43.360 align:middle line:90% I may have jumbled my words. 01:06:43.360 --> 01:07:03.580 align:middle line:84% C is strictly dominated by D, so I throw out C and keep D. 01:07:03.580 --> 01:07:05.520 align:middle line:84% And now I could go one more stage. 01:07:05.520 --> 01:07:08.300 align:middle line:84% But actually, here one way you always know you're done 01:07:08.300 --> 01:07:10.780 align:middle line:84% is if each player only has one strategy left. 01:07:10.780 --> 01:07:12.920 align:middle line:84% But let's make sure we didn't make a mistake. 01:07:12.920 --> 01:07:15.720 align:middle line:84% Let's do our best response thing at check at the end. 01:07:15.720 --> 01:07:18.733 align:middle line:90% Is D a best response to D? 01:07:18.733 --> 01:07:20.900 align:middle line:84% Well, we can actually go through our procedure here. 01:07:20.900 --> 01:07:23.760 align:middle line:90% 01:07:23.760 --> 01:07:26.080 align:middle line:84% So I'm just doing the underlining that we did before. 01:07:26.080 --> 01:07:30.220 align:middle line:84% And sure enough D is a best response to D for both players. 01:07:30.220 --> 01:07:32.700 align:middle line:84% And we know-- well, we don't know we've done it right 01:07:32.700 --> 01:07:36.280 align:middle line:84% but we at least know we didn't obviously do it wrong. 01:07:36.280 --> 01:07:39.480 align:middle line:90% So we're done there. 01:07:39.480 --> 01:07:41.860 align:middle line:90% Let's do one more example. 01:07:41.860 --> 01:07:44.440 align:middle line:90% 01:07:44.440 --> 01:07:48.820 align:middle line:84% Maybe I'll call top, bottom, left to right. 01:07:48.820 --> 01:07:52.620 align:middle line:84% So again, two players, two strategies each. 01:07:52.620 --> 01:07:59.520 align:middle line:90% 01:07:59.520 --> 01:08:06.840 align:middle line:84% Let's do it like this, and let's go through our procedure. 01:08:06.840 --> 01:08:08.580 align:middle line:90% So again, step zero is easy. 01:08:08.580 --> 01:08:11.950 align:middle line:84% We're just writing down the strategies the players have. 01:08:11.950 --> 01:08:14.910 align:middle line:84% Player one strategies are T and B. Player 01:08:14.910 --> 01:08:23.790 align:middle line:84% two strategies are L and R. So we have S10 equals TB, 01:08:23.790 --> 01:08:28.189 align:middle line:90% and S20 equals L. 01:08:28.189 --> 01:08:30.370 align:middle line:84% And I warn you when you look at this, 01:08:30.370 --> 01:08:33.149 align:middle line:84% it gets a little confusing when you see things like S11. 01:08:33.149 --> 01:08:36.229 align:middle line:84% But I'm always going to use the convention that the subscript is 01:08:36.229 --> 01:08:38.350 align:middle line:84% the player, the thing at the bottom, 01:08:38.350 --> 01:08:40.350 align:middle line:84% and the superscript is the round number, 01:08:40.350 --> 01:08:42.270 align:middle line:90% or the stage of elimination. 01:08:42.270 --> 01:08:45.288 align:middle line:84% So I'll call these things k at the top and i at the bottom. 01:08:45.288 --> 01:08:46.830 align:middle line:84% But if they take actual numbers, just 01:08:46.830 --> 01:08:48.750 align:middle line:84% remember the bottom is the player, 01:08:48.750 --> 01:08:51.550 align:middle line:84% and the top is the stage that we're at. 01:08:51.550 --> 01:08:54.550 align:middle line:90% So that's step one-- step zero. 01:08:54.550 --> 01:08:57.270 align:middle line:90% Now, let's go to step one. 01:08:57.270 --> 01:08:58.850 align:middle line:90% What do we throw out here? 01:08:58.850 --> 01:09:02.290 align:middle line:90% 01:09:02.290 --> 01:09:02.790 align:middle line:90% Yeah? 01:09:02.790 --> 01:09:04.590 align:middle line:84% STUDENT: Can a strategy [INAUDIBLE]. 01:09:04.590 --> 01:09:05.350 align:middle line:90% IAN BALL: Right. 01:09:05.350 --> 01:09:06.910 align:middle line:90% So we don't throw anything out. 01:09:06.910 --> 01:09:08.790 align:middle line:90% And how can we see that? 01:09:08.790 --> 01:09:11.010 align:middle line:84% Well, let's go through our best response thing. 