WEBVTT 00:01.930 --> 00:06.350 Professor Ben Polak: So last time we saw this, 00:06.353 --> 00:11.373 we saw an example of a mixed strategy which was to play 1/3, 00:11.372 --> 00:15.372 1/3, 1/3 in our rock, paper, scissors game. 00:15.370 --> 00:18.230 Today, we're going to be formal, we're going to define 00:18.233 --> 00:20.993 mixed strategies and we're going to talk about them, 00:20.988 --> 00:22.878 and it's going to take a while. 00:22.880 --> 00:29.240 So let's start with a formal definition: a mixed strategy 00:29.244 --> 00:34.704 (and I'll develop notation as I'm going along, 00:34.700 --> 00:40.020 so let me call it P_i, 00:40.020 --> 00:49.290 i being the person who's playing it) P_i is a 00:49.289 --> 00:56.669 randomization over i's pure strategies. 00:56.670 --> 01:00.700 So in particular, we're going to use the notation 01:00.696 --> 01:05.646 P_i (si) to be the probability that Player i plays 01:05.645 --> 01:09.835 si given that he's mixing using P_i. 01:09.840 --> 01:24.380 So P_i(si) is the probability that P_i 01:24.378 --> 01:33.048 assigns to the pure strategy si. 01:33.050 --> 01:35.430 Let's immediately refer that back to our example. 01:35.430 --> 01:39.800 So for example, if I'm playing 1/3,1/3,1/3 in 01:39.799 --> 01:45.559 rock, paper, scissors then P_i is 1/3,1/3,1/3 and 01:45.559 --> 01:51.219 P_i of rock--so P_i(R)--is a 1/3. 01:51.220 --> 01:54.700 So without belaboring it, that's all I'm doing here, 01:54.701 --> 01:56.751 is developing some notation. 01:56.750 --> 02:00.200 Let's immediately encounter two things you might have questions 02:00.204 --> 02:02.984 about. So the first is, 02:02.975 --> 02:10.385 that in principle P_i(si) could be zero. 02:10.389 --> 02:12.689 Just because I'm playing a mixed strategy, 02:12.686 --> 02:15.876 it doesn't mean I have to involve all of my strategies. 02:15.879 --> 02:18.959 I could be playing a mixed strategy on two of my strategies 02:18.955 --> 02:21.495 and leave the other one with zero probability. 02:21.500 --> 02:24.340 So, for example, again in rock, 02:24.343 --> 02:28.703 paper, scissors, we could think of the strategy 02:28.703 --> 02:32.593 1/2,1/2,0. In this strategy I assign--I 02:32.591 --> 02:37.261 play rock half the time, I play paper half the time, 02:37.257 --> 02:39.907 but I never play scissors. 02:39.910 --> 02:42.530 So everyone understand that? 02:42.530 --> 02:46.750 And while we're here let's look at the other extreme. 02:46.750 --> 02:52.100 The probability assigned by my mixed strategy to a particular 02:52.096 --> 02:55.696 si could be one. It could be that I assign all 02:55.702 --> 02:58.812 of the probability to a particular strategy. 02:58.810 --> 03:01.640 What would we call a mixed strategy that assigns 03:01.640 --> 03:04.410 probability 1 to one of the pure strategies? 03:04.410 --> 03:06.340 What's a good name for that? 03:06.340 --> 03:08.780 That's a "pure strategy." 03:08.780 --> 03:12.480 So notice that we can think of pure strategies as the special 03:12.479 --> 03:15.929 case of a mixed strategy that assigns all the weight to a 03:15.931 --> 03:17.721 particular pure strategy. 03:17.720 --> 03:21.190 So, for example, if Pi(R) was 1, 03:21.185 --> 03:26.995 that's equivalent to saying that I'm playing the pure 03:26.998 --> 03:29.568 strategy rock, i.e. 03:29.570 --> 03:33.820 a pure strategy. So there's nothing here. 03:33.819 --> 03:37.649 I'm just being a little bit nerdy about developing notation 03:37.645 --> 03:40.675 and making sure that everything is in place, 03:40.680 --> 03:44.130 and just to point out again, one consequence of this is 03:44.126 --> 03:47.566 we've now got our pure strategies embedded in our mixed 03:47.572 --> 03:51.832 strategies. When I've got a mixed strategy 03:51.830 --> 03:58.320 I really am including in those all of the pure strategies. 03:58.320 --> 04:07.020 So let's proceed. 04:07.020 --> 04:19.630 I'm going to push that up a little high, sorry. 04:19.629 --> 04:22.649 So now I want to think about what are the payoffs that I get 04:22.651 --> 04:24.821 from mixed strategies, and again, I'm going to go a 04:24.818 --> 04:26.828 little slowly because it's a little tricky at first and we'll 04:26.829 --> 04:28.249 get used to this, don't panic, 04:28.248 --> 04:31.738 we'll get used to this as we go on and as you see them in 04:31.741 --> 04:34.051 homework assignments and in class. 04:34.050 --> 04:48.430 So let's talk about the payoffs from a mixed strategy. 04:48.430 --> 04:51.120 In particular, what we're going to worry about 04:51.123 --> 04:52.503 are expected payoffs. 04:52.500 --> 04:59.020 So the expected payoff of the mixed strategy P, 04:59.017 --> 05:05.387 let's be consistent and call it P_i, 05:05.393 --> 05:11.773 the mixed strategy P_i is what? 05:11.769 --> 05:26.669 It's the weighted average--it's a weighted average or a weighted 05:26.674 --> 05:40.404 mixture if you like--of the expected payoffs of each of the 05:40.396 --> 05:47.726 pure strategies in the mix. 05:47.730 --> 05:49.970 So this is a long way of saying something again which I think is 05:49.973 --> 05:51.863 a little bit obvious, but let me just say it again. 05:51.860 --> 05:55.070 The way in which we figure out the expected payoff of a mixed 05:55.065 --> 05:57.945 strategy is, we take the appropriately weighted average 05:57.951 --> 06:01.101 of the expected payoffs I would get from the pure strategies 06:01.103 --> 06:02.603 over which I'm mixing. 06:02.600 --> 06:06.520 So to make that less abstract let's immediately look at an 06:06.521 --> 06:09.841 example. So here's an example we'll come 06:09.840 --> 06:13.410 back to several times, but just once today, 06:13.405 --> 06:16.625 and this a game you've seen before. 06:16.629 --> 06:21.809 Here is the game Battle of the Sexes, in which Player A can 06:21.813 --> 06:25.213 choose--Player I can choose A and B, 06:25.209 --> 06:30.599 and Player II can choose a and b, and what I want to do is I 06:30.595 --> 06:35.975 want to figure out the payoff from particular strategies. 06:35.980 --> 06:47.120 So suppose that P is being played by Player I and P is 06:47.117 --> 06:51.737 let's say (1/5,4/5). 06:51.740 --> 06:52.900 So what do I mean by that? 06:52.899 --> 07:01.109 I mean that Player I is assigning 1/5 to playing A and 07:01.107 --> 07:04.047 4/5 to playing B. 07:04.050 --> 07:07.010 And suppose that Q--so I am going to use P and Q because 07:07.005 --> 07:10.335 it's convenient to do so rather than calling them P_1 07:10.337 --> 07:11.517 and P_2. 07:11.519 --> 07:15.389 So suppose that Q is the mixture that Player II is 07:19.346 --> 07:24.086 so she's putting a probability 1/2 on a and a probability 1/2 07:24.090 --> 07:26.270 on b. Just to notice I switched 07:26.273 --> 07:29.323 notation on you a little bit, for this example to keep life 07:29.319 --> 07:32.619 easy, I'm going to use P to be row's 07:32.615 --> 07:36.445 mixtures and Q to be column's mixtures. 07:36.449 --> 07:42.179 And the question I want to answer is what is the expected 07:42.184 --> 07:44.954 payoff in this case of P? 07:44.950 --> 07:52.560 What is P's expected payoff? 07:52.560 --> 07:56.810 The way I'm going to do that is, I'm first of all going to 07:56.810 --> 08:01.360 ask what is the expected payoff of each of the pure strategies 08:01.358 --> 08:06.458 that P involves, the pure strategies involved in 08:06.460 --> 08:10.570 P. So to start off--so the first 08:10.566 --> 08:18.856 step is ask what is the expected payoff for Player I of playing A 08:18.857 --> 08:26.107 against Q and what is the expected payoff for Player I of 08:26.111 --> 08:29.351 playing B against Q? 08:29.350 --> 08:32.500 That will be our first question and we'll come back and 08:32.498 --> 08:34.188 construct the payoff for P. 08:34.190 --> 08:36.910 So these are things we can do I think. 08:36.909 --> 08:41.199 So the expected payoff of A against Q is what? 08:41.200 --> 08:45.660 Well, half the time if you play A you're going to find your 08:45.656 --> 08:49.346 opponent is playing a, in which case you'll get 2, 08:49.349 --> 08:53.529 and half the time when you play A you'll find your opponent is 08:53.529 --> 08:56.269 playing b in which case you'll get 0. 08:56.270 --> 08:57.680 So let's just write that up. 08:57.679 --> 09:07.139 So I'm going to get 2 with probability 1/2 plus 0 with 09:07.137 --> 09:11.457 probability 1/2. Everyone happy with that? 09:11.460 --> 09:16.910 That gives me 1. Please correct my math in this. 09:16.909 --> 09:19.469 It's very easy at the board to make mistakes, 09:19.474 --> 09:21.344 but I think that one is right. 09:21.340 --> 09:23.120 Conversely, what if I played B? 09:23.120 --> 09:27.690 What's the expected payoff for the row player of playing B 09:27.692 --> 09:30.262 against Q, where Q is 1/2,1/2? 09:30.259 --> 09:36.189 So half the time when I play B, I'll meet a Player II playing a 09:36.186 --> 09:42.296 and I'll get 0 and half the time I'll find Player II is playing b 09:42.303 --> 09:45.713 and I'll get 1. So let's write that up. 09:45.710 --> 09:59.470 So I'll get 0 half the time and I'll get 1 half the time for an 09:59.467 --> 10:04.227 average of 1/2. That's the first thing I ask. 10:04.230 --> 10:08.600 And now to finish the job, I now want to figure out what 10:08.596 --> 10:13.276 is the expected payoff for Player I of using P against Q? 10:13.279 --> 10:16.819 That was the question I really wanted to start off with. 10:16.820 --> 10:18.290 What's the way to think about this? 10:18.289 --> 10:22.879 Well P is 1/5 of the time--according to P, 10:22.883 --> 10:29.163 1/5 of the time Player I is playing A and 4/5 of the time 10:29.156 --> 10:33.746 Player I is playing B, is that right? 