WEBVTT 00:00:00.000 --> 00:00:02.448 align:middle line:90% [SQUEAKING] 00:00:02.448 --> 00:00:03.909 align:middle line:90% [RUSTLING] 00:00:03.909 --> 00:00:09.885 align:middle line:90% [CLICKING] 00:00:09.885 --> 00:00:12.260 align:middle line:84% IAN BALL: So today, we're going to start analyzing games. 00:00:12.260 --> 00:00:19.652 align:middle line:90% 00:00:19.652 --> 00:00:21.360 align:middle line:84% So in my view, this is where things start 00:00:21.360 --> 00:00:22.760 align:middle line:90% getting a lot more interesting. 00:00:22.760 --> 00:00:25.740 align:middle line:84% So far, we've just said how to set up a game-- or first, 00:00:25.740 --> 00:00:27.980 align:middle line:84% how to set up an individual decision problem, 00:00:27.980 --> 00:00:29.937 align:middle line:84% and then how to set up or represent a game. 00:00:29.937 --> 00:00:31.520 align:middle line:84% And now, we're actually going to start 00:00:31.520 --> 00:00:33.320 align:middle line:84% thinking about analyzing the game 00:00:33.320 --> 00:00:35.120 align:middle line:84% and trying to make predictions about how 00:00:35.120 --> 00:00:37.400 align:middle line:90% people will play in the game. 00:00:37.400 --> 00:00:40.800 align:middle line:84% So I want to first go over the approach 00:00:40.800 --> 00:00:44.880 align:middle line:84% that we're usually going to take in this course. 00:00:44.880 --> 00:00:48.480 align:middle line:84% We'll usually start with some informal description 00:00:48.480 --> 00:00:49.615 align:middle line:90% of a strategic setting. 00:00:49.615 --> 00:00:51.740 align:middle line:84% That may be something that you get in a problem set 00:00:51.740 --> 00:00:54.080 align:middle line:84% or from thinking about the world. 00:00:54.080 --> 00:00:56.740 align:middle line:84% Then, based on the informal description, 00:00:56.740 --> 00:00:59.560 align:middle line:84% we'll usually write down the extensive form representation. 00:00:59.560 --> 00:01:06.500 align:middle line:90% 00:01:06.500 --> 00:01:09.060 align:middle line:84% Remember, this is the very detailed representation 00:01:09.060 --> 00:01:11.700 align:middle line:84% that we talked about a lot on Tuesday. 00:01:11.700 --> 00:01:16.180 align:middle line:84% And then we'll reduce this extensive form representation 00:01:16.180 --> 00:01:20.580 align:middle line:84% to the strategic form representation. 00:01:20.580 --> 00:01:26.380 align:middle line:90% 00:01:26.380 --> 00:01:27.940 align:middle line:84% And later on in the course, we will 00:01:27.940 --> 00:01:29.700 align:middle line:84% discuss some issues that can only 00:01:29.700 --> 00:01:33.487 align:middle line:84% be perceived in the extensive form representation. 00:01:33.487 --> 00:01:35.820 align:middle line:84% But for the beginning of the course-- and in particular, 00:01:35.820 --> 00:01:36.820 align:middle line:90% in today-- 00:01:36.820 --> 00:01:38.500 align:middle line:84% everything we're going to say is going 00:01:38.500 --> 00:01:41.300 align:middle line:84% to be about a strategic form representation. 00:01:41.300 --> 00:01:49.820 align:middle line:84% So today, our starting point is going to be 3. 00:01:49.820 --> 00:01:52.740 align:middle line:84% We're going to imagine that we're given a strategic form 00:01:52.740 --> 00:01:53.320 align:middle line:90% game. 00:01:53.320 --> 00:01:55.180 align:middle line:84% But you can imagine, in the background, 00:01:55.180 --> 00:01:58.250 align:middle line:84% that we may have derived arrived this given strategic form 00:01:58.250 --> 00:02:02.530 align:middle line:84% game from an extensive form that we previously wrote down. 00:02:02.530 --> 00:02:06.330 align:middle line:84% So let's just recall the notation for a strategic form 00:02:06.330 --> 00:02:07.010 align:middle line:90% game. 00:02:07.010 --> 00:02:10.970 align:middle line:84% We have the players, i equals 1 through n 00:02:10.970 --> 00:02:15.770 align:middle line:84% And then for each player i, we have a strategy set associated 00:02:15.770 --> 00:02:17.050 align:middle line:90% with that player i. 00:02:17.050 --> 00:02:21.690 align:middle line:84% So we have big S1 up to big Sn, where 00:02:21.690 --> 00:02:26.310 align:middle line:84% S1 is the set of strategies that are available to player 1. 00:02:26.310 --> 00:02:28.410 align:middle line:84% Remember, the set can be quite complicated, 00:02:28.410 --> 00:02:31.850 align:middle line:84% if we've derived this set by evaluating 00:02:31.850 --> 00:02:34.690 align:middle line:84% all complete contingent plans in the extensive form 00:02:34.690 --> 00:02:36.330 align:middle line:90% representation. 00:02:36.330 --> 00:02:42.290 align:middle line:84% And then we have our payoffs, ui from S to R, where, remember, 00:02:42.290 --> 00:02:46.730 align:middle line:84% S is the set of strategy profiles. 00:02:46.730 --> 00:02:51.410 align:middle line:84% So S, mathematically, is a product set of S1 up to Sn. 00:02:51.410 --> 00:02:54.890 align:middle line:84% But the key thing is that what lives in S are profiles 00:02:54.890 --> 00:02:57.990 align:middle line:84% files specifying a strategy for each of the n players. 00:02:57.990 --> 00:02:59.490 align:middle line:84% And we have a function like this, 00:02:59.490 --> 00:03:04.670 align:middle line:84% again for each of the n players, that specifies each player i's 00:03:04.670 --> 00:03:08.550 align:middle line:84% utility as a function of the profile of strategies 00:03:08.550 --> 00:03:09.690 align:middle line:90% that the players play. 00:03:09.690 --> 00:03:12.790 align:middle line:90% 00:03:12.790 --> 00:03:18.790 align:middle line:84% Now, what is our goal when we analyze a game? 00:03:18.790 --> 00:03:22.230 align:middle line:84% I think we're asking the following fundamental question-- 00:03:22.230 --> 00:03:25.030 align:middle line:84% given a game, we want to ask, how do we 00:03:25.030 --> 00:03:28.410 align:middle line:84% think players will play or should play the game? 00:03:28.410 --> 00:03:46.430 align:middle line:90% 00:03:46.430 --> 00:03:50.270 align:middle line:84% Now, notice, I think the distinction between will play 00:03:50.270 --> 00:03:53.610 align:middle line:84% and should play gets into two different interpretations 00:03:53.610 --> 00:03:55.250 align:middle line:90% of the analysis that we do. 00:03:55.250 --> 00:03:58.530 align:middle line:84% So the answers that we get and the formal analysis we do 00:03:58.530 --> 00:04:00.710 align:middle line:84% won't really depend on which interpretation 00:04:00.710 --> 00:04:01.870 align:middle line:90% you want to keep in mind. 00:04:01.870 --> 00:04:03.270 align:middle line:84% But depending on the application, 00:04:03.270 --> 00:04:05.570 align:middle line:84% one may be more or less compelling. 00:04:05.570 --> 00:04:08.630 align:middle line:84% So will play-- if we want to think about will play, 00:04:08.630 --> 00:04:12.290 align:middle line:84% this is what sometimes called positive analysis 00:04:12.290 --> 00:04:16.130 align:middle line:90% or descriptive analysis. 00:04:16.130 --> 00:04:19.130 align:middle line:84% And the idea of this is that, if we observe some game, 00:04:19.130 --> 00:04:21.970 align:middle line:84% we want to be able to make a prediction about how 00:04:21.970 --> 00:04:24.730 align:middle line:84% rational players will choose to play this game. 00:04:24.730 --> 00:04:29.810 align:middle line:84% So the goal here is really a prediction, 00:04:29.810 --> 00:04:31.690 align:middle line:84% which will, of course, only be valid 00:04:31.690 --> 00:04:34.450 align:middle line:84% if we think that the utility functions that we write down 00:04:34.450 --> 00:04:38.210 align:middle line:84% and the rationality assumptions that we make are compelling. 00:04:38.210 --> 00:04:40.010 align:middle line:84% Another interpretation of game theory 00:04:40.010 --> 00:04:43.770 align:middle line:84% is to put yourself in the role of an advisor. 00:04:43.770 --> 00:04:46.570 align:middle line:84% Someone comes to you and says, how should I play in this game? 00:04:46.570 --> 00:04:48.933 align:middle line:84% What is your recommendation for how I should play? 00:04:48.933 --> 00:04:50.850 align:middle line:84% And companies do sometimes hire game theorists 00:04:50.850 --> 00:04:53.033 align:middle line:84% to consult and make recommendations for maybe 00:04:53.033 --> 00:04:54.450 align:middle line:84% how they should bid in an auction. 00:04:54.450 --> 00:04:55.670 align:middle line:90% That would be one example. 00:04:55.670 --> 00:05:00.550 align:middle line:84% So here, this is sometimes called normative 00:05:00.550 --> 00:05:05.710 align:middle line:90% or prescriptive analysis. 00:05:05.710 --> 00:05:08.750 align:middle line:84% And the goal of our analysis is to provide 00:05:08.750 --> 00:05:17.310 align:middle line:84% what I might say is a recommendation for how players 00:05:17.310 --> 00:05:22.250 align:middle line:84% should play if they want to achieve their objectives 00:05:22.250 --> 00:05:25.050 align:middle line:84% and they recognize that other players are rational as well 00:05:25.050 --> 00:05:27.495 align:middle line:84% and maybe also are getting advice from game theorists 00:05:27.495 --> 00:05:29.870 align:middle line:84% or from other informed parties about how they should play 00:05:29.870 --> 00:05:30.910 align:middle line:90% in the game. 00:05:30.910 --> 00:05:33.750 align:middle line:84% But I just want you to keep this in mind, in the background, 00:05:33.750 --> 00:05:34.830 align:middle line:90% throughout the course. 00:05:34.830 --> 00:05:37.247 align:middle line:84% But when you're formally asked to solve for an equilibrium 00:05:37.247 --> 00:05:39.830 align:middle line:84% or to analyze a game, your answer 00:05:39.830 --> 00:05:42.510 align:middle line:84% won't really depend on your particular interpretation 00:05:42.510 --> 00:05:45.550 align:middle line:90% in that context. 00:05:45.550 --> 00:05:49.860 align:middle line:84% So I want to start with an example of a game, an example 00:05:49.860 --> 00:05:51.243 align:middle line:90% of a strategic form game. 00:05:51.243 --> 00:05:53.660 align:middle line:84% And this is the most famous game in game theory, probably. 00:05:53.660 --> 00:05:54.600 align:middle line:90% You may have heard of it. 00:05:54.600 --> 00:05:56.080 align:middle line:84% It's called the prisoner's dilemma. 00:05:56.080 --> 00:06:00.620 align:middle line:90% 00:06:00.620 --> 00:06:02.440 align:middle line:90% Make sure to [INAUDIBLE]. 00:06:02.440 --> 00:06:06.460 align:middle line:90% 00:06:06.460 --> 00:06:08.860 align:middle line:84% Now, the first thing I want to write down 00:06:08.860 --> 00:06:12.000 align:middle line:84% is think about how we can represent this game. 00:06:12.000 --> 00:06:14.500 align:middle line:84% It's kind of ugly to think about all these functions and all 00:06:14.500 --> 00:06:15.540 align:middle line:90% these sets. 00:06:15.540 --> 00:06:18.700 align:middle line:84% So normally, we represent strategic form games 00:06:18.700 --> 00:06:21.020 align:middle line:84% using what's called a by matrix, just a box 00:06:21.020 --> 00:06:23.220 align:middle line:90% with some numbers in it. 00:06:23.220 --> 00:06:26.100 align:middle line:84% So in this game, each player is going 00:06:26.100 --> 00:06:30.100 align:middle line:84% to have two strategies, which we'll call M and F. So 00:06:30.100 --> 00:06:31.520 align:middle line:90% let me tell the story here. 00:06:31.520 --> 00:06:32.840 align:middle line:90% So we have two prisoners. 00:06:32.840 --> 00:06:36.620 align:middle line:90% 00:06:36.620 --> 00:06:40.180 align:middle line:84% And we assume that they're placed in separate cells. 00:06:40.180 --> 00:06:42.260 align:middle line:84% Again, this is going to be a colorful story 00:06:42.260 --> 00:06:44.360 align:middle line:84% to help motivate this particular game. 00:06:44.360 --> 00:06:46.272 align:middle line:84% But the lessons from this game, I 00:06:46.272 --> 00:06:48.480 align:middle line:84% think a lot of game theorists and economists believe, 00:06:48.480 --> 00:06:51.280 align:middle line:84% are much broader than this kind of silly story 00:06:51.280 --> 00:06:52.760 align:middle line:90% that we're going to tell here. 00:06:52.760 --> 00:06:54.700 align:middle line:90% So we have two prisoners. 00:06:54.700 --> 00:06:56.640 align:middle line:84% And they've been accused of a crime. 00:06:56.640 --> 00:06:59.040 align:middle line:84% And each prisoner has two options. 00:06:59.040 --> 00:07:02.760 align:middle line:84% We'll call them M and F. So M is to stay mum-- 00:07:02.760 --> 00:07:05.360 align:middle line:84% that is, don't admit to anything. 00:07:05.360 --> 00:07:09.265 align:middle line:84% And F is sometimes the word is used fink or tattle. 00:07:09.265 --> 00:07:11.140 align:middle line:84% It's to tell on the other person-- say, yeah, 00:07:11.140 --> 00:07:12.760 align:middle line:90% the other guy did it. 00:07:12.760 --> 00:07:15.640 align:middle line:84% So the prosecutor comes by and says, 00:07:15.640 --> 00:07:19.120 align:middle line:84% do you want to accuse your co-conspirator 00:07:19.120 --> 00:07:20.640 align:middle line:90% of committing the crime? 00:07:20.640 --> 00:07:23.420 align:middle line:84% And the prosecutor says, look, we'll give you a good deal. 00:07:23.420 --> 00:07:27.680 align:middle line:84% So if you're willing to testify against the other prisoner, 00:07:27.680 --> 00:07:29.888 align:middle line:84% we'll give you a sentence reduction. 00:07:29.888 --> 00:07:32.180 align:middle line:84% But now, we're going to be able to catch the other guy. 00:07:32.180 --> 00:07:34.463 align:middle line:84% And the other guy is going to get a bigger sentence. 00:07:34.463 --> 00:07:35.880 align:middle line:84% So now, what we want to do is fill 00:07:35.880 --> 00:07:38.560 align:middle line:90% in the payoffs in this matrix. 00:07:38.560 --> 00:07:43.800 align:middle line:84% And what we imagine is, if both players stay silent, then 00:07:43.800 --> 00:07:47.060 align:middle line:90% both players get a payoff of 2. 00:07:47.060 --> 00:07:49.340 align:middle line:84% They both get convicted of, say, a minor crime, 00:07:49.340 --> 00:07:50.820 align:middle line:84% because they didn't have testimony 00:07:50.820 --> 00:07:52.540 align:middle line:90% to convict either of them. 00:07:52.540 --> 00:07:54.460 align:middle line:90% Now, let's say I'm player 1. 00:07:54.460 --> 00:08:01.860 align:middle line:84% So the way I'm writing this here is player 1 is the column-- 00:08:01.860 --> 00:08:03.540 align:middle line:90% sorry, the row player. 00:08:03.540 --> 00:08:06.440 align:middle line:90% Player 1 is choosing the row. 00:08:06.440 --> 00:08:09.460 align:middle line:84% So they're choosing between M and F. Player 2 00:08:09.460 --> 00:08:10.940 align:middle line:90% is the column player. 00:08:10.940 --> 00:08:13.660 align:middle line:84% They're choosing between the column m and F. 00:08:13.660 --> 00:08:16.500 align:middle line:84% So the strategy set-- maybe this is helpful here-- 00:08:16.500 --> 00:08:22.460 align:middle line:84% for each player S1 equals S2 equals M comma F. 00:08:22.460 --> 00:08:26.180 align:middle line:84% So if we want to formally connect it to our strategic form 00:08:26.180 --> 00:08:30.080 align:middle line:84% representation over here, each player has two strategies. 00:08:30.080 --> 00:08:31.800 align:middle line:84% They each have the same two strategies, 00:08:31.800 --> 00:08:35.179 align:middle line:84% simply staying silent, M, or testifying 00:08:35.179 --> 00:08:38.260 align:middle line:84% against the other player, finking, F. 00:08:38.260 --> 00:08:41.940 align:middle line:84% And then when we write 2, 2 in this box, the first number 00:08:41.940 --> 00:08:46.330 align:middle line:84% represents player 1's utility from this strategy profile 00:08:46.330 --> 00:08:47.930 align:middle line:90% corresponding to this box. 00:08:47.930 --> 00:08:52.050 align:middle line:84% So this first box corresponds to the strategy profile where 00:08:52.050 --> 00:08:55.650 align:middle line:84% player one chooses M and player 2 chooses M. 00:08:55.650 --> 00:08:58.890 align:middle line:84% And then in the box, we're writing player 1'2 payoff from 00:08:58.890 --> 00:09:01.450 align:middle line:84% that consequence or that strategy profile, 00:09:01.450 --> 00:09:04.690 align:middle line:84% and then player 2's payoff from that strategy profile. 00:09:04.690 --> 00:09:05.710 align:middle line:90% Now, let's go down here. 00:09:05.710 --> 00:09:08.410 align:middle line:84% We want to say what happens if player 1 instead 00:09:08.410 --> 00:09:11.690 align:middle line:84% testifies or finks against the other prisoner. 00:09:11.690 --> 00:09:13.890 align:middle line:84% What this should do is it should increase 00:09:13.890 --> 00:09:17.610 align:middle line:84% the utility of the prisoner who finks, because that prisoner 00:09:17.610 --> 00:09:19.570 align:middle line:90% gets a sentence reduction. 00:09:19.570 --> 00:09:22.635 align:middle line:84% But it decreases the utility of the other prisoner, 00:09:22.635 --> 00:09:24.010 align:middle line:84% because the other prisoner is now 00:09:24.010 --> 00:09:26.650 align:middle line:84% going to be convicted of a more serious crime. 00:09:26.650 --> 00:09:29.870 align:middle line:84% So the exact numbers we'll use are 3 and negative 1. 00:09:29.870 --> 00:09:34.690 align:middle line:90% 00:09:34.690 --> 00:09:37.630 align:middle line:84% So relative to the case where they both stay silent, 00:09:37.630 --> 00:09:42.430 align:middle line:84% if I fink on the other prisoner, my utility goes up by 1 util. 00:09:42.430 --> 00:09:47.430 align:middle line:84% But my opponent's utility goes down by 3 utils. 00:09:47.430 --> 00:09:50.390 align:middle line:84% And now, we're going to put the symmetric payoffs over here. 00:09:50.390 --> 00:09:54.750 align:middle line:84% But remember, over here, it's prisoner 2 who's testifying. 00:09:54.750 --> 00:09:57.150 align:middle line:84% And therefore, prisoner 2 gets the higher payoff. 00:09:57.150 --> 00:09:59.150 align:middle line:84% And prisoner 1 gets the lower payoff. 00:09:59.150 --> 00:10:03.923 align:middle line:84% So here, we're going to put negative 1, 3 in this box. 00:10:03.923 --> 00:10:05.590 align:middle line:84% And then finally, we have to decide what 00:10:05.590 --> 00:10:08.430 align:middle line:90% happens if they both testify. 