01:09:11.010 --> 01:09:13.290 align:middle line:84% If I'm player one, if my opponent plays L, 01:09:13.290 --> 01:09:16.090 align:middle line:90% I'd rather play T. 01:09:16.090 --> 01:09:17.729 align:middle line:84% If my opponent plays R-- we haven't 01:09:17.729 --> 01:09:19.229 align:middle line:90% gotten to a case like this. 01:09:19.229 --> 01:09:21.588 align:middle line:84% But here, I have multiple best responses. 01:09:21.588 --> 01:09:24.130 align:middle line:84% And it's really crucial when you have multiple best responses 01:09:24.130 --> 01:09:25.838 align:middle line:84% to underline both of them because they're 01:09:25.838 --> 01:09:28.010 align:middle line:84% both reasonable ways you could respond. 01:09:28.010 --> 01:09:31.810 align:middle line:84% So given my opponent plays R, I get 0 from both T and B 01:09:31.810 --> 01:09:34.012 align:middle line:84% So I'm going to underline both of them. 01:09:34.012 --> 01:09:35.970 align:middle line:84% And then we'll do the same thing the other way. 01:09:35.970 --> 01:09:38.170 align:middle line:84% If player one plays T, player two 01:09:38.170 --> 01:09:41.330 align:middle line:84% wants to play L. If player one plays B, 01:09:41.330 --> 01:09:43.569 align:middle line:84% player two is indifferent between L and R, 01:09:43.569 --> 01:09:44.990 align:middle line:90% so we underline both of these. 01:09:44.990 --> 01:09:48.170 align:middle line:90% 01:09:48.170 --> 01:09:49.930 align:middle line:84% And if we look at our underlining, 01:09:49.930 --> 01:09:54.250 align:middle line:84% we see that T is indeed a best response for player one. 01:09:54.250 --> 01:09:58.210 align:middle line:84% It's a best response to L, and actually also to R. 01:09:58.210 --> 01:09:59.810 align:middle line:90% But L is enough. 01:09:59.810 --> 01:10:01.210 align:middle line:90% What about B? 01:10:01.210 --> 01:10:03.090 align:middle line:84% Is B a best response to anything? 01:10:03.090 --> 01:10:03.930 align:middle line:90% Yes. 01:10:03.930 --> 01:10:08.430 align:middle line:84% B is a best response to R. It's underlined here. 01:10:08.430 --> 01:10:13.550 align:middle line:84% So both T and B are best responses to some belief. 01:10:13.550 --> 01:10:17.910 align:middle line:84% And similarly, both L and R are best responses to some belief. 01:10:17.910 --> 01:10:23.860 align:middle line:90% 01:10:23.860 --> 01:10:27.630 align:middle line:84% L is a best response to both T and B 01:10:27.630 --> 01:10:30.470 align:middle line:84% because we can see L underlined twice. 01:10:30.470 --> 01:10:35.890 align:middle line:84% And R is a best response to B, but not to T. But that's OK. 01:10:35.890 --> 01:10:37.390 align:middle line:84% It's a best response to something. 01:10:37.390 --> 01:10:46.937 align:middle line:84% So we get S11 equals TB, and S21 equals LR. 01:10:46.937 --> 01:10:48.770 align:middle line:84% So do we have to keep going, or are we done? 01:10:48.770 --> 01:10:52.310 align:middle line:90% 01:10:52.310 --> 01:10:53.550 align:middle line:90% So now we're done. 01:10:53.550 --> 01:10:57.070 align:middle line:84% But the reason we're done is that player one's strategy 01:10:57.070 --> 01:10:59.667 align:middle line:84% set didn't change, and player two's strategy set 01:10:59.667 --> 01:11:00.250 align:middle line:90% didn't change. 