10:33.750 --> 10:38.060 So to work out the expected payoff what we're going to do is 10:38.062 --> 10:40.842 we're going to take 1/5 of the time, 10:40.840 --> 10:45.250 and at which case he's playing A and he'll get the expected 10:45.253 --> 10:49.213 payoff he would have got from playing A against Q, 10:49.210 --> 10:56.970 and 4/5 of the time he's going to be playing B in which case 10:56.971 --> 11:04.471 he'll get the expected payoff from playing B against Q. 11:04.470 --> 11:07.840 Now just plugging in some numbers to that from above, 11:07.838 --> 11:11.598 so we've got 1/5 of the time he's doing the expected payoff 11:11.595 --> 11:15.865 from A against Q and that's this number we worked out already. 11:15.870 --> 11:25.540 So this number here can come down here, 1. 11:25.539 --> 11:32.359 And 4/5 of the time he's playing B against Q, 11:32.358 --> 11:38.708 in which case his expected payoff was 1/2, 11:38.711 --> 11:43.361 so this 1/2 comes in here. 11:43.360 --> 11:46.400 Everyone okay so far, how I constructed it so far? 11:46.399 --> 11:48.389 Is this podium in the way of you guys, are you okay? 11:48.390 --> 11:52.460 Let me push it slightly. 11:52.460 --> 11:54.920 So the total here is what? 12:01.360 --> 12:07.630 4/5 of 1/2 is 2/5, so I've got a total of 3/5. 12:07.630 --> 12:11.960 So the total here is 3/5. 12:11.960 --> 12:15.850 Everyone understand how I did that? 12:15.850 --> 12:19.380 Now while it's here let's notice something. 12:19.379 --> 12:23.899 When I played P, some of the time I played A and 12:23.902 --> 12:26.792 some of the time I played B. 12:26.789 --> 12:29.829 And when I ended up playing A, I got A's expected payoff. 12:29.830 --> 12:32.350 And when I played B, I got B's expected payoff. 12:32.350 --> 12:38.370 So the number I ended up with 3/5 must lie between the payoff 12:38.365 --> 12:44.575 I would have got from A which is 1, and the payoff I would have 12:44.582 --> 12:47.392 got from B which is 1/2. 12:47.390 --> 12:54.440 Is that right? So 3/5 lies between 1/2 and 1. 12:54.440 --> 12:57.460 Everyone okay with that? 12:57.460 --> 13:02.890 Now that's a simple but very general and very useful idea it 13:02.886 --> 13:06.076 turns out. The idea here is that the 13:06.084 --> 13:11.284 payoff I'm going to get must lie between the expected payoffs I 13:11.277 --> 13:14.877 would have got from the pure strategies. 13:14.880 --> 13:15.630 Let me say it again. 13:15.629 --> 13:20.509 In general, when I play a mixed strategy the expected payoff I 13:20.506 --> 13:25.296 get, is a weighted average of the expected payoffs of each of 13:25.302 --> 13:31.302 the pure strategies in the mix, and weighted averages always 13:31.300 --> 13:37.360 lie inside the payoffs that are involved in the mix. 13:37.360 --> 13:42.870 So let me try and push that simple idea a little harder. 13:42.870 --> 13:47.250 Suppose I was going to take the average height in the 13:47.249 --> 13:50.449 class--average height in this class. 13:50.450 --> 13:52.280 So let me just, rather than use the class, 13:52.278 --> 13:53.838 let me just use some T.A.'s here. 13:53.840 --> 13:59.700 So let me get these three T.A.'s to stand up a second. 13:59.700 --> 14:03.930 Suppose I want to figure out the average height of these 14:03.934 --> 14:06.194 three T.A.'s. So stand up close together so I 14:06.188 --> 14:07.568 can at least see what's going on here. 14:07.570 --> 14:10.760 So I think, from where I'm standing, I've got that Ale is 14:10.758 --> 14:13.888 the tallest and Myrto is the smallest, is that right? 14:13.889 --> 14:18.079 So I don't know instantaneously what this average would be, 14:18.084 --> 14:22.644 but I claim that any weighted average of their three heights, 14:22.639 --> 14:26.169 is going to give me a number that's somewhere between the 14:26.167 --> 14:29.607 smallest height of the three, which is Myrto's height, 14:29.611 --> 14:33.341 and the tallest height of the three, which is Ale's height, 14:33.340 --> 14:35.800 is that right? Is that correct? 14:35.800 --> 14:36.960 So that's a pretty general idea. 14:36.960 --> 14:40.500 Thanks guys I'll come back to you in a second. 14:40.500 --> 14:42.320 Let's think about this somewhere else, 14:42.320 --> 14:44.780 let's think about the batting average of a team. 14:44.779 --> 14:48.759 The team batting average in baseball, let's use the Yankees, 14:48.756 --> 14:51.286 for example. We know that the team batting 14:51.292 --> 14:54.592 average, the average batting average of the Yankee's--I don't 14:54.586 --> 14:56.976 know what it is, I didn't look it up this 14:56.982 --> 15:00.312 morning--but I know it lies somewhere between the player who 15:00.308 --> 15:03.858 has the highest batting average which I'm guessing is Jeter, 15:03.860 --> 15:05.350 I'm guessing, and the lowest, 15:05.345 --> 15:08.525 the person on the team who has the lowest batting average, 15:08.529 --> 15:10.599 who is probably one of the pitchers who played, 15:10.604 --> 15:13.314 who batted a few times in one of those inter-league games. 15:13.309 --> 15:16.829 (It would have been better if I'd used the Mets but I feel I 15:16.830 --> 15:20.470 should take pity on Mets fans this week and not mention them.) 15:20.471 --> 15:23.851 So this is a very simple idea, it's deceptively simple. 15:23.850 --> 15:26.040 It says averages, weighted averages, 15:26.039 --> 15:29.789 lie between the highest thing over which you're averaging and 15:29.794 --> 15:32.864 the lowest thing over which you're averaging. 15:32.860 --> 15:35.400 Everyone okay with that idea? 15:35.399 --> 15:39.359 Now this very simple idea is going to have an enormous 15:39.359 --> 15:43.169 consequence, and here's the enormous consequence. 15:43.170 --> 15:49.500 Simple idea, big consequence. 15:49.500 --> 15:52.830 So there's going to be a lesson that follows from this 15:52.826 --> 15:55.836 incredibly simple idea and this is the lesson. 15:55.840 --> 16:05.590 If a mixed strategy is a best response, so if a mixed strategy 16:05.593 --> 16:11.513 is the best thing you can be doing, 16:11.509 --> 16:19.289 then each of the pure strategies in the mix--I'm being 16:19.286 --> 16:26.766 a little bit loose here but I mean assigned positive 16:26.768 --> 16:34.288 probability in the mix, for those people who are nerdy 16:34.290 --> 16:41.290 enough to worry about it--each of the pure strategies in the 16:41.293 --> 16:46.163 mix must themselves be best responses. 16:46.159 --> 17:00.029 So, in particular, each must yield the same 17:00.033 --> 17:07.913 expected payoff. So here's a big conclusion that 17:07.912 --> 17:11.652 follows from that incredibly simple idea about averages lying 17:11.652 --> 17:14.522 between the highest one and the lowest one. 17:14.519 --> 17:19.359 Let's draw ourselves from that lesson to this big conclusion. 17:19.360 --> 17:20.320 What is the conclusion? 17:20.319 --> 17:23.419 The conclusion is if a mixed strategy is a best response, 17:23.417 --> 17:26.567 if the best thing I can do is to play a mixed strategy, 17:26.569 --> 17:29.449 then each of the pure strategies which I'm playing in 17:29.451 --> 17:32.781 that mix, which I'm assigning positive probability to in that 17:32.776 --> 17:36.186 mix, must themselves be best 17:36.189 --> 17:38.389 responses. In particular, 17:38.391 --> 17:41.741 each of them therefore must yield the same expected payoff. 17:41.740 --> 17:43.210 So let's go back to our example. 17:43.210 --> 17:48.030 Can I steal my three T.A.'s again? 17:48.029 --> 17:49.689 Suppose the game, suppose the thing I'm involved 17:49.686 --> 17:51.126 in--I should have made this easier before, 17:51.131 --> 17:52.331 let me come down a little bit. 17:52.329 --> 17:54.679 I'll stand above here, this is good. 17:54.680 --> 17:58.550 So suppose the game I'm involved in, the payoff in the 17:58.547 --> 18:02.997 game is, a game in which I have to choose the tallest group of 18:02.998 --> 18:06.228 my T.A.'s. So my payoff is going to be the 18:06.228 --> 18:10.428 average height of whichever subgroup of my T.A.'s I pick and 18:10.432 --> 18:12.572 these are my three choices. 18:12.569 --> 18:16.099 So if I pick more than one of them I'm going to get a weighted 18:16.102 --> 18:18.132 average, that's a mixed strategy. 18:18.130 --> 18:23.850 My aim here is to maximize the height of whatever subgroup I 18:23.851 --> 18:25.391 pick. So in this game, 18:25.386 --> 18:27.796 here's my three pure strategies: my three pure 18:27.796 --> 18:30.486 strategies are to pick Myrto; Ale; 18:30.490 --> 18:33.350 or Jake. Those are my three pure 18:33.353 --> 18:34.823 strategies. And my mixture, 18:34.821 --> 18:37.381 I could mix these two, I could mix these two, 18:37.377 --> 18:38.827 I could mix all three. 18:38.829 --> 18:42.659 But remember my payoff here is to get the group, 18:42.662 --> 18:45.192 the average as high as I can. 18:45.190 --> 18:52.460 So how am I going to get the average as high as I can? 18:52.460 --> 18:55.940 I get the average as high I as I can, I'm going to kick out 18:55.935 --> 18:59.225 Myrto for a start because Myrto's just bringing down the 18:59.231 --> 19:00.851 average, is that right? 