WEBVTT 00:01.790 --> 00:03.770 Professor Ben Polak: All right, 00:03.768 --> 00:06.498 so last time we did something I think substantially harder than 00:06.495 --> 00:08.425 anything we've done in the class so far. 00:08.430 --> 00:13.430 We looked at mixed strategies, and in particular, 00:13.429 --> 00:17.699 we looked at mixed-strategy equilibria. 00:17.700 --> 00:19.840 There was a big idea last time. 00:19.840 --> 00:24.260 The big idea was if a player is playing a mixed strategy in 00:24.259 --> 00:28.679 equilibrium, then every pure strategy in the mix--that's to 00:28.678 --> 00:32.948 say every pure strategy on which they place some positive 00:32.945 --> 00:37.355 weight--must also be a best response to what the other side 00:37.364 --> 00:40.474 is doing. Then we used that trick. 00:40.470 --> 00:44.420 We used it in this game here, to help us find Nash Equilibria 00:44.421 --> 00:48.241 and the way it allowed us to find the Nash Equilibria is we 00:48.240 --> 00:50.340 knew that if, in this case, 00:50.340 --> 00:53.880 Venus Williams is mixing between left and right, 00:53.883 --> 00:58.483 it must be this case that her payoff is equal to that of right 00:58.483 --> 01:01.653 and we use that to find Serena's mix. 01:01.649 --> 01:06.109 Conversely, since we knew that Serena is mixing again between l 01:06.105 --> 01:10.195 and r, we knew she must be indifferent between l and r and 01:10.202 --> 01:12.792 we used that to find Venus' mix. 01:12.790 --> 01:15.610 So I want to go back to this example just for a few moments 01:15.606 --> 01:18.176 just to make one more point and then we'll move on, 01:18.180 --> 01:20.420 but we'll still be talking about mixed strategies 01:20.415 --> 01:21.295 throughout today. 01:21.299 --> 01:24.979 So this was the mix that we found before we changed the 01:24.980 --> 01:28.730 payoffs, we found that Venus' equilibrium mix was .7, 01:28.730 --> 01:36.450 .3 and Serena's equilibrium mix was .6, .4. 01:36.450 --> 01:39.450 And a reasonable question at this point would be, 01:39.447 --> 01:42.317 how do we know that's really an equilibrium? 01:42.319 --> 01:44.779 We kind of found it but we didn't kind of go back and 01:44.781 --> 01:46.331 check. So what I want to do now is 01:46.332 --> 01:48.072 actually do that, do that missing step. 01:48.069 --> 01:50.009 We rushed it a bit last time because we wanted to get through 01:50.014 --> 01:50.634 all the material. 01:50.629 --> 01:56.389 Let's actually check that in fact P* is a best response to 01:56.394 --> 02:00.674 Q*. So what I want to do is I want 02:00.666 --> 02:08.566 to check that Venus' mix P* is a best response for Venus against 02:08.574 --> 02:13.014 Serena's mix Q*. The way I'm going to do that is 02:13.007 --> 02:16.897 I'm going to look at payoffs that Venus gets now she knows - 02:16.904 --> 02:20.674 or rather now we know she's playing against Q*. 02:20.669 --> 02:23.009 So let's look at Venus' payoffs.. 02:23.009 --> 02:26.299 I'm going to figure out her payoffs for L, 02:26.297 --> 02:29.987 her payoffs for R, and also her payoff for what 02:29.985 --> 02:32.225 she's actually doing P*. 02:32.229 --> 02:37.049 So Venus' payoffs, if she chooses L against Q* 02:37.052 --> 02:43.382 then she gets--very similar to what we had on the board last 02:43.375 --> 02:46.675 week, but now I'm going to put in 02:46.677 --> 02:50.657 what Q* is explicitly--she gets 50 times .6.. 02:50.660 --> 02:56.040 [This is Q* and this is 1-Q*.]. 02:56.039 --> 03:07.879 So she gets 50 times .6 and 80 times 1 minus .6 which is .4,80 03:07.875 --> 03:11.335 times .4. We can work this out, 03:11.335 --> 03:14.425 and I worked it out at home, but if somebody has a 03:14.428 --> 03:16.888 calculator they can please check me. 03:16.890 --> 03:20.690 I think this comes to .62. 03:20.690 --> 03:24.120 Somebody should just check that. 03:24.120 --> 03:29.960 If Venus chose R--remember R here means shooting to Serena's 03:29.961 --> 03:36.201 right, to Serena's forehand--if she chose R then her payoffs are 03:36.199 --> 03:41.439 90 Q*. So 90(.6) plus 20(1-Q*) so 03:41.440 --> 03:51.250 20(.4), so 90(.6) plus 20(.4), and again I worked that out at 03:51.249 --> 03:59.749 home, and fortunately that also comes out at .62. 03:59.750 --> 04:04.010 So what's Venus' payoff for P*? 04:04.009 --> 04:09.289 We've got her payoff for both her pure strategies, 04:09.291 --> 04:14.681 so her payoff from actually choosing P* is what? 04:14.680 --> 04:19.230 Well, P* is .7, so .7 of the time she will 04:19.228 --> 04:24.218 actually be playing L and when she plays L, 04:24.220 --> 04:34.650 she'll get a payoff of .62, and .3 of the time she'll be 04:34.650 --> 04:44.540 playing R, and once again, she'll be getting a payoff of 04:44.539 --> 04:49.749 .62 and--do I have a calculator? 04:49.750 --> 04:50.520 Sorry, thank you. 04:50.519 --> 04:53.789 So P* is .7, yes, you're absolutely right, 04:53.786 --> 04:58.006 so this is P* and 1-P*, So let's make that clearer. 04:58.009 --> 05:03.349 I'll show you what the equilibrium is but P* itself is 05:03.346 --> 05:05.566 .7. So when Venus plays L with 05:05.566 --> 05:08.736 probability of .7, then .7 of the time she'll get 05:08.735 --> 05:12.495 the expected payoff of .62 and .3 of the time she'll get a 05:12.498 --> 05:16.588 payoff again of .62 and that's the kind of math I don't have to 05:16.591 --> 05:21.301 do at home, that's going to come out at .62. 05:21.300 --> 05:23.700 Again, assuming my math is correct. 05:23.699 --> 05:26.639 So all I've really done here is confirm what we did already last 05:26.642 --> 05:30.052 time. We knew--we in fact chose 05:30.051 --> 05:37.201 Serena's mix Q to make Venus indifferent between L and R. 05:37.199 --> 05:41.809 And that's exactly what we found here, going left it's .62, 05:41.812 --> 05:45.632 going right it gets .62 and hence P* gets .62. 05:45.629 --> 05:49.219 But I claim we can now see something a little bit else. 05:49.220 --> 05:52.640 We can now ask the question, is P* in fact the best 05:52.638 --> 05:55.168 response? Well, for it not to be a best 05:55.168 --> 05:59.028 response, for this not to be an equilibrium, there would have to 05:59.030 --> 06:02.590 be some deviation that Venus could make that would make her 06:02.587 --> 06:04.117 strictly better off. 06:04.120 --> 06:05.060 Let me repeat that. 06:05.060 --> 06:08.890 If this were not an equilibrium, there would have to 06:08.889 --> 06:13.089 be some deviation for Venus, that would make her strictly 06:13.093 --> 06:16.823 better off. By playing P* she's getting a 06:16.823 --> 06:20.693 return of .62. So one thing she could deviate 06:20.692 --> 06:23.302 to, is playing L all the time. 06:23.300 --> 06:26.820 If she deviates to playing L all the time, 06:26.821 --> 06:31.891 her payoff is still .62 so she's not strictly better off. 06:31.889 --> 06:35.289 That's not a strictly profitable deviation. 06:35.290 --> 06:39.300 Another thing she could deviate to, is she could deviate to 06:39.300 --> 06:42.710 playing R. If she deviates to playing R, 06:42.712 --> 06:44.852 her payoff will be .62. 06:44.850 --> 06:48.730 Once again, she's not strictly better off: she's the same as 06:48.725 --> 06:51.285 she was before, so that's not a strictly 06:51.287 --> 06:52.927 profitable deviation. 06:52.930 --> 06:54.410 So what have I shown so far? 06:54.410 --> 06:57.810 I've shown that P* is as good as playing L, 06:57.809 --> 07:00.479 and P* is as good as playing R. 07:00.480 --> 07:02.640 In fact that's how we constructed it. 07:02.639 --> 07:07.179 So deviating to L is not a strictly profitable deviation 07:07.183 --> 07:12.143 and deviating to R is not a strictly profitable deviation. 07:12.139 --> 07:15.579 But at this point, somebody might ask and say, 07:15.578 --> 07:19.858 okay, you've shown me that there's no way to deviate to a 07:19.858 --> 07:23.448 pure strategy in a strictly profitable way, 07:23.449 --> 07:28.669 but how about deviating to another mixed strategy? 07:28.670 --> 07:43.790 So, so far we've shown--we've shown just up here--we can see 07:43.789 --> 07:56.599 that Venus has no strictly profitable pure-strategy 07:56.602 --> 08:02.412 deviation. She has no strictly profitable 08:02.412 --> 08:06.712 pure-strategy deviation because each of her pure strategies 08:06.708 --> 08:10.558 yields the same payoff as did her mixed strategy, 08:10.560 --> 08:12.520 yields the same as P*. 08:12.519 --> 08:16.589 But how do we know that she doesn't have a mixed strategy 08:16.592 --> 08:18.922 that would be strictly better? 08:18.920 --> 08:20.690 How do we know that? 08:20.690 --> 08:26.990 Anybody? No hands going up; 08:26.990 --> 08:29.220 oh, there was a hand up, good. 08:29.220 --> 08:32.410 Student: Any mix between left and right will still yield 08:32.411 --> 08:34.911 .62. Professor Ben Polak: 08:34.909 --> 08:39.209 Good, so any mix that Venus deviates to, will be a mix 08:39.208 --> 08:43.708 between L and R, and any mix between L and R 08:43.707 --> 08:50.277 will be a mix between .62 and .62 and hence will yield .62. 08:50.279 --> 08:53.279 So we're going to use again, this fact we developed last 08:53.278 --> 08:56.098 week. The fact we developed last week 08:56.097 --> 09:00.187 was that any mixed strategy yields a payoff that is a 09:00.186 --> 09:04.036 weighted average of the pure strategy payoffs, 09:04.039 --> 09:06.569 the payoffs to the pure strategies in the mix. 09:06.570 --> 09:09.270 Any mixed strategy yields a payoff that is a weighted 09:09.274 --> 09:12.294 average of the payoff to the pure strategies in the mix. 09:12.290 --> 09:14.020 That was our key fact last week. 09:14.019 --> 09:17.999 So here if we've shown that there's no pure-strategy 09:17.998 --> 09:21.038 deviation that's strictly profitable, 09:21.039 --> 09:25.159 then there can't be any mixed strategy deviation that's 09:25.161 --> 09:26.841 strictly profitable. 09:26.840 --> 09:29.160 Why? Because the mixed strategy 09:29.156 --> 09:33.366 deviations must yield payoffs that lie among the pure strategy 09:33.372 --> 09:36.312 deviations. So this is a great fact for us. 09:36.310 --> 09:39.310 What's the lesson here? 09:39.309 --> 10:02.819 The lesson is we only ever have to check for strictly profitable 10:02.822 --> 10:12.902 pure-strategy deviations. 10:12.900 --> 10:13.720 That's a good job. 10:13.720 --> 10:15.580 Why? Because if we had to check for 10:15.583 --> 10:18.943 mixed strategy deviations one by one, we'd be here all night, 10:18.940 --> 10:20.910 because there's an infinite number of possible 10:20.912 --> 10:22.142 mixed-strategy deviations. 10:22.139 --> 10:23.849 But there aren't so many pure strategy deviations we have to 10:23.848 --> 10:25.818 check. Let's just repeat the idea. 10:25.820 --> 10:29.700 Suppose there isn't any pure-strategy deviation that's 10:29.704 --> 10:34.034 profitable, then there can't be any mixed strategy deviation 10:34.027 --> 10:37.637 that's profitable, because the highest expected 10:37.639 --> 10:41.139 return you could ever get from a mixed strategy, 10:41.139 --> 10:44.349 is one of the pure strategies in the mix, and you've already 10:44.347 --> 10:46.737 checked that none of those are profitable. 