01:11:00.250 --> 01:11:03.150 align:middle line:84% But remember, we're only done because every single player's 01:11:03.150 --> 01:11:04.650 align:middle line:90% strategy set stayed the same. 01:11:04.650 --> 01:11:08.380 align:middle line:84% It's not enough for just one to stay the same. 01:11:08.380 --> 01:11:09.560 align:middle line:90% So we're done here. 01:11:09.560 --> 01:11:10.620 align:middle line:90% We're done here. 01:11:10.620 --> 01:11:14.380 align:middle line:84% And these two examples illustrate that, in some cases, 01:11:14.380 --> 01:11:16.820 align:middle line:84% this rationalizability procedure gives us 01:11:16.820 --> 01:11:19.820 align:middle line:84% a very strong prediction about what we'll play in the game. 01:11:19.820 --> 01:11:21.900 align:middle line:84% But in other cases, it doesn't really give us 01:11:21.900 --> 01:11:23.640 align:middle line:90% any sharp prediction at all. 01:11:23.640 --> 01:11:26.700 align:middle line:84% We're not able to rule out any strategies using this concept 01:11:26.700 --> 01:11:30.140 align:middle line:90% of rationalizability alone. 01:11:30.140 --> 01:11:31.280 align:middle line:90% Let me check the time. 01:11:31.280 --> 01:11:34.080 align:middle line:90% 01:11:34.080 --> 01:11:34.580 align:middle line:90% Great. 01:11:34.580 --> 01:11:36.820 align:middle line:84% So let me do one maybe more substantive example 01:11:36.820 --> 01:11:41.940 align:middle line:84% to finish that actually has continuous strategies. 01:11:41.940 --> 01:11:45.820 align:middle line:84% So let's go back to the beauty contest game 01:11:45.820 --> 01:11:49.748 align:middle line:84% that we played on the first day of class. 01:11:49.748 --> 01:11:51.540 align:middle line:84% And I think some people were starting to go 01:11:51.540 --> 01:11:52.860 align:middle line:90% through this kind of reasoning. 01:11:52.860 --> 01:11:57.260 align:middle line:84% Here, we can formally solve this game. 01:11:57.260 --> 01:11:59.240 align:middle line:84% So let's go to the beauty contest. 01:11:59.240 --> 01:12:03.260 align:middle line:90% 01:12:03.260 --> 01:12:05.040 align:middle line:84% But this time, it's a bit easier to say 01:12:05.040 --> 01:12:08.440 align:middle line:84% that you get to choose a number, any number, any real number 01:12:08.440 --> 01:12:09.820 align:middle line:90% between 0 and 100. 01:12:09.820 --> 01:12:12.080 align:middle line:84% So you're not restricted to integers here. 01:12:12.080 --> 01:12:13.580 align:middle line:90% So each player gets to choose. 01:12:13.580 --> 01:12:17.640 align:middle line:84% So we have players i equals 1 through n. 01:12:17.640 --> 01:12:23.372 align:middle line:84% And each player i gets to choose a number between 0 and 100. 01:12:23.372 --> 01:12:25.080 align:middle line:84% So I'm following the notes here and using 01:12:25.080 --> 01:12:31.200 align:middle line:84% x to be the number that you choose instead of s. 01:12:31.200 --> 01:12:33.160 align:middle line:90% Let's write down our payoffs. 01:12:33.160 --> 01:12:39.280 align:middle line:84% So let's say I'm player i, and the numbers chosen are x1, 01:12:39.280 --> 01:12:44.080 align:middle line:90% x2, all the way to xn. 01:12:44.080 --> 01:12:48.720 align:middle line:84% So we're going to consider here the alpha beauty contest. 01:12:48.720 --> 01:12:54.040 align:middle line:84% So we're going to have some number alpha 01:12:54.040 --> 01:12:55.840 align:middle line:90% that's between 0 and 1. 01:12:55.840 --> 01:13:00.280 align:middle line:84% And your goal is for your guess to match alpha times the average 01:13:00.280 --> 01:13:02.280 align:middle line:90% of everyone's guess. 01:13:02.280 --> 01:13:04.412 align:middle line:84% So on the first day of class, we played. 01:13:04.412 --> 01:13:06.870 align:middle line:84% And the second day, we played the beauty contest with alpha 01:13:06.870 --> 01:13:10.470 align:middle line:90% equals 2/3, I think. 01:13:10.470 --> 01:13:14.870 align:middle line:84% And to make things a bit easier, instead of just saying 01:13:14.870 --> 01:13:16.590 align:middle line:84% you either win or you don't, we're 01:13:16.590 --> 01:13:19.630 align:middle line:84% going to say that your payoff depends on how close you 01:13:19.630 --> 01:13:22.230 align:middle line:90% are to alpha times the average. 01:13:22.230 --> 01:13:23.710 align:middle line:84% You get the highest possible payoff 01:13:23.710 --> 01:13:26.350 align:middle line:84% if you're exactly alpha times the average. 