19:00.849 --> 19:04.689 Average height I should say, there's nothing--and actually I 19:04.690 --> 19:08.010 think I'm going to kick out Jake as well I think, 19:08.009 --> 19:11.669 I'm probably going to kick out Jake as well because that way I 19:11.670 --> 19:14.010 just have Ale. So if it was the case that I 19:14.006 --> 19:16.656 was picking both of them, it would have to be they were 19:16.662 --> 19:19.172 equally tall but since they're not equally tall, 19:19.170 --> 19:21.570 I should just pick the best one. 19:21.569 --> 19:24.689 Let's go back to my Yankee's example, if I want to pick a 19:24.693 --> 19:27.723 sub-team of the Yankee's, I'm allowed to pick any number 19:27.717 --> 19:30.597 of people, to have the highest average, batting average, 19:30.600 --> 19:32.050 in that sub-team. 19:32.049 --> 19:36.089 The way to do it is to find the Yankee who has the highest 19:36.089 --> 19:38.639 batting average and just pick him. 19:38.640 --> 19:40.050 Let's do one more example. 19:40.049 --> 19:43.209 Let me use the front row of students here, 19:43.211 --> 19:45.991 so here's my, can I get this front of 19:45.988 --> 19:48.608 students to stand up a second? 19:48.610 --> 19:51.020 This is a part of the row. 19:51.019 --> 19:57.679 And suppose my aim in life is to construct the highest average 19:57.675 --> 19:59.575 GPA. I'm not going to embarrass 19:59.583 --> 20:01.803 these guys and ask them what their GPA's are. 20:01.799 --> 20:05.849 So my aim in life here is to pick some sub-group of these 20:05.853 --> 20:07.593 one, two, three, four, 20:07.589 --> 20:10.599 five, six, seven, eight students, 20:10.599 --> 20:16.059 such that the average GPA of that sub-group is as high as I 20:16.055 --> 20:19.145 can make it. So what will I do here? 20:19.150 --> 20:24.310 So this being Yale I'll just find the people who have the 4.0 20:24.308 --> 20:26.628 GPA's and just pick them. 20:26.630 --> 20:29.330 Is that right? You might think well why not 20:29.330 --> 20:31.260 include somebody who has a 3.9 GPA? 20:31.260 --> 20:32.530 That's pretty good. 20:32.530 --> 20:35.480 So why not? Because if there's anybody in 20:35.482 --> 20:39.292 this group who has a 4.0 GPA, I'd do better just to pick that 20:39.294 --> 20:41.824 person. The 3.9 person would just be 20:41.822 --> 20:43.582 pulling down the average. 20:43.579 --> 20:47.879 Now suppose there's nobody with a 4.0 GPA and suppose it's the 20:47.884 --> 20:52.124 case that three of these people, let's say these three people 20:52.119 --> 20:55.519 have a 3.9 GPA. So these three have 3.9 GPA, 20:55.517 --> 20:58.677 imagine that, and these other people they've 20:58.677 --> 21:01.687 got horrible grades like B+ somewhere. 21:01.690 --> 21:04.510 These are our future law school students and these are the 21:04.511 --> 21:07.381 people--who knows what they're going to end up doing--being 21:07.381 --> 21:08.521 President probably. 21:08.519 --> 21:11.889 So to construct the group with the highest average GPA, 21:11.891 --> 21:13.391 what am I going to do? 21:13.390 --> 21:16.420 Well first I'll throw out all these guys with low GPA's, 21:16.421 --> 21:19.561 so they can all sit down and I'll look at these last three 21:19.562 --> 21:23.232 and these last three, if they're all in the group 21:23.226 --> 21:26.186 they better all have the same GPA. 21:26.190 --> 21:29.150 Why on earth? If I'm trying to maximize the 21:29.154 --> 21:32.874 average of my group, if any of them had a lower GPA 21:32.868 --> 21:36.198 I should kick them out, and if one of them has a higher 21:36.196 --> 21:38.396 GPA than the other two, I should kick out both the 21:38.404 --> 21:40.334 other two. So if I'm including all three 21:40.327 --> 21:42.997 of them, in my constructing of the average all of them must 21:43.001 --> 21:45.551 have the same GPA, which I'm going to assume is 21:45.554 --> 21:48.444 3.9, to assume you can still make into law school. 21:48.440 --> 21:49.640 Everyone understand that? 21:49.640 --> 21:54.860 Yeah? Okay, thanks guys. 21:54.859 --> 21:58.679 So that's the way I want to think about this. 21:58.680 --> 22:03.930 So the idea here is if I'm using a mixed strategy as a best 22:03.928 --> 22:09.088 response, it must be the case that everything on which I'm 22:09.087 --> 22:11.437 mixing is itself best. 22:11.440 --> 22:14.480 And the reason is, if it wasn't, 22:14.477 --> 22:20.647 kick out the thing that isn't best and my average will go up. 22:20.650 --> 22:24.840 So that leads us to the next idea, but before I do just for 22:24.844 --> 22:27.524 formality, let me add a definition. 22:27.519 --> 22:33.769 The definition is this, a mixed strategy profile--what 22:33.770 --> 22:40.610 I'm going to do now is I'm going to define Nash Equilibrium 22:40.610 --> 22:42.940 again, just so we have it in our notes 22:42.941 --> 22:44.861 somewhere. So a mixed strategy 22:44.855 --> 22:49.445 profile--there should be a hyphen there--(P_1*, 22:49.450 --> 22:56.020 P_2*, …all the way up to 22:56.021 --> 23:02.591 P_N*), is a mixed strategy Nash 23:02.592 --> 23:12.282 Equilibrium if for each Player i--so for each Player i--that 23:12.284 --> 23:20.014 player's mixed strategy P_i* is a best 23:20.005 --> 23:29.035 response for Player i to the strategies everyone else is 23:29.040 --> 23:34.790 picking P _-i*. 23:34.789 --> 23:41.849 So I'm exploiting, by now, a well developed 23:41.846 --> 23:47.386 notation for player strategies. 23:47.390 --> 23:50.550 So this definition of Nash Equilibrium, it's exactly the 23:50.547 --> 23:54.107 same as the definition of Nash Equilibrium we've been using now 23:54.106 --> 23:57.466 for several weeks, except everywhere where before 23:57.466 --> 24:00.116 we saw a pure strategy, which was an S, 24:00.123 --> 24:02.293 I have replaced it with a P. 24:02.289 --> 24:05.519 So the same definition except I'm using mixed strategies 24:05.524 --> 24:07.234 instead of pure strategies. 24:07.230 --> 24:11.870 But an implication of our lesson is what? 24:11.869 --> 24:16.199 It's that if P_i* is part of a Nash Equilibrium--so 24:16.201 --> 24:20.461 if Pi* is a best response to what everyone else is doing, 24:20.460 --> 24:24.370 P_-i* --, then each of the pure 24:24.366 --> 24:29.536 strategies involved in P_i* must itself be a 24:29.542 --> 24:35.322 best response. So an implication of the lesson 24:35.321 --> 24:40.251 is, the lesson implies the following. 24:40.250 --> 24:45.600 If P_i* of a particular strategy is positive, 24:45.600 --> 24:50.260 so in other words, I'm using this strategy in my 24:50.257 --> 24:57.167 mix, then that strategy is also a 24:57.168 --> 25:07.218 best response to what everyone else is doing. 25:07.220 --> 25:15.030 Okay, so from a math point of view this is the big idea of the 25:15.026 --> 25:18.236 day, this board. If you're having trouble 25:18.243 --> 25:20.303 reading this at the back, trust me I've written that up 25:20.298 --> 25:22.618 on the handout that will appear magically on the computer, 25:22.620 --> 25:24.500 at the end of class. 25:24.500 --> 25:27.150 At the moment you're staring at this, it's all a bit new, 25:27.151 --> 25:30.051 and as well as being new, you're saying, 25:30.049 --> 25:34.789 okay but so what, why do I care about this 25:34.792 --> 25:37.802 seemingly mundane fact? 25:37.799 --> 25:41.729 The reason we're going to turn out to care about this seemingly 25:41.729 --> 25:46.119 mundane fact, is that this fact is going to 25:46.122 --> 25:51.942 make it remarkably easy to find Nash Equilibria. 25:51.940 --> 25:55.820 This fact, this lesson, this idea that if I'm playing a 25:55.823 --> 25:58.343 pure strategy as part of the mix, 25:58.339 --> 26:02.859 it must itself be a best response, that's going to be the 26:02.864 --> 26:07.234 trick we're going to use in finding mixed strategy Nash 26:07.228 --> 26:10.638 Equilibria. The only way I can illustrate 26:10.642 --> 26:14.742 that to you is to do it, so I'm going to spend the rest 26:14.743 --> 26:16.873 of today just doing that. 26:16.869 --> 26:20.309 I'm going to look at a game and we're going to go through this 26:20.314 --> 26:22.454 game. We'll discuss it a little bit 26:22.445 --> 26:25.495 because it's a fun game, and we're going to find the 26:25.496 --> 26:28.006 mixed-strategy equilibria of this game. 26:28.010 --> 26:29.970 Everyone know where we're going? 26:29.970 --> 26:33.530 I want to make sure before I go on, are people looking very sort 26:33.526 --> 26:35.046 of deer in the headlamps? 26:35.049 --> 26:39.629 That was a lot of formality to get through in a short period of 26:39.632 --> 26:40.822 time. Does anyone want to ask a 26:40.818 --> 26:41.458 question at this point? 26:41.460 --> 26:44.920 Are you okay? Okay to go on? 26:44.920 --> 26:47.650 So just remember that the conclusion here comes from this 26:47.652 --> 26:48.582 very simple idea. 26:48.579 --> 26:50.959 The simple idea is, the payoff to a weighted 26:50.957 --> 26:54.277 average must lie between the best and worst thing involved in 26:54.275 --> 26:57.145 the average, and therefore if I'm including 26:57.154 --> 26:59.954 things in there as part of a best response, 26:59.947 --> 27:01.607 they must all be good. 27:01.609 --> 27:04.799 That's the simple idea, this is the dramatic 27:04.796 --> 27:06.996 conclusion. So the only way to prove this 27:06.998 --> 27:09.728 to you and the only way to prove to you that this is useful is to 27:09.730 --> 27:10.670 go ahead and do it. 27:10.670 --> 27:17.590 So what I'm going to do is I'm going to clean these boards and 27:17.585 --> 27:22.115 I'm going to start showing an example. 