10:46.740 --> 10:50.710 So this simple idea, the simple idea we developed 10:50.705 --> 10:54.995 last time, not only helps us to find Nash Equilibria, 10:55.000 --> 10:58.470 but also to check for Nash Equilibria. 10:58.470 --> 11:02.540 Now a lot of people I gathered from feedback from sections were 11:02.535 --> 11:04.695 left pretty confused last time. 11:04.700 --> 11:06.210 It's a hard idea. 11:06.210 --> 11:09.260 Actually I looked at the tape over the weekend, 11:09.262 --> 11:12.052 I could see where it could be confusing. 11:12.049 --> 11:13.959 But it's actually, I think what's really confusing 11:13.957 --> 11:16.367 here--it wasn't so much--I think it wasn't so much that I could 11:16.370 --> 11:18.550 have been clearer though I'm sure I could have been. 11:18.549 --> 11:21.779 It's that this is really a hard idea, this idea of mixed 11:21.777 --> 11:23.827 strategies. So we're going to work on it 11:23.830 --> 11:26.460 again today, but I think one of the ideas that gets people 11:26.457 --> 11:28.067 confused, is the following idea. 11:28.070 --> 11:33.440 They say, look we found Venus' equilibrium mix by choosing a P 11:33.444 --> 11:36.884 and a 1-P to make Serena indifferent. 11:36.879 --> 11:40.969 We found Serena's equilibrium mix by finding a Q and a 1-Q to 11:40.965 --> 11:45.045 make Venus indifferent and a natural question you hear people 11:45.050 --> 11:48.910 ask then is, why is Venus "trying to make 11:48.905 --> 11:50.975 Serena indifferent?" 11:50.980 --> 11:55.020 Why is Serena "trying to make Venus indifferent?" 11:55.019 --> 11:57.329 That's not really the point here. 11:57.330 --> 12:01.410 It isn't that Venus is trying to make Serena indifferent. 12:01.409 --> 12:04.529 It's that in equilibrium, she is going to make Serena 12:04.529 --> 12:06.799 indifferent. It isn't her goal in life to 12:06.804 --> 12:09.934 make Serena indifferent between l and r, and it isn't Serena's 12:09.928 --> 12:12.898 goal in life to make Venus indifferent between L and R, 12:12.899 --> 12:16.979 but in equilibrium it ends up that they make each other 12:16.981 --> 12:20.241 indifferent. The way that we can see that is 12:20.239 --> 12:24.499 that if Venus puts--we said last time it's repeated--if Venus 12:24.497 --> 12:27.587 puts too much weight, more than .7 on L, 12:27.588 --> 12:31.288 then Serena just cheats to the left all the time, 12:31.289 --> 12:34.759 and that can't possibly be an equilibrium. 12:34.759 --> 12:37.929 And if Venus puts too much weight on R, then Serena cheats 12:37.928 --> 12:41.318 to the right all the time and that can't be an equilibrium. 12:41.320 --> 12:46.500 So it has to be that what Venus is doing is going to make Serena 12:46.498 --> 12:49.538 exactly indifferent and vice versa. 12:49.539 --> 12:52.959 Now let's see that idea in some other applications. 12:52.960 --> 12:56.680 Let's talk about this a bit before we move on. 12:56.679 --> 13:01.409 So it turns out that some very natural applications for 13:01.408 --> 13:06.048 mixed-strategy equilibria arise in games, in sport. 13:06.050 --> 13:08.130 So let's talk about a few now. 13:08.129 --> 13:12.259 Can anybody suggest some other places where we see 13:12.257 --> 13:16.977 randomization or at least mixed strategy in equilibria in 13:16.975 --> 13:21.715 sporting events? Let me actually grab the mike 13:21.724 --> 13:24.894 myself. Anybody here play football for 13:24.887 --> 13:28.587 example, and we're talking American football now, 13:28.594 --> 13:32.074 the gridiron game, not the civilized type. 13:32.070 --> 13:34.150 Anyone play? Yes, so some of you play 13:34.152 --> 13:38.082 football. So where is the mixing involved 13:38.075 --> 13:40.445 in playing football? 13:40.450 --> 13:42.470 Where in equilibrium would we expect to see mixed strategies? 13:42.470 --> 13:46.660 There's somebody down there can we go on and get them. 13:46.660 --> 13:49.340 So shout it out. Student: Running game 13:49.340 --> 13:49.880 and passing game. 13:49.879 --> 13:51.469 Professor Ben Polak: All right, the running game and the 13:51.466 --> 13:53.706 passing game. So a very simple idea whether 13:53.705 --> 13:57.535 to run or whether to pass when you have the ball is likely to 13:57.543 --> 14:00.233 end up as a mixed-strategy equilibrium. 14:00.230 --> 14:03.500 The defense is also randomizing between, for example, 14:03.498 --> 14:06.388 rushing the passer or playing a run defense. 14:06.389 --> 14:08.799 Is that right--this is not exactly a game I know a lot 14:08.797 --> 14:11.247 about, but I'm hoping I'm getting close enough here. 14:11.250 --> 14:14.180 It couldn't possibly be a pure-strategy equilibrium, 14:14.181 --> 14:16.711 other than very extreme parts of the game, 14:16.710 --> 14:19.430 like at the end of the game perhaps, but for most of the 14:19.433 --> 14:22.013 game, it's very unlikely to end up as a pure-strategy 14:22.007 --> 14:24.077 equilibrium. Much more likely that the 14:24.081 --> 14:26.391 offense is mixing between passing and running, 14:26.388 --> 14:28.898 and for that matter between going to the left, 14:28.899 --> 14:33.329 going to the right and going to the center, and the defense is 14:33.330 --> 14:36.890 also mixing between--over its types of defense. 14:36.889 --> 14:40.259 So we see that--for those people who were watching 14:40.261 --> 14:43.221 yesterday--we see that in football games. 14:43.220 --> 14:49.130 Where do we see it else in sport, some other sports? 14:49.129 --> 14:51.459 I can't have a room full of non-sports fans. 14:51.460 --> 14:53.250 How many of you ever watch any sports? 14:53.250 --> 14:58.220 Let's raise some hands here--some of you do. 14:58.220 --> 15:01.360 So this is baseball playoff season. 15:01.360 --> 15:04.100 How many of you have been watching the baseball playoffs? 15:04.100 --> 15:05.710 Raise your hands if you've been watching the baseball playoffs. 15:05.710 --> 15:07.110 I'll let you off, I know you should have been 15:07.112 --> 15:07.752 doing my homework. 15:07.750 --> 15:09.350 How many of you have been watching the playoffs instead? 15:09.350 --> 15:11.850 How many watched the Yankees game last night? 15:11.850 --> 15:14.020 Quite a few of you. 15:14.019 --> 15:15.799 So they haven't been very exciting yet but we're hoping 15:15.802 --> 15:17.092 that it's going to get more exciting. 15:17.090 --> 15:20.590 So when you're watching baseball what kind of things do 15:20.593 --> 15:24.623 you see where you just know that there must be mixed strategies 15:24.616 --> 15:26.816 involved? There must be randomization 15:26.824 --> 15:28.304 involved. Now I've got a few more hands 15:28.302 --> 15:29.992 out. Good, so you sir. 15:29.990 --> 15:31.230 Student: Choosing how to pitch the ball. 15:31.230 --> 15:32.460 Professor Ben Polak: Choosing how to pitch the ball. 15:32.460 --> 15:34.910 Enlarge a little bit more, say a bit more. 15:34.909 --> 15:36.559 Student: Fast ball versus slider, 15:36.562 --> 15:38.682 versus change up, all sorts of different things. 15:38.679 --> 15:41.169 Prof Ben Polak: All right, so there's different ways 15:41.171 --> 15:43.471 of throwing the ball, and there's going to be 15:43.470 --> 15:46.940 randomization from the pitcher, or at least it's going to look 15:46.944 --> 15:50.134 like there's randomization by the pitcher over whether to 15:50.134 --> 15:53.044 throw a fast ball or a curve ball or whatever. 15:53.039 --> 15:56.359 How is the hitter randomizing there? 15:56.360 --> 15:58.130 How is the hitter randomizing? 15:58.129 --> 15:59.009 Is the hitter randomizing at all? 15:59.009 --> 16:02.549 What's the hitter doing while this is going on? 16:02.550 --> 16:06.240 Anybody? Yeah. 16:06.240 --> 16:07.800 Student: He's choosing whether to swing or not to 16:07.802 --> 16:08.902 swing. Professor Ben Polak: 16:08.901 --> 16:10.581 Okay, he's choosing whether to swing or not to swing, 16:10.578 --> 16:12.348 although presumably he can do that just after the ball's 16:12.352 --> 16:14.072 thrown. So you sometimes hear the 16:14.065 --> 16:17.245 commentator say that that hitter was looking for a fast ball, 16:17.248 --> 16:19.448 is that right? Or looking for a curve ball. 16:19.450 --> 16:21.980 The hitter is trying to anticipate the pitch, 16:21.980 --> 16:23.900 is that right? This is not a game I played a 16:23.904 --> 16:25.224 lot of either--I played a little bit. 16:25.220 --> 16:28.870 You're trying to anticipate where the ball is going to be 16:28.872 --> 16:31.092 thrown. So the type of ball you throw 16:31.093 --> 16:34.593 in baseball and the way in which the pitch being anticipated by 16:34.586 --> 16:37.286 the hitter, is likely to be a mixed strategy. 16:37.289 --> 16:40.909 What else is likely to be a mixed strategy in baseball? 16:40.910 --> 16:46.080 What else? Anybody here on the Yale 16:46.075 --> 16:50.555 baseball team? Okay, I've got one volunteer 16:50.558 --> 16:51.988 here. So what else, 16:51.990 --> 16:53.160 stand up for a second. 16:53.159 --> 16:57.369 Let's have a Yale baseball team member, what's your name? 16:57.370 --> 16:57.960 Student: Chris. 16:57.960 --> 16:58.700 Professor Ben Polak: Where do you play? 16:58.700 --> 16:59.400 Student: I'm a pitcher. 16:59.399 --> 17:00.059 Professor Ben Polak: You're a pitcher, 17:00.055 --> 17:01.135 okay. So he's not going to get on 17:01.137 --> 17:02.727 base now, so he's not going to answer this. 17:02.730 --> 17:07.480 Suppose you did get on base, pitchers don't often get on 17:07.484 --> 17:09.964 base. Let's assume that happens, 17:09.960 --> 17:11.830 what might you randomize? 17:11.829 --> 17:13.479 There you are, you're standing on base, 17:13.483 --> 17:14.923 what might you randomize about? 17:14.920 --> 17:16.080 Student: Whether to steal second or not. 17:16.079 --> 17:17.709 Professor Ben Polak: Right, whether to steal or not, 17:17.706 --> 17:18.656 whether to try and steal or not. 17:18.660 --> 17:20.470 Stay up a second. 17:20.470 --> 17:24.450 So the decision whether to try and steal or not is likely to 17:24.447 --> 17:25.927 end up being random. 17:25.930 --> 17:31.150 If you're the pitcher, what can you do in response to 17:31.147 --> 17:32.967 that? Student: You can either 17:32.971 --> 17:34.431 choose to try to pick them off or not. 17:34.430 --> 17:35.150 Professor Ben Polak: What else? 17:35.150 --> 17:36.380 So one thing you can try and pick him off. 17:36.380 --> 17:37.300 What else? Student: You can be 17:37.302 --> 17:37.882 quicker to the plate. 17:37.880 --> 17:39.080 Professor Ben Polak: Quicker to the plate, 17:39.079 --> 17:40.109 what else? Student: You can pitch 17:40.105 --> 17:41.795 out. Professor Ben Polak: You 17:41.796 --> 17:43.106 can pitch out, what else? 17:43.109 --> 17:44.259 At least those three things, right? 17:44.260 --> 17:45.120 Student: Yeah. 17:45.119 --> 17:46.849 Professor Ben Polak: At least those three things okay, 17:46.845 --> 17:48.475 thank you. I have an expert here, 17:48.482 --> 17:50.012 I'm glad I had an expert. 