00:10:08.430 --> 00:10:11.070 align:middle line:90% And here, the payoff is 0, 0. 00:10:11.070 --> 00:10:13.510 align:middle line:84% So the exact numbers aren't so important. 00:10:13.510 --> 00:10:15.330 align:middle line:84% But the order is pretty important. 00:10:15.330 --> 00:10:17.030 align:middle line:90% So why is it 0 here? 00:10:17.030 --> 00:10:21.830 align:middle line:84% Well, the other person is testifying against me. 00:10:21.830 --> 00:10:24.670 align:middle line:84% So I get convicted of a pretty serious crime. 00:10:24.670 --> 00:10:27.530 align:middle line:84% But it's not quite as bad as the negative 1 over here, 00:10:27.530 --> 00:10:29.990 align:middle line:84% because I still get a little bit of a sentence reduction 00:10:29.990 --> 00:10:32.510 align:middle line:84% for testifying against the other guy. 00:10:32.510 --> 00:10:34.830 align:middle line:84% So you can see, there's an ordering 00:10:34.830 --> 00:10:36.010 align:middle line:90% between the four outcomes. 00:10:36.010 --> 00:10:37.510 align:middle line:84% And this is really what's important. 00:10:37.510 --> 00:10:41.890 align:middle line:90% 00:10:41.890 --> 00:10:46.170 align:middle line:84% So the highest payoff I get is if I testify and the other guy 00:10:46.170 --> 00:10:48.290 align:middle line:90% doesn't. 00:10:48.290 --> 00:10:51.590 align:middle line:84% The worst payoff is if the other prisoner testifies 00:10:51.590 --> 00:10:53.890 align:middle line:84% and I don't, because then I get convicted 00:10:53.890 --> 00:10:56.250 align:middle line:84% and I don't get any sentence reduction. 00:10:56.250 --> 00:10:58.450 align:middle line:84% And then in between, we have the payoff 00:10:58.450 --> 00:11:01.650 align:middle line:84% where we both stay silent, because we only get charged 00:11:01.650 --> 00:11:04.170 align:middle line:90% with a pretty small crime. 00:11:04.170 --> 00:11:05.810 align:middle line:84% And then the 0 here is where we both 00:11:05.810 --> 00:11:08.430 align:middle line:84% testify, we both get charged with a serious crime, 00:11:08.430 --> 00:11:12.210 align:middle line:84% but we get a sentence reduction because we each cooperated 00:11:12.210 --> 00:11:13.690 align:middle line:90% with the authorities. 00:11:13.690 --> 00:11:17.010 align:middle line:90% 00:11:17.010 --> 00:11:21.010 align:middle line:84% Now, I guess the question is, how would you play in this game? 00:11:21.010 --> 00:11:23.010 align:middle line:84% If you were here, you were one of the prisoners, 00:11:23.010 --> 00:11:25.550 align:middle line:84% you're alone in your cell, how would you think about this? 00:11:25.550 --> 00:11:28.310 align:middle line:84% And how would you expect the other person to play? 00:11:28.310 --> 00:11:34.130 align:middle line:90% 00:11:34.130 --> 00:11:35.290 align:middle line:90% STUDENT: [INAUDIBLE] go F. 00:11:35.290 --> 00:11:36.370 align:middle line:90% IAN BALL: Say again? 00:11:36.370 --> 00:11:36.890 align:middle line:90% STUDENT: [INAUDIBLE] go F. 00:11:36.890 --> 00:11:37.932 align:middle line:90% IAN BALL: You would go F? 00:11:37.932 --> 00:11:39.976 align:middle line:90% OK, tell me why. 00:11:39.976 --> 00:11:42.390 align:middle line:84% STUDENT: Because the other one doesn't, then you're 00:11:42.390 --> 00:11:45.070 align:middle line:90% just [INAUDIBLE]. 00:11:45.070 --> 00:11:47.550 align:middle line:84% IAN BALL: So you're saying you want 00:11:47.550 --> 00:11:50.110 align:middle line:84% to go F because you're really afraid of-- so let's say 00:11:50.110 --> 00:11:50.970 align:middle line:90% you're player 1. 00:11:50.970 --> 00:11:54.830 align:middle line:84% So you're going to say, if I go F, you're saying, well, 00:11:54.830 --> 00:11:56.250 align:middle line:90% I can either get 3 or 0. 00:11:56.250 --> 00:11:58.130 align:middle line:84% But I'm at least not going to get negative 1. 00:11:58.130 --> 00:11:59.163 align:middle line:90% So that's good. 00:11:59.163 --> 00:12:00.830 align:middle line:84% So certainly, the worst possible outcome 00:12:00.830 --> 00:12:04.890 align:middle line:84% is better if I go F. I think F is a good strategy. 00:12:04.890 --> 00:12:08.470 align:middle line:84% Any other thoughts about why I might want to go F? 00:12:08.470 --> 00:12:09.070 align:middle line:90% Yeah? 00:12:09.070 --> 00:12:10.790 align:middle line:84% STUDENT: Regardless of what the other player chooses, 00:12:10.790 --> 00:12:11.650 align:middle line:90% you're better off. 00:12:11.650 --> 00:12:12.323 align:middle line:90% IAN BALL: Right. 00:12:12.323 --> 00:12:13.990 align:middle line:84% So this is a really critical observation 00:12:13.990 --> 00:12:15.390 align:middle line:90% to make in this game. 00:12:15.390 --> 00:12:17.110 align:middle line:90% So I'm player 1. 00:12:17.110 --> 00:12:19.830 align:middle line:84% I don't know whether the other prisoner 00:12:19.830 --> 00:12:21.543 align:middle line:90% is going to mum or fink. 00:12:21.543 --> 00:12:23.710 align:middle line:84% I don't know if they're going to testify against me. 00:12:23.710 --> 00:12:26.550 align:middle line:84% But let me reason separately by cases, OK? 00:12:26.550 --> 00:12:29.070 align:middle line:84% One possibility is the other prisoner 00:12:29.070 --> 00:12:30.750 align:middle line:90% is going to stay silent. 00:12:30.750 --> 00:12:33.350 align:middle line:84% If the other prisoner stays silent, well, what happens? 00:12:33.350 --> 00:12:36.580 align:middle line:84% Either I get a payoff of 2, if I stay silent, 00:12:36.580 --> 00:12:39.340 align:middle line:84% or I get a payoff of 3, if I testify. 00:12:39.340 --> 00:12:41.260 align:middle line:90% So I'm better off testifying. 00:12:41.260 --> 00:12:42.820 align:middle line:84% Now, let's look at the other case. 00:12:42.820 --> 00:12:46.040 align:middle line:84% What if the other prisoner is going to testify against me? 00:12:46.040 --> 00:12:47.360 align:middle line:90% That's certainly worse for me. 00:12:47.360 --> 00:12:48.660 align:middle line:90% I'm certainly worse off. 00:12:48.660 --> 00:12:51.280 align:middle line:84% But if the other prisoner testifies, well, 00:12:51.280 --> 00:12:54.180 align:middle line:84% if I stay silent, I get negative 1. 00:12:54.180 --> 00:12:56.820 align:middle line:84% And if I fink, I get 0, which is higher? 00:12:56.820 --> 00:13:00.580 align:middle line:84% So in both cases, I do strictly better if I fink. 00:13:00.580 --> 00:13:04.180 align:middle line:84% So often, we think that a prediction of this game 00:13:04.180 --> 00:13:07.660 align:middle line:84% might be we might be pretty confident that, 00:13:07.660 --> 00:13:08.640 align:middle line:90% for each player-- 00:13:08.640 --> 00:13:12.448 align:middle line:90% 00:13:12.448 --> 00:13:13.240 align:middle line:90% what are we saying? 00:13:13.240 --> 00:13:19.700 align:middle line:84% Well, F is better than M no matter 00:13:19.700 --> 00:13:20.920 align:middle line:90% what the other player does. 00:13:20.920 --> 00:13:34.520 align:middle line:90% 00:13:34.520 --> 00:13:37.780 align:middle line:84% And in a sense, games like this are a bit easier to analyze. 00:13:37.780 --> 00:13:40.240 align:middle line:84% Because, remember, we said what makes game theory hard 00:13:40.240 --> 00:13:42.920 align:middle line:84% is what we called strategic interdependence, where 00:13:42.920 --> 00:13:44.920 align:middle line:84% the best thing for me to do depends 00:13:44.920 --> 00:13:47.360 align:middle line:90% on what the other player does. 00:13:47.360 --> 00:13:49.760 align:middle line:84% In this case, it is an interactive problem. 00:13:49.760 --> 00:13:51.640 align:middle line:84% But the best thing for me to do is 00:13:51.640 --> 00:13:53.898 align:middle line:84% independent of what my opponent does. 00:13:53.898 --> 00:13:55.440 align:middle line:84% So there actually is not this element 00:13:55.440 --> 00:13:57.120 align:middle line:90% of strategic interdependence. 00:13:57.120 --> 00:13:59.480 align:middle line:84% F is better than M if my opponent 00:13:59.480 --> 00:14:04.240 align:middle line:84% plays M. And F is still better than M if my opponent plays F. 00:14:04.240 --> 00:14:07.080 align:middle line:84% So I don't really have to think too carefully about exactly 00:14:07.080 --> 00:14:10.320 align:middle line:84% how my opponent is going to play because, however they play, I 00:14:10.320 --> 00:14:14.200 align:middle line:90% know I'm better off if I play F. 00:14:14.200 --> 00:14:16.100 align:middle line:90% So why is this called a dilemma? 00:14:16.100 --> 00:14:18.640 align:middle line:84% Or sometimes, people think of it as a paradox. 00:14:18.640 --> 00:14:23.880 align:middle line:84% Well, if both players play F, then-- 00:14:23.880 --> 00:14:27.840 align:middle line:84% if I have chalk, but I'll just highlight this-- 00:14:27.840 --> 00:14:30.172 align:middle line:90% we both get a payoff of 0. 00:14:30.172 --> 00:14:32.380 align:middle line:84% But I think it's reasonable for someone to come along 00:14:32.380 --> 00:14:34.380 align:middle line:84% and say, wait a second, these prisoners 00:14:34.380 --> 00:14:36.380 align:middle line:90% must be doing something wrong. 00:14:36.380 --> 00:14:38.340 align:middle line:84% They're both getting a payoff of 0. 00:14:38.340 --> 00:14:42.300 align:middle line:84% But if they both did M, they'd both be strictly better off. 00:14:42.300 --> 00:14:43.508 align:middle line:90% So are they being irrational? 00:14:43.508 --> 00:14:44.800 align:middle line:90% Are they doing something wrong? 00:14:44.800 --> 00:14:46.160 align:middle line:90% In this game, what do you think? 00:14:46.160 --> 00:14:48.382 align:middle line:84% I mean, why is this a good prediction of the game? 00:14:48.382 --> 00:14:49.840 align:middle line:84% We said we're studying game theory. 00:14:49.840 --> 00:14:51.360 align:middle line:84% We said people are very rational. 00:14:51.360 --> 00:14:53.040 align:middle line:84% They act towards their objectives. 00:14:53.040 --> 00:14:54.060 align:middle line:84% And if they both do this, they're 00:14:54.060 --> 00:14:55.160 align:middle line:90% both strictly better off. 00:14:55.160 --> 00:14:56.900 align:middle line:90% So why don't they do that? 00:14:56.900 --> 00:14:58.400 align:middle line:84% Is something wrong with game theory? 00:14:58.400 --> 00:15:00.940 align:middle line:90% 00:15:00.940 --> 00:15:01.840 align:middle line:90% Yeah? 00:15:01.840 --> 00:15:05.020 align:middle line:84% STUDENT: If one player doesn't know what the other player is 00:15:05.020 --> 00:15:09.100 align:middle line:84% going to do, they're just operating in like 00:15:09.100 --> 00:15:14.300 align:middle line:84% just maximize my utility independent of what 00:15:14.300 --> 00:15:15.840 align:middle line:90% the other person chooses. 00:15:15.840 --> 00:15:16.540 align:middle line:90% IAN BALL: Right. 00:15:16.540 --> 00:15:17.915 align:middle line:84% And another way of saying it is I 00:15:17.915 --> 00:15:19.880 align:middle line:84% can't influence what the other person chooses. 00:15:19.880 --> 00:15:22.200 align:middle line:84% So when I framed it, I said, oh, we're down here. 00:15:22.200 --> 00:15:24.420 align:middle line:90% Wouldn't we like to be up here? 00:15:24.420 --> 00:15:25.920 align:middle line:90% But how do we get up here? 00:15:25.920 --> 00:15:28.602 align:middle line:84% Well, both players would have to behave differently. 00:15:28.602 --> 00:15:30.560 align:middle line:84% And I can't control what the other person does. 00:15:30.560 --> 00:15:32.480 align:middle line:90% I can only control what I do. 00:15:32.480 --> 00:15:36.360 align:middle line:84% And I think this observation was surprising to a lot of people 00:15:36.360 --> 00:15:36.900 align:middle line:90% early on. 00:15:36.900 --> 00:15:39.140 align:middle line:84% I mean, mathematically, it's not very deep. 00:15:39.140 --> 00:15:40.840 align:middle line:84% But it definitely surprised people. 00:15:40.840 --> 00:15:44.400 align:middle line:84% And I think the key interpretation of the prisoner's 00:15:44.400 --> 00:15:51.720 align:middle line:84% dilemma, or what we learned from it, is that, in games, 00:15:51.720 --> 00:15:56.280 align:middle line:84% rational behavior can lead to inefficiency. 00:15:56.280 --> 00:16:01.920 align:middle line:84% Or more precisely, rational behavior 00:16:01.920 --> 00:16:03.380 align:middle line:90% can make everyone worse off. 00:16:03.380 --> 00:16:14.600 align:middle line:90% 00:16:14.600 --> 00:16:16.143 align:middle line:84% Worse off than what, you might say? 00:16:16.143 --> 00:16:18.060 align:middle line:84% Well, what I mean is than some other behavior. 00:16:18.060 --> 00:16:26.920 align:middle line:90% 00:16:26.920 --> 00:16:29.790 align:middle line:84% And this is a novel implication or a novel feature 00:16:29.790 --> 00:16:31.670 align:middle line:90% of interactive decision making. 00:16:31.670 --> 00:16:34.830 align:middle line:84% In a single decision maker problem, this never happens. 00:16:34.830 --> 00:16:37.650 align:middle line:84% In a single agent problem, I just do what's best for me. 00:16:37.650 --> 00:16:39.630 align:middle line:84% And that gives me the best possible outcome. 00:16:39.630 --> 00:16:42.910 align:middle line:84% But with an interactive problem, even if we're both behaving, 00:16:42.910 --> 00:16:45.950 align:middle line:84% quote unquote, "rationally," the outcome may be very bad for both 00:16:45.950 --> 00:16:48.910 align:middle line:84% of us and may be worse than a different outcome that would be 00:16:48.910 --> 00:16:50.410 align:middle line:90% feasible for both of us. 00:16:50.410 --> 00:16:56.470 align:middle line:90% 00:16:56.470 --> 00:17:00.710 align:middle line:84% So I think today we want to continue what we started here. 00:17:00.710 --> 00:17:04.829 align:middle line:84% And we argued that this game was relatively 00:17:04.829 --> 00:17:08.030 align:middle line:84% easy to analyze because of this property, that the best 00:17:08.030 --> 00:17:10.727 align:middle line:84% thing for me is independent of what the other player does. 00:17:10.727 --> 00:17:13.310 align:middle line:84% And I think that's a guide for the way we're going to approach 00:17:13.310 --> 00:17:14.730 align:middle line:90% analysis in this class. 00:17:14.730 --> 00:17:17.630 align:middle line:84% So moving forward, we're going to start 00:17:17.630 --> 00:17:23.950 align:middle line:84% with minimal assumptions, minimal assumptions 00:17:23.950 --> 00:17:24.890 align:middle line:90% about behavior. 00:17:24.890 --> 00:17:31.570 align:middle line:90% 00:17:31.570 --> 00:17:34.090 align:middle line:84% And because these are minimal or weak assumptions, 00:17:34.090 --> 00:17:37.010 align:middle line:84% we're going to be quite confident in the implications 00:17:37.010 --> 00:17:38.170 align:middle line:90% of these assumptions. 00:17:38.170 --> 00:17:40.790 align:middle line:84% So this might make us confident about predictions. 00:17:40.790 --> 00:17:48.450 align:middle line:90% 00:17:48.450 --> 00:17:50.250 align:middle line:84% But what we'll see is that a lot of games 00:17:50.250 --> 00:17:52.010 align:middle line:84% are not like the prisoner's dilemma. 00:17:52.010 --> 00:17:55.250 align:middle line:84% And we're not able to make very sharp predictions only based 00:17:55.250 --> 00:17:56.610 align:middle line:90% on minimal assumptions. 00:17:56.610 --> 00:17:58.710 align:middle line:84% So we might be confident about our predictions. 00:17:58.710 --> 00:18:00.168 align:middle line:84% But these predictions are generally 00:18:00.168 --> 00:18:01.970 align:middle line:84% going to be very weak predictions. 00:18:01.970 --> 00:18:04.070 align:middle line:84% Or they could be very weak predictions. 00:18:04.070 --> 00:18:06.650 align:middle line:90% 00:18:06.650 --> 00:18:11.290 align:middle line:84% And here, notice, I'm using the interpretation of prediction 00:18:11.290 --> 00:18:13.490 align:middle line:84% rather than recommendation for now. 00:18:13.490 --> 00:18:15.370 align:middle line:84% In fact, sometimes, if we're only 00:18:15.370 --> 00:18:18.170 align:middle line:84% willing to make minimal assumptions about behavior, 00:18:18.170 --> 00:18:20.810 align:middle line:84% our only prediction might be anything could happen. 00:18:20.810 --> 00:18:22.350 align:middle line:84% That's not a very useful prediction. 00:18:22.350 --> 00:18:24.950 align:middle line:84% We might not be able to narrow things down very much. 00:18:24.950 --> 00:18:27.790 align:middle line:84% So what we want to do, over the course of the class, 00:18:27.790 --> 00:18:30.430 align:middle line:84% is gradually strengthen the assumptions 00:18:30.430 --> 00:18:32.110 align:middle line:90% that we make about behavior. 00:18:32.110 --> 00:18:34.550 align:middle line:84% We might gradually lose some confidence 00:18:34.550 --> 00:18:36.330 align:middle line:84% in the accuracy of these predictions. 00:18:36.330 --> 00:18:38.705 align:middle line:84% But at the same time, we'll be able to sharpen and refine 00:18:38.705 --> 00:18:39.913 align:middle line:90% the predictions that we make. 00:18:39.913 --> 00:18:41.710 align:middle line:84% And that's the direction we're going to go. 00:18:41.710 --> 00:18:43.377 align:middle line:84% So today, we're going to start with what 00:18:43.377 --> 00:18:45.670 align:middle line:84% I think of as the most basic or most minimal 00:18:45.670 --> 00:18:47.190 align:middle line:90% assumption about behavior. 00:18:47.190 --> 00:18:50.130 align:middle line:84% And later on, we'll move to more complicated assumptions. 00:18:50.130 --> 00:18:54.990 align:middle line:90% 00:18:54.990 --> 00:18:57.750 align:middle line:84% So the first thing I need to do is just introduce some notation 00:18:57.750 --> 00:19:00.910 align:middle line:90% for strategic form games. 00:19:00.910 --> 00:19:03.990 align:middle line:84% And then I can formally define and capture 00:19:03.990 --> 00:19:05.990 align:middle line:90% what we described in this game. 00:19:05.990 --> 00:19:09.533 align:middle line:84% So strategic form games often have many players. 00:19:09.533 --> 00:19:10.950 align:middle line:84% But we're often interested in what 00:19:10.950 --> 00:19:13.770 align:middle line:84% happens when a single player modifies their behavior, 00:19:13.770 --> 00:19:15.070 align:middle line:90% modifies their strategy. 