WEBVTT 00:01.540 --> 00:05.360 Professor Ben Polak: So this is what we did last time: 00:05.358 --> 00:09.238 we looked at a game involving an entrant and an incumbent in a 00:09.240 --> 00:11.980 market; and the entrant had to decide 00:11.978 --> 00:14.578 whether to enter that market or not; 00:14.580 --> 00:17.730 and if they stayed out the incumbent remained a monopolist; 00:17.730 --> 00:20.850 and the monopolist made 3 million in profit. 00:20.850 --> 00:23.950 If the entrant goes in, then the incumbent can decide 00:23.952 --> 00:27.112 whether to accommodate the entrant and just settle for 00:27.114 --> 00:29.684 duopoly profits, making a million each; 00:29.680 --> 00:33.350 or the incumbent can fight, in which case the incumbent 00:33.354 --> 00:37.714 makes no money at all and the entrant loses a million dollars. 00:37.710 --> 00:40.060 We pointed out a number of things about this game. 00:40.060 --> 00:44.480 One was that when we analyzed it in a matrix form we quickly 00:44.483 --> 00:48.313 found that there were two Nash Equilibria, that Nash 00:48.306 --> 00:51.226 Equilibrium were: in and not fight; 00:51.230 --> 00:53.740 and out and fight. 00:53.740 --> 00:57.200 But we argued that backward induction tells us that the 00:57.197 --> 00:59.627 sensible answer is in and not fight. 00:59.630 --> 01:03.380 Once the incumbent knows the entrant is in they're not going 01:03.380 --> 01:06.620 to fight because 1 is bigger than 0, and the entrant 01:06.622 --> 01:08.722 anticipating this will enter. 01:08.719 --> 01:11.479 When we talked a little bit more we said this other 01:11.482 --> 01:14.142 equilibrium, this out fight equilibrium--it is an 01:14.135 --> 01:17.335 equilibrium because if the entrant believes the incumbent's 01:17.339 --> 01:20.599 going to fight then the entrant is going to stay out, 01:20.599 --> 01:24.399 and it's costless for the incumbent to "fight" if in fact 01:24.395 --> 01:28.055 the entrant does stay out because they never get called 01:28.055 --> 01:29.745 upon to fight anyway. 01:29.750 --> 01:33.700 So the idea of this was that for the incumbent to say they're 01:33.699 --> 01:36.529 going to fight is an "incredible threat." 01:36.530 --> 01:38.030 That's terrible English. 01:38.030 --> 01:39.580 It's the way it is always taught in the textbooks. 01:39.580 --> 01:43.010 It needs to be called a "not credible" threat. 01:43.010 --> 01:45.070 And that "not credible" threat is: he's not really going to 01:45.072 --> 01:47.242 fight if the entrant comes in, and therefore, 01:47.240 --> 01:51.360 the entrant should come in and in fact the incumbent will 01:51.364 --> 01:54.064 accommodate it. So what we've shown here is 01:54.060 --> 01:56.850 that, if we believe this argument, then the entrant will 01:56.851 --> 01:59.491 come in and the incumbent is going to let him in. 01:59.489 --> 02:02.689 At the end, we started talking about this in a slightly more 02:02.685 --> 02:05.115 elaborate setting, so let's just remind you of 02:05.123 --> 02:07.293 what that more elaborate setting is. 02:07.290 --> 02:12.300 The more elaborate setting is suppose that there is one firm, 02:12.300 --> 02:15.810 one monopolist, and that monopolist holds a 02:15.806 --> 02:18.976 monopoly in ten different markets. 02:18.979 --> 02:21.799 So we'll have our monopolist be Ale. 02:21.800 --> 02:24.560 So here's Ale. He's our monopolist, 02:24.561 --> 02:28.831 and he owns pizzeria monopolies in ten different markets. 02:28.830 --> 02:31.590 And each of these ten different markets are separate, 02:31.594 --> 02:33.034 they are different towns. 02:33.030 --> 02:36.130 And in each of those ten markets he thinks--he knows he's 02:36.125 --> 02:39.435 going to face an entrant and those entrants are going to come 02:39.441 --> 02:40.961 in order. So let's just talk about who 02:40.958 --> 02:41.868 those entrants are going to be. 02:41.870 --> 02:44.720 The entrants are going to be this person, this person and so 02:44.718 --> 02:45.908 on. Let's find out who they are, 02:45.912 --> 02:47.002 so your name is? Student: Isabella 02:47.000 --> 02:47.820 Professor Ben Polak: Where are you from? 02:47.819 --> 02:48.479 Student: Miami. 02:48.479 --> 02:50.159 Professor Ben Polak: Miami, so Miami is one of the 02:50.160 --> 02:51.240 markets. Your name is? 02:51.244 --> 02:52.584 Student: Scott. 02:52.580 --> 02:53.530 Professor Ben Polak: From where? 02:53.525 --> 02:54.165 Student: Wisconsin. 02:54.171 --> 02:55.441 Professor Ben Polak: Where in Wisconsin? 02:55.440 --> 02:56.150 Student: Madison. 02:56.151 --> 02:57.871 Professor Ben Polak: Madison: we've got two towns. 02:57.870 --> 02:58.890 We're just going to do towns now. 02:58.889 --> 02:59.799 Student: My name is Lang. 02:59.801 --> 03:00.831 I'm from Bridgeport, Connecticut. 03:00.830 --> 03:02.320 Professor Ben Polak: Okay, we've got three towns. 03:02.319 --> 03:03.249 Student: I'm from Miami too. 03:03.253 --> 03:04.793 Professor Ben Polak: Talk about Yale diversity. 03:04.789 --> 03:06.229 Well we'll pretend you're from somewhere else. 03:06.229 --> 03:08.119 Put him in New Orleans or something. 03:08.120 --> 03:08.890 Student: Chris from Boston. 03:08.887 --> 03:10.117 Professor Ben Polak: From Boston, all right. 03:10.120 --> 03:11.350 Student: From Orange, Connecticut. 03:11.351 --> 03:13.171 Professor Ben Polak: From Orange, Connecticut so just 03:13.167 --> 03:14.177 down the road. Student: St. 03:14.182 --> 03:15.532 Louis, Missouri. Professor Ben Polak: All 03:15.527 --> 03:16.127 right, have we got ten yet? 03:16.130 --> 03:16.740 I'm not quite at ten. 03:16.740 --> 03:19.600 One, two, three, four, five, six, 03:19.604 --> 03:20.764 seven. Student: Saffron, 03:20.757 --> 03:21.487 New York. Professor Ben Polak: All 03:21.486 --> 03:22.296 right. Student: Hong Kong. 03:22.300 --> 03:23.700 Professor Ben Polak: Hong Kong, that's way away. 03:23.699 --> 03:24.329 Student: Long Island. 03:24.333 --> 03:25.263 Professor Ben Polak: Long Island. 03:25.259 --> 03:27.189 I think I've probably got ten markets here. 03:27.190 --> 03:29.060 So Ale owns a pizza shop. 03:29.060 --> 03:32.160 He's the monopoly pizza shop owner in each of these ten 03:32.156 --> 03:34.086 markets. And what we're going to see is 03:34.089 --> 03:36.269 we're going to see what happens as, sequentially, 03:36.265 --> 03:37.665 these entrants try to enter. 03:37.669 --> 03:40.199 The way that this game's going to work is that they're lined 03:40.202 --> 03:42.952 up--we know the order in which the entrants are going to come. 03:42.949 --> 03:44.649 They're going to start off, the first person who is going 03:44.653 --> 03:46.233 to have to make a decision is- Student: Enter. 03:46.234 --> 03:47.934 Isabella. Professor Ben Polak: Is 03:47.925 --> 03:50.695 Isabella, right? We're going to see how our 03:50.701 --> 03:52.351 monopolist responds. 03:52.350 --> 03:55.220 So let's have a look at this. 03:55.220 --> 03:56.790 So Isabella who is in which market again? 03:56.790 --> 03:57.050 Student: Miami. 03:57.050 --> 03:58.220 Professor Ben Polak: In Miami, okay what are you going 03:58.223 --> 03:59.073 to do? Student: Enter. 03:59.074 --> 04:00.784 Professor Ben Polak: What are you going to do? 04:00.780 --> 04:01.930 Student: I will fight. 04:01.930 --> 04:04.450 Professor Ben Polak: Oh dear, so you owe me a million 04:04.452 --> 04:06.682 dollars. So one person's down a million 04:06.684 --> 04:08.974 dollars, let's see what happens next. 04:08.969 --> 04:10.569 Student: I'm going to stay out. 04:10.574 --> 04:12.834 Professor Ben Polak: Which market was this? 04:12.830 --> 04:14.050 Student: Scott. Madison. 04:14.050 --> 04:15.430 Professor Ben Polak: Madison stayed out. 04:15.430 --> 04:16.280 Student: I'm going to stay out. 04:16.282 --> 04:17.252 Professor Ben Polak: Staying out. 04:17.250 --> 04:18.450 So Bridgeport stayed out. 04:18.449 --> 04:19.729 Student: I guess I'll stay out. 04:19.730 --> 04:20.730 Professor Ben Polak: Stayed out again. 04:20.730 --> 04:21.420 Student: Stay out. 04:21.420 --> 04:23.430 Professor Ben Polak: Which market are we up to now, 04:23.426 --> 04:24.796 somewhere in Orange County wasn't it? 04:24.800 --> 04:26.940 Where were we? Student Orange, 04:26.938 --> 04:28.378 Connecticut. Stay out. 04:28.379 --> 04:30.079 Professor Ben Polak: Stay out. 04:30.079 --> 04:31.869 Student: I think I'll stay in. 04:31.870 --> 04:33.030 Professor Ben Polak: You'll come in okay, 04:33.032 --> 04:33.702 and which market is this? 04:33.699 --> 04:34.229 Student: St. 04:34.225 --> 04:35.515 Louis, Missouri. Professor Ben Polak: St. 04:35.524 --> 04:38.054 Louis, Missouri. So you owe me a million dollars 04:38.051 --> 04:40.061 as well, okay. A couple of million dollars is 04:40.056 --> 04:41.376 a good class. We're going to have plenty of 04:41.378 --> 04:42.878 money for lunch. Student: I'm also going 04:42.882 --> 04:43.972 to fight. Professor Ben Polak: 04:43.970 --> 04:45.220 You're going to fight, which market is that? 04:45.220 --> 04:46.060 Student: Saffron, New York. 04:46.060 --> 04:46.850 Professor Ben Polak: Where abouts? 04:46.850 --> 04:47.630 Student: Saffron. 04:47.629 --> 04:49.019 Professor Ben Polak: Saffron, New York. 04:49.020 --> 04:51.040 Where are we at? One, two, three, 04:51.043 --> 04:53.863 four, five, six, seven, eight, 04:53.860 --> 04:55.500 Ale? TA: Fight. 04:55.500 --> 04:56.460 Professor Ben Polak: You fight? 04:56.461 --> 04:57.451 So you owe me a million dollars too. 04:57.450 --> 04:59.360 That was eight, nine? 04:59.360 --> 04:59.690 Student: Out. 04:59.690 --> 05:01.590 Professor Ben Polak: Out and ten? 05:01.590 --> 05:02.240 Student: Stay out. 05:02.240 --> 05:03.020 Professor Ben Polak: Stays out, okay. 05:03.019 --> 05:06.679 Now let's just notice something here, which was the tenth 05:06.681 --> 05:09.111 market? What town were you? 05:09.110 --> 05:09.640 Student: Long Island. 05:09.639 --> 05:11.179 Professor Ben Polak: Whereabouts in Long Island? 05:11.180 --> 05:11.580 Student: Huntington. 05:11.579 --> 05:13.379 Professor Ben Polak: So if Huntington, 05:13.382 --> 05:15.922 Long Island our last market had come in, suppose you'd said in, 05:15.922 --> 05:17.112 what would Ale have said? 05:17.110 --> 05:18.830 TA: I would not have fought. 05:18.829 --> 05:20.339 Professor Ben Polak: Would not have fought, 05:20.344 --> 05:22.804 aha! Okay, so what happened here? 05:22.800 --> 05:26.750 When we analyzed this last time as an individual market, 05:26.752 --> 05:30.992 we argued that each entrant should come in just as our first 05:30.992 --> 05:35.262 entrant came in, and our monopolist should not 05:35.259 --> 05:36.939 fight:. That's what we have up on the 05:36.942 --> 05:38.282 board. That's what backward induction 05:38.277 --> 05:41.137 suggests. But in fact, Ale fought. 05:41.139 --> 05:44.879 A whole bunch of people came in, and a whole bunch of them 05:44.881 --> 05:46.721 stayed out, is that right? 05:46.720 --> 05:48.460 Now why? Why was Ale fighting these guys 05:48.461 --> 05:49.891 and why were they staying out? 05:49.889 --> 05:52.449 Let's talk about why, and what market were you again? 05:52.449 --> 05:53.799 Student: Madison, Wisconsin. 05:53.800 --> 05:55.080 Professor Ben Polak: So why did Madison, 05:55.083 --> 05:55.673 Wisconsin stay out? 05:55.670 --> 05:58.310 Student: Well we talked about it last time how he has an 05:58.310 --> 06:00.700 incentive to fight now because there's more that just our 06:00.695 --> 06:03.285 analysis up there in terms of establishing that he'll fight to 06:03.293 --> 06:05.283 keep people out. Professor Ben Polak: All 06:05.276 --> 06:07.056 right, so it looks like there might be some reason for 06:07.064 --> 06:08.014 fighting to keep you out. 06:08.009 --> 06:09.739 So let's just talk about it a little bit more, 06:09.736 --> 06:10.806 let's go to the third guy. 06:10.810 --> 06:11.640 Which market are you again? 06:11.639 --> 06:12.489 Student: Bridgeport. 06:12.491 --> 06:14.321 Professor Ben Polak: Bridgeport, so why did you stay 06:14.319 --> 06:15.619 out? Student: Because I knew 06:15.621 --> 06:16.451 he was going to fight. 06:16.449 --> 06:17.219 Professor Ben Polak: You knew he was going to fight. 06:17.220 --> 06:19.260 Now how did you know he was going to fight? 06:19.259 --> 06:23.629 Student: Because he has an incentive to establish, 06:23.631 --> 06:28.081 he established that he was going to fight for every single 06:28.081 --> 06:31.361 market and so I was going to lose out. 06:31.360 --> 06:35.030 Professor Ben Polak: All right, so we know--we think we 06:35.029 --> 06:38.519 know that Ale is--you know he's this tough Italian pizzeria 06:38.519 --> 06:41.829 owner and we think he's going to try and establish what: 06:41.828 --> 06:45.498 a reputation as being a tough pizzeria owner by fighting these 06:45.497 --> 06:47.687 guys, perhaps fighting a few guys 06:47.692 --> 06:49.892 early on in order to keep these guys out. 06:49.889 --> 06:54.009 In fact, he had to fight the first person but he kept out 06:54.006 --> 06:58.046 2,3, 4,5, 6 and this person came in, so 7 and 8 came in, 06:58.050 --> 07:00.550 but then 9 and 10 he kept out. 07:00.550 --> 07:03.810 So he kept a lot of people out of the market by fighting early 07:03.812 --> 07:06.182 on. Now this argument sounds right: 07:06.181 --> 07:07.741 it seems to ring true. 07:07.740 --> 07:10.490 It's about establishing reputation, but now I want to 07:10.486 --> 07:13.176 show you that there's a worry with this argument. 