17:50.009 --> 17:53.969 So in this case we can see there's randomization going on 17:53.967 --> 17:57.847 from the runner whether he attempts to steal the base or 17:57.854 --> 18:00.204 not, and by the pitcher on whether 18:00.198 --> 18:03.508 he throws the pitch out or whether he tries to throw, 18:03.510 --> 18:06.780 to get to the plate faster. 18:06.780 --> 18:09.660 So we see this in sport. 18:09.660 --> 18:12.320 We don't see it well anticipated by sports 18:12.317 --> 18:14.797 commentators. Let me put this down a second. 18:14.799 --> 18:16.479 So in baseball, for example, 18:16.482 --> 18:19.662 you'll sometimes see quite sophisticated statistical 18:19.660 --> 18:23.210 analyses of baseball in which somebody will have looked at 18:23.212 --> 18:27.012 base stealers across the major leagues and they'll look at all 18:27.013 --> 18:30.693 the instances in which a player was on first base and in the 18:30.689 --> 18:33.929 position where you think they might steal, 18:33.930 --> 18:36.410 and they'll look at what happened on every attempt to 18:36.407 --> 18:39.167 steal, whether they were in fact caught stealing or not, 18:39.170 --> 18:41.410 and they'll try and measure the value of these things and 18:41.405 --> 18:43.595 they'll see, the conclusion they'll come to is something 18:43.601 --> 18:46.001 like this. They'll conclude that whether 18:46.001 --> 18:49.111 the guy stole or not, whether the guy attempted to 18:49.113 --> 18:51.773 steal or not, sorry, or whether he just sat 18:51.769 --> 18:54.709 on first base doesn't seem to make much difference, 18:54.710 --> 18:57.510 they'll say. They'll say that the payoff for 18:57.507 --> 19:01.307 even great base stealers are attempting to steal or not, 19:01.309 --> 19:04.959 when you take into account the pick offs versus just staying 19:04.956 --> 19:08.596 put, turns out the payoff in terms of the impact on the game 19:08.602 --> 19:10.992 is roughly equal, and then they'll draw-- these 19:10.990 --> 19:12.870 analysts will then draw the following conclusion. 19:12.869 --> 19:16.169 They'll say, oh look, speed or the ability 19:16.172 --> 19:20.362 to steal bases is therefore overrated in baseball. 19:20.360 --> 19:22.350 How have they made a mistake? 19:22.349 --> 19:25.609 What's the mistake they made there? 19:25.609 --> 19:28.279 So the premise was, let's give them the premise, 19:28.278 --> 19:31.458 the premise was that when a base stealer is attempting to 19:31.457 --> 19:35.027 steal or not the expected return in terms of outcome of the game 19:35.034 --> 19:38.214 is roughly equal, whether they attempt to steal 19:38.214 --> 19:39.964 or don't attempt to steal. 19:39.960 --> 19:43.350 The conclusion is, therefore stealing doesn't seem 19:43.346 --> 19:47.126 such a big deal. What's the mistake they've made? 19:47.130 --> 19:51.400 Yeah, let me borrow it again, sorry. 19:51.400 --> 19:54.800 Student: The pitcher has to react differently in pitching 19:54.799 --> 19:57.389 when he knows that there's a fast guy on base. 19:57.390 --> 19:58.790 Professor Ben Polak: Good, so our pitcher has to 19:58.791 --> 19:59.311 react differently. 19:59.309 --> 20:02.459 Let's talk to our pitcher again, so one thing our pitcher 20:02.463 --> 20:05.113 said was he wants to get to the plate faster. 20:05.109 --> 20:07.869 What does that mean getting to the plate faster? 20:07.869 --> 20:09.829 –Shout out so people can hear you. 20:09.829 --> 20:11.619 Student: It means just getting the ball to the catcher 20:11.620 --> 20:13.440 as fast as possible so he has the best chance to throw out the 20:13.441 --> 20:14.871 runner. Professor Ben Polak: 20:14.869 --> 20:17.269 Right, so you're going to pitch from, you're not going to do 20:17.270 --> 20:19.140 that funny windup thing, you're not, thank you, 20:19.138 --> 20:21.308 you're going to pitch from the stretch, I knew there was a term 20:21.305 --> 20:23.625 there somewhere. I'm learning American by being 20:23.628 --> 20:25.248 here. And you're more likely to throw 20:25.252 --> 20:27.462 a fast ball, there's some advantage in throwing a fast 20:27.463 --> 20:28.843 ball rather than a curve ball. 20:28.839 --> 20:31.719 Both actions of which, both having to move more 20:31.719 --> 20:35.349 towards fast balls and pitching to the stretch are actually 20:35.349 --> 20:37.039 costly for the pitcher. 20:37.039 --> 20:38.899 But we'll get there in a second, let's just back up a 20:38.901 --> 20:40.371 second, so that was good, that's right. 20:40.370 --> 20:41.780 But let's just back up a second. 20:41.779 --> 20:44.469 The premise of these commentators was what? 20:44.470 --> 20:48.540 It was that the return to stealing, attempting to steal, 20:48.536 --> 20:50.676 seems to be roughly a wash. 20:50.680 --> 20:53.440 It seems to be that the expected return when this great 20:53.440 --> 20:56.610 base runner attempts to steal a base is roughly the same as the 20:56.610 --> 20:59.320 return when they don't attempt to steal the base. 20:59.319 --> 21:01.719 But I claim we knew that was going to the case. 21:01.720 --> 21:03.680 We didn't have to go and look at the data. 21:03.680 --> 21:07.530 Why did we know that was going to be the case? 21:07.529 --> 21:12.739 How did we know that we were bound to find a return in that 21:12.744 --> 21:17.154 analysis that finds those things roughly equal? 21:17.150 --> 21:19.760 Yeah. Student: If he is 21:19.759 --> 21:21.819 randomizing that means that the returns will be equal. 21:21.819 --> 21:24.189 If they weren't equal he would just do one or the other all the 21:24.192 --> 21:25.812 time. Professor Ben Polak: 21:25.809 --> 21:28.969 Good, excellent. Since we're in a mixed strategy 21:28.974 --> 21:33.084 equilibrium, since he's randomizing, it must be the case 21:33.080 --> 21:35.320 that the returns are equal. 21:35.319 --> 21:38.199 That's the big idea here, that's the thing we learned 21:38.196 --> 21:39.496 last time. If the player, 21:39.495 --> 21:41.945 and these are professional baseball players doing this, 21:41.946 --> 21:45.216 they've been very well trained, a lot of money has been spent 21:45.215 --> 21:46.995 on getting the tactics right. 21:47.000 --> 21:49.910 There's people sitting there who are paid to get the tactics 21:49.914 --> 21:52.024 right. If it was the case that the 21:52.022 --> 21:55.832 return to base stealing wasn't roughly equal when you attempt 21:55.826 --> 21:58.296 to steal or didn't attempt to steal, 21:58.299 --> 22:00.229 then you shouldn't be randomizing. 22:00.230 --> 22:04.140 Since you are randomizing it must be the case that the 22:04.136 --> 22:06.196 returns are roughly equal. 22:06.200 --> 22:08.740 So that's the first thing to observe and the second thing to 22:08.742 --> 22:10.382 observe is what we just pointed out. 22:10.380 --> 22:15.010 In fact, the value of having a fast base stealer on the team 22:15.007 --> 22:19.947 doesn't show up in the expected return on the occasions on which 22:19.949 --> 22:23.449 he attempts to steal, or which he does not attempt to 22:23.451 --> 22:24.931 steal. It shows up where? 22:24.930 --> 22:28.240 It shows up in the fact that the pitching team changes their 22:28.236 --> 22:31.366 behavior to make it harder for this guy to steal by going 22:31.374 --> 22:34.104 faster to the plate, or throwing more fast balls. 22:34.099 --> 22:36.049 Where will that show up in the statistics? 22:36.049 --> 22:38.359 If you're just a statistician like me, you just look at the 22:38.363 --> 22:39.643 data, where will that show up? 22:39.640 --> 22:41.670 I mean suppose I can't keep track of every single pitch, 22:41.669 --> 22:43.439 I can't actually observe all these fast balls, 22:43.440 --> 22:46.710 where will I see the effect of all these extra fast balls in 22:46.707 --> 22:51.417 pitching and from the stretch, in the data? 22:51.420 --> 22:55.370 Somebody? It's going to show up in the 22:55.370 --> 23:00.180 batting average of the guy who's hitting behind the base stealer. 23:00.180 --> 23:02.760 The guy hitting behind the base stealer is going to have a 23:02.762 --> 23:05.482 higher batting average because he's going to get more pitches 23:05.480 --> 23:08.740 which are fast balls to hit, and more pitches out of the 23:08.744 --> 23:10.814 stretch. So if you ignore that effect, 23:10.814 --> 23:12.384 you're going to be in trouble. 23:12.380 --> 23:14.040 But we know, if we analyze this properly 23:14.043 --> 23:15.833 using Game Theory, we know we're in a mixed 23:15.833 --> 23:16.903 strategy equilibrium. 23:16.900 --> 23:19.350 We know, in fact, the pitching team must be 23:19.348 --> 23:22.458 reacting to it. We know there must be a cost in 23:22.463 --> 23:26.303 doing that, and the cost turns up in the hitter behind. 23:26.299 --> 23:28.509 So when you're watching the playoffs in the last-- now I'm 23:28.513 --> 23:30.613 giving you permission to watch a bit of TV at night, 23:30.609 --> 23:33.699 after you've done my homework assignment, but before anyone 23:33.704 --> 23:36.644 else's homework assignment--you can have a look at these 23:36.638 --> 23:39.888 baseball games and have a go at being a little bit better than 23:39.892 --> 23:42.402 the commentators who are working on them. 23:42.400 --> 23:45.990 So one application for mixed strategies is in sports, 23:45.989 --> 23:48.129 but not the only application. 23:48.130 --> 23:50.760 Let's just talk about another application, a slightly more 23:50.757 --> 23:51.677 scary application. 23:51.680 --> 23:55.640 So after 9/11 there was a lot of talk in the U.S. 23:55.640 --> 24:01.370 about the placement of baggage checking machines at airports. 24:01.369 --> 24:04.329 Actually there's still quite a lot of talk, but there was a lot 24:04.328 --> 24:07.048 of talk then about the placement of machines to search the 24:07.048 --> 24:08.478 luggage that goes onboard. 24:08.480 --> 24:10.720 The hand luggage was being searched anyway, 24:10.723 --> 24:13.183 but to search luggage going into the cabins. 24:13.180 --> 24:16.220 It was pointed out at the time, this has changed since, 24:16.218 --> 24:19.198 there weren't actually enough machines in the U.S., 24:19.200 --> 24:23.090 on the day after 9/11, to search every single bag that 24:23.089 --> 24:24.629 went into the hold. 24:24.630 --> 24:27.080 You'd hear discussions of the following type. 24:27.079 --> 24:31.019 You'd hear these experts on Nightline or whatever and they'd 24:31.020 --> 24:34.160 say: look there's no point trying to do this, 24:34.160 --> 24:38.080 because if we put all our baggage searching machines at 24:38.081 --> 24:40.711 Logan Airport in Boston, for example, 24:40.706 --> 24:44.506 then the terrorists will simply move their attack to O'Hare and 24:44.511 --> 24:47.571 if we put them at O'Hare, then they'll move their attack 24:47.565 --> 24:49.715 to Logan. If we have enough to do both 24:49.715 --> 24:53.365 Logan and O'Hare then they'll move their attack to some third 24:53.369 --> 24:55.729 airport. So there was a sense of doom in 24:55.729 --> 24:57.169 the air. It was kind of a depressing 24:57.171 --> 25:00.101 time anyway. There was a sense of doom in 25:00.095 --> 25:05.465 the air saying that if you put your baggage searching machines 25:05.465 --> 25:07.925 somewhere, all you do is cause the 25:07.933 --> 25:10.813 attempted terrorists, terrorists attempting to blow 25:10.811 --> 25:12.831 up the planes, to go elsewhere. 