01:13:26.350 --> 01:13:28.870 align:middle line:84% And the farther your guess is away from alpha times 01:13:28.870 --> 01:13:30.733 align:middle line:84% the average, the lower your payoff is. 01:13:30.733 --> 01:13:32.150 align:middle line:84% And we're going to specifically do 01:13:32.150 --> 01:13:34.470 align:middle line:84% that with what's called a quadratic loss function. 01:13:34.470 --> 01:13:39.750 align:middle line:84% So you lose the square of your distance from your ideal guess. 01:13:39.750 --> 01:13:42.070 align:middle line:90% So what is that going to be? 01:13:42.070 --> 01:13:45.270 align:middle line:84% It's going to be minus something squared. 01:13:45.270 --> 01:13:46.410 align:middle line:90% So the thing squared. 01:13:46.410 --> 01:13:49.477 align:middle line:84% This is your loss, and you're subtracting this loss. 01:13:49.477 --> 01:13:50.810 align:middle line:90% And what are we going to square? 01:13:50.810 --> 01:13:54.030 align:middle line:84% We want to square the distance or the difference between what 01:13:54.030 --> 01:13:58.230 align:middle line:84% you actually guess and the ideal guess, which is alpha times 01:13:58.230 --> 01:14:00.190 align:middle line:90% the average of the class. 01:14:00.190 --> 01:14:02.490 align:middle line:90% So this is player i's payoff. 01:14:02.490 --> 01:14:04.410 align:middle line:90% So I have xi here. 01:14:04.410 --> 01:14:06.570 align:middle line:90% That's what I actually guess. 01:14:06.570 --> 01:14:07.970 align:middle line:90% What's the best possible guess? 01:14:07.970 --> 01:14:11.090 align:middle line:84% It's alpha times the average of everyone else's guess-- 01:14:11.090 --> 01:14:13.690 align:middle line:84% or average of everyone's guess, I should say. 01:14:13.690 --> 01:14:19.970 align:middle line:84% So that's minus alpha times x1 to xn over n. 01:14:19.970 --> 01:14:22.410 align:middle line:84% And this is kind of a subtle aspect of this game, 01:14:22.410 --> 01:14:26.010 align:middle line:84% that my own guess contributes to the class average. 01:14:26.010 --> 01:14:29.910 align:middle line:84% So here, this is the class average. 01:14:29.910 --> 01:14:32.850 align:middle line:90% 01:14:32.850 --> 01:14:35.250 align:middle line:90% There's n people in the class. 01:14:35.250 --> 01:14:36.830 align:middle line:90% Each of them guesses a number. 01:14:36.830 --> 01:14:40.490 align:middle line:84% So I divide by n to get the average guess in the class. 01:14:40.490 --> 01:14:42.770 align:middle line:84% Then I multiply the average guess, which remember, 01:14:42.770 --> 01:14:45.090 align:middle line:90% I is player i contribute to. 01:14:45.090 --> 01:14:47.610 align:middle line:84% I multiply that average guess by alpha. 01:14:47.610 --> 01:14:49.590 align:middle line:90% That's the ideal guess. 01:14:49.590 --> 01:14:52.570 align:middle line:84% That would be the best possible guess I could make. 01:14:52.570 --> 01:14:55.790 align:middle line:84% If I guess exactly that, then my payoff is zero. 01:14:55.790 --> 01:14:56.710 align:middle line:90% That's great. 01:14:56.710 --> 01:14:59.330 align:middle line:84% But the farther my guess is away from my bliss point, 01:14:59.330 --> 01:15:02.090 align:middle line:84% or the ideal guess, the greater my losses, 01:15:02.090 --> 01:15:06.390 align:middle line:84% or the more negative my payoff is. 01:15:06.390 --> 01:15:08.110 align:middle line:90% It turns out because-- 01:15:08.110 --> 01:15:10.210 align:middle line:84% it's a little tricky to analyze this directly 01:15:10.210 --> 01:15:12.470 align:middle line:84% because when I think about my guess, 01:15:12.470 --> 01:15:14.990 align:middle line:84% I'm also changing the class average at the same time. 01:15:14.990 --> 01:15:17.490 align:middle line:84% So it's better to just factor out the xi. 01:15:17.490 --> 01:15:21.990 align:middle line:84% So let's write this a bit differently. 