00:19:15.070 --> 00:19:17.870 align:middle line:84% So we'd like to have some notation to capture that. 00:19:17.870 --> 00:19:26.500 align:middle line:84% And what we're going to write is a strategy profile will often 00:19:26.500 --> 00:19:28.180 align:middle line:90% decompose it in this way. 00:19:28.180 --> 00:19:32.040 align:middle line:84% So we'll say we can think of the profile as player i's strategy. 00:19:32.040 --> 00:19:35.140 align:middle line:90% 00:19:35.140 --> 00:19:38.040 align:middle line:84% And S negative i's specifies the strategies of everyone else. 00:19:38.040 --> 00:19:44.780 align:middle line:90% 00:19:44.780 --> 00:19:46.120 align:middle line:90% And this notation is important. 00:19:46.120 --> 00:19:48.780 align:middle line:84% So let's just go through an example with three players. 00:19:48.780 --> 00:19:54.020 align:middle line:84% If S equals S1, S2, S3, this strategy profile 00:19:54.020 --> 00:19:56.802 align:middle line:84% specifies a strategy for each of the three players. 00:19:56.802 --> 00:19:58.260 align:middle line:84% But there's three different ways we 00:19:58.260 --> 00:20:00.220 align:middle line:84% can express this depending on which player 00:20:00.220 --> 00:20:01.700 align:middle line:90% we want to focus on. 00:20:01.700 --> 00:20:06.300 align:middle line:84% So we could write this as S1, S negative 1, 00:20:06.300 --> 00:20:09.400 align:middle line:84% meaning we're going to put player 1's strategy by itself. 00:20:09.400 --> 00:20:12.540 align:middle line:84% And then we're going to group the other two players' strategy 00:20:12.540 --> 00:20:15.300 align:middle line:84% under this notation S negative 1. 00:20:15.300 --> 00:20:18.080 align:middle line:84% Here, negative 1 is not really the number negative 1. 00:20:18.080 --> 00:20:19.860 align:middle line:90% What we mean is not player 1. 00:20:19.860 --> 00:20:24.578 align:middle line:84% So think of the minus here as not rather than minus. 00:20:24.578 --> 00:20:26.620 align:middle line:84% But then there's another way we could write this. 00:20:26.620 --> 00:20:32.600 align:middle line:84% We could write it as S2 S negative 2 or as S3 S 00:20:32.600 --> 00:20:34.800 align:middle line:90% negative 3. 00:20:34.800 --> 00:20:37.467 align:middle line:84% And you might object that, well, S3 is supposed to come last. 00:20:37.467 --> 00:20:39.800 align:middle line:84% We're not really going to be consistent about the order. 00:20:39.800 --> 00:20:41.920 align:middle line:84% As long as we understand what this means, 00:20:41.920 --> 00:20:45.200 align:middle line:84% this is a strategy profile where player three 00:20:45.200 --> 00:20:48.200 align:middle line:84% chooses S3 and players 1 and 2 collectively 00:20:48.200 --> 00:20:49.308 align:middle line:90% choose S negative 3. 00:20:49.308 --> 00:20:50.100 align:middle line:90% It's just notation. 00:20:50.100 --> 00:20:52.500 align:middle line:84% But it's going to help us write things down more cleanly. 00:20:52.500 --> 00:20:53.000 align:middle line:90% Yeah? 00:20:53.000 --> 00:20:55.760 align:middle line:84% STUDENT: So I need to add more of the [INAUDIBLE] 00:20:55.760 --> 00:20:57.720 align:middle line:90% first, [INAUDIBLE]. 00:20:57.720 --> 00:21:00.040 align:middle line:84% IAN BALL: Sorry, I don't mean that. 00:21:00.040 --> 00:21:01.857 align:middle line:84% I'm not making any-- it's just notation, 00:21:01.857 --> 00:21:02.940 align:middle line:90% no behavioral assumptions. 00:21:02.940 --> 00:21:04.980 align:middle line:84% So literally, when I write S negative 3, 00:21:04.980 --> 00:21:09.000 align:middle line:90% what I mean is S1, S2. 00:21:09.000 --> 00:21:11.420 align:middle line:90% It's just notation. 00:21:11.420 --> 00:21:14.160 align:middle line:84% So I'm not claiming that they're going to decide together. 00:21:14.160 --> 00:21:15.660 align:middle line:84% Or I'm not doing anything like that. 00:21:15.660 --> 00:21:17.300 align:middle line:84% It's just to save space on the board. 00:21:17.300 --> 00:21:20.640 align:middle line:84% Instead of writing S1, S2, I'm going to write S negative 3. 00:21:20.640 --> 00:21:23.320 align:middle line:84% But they're still choosing independently of each other. 00:21:23.320 --> 00:21:24.640 align:middle line:90% That's a good clarification. 00:21:24.640 --> 00:21:27.020 align:middle line:90% Thank you. 00:21:27.020 --> 00:21:29.980 align:middle line:84% And then we need some notation for where these things live. 00:21:29.980 --> 00:21:31.780 align:middle line:90% So we're going to use notation. 00:21:31.780 --> 00:21:33.660 align:middle line:90% We've already introduced Si. 00:21:33.660 --> 00:21:35.820 align:middle line:84% But now, we're going to use S negative i 00:21:35.820 --> 00:21:38.900 align:middle line:84% to be the set of-- you might call them partial strategy 00:21:38.900 --> 00:21:39.800 align:middle line:90% profiles. 00:21:39.800 --> 00:21:41.700 align:middle line:84% These are the set of profiles specifying 00:21:41.700 --> 00:21:44.780 align:middle line:84% a strategy for every player except player i. 00:21:44.780 --> 00:21:47.820 align:middle line:90% So S little i lives in SI. 00:21:47.820 --> 00:21:51.260 align:middle line:84% And S negative i, or not i, lives in S0 00:21:51.260 --> 00:21:53.080 align:middle line:84% So that notation will be useful below. 00:21:53.080 --> 00:21:58.284 align:middle line:90% 00:21:58.284 --> 00:22:00.780 align:middle line:84% So now, we'd like to formalize what happened 00:22:00.780 --> 00:22:02.140 align:middle line:90% in the prisoner's dilemma. 00:22:02.140 --> 00:22:10.700 align:middle line:84% And the formal term for this is dominance, a natural term 00:22:10.700 --> 00:22:13.020 align:middle line:84% because F is better than M no matter 00:22:13.020 --> 00:22:14.540 align:middle line:90% what the other player does. 00:22:14.540 --> 00:22:17.988 align:middle line:84% We would say that strategy F dominates strategy M. 00:22:17.988 --> 00:22:20.280 align:middle line:84% But we have to be a little more careful because there's 00:22:20.280 --> 00:22:21.620 align:middle line:90% strict and weak dominance. 00:22:21.620 --> 00:22:25.400 align:middle line:84% So now, I'm going to define these two terms. 00:22:25.400 --> 00:22:31.440 align:middle line:90% So for player i-- 00:22:31.440 --> 00:22:33.080 align:middle line:90% so let's consider player i. 00:22:33.080 --> 00:22:36.820 align:middle line:84% And let's consider two strategies-- 00:22:36.820 --> 00:22:39.960 align:middle line:90% 00:22:39.960 --> 00:22:41.860 align:middle line:90% Si and Si prime. 00:22:41.860 --> 00:22:44.440 align:middle line:90% 00:22:44.440 --> 00:22:47.280 align:middle line:84% And I want to say, so I'm focusing on player i. 00:22:47.280 --> 00:22:50.760 align:middle line:84% I'm focusing on two strategies-- they could be the same, 00:22:50.760 --> 00:22:52.960 align:middle line:90% but two strategies of player i. 00:22:52.960 --> 00:22:55.440 align:middle line:90% This is a prime here. 00:22:55.440 --> 00:22:59.640 align:middle line:84% And I want to formally define what it means for strategy Si 00:22:59.640 --> 00:23:03.440 align:middle line:84% to strictly dominate strategy S negative i. 00:23:03.440 --> 00:23:07.580 align:middle line:84% So I'm going to say Si strictly dominates-- 00:23:07.580 --> 00:23:10.460 align:middle line:90% 00:23:10.460 --> 00:23:11.960 align:middle line:84% sorry, I may have said S negative i. 00:23:11.960 --> 00:23:13.000 align:middle line:90% I mean Si prime. 00:23:13.000 --> 00:23:15.550 align:middle line:90% 00:23:15.550 --> 00:23:18.230 align:middle line:84% Can anyone guess in words, strict dominates, 00:23:18.230 --> 00:23:21.150 align:middle line:84% what that might mean, just if we think through from the example, 00:23:21.150 --> 00:23:24.110 align:middle line:84% what it might mean for player i if strategy Si strictly 00:23:24.110 --> 00:23:27.460 align:middle line:90% dominates Si prime? 00:23:27.460 --> 00:23:28.210 align:middle line:90% Maybe at the back? 00:23:28.210 --> 00:23:28.910 align:middle line:90% Yeah? 00:23:28.910 --> 00:23:30.550 align:middle line:84% STUDENT: They have a better payoff 00:23:30.550 --> 00:23:32.910 align:middle line:90% regardless of S negative i. 00:23:32.910 --> 00:23:33.910 align:middle line:90% IAN BALL: Exactly right. 00:23:33.910 --> 00:23:36.590 align:middle line:90% And by better, strictly better. 00:23:36.590 --> 00:23:40.430 align:middle line:90% So Si dominates Si prime means-- 00:23:40.430 --> 00:23:44.330 align:middle line:84% sometimes I'll use this arrow just to mean a definition. 00:23:44.330 --> 00:23:47.790 align:middle line:84% So I'm saying this is exactly what this phrase means. 00:23:47.790 --> 00:23:51.510 align:middle line:84% It means that-- well, let me first just partially 00:23:51.510 --> 00:23:52.290 align:middle line:90% write it out. 00:23:52.290 --> 00:23:57.790 align:middle line:90% 00:23:57.790 --> 00:24:00.152 align:middle line:84% So I'm going to fill it in in a second. 00:24:00.152 --> 00:24:02.110 align:middle line:84% But the first thing is I'm looking-- let's just 00:24:02.110 --> 00:24:03.630 align:middle line:90% focus on what's here already. 00:24:03.630 --> 00:24:05.750 align:middle line:90% I'm focusing on player i. 00:24:05.750 --> 00:24:08.430 align:middle line:84% And player i is making a comparison between things 00:24:08.430 --> 00:24:10.430 align:middle line:90% that are in player i's control. 00:24:10.430 --> 00:24:13.330 align:middle line:84% Player i is comparing what happens 00:24:13.330 --> 00:24:17.810 align:middle line:84% if she plays Si to what happens if she plays Si prime. 00:24:17.810 --> 00:24:20.930 align:middle line:84% But what we want to argue is that this inequality holds 00:24:20.930 --> 00:24:23.290 align:middle line:90% whatever the opponents play. 00:24:23.290 --> 00:24:26.210 align:middle line:84% So we're going to fill in S negative i here 00:24:26.210 --> 00:24:27.610 align:middle line:90% and S negative i here. 00:24:27.610 --> 00:24:33.730 align:middle line:84% And we want this to hold for all S negative i in S negative I. 00:24:33.730 --> 00:24:36.250 align:middle line:90% So notice a key point here. 00:24:36.250 --> 00:24:38.970 align:middle line:84% S negative i can take any possible value. 00:24:38.970 --> 00:24:41.570 align:middle line:84% But in the inequality, the same S negative i 00:24:41.570 --> 00:24:43.130 align:middle line:90% appears on both sides. 00:24:43.130 --> 00:24:45.410 align:middle line:84% So fixing the way that my opponent is playing, 00:24:45.410 --> 00:24:50.010 align:middle line:84% I'm strictly preferred to play Si to playing Si prime. 00:24:50.010 --> 00:24:51.450 align:middle line:90% Why do we write it this way? 00:24:51.450 --> 00:24:53.890 align:middle line:84% Because S negative i's outside my control. 00:24:53.890 --> 00:24:55.790 align:middle line:84% I can't control what my opponents are doing. 00:24:55.790 --> 00:24:57.130 align:middle line:90% But they're doing something. 00:24:57.130 --> 00:25:01.378 align:middle line:84% So however they're behaving, this inequality should hold. 00:25:01.378 --> 00:25:03.670 align:middle line:84% Now, we want to give a slightly weaker version of this. 00:25:03.670 --> 00:25:04.850 align:middle line:90% This is very demanding. 00:25:04.850 --> 00:25:06.090 align:middle line:90% And it may not hold. 00:25:06.090 --> 00:25:09.370 align:middle line:84% So let's now say weakly dominates. 00:25:09.370 --> 00:25:14.790 align:middle line:84% I won't fill out everything else, but just copy these words. 00:25:14.790 --> 00:25:18.430 align:middle line:84% And now, we're going to weaken this to weak inequality. 00:25:18.430 --> 00:25:30.990 align:middle line:90% 00:25:30.990 --> 00:25:34.870 align:middle line:84% But this is a little too weak because this allows these two 00:25:34.870 --> 00:25:38.370 align:middle line:84% strategies to give me the same payoff whatever my opponents do. 00:25:38.370 --> 00:25:40.990 align:middle line:84% And we want there to be some reason that Si is better than Si 00:25:40.990 --> 00:25:41.670 align:middle line:90% prime. 00:25:41.670 --> 00:25:43.190 align:middle line:84% So we demand this weak inequality 00:25:43.190 --> 00:25:52.430 align:middle line:84% for every S negative i, but strict inequality 00:25:52.430 --> 00:25:53.890 align:middle line:90% for some S negative i. 00:25:53.890 --> 00:25:57.310 align:middle line:90% 00:25:57.310 --> 00:26:00.350 align:middle line:84% So however my opponents play, Si does 00:26:00.350 --> 00:26:02.190 align:middle line:90% weakly better than Si prime. 00:26:02.190 --> 00:26:04.430 align:middle line:84% But there's some, at least one particular way 00:26:04.430 --> 00:26:08.940 align:middle line:84% my opponents could play such that Si is just strictly better 00:26:08.940 --> 00:26:11.620 align:middle line:90% than Si prime. 00:26:11.620 --> 00:26:13.700 align:middle line:84% And in particular, the strict part of it 00:26:13.700 --> 00:26:18.300 align:middle line:84% means that no strategy can weakly dominate itself, 00:26:18.300 --> 00:26:21.740 align:middle line:84% because if we put the same strategy here for Si and Si 00:26:21.740 --> 00:26:24.400 align:middle line:84% prime, then we would never get a strict inequality. 00:26:24.400 --> 00:26:27.780 align:middle line:90% 00:26:27.780 --> 00:26:31.760 align:middle line:84% So far, these definitions have been pairwise comparisons. 00:26:31.760 --> 00:26:33.900 align:middle line:84% We've taken one strategy and we've compared it 00:26:33.900 --> 00:26:35.300 align:middle line:90% to another strategy. 00:26:35.300 --> 00:26:37.960 align:middle line:84% But in reality, the player just needs to choose one strategy. 00:26:37.960 --> 00:26:39.335 align:middle line:84% So what we'd like to know is when 00:26:39.335 --> 00:26:42.060 align:middle line:84% is one strategy better than everything else or dominates 00:26:42.060 --> 00:26:43.140 align:middle line:90% everything else. 00:26:43.140 --> 00:26:47.400 align:middle line:84% So we'll say a strategy Si is weakly dominant-- 00:26:47.400 --> 00:26:52.180 align:middle line:90% 00:26:52.180 --> 00:26:53.660 align:middle line:90% we could do weakly or strictly. 00:26:53.660 --> 00:27:00.525 align:middle line:90% 00:27:00.525 --> 00:27:04.940 align:middle line:84% Si is weakly dominant if Si weakly or strictly dominates 00:27:04.940 --> 00:27:07.360 align:middle line:84% every other strategy that player i has. 00:27:07.360 --> 00:27:10.000 align:middle line:84% So maybe I'll just do weakly to avoid confusion. 00:27:10.000 --> 00:27:22.560 align:middle line:84% We say Si is weakly dominant if Si weakly dominates 00:27:22.560 --> 00:27:27.162 align:middle line:90% every strategy Si prime. 00:27:27.162 --> 00:27:28.620 align:middle line:84% Now, I need to be a little careful. 00:27:28.620 --> 00:27:29.960 align:middle line:90% There's something wrong here. 00:27:29.960 --> 00:27:32.000 align:middle line:84% I can't say every strategy, because Si 00:27:32.000 --> 00:27:33.320 align:middle line:90% can't dominate itself. 00:27:33.320 --> 00:27:41.760 align:middle line:84% So every strategy Si prime that's not equal to C. 00:27:41.760 --> 00:27:44.840 align:middle line:84% So we think that, in this case, if player i has 00:27:44.840 --> 00:27:46.840 align:middle line:84% a weakly dominant strategy, that's 00:27:46.840 --> 00:27:50.120 align:middle line:84% a pretty safe prediction about how that player will 00:27:50.120 --> 00:27:52.940 align:middle line:84% play in the game, about which strategy that player will use. 00:27:52.940 --> 00:27:57.560 align:middle line:90% 00:27:57.560 --> 00:27:59.860 align:middle line:84% And we have one final definition. 00:27:59.860 --> 00:28:02.800 align:middle line:90% 00:28:02.800 --> 00:28:05.260 align:middle line:84% And that is, well, let's think about the simplest 00:28:05.260 --> 00:28:06.460 align:middle line:90% possible case. 00:28:06.460 --> 00:28:10.820 align:middle line:84% That's where every player has some dominant strategy, 00:28:10.820 --> 00:28:12.843 align:middle line:84% has some weakly dominant strategy. 00:28:12.843 --> 00:28:14.260 align:middle line:84% If that's the case, then we can be 00:28:14.260 --> 00:28:16.860 align:middle line:84% pretty confident our prediction in that game 00:28:16.860 --> 00:28:18.740 align:middle line:84% is that each player is going to play 00:28:18.740 --> 00:28:22.140 align:middle line:84% their corresponding weakly dominant strategy. 00:28:22.140 --> 00:28:24.000 align:middle line:84% So we can give a definition here, 00:28:24.000 --> 00:28:26.980 align:middle line:84% our first notion of an equilibrium. 00:28:26.980 --> 00:28:34.380 align:middle line:84% We say a profile S1 star up to Sn star 00:28:34.380 --> 00:28:36.640 align:middle line:84% is a dominant strategy equilibrium. 00:28:36.640 --> 00:28:45.735 align:middle line:90% 00:28:45.735 --> 00:28:47.860 align:middle line:84% And here, I'm just going to write dominant strategy 00:28:47.860 --> 00:28:48.840 align:middle line:90% equilibrium. 00:28:48.840 --> 00:28:52.140 align:middle line:84% But more precisely, we mean weakly dominant strategy 00:28:52.140 --> 00:28:53.020 align:middle line:90% equilibrium. 00:28:53.020 --> 00:28:54.480 align:middle line:90% But we'll just say dominant. 00:28:54.480 --> 00:28:56.080 align:middle line:84% And sometimes, we'll call this DSE-- 00:28:56.080 --> 00:29:00.740 align:middle line:90% 00:29:00.740 --> 00:29:02.120 align:middle line:90% if what? 00:29:02.120 --> 00:29:03.780 align:middle line:90% Well, this is a profile. 00:29:03.780 --> 00:29:05.855 align:middle line:84% So we basically made a prediction 00:29:05.855 --> 00:29:07.980 align:middle line:84% about how each player in the game is going to play. 00:29:07.980 --> 00:29:10.640 align:middle line:84% We've made a prediction about each player's strategy. 00:29:10.640 --> 00:29:13.320 align:middle line:84% And we say that this is a dominant strategy equilibrium 00:29:13.320 --> 00:29:16.240 align:middle line:84% if each player is playing a strategy that's 00:29:16.240 --> 00:29:19.120 align:middle line:84% a weakly dominant strategy for that player. 00:29:19.120 --> 00:29:29.440 align:middle line:84% So if, for each player i, strategy-- 00:29:29.440 --> 00:29:32.040 align:middle line:84% well, what is the strategy used by player i? 00:29:32.040 --> 00:29:33.440 align:middle line:90% It's Si star. 00:29:33.440 --> 00:29:37.700 align:middle line:84% So strategy Si star is weakly dominant. 