25:12.829 --> 25:15.089 And you hear the same things today about searching 25:15.094 --> 25:16.854 individuals as they go on the plane. 25:16.849 --> 25:19.869 For example, you'll hear a discussion that 25:19.870 --> 25:23.260 says, if we only search men traveling alone, 25:23.259 --> 25:27.779 let's say, then you'll quickly end up with all the people 25:27.780 --> 25:31.010 carrying bombs being couples or women. 25:31.009 --> 25:33.499 Again, there's this sense of doom, this sense that says it's 25:33.496 --> 25:35.456 hopeless. Whatever we do we're just going 25:35.459 --> 25:38.409 to force the terrorists to do something else but we won't have 25:38.410 --> 25:42.850 gained anything. So once again that's wrong. 25:42.850 --> 25:45.420 What's wrong about that one? 25:45.420 --> 25:47.100 What should we be doing in that setting? 25:47.100 --> 25:48.990 Let me come down again. 25:48.990 --> 25:53.090 What should they have done--in fact they did do--with those 25:53.092 --> 25:57.122 luggage/baggage searching machines when they were in short 25:57.123 --> 25:58.683 supply after 9/11? 25:58.680 --> 26:02.910 What do they do with searching people as they got on planes? 26:02.910 --> 26:08.410 What do they do? Well here's what they didn't 26:08.406 --> 26:11.136 do, they didn't just put them at certain airports and announce 26:11.144 --> 26:12.674 they're just at these airports. 26:12.670 --> 26:14.220 That would have been a crazy thing to do. 26:14.220 --> 26:17.340 That would have been hopeless--not entirely 26:17.341 --> 26:19.201 hopeless--but not wise. 26:19.200 --> 26:19.960 What should they have done? 26:19.960 --> 26:22.510 What did they do? 26:22.510 --> 26:26.590 Anybody want to guess? 26:26.590 --> 26:29.950 Yeah. Student: In name they 26:29.951 --> 26:31.291 randomized who they were checking. 26:31.289 --> 26:34.319 Professor Ben Polak: Right, so when they're checking 26:34.316 --> 26:37.286 passengers, they're going to randomly check passengers. 26:37.289 --> 26:39.969 When they're checking, when they think about the 26:39.965 --> 26:42.525 baggage machines, a sensible thing to do is to 26:42.527 --> 26:45.427 put a big metal box at every single airport and say: 26:45.431 --> 26:48.621 we're not going to tell you which of these boxes actually 26:48.619 --> 26:51.769 have baggage checking machines, which effectively is 26:51.767 --> 26:54.227 randomizing. From the point of view of the 26:54.230 --> 26:58.140 terrorists, they're not going to know where the baggage checks 26:58.143 --> 27:00.453 are going on. That's worth doing. 27:00.450 --> 27:02.570 It doesn't--It isn't going to perfectly eliminate, 27:02.572 --> 27:04.912 well unfortunately it isn't probably going to perfectly 27:04.910 --> 27:08.690 eliminate all terrorist attacks, but it does make it harder for 27:08.693 --> 27:11.703 the terrorists. So randomization there--whether 27:11.700 --> 27:14.520 it's literally randomizing over who is checked, 27:14.519 --> 27:18.349 or whether it's "as it were" randomizing, by concealing where 27:18.347 --> 27:22.427 in fact you have placed those machines--can be very effective. 27:22.430 --> 27:26.120 The hard thing, both in sports and in these 27:26.123 --> 27:29.383 military examples, is really mimicking 27:29.377 --> 27:32.817 randomization. It's very hard for us as humans 27:32.823 --> 27:35.953 to do it, and there's a famous story about a military 27:35.948 --> 27:38.708 commander, actually a English military 27:38.711 --> 27:43.121 commander during an insurgent war in, I think it was Malaysia 27:43.115 --> 27:46.415 after World War II, where again he had to worry 27:46.422 --> 27:49.182 about randomizing which convoys to protect. 27:49.180 --> 27:51.330 And the way in which…--He figured out 27:51.327 --> 27:54.317 that randomizing was the right thing to do to try and protect 27:54.323 --> 27:57.023 these convoys as well as he could with small numbers of 27:57.020 --> 27:58.640 troops. And the way in which he 27:58.639 --> 28:00.619 randomized was he literally randomized. 28:00.619 --> 28:04.229 Every morning he put a bit of paper in his hand and he had 28:04.227 --> 28:07.637 somebody, had one of his sergeants, pick which hand the 28:07.644 --> 28:10.324 paper was in. So we do actually see these 28:10.324 --> 28:11.754 random strategies used. 28:11.750 --> 28:14.970 The reason we have to literally randomize is because it's very 28:14.965 --> 28:18.335 difficult to do so unless you're a professional sports player. 28:18.339 --> 28:21.779 Okay, but it turns out that mixed strategy equilibria, 28:21.781 --> 28:23.991 and mixed strategies in general, 28:23.990 --> 28:27.520 are relevant beyond just these contexts in which you think of 28:27.517 --> 28:29.337 people literally randomizing. 28:29.339 --> 28:36.919 I want to look at a different context now. 28:36.920 --> 28:41.630 So I want to go back to a game we started a few weeks ago. 28:41.630 --> 28:42.580 This isn't the same game. 28:42.580 --> 28:44.710 It's a sequel. It's a follow up in our 28:44.714 --> 28:47.984 exciting adventure of our dating couple in the classroom. 28:47.980 --> 28:52.030 Who were our dating couple, do we still have them here? 28:52.029 --> 28:54.539 That was the guy, who is the--yeah, 28:54.536 --> 28:57.086 there they are. They're even sitting closer. 28:57.090 --> 28:59.980 What a success here. 28:59.980 --> 29:02.170 Can we get the camera on them a second? 29:02.170 --> 29:08.620 Stand up a second, thank you. 29:08.620 --> 29:11.810 Your name was? Student: David. 29:11.809 --> 29:16.199 Professor Ben Polak: David. 29:16.200 --> 29:16.800 And look at this. 29:16.800 --> 29:17.940 Is this romantic or what? 29:17.940 --> 29:20.080 David and your name is? 29:20.080 --> 29:20.980 Student: Nina. 29:20.980 --> 29:22.860 Professor Ben Polak: Nina and David, 29:22.857 --> 29:25.167 okay. I think we pretty much figured 29:25.168 --> 29:28.898 out last time that Nina's Player I and David's Player II, 29:28.897 --> 29:31.527 is that right? As we remember last time, 29:31.526 --> 29:34.936 I'll pick on you in a second, you can sit down a second. 29:34.940 --> 29:38.100 So we figured out last time that they were going to try and 29:38.097 --> 29:41.197 go on a date and they had arranged to go to the movies. 29:41.200 --> 29:43.520 They picked out two, in fact three movies, 29:43.520 --> 29:46.580 but two that remained viable, and the problem was being 29:46.576 --> 29:48.666 typical Economics majors who are, 29:48.670 --> 29:50.890 are you both Economics majors, I think we figured that out? 29:50.890 --> 29:52.860 They are, look at that, so being typical Economics 29:52.859 --> 29:55.229 majors who are just hopeless at dating they had forgotten to 29:55.230 --> 29:57.200 tell each other which movie they're going to. 29:57.200 --> 29:59.980 So that, I don't know if that worked out well or not, 29:59.977 --> 30:03.647 but now that life has moved on, they're going to try it again, 30:03.652 --> 30:07.162 but this time taking advantage of fall in New England, 30:07.160 --> 30:10.230 rather than go to a movie, they've decided on some new 30:10.227 --> 30:15.797 activities. So they might either go apple 30:15.796 --> 30:25.426 picking or they might go to the Yale Rep and see a play.. 30:25.430 --> 30:31.770 And so apple picking has its advantages: the fall weather, 30:31.768 --> 30:37.218 it's local flavor, it has certain undertones about 30:37.216 --> 30:41.216 the Garden of Eden or something. 30:41.220 --> 30:44.860 I don't know if you can use the term flavor, local or otherwise, 30:44.861 --> 30:47.001 for American apples but never mind. 30:47.000 --> 30:49.460 And the Yale Rep, Yale Rep is a good thing to do 30:49.460 --> 30:50.820 in New Haven, go to a play, 30:50.821 --> 30:53.701 I think it's Richard II is showing now, is that it? 30:53.700 --> 30:56.630 Probably not a great "date play" but Economists are trying 30:56.627 --> 30:58.987 to show they have culture, so there it goes. 30:58.990 --> 31:03.690 And let's assume the payoffs are like this. 31:03.690 --> 31:08.270 Much as they were before, whereby we mean that Nina wants 31:08.267 --> 31:12.747 to meet David but she would, given the choice she would 31:12.748 --> 31:15.788 rather meet David in the apple fields. 31:15.789 --> 31:19.239 And David who's a dark personality, likes the sort of 31:19.243 --> 31:21.173 darker side of Shakespeare. 31:21.170 --> 31:26.250 And he also wants to meet Nina but he would rather meet at the 31:26.253 --> 31:29.443 Yale Rep. If that's backwards I apologize 31:29.439 --> 31:31.049 to their preferences. 31:31.049 --> 31:33.639 But once again, because they're still 31:33.635 --> 31:37.945 incompetent Economics majors, they've again forgotten to tell 31:37.945 --> 31:40.455 each other where they're going. 31:40.460 --> 31:43.580 So let's analyze this game again, we've figured out this 31:43.575 --> 31:46.855 was a coordination game last time or several weeks ago., 31:46.859 --> 31:51.309 And we know in this game, we know what the pure strategy 31:51.309 --> 31:55.239 Nash Equilibria are, so no prizes to be able to spot 31:55.243 --> 31:58.953 them. One of the Nash Equilibria in 31:58.953 --> 32:03.273 pure strategies, let's put this in pure 32:03.269 --> 32:06.669 strategies, so one of the pure strategy 32:06.674 --> 32:10.914 Nash Equilibria is for them both to go apple picking and meet up 32:10.907 --> 32:16.647 in Bishop's Orchard or whatever, and another pure strategy 32:16.650 --> 32:23.160 equilibrium is for them both to choose the Rep. 32:23.160 --> 32:26.750 We'd figured out that if they were able to communicate, 32:26.753 --> 32:30.353 there's really a pretty good chance of them managing to 32:30.346 --> 32:34.136 coordinate at one of these equilibria but we suspect, 32:34.140 --> 32:37.040 I think, that this is not all that's going on here. 32:37.039 --> 32:40.829 It looks quite likely that come your next Saturday afternoon, 32:40.830 --> 32:43.610 when we send these guys out on their date, 32:43.609 --> 32:46.659 they're going to fail to meet, it's at least plausible. 32:46.660 --> 32:49.060 To test the plausibility of that, let's ask them, 32:49.059 --> 32:51.209 have you been, have you managed to meet on a 32:51.210 --> 32:52.890 date yet? No, haven't managed to meet on 32:52.886 --> 32:55.396 a date. See, so I'm proving the point 32:55.400 --> 32:59.650 that in fact they haven't managed to at least coordinate 32:59.650 --> 33:01.350 an equilibrium yet. 33:01.349 --> 33:04.519 So it seems at least plausible that they're going to fail to 33:04.521 --> 33:07.231 coordinate. It's plausible they're going to 33:07.231 --> 33:08.531 fail to coordinate. 