07:13.180 --> 07:16.640 The worry is this is a sequential game and like all 07:16.643 --> 07:20.523 sequential games of perfect information we've seen in the 07:20.521 --> 07:23.501 class, we should analyze this game how? 07:23.500 --> 07:25.570 Now that wasn't loud enough, how? 07:25.570 --> 07:27.320 Backward induction. 07:27.319 --> 07:28.469 So where's the back? 07:28.472 --> 07:30.262 Where's the back of this game? 07:30.259 --> 07:34.179 Way back here--sorry for the guys in the balcony. 07:34.180 --> 07:38.080 Way back here we have the last market in town--which was the 07:38.083 --> 07:41.023 last market? And if we look at this last 07:41.016 --> 07:45.026 market, we in fact saw that if the last market came in, 07:45.030 --> 07:46.740 Ale in fact gave in. 07:46.740 --> 07:49.420 Ale gave in. Now why did Ale give in, 07:49.416 --> 07:50.426 in the last market? 07:50.430 --> 07:53.160 Let's have a look back on the board. 07:53.160 --> 07:57.930 So on the board we can see what that last market looks like. 07:57.930 --> 08:01.220 With ten markets this is a very complicated game. 08:01.220 --> 08:04.020 This would be the first market, and then there's three versions 08:04.017 --> 08:06.627 of the second market depending on what Ale did in the first 08:06.633 --> 08:08.133 market, and so there's nine versions of 08:08.134 --> 08:10.074 the third one. The tree for this game is 08:10.065 --> 08:11.515 horrendous. But nevertheless, 08:11.522 --> 08:14.122 once we get to the end of the game, the tenth market--which 08:14.122 --> 08:15.712 was what? Bridgeport or something, 08:15.710 --> 08:18.030 I've forgotten where it was at now--wherever it was. 08:18.029 --> 08:22.029 Once we get to that last market this tree pretty well describes 08:22.034 --> 08:24.364 that last market--is that correct? 08:24.360 --> 08:27.700 There isn't another market afterwards. 08:27.700 --> 08:28.730 There's only ten markets. 08:28.730 --> 08:32.790 So, in this last market, what do we know Ale's going to 08:32.793 --> 08:34.663 do? In this last market if the 08:34.659 --> 08:38.209 entrant enters Ale is going to not fight, which is what exactly 08:38.211 --> 08:40.291 what Ale did do. So Ale is that right? 08:40.289 --> 08:43.499 So when in fact we discussed the tenth guy coming in, 08:43.503 --> 08:46.413 you chose to? TA: I would have chosen 08:46.414 --> 08:47.844 not to fight. Professor Ben Polak: 08:47.843 --> 08:48.443 Would have chosen not to fight. 08:48.440 --> 08:50.800 That's exactly what the model predicts. 08:50.799 --> 08:53.639 He has no incentive to establish a reputation for the 08:53.643 --> 08:56.763 eleventh market because there isn't an eleventh market. 08:56.760 --> 08:58.290 He's done at ten. 08:58.290 --> 09:01.250 Is that right? So we know that in the last 09:01.250 --> 09:05.550 market, the tenth market, Ale actually is not going to 09:05.549 --> 09:08.659 fight. Therefore, the person who's the 09:08.657 --> 09:13.177 entrant in the tenth market should know that they can safely 09:13.182 --> 09:15.792 enter and Ale won't fight them. 09:15.790 --> 09:16.890 But now we're in trouble. 09:16.890 --> 09:18.270 Why are we in trouble? 09:18.269 --> 09:21.339 Well let's go back to the ninth market, the second to last 09:21.335 --> 09:22.595 market. So I've forgotten where it was. 09:22.600 --> 09:24.180 Raise your hand, the second to the last market. 09:24.179 --> 09:26.919 Okay, the second to last market is this guy? 09:26.920 --> 09:28.560 You're the tenth market? 09:28.559 --> 09:32.869 So this guy who is in the Hong Kong market, he should know he's 09:32.865 --> 09:34.805 the second to last market. 09:34.809 --> 09:39.319 He knows that no matter what he does the tenth market's going to 09:39.316 --> 09:43.246 enter and Ale's going to give in to the tenth market. 09:43.250 --> 09:45.490 Ale's going to let the tenth entrant in. 09:45.490 --> 09:47.870 Is that right? So the ninth market actually 09:47.874 --> 09:51.474 knows that nothing Ale's going to do here is going to establish 09:51.472 --> 09:53.912 a reputation to keep the tenth guy out. 09:53.909 --> 09:55.889 So therefore in fact, he should what? 09:55.890 --> 09:57.240 He should come in, right? 09:57.240 --> 09:59.020 He should come in, and in fact if he had come in, 09:59.021 --> 10:01.251 Ale would have had to give in because there's no way that Ale 10:01.249 --> 10:02.399 can keep the tenth guy out. 10:02.399 --> 10:07.049 We just argued the tenth guy's coming in by backward induction. 10:07.049 --> 10:08.659 So since we know that the tenth guy's coming in anyway, 10:08.664 --> 10:10.044 and in fact, Ale's going to concede to them, 10:10.039 --> 10:13.619 there's no point Ale trying to scare off the tenth guy. 10:13.620 --> 10:20.340 So in fact, Ale's going to say no fight to the ninth guy. 10:20.340 --> 10:22.120 But now we go to the eighth guy. 10:22.120 --> 10:24.670 We've just argued that the tenth guy's coming in anyway and 10:24.672 --> 10:26.082 Ale's going to give in to him. 10:26.080 --> 10:27.840 We've argued the ninth guy's coming in, so Ale's going to 10:27.844 --> 10:29.614 give in to this guy as well because you can't put off the 10:29.608 --> 10:31.578 tenth guy. And therefore we know that once 10:31.579 --> 10:34.289 we get to the eighth guy, once again, he can safely come 10:34.294 --> 10:37.214 in because Ale knows by backward induction he can't keep the 10:37.205 --> 10:39.175 ninth and the tenth guy out anyway, 10:39.179 --> 10:41.509 and so this guy should come in as well, and if we do this 10:41.513 --> 10:47.853 argument all the way back, what do we get? 10:47.850 --> 10:48.370 He lets everybody in. 10:48.370 --> 10:52.670 Everybody should come in and he should let everybody in. 10:52.670 --> 10:54.230 So we have a problem here. 10:54.230 --> 10:55.240 We have a problem. 10:55.240 --> 10:58.980 Backward induction says, even with these ten markets, 10:58.978 --> 11:01.708 Ale in fact should let everybody in. 11:01.710 --> 11:05.130 Everyone should know that, so they should come in. 11:05.130 --> 11:06.770 So there's a disconnect here. 11:06.769 --> 11:09.579 There's a disconnect between what the theory is telling 11:09.579 --> 11:12.549 us--backward induction is telling us Ale can't keep people 11:12.545 --> 11:15.525 out by threatening to fight, by establishing a reputation 11:15.531 --> 11:17.891 --, and what we actually just saw, what happened, 11:17.889 --> 11:20.949 which was Ale did fight and did keep people out, 11:20.951 --> 11:24.471 and we know that other monopolist's do that as well. 11:24.470 --> 11:29.920 So how can we make rigorous this idea of reputation? 11:29.919 --> 11:32.479 It's not captured by what we've done so far in the class. 11:32.480 --> 11:35.790 So how can we bring back what must be true in some sense--it's 11:35.788 --> 11:37.498 intuition that, by fighting, 11:37.502 --> 11:41.602 Ale could keep people out and therefore will keep people out. 11:41.600 --> 11:47.010 So to make that idea work I want to introduce a new idea. 11:47.009 --> 11:51.119 And the new idea is that, with very small probability, 11:51.124 --> 11:53.924 let's say 1% chance, Ale is crazy. 11:53.919 --> 11:57.139 So stand up a second, so he looks like a normal kind 11:57.139 --> 12:00.989 of guy but there's just 1% chance that he's really bonkers. 12:00.990 --> 12:04.970 There's a 1% chance that he's actually Rahul. 12:04.970 --> 12:10.460 So now let's redo the analysis--And what do I mean by 12:10.455 --> 12:12.805 bonkers? By bonkers, I mean, 12:12.813 --> 12:16.383 with 1%, Ale is the kind of guy who likes to fight. 12:16.379 --> 12:19.349 So with 1% chance, he's actually not got these 12:19.348 --> 12:21.868 payoffs at all; he's actually got some 12:21.872 --> 12:24.872 different payoffs, which are payoffs of somebody 12:24.868 --> 12:28.628 who--okay he'll lose money--but he so much enjoys a fight he 12:28.628 --> 12:31.588 gets +10 here. That's the bonkers guy's payoff. 12:31.590 --> 12:35.080 But there's only 1% chance he's this bonkers guy. 12:35.080 --> 12:36.650 So now what happens? 12:36.650 --> 12:38.200 Let's just walk it through. 12:38.200 --> 12:42.070 With 1% chance, if there was only one market, 12:42.073 --> 12:44.013 not the ten markets. 12:44.009 --> 12:46.799 So there's only one market and this one market was--I've 12:46.802 --> 12:47.922 forgotten your name? 12:47.919 --> 12:48.249 Student: Isabella. 12:48.246 --> 12:48.936 Professor Ben Polak: Isabella, who was in which 12:48.939 --> 12:49.239 market, I've forgotten. 12:49.240 --> 12:49.580 Student: Miami. 12:49.580 --> 12:52.970 Professor Ben Polak: In Miami, then she doesn't really 12:52.973 --> 12:56.313 much care about the 1% chance that Ale is actually Rahul. 12:56.309 --> 12:58.149 That doesn't really bother her very much, why? 12:58.149 --> 13:01.899 Because with 99% chance Ale's going to give way and that's 13:01.899 --> 13:03.149 good enough odds. 13:03.149 --> 13:06.119 With 99% chance she's happy to come in. 13:06.120 --> 13:09.000 So if there was only one market here, we'd be done. 13:09.000 --> 13:12.020 But with ten markets things are a little different. 13:12.020 --> 13:13.230 Why? Let's see why. 13:13.230 --> 13:17.240 So suppose in fact that Isabella in Miami thinks that 13:17.238 --> 13:21.708 Ale--and everybody else thinks Ale is a pretty sane guy. 13:21.710 --> 13:25.670 With 99% probability he's a sane guy, and Isabella enters 13:25.671 --> 13:27.441 and everyone sees this. 13:27.440 --> 13:31.380 And to everyone's surprise, rather than doing the sane 13:31.383 --> 13:36.153 thing, which is letting Isabella in and switching to a duopoly in 13:36.145 --> 13:38.475 Miami, what happens, 13:38.481 --> 13:44.971 in fact, after Isabella comes in is that Ale fights. 13:44.970 --> 13:48.150 Okay, so now it's too late for Isabella, she's lost her money, 13:48.145 --> 13:50.795 but our second market is, what's your name again? 13:50.800 --> 13:51.380 Student: Scott. 13:51.379 --> 13:52.649 Professor Ben Polak: Scott, which market were you? 13:52.650 --> 13:53.460 Student: Madison. 13:53.460 --> 13:56.130 Professor Ben Polak: So Scott in Madison has observed 13:56.133 --> 13:58.633 what happened in Miami and initially he thought that Ale 13:58.626 --> 14:01.326 was Ale. 99% probability Ale was this 14:01.329 --> 14:03.719 sane, nice, calm, Italian guy. 14:03.720 --> 14:07.150 But on the other hand, he just saw this sane, 14:07.148 --> 14:11.118 calm, Italian guy fight, as he shouldn't have fought 14:11.121 --> 14:15.101 because of backward induction--fought the entrant in 14:15.095 --> 14:18.375 Miami. So now Scott thinks to himself: 14:18.376 --> 14:22.006 hmm, I'm not so sure that Ale is this sane guy. 14:22.009 --> 14:24.769 Maybe I should update my beliefs in the direction of 14:24.765 --> 14:27.625 thinking that Ale might actually be the insane guy. 14:27.629 --> 14:31.859 So maybe--we're up to maybe a probability 1/3 that Ale's 14:31.864 --> 14:33.984 actually insane. So he thinks: 14:33.983 --> 14:36.693 okay, probability 1/3 that's still not very much, 14:36.690 --> 14:38.440 I'll still risk it, he comes in, 14:38.438 --> 14:40.128 and Ale fights him again. 14:40.129 --> 14:42.929 It's a probability 1/3 he's sane. 14:42.930 --> 14:43.910 He's going to give in to me. 14:43.909 --> 14:45.979 He comes in--Ale fights him again. 14:45.980 --> 14:48.970 So now we're in the third market, which was which market? 14:48.970 --> 14:49.300 Student: Bridgeport. 14:49.300 --> 14:49.790 Professor Ben Polak: Bridgeport. 14:49.789 --> 14:53.959 And Bridgeport's seen this horrible fight going on in Miami 14:53.961 --> 14:57.271 and this horrible fight going on in Madison, 14:57.269 --> 14:59.889 and now he's getting pretty sure that this nice, 14:59.893 --> 15:02.293 calm, looking Ale is not nice, calm, looking Ale. 15:02.290 --> 15:04.160 He's crazy Rahul. 15:04.159 --> 15:06.729 There's lot of evidence that he's crazy Rahul. 15:06.730 --> 15:08.820 He's fought the last two markets making huge losses. 15:08.820 --> 15:10.100 It must be that Ale likes to fight. 