27:22.119 --> 27:24.439 Again don't panic, I think a lot of people at this 27:24.439 --> 27:26.569 part of the class have a tendency to panic, 27:26.569 --> 27:29.749 because it's a new idea, it seems like a lot of math 27:29.749 --> 27:31.689 around. None of it's very hard math, 27:31.685 --> 27:33.105 it's all kind of arithmetic. 27:33.109 --> 27:38.709 It's just this idea of not panicking. 27:38.710 --> 27:45.370 So the example I want to look at is going to be from tennis, 27:45.374 --> 27:50.914 and I'm going to consider a game within a game, 27:50.910 --> 28:01.330 played by two tennis players, and let's call them Venus and 28:01.327 --> 28:05.587 Serena Williams. So a couple of years ago we 28:05.591 --> 28:07.731 used to use Venus and Serena Williams for this example, 28:07.730 --> 28:10.500 and then for a while I worried, that you wouldn't even remember 28:10.498 --> 28:12.148 who Venus and Serena Williams were, 28:12.150 --> 28:16.700 and so we picked any two random Russians, but now we're back. 28:16.700 --> 28:19.880 Seems like we're back to picking Venus and Serena. 28:19.880 --> 28:25.420 So the game within the game is this, suppose that they're 28:25.416 --> 28:32.036 playing and Serena is at the net and the ball is on Venus' court, 28:32.039 --> 28:37.829 and Venus has reached the ball and Venus has to decide whether 28:37.826 --> 28:43.416 to try to hit a passing shot past Serena on Serena's left or 28:43.423 --> 28:45.513 on Serena's right. 28:45.509 --> 28:49.029 Notice I'm going to exclude the possibility of throwing up a lob 28:49.026 --> 28:51.256 for now, just to make this manageable. 28:51.259 --> 28:55.889 So basically the choice facing Venus is should she try to pass 28:55.887 --> 29:00.557 Serena to Serena's left, which is Serena's backhand side 29:00.556 --> 29:04.636 or to Serena's right, which is Serena's forehand 29:04.644 --> 29:06.764 side. People are familiar enough with 29:06.759 --> 29:08.819 tennis to understand what I'm talking about? 29:08.819 --> 29:10.229 So we're going to assume this is Wimbledon, 29:10.226 --> 29:12.266 otherwise no one would be at the net to start with I guess. 29:12.270 --> 29:14.490 So this is at Wimbledon. 29:14.490 --> 29:19.600 Let's try and put up some payoffs here. 29:19.599 --> 29:20.679 So these are going to be the payoffs. 29:20.680 --> 29:25.620 I think that this example is originally due to Dixit, 29:25.619 --> 29:28.089 but it's not a big deal. 29:28.089 --> 29:34.179 I think this example is due to Dixit and Skeath. 29:34.180 --> 29:38.350 So here's some numbers and I'll explain the numbers in a minute. 29:38.349 --> 29:52.879 So this is 50,50, 80,20, 90,10 and 20,80. 29:52.880 --> 29:54.610 So what are these numbers? 29:54.609 --> 29:58.459 So first of all let me just explain what the strategies are, 29:58.460 --> 30:02.380 so I'm assuming the row player is Venus and the column player 30:02.376 --> 30:06.076 is Serena. I'm assuming that if Venus 30:06.077 --> 30:11.437 chooses L that means she attempts to pass Serena to 30:11.437 --> 30:15.017 Serena's left, we'll orient things from 30:15.019 --> 30:18.609 Serena's point of view, and if she hits right that 30:18.613 --> 30:22.943 means she's attempting to pass Serena on Serena's right. 30:22.940 --> 30:26.900 If Serena chooses L that means she cheats slightly towards her 30:26.897 --> 30:30.397 left: not cheats in the sense of breaking the rules, 30:30.400 --> 30:33.080 but cheats in terms of where she's standing or leaning. 30:33.079 --> 30:36.039 And if she chooses right that means she cheats slightly 30:36.035 --> 30:37.125 towards her right. 30:37.130 --> 30:40.570 So this is cheating towards her backhand and this is cheating 30:40.568 --> 30:43.378 towards her forehand, assuming she's right handed, 30:43.377 --> 30:44.807 which she in fact is. 30:44.809 --> 30:46.939 Okay, what do these numbers mean? 30:46.940 --> 30:49.140 So let's start with the easy ones. 30:49.140 --> 30:54.950 So if Venus chooses left and Serena chooses right, 30:54.947 --> 30:58.737 then Serena has guessed wrong. 30:58.740 --> 31:04.690 Is that correct? In which case Venus wins the 31:04.691 --> 31:10.531 points 80% of the time and Serena wins it 20% of the time. 31:10.529 --> 31:15.889 Conversely, if Venus chooses right and Serena chooses left, 31:15.888 --> 31:21.428 then again, Serena has guessed wrong and this time Venus wins 31:21.432 --> 31:26.882 the points 90% of the time and Serena wins the points 10% of 31:26.884 --> 31:30.164 the time. This should be a familiar idea 31:30.164 --> 31:34.024 by now, but why is it the case these nineties and eighties are 31:34.016 --> 31:36.796 not a 100%? Why is it the case that if 31:36.795 --> 31:40.945 Serena guesses wrong Venus doesn't win 100% of the time? 31:40.950 --> 31:42.670 Anybody? Perhaps we can get a show of 31:42.668 --> 31:44.068 hands, get some mikes up. 31:44.070 --> 31:45.640 Why isn't it 100% here? 31:45.640 --> 31:48.210 Somebody? Patrick? 31:48.210 --> 31:48.990 Wait for the mike. 31:48.990 --> 31:50.860 Student: Sometimes she hits it out of bounds when she 31:50.864 --> 31:52.024 serves. Professor Ben Polak: 31:52.022 --> 31:53.992 Right, this isn't even a serve, this is a passing shot but the 31:53.993 --> 31:55.693 same is true. So sometimes you're 31:55.688 --> 31:59.098 successfully going to hit it past Serena but the ball is 31:59.097 --> 32:00.397 going to sail out. 32:00.400 --> 32:05.680 So that happens 10% of the time here and 20% of the time here. 32:05.680 --> 32:11.280 Look at the other two boxes, if Venus hits to Serena's left 32:11.279 --> 32:15.589 and Serena guesses left, then we're going to assume that 32:15.590 --> 32:18.610 Serena's going to reach the ball and make a volley, 32:18.609 --> 32:22.229 but her volley only manages to go in--go over the net and go 32:22.232 --> 32:25.112 in--half the time, so the payoffs are (50,50). 32:25.109 --> 32:29.179 Half the time Venus wins the point and half the time Serena 32:29.177 --> 32:32.157 wins the point. Conversely, if Venus hits the 32:32.159 --> 32:35.489 ball to Serena's right and Serena guesses correctly and 32:35.490 --> 32:38.020 chooses right, then we're in this box. 32:38.019 --> 32:41.439 Once again, Serena has guessed correctly and she's going to 32:41.435 --> 32:45.025 successfully reach the volley and this time she gets it in 80% 32:45.027 --> 32:49.107 of the time, so Venus wins the point 20% of 32:49.105 --> 32:53.805 the time and Serena wins it 80% of the time. 32:53.809 --> 32:57.979 So just to finish up the description of the game here, 32:57.976 --> 33:02.606 notice that we're assuming that Serena is a little better at 33:02.614 --> 33:07.414 volleying to her right than she is volleying to her left. 33:07.410 --> 33:11.300 So this is her forehand volley and we're going to assume that 33:11.295 --> 33:14.075 that's stronger than her backhand volley. 33:14.079 --> 33:18.369 Conversely, we're assuming that Venus' passing shot is a little 33:18.365 --> 33:22.165 better when she shoots it to Serena's left than when she 33:22.167 --> 33:24.377 shoots it to Serena's right. 33:24.380 --> 33:28.250 This is her cross court passing shot and this is her down the 33:28.250 --> 33:29.540 line passing shot. 33:29.539 --> 33:32.089 So none of that fine detail matters a great deal, 33:32.086 --> 33:35.156 but just if you're interested that's where the numbers come 33:35.162 --> 33:36.722 from. I'm not claiming this is true 33:36.722 --> 33:38.292 data by the way, I made up these numbers. 33:38.289 --> 33:40.709 Actually I think Dixit made up these numbers, 33:40.707 --> 33:42.517 I forget where I got them from. 33:42.519 --> 33:45.679 So okay, everyone understand the game? 33:45.680 --> 33:48.730 So now imagine, either imagine you are Venus or 33:48.730 --> 33:51.710 Serena, or imagine perhaps more realistically, 33:51.713 --> 33:54.833 that you've become Venus or Serena's coach. 33:54.829 --> 33:57.739 Do I have any members of the tennis team here? 33:57.740 --> 34:00.410 No. Well imagine you've become 34:00.414 --> 34:03.784 their coach, so you take this class and then you apply to 34:03.782 --> 34:06.492 replace their father as being their coach. 34:06.490 --> 34:09.590 That's a tough assignment I would think. 34:09.590 --> 34:12.800 So an obvious question is, you're coaching Venus before 34:12.804 --> 34:16.024 Wimbledon, you know this situation's going to arise and 34:16.018 --> 34:19.648 you might want to coach Venus on what should she do here? 34:19.650 --> 34:24.620 Should she try and pass Serena down the line or she should try 34:24.621 --> 34:29.431 and hit the cross court volley, cross court passing shot? 34:29.429 --> 34:33.079 Notice that this is a question of should you, 34:33.082 --> 34:38.062 Venus, play to your strength which is the cross court passing 34:38.062 --> 34:42.672 shot, or should you play to Serena's 34:42.665 --> 34:49.985 weakness, which would be to hit it to Serena's backhand. 