33:08.529 --> 33:11.329 We'd like to sort of capture that idea, and the way we're 33:11.329 --> 33:14.229 going to capture that idea is--let's see if there's another 33:14.230 --> 33:15.680 equilibrium in this game. 33:15.680 --> 33:17.430 Well, there certainly isn't another pure strategy 33:17.434 --> 33:18.754 equilibrium in this game is there? 33:18.750 --> 33:21.670 We know that.. So if there's another 33:21.672 --> 33:23.582 equilibrium it better be mixed. 33:23.579 --> 33:31.729 So let's try and find a mixed Nash Equilibrium in this game, 33:31.734 --> 33:38.924 and remember this game is called Battle of the Sexes, 33:38.921 --> 33:42.101 it's a famous game. 33:42.099 --> 33:46.279 This is Battle of the Sexes revisited. 33:46.279 --> 33:49.939 So how are we going to go about finding this mixed Nash 33:49.938 --> 33:51.698 Equilibrium in the game? 33:51.700 --> 33:55.600 We'll interpret it later but let's just work on finding it. 33:55.599 --> 34:02.449 So, in particular, I'm going to postulate the idea 34:02.447 --> 34:10.687 that Nina is going to mix P, 1 - P and David is going to mix 34:10.693 --> 34:15.153 Q, 1 - Q. So how do we go about finding 34:15.148 --> 34:18.148 David's equilibrium mix Q, 1--Q? 34:18.150 --> 34:20.560 What's our trick from last week? 34:20.559 --> 34:22.979 Should be able to cold call at this point, but let's not have 34:22.978 --> 34:25.728 to. How am I going to find Q, 34:25.725 --> 34:27.655 the equilibrium Q? 34:27.660 --> 34:32.840 Somebody? Thank you, they can use Venus' 34:32.840 --> 34:36.310 payoffs, good. So to find--it isn't Venus' 34:36.306 --> 34:38.466 payoffs--it's Nina's payoffs. 34:38.470 --> 34:39.490 Fair enough, sorry. 34:39.489 --> 34:49.349 So to find the Nash Equilibrium Q, to find the mix that David's 34:49.349 --> 34:54.119 using we use Nina's payoffs. 34:54.120 --> 35:02.740 So let's do that. 35:02.739 --> 35:09.509 So, in particular, for Nina, if she goes apple 35:09.510 --> 35:18.390 picking then her payoff is 2 with probability Q if she meets 35:18.387 --> 35:22.297 David and 0 otherwise. 35:22.300 --> 35:29.860 If she goes to the Rep then her payoff is 1 if she meets David, 35:29.859 --> 35:35.589 sorry, need to be careful, let's do it again. 35:35.590 --> 35:39.550 If she goes to the Rep her payoff is 0 if David goes apple 35:39.546 --> 35:43.296 picking with probability Q, and her payoff is 1 if she 35:43.300 --> 35:46.850 meets David at the Rep, which happens with probability 35:46.847 --> 35:51.087 1 - Q, is that correct? 35:51.090 --> 35:53.800 So this is her payoff from apple picking and this is her 35:53.795 --> 35:55.365 payoff from seeing Richard II. 35:55.369 --> 36:00.879 And what do we know if Nina is indeed mixing, 36:00.880 --> 36:06.140 what do we know about these two payoffs? 36:06.140 --> 36:07.440 They must be equal. 36:07.440 --> 36:12.060 If Nina is in fact mixing, then these two things must be 36:12.058 --> 36:16.338 equal. And that means: 36:16.340 --> 36:29.100 what we're saying is 2Q equals 1(1-Q) or Q equals 2/3, 36:29.097 --> 36:41.977 I guess it is. No it's 1/3 sorry. 36:41.980 --> 36:44.380 Is that right? Q is 1/3. 36:44.380 --> 36:48.630 Okay, so our guess is that if there's a mixed strategy 36:48.630 --> 36:53.120 equilibrium it must be the case that David is assigning a 36:53.121 --> 36:56.571 probability 1/3 to going apple picking, 36:56.570 --> 37:02.570 which means he's assigning probability 2/3 to his more 37:02.566 --> 37:08.446 favored activity which is going to see Richard II. 37:08.449 --> 37:17.549 What about, I'm going to pull these both down, 37:17.552 --> 37:24.432 okay, how do we find Nina's mix? 37:24.429 --> 37:35.079 So to find the Nash Equilibrium P, to find Nina's mix what do we 37:35.084 --> 37:37.374 do? What's the trick? 37:37.370 --> 37:41.730 Somebody? Use David's payoffs. 37:41.730 --> 37:48.030 So David's payoffs, if he goes apple picking then 37:48.031 --> 37:55.781 he gets a payoff of 1 if he meets Nina there and 0 otherwise 37:55.776 --> 38:03.126 and if he goes to the Rep he gets a payoff of 0 if Nina's 38:03.128 --> 38:10.708 gone apple picking, and he gets a payoff of 2 if he 38:10.709 --> 38:13.899 meets Nina at the Rep. 38:13.900 --> 38:17.750 Once again, if David is indifferent it must be that 38:17.745 --> 38:20.915 these are equal. So if these are-- if David is 38:20.920 --> 38:24.620 in fact mixing between apple picking and going to the Rep--it 38:24.623 --> 38:28.573 must be that these two are equal and if we set this out carefully 38:28.574 --> 38:32.594 we'll get, let's just see, 38:32.594 --> 38:44.564 we'll get 1(P) equals 2(1-P), which is P equals 2/3 and 1-P 38:44.561 --> 38:50.581 equals 1/3. So here we have Nina assigning 38:50.584 --> 38:56.024 2/3 to going apple picking, which in fact is her more 38:56.018 --> 39:00.718 favored thing and 1/3 to going to the Rep. 39:00.719 --> 39:05.329 Okay, so we just used the same trick as last time, 39:05.332 --> 39:10.042 let's check that this is in fact an equilibrium. 39:10.039 --> 39:15.199 So, in particular, let's check that it is in fact 39:15.199 --> 39:19.929 an equilibrium for Nina to choose 2/3,1/3. 39:19.930 --> 39:28.480 Let's check. So check that P equals 2/3 is 39:28.484 --> 39:34.974 in fact the best response for Nina. 39:34.970 --> 39:38.020 Let's go back to Nina's payoffs. 39:38.019 --> 39:43.799 For Nina, if she chose to go apple picking, 39:43.804 --> 39:52.074 her payoff now is 2 times Q but Q is equal to 1/3 plus 0(1-Q) 39:52.067 --> 40:00.327 and if she chooses to go to the Rep then her payoff is 0 with 40:00.330 --> 40:07.630 probability 1/3 and 1 with probability now 2/3. 40:07.630 --> 40:13.010 All I've done is I've taken the lines I had before and 40:13.012 --> 40:19.112 substituted in now what we know must be the correct Q and 1-Q 40:19.105 --> 40:24.585 and this gives her a payoff of 2/3 in either case. 40:24.590 --> 40:30.720 If she chooses P, her payoff to P will be 2/3 of 40:30.722 --> 40:38.682 the time she'll get the payoff from apple picking which is 2/3 40:38.681 --> 40:46.771 and 1/3 of the time she'll get the payoff from going to the Rep 40:46.771 --> 40:51.731 which is 2/3 for a total of 2/3. 40:51.730 --> 40:55.330 So Nina's payoff from either of her pure strategies is 2/3. 40:55.329 --> 40:59.839 Her payoff from our claimed equilibrium mixed strategy is 40:59.835 --> 41:04.655 2/3, so neither of her possible pure strategy deviations were 41:04.663 --> 41:07.003 profitable. She didn't lose her anything 41:07.000 --> 41:09.260 either, but they weren't profitable, and by the lesson we 41:09.260 --> 41:12.320 started the class with, that means there cannot be any 41:12.318 --> 41:15.128 strictly profitable mixed deviation either, 41:15.129 --> 41:19.819 so indeed, for Nina, P is a best response 41:19.824 --> 41:22.754 to Q. We can do the same for David 41:22.753 --> 41:25.693 but let's not bother, it's symmetric. 41:25.690 --> 41:29.960 So in this game we found another equilibrium. 41:29.960 --> 41:38.040 The other equilibrium, the new equilibrium is Nina 41:38.043 --> 41:47.453 mixed 2/3,1/3 and David mixed 1/3,2/3 and we also know the 41:47.446 --> 41:52.886 payoff from this equilibrium. 41:52.889 --> 42:02.669 The equilibrium from this payoff, for both players, 42:02.673 --> 42:06.003 was 2/3. There are three equilibria in 42:06.000 --> 42:07.930 this game. They managed to meet at apple 42:07.931 --> 42:10.221 picking in which case the payoffs are 2 and 1. 42:10.219 --> 42:12.929 They managed to meet at the Rep, that's the second pure 42:12.934 --> 42:15.994 strategy equilibrium, in which case the payoffs are 1 42:15.993 --> 42:19.313 and 2, or they mixed, both of them mixed in this way, 42:19.310 --> 42:24.950 and their payoffs are 2/3,2/3. 42:24.949 --> 42:30.849 Why is the payoff so bad in this mixed strategy equilibrium? 42:30.849 --> 42:33.839 Does everyone agree, this is a pretty lousy payoff? 42:33.840 --> 42:37.390 The other equilibrium payoffs the worst you got was 1 and you 42:37.393 --> 42:40.043 sometimes got 2, but now here you are playing a 42:40.038 --> 42:42.878 different equilibrium and at this different equilibrium 42:42.879 --> 42:44.299 you're only getting 2/3. 42:44.300 --> 42:46.230 Why are you only getting--what happened? 42:46.230 --> 42:49.150 Why have these payoffs got pushed down so far? 42:49.150 --> 42:52.270 What's happening to our poor hapless couple? 42:52.270 --> 42:53.880 Or not hapless I don't know. 42:53.880 --> 43:00.990 What's happening to our couple? 43:00.989 --> 43:02.439 Student: Sometimes they don't meet. 43:02.440 --> 43:04.620 Professor Ben Polak: Yeah, they're failing to meet. 43:04.619 --> 43:08.169 The reason, what's forcing these payoffs down is they're 43:08.167 --> 43:09.777 not meeting very often? 43:09.780 --> 43:12.560 How often are they actually meeting? 43:12.560 --> 43:15.360 How often are they meeting? 43:15.360 --> 43:15.890 Let's have a look. 43:15.889 --> 43:25.829 Let's go back to the previous board. 43:25.829 --> 43:31.409 Here it is..So they meet when they end up in this box or this 43:31.407 --> 43:33.357 box, is that right? 43:33.360 --> 43:35.430 So what's the probability of them ending in those boxes? 43:35.429 --> 43:42.979 Well ending up in this box is probability 2/3,1/3 and ending 43:42.975 --> 43:50.005 up in this box is probability 1/3,2/3, is that right? 43:50.010 --> 43:53.650 You end up meeting apple picking, the 2/3 of the time 43:53.645 --> 43:57.905 when Nina goes there times the 1/3 of the time when David goes 43:57.910 --> 44:00.530 there. And you end up meeting at the 44:00.526 --> 44:04.426 Rep the 1/3 of the time Nina goes there times the 2/3 of the 44:04.426 --> 44:06.406 time that David goes there. 44:06.409 --> 44:22.429 So this is the total probability of meeting and it's 44:22.434 --> 44:31.864 equal to 4/9, is that right? 44:31.860 --> 44:35.960 So 4/9 of the time they're meeting, but 5/9 of the 44:35.964 --> 44:40.404 time--more than half the time--they're screwing up and 44:40.404 --> 44:45.464 failing to meet. This is why I call them a 44:45.460 --> 44:48.570 hapless dating couple. 44:48.570 --> 44:51.680 So this is a very bad equilibrium, but it captures 44:51.678 --> 44:54.278 something which is true about the game. 44:54.280 --> 44:57.400 What is surely true about this game is that if they just played 44:57.397 --> 44:59.657 this game, they wouldn't meet all the time. 44:59.659 --> 45:03.889 In fact what we're arguing here is they'd meet less than half of 45:03.891 --> 45:05.941 the time. But certainly this idea that 45:05.942 --> 45:08.052 we're given from the pure strategy equilibria, 45:08.051 --> 45:10.721 that they would magically always manage to meet seems very 45:10.723 --> 45:13.073 unlikely, so this does seem to add a 45:13.073 --> 45:16.463 little bit of realism to this analysis of the game. 45:16.460 --> 45:20.810 However, it leads to a bit of an interpretation problem. 45:20.809 --> 45:24.279 You might ask the question why on Earth are they randomizing in 45:24.275 --> 45:26.375 this way. Why are they doing this? 45:26.380 --> 45:27.520 It's bad for everybody. 