15:10.100 --> 15:11.540 So what does he do? 15:11.539 --> 15:13.899 He says, I'm going to stay out of here. 15:13.899 --> 15:18.009 I'm convinced that this guy may be crazy, so I'll stay out. 15:18.009 --> 15:21.099 And all the subsequent markets, they think: oh well you know he 15:21.103 --> 15:23.893 fought these first two markets, that means he must be a crazy 15:23.888 --> 15:26.208 guy or at least there's a high probability that he's a crazy 15:26.205 --> 15:27.775 guy, so they all stay out which is 15:27.775 --> 15:29.735 exactly what happened until we got to here. 15:29.740 --> 15:33.600 And even here when they came in Ale acted liked crazy Rahul. 15:33.600 --> 15:37.180 So what made that argument possible was what? 15:37.179 --> 15:40.679 What made that argument possible was the small 15:40.684 --> 15:44.894 possibility, the 1% possibility that Ale is bonkers. 15:44.889 --> 15:46.459 But you know, how well do you know Ale? 15:46.460 --> 15:48.220 There's a 1% chance he's bonkers. 15:48.220 --> 15:51.810 How many of you think you're really sure that he's a sane 15:51.814 --> 15:53.534 guy? He supports Italian football 15:53.525 --> 15:55.555 teams, he's got to be pretty crazy, right? 15:55.560 --> 15:58.550 So what happened here? 15:58.549 --> 16:04.459 This small possibility that Ale is crazy allowed him to build up 16:04.455 --> 16:08.575 a reputation that kept all these guys out, 16:08.580 --> 16:11.160 but actually the argument is stronger than that. 16:11.159 --> 16:13.049 Let's try and push this argument harder. 16:13.049 --> 16:17.259 Suppose, in fact, that Ale is not crazy. 16:17.259 --> 16:20.019 Suppose that Ale is the sane Ale, the nice, 16:20.015 --> 16:22.175 calm, Ale we all know and love. 16:22.179 --> 16:26.839 But we've just argued that if Ale acts as if he's the crazy 16:26.838 --> 16:31.418 guy then you're going to be convinced that he is the crazy 16:31.417 --> 16:34.227 guy, so by acting crazy he might be 16:34.228 --> 16:38.488 able to convince you that he is crazy and therefore keep you 16:38.493 --> 16:40.643 out. So we argued before that, 16:40.635 --> 16:44.225 if there's some chance that Ale's crazy, by acting crazy 16:44.234 --> 16:46.374 early on, he's going to deter these late 16:46.365 --> 16:48.645 entrants from entering the market because they think 16:48.653 --> 16:51.393 they're fighting Rahul and they don't want to fight Rahul. 16:51.389 --> 16:55.689 But we said these early guys, they had probability .99 that 16:55.691 --> 16:57.901 he was sane; and .6 that he was sane; 16:57.899 --> 17:01.169 and maybe even .5 he was sane here--so they thought of coming 17:01.167 --> 17:03.897 in. But now we're arguing that even 17:03.899 --> 17:07.229 if Ale is sane, even if he's that sane guy, 17:07.231 --> 17:11.461 a rational guy, he should behave as if he's 17:11.455 --> 17:16.005 crazy in order to keep these late guys out. 17:16.009 --> 17:19.329 And these early players knowing that even the sane version of 17:19.334 --> 17:22.164 Ale is going to fight them, should also stay out. 17:22.160 --> 17:23.360 Now notice something's happened here. 17:23.359 --> 17:25.919 They're not staying out because they think Ale's crazy, 17:25.920 --> 17:28.530 they're staying out because they know that even the sane 17:28.528 --> 17:31.608 version of Ale is going to fight them in order to seem crazy, 17:31.610 --> 17:33.700 is that right? Everyone see that's a stronger 17:33.704 --> 17:35.554 argument? So now even these early guys 17:35.550 --> 17:37.560 are going to stay out of the market. 17:37.560 --> 17:40.280 Now we're almost there. 17:40.279 --> 17:43.659 What we've argued--let's just make sure we get the two pieces 17:43.655 --> 17:44.945 of this on the board. 17:44.950 --> 17:50.460 We've argued that if there's an epsilon chance, 17:50.457 --> 17:56.557 a very small chance, let's call it a 1% chance where 17:56.564 --> 18:06.844 Ale is crazy, then he can deter entry by 18:06.842 --> 18:16.672 fighting, i.e., seeming crazy. 18:16.670 --> 18:20.180 We argued that what really makes this argument strong is 18:20.180 --> 18:24.010 once we realize that the sane person's going to act crazy, 18:24.009 --> 18:25.749 we really know that everyone's going to act crazy and therefore 18:25.750 --> 18:26.340 we should stay out. 18:26.339 --> 18:28.789 Now that argument won't quite be right. 18:28.789 --> 18:31.129 So that's enough of the argument I want you to have for 18:31.127 --> 18:33.467 the purpose of the exam, but let me just point out that 18:33.465 --> 18:35.105 that argument isn't quite correct. 18:35.109 --> 18:36.889 That can't quite be an equilibrium. 18:36.890 --> 18:39.150 Now why can't that be an equilibrium? 18:39.150 --> 18:42.310 We've just argued that even the sane version of Ale--so this is 18:42.309 --> 18:45.469 a sort of slightly more subtle argument so just pay attention a 18:45.469 --> 18:47.799 second. We've argued that even the sane 18:47.797 --> 18:50.587 version of Ale is going to act like a crazy guy. 18:50.589 --> 18:57.589 So if anyone came in, he's going to act crazy anyway. 18:57.589 --> 18:59.789 So you're not going to update your belief as to whether he's 18:59.789 --> 19:01.989 crazy or sane because we know that the crazy guy is going to 19:01.988 --> 19:04.188 fight because he likes fighting and the sane guy is going to 19:04.187 --> 19:06.197 fight because he wants to seem like a crazy guy. 19:06.200 --> 19:08.150 So you're really learning nothing whether your observe him 19:08.145 --> 19:10.715 fighting or now. But now let's go back to our 19:10.720 --> 19:13.800 tenth market, way back in our tenth market. 19:13.799 --> 19:18.299 Our tenth market participant, whose name was Andy, 19:18.302 --> 19:21.612 hasn't learned anything about Ale. 19:21.609 --> 19:25.259 He hasn't learned anything because whether Ale was sane or 19:25.258 --> 19:27.048 crazy he's going to fight. 19:27.049 --> 19:29.849 So observing what his actions early on, if that was really an 19:29.854 --> 19:32.574 equilibrium, our tenth guy wouldn't have updated his belief 19:32.565 --> 19:33.845 at all, and therefore, 19:33.853 --> 19:36.723 would still believe with probability .99 that Ale was 19:36.724 --> 19:39.434 sane, in which case our tenth guy would enter. 19:39.430 --> 19:42.770 Once again, that argument would unravel from the back. 19:42.769 --> 19:47.009 So what I described before can't quite be an equilibrium. 19:47.009 --> 19:49.629 It can't be just as simple as all sane guys are going to act 19:49.626 --> 19:51.796 crazy because then you wouldn't learn anything. 19:51.799 --> 19:53.779 So it turns out that the equilibrium in this model is 19:53.779 --> 19:55.379 actually very subtle, and it involves mixed 19:55.378 --> 19:56.688 strategies, and mixed strategies are 19:56.692 --> 19:58.302 something we did in the first half of the semester, 19:58.299 --> 19:59.989 so we don't want to go back to it now. 19:59.990 --> 20:02.810 So trust me, you can solve this out with 20:02.812 --> 20:07.012 mixed strategies and the basic idea I gave you is right. 20:07.009 --> 20:10.299 The basic idea is sane guys are occasionally going to act like 20:10.295 --> 20:12.875 crazy guys in order to establish a reputation, 20:12.880 --> 20:17.390 and that reputation helps them down the tree. 20:17.390 --> 20:20.980 So this idea that even when there's a chain store, 20:20.982 --> 20:24.722 people will enter--even when Ale has ten monopolies, 20:24.721 --> 20:28.021 people will enter--this is a famous idea. 20:28.019 --> 20:31.899 It's called the Chain Store Paradox, and it's due to a guy 20:31.902 --> 20:35.242 called Selten who actually won the Nobel Prize. 20:35.240 --> 20:39.080 This is the Chain Store Paradox and this idea of establishing 20:39.081 --> 20:40.811 reputation is a big idea. 20:40.809 --> 20:45.999 The idea is once again you might want to behave as if 20:45.995 --> 20:51.675 you're someone else in order to deter people's actions, 20:51.680 --> 20:54.850 in order to affect people's actions down the tree. 20:54.849 --> 20:59.079 Okay, so what have we learned here? 20:59.080 --> 21:03.550 Let's just work it out. 21:03.549 --> 21:06.939 So the first thing we learned was kind of a nerdy point, 21:06.936 --> 21:08.656 but let me make it anyway. 21:08.660 --> 21:12.460 Introducing just a very, very small probability, 21:12.461 --> 21:17.071 just a tiny probability that Ale might be someone else--he 21:17.071 --> 21:19.921 might be a Rahul, he might be crazy, 21:19.918 --> 21:23.558 he must like fights--that very small probability radically 21:23.556 --> 21:25.786 changes the outcome of the game. 21:25.789 --> 21:29.809 If we were all 100% sure he was sane we'd be tied by backward 21:29.806 --> 21:32.146 induction and entry would follow. 21:32.150 --> 21:34.510 He wouldn't be able to keep anybody out. 21:34.509 --> 21:37.639 But that small probability allows us to get a very 21:37.641 --> 21:38.921 different outcome. 21:38.920 --> 21:41.010 That's the first point I want to draw into this. 21:41.009 --> 21:45.799 The second point I want to get out of this, is really this idea 21:45.795 --> 21:48.725 of reputation. There are lots of settings in 21:48.730 --> 21:52.430 society where reputation matters and one of them is a reputation 21:52.426 --> 21:55.596 to fight. How many of you have friends 21:55.597 --> 21:58.327 who have somewhat short fuses? 21:58.329 --> 22:00.439 You know people who have short fuses, right? 22:00.440 --> 22:03.560 When you're going out, choosing some movie to go to 22:03.555 --> 22:07.355 with these guys who have short fuses or trying to decide who's 22:07.356 --> 22:10.156 going to order something at a restaurant, 22:10.160 --> 22:12.770 or who's going to get to be which side when you're playing 22:12.771 --> 22:14.101 some game, some video game. 22:14.099 --> 22:17.079 I claim, is this true, that the people who have 22:17.076 --> 22:20.306 slightly short fuses, slightly more often get their 22:20.311 --> 22:21.801 way, is that right? 22:21.799 --> 22:24.939 If you have a sibling who has a short fuse, they slightly more 22:24.937 --> 22:27.557 often get their way and that's exactly this idea. 22:27.559 --> 22:29.869 They have short fuses, the fact that they tend to blow 22:29.871 --> 22:32.141 up and get angry at you gives them a little bit of an 22:32.139 --> 22:33.829 advantage. And notice that maybe they 22:33.833 --> 22:36.333 don't have a short fuse at all, maybe they're just pretending 22:36.332 --> 22:38.752 to have a short fuse because they know they're going to get 22:38.748 --> 22:40.538 their way over you softies more often. 22:40.539 --> 22:43.639 None of you have short fuses, you're all sane people right? 22:43.640 --> 22:46.140 So this idea, it should be a familiar idea to 22:46.144 --> 22:49.624 you, but it's not just an idea in the sort of trivial world of 22:49.616 --> 22:52.896 bargaining. Notice this idea of reputation 22:52.901 --> 22:55.091 occurs all over the place. 22:55.089 --> 22:58.429 So another place it occurs, somewhat grim place it occurs, 22:58.425 --> 23:00.995 is in the subject of hostage negotiations. 23:01.000 --> 23:03.670 In the subject of hostage negotiations, 23:03.671 --> 23:07.611 when some other country has seized some hostages from the 23:07.607 --> 23:09.247 U.S. or maybe some criminal 23:09.247 --> 23:12.087 organization has seized some members of your family or some 23:12.088 --> 23:14.878 members of your community and is holding the hostages, 23:14.880 --> 23:17.560 there's a well known idea which is what? 23:17.559 --> 23:22.109 Which is that you should never negotiate with hostage takers, 23:22.107 --> 23:24.547 is that right? Everyone's heard that idea? 23:24.549 --> 23:26.669 You should never negotiate with hostage takers. 23:26.670 --> 23:28.940 You never give in just because they have hostages. 23:28.940 --> 23:31.380 Why? It's the same idea: 23:31.379 --> 23:34.819 because you want to have a reputation for being somebody 23:34.820 --> 23:38.640 who doesn't give in to hostage takers in order to deter future 23:38.638 --> 23:41.828 potential hostage takers from taking hostages. 23:41.829 --> 23:45.439 This has grim consequences but sometimes it's worth bearing the 23:45.441 --> 23:48.941 cost of having your relatives come back in pieces in order to 23:48.936 --> 23:51.496 deter future relatives from being taken. 23:51.500 --> 23:54.460 So that's a somewhat macabre version of this. 23:54.460 --> 23:56.000 Let me give you one other version. 23:56.000 --> 23:58.920 There are whole areas of the economy where reputation is 23:58.923 --> 24:01.533 crucial, where if people played according to their 24:01.527 --> 24:04.247 backward-induction, sane incentives, 24:04.253 --> 24:06.323 we'd have a disaster. 24:06.319 --> 24:09.789 But having a reputation here isn't necessarily a reputation 24:09.792 --> 24:13.572 as a tough guy. It could be a reputation for 24:13.569 --> 24:16.229 somebody who's a nice guy. 24:16.230 --> 24:19.900 Could be that you want to have a reputation for being the sort 24:19.895 --> 24:22.835 of person who derives pleasure or utility from, 24:22.839 --> 24:25.759 (quote) "doing the right thing," from acting honestly. 