34:49.989 --> 34:53.079 Playing to your strength is to choose right and playing to 34:53.079 --> 34:55.139 Serena's weakness is to choose left. 34:55.139 --> 34:59.209 Conversely, for Serena, should you lean towards your 34:59.209 --> 35:03.679 strength, which I guess is leaning to the right or should 35:03.677 --> 35:06.627 you lean towards Venus' weakness, 35:06.630 --> 35:09.560 which I guess is leaning left? 35:09.559 --> 35:12.139 When you look at coaching manuals on this stuff, 35:12.144 --> 35:15.344 or you listen to the terrible guys who commentate on tennis 35:15.335 --> 35:18.575 for ESPN--oh no I'm getting in trouble again--very nice guys 35:18.579 --> 35:20.779 who commentate on tennis for ESPN, 35:20.780 --> 35:24.640 they say just incredibly dumb things at this point. 35:24.639 --> 35:27.409 They say things like, you should always play to your 35:27.411 --> 35:30.781 strengths and don't worry about the other person's weakness. 35:30.780 --> 35:35.350 I think it won't take much time today to figure out that's not 35:35.346 --> 35:38.196 great advice. But can people at least see 35:38.200 --> 35:41.210 that this is a difficult problem, this is not an 35:41.213 --> 35:44.293 immediately obvious problem, is that correct? 35:44.289 --> 35:48.309 One reason it's not immediately obvious is not only is no 35:48.312 --> 35:52.822 strategy dominated here, but there is no pure strategy 35:52.817 --> 35:57.907 Nash Equilibrium in this game, in this little sub game. 35:57.909 --> 36:01.499 There is no pure strategy Nash Equilibrium--and notice that I 36:01.496 --> 36:03.046 added the qualifier now. 36:03.050 --> 36:05.300 Previously I would just have said Nash Equilibrium, 36:05.297 --> 36:07.767 but now that we have mixed strategies in the picture, 36:07.769 --> 36:10.719 I'm going to talk about pure strategy Nash Equilibria to be 36:10.717 --> 36:13.357 those that are the only involving pure strategies. 36:13.360 --> 36:17.020 Okay, so why is there no pure strategy Nash Equilibrium? 36:17.020 --> 36:18.240 Well let's have a look. 36:18.239 --> 36:21.949 So if Venus--If Serena thought that Venus was going to choose 36:21.946 --> 36:25.146 left then her best response, not surprisingly, 36:25.150 --> 36:29.920 is to lean left and if Serena thought that Venus was going to 36:29.918 --> 36:33.868 choose right, then her best response is to 36:33.868 --> 36:37.978 cheat to the right, so 50 is bigger than 20, 36:37.981 --> 36:40.661 and 80 is bigger than 10. 36:40.659 --> 36:44.389 And conversely, if Venus thought that Serena 36:44.387 --> 36:49.757 was cheating a bit to the left then her best response is to hit 36:49.762 --> 36:54.792 it to Serena's right, and if Venus thought Serena was 36:54.788 --> 37:00.408 leaning to the right then Venus' best response is to hit it to 37:00.412 --> 37:03.112 Serena's left. So I think that's not at all 37:03.109 --> 37:05.489 surprising when you think about it, not at all surprising, 37:05.489 --> 37:07.919 you're going to get this little cycle like this, 37:07.916 --> 37:10.856 but we can see immediately that these best responses never 37:10.858 --> 37:23.298 coincide, so there is no pure strategy 37:23.296 --> 37:30.556 equilibrium. So that leaves us a bit stuck 37:30.556 --> 37:34.266 except I guess you know what the next question's going to be, 37:34.274 --> 37:37.314 and I shouldn't leave it in too much suspense. 37:37.309 --> 37:40.239 The next question's going to be, okay there's not pure 37:40.244 --> 37:43.404 strategy Nash Equilibrium, but we've just introduced a new 37:43.401 --> 37:44.731 idea which was what? 37:44.730 --> 37:47.990 It was Nash Equilibrium in mixed strategies. 37:47.989 --> 37:51.879 Maybe there's going to be a mixed strategy Nash Equilibrium. 37:51.880 --> 37:53.980 In fact, there is, there is going to be one. 37:53.980 --> 38:00.620 So our exercise now is, let's find a mixed strategy 38:00.624 --> 38:05.814 Nash Equilibrium, and before we find it, 38:05.807 --> 38:12.317 let's just interpret what it's going to mean. 38:12.320 --> 38:16.350 A mixed strategy Nash Equilibrium in this game, 38:16.354 --> 38:21.354 is going to be a mix for Venus between hitting the ball to 38:21.354 --> 38:24.604 Serena's left and Serena's right, 38:24.599 --> 38:30.029 and a mix for Serena between leaning left and leaning right, 38:30.029 --> 38:35.869 such that each person's mix, each person's randomization is 38:35.865 --> 38:41.295 a best response to the other person's randomization. 38:41.300 --> 38:44.320 Since these players are sisters and have played each other many, 38:44.317 --> 38:46.207 many times, not just in competition but 38:46.210 --> 38:48.660 probably in practice, it seems like a reasonable idea 38:48.658 --> 38:51.198 that they might have arrived in playing each other, 38:51.199 --> 38:54.949 at a mixed strategy Nash Equilibrium. 38:54.949 --> 39:01.509 That's what we're going to try and do, now how are we going to 39:01.513 --> 39:04.693 do that? So what we're going to do is 39:04.694 --> 39:08.454 we're going to exploit the trick that we have here, 39:08.452 --> 39:12.492 the lesson here. The lesson we have here says if 39:12.488 --> 39:16.948 players are playing a mixed strategy as part of a Nash 39:16.947 --> 39:20.667 Equilibrium, each of the pure strategies 39:20.666 --> 39:25.126 involved in the mix, each of their pure strategies 39:25.126 --> 39:28.216 must itself be a best response. 39:28.220 --> 39:33.820 We're going to use that idea. 39:33.820 --> 39:37.830 So let's try and do that. 39:37.829 --> 39:40.169 So I'm hoping that by doing this, I'm going to illustrate to 39:40.174 --> 39:43.904 you immediately, that this idea is actually 39:43.900 --> 39:50.620 useful, at least useful if you end up coaching the Williams 39:50.616 --> 39:53.236 sisters. Alright, I want to keep this so 39:53.244 --> 39:54.254 you can still read it. 39:54.250 --> 39:55.120 Ill bring it down a bit. 39:55.120 --> 39:59.100 Can people still read it? 39:59.099 --> 40:04.089 Okay, so what I want to do is, I want to find a mixture for 40:04.092 --> 40:08.742 Serena and a mixture for Venus that are equilibrium. 40:08.739 --> 40:11.279 Having put it up there let me bring it down again. 40:11.280 --> 40:13.010 This was not so intelligent of me. 40:13.010 --> 40:16.350 I actually want to bring in some notation, 40:16.348 --> 40:19.848 so as before, let's assume that Serena's mix 40:19.849 --> 40:23.609 is, let's use Q and (1-Q) to be 40:23.614 --> 40:30.364 Serena's mix and let's use P and (1-P) to be Venus' mix. 40:30.360 --> 40:40.830 Let's establish that notation. 40:40.829 --> 40:45.129 So here's the trick, So this is the slightly magic 40:45.131 --> 40:48.121 bit of the class, so pay attention, 40:48.117 --> 40:51.977 I'm about to pull a rabbit out of a hat. 40:51.980 --> 41:03.080 Trick, what should I do first, to find Serena's Nash 41:03.077 --> 41:10.037 Equilibrium mix, so that's (Q, 41:10.039 --> 41:15.449 (1-Q)), what I'm going to do is I'm going to look at 41:15.452 --> 41:18.002 Venus' payoffs. 41:18.000 --> 41:26.690 So to find Serena's Nash Equilibrium mix the trick is to 41:26.687 --> 41:35.687 look at Venus' payoffs, that's going to be my magic 41:35.690 --> 41:39.600 trick. Let's try and see why. 41:39.599 --> 41:50.239 So let's look at Venus' payoffs, Venus' payoffs against 41:50.241 --> 41:53.451 Q. So if Serena is choosing (Q, 41:53.445 --> 41:56.115 1-Q), what are Venus' payoffs? 41:56.119 --> 42:05.369 So if she chooses left then her payoff is 50 with probability 42:05.374 --> 42:11.544 Q--and I'm going to use the pointer here, 42:11.544 --> 42:18.644 and hope that the camera can see this too. 42:18.639 --> 42:33.859 She gets 50 with probability Q and she gets 80 with probability 42:33.857 --> 42:43.127 1-Q. If she chooses right then she 42:43.127 --> 42:59.527 gets 90 with probability Q and she gets 20 with probability of 42:59.527 --> 43:10.967 1-Q. I meant to point to that. 43:10.970 --> 43:14.240 So what? So what is this: 43:14.237 --> 43:18.277 we're looking for a mixed strategy Nash Equilibrium, 43:18.276 --> 43:22.406 so in particular, not only Serena is mixing but 43:22.409 --> 43:27.859 in this case what we're claiming is, Venus is mixing as well. 43:27.860 --> 43:34.870 So if Venus is mixing as well, that means that Venus is using 43:34.871 --> 43:41.181 the strategy left with some probability P and using the 43:41.181 --> 43:46.441 strategy right with some probability 1-P. 43:46.440 --> 43:53.090 Since Venus sometimes chooses left and sometimes chooses right 43:53.088 --> 43:59.088 as her best response to Q, her best response to Serena, 43:59.090 --> 44:05.940 what must be true of the payoff to left and the payoff to right? 44:05.940 --> 44:10.860 Let's go through it again, so we're going to assume that 44:10.859 --> 44:13.719 Venus is mixing. So sometimes she chooses left 44:13.720 --> 44:16.000 and sometimes she chooses right and she's going to be, 44:16.004 --> 44:18.254 she's in a Nash Equilibrium, so she's choosing a best 44:18.246 --> 44:21.136 response. So whatever that mix P, 44:21.136 --> 44:24.116 1-P is, it's a best response. 44:24.119 --> 44:28.539 Since she's playing a best response of P and that sometimes 44:28.544 --> 44:32.594 involves choosing left and sometimes involves choosing 44:32.587 --> 44:36.287 right, it must be the case that what? 44:36.289 --> 44:43.529 It must be the case that both left itself and right itself are 44:43.531 --> 44:47.331 both themselves best response. 44:47.329 --> 44:51.329 If she's mixing between them, it must be that both choosing 44:51.333 --> 44:55.133 left or choosing right are themselves best responses. 