45:27.520 --> 45:31.800 Why are they doing this? 45:31.800 --> 45:35.880 This leads us to think about a second interpretation for what 45:35.881 --> 45:38.671 we think mixed strategy equilibria are. 45:38.670 --> 45:42.430 Rather than thinking of them literally as randomizing, 45:42.427 --> 45:46.037 it's probably better in this case to think about the 45:46.043 --> 45:50.333 following idea. We need to think about David's 45:50.329 --> 45:56.209 mixture as being a statement about what Nina believes David's 45:56.210 --> 45:59.910 going to do. David may not be literally 45:59.913 --> 46:02.663 randomizing. But his mixture Q, 46:02.657 --> 46:07.897 1--Q, we could think of as Nina's belief about what David's 46:07.903 --> 46:11.693 going to do. Conversely, Nina may not 46:11.693 --> 46:14.333 literally be randomizing. 46:14.329 --> 46:19.049 But her P, 1 - P, we could think of as David's 46:19.049 --> 46:23.139 belief about what Nina's going to do. 46:23.139 --> 46:28.539 And what we've done is we've found the beliefs such that 46:28.542 --> 46:34.242 these players are exactly indifferent over what they do. 46:34.239 --> 46:37.789 We found the beliefs for David over what Nina's going to do, 46:37.790 --> 46:41.100 such that David doesn't really quite know what to do. 46:41.099 --> 46:44.649 And we found the beliefs that Nina holds about what David's 46:44.654 --> 46:48.274 going to do such that Nina doesn't quite know what to do. 46:48.270 --> 46:54.270 That make sense? So it's probably better here to 46:54.270 --> 46:58.950 think about this not as people literally randomizing but these 46:58.954 --> 47:02.874 mixed strategies being a statement about what people 47:02.870 --> 47:05.020 believe in equilibrium. 47:05.019 --> 47:08.589 We'll come back and look at this game some more later on, 47:08.593 --> 47:12.553 so our couple I'm afraid are not quite out of the woods yet. 47:12.550 --> 47:15.610 But I want to spend the rest of today looking at yet another 47:15.606 --> 47:17.986 interpretation of mixed strategy equilibria. 47:17.989 --> 47:22.529 So, so far we have two, we have people are literally 47:22.525 --> 47:25.355 randomizing. We have thinking of these as 47:25.358 --> 47:28.648 expressions about what people believe in equilibrium rather 47:28.650 --> 47:30.750 than what they're literally doing. 47:30.750 --> 47:44.660 And now I'm going to give you a third interpretation. 47:44.659 --> 47:55.639 So for now we can get rid of the Venus and Serena game. 47:55.639 --> 48:01.059 So to motivate this third idea I want to think about tax 48:01.062 --> 48:03.372 audits. So none of you here have ever, 48:03.369 --> 48:05.309 probably ever, had to fill out a tax form, 48:05.308 --> 48:08.188 except for the fact that there seems to be a lot of parents in 48:08.193 --> 48:10.273 the room today, is it parents weekend, 48:10.272 --> 48:11.562 is that what's going on? 48:11.559 --> 48:13.109 So where are the parents in the room? 48:13.110 --> 48:16.060 Wave your arms in the air if you're a parent here. 48:16.059 --> 48:19.149 So at least these guys at probably some point in their 48:19.149 --> 48:20.839 life filled out a tax form. 48:20.840 --> 48:23.980 So come tax day, the parents in the room face a 48:23.977 --> 48:28.067 choice, and the choice is are they going to honestly fill out 48:28.069 --> 48:31.069 their taxes, or are they going to cheat? 48:31.070 --> 48:35.700 I'm not going to ask them what they, well maybe I will, 48:35.700 --> 48:39.560 but for now I won't ask them what they did. 48:39.559 --> 48:42.059 So they can choose one of two things. 48:42.059 --> 48:46.339 They can choose to pay their taxes honestly--we'll call that 48:46.337 --> 48:51.727 H--or to cheat. This is the tax payer, 48:51.730 --> 48:56.220 the parent. And at the same time the audit 48:56.216 --> 48:59.246 office, the auditor, has to make a choice, 48:59.245 --> 49:03.525 and the auditor's choice is whether to audit you or not and 49:03.531 --> 49:08.041 it's not literally true because literally the auditor can wait 49:08.038 --> 49:12.618 until your tax return comes in and then decide whether to audit 49:12.619 --> 49:15.349 you. But for now let's think of 49:15.353 --> 49:18.013 these choices being made simultaneously, 49:18.013 --> 49:21.563 and we'll see why that makes it more interesting. 49:21.559 --> 49:27.849 So let me put down some payoffs here and then I'll explain them. 49:27.850 --> 49:36.410 So 2,0, 4, -10,4, 0 and 0,4. 49:36.410 --> 49:37.720 So how do we interpret this? 49:37.719 --> 49:40.509 Let's look at the auditor's payoffs first of all. 49:40.510 --> 49:45.480 So the auditor is very happy not having to audit your parents 49:45.480 --> 49:49.950 and having your parents pay taxes, so we'll give that a 49:49.954 --> 49:52.384 payoff of 4. It'll turn out, 49:52.375 --> 49:55.435 in this game, we've decided in the payoffs, 49:55.444 --> 49:59.464 that the auditor is equally happy if she actually audits 49:59.463 --> 50:02.973 your parents in the year that they cheated. 50:02.969 --> 50:07.639 We'll say that makes the auditor equally happy. 50:07.639 --> 50:13.619 Now the auditor is not so happy if she audits your parents when 50:13.615 --> 50:17.755 they're honest because audits are costly. 50:17.760 --> 50:24.110 The auditor is really unhappy if she fails to audit when the 50:24.112 --> 50:26.982 parents cheated. Let's look at the--I keep 50:26.982 --> 50:29.032 wanting to call them parents--I should stop calling them 50:29.033 --> 50:30.453 parents, let's call them taxpayers. 50:30.449 --> 50:33.009 So for the taxpayers, what are their payoffs? 50:33.010 --> 50:37.800 Well we'll normalize things, so if they're honest we'll give 50:37.796 --> 50:39.496 them a payoff of 0. 50:39.500 --> 50:42.050 That means they correctly fill in their tax form and pay what 50:42.053 --> 50:44.823 they're supposed to pay, but if they can conceal some of 50:44.815 --> 50:47.055 their income, they pretend to have whatever 50:47.059 --> 50:48.669 it is, a third child, 50:48.673 --> 50:52.723 then they might be in trouble if they're audited. 50:52.719 --> 50:56.059 If they're audited they're going to have to pay a big fine, 50:56.056 --> 50:58.296 maybe even go to jail, so that's -10. 50:58.300 --> 51:02.190 Of course if they're not audited they get to keep a chunk 51:02.194 --> 51:04.424 of money so we'll call that 4. 51:04.420 --> 51:07.420 Everyone understand the basic idea of this game? 51:07.420 --> 51:09.230 In reality, we could add more complications, 51:09.233 --> 51:11.683 we could think of different ways to cheat on your taxes, 51:11.679 --> 51:14.729 but I don't want to give tutorials on how to cheat on 51:14.725 --> 51:17.605 your taxes here. So it's not going to take long 51:17.614 --> 51:21.044 staring at this game to figure out that there are no pure 51:21.038 --> 51:23.238 strategy equilibria in this game. 51:23.239 --> 51:27.029 Let's just do that, so from the taxpayer's point of 51:27.032 --> 51:29.992 view, if they're going to be audited, 51:29.989 --> 51:33.709 then they'd rather pay their taxes than not, 51:33.714 --> 51:39.004 and if they're not going to be audited then according to these 51:38.998 --> 51:41.768 payoffs they'd rather cheat. 51:41.769 --> 51:45.099 From the auditor's point of view, if they knew everyone was 51:45.095 --> 51:48.265 going to pay taxes, then they wouldn't bother 51:48.265 --> 51:52.585 auditing and if they knew everyone was going to cheat, 51:52.590 --> 51:57.330 then they'd of course audit. 51:57.329 --> 52:01.489 So you can quickly see that there's no box in which the best 52:01.489 --> 52:04.869 responses coincide, there's no pure strategy Nash 52:04.872 --> 52:07.192 Equilibria. For those people who are 52:07.185 --> 52:10.265 thinking this is seeming other worldly, you will have to pay 52:10.270 --> 52:14.920 taxes in a couple of years, and trust me your parents are 52:14.918 --> 52:16.828 paying taxes now. 52:16.829 --> 52:20.139 So what we want to do here is we're going to solve out and 52:20.139 --> 52:22.229 find a mixed strategy equilibrium, 52:22.230 --> 52:25.320 but we're going to give it a different interpretation to the 52:25.320 --> 52:26.840 equilibria we found so far. 52:26.840 --> 52:29.180 But the basic, initial exercise is what? 52:29.179 --> 52:34.409 We're going to find--we're going to try and find the 52:34.411 --> 52:36.361 equilibrium here. 52:36.360 --> 52:41.280 So to find the Nash Equilibrium here we know it's going to be 52:41.283 --> 52:44.513 mixed. So to find the probability with 52:44.509 --> 52:49.149 which taxpayers pay their taxes--and let me already start 52:49.151 --> 52:54.211 getting ahead of myself and just say to find the proportion of 52:54.207 --> 52:59.427 taxpayers who are going to pay their taxes--what do we do? 52:59.429 --> 53:03.729 What must be true of that equilibrium proportion Q of 53:03.732 --> 53:06.382 taxpayers who pay their taxes? 53:06.380 --> 53:08.790 How am I going to find that Q? 53:08.790 --> 53:11.650 Shout it out somebody. 53:11.650 --> 53:13.870 Yeah look at the auditor's payoffs. 53:13.869 --> 53:24.139 So from the auditor's point of view, if the auditor audits, 53:24.135 --> 53:34.575 their payoff is 2Q plus 4(1-Q) and if they don't audit their 53:34.577 --> 53:39.707 payoff is 4Q plus 0(1-Q). 53:39.710 --> 53:44.570 Everyone see how I do this, this is 2Q plus 4(1-Q) and this 53:44.573 --> 53:46.253 is 4Q plus 0(1-Q). 53:46.250 --> 53:53.840 And if indeed the auditor is mixing, then these must be 53:53.836 --> 53:59.616 equal. And if they're equal, 53:59.619 --> 54:13.149 let's just do a little bit of algebra here and we'll find that 54:13.154 --> 54:24.474 2Q equals 4(1-Q) so Q equals 2/3, is that right? 54:24.469 --> 54:28.239 So our claim is to make the auditor exactly indifferent 54:28.237 --> 54:33.257 between whether to audit or not, it must be the case that 2/3 of 54:33.257 --> 54:38.207 the parents of the kids in the room, are going to be paying 54:38.210 --> 54:41.580 their taxes honestly, which means 1/3 aren't, 54:41.580 --> 54:44.050 which is kind of worrying, but never mind. 54:44.050 --> 54:46.280 Let's have a look at the taxpayer. 54:46.280 --> 54:51.340 To find, sorry. We found the taxpayer, 54:51.342 --> 54:53.942 we found the proportion of taxpayers who are paying their 54:53.943 --> 54:56.413 taxes, now I want to find out the probability of being 54:56.406 --> 54:58.866 audited. How do I figure out the 54:58.870 --> 55:03.400 equilibrium probability of being audited in this model? 55:03.400 --> 55:08.230 How do I work out the equilibrium probability of being 55:08.228 --> 55:11.498 audited? Shout it out. 55:11.500 --> 55:15.550 So the equilibrium probability of being audited are going to 55:15.553 --> 55:18.103 use P and 1-P, so P is going to be the 55:18.095 --> 55:21.525 probability of being audited, how do I find P? 55:21.530 --> 55:23.990 Yeah, I'm going to look at the taxpayer's payoffs. 