24:25.759 --> 24:28.999 So think about certain professions where the reputation 24:29.000 --> 24:31.700 of the person in the profession is crucial. 24:31.700 --> 24:35.130 Doctors, for example: it's crucial for a doctor that 24:35.134 --> 24:39.314 he or she has the reputation of someone who tells the truth. 24:39.309 --> 24:41.709 Otherwise, you'd stop going to that doctor. 24:41.710 --> 24:44.980 Accountants: accounting firms rely on having 24:44.977 --> 24:49.457 a reputation for being honest and not cheating the books. 24:49.460 --> 24:51.880 When they stop having that reputation for being 24:51.884 --> 24:55.204 honest--think of Arthur Anderson after the events in Enron--they 24:55.204 --> 24:57.474 pretty quickly cease to be in business. 24:57.470 --> 24:59.590 I gave that example a couple of years ago, and it was very 24:59.590 --> 25:01.820 embarrassing because it turned out Arthur Anderson was in the 25:01.822 --> 25:03.982 class--literally Arthur Anderson III was in the class. 25:03.980 --> 25:05.110 These things happen at Yale. 25:05.109 --> 25:08.099 But nevertheless, Arthur Anderson relied on his 25:08.099 --> 25:11.809 reputation, the firm relied on its reputation as an honest 25:11.805 --> 25:13.995 firm, and it was worth behaving 25:13.998 --> 25:17.728 honestly to maintain that reputation for future business. 25:17.730 --> 25:22.070 Reputation is a huge topic and my guess is that the next time 25:22.067 --> 25:26.547 there's a Nobel Prize in game theory it'll be for this idea. 25:26.550 --> 25:29.600 So that's my prediction. 25:29.599 --> 25:32.869 Now having said that, I want to spend the whole of 25:32.872 --> 25:36.882 the rest of today playing one game and analyzing one game. 25:36.880 --> 25:40.600 So we're going to play this game and for this game I need a 25:40.596 --> 25:42.066 couple of volunteers. 25:42.069 --> 25:44.609 So I'm going to pull out some volunteers. 25:44.610 --> 25:46.410 Anyone want to volunteer? 25:46.410 --> 25:50.160 I need two volunteers for this game. 25:50.160 --> 25:51.800 How about my guy from the football team, 25:51.799 --> 25:52.849 was that a raised hand? 25:52.849 --> 25:55.229 It wasn't a raised hand, how about my guy from the 25:55.232 --> 25:57.272 football team? Is it football team? 25:57.269 --> 26:00.029 Baseball team, that may be unfair in this 26:00.028 --> 26:03.148 particular game. Maybe I'll take someone who 26:03.150 --> 26:05.100 isn't on the baseball team. 26:05.099 --> 26:07.419 Anyone who's on the football team? 26:07.420 --> 26:09.120 These guys, you guys on the football team? 26:09.120 --> 26:10.890 Okay great, so you two guys. 26:10.890 --> 26:18.420 I need you at the front and your names are? 26:18.420 --> 26:19.110 Chevy and Patrick. 26:19.109 --> 26:21.419 So Chevy and Patrick are going to be our volunteers. 26:21.420 --> 26:24.170 Now the idea of this game is you guys provided the 26:24.165 --> 26:27.355 volunteers--wait down here a second--you guys provided the 26:27.359 --> 26:30.159 volunteers. This game involves two 26:30.160 --> 26:35.150 volunteers that you just provided, and two wet sponges. 26:35.150 --> 26:38.140 I will provide the wet sponges. 26:38.140 --> 26:43.010 26:43.009 --> 26:44.939 So I have here a couple of sponges and in a minute I'm 26:44.940 --> 26:46.540 going to wet them, and I'll tell you what the 26:46.543 --> 26:47.493 rules are in a second. 26:47.490 --> 26:52.320 Okay, so I'm going to give one of these sponges each to Chevy 26:52.320 --> 26:57.070 and Patrick, and then going to position Chevy and Patrick at 26:57.071 --> 27:00.051 either end of this central aisle, 27:00.049 --> 27:02.849 of this aisle here, and the game is going to be as 27:02.848 --> 27:04.678 follows. It's important that everyone 27:04.680 --> 27:07.290 listen to the rules here because I'm going to pick two more 27:07.289 --> 27:08.459 volunteers in a moment. 27:08.460 --> 27:10.670 So the game they're going to play is as follows. 27:10.670 --> 27:12.090 Each of them has one sponge. 27:12.089 --> 27:13.969 It's crucial they only have one sponge. 27:13.970 --> 27:18.040 And they're going to take turns. 27:18.039 --> 27:20.499 And when it's your turn, as long as you still have your 27:20.495 --> 27:22.355 sponge in your hand, you face a choice. 27:22.359 --> 27:27.299 You can either throw your sponge at your opponent, 27:27.296 --> 27:31.826 and if you hit your opponent you win the game, 27:31.830 --> 27:35.760 or you have to take a step forward. 27:35.759 --> 27:40.289 So either you throw the sponge or you take a step forward. 27:40.290 --> 27:43.510 Now there's a crucial rule here. 27:43.509 --> 27:48.139 Each player only has one sponge and, once they've thrown that 27:48.139 --> 27:51.379 sponge, they do not get the sponge back. 27:51.380 --> 27:53.340 Everyone understand that? 27:53.339 --> 27:56.729 Once you've thrown the sponge you do not get the sponge back. 27:56.730 --> 27:59.620 So once again, if you throw your sponge at 27:59.615 --> 28:03.695 your opponent and you hit your opponent then you've won the 28:03.696 --> 28:07.336 game. But if you throw your sponge at 28:07.343 --> 28:12.013 your opponent and you miss, the game continues. 28:12.009 --> 28:14.089 So let's make sure we understand that, 28:14.086 --> 28:17.226 if you throw your sponge and miss the game continues:. 28:17.230 --> 28:18.320 You still have to step forward. 28:18.319 --> 28:20.339 So, what's your opponent going to do at that point? 28:20.339 --> 28:21.339 What's your opponent going to do? 28:21.339 --> 28:25.509 Let's make sure that our football players understand 28:25.507 --> 28:29.417 this. It wasn't meant that way. 28:29.420 --> 28:31.010 They could have been soccer players, come on. 28:31.009 --> 28:32.599 Student: I didn't appreciate that very much 28:32.599 --> 28:34.319 Professor Ben Polak: I'm sorry I didn't mean it 28:34.318 --> 28:37.218 that way. So if your opponent whose name 28:37.223 --> 28:41.863 is Patrick throws and misses, what are you going to do? 28:41.859 --> 28:43.799 Student: I'll walk forward until I slam the sponge 28:43.796 --> 28:45.236 in his face. Professor Ben Polak: 28:45.243 --> 28:47.383 Great, you will walk forward until you politely put it on his 28:47.384 --> 28:48.784 head. Everyone understand? 28:48.779 --> 28:51.219 That, if in fact, you throw and miss you've lost 28:51.221 --> 28:54.291 the game, because the other guy can wait until he's standing 28:54.285 --> 28:57.295 right on top of you and just place the sponge gently on his 28:57.298 --> 28:59.148 head. Now for fairness sake, 28:59.150 --> 29:02.290 it's important that these sponges are equally weighted, 29:02.292 --> 29:05.442 and I'm going to weight them--I'm going to put water in 29:05.435 --> 29:07.945 them now. And--you know nothing but the 29:07.945 --> 29:11.385 best for Yale students--I'm going to provide Yale University 29:11.394 --> 29:13.654 spring water. Who knew that Yale University 29:13.645 --> 29:16.845 had a spring. That's kind of a strange one? 29:16.849 --> 29:21.409 If it makes you feel better you can think of this American beer. 29:21.410 --> 29:25.270 29:25.269 --> 29:29.149 I'm not going to make these too heavy, partly because it makes 29:29.145 --> 29:32.825 it too easy and partly because I don't want to get sued. 29:32.829 --> 29:39.219 So I'm going to squeeze these out somewhere away from the 29:39.217 --> 29:42.637 wires. We're going to get our judge 29:42.643 --> 29:46.053 here to weigh them, I need a mike here, 29:46.054 --> 29:47.944 let me get a mike. 29:47.940 --> 29:51.650 So I'll need you to hold those sponges in your hands and tell 29:51.649 --> 29:53.689 me if they're equally weighted. 29:53.690 --> 29:55.800 Pretty equal? Okay, they're pretty equal, 29:55.800 --> 29:57.410 everyone agrees. So how is this going to work? 29:57.410 --> 30:01.710 I'm going to give the blue sponge to Chevy and the green 30:01.714 --> 30:03.284 sponge to Patrick. 30:03.279 --> 30:06.319 And Chevy's going to stand here, and Patrick's going to 30:06.316 --> 30:08.956 stand as far back as I can get him on camera, 30:08.960 --> 30:11.200 which I'm going to be told how far back I can go. 30:11.200 --> 30:12.420 Don't go too far. 30:12.420 --> 30:14.820 Okay come back Patrick, you're too ambitious. 30:14.820 --> 30:23.040 Come back. Keep coming. stop. 30:23.039 --> 30:26.049 Okay, we're going to start here--start quite close 30:26.053 --> 30:28.453 actually. Everyone understand how we're 30:28.446 --> 30:29.626 going to play this? 30:29.630 --> 30:33.730 So Chevy is Player 1, Patrick is Player 2. 30:33.730 --> 30:37.440 Chevy has to decide whether to throw or to step. 30:37.440 --> 30:39.010 Student: I'll step. 30:39.009 --> 30:40.469 Professor Ben Polak: Okay, let's just hold the game a 30:40.467 --> 30:41.667 second. Now its Patrick's turn. 30:41.670 --> 30:44.340 Does anyone have any advice for Patrick at this point? 30:44.340 --> 30:47.430 30:47.430 --> 30:52.500 If you think throw, raise your hand. 30:52.500 --> 30:56.100 If you think step, raise your hand. 30:56.099 --> 30:57.219 There's a lot more steps than throws. 30:57.220 --> 30:58.740 I thought the Yale football team was good this year. 30:58.740 --> 31:03.750 Your choice, step or throw. 31:03.750 --> 31:06.100 I should announce two other rules. 31:06.099 --> 31:07.449 It's kind of important I should have said this. 31:07.450 --> 31:11.430 First, a step has to be a proper step, like a yard long; 31:11.430 --> 31:14.500 and second (I think this will work in America): 31:14.503 --> 31:16.043 gentleman never duck. 31:16.040 --> 31:18.620 No dodging the sponge okay. 31:18.620 --> 31:20.630 Chevy: your turn. 31:20.630 --> 31:24.620 Student: I don't really trust my arm. 31:24.620 --> 31:25.670 I'm going to step. 31:25.670 --> 31:29.100 Professor Ben Polak: All right, so you're stepping again. 31:29.100 --> 31:29.930 Let me go to Patrick. 31:29.930 --> 31:33.540 I feel like I'm in the line of fire here. 31:33.539 --> 31:34.519 Patrick what are you going to do? 31:34.519 --> 31:34.669 Student: I'm going to throw. 31:34.670 --> 31:36.610 Professor Ben Polak: Patrick's going to throw, 31:36.606 --> 31:38.316 I'm really going to get out of the way then. 31:38.320 --> 31:42.230 We'll see this in slow motion. 31:42.230 --> 31:45.850 31:45.849 --> 31:50.539 Continue, all right so you have to take a step forward, 31:50.540 --> 31:54.970 so Chevy's going to take a step forward I assume? 31:54.970 --> 31:57.430 Patrick's going to take a step forward. 31:57.430 --> 31:59.980 Chevy's going to take a step forward. 31:59.980 --> 32:02.470 Patrick's going to take a step forward. 32:02.470 --> 32:06.220 32:06.220 --> 32:12.880 Good, so a round of applause for our players, 32:12.875 --> 32:15.785 thank you. So I think we have time to do 32:15.792 --> 32:17.542 this once more, and then we're going to analyze 32:17.541 --> 32:18.951 it. So I want to get two women 32:18.954 --> 32:20.984 involved. It's too sexist otherwise. 32:20.980 --> 32:26.100 So can we have two women in the class please? 32:26.099 --> 32:28.559 Two volunteers, come on, you can volunteer. 32:28.560 --> 32:29.410 There's a volunteer. 32:29.410 --> 32:34.490 Thank you great. Okay great, thank you. 32:34.490 --> 32:36.540 So your name is? Student: Jessica. 32:36.535 --> 32:38.545 Professor Ben Polak: Jessica and your name is? 32:38.549 --> 32:39.649 Student: Clara-Elise. 32:39.652 --> 32:41.742 Professor Ben Polak: Clara-Elise and Jessica. 32:41.740 --> 32:44.060 We'll start at the same positions, we'll use the same 32:44.064 --> 32:46.614 sponges, and I just need to remind you where that position 32:46.612 --> 32:48.572 was. Just give me a thumbs up when 32:48.570 --> 32:50.060 I'm in the right position. 32:50.059 --> 32:54.019 Good, same rules, Clara-Elise and Jessica. 32:54.020 --> 32:55.630 We'll let Jessica be Player 1. 32:55.630 --> 32:58.160 So Jessica you can step or throw, what do you want to do? 32:58.160 --> 32:58.710 Student: I'm going to step. 32:58.710 --> 33:00.390 Professor Ben Polak: Going to step, 33:00.392 --> 33:01.912 okay. You know what might be a good 33:01.914 --> 33:04.174 idea: Ale why don't you put the mike on Clara-Elise. 33:04.170 --> 33:06.160 So we have a mike at either end rather than running to and fro, 33:06.160 --> 33:07.890 that's good. So Clara-Elise what are you 33:07.892 --> 33:09.432 going to do? Student: I'm going to 33:09.433 --> 33:10.493 step. Professor Ben Polak: 33:10.488 --> 33:12.498 You're going to step and Jessica what are you going to do? 33:12.500 --> 33:13.580 Student: I'm going to step. 33:13.580 --> 33:15.250 Professor Ben Polak: You're going to step. 33:15.250 --> 33:17.850 Ale and I are in danger here but never mind. 33:17.850 --> 33:20.370 Any votes now? Do people think that Jessica 33:20.366 --> 33:22.836 should throw? If you think she should throw 33:22.842 --> 33:23.982 raise your hands. 