44:55.130 --> 44:57.620 If they weren't she should just drop them out of the mix, 44:57.620 --> 44:59.310 that would raise her average payoff. 44:59.309 --> 45:04.569 Right, just like we dropped out the short T.A.'s to get a high 45:04.570 --> 45:09.660 height and we dropped out the failing Yale students to get a 45:09.658 --> 45:19.798 high GPA. So if Venus is mixing in this 45:19.799 --> 45:38.459 Nash Equilibrium then the payoff to left and to right must be 45:38.460 --> 45:42.860 equal, they must both be best 45:42.862 --> 45:46.942 responses, both left and right must be a best response, 45:46.940 --> 45:50.810 so in particular, the expected payoffs must be 45:50.805 --> 45:54.395 the same. Is that right, is that correct? 45:54.400 --> 45:56.930 So what does that allow me to do? 45:56.929 --> 46:04.659 It allows me to put an equals sign in here. 46:04.659 --> 46:07.349 Since left is a best response and right is a best response, 46:07.351 --> 46:09.811 since they're both best responses, they must yield the 46:09.810 --> 46:10.970 same expected payoff. 46:10.969 --> 46:13.429 Here's their expected payoffs, they must be the same. 46:13.429 --> 46:18.089 Now, I've got one equation and one unknown, and now I'm down to 46:18.088 --> 46:20.388 algebra. So let me do the algebra. 46:20.389 --> 46:24.279 I claim this expression is equal to that expression, 46:24.283 --> 46:29.023 so simplifying a bit I'm going to get--you should just watch to 46:29.016 --> 46:33.516 make sure I don't get this wrong--I'm going to get 40Q, 46:33.519 --> 46:41.479 so this implies 40Q is equal to 60(1-Q). 46:41.480 --> 46:49.080 So I took this 50 onto this side and this 20 onto that side, 46:49.080 --> 46:56.810 so I have 40Q is equal to 60(1- Q) and that implies that Q is 46:56.808 --> 47:02.258 equal to .6. So those last two steps were 47:02.256 --> 47:05.156 just algebra. So what was the trick here? 47:05.159 --> 47:11.679 The trick was I found Q, which is how Serena is mixing 47:11.684 --> 47:18.064 by looking at Venus' payoffs, knowing that Venus is mixing 47:18.056 --> 47:23.256 and hence I can set Venus' payoffs equal to one another. 47:23.260 --> 47:26.520 Say that again, I found the way in which Serena 47:26.524 --> 47:29.864 is mixing by knowing that if Venus is mixing, 47:29.860 --> 47:37.950 her expected payoffs must be equal and I solved out for 47:37.948 --> 47:43.488 Serena's mix, this is Serena's mix. 47:43.490 --> 47:47.780 Let's do it again. 47:47.780 --> 47:49.720 Here I'm wishing I had another board. 47:49.719 --> 47:55.119 I don't want to lose those numbers entirely, 47:55.119 --> 48:00.519 so I'm going to try and squeeze in a bit. 48:00.520 --> 48:01.320 I know what I can do. 48:01.320 --> 48:04.210 Let's get rid of this one entirely. 48:04.210 --> 48:08.950 There we go, that works. 48:08.949 --> 48:11.809 Let's get rid of this one entirely. 48:11.810 --> 48:15.140 I can still see my numbers. 48:15.140 --> 48:17.130 Let's do the converse. 48:17.130 --> 48:22.080 Let's do the trick again, this time what I'm going to do 48:22.079 --> 48:26.489 is I'm going to figure out how Venus is mixing. 48:26.489 --> 48:31.279 I know how Serena is mixing now, so now I'm going to work 48:31.284 --> 48:33.514 out how Venus is mixing. 48:33.510 --> 48:41.640 Now, to figure out how Serena was mixing, I used Venus' 48:41.635 --> 48:44.885 payoffs. So to find out how Venus is 48:44.892 --> 48:46.842 mixing what am I going to do? 48:46.840 --> 48:50.580 I'm going to use Serena's payoffs. 48:50.579 --> 48:58.659 So to find Venus' mix, which is P, 1-P, 48:58.662 --> 49:09.722 --let's be careful it's her Nash Equilibrium mix--use 49:09.723 --> 49:14.193 Serena's payoffs. 49:14.190 --> 49:22.770 Here we go, so if Serena chooses, this is S's payoffs, 49:22.765 --> 49:31.175 if Serena chooses L then her payoffs will be what? 49:31.179 --> 49:34.369 So again, just watch to make sure I don't get this wrong and 49:34.371 --> 49:37.401 I'll point to the things to try and help myself a bit. 49:37.400 --> 49:44.640 So with probability P she'll get 50. 49:44.639 --> 49:59.479 So 50 with probability P, and with probability 1-P she'll 49:59.475 --> 50:07.345 get 10. And if she chooses to lean to 50:07.354 --> 50:18.914 the right, to lean towards her forehand, then with probability 50:18.911 --> 50:29.901 P she'll get 20 and with probability 1-P she'll get 80. 50:29.900 --> 50:38.460 We know that Serena is mixing, so since Serena is mixing what 50:38.456 --> 50:43.586 must be true of these two payoffs? 50:43.590 --> 50:44.830 What must be true of the two payoffs? 50:44.829 --> 50:48.909 The payoff to l and the payoff to r, what must be true about 50:48.914 --> 50:53.004 them since Serena is using a mixture of these two strategies 50:52.998 --> 50:54.658 in Nash Equilibrium? 50:54.659 --> 50:58.969 It must be the case that both l is a best response and r is a 50:58.972 --> 51:02.062 best response, in which case the payoff must 51:02.062 --> 51:05.442 be, someone shout it out, equal, thank you. 51:05.440 --> 51:10.780 They must be equal, these must be equal. 51:10.780 --> 51:14.810 They must be equal since Serena is indifferent between choosing 51:14.809 --> 51:17.799 left or right and hence is mixing over them. 51:17.800 --> 51:21.130 So again, using the fact that they're equal reduces this to 51:21.130 --> 51:23.830 algebra, and again, I'll probably get this wrong 51:23.829 --> 51:30.189 but let me try. So I claim, let's take 20 away 51:30.191 --> 51:37.221 from here, I've got 30P equals 70(1-P). 51:37.219 --> 51:39.199 I hope that's right, that looks right. 51:39.199 --> 51:41.149 Again, this is just algebra at this point. 51:41.150 --> 51:46.770 So I took 20 away from here and 10 away from there, 51:46.766 --> 51:50.806 and this implies that P equals .7. 51:50.809 --> 51:56.149 So I claim I have now found the mixed strategy Nash Equilibrium. 51:56.150 --> 52:01.120 Here it is. The Nash Equilibrium is as 52:01.117 --> 52:05.087 follows. Let's be careful, 52:05.088 --> 52:08.708 this is Venus' mix. 52:08.710 --> 52:18.790 So if Venus is mixing .7, .3, .7 on left and .3 on right, 52:18.789 --> 52:28.329 and Serena is mixing .6, .4, so this is Venus' mix and 52:28.329 --> 52:32.289 this Serena's mix. 52:32.289 --> 52:41.109 Venus is shooting to the left of Serena with probability of .7 52:41.106 --> 52:49.196 and Serena is leaning that way with probability of .6. 52:49.199 --> 52:54.879 So we were able to find this Nash Equilibrium by using the 52:54.876 --> 52:58.826 trick before. Now let's just reinforce this a 52:58.830 --> 53:01.370 little bit by talking about it. 53:01.369 --> 53:07.559 So suppose it were the case that Serena, instead of leaning 53:07.558 --> 53:14.278 to the left .6 of the time leant to the left more than .6 of the 53:14.279 --> 53:19.239 time. So suppose you're Venus' coach, 53:19.235 --> 53:26.185 and suppose you know that Serena leans to the left more 53:26.194 --> 53:32.644 than .6 of the time, what would you advise Venus to 53:32.638 --> 53:34.528 do? Let me try it again. 53:34.530 --> 53:38.750 So suppose your Venus' coach and suppose you've observed the 53:38.747 --> 53:43.177 fact that Serena leans to the left more than .6 of the time, 53:43.179 --> 53:47.869 what would you advise Venus to do? 53:47.870 --> 53:50.240 Pass to the right, shout out. 53:50.239 --> 53:50.649 Student: Pass to the right. 53:50.650 --> 53:51.980 Professor Ben Polak: Pass to the right, 53:51.983 --> 53:55.113 exactly. So if Serena cheats to the left 53:55.107 --> 53:59.937 more than .6 of the time, then Venus' best response is 53:59.941 --> 54:02.861 always to shoot to the right. 54:02.860 --> 54:06.020 That maximizes her chance of winning the point. 54:06.019 --> 54:15.299 Conversely, if Serena leans to the left less than .6 of the 54:15.300 --> 54:20.740 time, then Venus should do what? 54:20.740 --> 54:24.150 Shoot to the left all the time. 54:24.150 --> 54:28.440 So if Serena doesn't choose exactly this mix, 54:28.435 --> 54:33.885 then Venus' best response is actually a pure strategy. 54:33.889 --> 54:36.969 Say it again, if Serena leans to the left too 54:36.966 --> 54:40.526 often, more than .6, then Venus should just go right 54:40.533 --> 54:43.823 and if Serena leans to the left too little, 54:43.820 --> 54:46.610 then Venus should always go left. 54:46.610 --> 54:49.770 We can do exactly the same the other way around. 54:49.769 --> 54:53.659 If Venus shoots to the right, so that's her cross hand 54:53.656 --> 54:56.586 passing shot more than .7 of the time, 54:56.590 --> 55:04.310 and you're Serena's coach, what should you tell Serena to 55:04.310 --> 55:08.400 do? Go that way all the time. 55:08.400 --> 55:12.460 So if Venus is hitting it to Serena's left more than .7 of 55:12.462 --> 55:16.242 the time, Serena should just always go to her left, 55:16.239 --> 55:20.319 and if Venus is hitting to the left less than .7 of the time, 55:20.317 --> 55:23.237 so to the right more than .3 of the time, 55:23.239 --> 55:28.179 then Serena should always go to the right. 55:28.179 --> 55:31.389 So that's how this kind of comes back into the sort of the 55:31.392 --> 55:33.142 coaching manuals if you like. 55:33.140 --> 55:35.450 Okay, so how am I doing so far? 55:35.449 --> 55:39.359 Have I lost everyone yet or are people still with me? 55:39.360 --> 55:42.160 How many of you play tennis, ever? 55:42.159 --> 55:47.699 So all your tennis is going to dramatically improve after 55:47.696 --> 55:50.806 today, right? So now let's make life more 55:50.813 --> 55:53.663 interesting. Let's go back to the start. 