55:23.989 --> 55:34.849 So from the taxpayer's point of view, if the taxpayer pays their 55:34.852 --> 55:44.342 taxes, their payoff is just 0, and if they cheat they're 55:44.335 --> 55:49.675 payoff is -10P plus 4(1-P). 55:49.679 --> 55:53.639 And if indeed the taxpayers are mixing--or in other words, 55:53.639 --> 55:57.529 we are saying that not all taxpayers are cheating and not 55:57.530 --> 56:01.420 all taxpayers are honestly paying their taxes--then these 56:01.420 --> 56:07.130 must be equal. So if these are equal I'm going 56:07.131 --> 56:14.521 to get 4P equals 14--no it didn't-- I'm going to get 4 56:14.518 --> 56:17.998 equals 14P, let's try again, 56:18.000 --> 56:21.540 4 equals 14P, that was a bit worrying, 56:21.540 --> 56:25.560 4 equals 14P, which is the same as saying P 56:25.559 --> 56:29.629 equals 2/7. If somebody can just check my 56:29.626 --> 56:32.246 algebra I think that's right. 56:32.250 --> 56:38.100 So my claim is that the equilibrium here is for 2/3 of 56:38.101 --> 56:43.731 the taxpayers to pay their taxes and for the audits, 56:43.732 --> 56:48.372 the auditor, to audit 2/7 of the time. 56:48.369 --> 56:50.939 Now we could go back in here and we could check, 56:50.942 --> 56:54.432 I could do what I did before, I could plug the Ps and Qs in 56:54.426 --> 56:57.626 here and check that in fact this is an equilibrium, 56:57.630 --> 57:02.550 but trust me that I've done that, trust me that it's okay. 57:02.550 --> 57:06.470 So here we have an equilibrium, let's just write down what it 57:06.473 --> 57:09.393 is. From the auditor's point of 57:09.385 --> 57:14.745 view it is that they audit 2/7 of the time, or 2/7 of the 57:14.749 --> 57:18.899 population, and from the taxpayers' point 57:18.897 --> 57:24.507 of view, it's that they pay their taxes honestly 2/3 of the 57:24.512 --> 57:27.032 time and not otherwise. 57:27.030 --> 57:31.150 Now without focusing too much on these exact numbers for a 57:31.150 --> 57:35.050 second, I want to focus first for a minute on how do we 57:35.054 --> 57:38.384 interpret this mixed strategy equilibrium. 57:38.380 --> 57:41.860 So from the point of view of the auditor we're really back 57:41.862 --> 57:45.592 where we were before with the base stealer or the person who's 57:45.589 --> 57:47.849 searching baggage at the airport. 57:47.849 --> 57:53.199 We could think of the auditor literally as randomizing. 57:53.199 --> 57:54.729 In fact, there's some truth to that. 57:54.730 --> 57:59.030 It actually is the case that by law, that the auditor's 57:59.030 --> 58:01.420 literally have to randomize. 58:01.420 --> 58:05.470 So this 2/7,5/7 this has the same interpretation as we had 58:05.470 --> 58:09.930 before. This is really a randomization. 58:09.929 --> 58:13.759 But this 2/3,1/3 has a different interpretation and a 58:13.760 --> 58:16.560 potentially exciting interpretation. 58:16.559 --> 58:20.899 It isn't that we think that your parents get to tax day, 58:20.896 --> 58:25.466 work out what their taxes would be and then toss a coin. 58:25.469 --> 58:27.709 They may be doing that, I'm looking at the parents and 58:27.712 --> 58:29.492 I don't think that's what they're doing. 58:29.489 --> 58:32.419 The interpretation here is that the parents, some parents are 58:32.422 --> 58:35.062 paying their taxes and some parents aren't paying their 58:35.061 --> 58:37.081 taxes. There's a lot of parents out 58:37.080 --> 58:39.590 there, a lot of potential taxpayers, and in the 58:39.594 --> 58:41.424 population, in equilibrium, 58:41.420 --> 58:45.520 if these numbers were true, 2/3, of parents would be paying 58:45.521 --> 58:48.421 their taxes and 1/3 would be cheating. 58:48.420 --> 58:56.050 So this is a randomization by a player, and this is a mixture in 58:56.050 --> 58:59.850 the population. The new interpretation here is, 58:59.853 --> 59:02.673 we could think of the mixed strategy not as players 59:02.666 --> 59:04.756 randomizing, but as a mix in a large 59:04.755 --> 59:07.895 population of which some people are doing one thing and the 59:07.895 --> 59:09.785 other group are doing the other. 59:09.789 --> 59:14.949 It's a proportion of people paying taxes. 59:14.949 --> 59:17.799 So I don't know if this 2/3,1/3 is an accurate number for the 59:17.804 --> 59:19.784 U.S. It's probably not very far off 59:19.777 --> 59:22.397 actually. For Italy I'm ashamed to say 59:22.403 --> 59:26.313 the number of people who pay taxes is more like 40%, 59:26.309 --> 59:29.749 maybe even lower now, and there are countries I think 59:29.749 --> 59:31.799 where it gets as high as 90%. 59:31.800 --> 59:35.800 I think the U.S. rate when they end up auditing 59:35.800 --> 59:40.100 is a little higher than this but not much. 59:40.099 --> 59:45.049 So again, we're going to think of this not as randomization but 59:45.051 --> 59:49.361 as a prediction of the proportion of American taxpayers 59:49.364 --> 59:52.324 who are going to pay their taxes. 59:52.320 --> 59:56.380 Now, I want to use this example in the time we have left, 59:56.377 --> 59:59.707 to actually think about a policy experiment. 59:59.710 --> 1:00:09.600 So let's put this up somewhere we can see it. 1:00:09.599 --> 1:00:13.109 Let's think about a new tax policy. 1:00:13.110 --> 1:00:16.790 So suppose that Congress gets fed up with all these newspaper 1:00:16.787 --> 1:00:20.217 reports about how 2/3 of American's don't pay their taxes 1:00:20.219 --> 1:00:22.609 or whatever the true proportion is, 1:00:22.610 --> 1:00:24.360 I think it's actually a little higher than that but never mind. 1:00:24.360 --> 1:00:28.510 They get fed up with all these reports and they say, 1:00:28.507 --> 1:00:32.327 this isn't fair, we should make people pay their 1:00:32.330 --> 1:00:36.560 taxes so we're going to change the law and instead of 1:00:36.559 --> 1:00:40.869 paying--instead of being in jail for ten years, 1:00:40.869 --> 1:00:46.549 or the equivalent of a fine of -10 if you're caught cheating, 1:00:46.554 --> 1:00:52.334 we're going to raise the fine or the time in jail so that it's 1:00:52.333 --> 1:00:57.263 now -20. So the policy experiment is 1:00:57.259 --> 1:01:05.819 let's raise the fine--to fine the cheating--to -20 and the aim 1:01:05.821 --> 1:01:12.841 of this policy is try to deter cheating, right? 1:01:12.840 --> 1:01:18.600 It seems a plausible thing for a government to want to do. 1:01:18.599 --> 1:01:28.459 Let's redraw the matrix, so here's the game - 2,0, 1:01:28.455 --> 1:01:36.495 4, -20,4, 0,0, 4 audit, not audit and pay 1:01:36.501 --> 1:01:41.531 honestly or cheating. 1:01:41.530 --> 1:01:44.770 So here's our new payoffs and let's ask the question, 1:01:44.770 --> 1:01:48.470 with this new fine in place, now we've raised the fine, 1:01:48.467 --> 1:01:52.537 to being caught not paying your taxes, in the long run once 1:01:52.543 --> 1:01:56.763 things have worked their way back into equilibrium again, 1:01:56.760 --> 1:02:01.100 after a few years, do we expect American taxpaying 1:02:01.095 --> 1:02:04.275 compliance to go up or to go down, 1:02:04.280 --> 1:02:06.830 or what do we expect? 1:02:06.829 --> 1:02:07.949 What do we think is going to happen? 1:02:07.949 --> 1:02:11.099 So who thinks it's going to go up? 1:02:11.099 --> 1:02:13.489 Who thinks it's going to go down? 1:02:13.489 --> 1:02:16.119 Who thinks it's going to say the same? 1:02:16.120 --> 1:02:17.680 Who's abstaining here? 1:02:17.679 --> 1:02:18.859 I notice the parents are abstaining. 1:02:18.860 --> 1:02:20.080 u're not really meant to abstain. 1:02:20.080 --> 1:02:22.620 You have to vote here. 1:02:22.619 --> 1:02:24.199 Well how are we going to figure this out? 1:02:24.199 --> 1:02:27.119 How are we going to figure out what's going to happen to 1:02:27.119 --> 1:02:38.099 compliance? What happens to tax compliance? 1:02:38.099 --> 1:02:43.409 Tax compliance, remember that was our P--no it 1:02:43.409 --> 1:02:46.949 wasn't, sorry, it was our Q. 1:02:46.949 --> 1:02:51.839 The only way we're going to figure this out is to work out, 1:02:51.843 --> 1:02:55.643 so let's work out the new Q in equilibrium. 1:02:55.639 --> 1:02:57.779 Let's do this, so to find out the new Q in 1:02:57.778 --> 1:03:00.588 equilibrium, once again, we're going to have to look at 1:03:00.594 --> 1:03:04.944 the auditor's payoffs, and the auditor's payoffs if 1:03:04.942 --> 1:03:10.182 they audit, they're going to get 2Q plus 4(1-Q), 1:03:10.179 --> 1:03:18.709 and if they don't audit they're going to get 4Q plus 0(1-Q), 1:03:18.707 --> 1:03:23.907 and if the auditor is indifferent, 1:03:23.909 --> 1:03:26.719 if they're mixing, it must still be the case that 1:03:26.717 --> 1:03:29.347 these are equal. And now I want to ask you a 1:03:29.352 --> 1:03:32.402 question, where have you seen that equation before? 1:03:32.400 --> 1:03:34.810 Yeah, it's still there right, I didn't delete it. 1:03:34.809 --> 1:03:40.029 It's the same equation that sits up there. 1:03:40.030 --> 1:03:43.790 Is that right? From the auditor's point of 1:03:43.787 --> 1:03:48.027 view, given the payoffs to the auditors nothing has changed, 1:03:48.030 --> 1:03:52.050 so the tax compliance rate that makes the auditor exactly 1:03:52.053 --> 1:03:56.153 indifferent between auditing your parents and not auditing 1:03:56.148 --> 1:04:01.208 your parents, is still exactly the same as it 1:04:01.213 --> 1:04:03.573 was before at 2/3. 1:04:03.570 --> 1:04:09.460 In equilibrium, tax compliance hasn't changed 1:04:09.456 --> 1:04:12.746 at all. Let me say that again, 1:04:12.751 --> 1:04:17.981 the policy was we're going to double the fines for being 1:04:17.978 --> 1:04:22.918 caught cheating and in equilibrium it made absolutely 1:04:22.921 --> 1:04:29.481 no difference whatsoever to the equilibrium tax compliance rate. 1:04:29.480 --> 1:04:32.950 Now why did it make no difference? 1:04:32.949 --> 1:04:35.469 Well let's have a techie answer and then a better, 1:04:35.465 --> 1:04:36.795 a more intuitive answer. 1:04:36.800 --> 1:04:42.360 The techie answer is this, what determines the equilibrium 1:04:42.363 --> 1:04:47.343 tax compliance rate, what determines the equilibrium 1:04:47.341 --> 1:04:51.051 mix for the column player is what? 1:04:51.050 --> 1:04:55.230 Is the row's payoffs. 1:04:55.230 --> 1:04:58.810 What determines the equilibrium mix for the column player are 1:04:58.806 --> 1:05:01.306 the row's payoffs--row player's payoffs. 1:05:01.309 --> 1:05:05.099 We didn't change the row player's payoffs, 1:05:05.104 --> 1:05:10.194 so we're not going to change the equilibrium mix for the 1:05:10.194 --> 1:05:13.814 column player. Say again, we changed one of 1:05:13.814 --> 1:05:17.724 the payoffs for the column player but the column player's 1:05:17.722 --> 1:05:21.702 equilibrium mix depends on the row player's payoffs and we 1:05:21.699 --> 1:05:24.909 haven't changed the row player's payoffs, 1:05:24.909 --> 1:05:28.959 so we won't change the equilibrium compliance rate, 1:05:28.963 --> 1:05:32.453 the equilibrium mix by the column player. 1:05:32.450 --> 1:05:35.160 What will have changed here? 1:05:35.159 --> 1:05:39.149 What will have changed in the new equilibrium? 