01:15:21.990 --> 01:15:24.470 align:middle line:90% So we have xi here. 01:15:24.470 --> 01:15:27.010 align:middle line:84% But then, there's also an xi here, 01:15:27.010 --> 01:15:29.870 align:middle line:84% and it's multiplied by alpha over n. 01:15:29.870 --> 01:15:37.550 align:middle line:84% So what I get is 1 minus alpha over n times xi. 01:15:37.550 --> 01:15:41.750 align:middle line:84% The 1 times xi comes from here, and the minus alpha over n times 01:15:41.750 --> 01:15:44.010 align:middle line:90% xi comes from over here. 01:15:44.010 --> 01:15:46.950 align:middle line:90% 01:15:46.950 --> 01:15:51.110 align:middle line:84% And then what I'm left with is alpha times the sum of everyone 01:15:51.110 --> 01:15:52.618 align:middle line:90% else's guess divided by n. 01:15:52.618 --> 01:15:54.910 align:middle line:84% But that's kind of awkward to deal with because there's 01:15:54.910 --> 01:15:57.470 align:middle line:84% only n minus 1 people, and I'm dividing by n 01:15:57.470 --> 01:15:59.200 align:middle line:84% So I'd rather divide by n minus 1. 01:15:59.200 --> 01:16:04.220 align:middle line:84% So let's write this as alpha times 01:16:04.220 --> 01:16:11.060 align:middle line:84% n minus 1 over n times x negative i bar, where 01:16:11.060 --> 01:16:19.980 align:middle line:84% this is the sum over everyone but me divided by n minus 1. 01:16:19.980 --> 01:16:22.480 align:middle line:84% So all I've done is multiplied and divided by n minus 1. 01:16:22.480 --> 01:16:23.605 align:middle line:90% It's a little algebra here. 01:16:23.605 --> 01:16:25.260 align:middle line:90% But I divided by n minus 1 here. 01:16:25.260 --> 01:16:26.760 align:middle line:84% I wasn't really supposed to do that. 01:16:26.760 --> 01:16:29.860 align:middle line:84% So I undid it by multiplying by n minus 1 over here. 01:16:29.860 --> 01:16:33.540 align:middle line:84% And I still have the n that I'm supposed to have on the bottom. 01:16:33.540 --> 01:16:35.820 align:middle line:84% And this is just nicer notation because this tells me, 01:16:35.820 --> 01:16:37.695 align:middle line:84% what is the average of everyone else's guess? 01:16:37.695 --> 01:16:39.987 align:middle line:84% And that's really the way you think about this problem. 01:16:39.987 --> 01:16:41.500 align:middle line:84% You think about your guess, and you 01:16:41.500 --> 01:16:43.280 align:middle line:84% think about the average of everyone else's guess. 01:16:43.280 --> 01:16:45.488 align:middle line:84% It's a bit weird to think of the class average, which 01:16:45.488 --> 01:16:47.540 align:middle line:84% partly reflects your guess and partly reflects 01:16:47.540 --> 01:16:48.640 align:middle line:90% everyone else's guess. 01:16:48.640 --> 01:16:51.160 align:middle line:84% So this is just a slightly nicer way of writing it, 01:16:51.160 --> 01:16:54.180 align:middle line:84% though it makes things a little messier. 01:16:54.180 --> 01:16:57.340 align:middle line:90% Let's go down here. 01:16:57.340 --> 01:17:16.040 align:middle line:90% 01:17:16.040 --> 01:17:18.360 align:middle line:84% So from this formula, it's also a bit easier 01:17:18.360 --> 01:17:20.020 align:middle line:90% to calculate my best response. 01:17:20.020 --> 01:17:22.000 align:middle line:84% So let's imagine, hypothetically, 01:17:22.000 --> 01:17:26.240 align:middle line:84% that I knew what the average of everyone else's guess was. 01:17:26.240 --> 01:17:31.660 align:middle line:84% Then I would want to choose my xi so that this equals this. 01:17:31.660 --> 01:17:33.580 align:middle line:90% And this should be squared. 01:17:33.580 --> 01:17:35.520 align:middle line:84% I want my loss as small as possible. 01:17:35.520 --> 01:17:37.320 align:middle line:90% I want this expression to be 0. 