33:23.980 --> 33:25.560 There's a large majority for step. 33:25.559 --> 33:27.649 So up to you, what are you going to do? 33:27.650 --> 33:30.190 Student: I'm going to step. 33:30.190 --> 33:31.800 Professor Ben Polak: Going to step okay. 33:31.800 --> 33:34.590 Clara-Elise any decisions? 33:34.589 --> 33:38.049 It's a pretty light sponge, It's pretty hard to throw the 33:38.049 --> 33:40.149 sponge, because we've seen that. 33:40.150 --> 33:43.110 Okay, stepping again. 33:43.109 --> 33:46.459 Student: I'm going to throw. 33:46.460 --> 33:48.670 Professor Ben Polak: You're going to throw okay, 33:48.665 --> 33:50.295 here we go, let me get out of the way. 33:50.300 --> 33:54.390 Oh okay, continue please. 33:54.390 --> 33:55.560 Clara-Elise's turn. 33:55.560 --> 33:56.930 Student: I'll step. 33:56.930 --> 33:58.290 Professor Ben Polak: You'll step. 33:58.289 --> 34:00.889 Jessica has to step, Clara-Elise has to step, 34:00.886 --> 34:03.916 all right good. So we've seen how the game 34:03.917 --> 34:07.657 works, everyone understands how the game works. 34:07.660 --> 34:11.920 I want to spend the rest of today analyzing this game. 34:11.920 --> 34:16.880 Before I do so, we should just talk about what 34:16.877 --> 34:22.377 this game is. Let me get some new boards down 34:22.384 --> 34:23.134 here. 34:23.130 --> 34:28.110 34:28.110 --> 34:30.630 So one quick announcement: I'm going to analyze this, 34:30.629 --> 34:33.389 and we're going to spend the rest of today analyzing this, 34:33.390 --> 34:35.280 but this is going to be quite hard. 34:35.280 --> 34:38.240 So I'm going to provide you a handout that I'll put on the web 34:38.235 --> 34:40.605 probably tomorrow that goes over this argument. 34:40.610 --> 34:42.520 So you don't have to take detailed notes now, 34:42.517 --> 34:44.947 I want you to pay attention and see if you can follow the 34:44.946 --> 34:48.016 argument. So this game is called duel, 34:48.017 --> 34:52.227 not surprisingly, and you may wonder what are we 34:52.227 --> 34:57.507 doing--as are my colleagues that are here--what are we doing 34:57.512 --> 35:00.022 having a duel in class. 35:00.019 --> 35:01.729 Of course one answer to that is, it's kind of fun watching 35:01.725 --> 35:03.635 the future leaders of America throw wet sponges at each other. 35:03.639 --> 35:04.739 That's probably reason in itself. 35:04.740 --> 35:06.570 But there are other reasons. 35:06.570 --> 35:08.950 Duel is a real game. 35:08.949 --> 35:13.489 So those of you who are well versed in Russian literature 35:13.489 --> 35:18.109 will have seen duals before, or at least read of duals. 35:18.110 --> 35:19.990 There are some famous duels in Russian literature. 35:19.989 --> 35:22.549 Anyone want to tell me some famous duels in Russian 35:22.553 --> 35:25.293 literature? Any Russian majors here? 35:25.289 --> 35:27.569 No, nobody want to give me a shot at this? 35:27.570 --> 35:31.220 Really nobody. This is Yale, come on. 35:31.219 --> 35:32.719 Well, how about in War and Peace okay. 35:32.719 --> 35:35.449 There's a duel like this in War and Peace, and in War and Peace 35:35.445 --> 35:37.855 and without giving away the ending--actually it's in the 35:37.863 --> 35:39.933 middle of the book and it's 800 pages long, 35:39.930 --> 35:41.030 so it isn't exactly the ending. 35:41.030 --> 35:44.400 But in War and Peace I think we're led to believe that the 35:44.404 --> 35:47.934 hero, the protagonist Pierre, shoots his gun--in War and 35:47.933 --> 35:51.583 Peace it's a gun and not a sponge, no surprise--he shoots 35:51.581 --> 35:54.321 his gun too early, we're led to believe. 35:54.320 --> 35:56.830 There's a famous one in Eugene Onegin, in Pushkin's Eugene 35:56.830 --> 35:58.810 Onegin, and there are lots of others actually, 35:58.812 --> 36:00.312 so there's lots in literature. 36:00.309 --> 36:03.459 There are also settings which aren't exactly out of 36:03.460 --> 36:07.450 literature. So one example would be in a 36:07.453 --> 36:10.343 bike race. How many of you ever watch the 36:10.343 --> 36:11.953 Tour de France? Everyone know what I mean by 36:11.954 --> 36:12.484 the Tour de France. 36:12.480 --> 36:15.490 So this is a bike race that goes around France. 36:15.490 --> 36:18.990 It takes stages. And in the Tour de France, 36:18.992 --> 36:20.492 there's a key decision. 36:20.489 --> 36:23.299 There's a game within the game, and the game within the 36:23.300 --> 36:26.370 game--I'm looking at Jake who's a real cyclist here--but the 36:26.372 --> 36:29.552 game within the game is when do you try to break away from the 36:29.547 --> 36:31.567 pack, which is called the peloton. 36:31.570 --> 36:34.080 And if you break away too early from the peloton, 36:34.077 --> 36:36.687 it turns out that you're going to get reeled in. 36:36.690 --> 36:39.420 It turns out that over the long haul the peloton can go much 36:39.417 --> 36:41.457 faster than you. So if you break too early 36:41.459 --> 36:42.959 they're going to catch you up. 36:42.960 --> 36:44.620 On the other hand, if you break too late, 36:44.620 --> 36:46.860 then you're going to lose because there are going to be 36:46.862 --> 36:49.522 some people in the peloton who are just excellent sprinters. 36:49.519 --> 36:52.879 So if you break too late the sprinters are going to win the 36:52.875 --> 36:55.195 race. So you have to decide when to 36:55.200 --> 36:56.790 break from the peloton. 36:56.789 --> 37:00.089 This is the second most important game within a game in 37:00.085 --> 37:03.075 the Tour de France, the most important game within 37:03.076 --> 37:05.636 a game is where to hide your steroids. 37:05.639 --> 37:10.599 So let me give you one other example. 37:10.599 --> 37:13.329 Let me give you a more economic example--it's meant to be an 37:13.327 --> 37:15.677 economics class. Imagine there's two firms, 37:15.683 --> 37:18.443 and these two firms are both engaged in R&D, 37:18.438 --> 37:20.078 research and development. 37:20.079 --> 37:23.579 And they're trying to develop a new product, and they're going 37:23.578 --> 37:26.158 to launch this new product onto the market. 37:26.159 --> 37:28.859 But the nature of this market is--maybe it's a network 37:28.861 --> 37:32.021 good--the nature of this market is there's only going to be one 37:32.021 --> 37:33.551 successful good out there. 37:33.550 --> 37:35.530 So essentially there's going to be one standard, 37:35.525 --> 37:37.665 let's say of a software or a technological standard, 37:37.668 --> 37:39.558 and only one of them is going to survive. 37:39.559 --> 37:42.509 The problem is you haven't perfected your product yet. 37:42.510 --> 37:46.130 If you launch your product too early it may not work, 37:46.125 --> 37:50.015 and then consumers are never going to trust you again. 37:50.019 --> 37:52.449 But if you launch it too late the other side will have 37:52.448 --> 37:54.648 launched already, they will have got a toehold in 37:54.647 --> 37:56.157 the market, and you're toast. 37:56.159 --> 38:00.389 So that game--that game about product launch--is like duel, 38:00.391 --> 38:04.481 except you're launching a product rather than launching a 38:04.477 --> 38:06.797 sponge. Is that right? 38:06.800 --> 38:09.300 Now all of these games have a common feature, 38:09.301 --> 38:11.121 and it's a new feature for us. 38:11.119 --> 38:13.639 It's about the nature of the strategic decision. 38:13.639 --> 38:16.829 In most of the games we've looked at in the course so far 38:16.825 --> 38:20.115 the strategic decision has been of the form: where should I 38:20.124 --> 38:22.124 locate, how much should I do, 38:22.117 --> 38:25.627 what price should I set, should I stand for election or 38:25.634 --> 38:29.074 not? Here the strategic decision is 38:29.066 --> 38:32.436 not of the form what should I do. 38:32.440 --> 38:33.970 It's of the form what? 38:33.970 --> 38:37.340 When? It's of the form when am 38:37.343 --> 38:39.083 I going to launch the sponge? 38:39.079 --> 38:41.009 We know perfectly well what you're going to do. 38:41.010 --> 38:42.120 You're going to throw the sponge. 38:42.119 --> 38:44.869 The strategic issue in question is when. 38:44.870 --> 38:47.570 So the issue here is when. 38:47.570 --> 38:53.050 38:53.050 --> 38:54.990 So to analyze this, I'm going to need a little bit 38:54.985 --> 38:56.995 of notation, and let me put that notation up now. 38:57.000 --> 39:02.410 So in particular, I want to use the notation, 39:02.412 --> 39:04.752 Pi[d] to be what? 39:04.750 --> 39:18.080 Let Pi[d] be Player i's probability of 39:18.075 --> 39:31.035 hitting if i shoots at distance d. 39:31.039 --> 39:33.889 So Pi[d] is the probability that i will 39:33.894 --> 39:36.904 hit if he or she shoots at distance d. 39:36.900 --> 39:39.170 Everyone happy with that? 39:39.170 --> 39:41.450 That's the only notation I'm going to use today. 39:41.449 --> 39:44.109 And I'm going to make some assumptions about the nature of 39:44.112 --> 39:45.002 this probability. 39:45.000 --> 39:46.960 Two of the assumptions are pretty innocent. 39:46.960 --> 39:50.290 So let's draw a picture. 39:50.289 --> 39:53.789 So the picture is going to look like this. 39:53.789 --> 39:57.109 Here's a graph, and on the horizontal axis I'm 39:57.114 --> 40:00.564 going to put d. This is the distance apart of 40:00.563 --> 40:04.743 the two players. And on the vertical axis I'm 40:04.741 --> 40:11.041 going to put P which is the probability, Pr the probability. 40:11.039 --> 40:13.879 So here they're at distance 0, and I'm going to make an 40:13.875 --> 40:16.655 assumption about what the probability of hitting is if 40:16.657 --> 40:17.967 you're at distance 0. 40:17.970 --> 40:19.200 What's the sensible assumption? 40:19.199 --> 40:21.919 What's the probability of hitting somebody with your 40:21.916 --> 40:23.776 sponge if you're 0 distance away? 40:23.780 --> 40:26.480 1, okay, I agree, so 1. 40:26.480 --> 40:28.320 So the first assumption I'm going to make is, 40:28.317 --> 40:30.947 if they're right on top of each other, they're going to hit with 40:30.948 --> 40:34.428 probability 1. Now the second assumption I'm 40:34.427 --> 40:38.657 going to make is: as you get further away this 40:38.658 --> 40:41.008 probability decreases. 40:41.010 --> 40:42.690 It doesn't have to look exactly like this but something like 40:42.690 --> 40:44.230 that. That also I think--Is that an 40:44.232 --> 40:46.982 okay assumption? As you're further away there's 40:46.979 --> 40:49.069 a lower probability of hitting. 40:49.070 --> 40:52.550 Now I'm not going to assume that these two players have 40:52.550 --> 40:54.960 equal abilities. For example, I don't know; 40:54.960 --> 40:57.540 I didn't ask them but one of our two football players might 40:57.537 --> 41:00.197 be a quarterback and the other one might be a linebacker or a 41:00.204 --> 41:02.324 running back. And I'm assuming the 41:02.324 --> 41:06.014 quarterback is probably more accurate, is that right? 41:06.010 --> 41:08.760 So I'm not going to assume that they're equally good, 41:08.758 --> 41:11.558 so it could be that their abilities look like this. 41:11.559 --> 41:18.339 This could be P1[d], and this could be P2[d]. 41:18.340 --> 41:20.350 Everyone okay with that? 41:20.349 --> 41:23.499 So shout this out, in this picture, 41:23.497 --> 41:28.587 who is the better shot and who is the less good shot? 41:28.590 --> 41:30.620 Who is the better shot? 41:30.619 --> 41:34.849 1 is the better shot because at every distance if Player 1 were 41:34.847 --> 41:39.207 to throw, Player 1's probability of hitting is higher than Player 41:39.212 --> 41:41.942 2's probability of hitting as drawn. 41:41.940 --> 41:43.960 Now, I don't even need to assume this. 41:43.960 --> 41:47.720 It could well be that these probabilities cross. 41:47.719 --> 41:51.009 It could be that these curves cross. 41:51.010 --> 41:53.650 So it could be that Player 1 is better at close distances, 41:53.646 --> 41:55.586 but Player 2 is better at far distances. 41:55.590 --> 41:57.260 That's fine. We'll assume it's like this 41:57.262 --> 41:58.542 today but I'm not going to use that. 41:58.540 --> 42:01.130 I could do away with that. 42:01.130 --> 42:04.330 As drawn, Player 1 is the better shot. 42:04.329 --> 42:06.369 Now I'm going to make one assumption that matters, 42:06.367 --> 42:08.027 and it's really a critical assumption. 42:08.030 --> 42:10.730 I'm going to make the assumption because it keeps the 42:10.725 --> 42:11.965 math simple for today. 42:11.969 --> 42:13.469 We have enough math to do anyway. 42:13.469 --> 42:18.509 I'm going to assume that these abilities are known. 42:18.510 --> 42:22.840 I'm going to assume that not only do you know your own 42:22.841 --> 42:27.011 ability of hitting your opponent at any distance. 42:27.010 --> 42:30.480 I'm going to assume you also know the ability of your 42:30.481 --> 42:31.951 opponent to hit you. 42:31.950 --> 42:35.170 42:35.170 --> 42:36.650 Now let's look at this a second. 