00:29:37.700 --> 00:29:45.463 align:middle line:90% 00:29:45.463 --> 00:29:46.880 align:middle line:84% But notice an important thing here 00:29:46.880 --> 00:29:49.920 align:middle line:84% that, when we make a prediction about a game, 00:29:49.920 --> 00:29:52.977 align:middle line:84% it's not enough just to predict one player's strategy. 00:29:52.977 --> 00:29:54.560 align:middle line:84% We have to make a prediction about how 00:29:54.560 --> 00:29:56.280 align:middle line:90% every player is going to play. 00:29:56.280 --> 00:29:58.440 align:middle line:84% So our prediction here is a profile 00:29:58.440 --> 00:30:01.070 align:middle line:84% of strategies specifying a strategy for each 00:30:01.070 --> 00:30:02.230 align:middle line:90% of the n players. 00:30:02.230 --> 00:30:07.090 align:middle line:84% And it's a dominant strategy equilibrium if, for each player, 00:30:07.090 --> 00:30:09.990 align:middle line:84% their strategy is weakly dominant for them, 00:30:09.990 --> 00:30:13.830 align:middle line:84% meaning that strategy, Si star-- if we go back to our definition, 00:30:13.830 --> 00:30:18.510 align:middle line:84% Si star weakly dominates every other strategy by player i. 00:30:18.510 --> 00:30:19.410 align:middle line:90% Any question on this? 00:30:19.410 --> 00:30:20.110 align:middle line:90% Yes? 00:30:20.110 --> 00:30:26.190 align:middle line:84% STUDENT: Is it possible to have multiple weakly dominant 00:30:26.190 --> 00:30:28.450 align:middle line:84% strategies, like one better than a different one? 00:30:28.450 --> 00:30:29.170 align:middle line:84% IAN BALL: That's a great question. 00:30:29.170 --> 00:30:30.390 align:middle line:90% So let's put this here. 00:30:30.390 --> 00:30:31.590 align:middle line:84% Let's write this on the board because I 00:30:31.590 --> 00:30:32.548 align:middle line:90% was going to get there. 00:30:32.548 --> 00:30:33.870 align:middle line:90% That's a great question. 00:30:33.870 --> 00:30:37.470 align:middle line:84% Is it possible for there to be multiple weakly dominant 00:30:37.470 --> 00:30:38.010 align:middle line:90% strategies? 00:30:38.010 --> 00:30:40.890 align:middle line:90% So maybe I'll say WD strategies. 00:30:40.890 --> 00:30:43.550 align:middle line:90% 00:30:43.550 --> 00:30:46.350 align:middle line:90% It's a great question. 00:30:46.350 --> 00:30:47.290 align:middle line:90% Any ideas? 00:30:47.290 --> 00:30:49.950 align:middle line:90% I'll put it to the class. 00:30:49.950 --> 00:30:51.295 align:middle line:90% Yeah? 00:30:51.295 --> 00:30:52.630 align:middle line:90% STUDENT: [INAUDIBLE] 00:30:52.630 --> 00:30:54.040 align:middle line:90% IAN BALL: Say it again? 00:30:54.040 --> 00:30:54.873 align:middle line:90% STUDENT: [INAUDIBLE] 00:30:54.873 --> 00:30:56.995 align:middle line:90% IAN BALL: OK. 00:30:56.995 --> 00:30:58.370 align:middle line:84% So what you have in mind is, what 00:30:58.370 --> 00:31:00.953 align:middle line:84% if there's two strategies that give me exactly the same payoff 00:31:00.953 --> 00:31:02.410 align:middle line:90% however my opponent plays? 00:31:02.410 --> 00:31:04.090 align:middle line:84% Now, we have to be really careful. 00:31:04.090 --> 00:31:07.490 align:middle line:84% The key thing, though, is this part of the definition. 00:31:07.490 --> 00:31:09.713 align:middle line:84% So if we didn't have this part of the definition, 00:31:09.713 --> 00:31:12.130 align:middle line:84% we could have two weak-- we could have two strategies that 00:31:12.130 --> 00:31:15.210 align:middle line:84% were both weakly dominant because they 00:31:15.210 --> 00:31:18.090 align:middle line:84% give the same payoff however the opponents play. 00:31:18.090 --> 00:31:20.050 align:middle line:84% But two strategies that give the same payoff 00:31:20.050 --> 00:31:23.730 align:middle line:84% are not going to satisfy strict inequality for any way 00:31:23.730 --> 00:31:24.970 align:middle line:90% my opponents are playing. 00:31:24.970 --> 00:31:28.370 align:middle line:84% So it's not possible for two strategies-- for one strategy 00:31:28.370 --> 00:31:30.770 align:middle line:84% to weakly dominate another and for that strategy 00:31:30.770 --> 00:31:32.350 align:middle line:90% to weakly dominate the first. 00:31:32.350 --> 00:31:34.090 align:middle line:84% And for that reason, it's impossible 00:31:34.090 --> 00:31:36.590 align:middle line:84% for there to be multiple weakly dominant strategies. 00:31:36.590 --> 00:31:39.170 align:middle line:90% So the answer is no. 00:31:39.170 --> 00:31:40.670 align:middle line:84% Let's go through the argument again. 00:31:40.670 --> 00:31:44.030 align:middle line:84% Suppose I had two strategies that were both weakly dominant. 00:31:44.030 --> 00:31:46.770 align:middle line:90% Let's call them SA and SB. 00:31:46.770 --> 00:31:50.370 align:middle line:84% Well, that means SA must weakly dominate SB. 00:31:50.370 --> 00:31:53.890 align:middle line:90% And SB must weakly dominate SA. 00:31:53.890 --> 00:31:56.750 align:middle line:84% But that means SB must do weakly weekly better than SA no matter 00:31:56.750 --> 00:32:00.230 align:middle line:84% how my opponents do, but sometimes strictly better. 00:32:00.230 --> 00:32:02.750 align:middle line:84% But if SB sometimes does strictly better, 00:32:02.750 --> 00:32:06.790 align:middle line:84% then it can't be the case that SA weakly dominates SB. 00:32:06.790 --> 00:32:09.227 align:middle line:84% Maybe a bit fast, it might be a good little exercise 00:32:09.227 --> 00:32:09.810 align:middle line:90% to write down. 00:32:09.810 --> 00:32:10.977 align:middle line:90% But that's a great question. 00:32:10.977 --> 00:32:12.290 align:middle line:90% And it's good to keep in mind. 00:32:12.290 --> 00:32:14.470 align:middle line:84% And that's one of our motivations 00:32:14.470 --> 00:32:17.470 align:middle line:84% for demanding strict inequality for some S negative i, 00:32:17.470 --> 00:32:20.070 align:middle line:84% to rule out this case of multiple weakly dominant 00:32:20.070 --> 00:32:21.510 align:middle line:90% strategies. 00:32:21.510 --> 00:32:22.010 align:middle line:90% Great. 00:32:22.010 --> 00:32:22.890 align:middle line:90% Any other questions? 00:32:22.890 --> 00:32:23.670 align:middle line:90% Yeah? 00:32:23.670 --> 00:32:26.790 align:middle line:84% STUDENT: Just to clarify, for the [INAUDIBLE] for strictly 00:32:26.790 --> 00:32:31.230 align:middle line:84% dominates, is it just that the [INAUDIBLE]? 00:32:31.230 --> 00:32:35.270 align:middle line:90% 00:32:35.270 --> 00:32:35.990 align:middle line:90% IAN BALL: Yeah. 00:32:35.990 --> 00:32:39.190 align:middle line:90% So what I mean is let's list-- 00:32:39.190 --> 00:32:41.310 align:middle line:84% for each S negative i, we have an inequality 00:32:41.310 --> 00:32:42.397 align:middle line:90% that looks like this. 00:32:42.397 --> 00:32:43.730 align:middle line:90% So let's say we have 10 of them. 00:32:43.730 --> 00:32:45.980 align:middle line:84% And I'm just going to go through each of them and say, 00:32:45.980 --> 00:32:47.790 align:middle line:84% does equality hold in every single one 00:32:47.790 --> 00:32:49.030 align:middle line:90% of those inequalities? 00:32:49.030 --> 00:32:53.350 align:middle line:84% And to get weak dominance, equality cannot hold in every 00:32:53.350 --> 00:32:54.070 align:middle line:90% single one. 00:32:54.070 --> 00:32:55.570 align:middle line:84% So another way of saying it is there 00:32:55.570 --> 00:32:57.510 align:middle line:84% must be one inequality where it's strict. 00:32:57.510 --> 00:32:59.710 align:middle line:84% But it could be equality in all but one. 00:32:59.710 --> 00:33:00.450 align:middle line:90% That's right. 00:33:00.450 --> 00:33:06.810 align:middle line:84% Yeah, so when I say strict, I'm referring to that being strict. 00:33:06.810 --> 00:33:07.310 align:middle line:90% Great. 00:33:07.310 --> 00:33:08.650 align:middle line:90% Any other questions? 00:33:08.650 --> 00:33:13.720 align:middle line:90% 00:33:13.720 --> 00:33:16.050 align:middle line:84% So now, let's look at a slightly more complicated game 00:33:16.050 --> 00:33:19.510 align:middle line:84% than the prisoner's dilemma and see how we would approach this. 00:33:19.510 --> 00:33:27.818 align:middle line:90% 00:33:27.818 --> 00:33:31.370 align:middle line:84% Let me make sure I get [INAUDIBLE] two on top. 00:33:31.370 --> 00:33:32.910 align:middle line:90% So we have a two player game. 00:33:32.910 --> 00:33:41.890 align:middle line:90% 00:33:41.890 --> 00:33:44.950 align:middle line:84% So remember, we, always put player 1 here. 00:33:44.950 --> 00:33:47.170 align:middle line:84% Player 1 is always choosing rows. 00:33:47.170 --> 00:33:49.530 align:middle line:84% Player 2 is always choosing columns. 00:33:49.530 --> 00:33:51.450 align:middle line:84% And when we fill in the payoffs, we always 00:33:51.450 --> 00:33:54.040 align:middle line:90% put player 1's payoffs first. 00:33:54.040 --> 00:33:56.330 align:middle line:84% So here, player 1 has three strategies. 00:33:56.330 --> 00:33:58.080 align:middle line:84% We call them top, middle, and bottom, just 00:33:58.080 --> 00:34:00.080 align:middle line:90% to make it easy to remember. 00:34:00.080 --> 00:34:02.080 align:middle line:84% And player 2 also has two strategies, 00:34:02.080 --> 00:34:03.760 align:middle line:90% which we call left and right. 00:34:03.760 --> 00:34:05.200 align:middle line:84% So for what I'm going to do here, 00:34:05.200 --> 00:34:07.500 align:middle line:84% I really only care about player 1's payoffs. 00:34:07.500 --> 00:34:10.139 align:middle line:90% So to avoid just distractions. 00:34:10.139 --> 00:34:13.300 align:middle line:84% I'm only going to fill in player 1's payoffs. 00:34:13.300 --> 00:34:16.802 align:middle line:90% 00:34:16.802 --> 00:34:18.219 align:middle line:84% So there should be something here. 00:34:18.219 --> 00:34:20.318 align:middle line:84% Maybe I'll put this to indicate, if I 00:34:20.318 --> 00:34:21.860 align:middle line:84% wanted to finish filling in the game, 00:34:21.860 --> 00:34:23.402 align:middle line:84% I'd have to put in all those numbers. 00:34:23.402 --> 00:34:24.880 align:middle line:84% But for what I'm doing now, I only 00:34:24.880 --> 00:34:26.500 align:middle line:90% care about these first numbers. 00:34:26.500 --> 00:34:30.480 align:middle line:90% 00:34:30.480 --> 00:34:35.753 align:middle line:84% So let's ask, in this game, does any strategy dominate any other? 00:34:35.753 --> 00:34:37.920 align:middle line:84% Does it strictly dominate, weakly dominate any other 00:34:37.920 --> 00:34:38.780 align:middle line:90% for player 1? 00:34:38.780 --> 00:34:43.679 align:middle line:90% 00:34:43.679 --> 00:34:44.500 align:middle line:90% What do you think? 00:34:44.500 --> 00:34:50.487 align:middle line:90% 00:34:50.487 --> 00:34:51.820 align:middle line:90% How do you feel about this game? 00:34:51.820 --> 00:34:53.520 align:middle line:84% Would you be happy to play any of these strategies? 00:34:53.520 --> 00:34:55.800 align:middle line:84% Or do you think some of them seem worse than others? 00:34:55.800 --> 00:35:03.900 align:middle line:90% 00:35:03.900 --> 00:35:05.560 align:middle line:90% OK, maybe we'll go on. 00:35:05.560 --> 00:35:08.320 align:middle line:84% But the answer is nothing dominates anything else. 00:35:08.320 --> 00:35:09.380 align:middle line:90% Let's go through it. 00:35:09.380 --> 00:35:11.420 align:middle line:90% Does T dominate M? 00:35:11.420 --> 00:35:13.980 align:middle line:84% Well, you might think so because 2 is greater than 0. 00:35:13.980 --> 00:35:16.180 align:middle line:84% But over here, we have negative 1 and 0. 00:35:16.180 --> 00:35:21.020 align:middle line:84% So M does better than T if my opponent plays R. 00:35:21.020 --> 00:35:25.020 align:middle line:84% So it can't be the case that T dominates M. And similarly, 00:35:25.020 --> 00:35:26.860 align:middle line:90% does M dominate B? 00:35:26.860 --> 00:35:32.100 align:middle line:84% Well, even though M does better than B when my opponent plays L, 00:35:32.100 --> 00:35:34.900 align:middle line:84% M does worse than B when my opponent plays R. 00:35:34.900 --> 00:35:36.900 align:middle line:84% And you can go through each of these comparisons 00:35:36.900 --> 00:35:41.020 align:middle line:84% and see that no strategy weakly dominates any other strategy. 00:35:41.020 --> 00:35:43.400 align:middle line:84% So now, we have a bit of a conundrum. 00:35:43.400 --> 00:35:44.780 align:middle line:84% How should you play in this game? 00:35:44.780 --> 00:35:45.530 align:middle line:90% It's not so clear. 00:35:45.530 --> 00:35:46.760 align:middle line:90% Before, it was pretty easy. 00:35:46.760 --> 00:35:49.052 align:middle line:84% In the prisoner's dilemma, we said, if you're rational, 00:35:49.052 --> 00:35:51.460 align:middle line:84% play F. You don't have to worry about your opponent. 00:35:51.460 --> 00:35:54.120 align:middle line:84% However they play, you're strictly better off playing F. 00:35:54.120 --> 00:35:56.655 align:middle line:90% But here, it's not so clear. 00:35:56.655 --> 00:35:58.280 align:middle line:84% So how would you approach this problem, 00:35:58.280 --> 00:35:59.570 align:middle line:90% if you were player 1 here? 00:35:59.570 --> 00:36:00.820 align:middle line:90% What would be relevant to you? 00:36:00.820 --> 00:36:02.278 align:middle line:84% What would you need to think about? 00:36:02.278 --> 00:36:05.120 align:middle line:90% 00:36:05.120 --> 00:36:05.800 align:middle line:90% Yeah? 00:36:05.800 --> 00:36:09.160 align:middle line:84% STUDENT: You want to know what the chances are the other player 00:36:09.160 --> 00:36:10.080 align:middle line:90% plays [INAUDIBLE]. 00:36:10.080 --> 00:36:10.480 align:middle line:90% IAN BALL: Exactly. 00:36:10.480 --> 00:36:12.105 align:middle line:84% So you're going to have to form beliefs 00:36:12.105 --> 00:36:13.500 align:middle line:84% about how the other player plays, 00:36:13.500 --> 00:36:16.200 align:middle line:84% just like on the first class when we said whether you walk 00:36:16.200 --> 00:36:19.480 align:middle line:84% home or take the T depends on, after checking your phone, what 00:36:19.480 --> 00:36:21.990 align:middle line:84% you think the probability is that it's going to rain. 00:36:21.990 --> 00:36:24.240 align:middle line:84% In the same way, we're going to form beliefs or assign 00:36:24.240 --> 00:36:28.360 align:middle line:84% probabilities to the strategies that my opponent is playing. 00:36:28.360 --> 00:36:35.240 align:middle line:84% So in this context, maybe we'll assign a probability of p to L 00:36:35.240 --> 00:36:40.560 align:middle line:84% and 1 minus p to R. So now, let's compute my payoffs 00:36:40.560 --> 00:36:44.040 align:middle line:90% I'm going to write u of L, p. 00:36:44.040 --> 00:36:45.630 align:middle line:90% So a few things here-- 00:36:45.630 --> 00:36:48.270 align:middle line:84% because my opponent only has two strategies, 00:36:48.270 --> 00:36:51.390 align:middle line:84% if I want to specify my belief about their strategy, 00:36:51.390 --> 00:36:53.483 align:middle line:84% it's enough to specify a single number p. 00:36:53.483 --> 00:36:54.650 align:middle line:90% And I want to be clear here. 00:36:54.650 --> 00:36:55.770 align:middle line:90% This is a number p. 00:36:55.770 --> 00:36:56.910 align:middle line:90% It's not a vector. 00:36:56.910 --> 00:36:58.030 align:middle line:90% It's just a number. 00:36:58.030 --> 00:37:00.950 align:middle line:84% Once I know how much probability I assigned to L, 00:37:00.950 --> 00:37:04.350 align:middle line:84% then the amount of probability I assign to R is always just 1 00:37:04.350 --> 00:37:05.770 align:middle line:90% minus that probability. 00:37:05.770 --> 00:37:08.030 align:middle line:90% So this is enough information. 00:37:08.030 --> 00:37:10.290 align:middle line:84% And then here, remember, technically-- 00:37:10.290 --> 00:37:11.450 align:middle line:90% I'll put 1 here-- 00:37:11.450 --> 00:37:15.310 align:middle line:84% we only define the utility u1 of a strategy profile. 00:37:15.310 --> 00:37:17.790 align:middle line:84% So technically, we haven't defined what this means yet. 00:37:17.790 --> 00:37:19.790 align:middle line:84% But we're going to extend our notation 00:37:19.790 --> 00:37:23.070 align:middle line:84% to define what my expected payoff is 00:37:23.070 --> 00:37:26.550 align:middle line:84% if I play L and I believe that my opponent is going 00:37:26.550 --> 00:37:30.230 align:middle line:84% to play L with probability p and R with probability 1 minus p. 00:37:30.230 --> 00:37:30.870 align:middle line:90% Yes? 00:37:30.870 --> 00:37:33.063 align:middle line:84% STUDENT: You're playing T or M or R [? instead. ?] 00:37:33.063 --> 00:37:34.230 align:middle line:90% IAN BALL: Ah, yes, T, sorry. 00:37:34.230 --> 00:37:35.190 align:middle line:90% Thank you. 00:37:35.190 --> 00:37:36.950 align:middle line:90% T, yes. 00:37:36.950 --> 00:37:38.590 align:middle line:90% Great. 00:37:38.590 --> 00:37:39.550 align:middle line:90% So here we are. 00:37:39.550 --> 00:37:42.710 align:middle line:84% So what is it/ Well, with probability p, 00:37:42.710 --> 00:37:47.910 align:middle line:84% my opponent plays L. And therefore, my payoff is u1 of T, 00:37:47.910 --> 00:37:55.090 align:middle line:84% L. But with probability 1 minus p, my opponent plays R. 00:37:55.090 --> 00:37:58.650 align:middle line:84% I'm still playing T. Thanks for the clarification. 00:37:58.650 --> 00:38:04.650 align:middle line:90% So we have T, R. 00:38:04.650 --> 00:38:06.290 align:middle line:84% So notice, we're treating my beliefs 00:38:06.290 --> 00:38:08.970 align:middle line:84% about my opponent's strategy just like we treated lotteries 00:38:08.970 --> 00:38:09.770 align:middle line:90% before. 00:38:09.770 --> 00:38:12.090 align:middle line:84% When I have beliefs about something, 00:38:12.090 --> 00:38:14.850 align:middle line:84% I think about my choices as corresponding 00:38:14.850 --> 00:38:17.450 align:middle line:90% to lotteries over outcomes. 00:38:17.450 --> 00:38:20.730 align:middle line:84% And then I evaluate expected utility over those outcomes. 00:38:20.730 --> 00:38:22.630 align:middle line:84% So here, we can just plug in the numbers. 00:38:22.630 --> 00:38:24.730 align:middle line:84% We're going to get this is going to be a 2. 