55:53.659 --> 55:56.599 We've figured out this is an equilibrium, this is how Venus 55:56.598 --> 55:58.668 and Serena play, Venus and Serena know each 55:58.673 --> 56:00.873 other perfectly well, they know that they mix this 56:00.866 --> 56:02.446 way, they're going to best respond 56:02.448 --> 56:04.458 to it, this is going to be where they end up. 56:04.460 --> 56:09.130 But in the meantime, Serena hires a new coach and 56:09.125 --> 56:15.025 Serena's new coach is just very, very good at teaching Serena 56:15.028 --> 56:18.828 how to play at the net, and in particular, 56:18.834 --> 56:21.994 how to hit the backhand volley. 56:21.989 --> 56:25.359 So Serena's new coach, let's say it's Tony Roche or 56:25.359 --> 56:29.539 somebody, it's just a brilliant coach and Tony Roche is able to 56:29.537 --> 56:34.117 improve Serena's backhand volley and that changes these payoffs. 56:34.119 --> 56:38.199 So you should rewrite the whole matrix but I'm going to cheat. 56:38.199 --> 56:43.779 So the new game is exactly the same as it was everywhere else, 56:43.775 --> 56:48.705 except for now when Serena gets to the backhand volley, 56:48.710 --> 56:51.910 she gets in it 70% of the time. 56:51.909 --> 56:59.689 So there used to 50,50 in that box and now it's 30,70. 56:59.690 --> 57:04.110 So the game has changed because Serena has got better at hitting 57:04.107 --> 57:05.437 backhand volleys. 57:05.440 --> 57:13.700 We want to figure out how is this going to affect play at 57:13.699 --> 57:17.199 Wimbledon? Now it doesn't take much to 57:17.203 --> 57:21.123 check that there is still no pure strategy Nash Equilibrium. 57:21.119 --> 57:24.459 It's still the case, in fact even more so, 57:24.463 --> 57:29.363 that Serena's best response to Venus choosing left is to lean 57:29.355 --> 57:33.485 to the left. So it's still the case that the 57:33.490 --> 57:39.150 best responses do not coincide, there is still no pure strategy 57:39.150 --> 57:41.940 equilibrium. What we're going to do of 57:41.942 --> 57:45.242 course is we're going to find a mixed strategy equilibrium, 57:45.242 --> 57:47.632 but before we do so, let's think about this 57:47.632 --> 57:51.572 intuitively. Let's see if we can intuit an 57:51.570 --> 57:53.780 answer. I'm guessing we can't, 57:53.778 --> 57:56.428 but let's see if we can intuit an answer. 57:56.429 --> 58:01.149 So Serena has improved her backhand volley, 58:01.150 --> 58:07.670 and hence when she reaches it she gets it in more often. 58:07.670 --> 58:10.140 So one effect, you might think, 58:10.140 --> 58:14.750 is what we might want to call a direct effect and I think 58:14.751 --> 58:17.141 there's two effects here. 58:17.139 --> 58:21.869 There are two effects, one of these I'm going to call 58:21.867 --> 58:25.047 the direct effect, and by effect, 58:25.050 --> 58:30.500 I mean in particular an effect on how Serena should play the 58:30.503 --> 58:33.723 game. So since Serena has improved 58:33.715 --> 58:38.105 her backhand volley, when she reaches that volley 58:38.113 --> 58:42.473 she gets it in more often, so one might say in that 58:42.471 --> 58:46.801 case--your Serena's coach--in that case you should lean to the 58:46.795 --> 58:49.555 left more often than you did before, 58:49.559 --> 58:52.019 because at least when you get that backhand volley you're 58:52.022 --> 58:53.432 going to get it in more often. 58:53.429 --> 59:03.369 So the direct effect says Serena should lean left more, 59:03.373 --> 59:09.453 in other words, Q should go up. 59:09.450 --> 59:11.490 Is that right? So Serena's now better at 59:11.485 --> 59:15.085 playing this backhand volley, so she may as well favor it a 59:15.092 --> 59:17.272 bit more and hence Q will go up. 59:17.269 --> 59:21.899 So that's the direct effect, but of course there's a "but" 59:21.899 --> 59:23.839 coming. What's the but? 59:23.840 --> 59:25.730 Again, let's see my tennis players here, 59:25.729 --> 59:27.569 raise your hands if you play tennis. 59:27.570 --> 59:30.030 Suddenly nobody plays tennis, come on raise your hands okay. 59:30.030 --> 59:31.260 What's the but here? 59:31.260 --> 59:37.670 We think Serena's backhand has improved so she might be tempted 59:37.666 --> 59:42.316 to play towards her backhand a bit more often, 59:42.316 --> 59:46.046 what's the but? So I claim the but is this--you 59:46.053 --> 59:49.493 tell me if I'm wrong--the but is that Venus (she's her sister 59:49.486 --> 59:52.676 after all, right, so Venus knows that 59:52.676 --> 59:58.416 Serena's backhand has improved) so Venus is going to hit it to 59:58.416 --> 1:00:02.176 Serena's left less often than before. 1:00:02.180 --> 1:00:05.400 Is that right? So since Serena's backhand has 1:00:05.396 --> 1:00:09.846 improved, Venus is going to hit it to Serena's backhand less 1:00:09.853 --> 1:00:13.543 often than before, and that might make Serena less 1:00:13.539 --> 1:00:17.399 inclined to cheat towards her backhand because the ball is 1:00:17.398 --> 1:00:19.428 coming that way less often. 1:00:19.429 --> 1:00:24.459 So this is a indirect or a strategic effect. 1:00:24.460 --> 1:00:32.430 The strategic effect is Venus hits L less often, 1:00:32.430 --> 1:00:42.100 so Serena should reduce the number of times that she leans 1:00:42.096 --> 1:00:52.946 to the left because the ball is coming that way fewer times. 1:00:52.949 --> 1:00:57.239 Now notice that these two effects go in opposite 1:00:57.242 --> 1:00:59.802 directions, is that right? 1:00:59.800 --> 1:01:02.470 One of them tends to argue that Q would go up, 1:01:02.469 --> 1:01:06.029 that's the direct effect and the other one is more subtle, 1:01:06.030 --> 1:01:09.720 it says we now think about not just how my play has improved, 1:01:09.719 --> 1:01:13.589 but also how the other person's going to respond to knowing that 1:01:13.592 --> 1:01:16.832 my play has improved, that's the more subtle effect 1:01:16.832 --> 1:01:18.772 and that's going to push Q down. 1:01:18.769 --> 1:01:22.699 That's going to make it less likely, that's an argument 1:01:22.697 --> 1:01:24.877 against leaning to the left. 1:01:24.880 --> 1:01:26.900 So imagine you're going to be Serena's coach, 1:01:26.899 --> 1:01:29.239 which of these effects do you think is going to win, 1:01:29.240 --> 1:01:30.250 let's have a poll. 1:01:30.250 --> 1:01:32.390 Which of these effects do you think is going to win? 1:01:32.389 --> 1:01:34.359 The direct effect or the indirect effect? 1:01:34.360 --> 1:01:35.910 The direct effect or the strategic effect? 1:01:35.910 --> 1:01:38.200 Who thinks the direct effect? 1:01:38.199 --> 1:01:40.759 Who thinks Serena, who'd advise Serena to play to 1:01:40.763 --> 1:01:43.333 her strength a bit more and lean left a bit more, 1:01:43.327 --> 1:01:45.087 who thinks the direct effect? 1:01:45.090 --> 1:01:47.690 Raise your hands, let's have a poll. 1:01:47.690 --> 1:01:51.710 Who thinks the indirect effect, the effect of Serena hitting it 1:01:51.713 --> 1:01:54.183 that way less often is going to win? 1:01:54.179 --> 1:01:56.729 Who's abstaining and basically refusing to be a coach? 1:01:56.730 --> 1:01:58.850 Quite a number of you, all right. 1:01:58.849 --> 1:02:10.789 Well we're going to find out by re-solving for the Nash 1:02:10.785 --> 1:02:17.275 Equilibrium. What we're going to do is redo 1:02:17.275 --> 1:02:23.155 the calculation we did before starting with Serena. 1:02:23.159 --> 1:02:26.319 So to find Serena's mix, to find Serena's new 1:02:26.317 --> 1:02:29.257 equilibrium mix, what do we have to do? 1:02:29.260 --> 1:02:31.280 The question is, in equilibrium, 1:02:31.282 --> 1:02:35.262 is Serena going to lean to the left more (so Q is going go up) 1:02:35.261 --> 1:02:37.741 or less (so Q's going to do down). 1:02:37.739 --> 1:02:41.009 So I need to find out what is Serena's new equilibrium mix. 1:02:41.010 --> 1:02:42.600 What's the new Q? 1:02:42.599 --> 1:02:44.989 How do I go about finding Serena's equilibrium Q, 1:02:44.993 --> 1:02:46.193 what's the trick here? 1:02:46.190 --> 1:02:51.470 Shout it out. Use Venus' payoffs. 1:02:51.469 --> 1:03:08.699 So to find the new Q for Serena, use Venus' payoffs. 1:03:08.700 --> 1:03:11.610 Now let's do that. 1:03:11.610 --> 1:03:18.170 So from Venus' point of view, if she chooses left then her 1:03:18.167 --> 1:03:22.997 payoffs are now, and again I should use the 1:03:22.999 --> 1:03:27.759 pointer, 30 with probability Q, 1:03:27.760 --> 1:03:34.670 this is the new Q and 80 with probability 1-Q, 1:03:34.673 --> 1:03:42.973 30 with probability Q plus 80 with probability 1-Q. 1:03:42.969 --> 1:03:46.209 Again, this is the new Q, I should really give it, 1:03:46.205 --> 1:03:48.775 put Q prime or something but I won't. 1:03:48.780 --> 1:03:55.520 If she chooses right then her payoff is what? 1:03:55.519 --> 1:04:09.349 It's going to be 90 with probability Q and 20 with 1:04:09.349 --> 1:04:16.359 probability 1-Q. What do we know about these two 1:04:16.361 --> 1:04:19.041 payoffs if Venus is mixing in equilibrium? 1:04:19.039 --> 1:04:21.389 We know she's mixing in equilibrium because we saw there 1:04:21.386 --> 1:04:22.876 was no pure strategy equilibrium, 1:04:22.880 --> 1:04:27.240 so what we do know about these two payoffs since Venus is using 1:04:27.238 --> 1:04:29.978 both these strategies in equilibrium? 