42:36.650 --> 42:39.890 We've got a little bit of notation on the board, 42:39.885 --> 42:41.945 let's discuss this a second. 42:41.949 --> 42:43.589 What do we think is going to happen here? 42:43.590 --> 42:47.310 In this particular example we have a good shot and a less good 42:47.312 --> 42:49.522 shot. Who do we think is going to 42:49.515 --> 42:53.415 shoot first? Let's try and cold call some 42:53.418 --> 42:56.838 people. So you sir what's your name? 42:56.840 --> 42:57.620 Student: Frank. 42:57.615 --> 42:59.585 Professor Ben Polak: Frank, so who do you think is 42:59.589 --> 43:01.319 going to shoot first, the better shot or the worse 43:01.317 --> 43:04.047 shot? Student: The better shot 43:04.049 --> 43:06.889 but also depends on who steps first. 43:06.889 --> 43:09.209 Professor Ben Polak: Okay, let's assume Player 1 is 43:09.213 --> 43:10.113 going to step first. 43:10.110 --> 43:11.130 Student: Player 1. 43:11.130 --> 43:13.370 Professor Ben Polak: So Frank thinks Player 1 is going 43:13.365 --> 43:15.335 to shoot first because Player 1 is the better shot. 43:15.340 --> 43:19.340 43:19.340 --> 43:20.640 Let's see, so what's your name? 43:20.640 --> 43:21.490 Student: Nick. 43:21.489 --> 43:22.699 Professor Ben Polak: Nick, who do you think is going 43:22.696 --> 43:24.156 to shoot first? Student: I think Player 43:24.155 --> 43:24.925 2 would shoot first. 43:24.929 --> 43:26.789 Professor Ben Polak: All right, so let's talk why. 43:26.789 --> 43:29.129 Why do you think Player 1 was going to shoot first? 43:29.130 --> 43:33.380 Let's do a poll. How many of you think the 43:33.382 --> 43:35.692 better shot's going to shoot first? 43:35.690 --> 43:39.130 How many people think the worse shot's going to shoot first? 43:39.130 --> 43:41.150 How many people of you are being chickens and abstaining? 43:41.150 --> 43:42.980 Quite a few okay. 43:42.980 --> 43:47.080 So why do we think the better shot might shoot first? 43:47.079 --> 43:50.229 Student: At equal distance, he has a better chance 43:50.228 --> 43:52.078 of hitting. Professor Ben Polak: 43:52.082 --> 43:54.662 Because he has a better chance of hitting, but why do you think 43:54.659 --> 43:56.529 that the less good shot might shoot first? 43:56.530 --> 44:01.090 Student: He knows that if P1 gets too close he's going 44:01.094 --> 44:04.374 to win anyway, so he may as well take a shot 44:04.365 --> 44:06.795 with lower chance before P1. 44:06.800 --> 44:09.470 Professor Ben Polak: All right, okay so you have two 44:09.472 --> 44:11.592 arguments here, the first argument is maybe the 44:11.591 --> 44:13.481 better shot will shoot first because, 44:13.480 --> 44:15.990 after all, he has a higher chance of hitting. 44:15.989 --> 44:19.259 And the other argument says maybe the worse shot will shoot 44:19.261 --> 44:22.291 first, to what? To pre-empt the better shot 44:22.286 --> 44:23.736 from shooting him. 44:23.739 --> 44:26.419 But now we get more complicated, because after all, 44:26.415 --> 44:29.465 if you're the better shot and you know that the worse shot 44:29.465 --> 44:32.515 maybe going to try and shoot first to try and pre-empt you 44:32.515 --> 44:35.245 from shooting him, you might be tempted to shoot 44:35.245 --> 44:38.135 before the worse shot shoots to preempt the worse shot from 44:38.135 --> 44:39.975 pre-empting you from shooting him. 44:39.980 --> 44:42.590 And if you're the worse shot maybe you're going to try and 44:42.588 --> 44:45.008 shoot first, even earlier, to pre-empt the better shot 44:45.006 --> 44:47.486 from pre-empting the worse shot, from pre-empting the better 44:47.491 --> 44:49.471 shot from shooting the worse shot and so on. 44:49.469 --> 44:53.669 So what's clear is that this game has a lot to do with 44:53.670 --> 44:57.760 pre-emption. Pre-emption's a big idea here, 44:57.756 --> 45:03.206 but I claim it's not at all obvious who's going to shoot 45:03.214 --> 45:07.584 first, the better shot or the worse shot. 45:07.580 --> 45:09.710 Is that right? So those people who abstained 45:09.706 --> 45:10.576 raise your hands again. 45:10.579 --> 45:12.579 Those people who abstained before, it seemed like it was a 45:12.578 --> 45:13.768 pretty sensible time to abstain. 45:13.769 --> 45:16.839 It's not obvious at all to me who's going to shoot first here. 45:16.840 --> 45:21.350 Are people convinced at least that it's a hard problem? 45:21.350 --> 45:23.480 Yes or no, people convinced? 45:23.480 --> 45:26.030 Yeah okay good. It's a hard problem. 45:26.030 --> 45:30.500 So what I want to do is, as a class, as a group, 45:30.502 --> 45:34.502 what I want us to do is solve this game; 45:34.500 --> 45:38.230 and I want to solve not just who is going to shoot first, 45:38.230 --> 45:42.160 I want to figure out exactly when they're going to shoot. 45:42.159 --> 45:45.509 So we're going to do this in the next half hour and we're 45:45.514 --> 45:48.754 going to do it as a class, so you're going to do it. 45:48.750 --> 45:51.720 So we're going to nail this problem basically, 45:51.717 --> 45:55.407 and we're going to do it using two kinds of arguments. 45:55.409 --> 45:58.869 One kind of argument is an argument we learned the very 45:58.873 --> 46:02.533 first day of the class and that's a dominance argument, 46:02.530 --> 46:05.460 and the second kind of argument is an argument we've been using 46:05.456 --> 46:08.286 a little bit recently, and what kind of argument is 46:08.294 --> 46:10.354 that? What is it? 46:10.350 --> 46:11.690 Backward induction. 46:11.690 --> 46:13.760 So we're going to use dominance arguments and backward 46:13.755 --> 46:16.085 induction, and we're going to figure out not just whether the 46:16.092 --> 46:18.082 better shot or the worse shot's going to shoot, 46:18.079 --> 46:22.469 but exactly who's going to shoot when. 46:22.469 --> 46:25.559 Let's keep our picture handy, get rid of, well I can get this 46:25.557 --> 46:26.327 down I guess. 46:26.330 --> 46:31.440 46:31.440 --> 46:36.090 Can you still see the picture? 46:36.090 --> 46:41.040 All right, let's proceed with this argument. 46:41.039 --> 46:46.229 So to do this argument, I first of all want to 46:46.230 --> 46:49.690 establish a couple of facts. 46:49.690 --> 46:57.830 I want to establish two facts, and we'll call the first fact: 46:57.829 --> 47:00.999 Fact A. Let's go back to our two 47:00.995 --> 47:02.855 players we had before. 47:02.860 --> 47:04.570 In fact, maybe it would be helpful to have our players. 47:04.570 --> 47:06.340 Can I use our first two players as props. 47:06.340 --> 47:10.110 Can I have you guys on stage? 47:10.110 --> 47:13.670 While they're coming up--sorry guys, I'm exploiting you a bit 47:13.674 --> 47:15.814 today. I hope you both signed your 47:15.809 --> 47:17.099 legal release forms. 47:17.099 --> 47:20.519 Why don't you guys sit here a second so I can use you as 47:20.522 --> 47:24.052 props. So imagine that these two guys 47:24.052 --> 47:26.682 still have their sponges. 47:26.680 --> 47:28.220 Let's actually set this up. 47:28.219 --> 47:33.679 So suppose that Chevy still has a sponge, and Patrick still has 47:33.684 --> 47:36.974 his sponge. And suppose it's Chevy's turn, 47:36.966 --> 47:40.666 and suppose that Chevy is trying to decide whether he 47:40.668 --> 47:43.088 should throw his sponge or not. 47:43.090 --> 47:47.240 47:47.239 --> 47:49.609 Let me give you a mike each so you have them for future 47:49.612 --> 47:51.842 reference. So Chevy is trying to decide 47:51.837 --> 47:54.017 whether to throw his sponge or not. 47:54.019 --> 48:01.579 Now suppose that Chevy knows that Patrick is not going to 48:01.576 --> 48:06.566 shoot next turn when it's his turn. 48:06.570 --> 48:09.760 So Chevy's trying to decide whether to shoot and he knows 48:09.760 --> 48:13.010 that Patrick is not going to shoot next turn when it's his 48:13.007 --> 48:14.347 turn. What should Chevy do? 48:14.350 --> 48:15.500 Chevy what should you do? 48:15.500 --> 48:16.080 Student: Take a step. 48:16.079 --> 48:17.859 Professor Ben Polak: Take a step, that's right. 48:17.860 --> 48:18.870 Why should he take a step? 48:18.869 --> 48:19.229 What's the argument? 48:19.231 --> 48:19.721 Why he should take a step? 48:19.719 --> 48:21.649 Well let's find out: what's the argument? 48:21.650 --> 48:27.620 Student: Because I'll just be one step closer and I'll 48:27.615 --> 48:31.985 be able to make the same choice next time. 48:31.989 --> 48:34.179 Professor Ben Polak: Good, hold the mike up to you. 48:34.180 --> 48:34.820 You're a rock star now. 48:34.820 --> 48:39.310 All right good, he's correctly saying he should 48:39.306 --> 48:41.936 wait, why should he wait? 48:41.940 --> 48:46.600 Because he's going to be closer next time. 48:46.599 --> 48:59.599 So the first fact is: assuming no one has thrown yet, 48:59.603 --> 49:13.113 if Player i knows (at (say) distance d) that j will not 49:13.107 --> 49:26.177 shoot--let me call it tomorrow; and tomorrow he'll be closer, 49:26.181 --> 49:34.081 he'll be at distance d - 1--then Chevy correctly says, 49:34.077 --> 49:38.247 I should not shoot today. 49:38.250 --> 49:41.590 Again, recall the argument, the argument is you'll get a 49:41.585 --> 49:44.795 better shot, a closer shot, the day after tomorrow. 49:44.800 --> 49:49.920 Now, let's turn things around. 49:49.920 --> 49:53.750 Suppose, conversely--once again we're picking on Chevy a 49:53.745 --> 49:56.105 second--so Chevy has his sponge. 49:56.110 --> 50:03.080 No one has thrown yet, and suppose Chevy knows that 50:03.083 --> 50:09.223 Patrick is going to throw tomorrow. 50:09.220 --> 50:11.150 Now what should Chevy do? 50:11.150 --> 50:16.240 Well that's a harder decision. 50:16.240 --> 50:16.830 What should Chevy do? 50:16.829 --> 50:19.989 He knows Patrick's going to shoot tomorrow. 50:19.990 --> 50:20.600 What should he do? 50:20.599 --> 50:23.679 Should he shoot or what, what's the answer this time? 50:23.680 --> 50:24.540 What do you reckon? 50:24.540 --> 50:25.150 Student: It depends. 50:25.150 --> 50:25.810 Professor Ben Polak: It depends. 50:25.809 --> 50:29.049 I think that's the right answer: it depends. 50:29.050 --> 50:31.160 Good, so the question is--someone else, 50:31.162 --> 50:34.442 I don't want to pick entirely on these guys--so what does it 50:34.441 --> 50:37.641 depend on? It's right that it depends. 50:37.640 --> 50:38.450 What does it depend on? 50:38.449 --> 50:41.419 Student: If the other guy's chances are greater than 50:41.417 --> 50:42.387 or less than 50%. 50:42.389 --> 50:45.389 Professor Ben Polak: All right, so it might depend on the 50:45.393 --> 50:48.113 other side's chances being less than or great than 50%. 50:48.110 --> 50:51.560 It certainly depends on the other guy's ability and on my 50:51.563 --> 50:53.693 ability. Everyone clear on that? 50:53.690 --> 50:56.660 Everyone agrees whether I should shoot now if I know the 50:56.657 --> 50:59.347 other guy is going to shoot tomorrow depends on our 50:59.354 --> 51:02.054 abilities, but how exactly does it depend 51:02.053 --> 51:03.223 on our abilities? 51:03.219 --> 51:07.929 Student: It depends on: if you're probability to hit is 51:07.929 --> 51:10.939 greater than his probability to miss. 51:10.940 --> 51:12.370 Professor Ben Polak: Good, your name is? 51:12.369 --> 51:13.469 Student: Osmont Professor Ben Polak: 51:13.468 --> 51:16.138 Osmont. So Osmont is saying--let's be 51:16.139 --> 51:19.289 careful here: it depends on whether my 51:19.289 --> 51:23.969 probability of hitting if I throw now is bigger than his 51:23.973 --> 51:27.723 probability of missing tomorrow. 51:27.719 --> 51:31.009 And why is that the right comparison? 51:31.010 --> 51:33.820 That's the right comparison because, if I throw now, 51:33.822 --> 51:37.082 my probability of winning the game is the probability that I 51:37.076 --> 51:40.486 hit my opponent. If I wait and take a step, 51:40.485 --> 51:45.975 then my probability of winning the game is the probability that 51:45.983 --> 51:48.203 he misses me tomorrow. 51:48.199 --> 51:50.709 So I have to compare winning probabilities with winning 51:50.713 --> 51:52.713 probabilities: I have to compare apples with 51:52.714 --> 51:54.394 apples, not apples with oranges. 51:54.390 --> 51:55.620 Everyone see that? 51:55.620 --> 51:57.620 Okay, so let's put that up. 51:57.619 --> 52:04.859 So the same assumption: assuming no one has thrown, 52:04.856 --> 52:12.666 if i knows (at d) that j will shoot tomorrow (at 52:12.672 --> 52:18.922 d--1), then i should shoot if--need a 52:18.916 --> 52:28.986 gap here--if i's probability of hitting at d--and let me leave a 52:28.988 --> 52:37.618 gap here--is bigger than or equal to--it doesn't really 52:37.621 --> 52:47.051 matter about the equal case--is greater than or equal to j's 52:47.054 --> 52:53.294 probability of missing tomorrow. 