00:38:24.730 --> 00:38:28.170 align:middle line:84% And this is going to be a negative 1. 00:38:28.170 --> 00:38:30.270 align:middle line:90% So what do we get? 00:38:30.270 --> 00:38:37.850 align:middle line:84% We get 2p minus 1 plus p, if we do the math right, 00:38:37.850 --> 00:38:40.990 align:middle line:90% which equals 3p minus 1. 00:38:40.990 --> 00:38:43.550 align:middle line:90% 00:38:43.550 --> 00:38:46.750 align:middle line:84% And it's often good to check if this makes sense. 00:38:46.750 --> 00:38:48.510 align:middle line:90% The higher is p-- 00:38:48.510 --> 00:38:49.510 align:middle line:90% yes? 00:38:49.510 --> 00:38:51.430 align:middle line:84% STUDENT: So I just [INAUDIBLE] question-- 00:38:51.430 --> 00:38:54.110 align:middle line:84% the utility, in this case, is specifically only 00:38:54.110 --> 00:38:56.330 align:middle line:84% for the lottery, which is represented by the value p. 00:38:56.330 --> 00:38:58.430 align:middle line:84% But we also have to include strategy, right? 00:38:58.430 --> 00:38:59.310 align:middle line:90% IAN BALL: Right. 00:38:59.310 --> 00:39:01.018 align:middle line:84% So if we want to be a little more formal, 00:39:01.018 --> 00:39:02.950 align:middle line:84% what is this/ This is the lottery. 00:39:02.950 --> 00:39:04.890 align:middle line:90% Well, this is a two player game. 00:39:04.890 --> 00:39:08.690 align:middle line:84% So this is a lottery over the strategies of my opponents. 00:39:08.690 --> 00:39:14.830 align:middle line:84% So this lives in delta of, in this case, S2, 00:39:14.830 --> 00:39:19.070 align:middle line:84% but, more generally, S negative 1, because what I need to do 00:39:19.070 --> 00:39:21.090 align:middle line:84% is I need to say, more generally, 00:39:21.090 --> 00:39:23.790 align:middle line:84% of all the possible profiles of strategies 00:39:23.790 --> 00:39:25.670 align:middle line:84% that my opponents could play, I need 00:39:25.670 --> 00:39:29.070 align:middle line:84% to form a belief that specifies the probability on each of them. 00:39:29.070 --> 00:39:32.010 align:middle line:84% In this case, I can represent that belief by a single number. 00:39:32.010 --> 00:39:34.210 align:middle line:84% But more generally, it's going to be a more complicated object. 00:39:34.210 --> 00:39:36.293 align:middle line:84% STUDENT: So that means utilities usually calculate 00:39:36.293 --> 00:39:40.250 align:middle line:84% the cross-product of a strategy [INAUDIBLE]. 00:39:40.250 --> 00:39:41.350 align:middle line:90% IAN BALL: Yes, yes. 00:39:41.350 --> 00:39:43.650 align:middle line:84% And I'll be more formal in a second, yeah. 00:39:43.650 --> 00:39:44.552 align:middle line:90% Other question, yeah? 00:39:44.552 --> 00:39:46.010 align:middle line:84% STUDENT: So is the utility function 00:39:46.010 --> 00:39:48.050 align:middle line:90% on the left side [INAUDIBLE]? 00:39:48.050 --> 00:39:51.375 align:middle line:90% 00:39:51.375 --> 00:39:52.750 align:middle line:84% IAN BALL: Good point, good point. 00:39:52.750 --> 00:39:56.123 align:middle line:84% People are being very, very precise here. 00:39:56.123 --> 00:39:57.790 align:middle line:84% You might say it's an abuse of notation. 00:39:57.790 --> 00:39:59.370 align:middle line:84% Or you might say we're extending the notation 00:39:59.370 --> 00:40:00.328 align:middle line:90% on the right-hand side. 00:40:00.328 --> 00:40:02.410 align:middle line:84% So originally, we defined utilities only 00:40:02.410 --> 00:40:04.290 align:middle line:90% for strategy profiles. 00:40:04.290 --> 00:40:06.090 align:middle line:84% And now, we're going to define-- formally, 00:40:06.090 --> 00:40:08.382 align:middle line:84% what we're doing is we're going to extend this function 00:40:08.382 --> 00:40:09.510 align:middle line:90% to a larger domain. 00:40:09.510 --> 00:40:11.810 align:middle line:84% So we're going to imagine that it has the same meaning 00:40:11.810 --> 00:40:12.910 align:middle line:90% on the original domain. 00:40:12.910 --> 00:40:15.370 align:middle line:84% But now, we're going to define it more broadly. 00:40:15.370 --> 00:40:18.490 align:middle line:84% So maybe if I was being very pedantic, 00:40:18.490 --> 00:40:20.390 align:middle line:84% I would use different notation here. 00:40:20.390 --> 00:40:22.738 align:middle line:84% But we're going to be a little less pedantic. 00:40:22.738 --> 00:40:24.530 align:middle line:84% And we're going to think of these utilities 00:40:24.530 --> 00:40:26.540 align:middle line:84% as being defined on a bit more general space. 00:40:26.540 --> 00:40:28.290 align:middle line:84% But you're right that, conceptually, there 00:40:28.290 --> 00:40:29.510 align:middle line:90% is a distinction here. 00:40:29.510 --> 00:40:31.450 align:middle line:84% Though, one way to think about it is, 00:40:31.450 --> 00:40:33.290 align:middle line:90% how do I interpret u1 of T, L? 00:40:33.290 --> 00:40:35.730 align:middle line:84% One interpretation is this is what happens when I play T 00:40:35.730 --> 00:40:39.320 align:middle line:84% and I hold the belief that my opponent is certain to play L. 00:40:39.320 --> 00:40:42.378 align:middle line:84% So you can think of this as a special case where p equals 1. 00:40:42.378 --> 00:40:43.920 align:middle line:84% That would be one way you can see how 00:40:43.920 --> 00:40:45.180 align:middle line:90% these functions fit together. 00:40:45.180 --> 00:40:45.680 align:middle line:90% Yeah. 00:40:45.680 --> 00:40:46.280 align:middle line:90% Any other questions? 00:40:46.280 --> 00:40:46.680 align:middle line:90% Yes? 00:40:46.680 --> 00:40:49.263 align:middle line:84% STUDENT: Can you just use big U on the left-hand side and then 00:40:49.263 --> 00:40:50.920 align:middle line:90% little u on the right-hand side? 00:40:50.920 --> 00:40:52.460 align:middle line:90% IAN BALL: I could do that. 00:40:52.460 --> 00:40:53.540 align:middle line:90% I'm not doing that here. 00:40:53.540 --> 00:40:54.140 align:middle line:90% I could do it. 00:40:54.140 --> 00:40:57.660 align:middle line:84% But eventually, we just find it easier to always do this. 00:40:57.660 --> 00:41:00.800 align:middle line:84% So from now on, we're going to use little u. 00:41:00.800 --> 00:41:03.840 align:middle line:84% And we can put in a belief or a strategy over here. 00:41:03.840 --> 00:41:05.880 align:middle line:84% And if we put in a strategy, the interpretation 00:41:05.880 --> 00:41:08.040 align:middle line:84% is that's a belief that puts all the probability 00:41:08.040 --> 00:41:09.560 align:middle line:90% on that particular strategy. 00:41:09.560 --> 00:41:10.360 align:middle line:90% Yeah. 00:41:10.360 --> 00:41:12.140 align:middle line:90% Any other questions here? 00:41:12.140 --> 00:41:12.640 align:middle line:90% Good. 00:41:12.640 --> 00:41:14.280 align:middle line:90% OK. 00:41:14.280 --> 00:41:15.748 align:middle line:90% And we could similarly compute-- 00:41:15.748 --> 00:41:16.540 align:middle line:90% I won't do it here. 00:41:16.540 --> 00:41:24.080 align:middle line:84% But we could compute u1 of M, p and u1 of B, p. 00:41:24.080 --> 00:41:25.762 align:middle line:84% Then I guess the question suggested 00:41:25.762 --> 00:41:27.720 align:middle line:84% we need to be a little more rigorous about what 00:41:27.720 --> 00:41:29.260 align:middle line:84% these beliefs are and what we mean. 00:41:29.260 --> 00:41:31.360 align:middle line:84% So let me bring this down and erase. 00:41:31.360 --> 00:41:43.060 align:middle line:90% 00:41:43.060 --> 00:41:45.720 align:middle line:84% So let's be a little more precise about what a belief is. 00:41:45.720 --> 00:41:48.660 align:middle line:90% 00:41:48.660 --> 00:41:53.060 align:middle line:84% Well, a belief for player i is just a probability distribution 00:41:53.060 --> 00:41:56.820 align:middle line:84% over the strategies that the other players are playing. 00:41:56.820 --> 00:42:04.500 align:middle line:84% So I'll say, for player i, it's a function. 00:42:04.500 --> 00:42:06.403 align:middle line:90% Maybe we'll say beta negative i. 00:42:06.403 --> 00:42:08.820 align:middle line:84% We're going to use the negative i because even though it's 00:42:08.820 --> 00:42:10.278 align:middle line:84% the belief that's held by player i, 00:42:10.278 --> 00:42:13.160 align:middle line:84% it's a belief about everyone else's strategies. 00:42:13.160 --> 00:42:15.220 align:middle line:84% And that's why we use the negative i here. 00:42:15.220 --> 00:42:15.880 align:middle line:90% And what is it? 00:42:15.880 --> 00:42:19.140 align:middle line:84% Well, it's a function from S negative i 00:42:19.140 --> 00:42:23.180 align:middle line:84% to 0, 1, if you want to be really formal, where it says, 00:42:23.180 --> 00:42:26.060 align:middle line:84% for each profile of strategies, or partial profile 00:42:26.060 --> 00:42:28.700 align:middle line:84% of strategies, that my opponents play, 00:42:28.700 --> 00:42:30.500 align:middle line:84% what is the probability that I assign 00:42:30.500 --> 00:42:33.280 align:middle line:84% to that particular profile of strategies? 00:42:33.280 --> 00:42:36.080 align:middle line:84% How likely is it that they played that particular profile? 00:42:36.080 --> 00:42:38.400 align:middle line:84% Now, what we need-- we have to be careful. 00:42:38.400 --> 00:42:41.540 align:middle line:84% For this to make sense, when we sum up all these probabilities, 00:42:41.540 --> 00:42:43.520 align:middle line:90% we have to get 1. 00:42:43.520 --> 00:42:49.640 align:middle line:84% So it's a function like this that's satisfying what? 00:42:49.640 --> 00:42:52.260 align:middle line:84% Well, the total probability is 1. 00:42:52.260 --> 00:42:53.880 align:middle line:90% So let me take a summation. 00:42:53.880 --> 00:42:55.360 align:middle line:90% What am I summing over? 00:42:55.360 --> 00:42:57.640 align:middle line:84% I'm summing over all the partial strategy 00:42:57.640 --> 00:42:59.480 align:middle line:90% profiles of my opponent. 00:42:59.480 --> 00:43:04.200 align:middle line:84% So I'm going to sum over s negative i in big S negative i. 00:43:04.200 --> 00:43:06.640 align:middle line:84% If you're not used to seeing summations that aren't just 00:43:06.640 --> 00:43:08.720 align:middle line:84% numbers 1 to n, think of this as just I'm 00:43:08.720 --> 00:43:15.560 align:middle line:84% writing out addition with one term for each strategy profile 00:43:15.560 --> 00:43:18.480 align:middle line:90% S negative i. 00:43:18.480 --> 00:43:22.980 align:middle line:84% And we have beta negative i of S negative i equals 1. 00:43:22.980 --> 00:43:26.680 align:middle line:90% 00:43:26.680 --> 00:43:28.960 align:middle line:84% So this is formally what a belief is. 00:43:28.960 --> 00:43:32.510 align:middle line:84% And now, let me formally define the function that people rightly 00:43:32.510 --> 00:43:33.810 align:middle line:90% challenged me on a little bit. 00:43:33.810 --> 00:43:36.490 align:middle line:84% So if I want to formally extend this function, 00:43:36.490 --> 00:43:43.310 align:middle line:84% we're going to define ui of Si beta negative i. 00:43:43.310 --> 00:43:47.070 align:middle line:84% So remember, so far, we only defined player i's utility 00:43:47.070 --> 00:43:48.870 align:middle line:90% from a strategy profile. 00:43:48.870 --> 00:43:51.550 align:middle line:84% Now, we're going to think about what is player i's expected 00:43:51.550 --> 00:43:55.950 align:middle line:84% utility if player i plays strategy Si and player 00:43:55.950 --> 00:44:00.270 align:middle line:84% i holds this belief about everyone else's strategies. 00:44:00.270 --> 00:44:02.030 align:middle line:84% We're going to take expectations. 00:44:02.030 --> 00:44:04.870 align:middle line:84% So we're going to need to do a summation. 00:44:04.870 --> 00:44:07.070 align:middle line:84% We always sum over things we don't know. 00:44:07.070 --> 00:44:09.750 align:middle line:84% What we don't know is how my opponents play. 00:44:09.750 --> 00:44:13.150 align:middle line:84% So this is going to be a sum over s negative i in S 00:44:13.150 --> 00:44:15.150 align:middle line:90% negative i. 00:44:15.150 --> 00:44:16.830 align:middle line:90% And we're taking an expectation. 00:44:16.830 --> 00:44:19.630 align:middle line:84% So for each way that my opponents could play, 00:44:19.630 --> 00:44:22.390 align:middle line:84% I want to consider the probability that that happens. 00:44:22.390 --> 00:44:24.670 align:middle line:84% And then I want to multiply that by the utility 00:44:24.670 --> 00:44:26.670 align:middle line:90% I get if that does happen. 00:44:26.670 --> 00:44:31.610 align:middle line:84% So what's the probability that S negative i happens? 00:44:31.610 --> 00:44:36.090 align:middle line:84% It's beta negative i of S negative i. 00:44:36.090 --> 00:44:38.050 align:middle line:84% And I want to multiply this by the utility 00:44:38.050 --> 00:44:39.690 align:middle line:90% I get if that happens. 00:44:39.690 --> 00:44:41.010 align:middle line:90% Well, what's my utility? 00:44:41.010 --> 00:44:43.370 align:middle line:90% Well, I'm still playing Si. 00:44:43.370 --> 00:44:46.210 align:middle line:84% If it turns out my opponents play S negative i, 00:44:46.210 --> 00:44:52.490 align:middle line:84% then my utility is ui of Si, S negative i. 00:44:52.490 --> 00:44:54.750 align:middle line:84% Let me maybe draw a line here to make it clear 00:44:54.750 --> 00:44:56.810 align:middle line:90% this is a separate line. 00:44:56.810 --> 00:44:59.050 align:middle line:84% And this is the formal definition-- 00:44:59.050 --> 00:45:01.130 align:middle line:84% or the formal extension of this function 00:45:01.130 --> 00:45:04.490 align:middle line:90% to strategies and beliefs. 00:45:04.490 --> 00:45:06.810 align:middle line:84% Now, we can also see here what we mentioned earlier. 00:45:06.810 --> 00:45:09.530 align:middle line:84% What if beta negative i is the special belief 00:45:09.530 --> 00:45:12.210 align:middle line:84% that puts probability 1 on a particular strategy 00:45:12.210 --> 00:45:14.090 align:middle line:90% of my opponents? 00:45:14.090 --> 00:45:16.330 align:middle line:84% If that happens, then all of these terms 00:45:16.330 --> 00:45:19.010 align:middle line:90% become 0 except one of them. 00:45:19.010 --> 00:45:21.970 align:middle line:84% And I simply get the same utility function 00:45:21.970 --> 00:45:24.490 align:middle line:84% that I started with, where S negative i is exactly 00:45:24.490 --> 00:45:29.830 align:middle line:84% the strategy that my belief puts probability 1 on. 00:45:29.830 --> 00:45:32.810 align:middle line:90% Any questions about this? 00:45:32.810 --> 00:45:33.310 align:middle line:90% Yeah? 00:45:33.310 --> 00:45:35.410 align:middle line:84% STUDENT: So in this case, just for the left side, 00:45:35.410 --> 00:45:38.490 align:middle line:84% we have the cross-product of [INAUDIBLE] strategy 00:45:38.490 --> 00:45:40.750 align:middle line:84% [INAUDIBLE] while the ui on the right side 00:45:40.750 --> 00:45:42.890 align:middle line:90% is just purely [INAUDIBLE]. 00:45:42.890 --> 00:45:43.890 align:middle line:90% IAN BALL: Exactly right. 00:45:43.890 --> 00:45:50.150 align:middle line:84% So this lives in S. And this lives in Si 00:45:50.150 --> 00:45:53.930 align:middle line:90% cross delta of S negative i. 00:45:53.930 --> 00:45:56.650 align:middle line:84% But I don't want people to be scared by the math. 00:45:56.650 --> 00:46:02.430 align:middle line:84% But yeah, that's where these things live. 00:46:02.430 --> 00:46:03.350 align:middle line:90% Great. 00:46:03.350 --> 00:46:05.190 align:middle line:90% OK. 00:46:05.190 --> 00:46:07.310 align:middle line:84% So now we've talked about beliefs, 00:46:07.310 --> 00:46:11.150 align:middle line:84% now we can talk about a best response to a belief. 00:46:11.150 --> 00:46:14.510 align:middle line:84% And this is, I think, one of the most central ideas 00:46:14.510 --> 00:46:16.630 align:middle line:90% in game theory. 00:46:16.630 --> 00:46:18.130 align:middle line:84% Mathematically, it's not very deep. 00:46:18.130 --> 00:46:21.230 align:middle line:84% But conceptually, it's really, I think, a step forward. 00:46:21.230 --> 00:46:22.730 align:middle line:84% In interactive decision problems, 00:46:22.730 --> 00:46:24.950 align:middle line:84% the idea is that there's two steps. 00:46:24.950 --> 00:46:28.490 align:middle line:84% First, I form a belief about how my opponents are going to play. 00:46:28.490 --> 00:46:30.450 align:middle line:84% And then I choose a strategy that's 00:46:30.450 --> 00:46:33.610 align:middle line:84% best given the belief that I form about everyone else. 00:46:33.610 --> 00:46:36.370 align:middle line:84% So we want to formally define that. 00:46:36.370 --> 00:46:53.770 align:middle line:84% So we say strategy Si is a best response to belief beta 00:46:53.770 --> 00:46:54.470 align:middle line:90% negative. 00:46:54.470 --> 00:46:56.550 align:middle line:84% And remember, for any of this to make sense, 00:46:56.550 --> 00:46:58.930 align:middle line:84% we have some game in the background. 00:46:58.930 --> 00:47:00.430 align:middle line:84% We have some game in the background. 00:47:00.430 --> 00:47:01.428 align:middle line:90% We have some player. 00:47:01.428 --> 00:47:02.970 align:middle line:84% And if I wanted to be really precise, 00:47:02.970 --> 00:47:06.650 align:middle line:84% I would say, for player i, strategy Si 00:47:06.650 --> 00:47:09.850 align:middle line:84% is a best response to belief beta negative i, 00:47:09.850 --> 00:47:11.490 align:middle line:84% because we're evaluating things here 00:47:11.490 --> 00:47:13.490 align:middle line:84% from the perspective of player i. 00:47:13.490 --> 00:47:16.130 align:middle line:84% Player i chooses a strategy for herself 00:47:16.130 --> 00:47:20.010 align:middle line:84% and forms beliefs about everyone else's strategy. 00:47:20.010 --> 00:47:23.010 align:middle line:90% Well, what do we need? 00:47:23.010 --> 00:47:27.240 align:middle line:84% Well, we want Si to do better than any other strategy given 00:47:27.240 --> 00:47:28.240 align:middle line:90% the belief. 00:47:28.240 --> 00:47:30.940 align:middle line:90% So we need ui of Si. 00:47:30.940 --> 00:47:42.282 align:middle line:90% 00:47:42.282 --> 00:47:43.240 align:middle line:90% So what am I going to-- 00:47:43.240 --> 00:47:44.520 align:middle line:90% I'll fill it in a second. 