1:05:39.150 --> 1:05:41.320 So we've pretty much established that people are 1:05:41.323 --> 1:05:43.823 cheating as much as they were before in equilibrium. 1:05:43.820 --> 1:05:45.320 Rahul, can I get Henry here? 1:05:45.320 --> 1:05:47.050 Student: Probability has changed. 1:05:47.050 --> 1:05:47.590 Professor Ben Polak: Say again. 1:05:47.590 --> 1:05:48.950 Student: The probability of audit would have changed. 1:05:48.949 --> 1:05:50.379 Professor Ben Polak: The probability of audit will have 1:05:50.384 --> 1:05:52.644 changed. What's going to change is not 1:05:52.640 --> 1:05:55.570 the Q but the P, the probability with which 1:05:55.566 --> 1:05:59.116 you're audited is going to change in this model. 1:05:59.119 --> 1:06:02.399 Let's just check it, to find the new P, 1:06:02.401 --> 1:06:06.461 I need to look at the taxpayer's payoffs and the 1:06:06.460 --> 1:06:10.960 taxpayer's payoffs are now 0, –sorry, 1:06:10.962 --> 1:06:17.062 if they pay their taxes honestly then they get 0, 1:06:17.062 --> 1:06:24.562 and if they cheat they get -20 with probability P and 4 with 1:06:24.561 --> 1:06:27.991 probability 1-P. If they're mixing, 1:06:27.986 --> 1:06:30.126 if some of them are paying and some of them are not, 1:06:30.129 --> 1:06:35.659 this must be the same, and I'm being more careful than 1:06:35.664 --> 1:06:43.334 I was last time I hope, this gives me 24 P is equal to 1:06:43.331 --> 1:06:46.371 4 or P equals 1/6. 1:06:46.369 --> 1:06:53.409 So the audit rate has gone down from 2/7 to 1/6. 1:06:53.409 --> 1:06:56.489 I'm guessing that probably wasn't the goal of the policy 1:06:56.487 --> 1:06:58.947 although it isn't necessarily a bad thing. 1:06:58.949 --> 1:07:00.969 There is some benefit for society here, 1:07:00.966 --> 1:07:03.936 because audits are costly, both to do for the auditor and 1:07:03.937 --> 1:07:05.897 they're unpleasant to be audited, 1:07:05.900 --> 1:07:10.960 so the fact that we've managed to lower the audit rate from 2/7 1:07:10.959 --> 1:07:18.699 to 1/6 is a good thing, but we didn't manage to raise 1:07:18.695 --> 1:07:22.815 the compliance rate. 1:07:22.820 --> 1:07:25.560 So I don't want to take this model too literally, 1:07:25.556 --> 1:07:28.656 because it's just a toy model, but nevertheless, 1:07:28.657 --> 1:07:32.637 let's try and draw out some lessons from this model. 1:07:32.639 --> 1:07:37.359 So here what we did was we changed the payoff to cheating, 1:07:37.364 --> 1:07:38.944 we made it worse. 1:07:38.940 --> 1:07:42.990 But a different kind of change is we could have changed, 1:07:42.991 --> 1:07:47.041 sorry,- we changed the payoff negatively to being caught 1:07:47.042 --> 1:07:49.812 cheating. But a different change we could 1:07:49.811 --> 1:07:53.321 have done is we could have left the -10 in place and we could 1:07:53.316 --> 1:07:56.876 have raised the payoff to cheating and not getting caught. 1:07:56.880 --> 1:08:01.900 We could have left this 10 in place and changed this 4 let's 1:08:01.902 --> 1:08:03.692 say to a 6 or an 8. 1:08:03.690 --> 1:08:08.020 We've increased the benefits to cheating if you're not caught. 1:08:08.019 --> 1:08:13.229 What would that have done in equilibrium? 1:08:13.230 --> 1:08:16.920 So I claim, once again, that would have done nothing in 1:08:16.916 --> 1:08:21.146 equilibrium to the probability of people paying their taxes, 1:08:21.149 --> 1:08:25.149 but that would have done what to the audit rate? 1:08:25.149 --> 1:08:28.139 The audit rate would have gone up, the equilibrium audit rate 1:08:28.142 --> 1:08:29.192 would have gone up. 1:08:29.190 --> 1:08:31.390 Let's tell that story a second. 1:08:31.390 --> 1:08:33.930 So rich people, people who are well paid, 1:08:33.928 --> 1:08:37.798 have a little bit more to gain from cheating on their taxes if 1:08:37.800 --> 1:08:41.310 they're not caught, there's more money at stake. 1:08:41.310 --> 1:08:44.100 So my colleagues who are finance professors in the 1:08:44.095 --> 1:08:47.785 business school have more money on their tax returns than I do, 1:08:47.789 --> 1:08:52.169 so in principle, they gain more if they cheat. 1:08:52.170 --> 1:08:56.910 Does that mean that they cheat more than me in equilibrium? 1:08:56.909 --> 1:08:58.429 No, it doesn't mean that they cheat more than me in 1:08:58.425 --> 1:09:01.315 equilibrium. What does it mean? 1:09:01.319 --> 1:09:04.909 It means they get audited more often. 1:09:04.909 --> 1:09:07.479 In equilibrium, richer people aren't 1:09:07.482 --> 1:09:11.672 necessarily going to cheat more, but they are going to get 1:09:11.671 --> 1:09:14.171 audited more, and that's true. 1:09:14.170 --> 1:09:17.620 The federal audit rates are designed so they audit the rich 1:09:17.617 --> 1:09:19.517 more than they audit the poor. 1:09:19.520 --> 1:09:22.000 Again, it's not because they think the rich are inherently 1:09:22.000 --> 1:09:24.350 less honest, or the poor are inherently more honest, 1:09:24.350 --> 1:09:26.500 or anything like that, it's simply that the gains to 1:09:26.504 --> 1:09:29.044 cheating and not getting caught are bigger if you're rich, 1:09:29.039 --> 1:09:35.039 so you need to audit more to push back into equilibrium. 1:09:35.039 --> 1:09:39.829 Now, suppose we did in fact want to use the policy of 1:09:39.827 --> 1:09:45.167 raising fines to push down, to push up the compliance rate, 1:09:45.166 --> 1:09:47.556 to push down cheating. 1:09:47.560 --> 1:09:50.640 How would we change the law? 1:09:50.640 --> 1:09:53.460 Suppose we want to raise the fines for cheating, 1:09:53.455 --> 1:09:56.685 we don't like people cheating so we raise the fines, 1:09:56.689 --> 1:10:00.079 but we're worried about this result that didn't push up 1:10:00.078 --> 1:10:03.028 compliance rates, how could we change the law or 1:10:03.027 --> 1:10:06.667 change the incentives in the game so that it actually would 1:10:06.667 --> 1:10:08.547 change compliance rates? 1:10:08.550 --> 1:10:11.330 What could we do? 1:10:11.330 --> 1:10:16.720 Yeah. Student: If we changed 1:10:16.720 --> 1:10:20.620 the payoffs of auditing to 4, from 2 to 4. 1:10:20.619 --> 1:10:24.359 Professor Ben Polak: Good, if we want to change the 1:10:24.358 --> 1:10:27.698 compliance rates we should change the payoffs to the 1:10:27.703 --> 1:10:30.513 auditor. The problem with the way the 1:10:30.511 --> 1:10:34.841 auditor is paid here is that the auditor is paid more if they 1:10:34.842 --> 1:10:38.382 manage to catch people, but audits are costly. 1:10:38.380 --> 1:10:40.870 The problem with that is when you raise the fine on the other 1:10:40.865 --> 1:10:43.265 side, all that happens is the auditor's audit less often in 1:10:43.267 --> 1:10:45.597 equilibrium. So if you want to get a higher 1:10:45.598 --> 1:10:47.878 compliance rate, one thing you could do is 1:10:47.878 --> 1:10:50.878 change the payoffs to the auditor to make auditing less 1:10:50.882 --> 1:10:54.132 costly for them, or making catching people nicer 1:10:54.127 --> 1:10:58.147 for them, give them a reward, or you could simply take it out 1:10:58.146 --> 1:11:00.086 of Game Theory altogether. 1:11:00.090 --> 1:11:03.230 You could enforce, you could have congressional 1:11:03.232 --> 1:11:06.992 law that sets the audit rates outside of equilibrium, 1:11:06.989 --> 1:11:09.799 and that's been much discussed in Congress over the last five 1:11:09.804 --> 1:11:11.554 years. Somebody setting audit rates, 1:11:11.550 --> 1:11:13.300 as it were, exogenously by Congress. 1:11:13.300 --> 1:11:15.200 Why might that not be a great idea? 1:11:15.199 --> 1:11:17.909 Leaving aside Economic Theory for a second, 1:11:17.908 --> 1:11:21.328 leaving aside Game Theory, why might it not be a great 1:11:21.326 --> 1:11:24.996 idea to have Congress set the audit rates rather than some 1:11:25.001 --> 1:11:27.151 office? Student: Most members in 1:11:27.154 --> 1:11:29.824 Congress have a lot of money so they're going to lower the audit 1:11:29.824 --> 1:11:31.524 rates so that they don't get audited. 1:11:31.520 --> 1:11:33.450 Professor Ben Polak: So the lady in the front is saying 1:11:33.454 --> 1:11:34.854 that a lot of Congressmen are rather rich, 1:11:34.850 --> 1:11:36.330 so maybe they have particular incentives here. 1:11:36.329 --> 1:11:39.209 I don't want to take a particular political stance 1:11:39.205 --> 1:11:42.725 here, but it could be whatever side of the political spectrum 1:11:42.726 --> 1:11:45.676 you guys sit on, it could be that you might not 1:11:45.676 --> 1:11:47.686 trust Congress to get this right. 1:11:47.689 --> 1:11:50.949 You might think they're going to be political considerations 1:11:50.951 --> 1:11:54.211 going on in Congress other than just having an efficient tax 1:11:54.212 --> 1:11:56.852 system. Okay, so what do I want to draw 1:11:56.846 --> 1:11:58.216 out as lessons here? 1:11:58.220 --> 1:12:01.820 The big lessons from this class are there are three different 1:12:01.824 --> 1:12:04.834 ways to think about randomization in equilibrium or 1:12:04.828 --> 1:12:06.208 out of equilibrium. 1:12:06.210 --> 1:12:08.100 One is it's genuinely randomization, 1:12:08.099 --> 1:12:10.529 another is it could be something about peoples 1:12:10.528 --> 1:12:12.918 belief's, and a third way and a very 1:12:12.920 --> 1:12:16.690 important way is it could be telling us something about the 1:12:16.692 --> 1:12:20.532 proportion of people who are doing something in society, 1:12:20.529 --> 1:12:26.479 in this case the proportion of people who are paying tax. 1:12:26.479 --> 1:12:29.609 A second important lesson I want to draw out here, 1:12:29.605 --> 1:12:33.565 beyond just finding equilibria, two other things we drew out 1:12:33.572 --> 1:12:37.232 today, one lesson was when you're checking equilibria, 1:12:37.229 --> 1:12:41.089 checking mixed strategy equilibria, you only have to 1:12:41.089 --> 1:12:43.889 check for pure strategy deviations. 1:12:43.890 --> 1:12:47.020 Be careful, you have to check for all possible pure strategy 1:12:47.023 --> 1:12:50.263 deviations, not just the pure strategies that were involved in 1:12:50.262 --> 1:12:52.892 the mix. If the guy has seven strategies 1:12:52.888 --> 1:12:56.418 and is only mixing on two, you have to remember to check 1:12:56.415 --> 1:12:59.095 the other five. The third lesson I want to draw 1:12:59.100 --> 1:13:01.570 out today is because of the way equilibria works, 1:13:01.569 --> 1:13:06.719 mixed strategy equilibria work, if I change the column player's 1:13:06.717 --> 1:13:11.197 payoffs it changes the row player's equilibrium mix, 1:13:11.199 --> 1:13:14.219 and if I change the row player's payoffs, 1:13:14.215 --> 1:13:17.905 it changes the column player's equilibrium mix. 1:13:17.909 --> 1:13:20.939 Next time, we're going to pick up this idea that mixed 1:13:20.939 --> 1:13:23.909 strategies can be about proportions of people playing 1:13:23.912 --> 1:13:27.002 things and take it to a totally different setting, 1:13:27.000 --> 1:13:28.740 namely evolution. 1:13:28.739 --> 1:13:31.999 So on Wednesday we'll start talking about evolution.