52:53.289 --> 52:57.739 Because this is the probability that you'll win if you throw, 52:57.739 --> 53:02.039 and this is the probability that you'll win if you wait. 53:02.039 --> 53:06.599 Okay, so let's put in what those things are. 53:06.599 --> 53:10.589 So the probability that i will hit at distance D, 53:10.591 --> 53:14.671 that's not that hard, that's Pi[d]--everyone happy 53:14.665 --> 53:19.365 with that? What's the probability that j 53:19.367 --> 53:23.267 will miss tomorrow if j throws? 53:23.269 --> 53:24.499 What's the probability that j will miss? 53:24.500 --> 53:29.850 Somebody? Let's be careful, 53:29.853 --> 53:40.573 so it's 1--Pj--but what distance will they be at--d--1: 53:40.569 --> 53:45.529 so it's (1--Pj[d--1]). 53:45.530 --> 53:50.990 So this is the key rule, if Chevy knows that Patrick's 53:50.987 --> 53:56.437 going to shoot tomorrow then Chevy should shoot if his 53:56.444 --> 54:01.804 probability of hitting Pi[d] is bigger than Patrick's 54:01.799 --> 54:06.329 probability of missing (1 - Pj[d--1]). 54:06.329 --> 54:08.159 Now I want to do one piece of math. 54:08.159 --> 54:11.409 This is the only math in this proof. 54:11.409 --> 54:13.589 So everyone who is math phobic, which I know there is a lot of 54:13.586 --> 54:15.046 you, can you just hold onto your seats? 54:15.050 --> 54:17.140 Don't panic: a little bit of math coming. 54:17.139 --> 54:22.169 This is the math, I want to add Pj[d--1] 54:22.173 --> 54:26.693 to both sides of this inequality. 54:26.690 --> 54:30.840 That's it, so what does that tell me? 54:30.840 --> 54:35.030 If add Pj[d -1] to this side, 54:35.028 --> 54:41.608 I get +Pj[d--1], everyone happy with that? 54:41.610 --> 54:49.070 On the other side if I add Pj[d--1], I get just 1. 54:49.070 --> 54:51.640 Everyone happy with that? 54:51.639 --> 54:53.769 So here's our rule, our rule is, 54:53.769 --> 54:57.129 let's flip it around, if Patrick hasn't thrown yet 54:57.134 --> 55:01.194 and thinks that Chevy's going to shoot tomorrow then Patrick 55:01.186 --> 55:04.896 should shoot now if his probability of hitting now plus 55:04.895 --> 55:09.355 Chevy's probability of hitting tomorrow is bigger than 1. 55:09.360 --> 55:14.580 Let's call this * and let's put this stuff up somewhere where we 55:14.575 --> 55:17.385 can use it for future reference. 55:17.390 --> 55:22.840 55:22.840 --> 55:24.860 Sorry guys, I'll use you again in a minute. 55:24.860 --> 55:28.590 I know you're feeling self conscious up there. 55:28.590 --> 55:30.580 Believe me, I'm self conscious up here too. 55:30.579 --> 55:34.139 So let's look at that * inequality up there. 55:34.139 --> 55:39.189 Now way out here, is that * inequality met or not 55:39.188 --> 55:39.818 met? 55:39.820 --> 55:43.010 55:43.010 --> 55:45.290 It's not met because way out here these two probabilities are 55:45.289 --> 55:46.619 small, so the sum is less than 1. 55:46.619 --> 55:51.869 In here, is the * inequality met or not met? 55:51.869 --> 55:56.749 Let me pick on you guys, so Patrick is it met or not met 55:56.752 --> 55:59.062 in here? Shout into your microphone. 55:59.060 --> 55:59.670 Student: It's met. 55:59.670 --> 56:01.100 Professor Ben Polak: It's met, thank you okay. 56:01.099 --> 56:03.649 So, in here, the inequality is met: 56:03.652 --> 56:07.502 the sum is bigger than 1; and out here it's less than 1. 56:07.500 --> 56:10.180 If we put in all the steps here--here they are getting 56:10.182 --> 56:11.652 closer and closer together. 56:11.650 --> 56:13.920 Here's the steps, as they get closer and closer 56:13.917 --> 56:17.237 together. We put these steps in. 56:17.239 --> 56:21.439 There's going to be some step where, for the first time, 56:21.436 --> 56:23.416 the * inequality is met. 56:23.420 --> 56:26.040 Notice that they start out here, they get closer and 56:26.037 --> 56:27.677 closer, it's not met, not met, 56:27.679 --> 56:29.149 not met, not met, not met, not met, 56:29.150 --> 56:30.620 then suddenly it's going to met. 56:30.620 --> 56:31.660 Maybe around here. 56:31.659 --> 56:36.669 Let's just try and pick it out, maybe it's here. 56:36.670 --> 56:41.980 So this might be the first time that this * inequality is met, 56:41.984 --> 56:43.644 let's call it d*. 56:43.640 --> 56:45.320 Everyone understand what D* is? 56:45.320 --> 56:48.540 At every one of these steps to the right of d*, 56:48.543 --> 56:51.633 when we take the sum of the probability Pi[d] 56:51.626 --> 56:54.776 + Pj[d--1], we get something less than 1. 56:54.780 --> 56:57.340 But to the left of d* or closer in than d*--the game is 56:57.342 --> 57:00.002 proceeding this way because it's moving right to left, 57:00.000 --> 57:03.140 they're getting closer and closer--to the left of d* the 57:03.142 --> 57:05.772 sum of those probabilities is bigger than 1. 57:05.769 --> 57:09.459 Let's say that again, d* is the first step at which 57:09.455 --> 57:12.915 the sum of those two probabilities exceeds 1. 57:12.920 --> 57:15.550 Everyone okay about what d* is? 57:15.550 --> 57:17.300 Anyone want to ask me a question? 57:17.300 --> 57:19.430 Okay, people should feel free to ask questions on this, 57:19.426 --> 57:21.156 I want to make sure everyone is following. 57:21.160 --> 57:22.740 Is everyone following so far? 57:22.740 --> 57:25.260 Yeah? I need to see all your eyes. 57:25.260 --> 57:27.100 You don't look like you're stuck in the headlamps like you 57:27.100 --> 57:29.060 were on Monday. I think we're better off than 57:29.063 --> 57:29.993 we were on Monday. 57:29.990 --> 57:33.490 Good okay. Okay, so here's our picture, 57:33.492 --> 57:38.142 and I actually want a bit more space--I'm not going to have 57:38.138 --> 57:39.658 much more space. 57:39.660 --> 57:45.640 57:45.639 --> 57:48.449 So now I'm going to tell you the solution. 57:48.449 --> 57:52.189 The solution to this game is this. 57:52.190 --> 57:59.080 I claim that the first shot should occur at d*. 57:59.080 --> 58:00.880 So that's my claim. 58:00.880 --> 58:06.050 58:06.050 --> 58:13.450 No one should shoot until you get to d*, and whoever's turn it 58:13.446 --> 58:20.476 is--whether it's Chevy's turn or Patrick's turn at d*--that 58:20.479 --> 58:23.389 person should shoot. 58:23.389 --> 58:26.039 That's my claim, and that's what we're going to 58:26.039 --> 58:28.199 prove. Everyone understand the claim? 58:28.199 --> 58:31.669 It says: nobody shoots, nobody shoots, 58:31.670 --> 58:34.300 nobody shoots, nobody shoots, 58:34.297 --> 58:36.917 nobody shoots, nobody shoots, 58:36.924 --> 58:38.904 shoot. Let's prove it, 58:38.897 --> 58:40.477 everyone ready to prove it? 58:40.480 --> 58:42.020 Yeah, people should be awake. 58:42.019 --> 58:44.829 If your neighbor's not awake, nudge them hard. 58:44.829 --> 58:50.749 Good, all right, let's start this analysis way 58:50.745 --> 58:53.895 out here, miles apart. 58:53.900 --> 58:57.700 These guys are miles apart and I want to use them as props. 58:57.699 --> 58:59.549 So I've got these guys--stay where you are Patrick but stand 58:59.554 --> 59:01.284 up, and Chevy over here somewhere, just where that black 59:01.282 --> 59:03.312 line is. Maybe they're even further than 59:03.310 --> 59:05.170 this. They're really far apart. 59:05.170 --> 59:08.370 And here they are miles away, and let's say it's Chevy's 59:08.372 --> 59:10.532 turn. He's way out here: 59:10.533 --> 59:14.043 perhaps the first step of the game. 59:14.039 --> 59:16.449 Imagine it's even further because it was even further, 59:16.445 --> 59:19.025 and let's think through what should be going on in Chevy's 59:19.031 --> 59:20.801 head. There are two possible things 59:20.800 --> 59:22.420 going on. He's going to think about what 59:22.424 --> 59:23.314 Patrick's going to do. 59:23.310 --> 59:24.090 So this is Chevy's turn. 59:24.090 --> 59:26.780 Here he is. And he should think: 59:26.776 --> 59:29.446 tomorrow, it's going to be Patrick's turn, 59:29.454 --> 59:31.614 and there's two possibilities. 59:31.610 --> 59:37.230 One possibility is that Patrick is not going to shoot tomorrow. 59:37.230 --> 59:41.720 And if we think that Patrick is not going to shoot tomorrow, 59:41.724 --> 59:44.014 which fact should Chevy use? 59:44.010 --> 59:46.720 Should he use Fact A or Fact B? 59:46.719 --> 59:48.019 Chevy which fact should you use? 59:48.015 --> 59:49.055 Student: Fact A. 59:49.059 --> 59:49.899 Professor Ben Polak: Fact A, okay. 59:49.900 --> 59:53.150 So, using Fact A, he should not shoot. 59:53.150 --> 59:56.070 Alternatively, he could think that Patrick's 59:56.067 --> 59:58.847 going to shoot tomorrow, is that right? 59:58.849 --> 1:00:00.139 He could think Patrick's going to shoot tomorrow. 1:00:00.139 --> 1:00:02.599 If he thinks Patrick's going to shoot tomorrow, 1:00:02.603 --> 1:00:03.943 which fact should he use? 1:00:03.942 --> 1:00:06.112 B. He should use Fact B, 1:00:06.107 --> 1:00:11.977 in which case he has to look at this inequality up here and say, 1:00:11.980 --> 1:00:15.960 I'll shoot if my probability of hitting now plus his probability 1:00:15.957 --> 1:00:18.417 of hitting tomorrow is bigger than 1. 1:00:18.420 --> 1:00:20.230 Well let's have a look. 1:00:20.230 --> 1:00:26.500 This is Patrick's probability of hitting today and this is 1:00:26.495 --> 1:00:31.435 Chevy's probability of--sorry--this is Chevy's 1:00:31.442 --> 1:00:35.792 probability of hitting today, and this is Patrick's 1:00:35.785 --> 1:00:36.905 probability of hitting tomorrow. 1:00:36.909 --> 1:00:40.429 And is the sum of them bigger than 1 or not? 1:00:40.430 --> 1:00:41.830 Is it bigger than 1 or not? 1:00:41.830 --> 1:00:42.800 It's not bigger than 1. 1:00:42.800 --> 1:00:45.200 So what should Chevy do? 1:00:45.200 --> 1:00:47.940 He should step. So he'd step. 1:00:47.940 --> 1:00:50.900 Now it's Patrick's turn, and once again, 1:00:50.902 --> 1:00:54.172 imagine this distance is still pretty large, 1:00:54.169 --> 1:00:57.739 and there's two things Patrick could think. 1:00:57.739 --> 1:01:02.749 Patrick could think that Chevy's not going to shoot 1:01:02.748 --> 1:01:05.078 tomorrow. So here's Patrick, 1:01:05.084 --> 1:01:08.994 he's looking forward to Chevy tomorrow, and he could think 1:01:08.992 --> 1:01:12.012 that Chevy's not going to shoot tomorrow. 1:01:12.010 --> 1:01:13.700 If he thinks Chevy's not going to shoot tomorrow which fact 1:01:13.704 --> 1:01:14.264 should he choose? 1:01:14.260 --> 1:01:14.640 Student: A. 1:01:14.637 --> 1:01:15.637 Professor Ben Polak: Should choose Fact A, 1:01:15.642 --> 1:01:18.412 okay. If he's using Fact A, 1:01:18.409 --> 1:01:21.009 he should not shoot. 1:01:21.010 --> 1:01:22.980 Alternatively, he could think that Chevy is 1:01:22.975 --> 1:01:25.455 going to shoot tomorrow, in which case he uses Fact B, 1:01:25.456 --> 1:01:26.576 and what does he do? 1:01:26.579 --> 1:01:30.429 He adds up his probability, Patrick's probability of 1:01:30.427 --> 1:01:34.797 hitting today plus Chevy's probability of hitting tomorrow: 1:01:34.803 --> 1:01:37.673 he asks is that sum bigger than 1, 1:01:37.670 --> 1:01:41.590 and he concludes, no. 1:01:41.590 --> 1:01:45.950 So we have no shot here and no shot here, and notice that both 1:01:45.951 --> 1:01:49.241 of those arguments were dominance arguments. 1:01:49.239 --> 1:01:51.959 In each case, whether Chevy thought that 1:01:51.964 --> 1:01:54.904 Patrick was going to shoot tomorrow or not, 1:01:54.898 --> 1:01:57.758 in either case, he concluded he should not 1:01:57.762 --> 1:02:00.202 shoot today. When it was Patrick's turn, 1:02:00.196 --> 1:02:02.456 whether Patrick thought that Chevy was going to shoot 1:02:02.462 --> 1:02:04.062 tomorrow or not, in either case, 1:02:04.064 --> 1:02:06.134 he concluded he should not shoot today. 1:02:06.130 --> 1:02:07.630 So he takes a step forward. 1:02:07.630 --> 1:02:11.430 This argument continues, it'll be Chevy's turn next, 1:02:11.430 --> 1:02:15.530 and once again he'll look at these two possibilities. 1:02:15.530 --> 1:02:17.120 If he thinks Patrick's not shooting tomorrow, 1:02:17.117 --> 1:02:18.777 he wants to step, if he thinks Patrick is going 1:02:18.776 --> 1:02:20.846 to shoot tomorrow, he's again going to want to 1:02:20.852 --> 1:02:22.042 step the way it's drawn. 1:02:22.039 --> 1:02:24.149 And, once again, we'll conclude step. 1:02:24.150 --> 1:02:27.010 We'll go on doing this argument, and everyone see that 1:02:27.005 --> 1:02:29.695 in each case this dominance argument will apply. 1:02:29.699 --> 1:02:32.439 It won't matter whether I think you should shoot tomorrow or 1:02:32.438 --> 1:02:35.188 not, in either case, it'll turn out that I should 1:02:35.194 --> 1:02:37.554 step forward: whether Fact A applies or 1:02:37.549 --> 1:02:40.999 whether Fact B applies, So we'll go on going forward 1:02:40.997 --> 1:02:43.097 and we'll have: no shot, no shot, 1:02:43.097 --> 1:02:45.967 no shot, no shot, no shot, no shot, 1:02:45.973 --> 1:02:48.183 and we'll arrive at d*. 1:02:48.180 --> 1:02:54.890 1:02:54.889 --> 1:02:58.659 So it turns out that d* is going to be Chevy's turn again. 