00:47:44.520 --> 00:47:45.680 align:middle line:90% I'm player i. 00:47:45.680 --> 00:47:49.520 align:middle line:84% I'm comparing the utility that I get from Si to the utility 00:47:49.520 --> 00:47:51.760 align:middle line:90% I get from Si prime. 00:47:51.760 --> 00:47:56.360 align:middle line:84% And I want Si to be better than Si prime for any Si prime 00:47:56.360 --> 00:47:57.920 align:middle line:90% that I contemplate. 00:47:57.920 --> 00:47:59.020 align:middle line:90% But what goes in here? 00:47:59.020 --> 00:48:02.320 align:middle line:84% Well, what goes in here is the fixed belief that I have. 00:48:02.320 --> 00:48:04.320 align:middle line:90% So this is beta negative i. 00:48:04.320 --> 00:48:05.660 align:middle line:90% And this is beta negative i. 00:48:05.660 --> 00:48:10.640 align:middle line:90% 00:48:10.640 --> 00:48:13.100 align:middle line:84% So given the belief beta negative i, 00:48:13.100 --> 00:48:17.840 align:middle line:84% I can compute my expected utility from any strategy. 00:48:17.840 --> 00:48:22.340 align:middle line:84% And we say that strategy Si is a best response to my belief 00:48:22.340 --> 00:48:24.860 align:middle line:84% if the expected utility I get from Si 00:48:24.860 --> 00:48:27.540 align:middle line:84% is weakly larger than the expected utility 00:48:27.540 --> 00:48:30.743 align:middle line:84% I get from any other strategy Si prime. 00:48:30.743 --> 00:48:33.160 align:middle line:84% Now, let's go back to the question that was asked earlier. 00:48:33.160 --> 00:48:36.580 align:middle line:84% Can I have multiple best responses to a belief? 00:48:36.580 --> 00:48:37.660 align:middle line:90% Is that possible? 00:48:37.660 --> 00:48:41.320 align:middle line:84% Can two strategies both be best responses to the same belief? 00:48:41.320 --> 00:48:47.180 align:middle line:90% 00:48:47.180 --> 00:48:47.820 align:middle line:90% Yes. 00:48:47.820 --> 00:48:49.940 align:middle line:84% So this goes back to the suggestion before. 00:48:49.940 --> 00:48:52.580 align:middle line:84% If I had two strategies, Si and Si prime, 00:48:52.580 --> 00:48:55.020 align:middle line:84% that gave me exactly the same utility no matter what 00:48:55.020 --> 00:48:57.140 align:middle line:84% my opponents did, then it's possible for them 00:48:57.140 --> 00:48:58.880 align:middle line:90% both to be best responses. 00:48:58.880 --> 00:49:01.000 align:middle line:84% In fact, they could both be best responses 00:49:01.000 --> 00:49:03.720 align:middle line:84% even if they didn't give me exactly the same utility, 00:49:03.720 --> 00:49:05.900 align:middle line:84% but the expected utility was the same. 00:49:05.900 --> 00:49:10.580 align:middle line:90% So let's note here-- 00:49:10.580 --> 00:49:19.240 align:middle line:84% there can be multiple best responses to the same belief. 00:49:19.240 --> 00:49:28.280 align:middle line:90% 00:49:28.280 --> 00:49:29.860 align:middle line:84% If we wanted there to only be one, 00:49:29.860 --> 00:49:32.705 align:middle line:84% we sometimes might use the phrase strict best response 00:49:32.705 --> 00:49:35.080 align:middle line:84% to say this is a best response and nothing else is a best 00:49:35.080 --> 00:49:36.200 align:middle line:90% response. 00:49:36.200 --> 00:49:40.000 align:middle line:84% But we'll generally just use the weak sense. 00:49:40.000 --> 00:49:40.500 align:middle line:90% OK. 00:49:40.500 --> 00:49:45.000 align:middle line:84% Let's try to understand this graphically. 00:49:45.000 --> 00:49:47.520 align:middle line:90% I think there's space. 00:49:47.520 --> 00:49:49.140 align:middle line:90% Maybe let's do this over here. 00:49:49.140 --> 00:49:55.580 align:middle line:90% So let me just erase this. 00:49:55.580 --> 00:49:58.240 align:middle line:90% STUDENT: [INAUDIBLE] 00:49:58.240 --> 00:50:00.560 align:middle line:90% 00:50:00.560 --> 00:50:01.720 align:middle line:90% IAN BALL: Yeah. 00:50:01.720 --> 00:50:05.560 align:middle line:84% Yes, so delta is the space of lotteries over the set, yeah. 00:50:05.560 --> 00:50:09.517 align:middle line:90% 00:50:09.517 --> 00:50:11.600 align:middle line:84% And let me actually just erase this to give myself 00:50:11.600 --> 00:50:12.580 align:middle line:90% a bit more space. 00:50:12.580 --> 00:50:16.040 align:middle line:90% 00:50:16.040 --> 00:50:20.940 align:middle line:84% So let's try to understand best responses in this game. 00:50:20.940 --> 00:50:22.240 align:middle line:90% Best responses to what? 00:50:22.240 --> 00:50:26.180 align:middle line:84% Well, best responses for player 1 to player 2's-- 00:50:26.180 --> 00:50:30.380 align:middle line:84% to player 1's belief about player 2's strategy. 00:50:30.380 --> 00:50:33.820 align:middle line:84% So what we want to do is we're going to draw some payoffs here. 00:50:33.820 --> 00:50:36.340 align:middle line:90% 00:50:36.340 --> 00:50:41.680 align:middle line:84% I'm going to draw a graph here where, on the horizontal axis, 00:50:41.680 --> 00:50:42.180 align:middle line:90% I'm-- 00:50:42.180 --> 00:50:44.780 align:middle line:90% 00:50:44.780 --> 00:50:47.780 align:middle line:84% maybe it's better if I put p on R. 00:50:47.780 --> 00:50:53.800 align:middle line:84% So let me just switch this so that we're geometrically better. 00:50:53.800 --> 00:51:00.580 align:middle line:90% 00:51:00.580 --> 00:51:03.580 align:middle line:84% So I can describe my belief by my probability on L 00:51:03.580 --> 00:51:05.340 align:middle line:84% or my probability on R. It's more 00:51:05.340 --> 00:51:07.550 align:middle line:84% convenient to put the probability on R 00:51:07.550 --> 00:51:09.800 align:middle line:84% because then the graph is going to move left to right. 00:51:09.800 --> 00:51:11.300 align:middle line:84% And that's going to be a bit better. 00:51:11.300 --> 00:51:15.660 align:middle line:84% So p down here is the probability on R. 00:51:15.660 --> 00:51:19.650 align:middle line:84% And I'm going to draw this graph from 0 to 1. 00:51:19.650 --> 00:51:22.570 align:middle line:84% So if p equals 1, then I'm certain that my opponent is 00:51:22.570 --> 00:51:26.850 align:middle line:84% playing R. If p equals 0, then I'm certain that my opponent is 00:51:26.850 --> 00:51:31.570 align:middle line:84% playing L. And say if p is 1/2, then I think it's equally likely 00:51:31.570 --> 00:51:34.310 align:middle line:84% that my opponent plays L and that my opponent plays R. 00:51:34.310 --> 00:51:36.750 align:middle line:84% You can see why I switch things so that we move from left 00:51:36.750 --> 00:51:37.250 align:middle line:90% to right. 00:51:37.250 --> 00:51:40.050 align:middle line:84% I think it's a bit more intuitive. 00:51:40.050 --> 00:51:43.370 align:middle line:84% And now, to think about best responses, 00:51:43.370 --> 00:51:46.010 align:middle line:84% I basically want to plot each of these functions. 00:51:46.010 --> 00:51:50.410 align:middle line:84% I want to say, if I play T, what is my utility from T 00:51:50.410 --> 00:51:54.250 align:middle line:84% as a function of the belief that I have about my opponents? 00:51:54.250 --> 00:51:57.730 align:middle line:90% So let's start. 00:51:57.730 --> 00:52:03.450 align:middle line:90% Let's put negative 1, 0, 1, 2. 00:52:03.450 --> 00:52:07.150 align:middle line:84% If I play T, well, if my opponent plays L, 00:52:07.150 --> 00:52:09.010 align:middle line:90% I get a payoff of 2. 00:52:09.010 --> 00:52:11.210 align:middle line:84% So that means, if I'm certain that my opponent is 00:52:11.210 --> 00:52:13.630 align:middle line:84% going to play L, if that's my belief, 00:52:13.630 --> 00:52:15.390 align:middle line:90% then my payoff better be 2. 00:52:15.390 --> 00:52:18.150 align:middle line:84% So we're going to get a point up here. 00:52:18.150 --> 00:52:20.630 align:middle line:84% And if I'm certain that my opponent is playing R, 00:52:20.630 --> 00:52:22.910 align:middle line:84% then I get a payoff of negative 1. 00:52:22.910 --> 00:52:27.910 align:middle line:84% So it better be that we get a point like this. 00:52:27.910 --> 00:52:29.890 align:middle line:84% And it turns out-- we could calculate it. 00:52:29.890 --> 00:52:31.830 align:middle line:84% But it turns out expected utilities are always 00:52:31.830 --> 00:52:33.190 align:middle line:90% linear in my beliefs. 00:52:33.190 --> 00:52:36.710 align:middle line:84% So once we know these two points, we can just fill it in. 00:52:36.710 --> 00:52:40.110 align:middle line:84% It's going to be a line like this. 00:52:40.110 --> 00:52:46.532 align:middle line:90% So this is u1 of T, p. 00:52:46.532 --> 00:52:48.990 align:middle line:84% And remember, if you did the formula earlier in your notes, 00:52:48.990 --> 00:52:52.030 align:middle line:90% we've switched p to 1 minus p. 00:52:52.030 --> 00:52:55.070 align:middle line:84% So the formula in your notes was for the old convention. 00:52:55.070 --> 00:52:58.390 align:middle line:90% So that no longer applies. 00:52:58.390 --> 00:53:00.710 align:middle line:90% What about B? 00:53:00.710 --> 00:53:02.450 align:middle line:90% Well, B is symmetric. 00:53:02.450 --> 00:53:05.550 align:middle line:84% If I play B, then if I know my opponent is playing left, 00:53:05.550 --> 00:53:08.310 align:middle line:90% I get negative 1. 00:53:08.310 --> 00:53:12.143 align:middle line:84% And if I know my opponent is playing right, I get 2. 00:53:12.143 --> 00:53:15.290 align:middle line:84% Let me test my drawing skills here. 00:53:15.290 --> 00:53:18.130 align:middle line:90% And it's going to be linear. 00:53:18.130 --> 00:53:21.210 align:middle line:84% So we're going to get something like this. 00:53:21.210 --> 00:53:21.760 align:middle line:90% What about M? 00:53:21.760 --> 00:53:23.510 align:middle line:84% What is the graph of M going to look like? 00:53:23.510 --> 00:53:27.050 align:middle line:84% It's going to look a bit different, right? 00:53:27.050 --> 00:53:28.750 align:middle line:90% What will that line look like? 00:53:28.750 --> 00:53:29.590 align:middle line:90% Yeah? 00:53:29.590 --> 00:53:30.600 align:middle line:90% STUDENT: [INAUDIBLE] 00:53:30.600 --> 00:53:32.350 align:middle line:84% IAN BALL: It's just a horizontal line at 0 00:53:32.350 --> 00:53:34.950 align:middle line:84% because, if I play M, whatever my opponent does, 00:53:34.950 --> 00:53:36.160 align:middle line:90% I get a payoff of 0. 00:53:36.160 --> 00:53:37.910 align:middle line:84% So whatever I believe my opponent's doing, 00:53:37.910 --> 00:53:39.730 align:middle line:90% I'm going to get a payoff of 0. 00:53:39.730 --> 00:53:43.330 align:middle line:84% So now, we have a horizontal line here at 0. 00:53:43.330 --> 00:53:48.090 align:middle line:84% And if we're careful, this intersection should be at 1/2. 00:53:48.090 --> 00:53:50.690 align:middle line:90% You can check that. 00:53:50.690 --> 00:53:52.830 align:middle line:84% So let me finish writing these out. 00:53:52.830 --> 00:54:03.370 align:middle line:90% 00:54:03.370 --> 00:54:06.330 align:middle line:84% So to be clear, each of these is a different function of p. 00:54:06.330 --> 00:54:08.270 align:middle line:84% We have three different functions of p, 00:54:08.270 --> 00:54:10.370 align:middle line:84% where p is a number between 0 and 1. 00:54:10.370 --> 00:54:13.520 align:middle line:84% And I've graphed all three of the functions here. 00:54:13.520 --> 00:54:15.440 align:middle line:84% So from the graph, we should be able to say 00:54:15.440 --> 00:54:18.000 align:middle line:90% a lot about best responses. 00:54:18.000 --> 00:54:21.180 align:middle line:84% Can anyone read off from the graph what my best response is? 00:54:21.180 --> 00:54:25.200 align:middle line:90% 00:54:25.200 --> 00:54:26.800 align:middle line:90% Let's go through it, right? 00:54:26.800 --> 00:54:28.600 align:middle line:90% Let's fix a belief. 00:54:28.600 --> 00:54:31.440 align:middle line:84% A belief is some point on the horizontal axis. 00:54:31.440 --> 00:54:33.000 align:middle line:84% Now, once I fix that belief, if I 00:54:33.000 --> 00:54:36.060 align:middle line:84% want to understand my payoff from my different strategies, 00:54:36.060 --> 00:54:40.240 align:middle line:84% I'm going to draw a vertical line upwards. 00:54:40.240 --> 00:54:42.380 align:middle line:84% And as I pass up through that vertical line, 00:54:42.380 --> 00:54:46.520 align:middle line:84% each curve I intersect is telling me my expected payoff 00:54:46.520 --> 00:54:49.743 align:middle line:84% from that particular strategy given this belief. 00:54:49.743 --> 00:54:51.160 align:middle line:84% So at this point, what it tells me 00:54:51.160 --> 00:54:56.680 align:middle line:84% is my lowest payoff is from B. M gives me my middle payoff. 00:54:56.680 --> 00:55:01.360 align:middle line:84% And then T gives me the highest payoff. 00:55:01.360 --> 00:55:05.440 align:middle line:84% So at any beliefs where the highest curve is T, 00:55:05.440 --> 00:55:07.720 align:middle line:90% T is my best response. 00:55:07.720 --> 00:55:10.280 align:middle line:84% At any beliefs where B is the highest curve, 00:55:10.280 --> 00:55:12.100 align:middle line:90% B is my best response. 00:55:12.100 --> 00:55:14.100 align:middle line:90% So let's look at this. 00:55:14.100 --> 00:55:16.200 align:middle line:84% I should have gotten out the colored chalk. 00:55:16.200 --> 00:55:19.140 align:middle line:90% 00:55:19.140 --> 00:55:20.800 align:middle line:90% That'd be nice if I have it. 00:55:20.800 --> 00:55:23.820 align:middle line:90% 00:55:23.820 --> 00:55:26.240 align:middle line:90% Maybe no colored chalk today. 00:55:26.240 --> 00:55:27.100 align:middle line:90% Ah, yes. 00:55:27.100 --> 00:55:30.883 align:middle line:90% 00:55:30.883 --> 00:55:33.300 align:middle line:84% So what I'm actually going to do-- the easiest thing to do 00:55:33.300 --> 00:55:35.460 align:middle line:84% is I'm going to shade in what's sometimes 00:55:35.460 --> 00:55:36.680 align:middle line:90% called the upper envelope. 00:55:36.680 --> 00:55:37.820 align:middle line:84% I'm going to go through all these curves. 00:55:37.820 --> 00:55:39.620 align:middle line:84% And I'm going to look at what's highest. 00:55:39.620 --> 00:55:42.590 align:middle line:84% So if I'm here, this is the highest curve. 00:55:42.590 --> 00:55:44.220 align:middle line:90% It comes down. 00:55:44.220 --> 00:55:45.120 align:middle line:90% Here, it changes. 00:55:45.120 --> 00:55:48.900 align:middle line:84% Now, this becomes the highest curve. 00:55:48.900 --> 00:55:51.900 align:middle line:90% And I get something like this. 00:55:51.900 --> 00:55:55.700 align:middle line:90% So now, you can see down here-- 00:55:55.700 --> 00:55:59.380 align:middle line:84% can anyone now read off from the graph what my best responses are 00:55:59.380 --> 00:56:01.420 align:middle line:90% as a function of my beliefs? 00:56:01.420 --> 00:56:03.660 align:middle line:90% So when is T a best response? 00:56:03.660 --> 00:56:05.900 align:middle line:84% For which beliefs is T a best response? 00:56:05.900 --> 00:56:06.400 align:middle line:90% Yeah? 00:56:06.400 --> 00:56:08.880 align:middle line:84% STUDENT: From 0 to 1/2 probability of choosing R. 00:56:08.880 --> 00:56:12.860 align:middle line:84% And then from 1/2 to 1, it's better to use B instead. 00:56:12.860 --> 00:56:14.860 align:middle line:90% [INAUDIBLE] 00:56:14.860 --> 00:56:16.380 align:middle line:84% IAN BALL: Great, so exactly right. 00:56:16.380 --> 00:56:18.940 align:middle line:84% And a key point to make at the end, these are inclusive. 00:56:18.940 --> 00:56:20.960 align:middle line:84% So at 1/2, I actually have two best responses. 00:56:20.960 --> 00:56:24.320 align:middle line:84% T and B are both best responses at 1/2, as you pointed out. 00:56:24.320 --> 00:56:27.800 align:middle line:84% And if my belief is below 1/2, then T is the unique best 00:56:27.800 --> 00:56:30.080 align:middle line:84% response, or the strict best response. 00:56:30.080 --> 00:56:33.640 align:middle line:84% And if p is above 1/2, then B is the unique best response 00:56:33.640 --> 00:56:36.540 align:middle line:90% or the strict best response. 00:56:36.540 --> 00:56:37.040 align:middle line:90% Great. 00:56:37.040 --> 00:56:41.000 align:middle line:84% Does this make sense/ Well, yes, because T does really well 00:56:41.000 --> 00:56:43.240 align:middle line:90% if my opponent plays left. 00:56:43.240 --> 00:56:45.823 align:middle line:84% So if my probability on right is small enough 00:56:45.823 --> 00:56:48.240 align:middle line:84% and I think it's likely enough that my opponent is playing 00:56:48.240 --> 00:56:51.120 align:middle line:90% left, then T is the best option. 00:56:51.120 --> 00:56:53.960 align:middle line:84% If it's likely enough that my opponent is playing R, 00:56:53.960 --> 00:56:55.560 align:middle line:90% then B is the best option. 00:56:55.560 --> 00:56:59.840 align:middle line:84% And we see that B is the best response-- is a best response. 00:56:59.840 --> 00:57:04.680 align:middle line:84% But now we have a predicament, which is, what about M? 00:57:04.680 --> 00:57:09.620 align:middle line:90% 00:57:09.620 --> 00:57:12.180 align:middle line:84% M is not a best response to any belief. 00:57:12.180 --> 00:57:13.640 align:middle line:84% Does this mean we shouldn't play M? 00:57:13.640 --> 00:57:15.740 align:middle line:90% What do you think about M? 00:57:15.740 --> 00:57:16.840 align:middle line:90% Would you ever play it? 00:57:16.840 --> 00:57:20.660 align:middle line:90% 00:57:20.660 --> 00:57:21.620 align:middle line:90% Yeah? 00:57:21.620 --> 00:57:23.940 align:middle line:84% STUDENT: If you're super uncertain about your beliefs 00:57:23.940 --> 00:57:27.780 align:middle line:84% [INAUDIBLE], you might consider M. 00:57:27.780 --> 00:57:31.305 align:middle line:84% IAN BALL: We have to be really careful with those words. 00:57:31.305 --> 00:57:33.180 align:middle line:84% What you're describing is something important 00:57:33.180 --> 00:57:34.460 align:middle line:84% that we're not going to cover in this class, which 00:57:34.460 --> 00:57:35.680 align:middle line:90% is called ambiguity aversion. 00:57:35.680 --> 00:57:40.360 align:middle line:84% So I don't want to go too far down this path. 00:57:40.360 --> 00:57:42.640 align:middle line:84% But we're going to stick with expected utility. 