1:04:29.980 --> 1:04:30.930 They must be the same. 1:04:30.929 --> 1:04:32.989 Since she's using both these strategies, these strategies 1:04:32.992 --> 1:04:33.842 must be equally good. 1:04:33.840 --> 1:04:38.790 They must both be best responses so these two payoffs 1:04:38.788 --> 1:04:41.818 are equal. Since they're equal all I have 1:04:41.821 --> 1:04:44.471 to do is solve out for Q, so let's do it. 1:04:44.469 --> 1:04:57.229 So I'm going to get 90 minus 30 is 60Q, is equal to 80 minus 20 1:04:57.227 --> 1:05:04.427 which is 60(1-Q), so Q equals .5. 1:05:04.429 --> 1:05:07.789 If I did the algebra too quickly just trust me, 1:05:07.786 --> 1:05:09.606 I think I got it right. 1:05:09.610 --> 1:05:12.820 From here on in, it was just algebra. 1:05:12.820 --> 1:05:15.100 So what have I found out? 1:05:15.100 --> 1:05:18.380 Did Q go up or go down? 1:05:18.380 --> 1:05:21.370 Well it used to be, Q used to be what? 1:05:21.369 --> 1:05:24.819 .6 and now its .5, so let me ask what I think is 1:05:24.820 --> 1:05:27.830 an easy question, did it go up or down? 1:05:27.830 --> 1:05:32.970 It went down. Q went down, 1:05:32.973 --> 1:05:37.793 the equilibrium Q went down. 1:05:37.789 --> 1:05:40.039 So which effect turned out to be bigger? 1:05:40.039 --> 1:05:43.269 The direct effect of playing more to your strength or the 1:05:43.267 --> 1:05:46.667 indirect effect of taking into account that your opponent is 1:05:46.668 --> 1:05:49.318 going to play less often to your strength. 1:05:49.320 --> 1:05:52.280 Which effect turned out to be the bigger effect? 1:05:52.280 --> 1:05:54.160 The indirect effect, the strategic effect. 1:05:54.159 --> 1:05:58.219 Of course I really did want the strategic effect to be bigger 1:05:58.220 --> 1:06:02.080 because this is a course about strategy, but the strategic 1:06:02.077 --> 1:06:04.037 effect actually won here. 1:06:04.039 --> 1:06:12.239 The strategic effect, the indirect effect is bigger. 1:06:12.239 --> 1:06:15.499 That's good news for me because it says the slightly dumb coach 1:06:15.502 --> 1:06:18.552 who didn't bother to take Game Theory would have stopped at 1:06:18.554 --> 1:06:21.764 this direct effect and they'd have told Serena to go the wrong 1:06:21.763 --> 1:06:23.683 way, but the smart coach who takes 1:06:23.683 --> 1:06:26.473 my class, and therefore somehow contributes to my salary, 1:06:26.469 --> 1:06:33.839 in an extraordinarily indirect way, gets it right. 1:06:33.840 --> 1:06:38.370 Now we can also solve out for Venus' new mix and we'll do it 1:06:38.374 --> 1:06:40.914 in a second. But before I do it, 1:06:40.914 --> 1:06:45.234 let me just point out that we actually, we really can now 1:06:45.230 --> 1:06:47.080 intuit Venus' effect. 1:06:47.079 --> 1:06:49.359 It may not be exact numbers but we can intuit here. 1:06:49.360 --> 1:06:52.820 As I claim, I claim if we think this through carefully, 1:06:52.817 --> 1:06:56.207 we know whether Venus is shooting more to the left, 1:06:56.210 --> 1:06:59.190 than she was before, or less to the left, 1:06:59.190 --> 1:07:00.830 than she was before. 1:07:00.829 --> 1:07:04.639 Notice that in the new equilibrium Serena is going less 1:07:04.636 --> 1:07:08.576 often to her left even though she's better at hitting the 1:07:08.583 --> 1:07:10.823 backhand, she's better at hitting the 1:07:10.819 --> 1:07:12.089 ball when she gets there. 1:07:12.090 --> 1:07:17.240 So since Serena is leaning left less often what must be true 1:07:17.235 --> 1:07:20.545 about Venus in this new equilibrium? 1:07:20.550 --> 1:07:23.620 It must be the case that Venus is hitting the ball to the left 1:07:23.616 --> 1:07:25.286 less often. Does that make sense? 1:07:25.289 --> 1:07:32.149 We have enough information already on the board to tell us 1:07:32.147 --> 1:07:36.957 that, nevertheless, let's do the math. 1:07:36.960 --> 1:07:51.560 Let's go and retrieve a board to do the math. 1:07:51.559 --> 1:08:02.489 Just to complete this, let's figure out exactly what 1:08:02.490 --> 1:08:07.850 Venus does do. So to figure out what Venus is 1:08:07.847 --> 1:08:09.837 going to do, what's our trick? 1:08:09.840 --> 1:08:11.690 I want to figure out how Venus is going to mix. 1:08:11.690 --> 1:08:17.200 I'm going to find out Venus' new P, how do I find out Venus' 1:08:17.196 --> 1:08:19.246 new equilibrium mix? 1:08:19.250 --> 1:08:22.270 I look at Serena's payoffs. 1:08:22.270 --> 1:08:25.400 So if Serena chooses left, her payoff is, 1:08:25.400 --> 1:08:28.610 and I'll read it off quickly this time, 1:08:28.609 --> 1:08:36.549 is 70P plus 10(1-P) and if Serena chooses right her payoff 1:08:36.550 --> 1:08:44.070 is 20P plus 80(1-P) and I'm praying that the T.A.'s are 1:08:44.072 --> 1:08:50.622 going to catch me if I make a mistake here, 1:08:50.619 --> 1:08:56.579 and I know these have to be equal because I know that in 1:08:56.579 --> 1:09:00.859 fact Venus is mixing--sorry, I know that Serena is mixing, 1:09:00.859 --> 1:09:01.959 so I know these must be equal. 1:09:01.960 --> 1:09:12.370 So since they're equal I can solve out and hope that I've got 1:09:12.366 --> 1:09:19.646 this right, so I've got 50P equals 70(1-P), 1:09:19.650 --> 1:09:24.160 so P is equal to 7/12. 1:09:24.159 --> 1:09:26.369 So again, that's just algebra, I rushed it a bit, 1:09:26.369 --> 1:09:27.289 it's just algebra. 1:09:27.290 --> 1:09:28.930 Same idea, just algebra. 1:09:28.930 --> 1:09:35.420 So 7/12 is indeed smaller than what it used to be, 1:09:35.419 --> 1:09:43.099 because it used to be 7/10, so that confirms our result. 1:09:43.100 --> 1:09:45.060 So the strategic effect dominated. 1:09:45.060 --> 1:09:49.850 Venus shot to Serena's backhand less often, and as a 1:09:49.848 --> 1:09:54.898 consequence, so much so, that Serena actually found it 1:09:54.899 --> 1:10:00.299 worthwhile going more to the right than she used to before. 1:10:00.300 --> 1:10:02.310 Now let's just talk this through one more time. 1:10:02.310 --> 1:10:04.780 This was a comparative statics exercise. 1:10:04.779 --> 1:10:08.019 We looked at a game, we found an equilibrium, 1:10:08.019 --> 1:10:11.699 we changed something fundamental about the game, 1:10:11.699 --> 1:10:14.499 and we looked again to look at the new equilibrium, 1:10:14.502 --> 1:10:16.522 that's called comparative statics. 1:10:16.520 --> 1:10:18.770 Let's talk through the intuition. 1:10:18.770 --> 1:10:27.640 Before we made any changes Venus was indifferent. 1:10:27.640 --> 1:10:31.410 She was indifferent between shooting to the left and 1:10:31.407 --> 1:10:33.177 shooting to the right. 1:10:33.180 --> 1:10:38.800 Then we improved Serena's ability to hit the volley to her 1:10:38.797 --> 1:10:42.737 left, we improved her backhand volley. 1:10:42.739 --> 1:10:50.709 If we had not changed the way Serena played then what would 1:10:50.707 --> 1:10:56.397 Venus have done? So suppose in fact Serena's Q 1:10:56.395 --> 1:11:01.335 had not changed. If Serena's Q had not changed, 1:11:01.342 --> 1:11:05.942 remembering that Venus was indifferent before, 1:11:05.935 --> 1:11:10.115 how would Venus have changed her play? 1:11:10.120 --> 1:11:13.090 Somebody? If we started from the old Q 1:11:13.085 --> 1:11:17.085 and then we improved Serena's ability to play the backhand 1:11:17.087 --> 1:11:21.437 volley, and if Q didn't change, what would Venus have done? 1:11:21.439 --> 1:11:24.409 She'd never, ever have shot to the left 1:11:24.405 --> 1:11:29.235 anymore, she'd only have shot to the right which can't possibly 1:11:29.243 --> 1:11:30.963 be an equilibrium. 1:11:30.960 --> 1:11:34.940 So something about Serena's play has to bring Venus back 1:11:34.943 --> 1:11:38.423 into equilibrium, it brings Venus back into being 1:11:38.419 --> 1:11:40.809 indifferent, and what was it? 1:11:40.810 --> 1:11:47.590 It was Serena moving to the left less often and moving to 1:11:47.593 --> 1:11:50.383 the right more often. 1:11:50.380 --> 1:11:52.870 To say it again, if we didn't change Q, 1:11:52.871 --> 1:11:56.741 Venus would only go to the right, so we need to reduce Q, 1:11:56.739 --> 1:12:00.959 have Serena go to the right, to bring Venus back into 1:12:00.955 --> 1:12:03.665 equilibrium. Conversely, if Venus hadn't 1:12:03.670 --> 1:12:06.600 changed her behavior, if Venus had gone on shooting 1:12:06.595 --> 1:12:09.705 exactly the same as she was, P and 1-P as before, 1:12:09.713 --> 1:12:13.493 then Serena would have only gone to the left and that can't 1:12:13.491 --> 1:12:14.861 be an equilibrium. 1:12:14.859 --> 1:12:17.509 So it must be something about Venus' play that brings Serena 1:12:17.512 --> 1:12:19.312 back into equilibrium, and what is it? 1:12:19.310 --> 1:12:24.320 It's that Venus starts shooting to the right more often. 1:12:24.319 --> 1:12:27.879 So just two reminders, before you leave two reminders. 1:12:27.880 --> 1:12:28.610 Wait, wait, wait. 1:12:28.609 --> 1:12:32.479 First, in about five minutes time a handout will magically 1:12:32.477 --> 1:12:36.817 appear on the website that goes through these arguments again, 1:12:36.819 --> 1:12:40.779 all of them in two other games, so you can have a look at the 1:12:40.779 --> 1:12:42.279 handout. Second thing, 1:12:42.276 --> 1:12:45.486 a problem set has already appeared by magic on that 1:12:45.492 --> 1:12:49.612 website that gives you lots of examples like this to work on. 1:12:49.609 --> 1:12:52.419 Play tennis over the weekend for practice and we'll see you 1:12:52.418 --> 1:12:52.998 on Monday.