01:17:37.320 --> 01:17:39.720 align:middle line:90% So I want this to equal that. 01:17:39.720 --> 01:17:41.480 align:middle line:84% And now I'm going to solve for xi. 01:17:41.480 --> 01:17:45.680 align:middle line:84% So I want xi to be this divided by this. 01:17:45.680 --> 01:17:48.223 align:middle line:84% Just a little bit of algebra, which you can work through 01:17:48.223 --> 01:17:49.140 align:middle line:90% at home if you'd like. 01:17:49.140 --> 01:17:52.360 align:middle line:84% And what you'll find is that the kind of best response, 01:17:52.360 --> 01:17:56.100 align:middle line:84% if I solve for this, is I get xi equals-- 01:17:56.100 --> 01:17:59.140 align:middle line:84% well, I get the right side divided by 1 minus alpha over n. 01:17:59.140 --> 01:18:00.880 align:middle line:90% And the n's cancel nicely. 01:18:00.880 --> 01:18:12.020 align:middle line:84% So I get alpha times n minus 1 over n minus alpha times 01:18:12.020 --> 01:18:13.440 align:middle line:90% x negative i bar. 01:18:13.440 --> 01:18:19.300 align:middle line:90% 01:18:19.300 --> 01:18:23.340 align:middle line:84% So what this says is if I knew that this 01:18:23.340 --> 01:18:26.440 align:middle line:84% is what everyone else was going to guess, or more specifically, 01:18:26.440 --> 01:18:28.860 align:middle line:84% this is the average of everyone else's guess, 01:18:28.860 --> 01:18:31.200 align:middle line:90% then this is my best response. 01:18:31.200 --> 01:18:32.440 align:middle line:90% This is my best guess. 01:18:32.440 --> 01:18:33.940 align:middle line:84% Now, at first you might have thought 01:18:33.940 --> 01:18:35.820 align:middle line:84% you would want to guess alpha times what 01:18:35.820 --> 01:18:37.903 align:middle line:84% everyone else guesses, but that's not what we get. 01:18:37.903 --> 01:18:42.620 align:middle line:84% Why do we get this kind of messy term here? 01:18:42.620 --> 01:18:44.480 align:middle line:84% Well, first, let's look at this number. 01:18:44.480 --> 01:18:47.400 align:middle line:84% Is this number bigger than 1 or less than 1? 01:18:47.400 --> 01:18:49.522 align:middle line:84% Let's assume n is greater than 1. 01:18:49.522 --> 01:18:51.480 align:middle line:84% Otherwise, this is not a very interesting game. 01:18:51.480 --> 01:18:55.290 align:middle line:90% 01:18:55.290 --> 01:19:00.890 align:middle line:84% This is less than 1 because I'm subtracting 1 on the top, 01:19:00.890 --> 01:19:03.690 align:middle line:84% and I'm subtracting something smaller on the bottom. 01:19:03.690 --> 01:19:05.690 align:middle line:84% So that means the bottom is bigger than the top. 01:19:05.690 --> 01:19:09.650 align:middle line:90% So this is less than 1. 01:19:09.650 --> 01:19:15.250 align:middle line:84% So that means it's best for me to guess 01:19:15.250 --> 01:19:18.690 align:middle line:84% not alpha times the average of everyone else, but strictly less 01:19:18.690 --> 01:19:20.170 align:middle line:90% than that. 01:19:20.170 --> 01:19:25.470 align:middle line:84% So this is less than alpha x minus i bar. 01:19:25.470 --> 01:19:31.578 align:middle line:90% Does anyone have any intuition? 01:19:31.578 --> 01:19:32.370 align:middle line:90% It's a bit strange. 01:19:32.370 --> 01:19:33.370 align:middle line:84% If you thought about this game, you 01:19:33.370 --> 01:19:35.370 align:middle line:84% might think you just want to guess exactly alpha 01:19:35.370 --> 01:19:36.970 align:middle line:90% times everyone else's guess. 01:19:36.970 --> 01:19:37.690 align:middle line:90% Yeah? 01:19:37.690 --> 01:19:38.490 align:middle line:84% STUDENT: Because your guess is also 01:19:38.490 --> 01:19:40.610 align:middle line:84% going to be part of the average, and you're bringing it down. 01:19:40.610 --> 01:19:41.650 align:middle line:90% IAN BALL: Exactly right. 01:19:41.650 --> 01:19:45.810 align:middle line:84% So if I said OK, everyone else, the average is x negative i bar. 01:19:45.810 --> 01:19:49.330 align:middle line:84% Let's say I guessed alpha times x negative i bar. 01:19:49.330 --> 01:19:50.510 align:middle line:90% Well, that sounds good. 01:19:50.510 --> 01:19:52.170 align:middle line:84% I'm guessing alpha times the average. 