00:57:42.640 --> 00:57:44.452 align:middle line:84% And risk aversion would not actually 00:57:44.452 --> 00:57:47.160 align:middle line:84% cause you to choose M. Something called ambiguity aversion would. 00:57:47.160 --> 00:57:48.785 align:middle line:84% But we're not going to talk about that. 00:57:48.785 --> 00:57:51.420 align:middle line:84% So we're imagining that we have our beliefs well-defined 00:57:51.420 --> 00:57:53.620 align:middle line:84% about how the other person is doing. 00:57:53.620 --> 00:57:55.860 align:middle line:84% So I think it's actually fair to say I don't really 00:57:55.860 --> 00:57:59.780 align:middle line:84% want to play M. There's no belief against which M is a best 00:57:59.780 --> 00:58:00.340 align:middle line:90% response. 00:58:00.340 --> 00:58:04.850 align:middle line:84% But I think it's natural to think back to dominance. 00:58:04.850 --> 00:58:07.450 align:middle line:84% What we said before is you might want to throw out a strategy 00:58:07.450 --> 00:58:09.530 align:middle line:84% if it's dominated by another strategy. 00:58:09.530 --> 00:58:12.130 align:middle line:84% But now, we're in this weird middle ground, where 00:58:12.130 --> 00:58:15.610 align:middle line:84% strategy M is not a best response to any belief, 00:58:15.610 --> 00:58:18.810 align:middle line:84% but it's also not dominated by any other strategy. 00:58:18.810 --> 00:58:21.330 align:middle line:90% And that is a weird conundrum. 00:58:21.330 --> 00:58:23.682 align:middle line:84% But we're going to show that, in fact, M is dominated. 00:58:23.682 --> 00:58:25.390 align:middle line:84% We just have to be a little more careful. 00:58:25.390 --> 00:58:28.893 align:middle line:84% Can anyone see a sense in which M is dominated? 00:58:28.893 --> 00:58:30.810 align:middle line:84% Is there anything I could do that would always 00:58:30.810 --> 00:58:32.390 align:middle line:90% do better than M? 00:58:32.390 --> 00:58:32.890 align:middle line:90% Yeah? 00:58:32.890 --> 00:58:35.290 align:middle line:84% STUDENT: [INAUDIBLE] strategy that basically 00:58:35.290 --> 00:58:37.250 align:middle line:90% varies on whatever [INAUDIBLE]. 00:58:37.250 --> 00:58:41.210 align:middle line:90% 00:58:41.210 --> 00:58:43.647 align:middle line:84% otherwise, that strategy [INAUDIBLE]. 00:58:43.647 --> 00:58:45.730 align:middle line:84% IAN BALL: OK, so I like the first part of what you 00:58:45.730 --> 00:58:46.870 align:middle line:90% said about a mixed strategy. 00:58:46.870 --> 00:58:48.412 align:middle line:84% But crucially, with a mixed strategy, 00:58:48.412 --> 00:58:52.270 align:middle line:84% we can't vary our mixing probability based on our belief. 00:58:52.270 --> 00:58:53.590 align:middle line:90% But the mixed idea is crucial. 00:58:53.590 --> 00:58:56.908 align:middle line:84% So what if I said, instead of choosing strategy M, 00:58:56.908 --> 00:58:57.950 align:middle line:90% I'm going to flip a coin. 00:58:57.950 --> 00:58:59.050 align:middle line:90% Do you want to add something? 00:58:59.050 --> 00:58:59.550 align:middle line:90% STUDENT: [INAUDIBLE] 00:58:59.550 --> 00:59:00.092 align:middle line:90% IAN BALL: OK. 00:59:00.092 --> 00:59:01.710 align:middle line:90% Imagine I flipped a coin. 00:59:01.710 --> 00:59:04.750 align:middle line:84% And if the coin is heads, I'm going to play T. 00:59:04.750 --> 00:59:09.917 align:middle line:84% So consider the strategy which we'll call 1/2 T plus 1/2 B. 00:59:09.917 --> 00:59:10.750 align:middle line:90% What does this mean? 00:59:10.750 --> 00:59:12.118 align:middle line:90% I'm going to flip a coin. 00:59:12.118 --> 00:59:13.910 align:middle line:84% If it's heads, I play-- well, if it's heads 00:59:13.910 --> 00:59:15.118 align:middle line:90% I play T. That's a bit weird. 00:59:15.118 --> 00:59:17.470 align:middle line:84% If it's tails I play T. If it's heads, I play B. 00:59:17.470 --> 00:59:20.910 align:middle line:90% If I were to do that-- 00:59:20.910 --> 00:59:22.970 align:middle line:84% let's add this strategy down here. 00:59:22.970 --> 00:59:32.350 align:middle line:84% So this is 1/2 T plus 1/2 B. If my opponent plays L, 00:59:32.350 --> 00:59:35.470 align:middle line:84% what is my payoff from this mixed strategy where I flip 00:59:35.470 --> 00:59:36.010 align:middle line:90% a coin? 00:59:36.010 --> 00:59:41.470 align:middle line:90% 00:59:41.470 --> 00:59:42.790 align:middle line:90% What's my expected payoff? 00:59:42.790 --> 00:59:48.750 align:middle line:84% Well, with probability 1/2 I play T and I get 2. 00:59:48.750 --> 00:59:51.910 align:middle line:84% With probability 1/2 I play B and I get negative 1. 00:59:51.910 --> 00:59:55.430 align:middle line:84% So my expected payoff is 1/2 of 2, which is 1, 00:59:55.430 --> 00:59:57.850 align:middle line:84% plus 1/2 negative 1, which is minus 1/2. 00:59:57.850 --> 01:00:01.930 align:middle line:90% And if I add it up, I get 1/2. 01:00:01.930 --> 01:00:05.730 align:middle line:90% And now, let's go to this side. 01:00:05.730 --> 01:00:09.290 align:middle line:84% If I do my same coin flip, with probability 1/2, 01:00:09.290 --> 01:00:11.370 align:middle line:90% I'm going to get negative 1. 01:00:11.370 --> 01:00:14.290 align:middle line:84% With probability 1/2, I'm going to get 2. 01:00:14.290 --> 01:00:16.190 align:middle line:84% So maybe I'll do the addition here. 01:00:16.190 --> 01:00:21.750 align:middle line:84% It's 1/2 times 2 plus 1/2 times negative 1. 01:00:21.750 --> 01:00:24.750 align:middle line:84% It's easier to distribute and say, well, 2 minus 1 is 1. 01:00:24.750 --> 01:00:26.490 align:middle line:84% So it's 1/2 times 1, which is 1/2. 01:00:26.490 --> 01:00:31.130 align:middle line:90% 01:00:31.130 --> 01:00:34.850 align:middle line:90% Well, now, is M dominated? 01:00:34.850 --> 01:00:37.530 align:middle line:84% So in this case, we see that M is not 01:00:37.530 --> 01:00:39.730 align:middle line:90% dominated by any pure strategy. 01:00:39.730 --> 01:00:42.330 align:middle line:84% But it is dominated by what's called a mixed strategy. 01:00:42.330 --> 01:00:45.490 align:middle line:84% There's a way I can flip a coin so that, 01:00:45.490 --> 01:00:49.410 align:middle line:84% if I flip a coin in this way, well, if my opponent plays L, 01:00:49.410 --> 01:00:52.570 align:middle line:84% I get 1/2 from this mixed strategy and I only get 0 from 01:00:52.570 --> 01:00:53.930 align:middle line:90% m. 01:00:53.930 --> 01:00:57.680 align:middle line:84% And if my opponent plays R, I get 1/2 from this mixed strategy 01:00:57.680 --> 01:00:59.920 align:middle line:90% and I only get 0 from M. 01:00:59.920 --> 01:01:03.360 align:middle line:84% And in fact, whatever belief I have about however my opponents 01:01:03.360 --> 01:01:05.933 align:middle line:84% play, I do strictly better under this mixture. 01:01:05.933 --> 01:01:08.100 align:middle line:84% We could even write down the payoff of this mixture. 01:01:08.100 --> 01:01:10.600 align:middle line:84% It's actually going to look exactly like this. 01:01:10.600 --> 01:01:13.640 align:middle line:84% If we wanted to say, what is my expected payoff 01:01:13.640 --> 01:01:16.633 align:middle line:84% from this mixed strategy as a function of my belief? 01:01:16.633 --> 01:01:18.800 align:middle line:84% So what I have here-- let me put it a bit separate-- 01:01:18.800 --> 01:01:25.200 align:middle line:84% is u1 of 1/2 T plus 1/2 B comma p. 01:01:25.200 --> 01:01:28.600 align:middle line:84% So this says, if I believe my opponent is playing right with 01:01:28.600 --> 01:01:33.520 align:middle line:84% probability p and I flip a coin and play T with probability 1/2 01:01:33.520 --> 01:01:36.280 align:middle line:84% and B with probability 1/2, what is my payoff going to be? 01:01:36.280 --> 01:01:39.240 align:middle line:84% And you can see the dominance relation because this is always 01:01:39.240 --> 01:01:41.760 align:middle line:90% higher than this. 01:01:41.760 --> 01:01:44.080 align:middle line:84% So do we think people really play mixed strategies 01:01:44.080 --> 01:01:45.360 align:middle line:90% in practice? 01:01:45.360 --> 01:01:46.450 align:middle line:90% We can debate about it. 01:01:46.450 --> 01:01:48.200 align:middle line:84% But I want to point out that this resolves 01:01:48.200 --> 01:01:51.260 align:middle line:84% this seeming paradox at first where, on the one hand, 01:01:51.260 --> 01:01:53.620 align:middle line:84% M was never a best response to any belief. 01:01:53.620 --> 01:01:55.600 align:middle line:84% So it seemed like a really bad strategy. 01:01:55.600 --> 01:01:57.340 align:middle line:84% But we couldn't directly show what 01:01:57.340 --> 01:01:58.900 align:middle line:90% would always be better than it. 01:01:58.900 --> 01:02:00.720 align:middle line:84% By using mixed strategies, we can. 01:02:00.720 --> 01:02:03.260 align:middle line:84% We see that, in fact, M is dominated. 01:02:03.260 --> 01:02:05.080 align:middle line:84% It's, in fact, strictly dominated, 01:02:05.080 --> 01:02:06.760 align:middle line:90% just not by a pure strategy. 01:02:06.760 --> 01:02:08.940 align:middle line:84% It's strictly dominated by a mixed strategy. 01:02:08.940 --> 01:02:11.060 align:middle line:84% So let's formally define a mixed strategy. 01:02:11.060 --> 01:02:14.800 align:middle line:84% It's what you would think, but be a little careful. 01:02:14.800 --> 01:02:26.300 align:middle line:90% 01:02:26.300 --> 01:02:28.020 align:middle line:90% So what is a mixed strategy? 01:02:28.020 --> 01:02:30.560 align:middle line:84% It says, well, just like a belief, 01:02:30.560 --> 01:02:32.980 align:middle line:84% it's a probability distribution, or a lottery. 01:02:32.980 --> 01:02:35.520 align:middle line:84% But it's not a lottery about other people's strategies. 01:02:35.520 --> 01:02:37.360 align:middle line:84% It's a lottery about my own strategy. 01:02:37.360 --> 01:02:45.470 align:middle line:84% So a mixed strategy for player i is a function-- 01:02:45.470 --> 01:02:47.600 align:middle line:84% or is a probability distribution. 01:02:47.600 --> 01:02:50.540 align:middle line:90% 01:02:50.540 --> 01:02:53.320 align:middle line:84% We often use the notation sigma i. 01:02:53.320 --> 01:02:55.840 align:middle line:84% So we use Si for a pure strategy. 01:02:55.840 --> 01:02:58.180 align:middle line:84% And sigma i, it's like the Greek version of S. 01:02:58.180 --> 01:02:59.920 align:middle line:84% We use that for a mixed strategy. 01:02:59.920 --> 01:03:02.800 align:middle line:84% Well, now, it's going to be a function from Si 01:03:02.800 --> 01:03:08.840 align:middle line:84% to 0, 1 that says, for each possible strategy, what 01:03:08.840 --> 01:03:11.277 align:middle line:84% is the probability that I play that strategy? 01:03:11.277 --> 01:03:13.360 align:middle line:84% But once again, we need another condition for this 01:03:13.360 --> 01:03:14.080 align:middle line:90% to make sense. 01:03:14.080 --> 01:03:16.440 align:middle line:84% The probabilities have to sum up to 1. 01:03:16.440 --> 01:03:18.500 align:middle line:90% So it's a function satisfying-- 01:03:18.500 --> 01:03:21.320 align:middle line:90% 01:03:21.320 --> 01:03:26.120 align:middle line:84% well, if I sum over all my strategies, 01:03:26.120 --> 01:03:29.500 align:middle line:84% the probability I put on that strategy, that better sum to 1. 01:03:29.500 --> 01:03:35.240 align:middle line:90% 01:03:35.240 --> 01:03:37.960 align:middle line:84% And I asked before, do people ever play mixed strategies? 01:03:37.960 --> 01:03:39.940 align:middle line:84% In a lot of economic contexts, they don't. 01:03:39.940 --> 01:03:41.482 align:middle line:84% But there's actually one domain where 01:03:41.482 --> 01:03:43.100 align:middle line:90% mixed strategies are crucial. 01:03:43.100 --> 01:03:45.920 align:middle line:84% Does anyone know what's a domain where 01:03:45.920 --> 01:03:49.100 align:middle line:84% people are very careful to play mixed strategies? 01:03:49.100 --> 01:03:51.947 align:middle line:90% 01:03:51.947 --> 01:03:54.280 align:middle line:84% Or are there times where you've played a mixed strategy? 01:03:54.280 --> 01:03:57.320 align:middle line:90% 01:03:57.320 --> 01:03:57.820 align:middle line:90% Yeah? 01:03:57.820 --> 01:03:59.112 align:middle line:90% STUDENT: Rock, paper, scissors. 01:03:59.112 --> 01:04:01.340 align:middle line:84% IAN BALL: Rock, paper, scissors is a good one. 01:04:01.340 --> 01:04:03.320 align:middle line:84% And if you played in rock, paper, scissors in competitions, 01:04:03.320 --> 01:04:04.760 align:middle line:84% yeah, that would probably be really important. 01:04:04.760 --> 01:04:06.460 align:middle line:84% There's another domain where people make a lot of money 01:04:06.460 --> 01:04:08.460 align:middle line:84% where randomization is also important. 01:04:08.460 --> 01:04:09.020 align:middle line:90% Yeah? 01:04:09.020 --> 01:04:09.480 align:middle line:90% STUDENT: Poker. 01:04:09.480 --> 01:04:10.688 align:middle line:90% IAN BALL: Professional poker. 01:04:10.688 --> 01:04:12.540 align:middle line:84% So professional poker players are extremely 01:04:12.540 --> 01:04:13.457 align:middle line:90% good at randomization. 01:04:13.457 --> 01:04:15.325 align:middle line:84% And they train to randomize well. 01:04:15.325 --> 01:04:17.200 align:middle line:84% Humans are very bad at randomizing naturally. 01:04:17.200 --> 01:04:19.380 align:middle line:84% And there's all these techniques professional poker players make. 01:04:19.380 --> 01:04:21.703 align:middle line:84% 15 years ago, you could play poker and win money 01:04:21.703 --> 01:04:23.120 align:middle line:84% without being good at randomizing. 01:04:23.120 --> 01:04:24.602 align:middle line:84% Today, you would just get destroyed 01:04:24.602 --> 01:04:27.060 align:middle line:84% because the best players are extremely good at randomizing. 01:04:27.060 --> 01:04:29.060 align:middle line:90% And they study game theory. 01:04:29.060 --> 01:04:31.540 align:middle line:84% And they randomize extremely carefully. 01:04:31.540 --> 01:04:33.043 align:middle line:84% And basically, the people who are 01:04:33.043 --> 01:04:35.460 align:middle line:84% better at randomizing with exactly the right probabilities 01:04:35.460 --> 01:04:36.500 align:middle line:90% win a lot of money. 01:04:36.500 --> 01:04:38.740 align:middle line:84% So poker is a great example of mixed strategies 01:04:38.740 --> 01:04:39.538 align:middle line:90% that really happen. 01:04:39.538 --> 01:04:41.080 align:middle line:84% If I taught this course 20 years ago, 01:04:41.080 --> 01:04:43.300 align:middle line:84% it would be hard to come up with an example of people 01:04:43.300 --> 01:04:44.360 align:middle line:90% playing mixed strategies. 01:04:44.360 --> 01:04:47.277 align:middle line:84% And now, the reality has caught up to the game theory. 01:04:47.277 --> 01:04:49.610 align:middle line:84% And now, there's a lot of people who make a lot of money 01:04:49.610 --> 01:04:51.770 align:middle line:90% by mixing. 01:04:51.770 --> 01:04:52.270 align:middle line:90% OK. 01:04:52.270 --> 01:04:55.450 align:middle line:84% So now, let's move on a little bit 01:04:55.450 --> 01:04:57.990 align:middle line:84% and try to understand how general this phenomenon is. 01:04:57.990 --> 01:05:00.290 align:middle line:84% So I want to make one observation here. 01:05:00.290 --> 01:05:02.730 align:middle line:84% And then we want to try to see how general that is. 01:05:02.730 --> 01:05:06.930 align:middle line:84% So if you're approaching a game like this and you're player 1, 01:05:06.930 --> 01:05:08.830 align:middle line:84% there's two ways you could go about it. 01:05:08.830 --> 01:05:12.410 align:middle line:84% If I were to ask you what is a reasonable strategy to play, 01:05:12.410 --> 01:05:14.330 align:middle line:90% you could say, well-- 01:05:14.330 --> 01:05:18.690 align:middle line:84% so let me say, let's look at player 1. 01:05:18.690 --> 01:05:20.430 align:middle line:84% And I'll just say, which strategies? 01:05:20.430 --> 01:05:23.902 align:middle line:90% 01:05:23.902 --> 01:05:25.610 align:middle line:84% Well, one thing you could do is you could 01:05:25.610 --> 01:05:27.610 align:middle line:90% start with the good strategies. 01:05:27.610 --> 01:05:30.110 align:middle line:84% And maybe I'll call them the reasonable strategies. 01:05:30.110 --> 01:05:33.935 align:middle line:90% 01:05:33.935 --> 01:05:35.310 align:middle line:84% This is not really a formal term. 01:05:35.310 --> 01:05:37.730 align:middle line:84% But I'll just say reasonable strategies. 01:05:37.730 --> 01:05:39.250 align:middle line:84% And these would be the strategies 01:05:39.250 --> 01:05:41.590 align:middle line:84% that are best responses to some belief. 01:05:41.590 --> 01:05:52.043 align:middle line:90% 01:05:52.043 --> 01:05:54.710 align:middle line:84% So I don't want you to think too much, for the moment, about how 01:05:54.710 --> 01:05:55.650 align:middle line:90% you form your beliefs. 01:05:55.650 --> 01:05:58.630 align:middle line:84% But let's just say, is there some belief 01:05:58.630 --> 01:06:00.870 align:middle line:84% I could have that would make this a best response? 01:06:00.870 --> 01:06:02.895 align:middle line:84% So in this game, we saw in the graph 01:06:02.895 --> 01:06:04.270 align:middle line:84% which strategies in this game are 01:06:04.270 --> 01:06:06.050 align:middle line:90% best responses to some belief. 01:06:06.050 --> 01:06:08.290 align:middle line:84% So which strategies are reasonable in this sense? 01:06:08.290 --> 01:06:11.630 align:middle line:90% 01:06:11.630 --> 01:06:12.910 align:middle line:90% Yeah? 01:06:12.910 --> 01:06:13.830 align:middle line:90% STUDENT: T and B. 01:06:13.830 --> 01:06:16.190 align:middle line:90% IAN BALL: T and B, right? 01:06:16.190 --> 01:06:18.830 align:middle line:84% So I might say, look, I have three strategies. 01:06:18.830 --> 01:06:21.903 align:middle line:84% But the reasonable ones are T and B. Another way 01:06:21.903 --> 01:06:24.070 align:middle line:84% you could approach it, though, is instead of looking 01:06:24.070 --> 01:06:25.430 align:middle line:84% for what you should do, you could 01:06:25.430 --> 01:06:27.770 align:middle line:84% try to look for what you don't want to do and avoid that. 01:06:27.770 --> 01:06:30.010 align:middle line:84% So I could say, what are the unreasonable strategies? 01:06:30.010 --> 01:06:35.030 align:middle line:90% 01:06:35.030 --> 01:06:37.150 align:middle line:84% And by an unreasonable strategy, I 01:06:37.150 --> 01:06:40.390 align:middle line:84% mean a strategy that's strictly dominated 01:06:40.390 --> 01:06:42.970 align:middle line:84% by some other strategy, possibly mixed. 01:06:42.970 --> 01:07:00.770 align:middle line:90% 01:07:00.770 --> 01:07:04.550 align:middle line:84% Now, technically, I only defined dominance for pure strategies. 01:07:04.550 --> 01:07:06.090 align:middle line:84% But everything makes sense if I just 01:07:06.090 --> 01:07:09.610 align:middle line:84% replace the Si's with the sigma i's and I 01:07:09.610 --> 01:07:13.060 align:middle line:84% interpret these utilities as expected utilities. 01:07:13.060 --> 01:07:15.310 align:middle line:84% So in this game, what are the unreasonable strategies? 01:07:15.310 --> 01:07:16.810 align:middle line:84% What are the strategies I could throw out 01:07:16.810 --> 01:07:18.750 align:middle line:84% by saying they're dominated by something else? 01:07:18.750 --> 01:07:21.450 align:middle line:90% 01:07:21.450 --> 01:07:22.310 align:middle line:90% Yeah? 01:07:22.310 --> 01:07:22.890 align:middle line:90% STUDENT: M. 01:07:22.890 --> 01:07:25.180 align:middle line:90% IAN BALL: Just M, right? 01:07:25.180 --> 01:07:26.930 align:middle line:84% So notice here, there's two different ways 01:07:26.930 --> 01:07:28.450 align:middle line:90% to get to the same answer. 01:07:28.450 --> 01:07:31.090 align:middle line:84% I could say, let me only consider 01:07:31.090 --> 01:07:33.970 align:middle line:84% strategies that are best responses to some beliefs. 01:07:33.970 --> 01:07:35.110 align:middle line:90% I would go through. 01:07:35.110 --> 01:07:38.390 align:middle line:84% And I would just get T and B. Or I could do the inverse. 01:07:38.390 --> 01:07:41.050 align:middle line:84% I could say, let me throw out all the strategies that 01:07:41.050 --> 01:07:43.870 align:middle line:84% are dominated, that are strictly dominated by something. 01:07:43.870 --> 01:07:48.590 align:middle line:84% I would throw out M. And I'd be left with just T and B. 01:07:48.590 --> 01:07:50.450 align:middle line:84% And the question is, is this a coincidence? 01:07:50.450 --> 01:07:52.310 align:middle line:84% Is this a special property of this game? 01:07:52.310 --> 01:07:54.550 align:middle line:84% Or is this more generally true, that we can always 01:07:54.550 --> 01:07:57.310 align:middle line:84% split our strategies into two groups, those 01:07:57.310 --> 01:07:59.950 align:middle line:84% that are best responses to some belief and those 01:07:59.950 --> 01:08:03.990 align:middle line:84% that are strictly dominated by some mixed strategy? 01:08:03.990 --> 01:08:06.130 align:middle line:84% And it turns out this is a very general property. 01:08:06.130 --> 01:08:10.150 align:middle line:84% And that'll be the first theorem that we state in this course. 01:08:10.150 --> 01:08:12.690 align:middle line:90% So let me state the theorem. 01:08:12.690 --> 01:08:30.990 align:middle line:90% 01:08:30.990 --> 01:08:34.029 align:middle line:90% So here's our theorem. 01:08:34.029 --> 01:08:35.170 align:middle line:90% We need some assumptions. 01:08:35.170 --> 01:08:37.529 align:middle line:84% So let's say we're in a finite strategic form game. 01:08:37.529 --> 01:08:40.580 align:middle line:90% 01:08:40.580 --> 01:08:43.380 align:middle line:84% Finite just means the set of strategies for every player 01:08:43.380 --> 01:08:44.100 align:middle line:90% is finite. 01:08:44.100 --> 01:08:45.520 align:middle line:84% And there's finitely many players. 01:08:45.520 --> 01:08:58.220 align:middle line:90% 01:08:58.220 --> 01:09:03.420 align:middle line:84% So for each player i and each strategy Si-- 01:09:03.420 --> 01:09:11.747 align:middle line:90% 01:09:11.747 --> 01:09:13.580 align:middle line:84% so before I write it, let's just understand. 01:09:13.580 --> 01:09:17.620 align:middle line:84% We fixed our finite strategic form game. 01:09:17.620 --> 01:09:19.899 align:middle line:84% We're considering a particular player 01:09:19.899 --> 01:09:23.939 align:middle line:90% i and a particular strategy Si. 01:09:23.939 --> 01:09:26.540 align:middle line:84% And what we want to argue, what we want to say, 01:09:26.540 --> 01:09:29.859 align:middle line:84% is that this strategy falls in exactly one of these two 01:09:29.859 --> 01:09:31.859 align:middle line:90% categories. 01:09:31.859 --> 01:09:33.500 align:middle line:90% It can't be in both. 01:09:33.500 --> 01:09:35.460 align:middle line:90% And it can't be in Neither. 01:09:35.460 --> 01:09:39.140 align:middle line:84% So what we say is exactly one of the following holds. 01:09:39.140 --> 01:09:42.000 align:middle line:90% 01:09:42.000 --> 01:09:44.979 align:middle line:84% Exactly one means not both and not neither. 01:09:44.979 --> 01:09:57.120 align:middle line:90% 01:09:57.120 --> 01:10:01.840 align:middle line:84% And one is that this strategy Si is reasonable in this sense. 01:10:01.840 --> 01:10:04.780 align:middle line:84% Meaning what/ Si is a best response to some belief. 01:10:04.780 --> 01:10:17.975 align:middle line:90% 01:10:17.975 --> 01:10:20.100 align:middle line:84% And when I say belief, what is that a belief about? 01:10:20.100 --> 01:10:25.140 align:middle line:84% It's a belief-- maybe I'll say over S negative i. 01:10:25.140 --> 01:10:27.800 align:middle line:84% Let's remind ourselves, when player i forms beliefs, 01:10:27.800 --> 01:10:29.400 align:middle line:84% they form beliefs over the strategy 01:10:29.400 --> 01:10:32.200 align:middle line:90% profiles of the other players. 01:10:32.200 --> 01:10:35.420 align:middle line:84% And then the other case is Si is strictly dominated. 01:10:35.420 --> 01:10:43.260 align:middle line:90% 01:10:43.260 --> 01:10:44.180 align:middle line:90% By what? 01:10:44.180 --> 01:10:46.280 align:middle line:90% Well, by some mixed strategy. 01:10:46.280 --> 01:10:47.700 align:middle line:90% And we have to allow mixed. 01:10:47.700 --> 01:10:49.287 align:middle line:84% If we only allowed pure strategies, 01:10:49.287 --> 01:10:51.620 align:middle line:84% we already saw up here that the theorem wouldn't be true 01:10:51.620 --> 01:10:54.700 align:middle line:84% because there are strategies that are never best responses, 01:10:54.700 --> 01:10:57.000 align:middle line:84% but are not dominated by any pure strategy. 01:10:57.000 --> 01:10:59.060 align:middle line:84% So we have to allow for mixed strategies. 01:10:59.060 --> 01:11:01.960 align:middle line:84% Si is strictly dominated by some mixed strategy. 01:11:01.960 --> 01:11:09.140 align:middle line:90% 01:11:09.140 --> 01:11:11.020 align:middle line:84% And of course, when I say strategy, 01:11:11.020 --> 01:11:13.260 align:middle line:84% that has to be a strategy of player i 01:11:13.260 --> 01:11:17.180 align:middle line:84% because player i compares two different strategies when 01:11:17.180 --> 01:11:18.880 align:middle line:90% they look at strict dominance. 01:11:18.880 --> 01:11:21.140 align:middle line:84% But wait a second, I said Si is strictly dominated 01:11:21.140 --> 01:11:22.360 align:middle line:90% by some mixed strategy. 01:11:22.360 --> 01:11:25.980 align:middle line:90% What about pure strategies? 01:11:25.980 --> 01:11:27.040 align:middle line:90% Am I including those? 01:11:27.040 --> 01:11:28.060 align:middle line:90% STUDENT: [INAUDIBLE] 01:11:28.060 --> 01:11:29.433 align:middle line:90% IAN BALL: Why? 01:11:29.433 --> 01:11:31.220 align:middle line:90% STUDENT: [INAUDIBLE] 01:11:31.220 --> 01:11:35.150 align:middle line:90% 01:11:35.150 --> 01:11:36.150 align:middle line:90% IAN BALL: Exactly right. 01:11:36.150 --> 01:11:39.450 align:middle line:84% So every pure strategy is a mixed strategy. 01:11:39.450 --> 01:11:42.770 align:middle line:84% But not every mixed strategy is a pure strategy. 01:11:42.770 --> 01:11:45.150 align:middle line:84% So as long as I cover mixed strategies-- in particular, 01:11:45.150 --> 01:11:47.730 align:middle line:84% I'm including the mixtures that put all the probability 01:11:47.730 --> 01:11:51.450 align:middle line:84% on a single strategy, which are simply pure strategies. 01:11:51.450 --> 01:11:53.910 align:middle line:84% So if we want to say that I'm not doing that, 01:11:53.910 --> 01:11:57.490 align:middle line:84% sometimes we say properly mixed strategy 01:11:57.490 --> 01:11:59.190 align:middle line:90% or nonpure mixed strategy. 01:11:59.190 --> 01:12:02.710 align:middle line:84% But here, we're allowing all kinds of mixed strategies, 01:12:02.710 --> 01:12:05.470 align:middle line:84% including the ones that are just pure strategies in disguise. 01:12:05.470 --> 01:12:09.370 align:middle line:90% 01:12:09.370 --> 01:12:11.930 align:middle line:84% And again, I think the interpretation of this 01:12:11.930 --> 01:12:15.610 align:middle line:84% is that there are two very natural ways of thinking 01:12:15.610 --> 01:12:18.390 align:middle line:84% about a game and which strategies someone might play. 01:12:18.390 --> 01:12:20.370 align:middle line:84% I think it would be very natural to say, 01:12:20.370 --> 01:12:22.530 align:middle line:84% it's reasonable to play anything that's 01:12:22.530 --> 01:12:25.210 align:middle line:90% a best response to some belief. 01:12:25.210 --> 01:12:28.470 align:middle line:84% And it would also be natural to say, here's what you should do. 01:12:28.470 --> 01:12:30.450 align:middle line:84% Just don't play anything that's strictly 01:12:30.450 --> 01:12:33.070 align:middle line:90% dominated by something else. 01:12:33.070 --> 01:12:35.320 align:middle line:84% And if those two approaches gave us different answers, 01:12:35.320 --> 01:12:37.070 align:middle line:84% we'd have to really think hard about which 01:12:37.070 --> 01:12:38.630 align:middle line:90% one was more reasonable. 01:12:38.630 --> 01:12:41.230 align:middle line:84% But it turns out these are literally going 01:12:41.230 --> 01:12:42.830 align:middle line:90% to give us the same answer. 01:12:42.830 --> 01:12:44.950 align:middle line:84% So you can think positively about what's 01:12:44.950 --> 01:12:47.230 align:middle line:84% the best response to something or you 01:12:47.230 --> 01:12:49.430 align:middle line:84% can think negatively about what's strictly 01:12:49.430 --> 01:12:51.190 align:middle line:90% dominated by something. 01:12:51.190 --> 01:12:54.030 align:middle line:84% And either way, you're going to get the same two 01:12:54.030 --> 01:12:56.530 align:middle line:84% groups of strategies-- the good ones and the bad ones. 01:12:56.530 --> 01:12:59.110 align:middle line:90% 01:12:59.110 --> 01:13:02.510 align:middle line:84% Let me say a little bit about how we prove this. 01:13:02.510 --> 01:13:06.810 align:middle line:84% The proof won't be covered on exams. 01:13:06.810 --> 01:13:08.850 align:middle line:84% But I want to give you some idea. 01:13:08.850 --> 01:13:13.550 align:middle line:90% 01:13:13.550 --> 01:13:15.790 align:middle line:90% So what's the argument? 01:13:15.790 --> 01:13:17.310 align:middle line:90% Proof? 01:13:17.310 --> 01:13:20.350 align:middle line:84% Well, how do you show that exactly one thing happens? 01:13:20.350 --> 01:13:23.650 align:middle line:84% You have to show that both can't happen and neither can't happen. 01:13:23.650 --> 01:13:26.210 align:middle line:84% So another way of saying it is at most one-- 01:13:26.210 --> 01:13:29.656 align:middle line:90% 01:13:29.656 --> 01:13:32.690 align:middle line:84% so at most one of these two things can be true 01:13:32.690 --> 01:13:33.670 align:middle line:90% and at least one. 01:13:33.670 --> 01:13:38.570 align:middle line:90% 01:13:38.570 --> 01:13:40.250 align:middle line:84% So to say at most one of the statements 01:13:40.250 --> 01:13:45.250 align:middle line:84% is true is another way of saying both statements can't be true-- 01:13:45.250 --> 01:13:48.330 align:middle line:90% at most one, not two. 01:13:48.330 --> 01:13:50.890 align:middle line:84% At least one is a way of saying it 01:13:50.890 --> 01:13:53.290 align:middle line:84% can't be the case that neither is true, 01:13:53.290 --> 01:13:56.490 align:middle line:84% at least one, either one or two, but not zero. 01:13:56.490 --> 01:13:59.090 align:middle line:84% And by one or two, I mean the number-- maybe 01:13:59.090 --> 01:14:01.030 align:middle line:84% I'll call these A and B to avoid confusion. 01:14:01.030 --> 01:14:03.730 align:middle line:90% 01:14:03.730 --> 01:14:05.865 align:middle line:84% I'm talking about at most one, meaning 01:14:05.865 --> 01:14:07.490 align:middle line:84% the number of statements that are true, 01:14:07.490 --> 01:14:08.990 align:middle line:84% not the labels of the statements. 01:14:08.990 --> 01:14:11.770 align:middle line:90% 01:14:11.770 --> 01:14:15.250 align:middle line:84% So one of these directions is a bit easier. 01:14:15.250 --> 01:14:17.097 align:middle line:90% One of these is quite tricky. 01:14:17.097 --> 01:14:19.430 align:middle line:84% Does anyone see one that they think is kind of intuitive 01:14:19.430 --> 01:14:20.478 align:middle line:90% and they could maybe-- 01:14:20.478 --> 01:14:22.770 align:middle line:84% I won't ask you for the proof, but could reason through 01:14:22.770 --> 01:14:23.350 align:middle line:90% on their own? 01:14:23.350 --> 01:14:26.610 align:middle line:90% 01:14:26.610 --> 01:14:27.550 align:middle line:90% Yeah? 01:14:27.550 --> 01:14:27.850 align:middle line:90% STUDENT: At most one. 01:14:27.850 --> 01:14:28.767 align:middle line:90% IAN BALL: At most one. 01:14:28.767 --> 01:14:31.190 align:middle line:84% So this is the easier direction-- not easy 01:14:31.190 --> 01:14:32.550 align:middle line:90% maybe, but easier. 01:14:32.550 --> 01:14:34.630 align:middle line:84% And what's your intuitive argument? 01:14:34.630 --> 01:14:36.920 align:middle line:90% We won't give a formal one. 01:14:36.920 --> 01:14:38.270 align:middle line:90% STUDENT: [INAUDIBLE] 01:14:38.270 --> 01:14:39.855 align:middle line:90% 01:14:39.855 --> 01:14:41.730 align:middle line:84% IAN BALL: We're going to get a contradiction. 01:14:41.730 --> 01:14:44.070 align:middle line:84% So if both were true, you're saying 01:14:44.070 --> 01:14:46.028 align:middle line:84% this is the best we can do against some belief. 01:14:46.028 --> 01:14:48.570 align:middle line:84% But there's something else that does strictly better whatever 01:14:48.570 --> 01:14:49.430 align:middle line:90% my opponent does. 01:14:49.430 --> 01:14:50.490 align:middle line:84% Well, if it does strictly better, 01:14:50.490 --> 01:14:52.698 align:middle line:84% whatever my opponent does, it should also do strictly 01:14:52.698 --> 01:14:54.110 align:middle line:90% better given my belief. 01:14:54.110 --> 01:14:56.755 align:middle line:84% But that means the other thing couldn't be a best response. 01:14:56.755 --> 01:14:58.130 align:middle line:84% So I think it's pretty intuitive. 01:14:58.130 --> 01:15:00.770 align:middle line:84% I'd encourage you to work out the algebra on your own. 01:15:00.770 --> 01:15:04.150 align:middle line:84% I think this is a good exercise to do at home. 01:15:04.150 --> 01:15:05.650 align:middle line:84% The at least one is a lot harder. 01:15:05.650 --> 01:15:07.470 align:middle line:84% And that I wouldn't expect you to be 01:15:07.470 --> 01:15:08.690 align:middle line:90% able to figure out at home. 01:15:08.690 --> 01:15:11.882 align:middle line:84% There is a proof given in the appendix of the lecture notes. 01:15:11.882 --> 01:15:13.090 align:middle line:90% And actually, let me clarify. 01:15:13.090 --> 01:15:14.632 align:middle line:84% I said I won't test you on the proof. 01:15:14.632 --> 01:15:16.770 align:middle line:84% But I could test you on this part of the proof. 01:15:16.770 --> 01:15:18.570 align:middle line:90% I think the top is fair game. 01:15:18.570 --> 01:15:19.410 align:middle line:90% The bottom is not. 01:15:19.410 --> 01:15:20.950 align:middle line:90% Let me be clear about that. 01:15:20.950 --> 01:15:25.910 align:middle line:84% So the bottom, for the harder part, see appendix. 01:15:25.910 --> 01:15:28.860 align:middle line:84% And it uses a mathematical result 01:15:28.860 --> 01:15:33.460 align:middle line:84% called either the separating hyperplane theorem, which 01:15:33.460 --> 01:15:37.780 align:middle line:84% you may have heard of, or something 01:15:37.780 --> 01:15:43.100 align:middle line:84% called Farkas' lemma is also my preferred way of proving it. 01:15:43.100 --> 01:15:46.740 align:middle line:84% And just to see why this is harder, well, if Si-- 01:15:46.740 --> 01:15:48.640 align:middle line:84% we want to show at least one is true. 01:15:48.640 --> 01:15:51.940 align:middle line:84% So we have to say if Si is not a best response to some belief, 01:15:51.940 --> 01:15:53.965 align:middle line:84% then it's strictly dominated by something. 01:15:53.965 --> 01:15:55.340 align:middle line:84% But that means the proof requires 01:15:55.340 --> 01:15:56.640 align:middle line:90% you to construct something. 01:15:56.640 --> 01:15:57.660 align:middle line:90% And that's what's hard. 01:15:57.660 --> 01:16:00.140 align:middle line:84% All we know is it's not a best response to any belief. 01:16:00.140 --> 01:16:02.300 align:middle line:84% But how do we know what dominates it? 01:16:02.300 --> 01:16:03.920 align:middle line:84% Well, how do we come up with that. 01:16:03.920 --> 01:16:04.800 align:middle line:90% That's pretty hard. 01:16:04.800 --> 01:16:07.220 align:middle line:84% So this theorem gives you a way of constructing 01:16:07.220 --> 01:16:10.337 align:middle line:84% either the belief you need or the dominating strategy 01:16:10.337 --> 01:16:10.920 align:middle line:90% that you need. 01:16:10.920 --> 01:16:12.295 align:middle line:90% And that's much more involved. 01:16:12.295 --> 01:16:13.920 align:middle line:84% So that won't be covered in the course. 01:16:13.920 --> 01:16:15.587 align:middle line:84% But I encourage you to read the appendix 01:16:15.587 --> 01:16:16.860 align:middle line:90% if you want to see that. 01:16:16.860 --> 01:16:18.140 align:middle line:90% And let me stop there. 01:16:18.140 --> 01:16:19.620 align:middle line:90% And I'll see everyone next week. 01:16:19.620 --> 01:16:22.250 align:middle line:90% Good luck on the quiz tomorrow. 01:16:22.250 --> 01:16:26.000 align:middle line:90%