01:19:52.170 --> 01:19:54.110 align:middle line:84% But now the average is strictly smaller 01:19:54.110 --> 01:19:56.030 align:middle line:84% because my guess of alpha times the average 01:19:56.030 --> 01:19:57.570 align:middle line:90% is smaller than the average. 01:19:57.570 --> 01:20:00.098 align:middle line:84% So exactly as you said, I've pulled down the class average, 01:20:00.098 --> 01:20:01.890 align:middle line:84% and now I want to guess a little bit lower. 01:20:01.890 --> 01:20:03.682 align:middle line:84% And that's exactly what this is reflecting, 01:20:03.682 --> 01:20:07.110 align:middle line:84% the fact that my guess directly affects the class average. 01:20:07.110 --> 01:20:10.570 align:middle line:84% And when I'm trying to guess a fraction of the class average, 01:20:10.570 --> 01:20:13.990 align:middle line:84% my guess is always going to pull it down. 01:20:13.990 --> 01:20:15.810 align:middle line:90% But not so important. 01:20:15.810 --> 01:20:17.070 align:middle line:90% Let's just give this a name. 01:20:17.070 --> 01:20:18.830 align:middle line:90% We'll call this alpha n. 01:20:18.830 --> 01:20:24.390 align:middle line:84% So technically it depends on this parameter, alpha 01:20:24.390 --> 01:20:26.178 align:middle line:84% and the number of players we have. 01:20:26.178 --> 01:20:27.970 align:middle line:84% And it's the number of players grows large, 01:20:27.970 --> 01:20:29.110 align:middle line:84% it's going to get really close to alpha 01:20:29.110 --> 01:20:30.550 align:middle line:84% because your effect on the average 01:20:30.550 --> 01:20:34.050 align:middle line:90% is going to become very small. 01:20:34.050 --> 01:20:39.370 align:middle line:84% But now let's try to go through our algorithm in this game. 01:20:39.370 --> 01:20:43.310 align:middle line:84% OK, running out of time, so let's just do it quickly. 01:20:43.310 --> 01:20:48.270 align:middle line:90% Si0 equals 0 to 100. 01:20:48.270 --> 01:20:55.500 align:middle line:84% Si1-- well, the highest my opponents could guess 01:20:55.500 --> 01:20:57.140 align:middle line:90% would be they all guessed 100. 01:20:57.140 --> 01:21:00.100 align:middle line:84% And in that case, I'd want to guess alpha n times 100. 01:21:00.100 --> 01:21:02.920 align:middle line:84% The lowest I'd want to guess is 0, if everyone else guesses 0. 01:21:02.920 --> 01:21:07.740 align:middle line:84% So we're going to get 0 to 100 alpha n. 01:21:07.740 --> 01:21:09.140 align:middle line:90% And you can see the pattern. 01:21:09.140 --> 01:21:14.340 align:middle line:84% Si to the k is going to be 0 times 100 alpha n to the k. 01:21:14.340 --> 01:21:17.260 align:middle line:84% Because each time, if I know that my opponents are only 01:21:17.260 --> 01:21:19.060 align:middle line:84% guessing up to 100 times alpha n, 01:21:19.060 --> 01:21:22.140 align:middle line:84% I want to take that and multiply it by alpha n again. 01:21:22.140 --> 01:21:27.060 align:middle line:84% So in the limit, what we find is that Si infinity 01:21:27.060 --> 01:21:30.300 align:middle line:90% equals 0 for all i. 01:21:30.300 --> 01:21:32.120 align:middle line:84% So this game is dominant solvable. 01:21:32.120 --> 01:21:34.460 align:middle line:84% And the unique rationalizable strategy for every player 01:21:34.460 --> 01:21:36.220 align:middle line:90% is to play exactly 0. 01:21:36.220 --> 01:21:38.227 align:middle line:84% That was a bit quick, but a variant of this 01:21:38.227 --> 01:21:40.060 align:middle line:84% is on the homework, so I didn't want to make 01:21:40.060 --> 01:21:41.260 align:middle line:90% it too easy for the homework. 01:21:41.260 --> 01:21:42.968 align:middle line:84% So you'll get to work on the problem set. 01:21:42.968 --> 01:21:45.450 align:middle line:90% OK, I'll see everyone next week. 01:21:45.450 --> 01:21:50.000 align:middle line:90%