1:02:58.659 --> 1:03:02.449 At d* we try exactly the same reasoning. 1:03:02.449 --> 1:03:05.589 At d* he says, if I think Patrick is not going 1:03:05.594 --> 1:03:08.184 to shoot tomorrow what should I do? 1:03:08.179 --> 1:03:09.359 What should I do if I think Patrick's not going to shoot 1:03:09.363 --> 1:03:10.123 tomorrow? Student: Not shoot. 1:03:10.116 --> 1:03:11.016 Professor Ben Polak: Not shoot. 1:03:11.019 --> 1:03:15.009 But now something different occurs. 1:03:15.010 --> 1:03:18.180 Now he says, if I think Patrick is going to 1:03:18.178 --> 1:03:20.968 shoot tomorrow, then when I look at my 1:03:20.969 --> 1:03:24.069 inequality up there, my * inequality, 1:03:24.070 --> 1:03:28.590 and add up my probability of hitting today--which is this 1:03:28.587 --> 1:03:32.377 line here--plus Patrick's probability of hitting 1:03:32.378 --> 1:03:36.978 tomorrow--which is this line here--suddenly he finds it is 1:03:36.976 --> 1:03:41.006 bigger than 1. So now if Chevy thinks that 1:03:41.014 --> 1:03:46.054 Patrick is going to shoot tomorrow, what should Chevy do? 1:03:46.050 --> 1:03:48.940 He should shoot. So up until this point, 1:03:48.944 --> 1:03:53.144 a dominance argument has told us no one should shoot, 1:03:53.138 --> 1:03:55.878 but suddenly we have a dilemma. 1:03:55.880 --> 1:03:58.060 The dilemma is: if Chevy thinks Patrick's not 1:03:58.061 --> 1:04:00.951 shooting, he should step; and if Chevy thinks Patrick is 1:04:00.947 --> 1:04:02.337 shooting, he should shoot. 1:04:02.340 --> 1:04:03.810 Everyone with me so far? 1:04:03.810 --> 1:04:05.370 So what have we shown so far? 1:04:05.369 --> 1:04:09.179 We've shown that no one should shoot until d* but we're stuck 1:04:09.183 --> 1:04:11.793 because we don't know what to do at d*; 1:04:11.789 --> 1:04:14.369 because we don't know what Chevy should believe at d*. 1:04:14.369 --> 1:04:17.739 We don't know whether Chevy should believe that Patrick's 1:04:17.742 --> 1:04:21.362 going to shoot or whether Chevy should believe that Patrick's 1:04:21.355 --> 1:04:22.735 not going to shoot. 1:04:22.739 --> 1:04:27.629 So how do we figure out what Chevy should believe Patrick's 1:04:27.629 --> 1:04:30.899 going to do? Wait, wake up the guy in orange 1:04:30.896 --> 1:04:34.326 there, the guy with the ginger hair, that's right. 1:04:34.329 --> 1:04:38.029 What's the answer to that question? 1:04:38.030 --> 1:04:39.880 Good, the answer to the question is backward induction. 1:04:39.880 --> 1:04:45.410 Round of applause for remembering the answer. 1:04:45.409 --> 1:04:49.309 Good, backward induction is the answer to all questions, 1:04:49.313 --> 1:04:52.013 especially when you're asleep right. 1:04:52.010 --> 1:04:54.780 Okay, so now we're going to use backward induction, 1:04:54.783 --> 1:04:57.393 but where does backward induction start here? 1:04:57.389 --> 1:05:02.129 Backward induction starts at the back of the game, 1:05:02.134 --> 1:05:05.914 and what's the back of the game here? 1:05:05.910 --> 1:05:07.560 The back of the game is where? 1:05:07.559 --> 1:05:10.329 It's when these two guys, neither of them have thrown 1:05:10.333 --> 1:05:12.523 their sponge, and they've reached here. 1:05:12.519 --> 1:05:15.499 So come here a second, step, step. 1:05:15.500 --> 1:05:19.430 Let's assume it's Patrick's turn, and they're absurdly 1:05:19.427 --> 1:05:22.167 close: they're uncomfortably close. 1:05:22.170 --> 1:05:23.970 If they had longer noses they'd be touching. 1:05:23.970 --> 1:05:27.010 They're at distance 0. 1:05:27.010 --> 1:05:31.810 And at distance 0, at d = 0, let's suppose it's 1:05:31.807 --> 1:05:34.187 Patrick's turn. So at d = 0, 1:05:34.190 --> 1:05:36.020 no one has shot, it's Patrick's turn, 1:05:36.020 --> 1:05:38.410 he's got the sponge, what should Patrick do? 1:05:38.410 --> 1:05:39.560 Shout it out Patrick. 1:05:39.559 --> 1:05:40.359 Student: I should shoot. 1:05:40.357 --> 1:05:41.537 Professor Ben Polak: You should shoot. 1:05:41.540 --> 1:05:42.840 Patrick should shoot, right? 1:05:42.840 --> 1:05:48.340 At d = 0, say it's 2's turn, and the answer is he should 1:05:48.344 --> 1:05:54.854 shoot because the probability of it hitting at distance 0 is 1. 1:05:54.849 --> 1:05:57.179 Let's just move you to the side a bit so that people can see the 1:05:57.183 --> 1:05:59.103 board. I know it's an awkward dance 1:05:59.100 --> 1:06:01.590 but here you are--stop there: that's good. 1:06:01.590 --> 1:06:06.950 So at distance 0 they should certainly shoot. 1:06:06.949 --> 1:06:10.089 So now let's go back a step in the backward induction. 1:06:10.090 --> 1:06:12.150 So we're at distance 1, just take a step back. 1:06:12.150 --> 1:06:19.890 So take a step back, it's Chevy's turn at distance 1:06:19.888 --> 1:06:23.038 1. And what does Chevy know at 1:06:23.039 --> 1:06:24.819 distance 1? Chevy what do you know? 1:06:24.820 --> 1:06:26.900 Shout it out. Student: That Patrick 1:06:26.898 --> 1:06:28.058 will shoot next turn. 1:06:28.059 --> 1:06:30.739 Professor Ben Polak: Right, so now Chevy knows that 1:06:30.744 --> 1:06:32.444 Patrick's going to shoot tomorrow. 1:06:32.440 --> 1:06:36.540 So which fact should Chevy use in deciding whether he should 1:06:36.537 --> 1:06:37.867 shoot today? B. 1:06:37.868 --> 1:06:46.828 He should use Fact B, and that tells us--so 1 knows 1:06:46.831 --> 1:06:56.941 that 2 will shoot tomorrow, so by B, 1 should shoot if his 1:06:56.937 --> 1:07:04.817 probability of hitting at distance 1 plus Patrick's 1:07:04.816 --> 1:07:11.116 probability of hitting at distance 0, 1:07:11.120 --> 1:07:14.840 if that is bigger than 1. 1:07:14.840 --> 1:07:16.270 Well is it bigger than 1? 1:07:16.270 --> 1:07:17.640 Well let's have a look. 1:07:17.639 --> 1:07:20.789 We had a shot in here already, we put a shot in here, 1:07:20.790 --> 1:07:22.850 and we're looking at distance 1. 1:07:22.849 --> 1:07:26.889 And so we're looking at this distance here plus this distance 1:07:26.888 --> 1:07:29.488 here. Is it bigger than 1? 1:07:29.490 --> 1:07:31.340 Yeah, it's bigger than 1. 1:07:31.340 --> 1:07:32.320 Here's a bit more math. 1:07:32.320 --> 1:07:33.420 I lied to you before. 1:07:33.420 --> 1:07:35.310 1 plus something is bigger than 1. 1:07:35.310 --> 1:07:38.930 So this is bigger than 1. 1:07:38.930 --> 1:07:44.280 So shoot. Let's put it on our chart as 1:07:44.284 --> 1:07:45.674 shoot. Let's go back a 1:07:45.674 --> 1:07:47.734 step--sorry--I'll have to have Chevy do all the going 1:07:47.734 --> 1:07:49.594 backwards. Now we're at distance what? 1:07:49.590 --> 1:07:53.770 We're at distance 2, we're at d = 2, 1:07:53.768 --> 1:08:00.338 and it's Patrick's turn, 2's turn, and what does Patrick 1:08:00.335 --> 1:08:02.425 know? Shout it out Patrick. 1:08:02.429 --> 1:08:04.249 Student: I know that Chevy's going to shoot next 1:08:04.253 --> 1:08:05.993 turn. Professor Ben Polak: 1:08:05.985 --> 1:08:08.805 Patrick knows that Chevy's going to shoot next turn, 1:08:08.811 --> 1:08:11.361 so Patrick therefore should use which fact? 1:08:11.360 --> 1:08:14.330 Fact B. And Fact B tells him that he 1:08:14.330 --> 1:08:17.290 should shoot--let's just put this in. 1:08:17.289 --> 1:08:22.839 So 2 knows that 1 will shoot tomorrow, so by B it's all the 1:08:22.836 --> 1:08:28.146 same thing. We know that 2 should shoot if 1:08:28.145 --> 1:08:29.765 P2[2] + P1[1] 1:08:29.771 --> 1:08:35.731 is bigger than 1, and if we look at it on the 1:08:35.734 --> 1:08:41.024 board--here we are--it's 2's turn, 1:08:41.020 --> 1:08:43.540 he's looking at this distance plus this distance, 1:08:43.536 --> 1:08:44.896 and is it bigger than 1? 1:08:44.900 --> 1:08:46.820 It is bigger than 1. 1:08:46.820 --> 1:08:49.810 So he should shoot. 1:08:49.810 --> 1:08:53.530 And we can go on doing this argument backwards., 1:08:53.526 --> 1:08:57.946 We'll find that Chevy should shoot here because this plus 1:08:57.954 --> 1:09:00.014 this is bigger than 1. 1:09:00.010 --> 1:09:02.300 And we'll know that here Patrick once again will know 1:09:02.301 --> 1:09:04.021 that Chevy's going to shoot tomorrow, 1:09:04.020 --> 1:09:05.810 so he should use Fact B, so he should shoot provided 1:09:05.805 --> 1:09:07.375 this plus this is bigger than 1, but it is. 1:09:07.380 --> 1:09:10.800 Now, we're back at d* and the question we had at d*-- the 1:09:10.795 --> 1:09:13.535 question we'd left hanging at d*--was what? 1:09:13.539 --> 1:09:17.379 At d* we knew already that Chevy would not shoot if he 1:09:17.383 --> 1:09:21.013 thought Patrick was not going to shoot tomorrow, 1:09:21.010 --> 1:09:23.190 but he should shoot if thinks Patrick is going to shoot 1:09:23.192 --> 1:09:24.932 tomorrow, but what does Chevy know at d*? 1:09:24.930 --> 1:09:26.230 Student: I know Patrick is going to shoot. 1:09:26.229 --> 1:09:28.209 Professor Ben Polak: He knows Patrick's going to shoot, 1:09:28.208 --> 1:09:28.888 so he should shoot. 1:09:28.890 --> 1:09:32.190 Is that right? He knows Patrick's going to 1:09:32.188 --> 1:09:34.798 shoot by backward induction so he should shoot. 1:09:34.800 --> 1:09:35.710 So we just solved this. 1:09:35.710 --> 1:09:37.460 What did we actually show? 1:09:37.460 --> 1:09:42.770 We showed, have seats gentlemen--sorry to keep you up 1:09:42.772 --> 1:09:48.602 here--we know that prior to d* no one will shoot--will not 1:09:48.596 --> 1:09:54.066 shoot--and we know that at d*, and in fact at any point 1:09:54.066 --> 1:09:56.476 further on, we should shoot. 1:09:56.479 --> 1:09:58.679 That's horrible writing but it says shoot. 1:09:58.680 --> 1:10:04.010 So we've shown more than we claimed. 1:10:04.010 --> 1:10:05.520 We claimed that the first shot should occur at d*, 1:10:05.516 --> 1:10:07.386 and we've actually shown more than that: we've shown that even 1:10:07.390 --> 1:10:10.030 if you went beyond d*, and if somebody had forgotten 1:10:10.029 --> 1:10:13.009 to shoot at d*, at least you should shoot now. 1:10:13.010 --> 1:10:14.670 Give me like two more minutes or three more minutes to finish 1:10:14.665 --> 1:10:15.875 this up because we're at a high point now. 1:10:15.880 --> 1:10:17.510 Everyone okay to wait a couple of minutes? 1:10:17.510 --> 1:10:20.700 Okay, so what did we prove here? 1:10:20.699 --> 1:10:28.949 We proved that the first shot occurs at d* whoever's turn it 1:10:28.951 --> 1:10:32.251 is at d*. It wasn't that the best guy, 1:10:32.248 --> 1:10:34.448 the better shot, should shoot first, 1:10:34.447 --> 1:10:36.957 or the worse shot should shoot first. 1:10:36.960 --> 1:10:39.470 It turned out that, given their abilities, 1:10:39.465 --> 1:10:43.065 there was a critical distance at which they should shoot. 1:10:43.069 --> 1:10:45.829 If you go back to the eighteenth century military 1:10:45.832 --> 1:10:49.172 strategy, you should shoot when you see the whites of their 1:10:49.171 --> 1:10:50.611 eyes, which is at d*. 1:10:50.609 --> 1:10:56.119 But we learned something else on the way. 1:10:56.119 --> 1:10:59.169 I claimed we learned that if you're patient and you go 1:10:59.167 --> 1:11:01.787 through things carefully, that the arguments we've 1:11:01.791 --> 1:11:04.201 learned from the course so far, dominance arguments and 1:11:04.201 --> 1:11:08.061 backward induction arguments, can solve out a really quite 1:11:08.063 --> 1:11:11.223 hard problem. This was hard. 1:11:11.220 --> 1:11:13.820 It would have been useful for the guy in War and Peace, 1:11:13.817 --> 1:11:16.557 or in Onegin or the guys cycling in the Tour de France, 1:11:16.560 --> 1:11:17.910 or you guys with your sponges to know this. 1:11:17.909 --> 1:11:22.069 And we can solve this exactly using backward induction, 1:11:22.069 --> 1:11:24.919 and everyone in the room can do it. 1:11:24.920 --> 1:11:29.020 Let me just push the argument a tiny bit further. 1:11:29.020 --> 1:11:32.060 One thing we've always asked in this class, is okay that's fine 1:11:32.058 --> 1:11:34.258 if everyone knows what's going on in the game: 1:11:34.264 --> 1:11:37.064 here we have our smart Yale football players and they know 1:11:37.057 --> 1:11:39.927 how to play this game, so they're going to shoot at 1:11:39.925 --> 1:11:41.775 the right time. But what happens if, 1:11:41.779 --> 1:11:44.429 instead of playing another smart Yale football player, 1:11:44.430 --> 1:11:47.530 they're playing some uneducated probably simple-minded football 1:11:47.529 --> 1:11:51.749 player from, say, Harvard. 1:11:51.750 --> 1:11:55.360 Now that changes things a bit doesn't it because we know that 1:11:55.361 --> 1:11:58.011 the Yale football player is sophisticated, 1:11:58.010 --> 1:11:59.790 has taken my class, and knows that he should shoot 1:11:59.790 --> 1:12:01.970 at d*, but the Harvard guy doesn't learn anything anymore, 1:12:01.970 --> 1:12:04.120 so they're stuck. 1:12:04.119 --> 1:12:07.919 So if you're the Yale guy playing the Harvard guy how does 1:12:07.922 --> 1:12:09.792 that change your decision? 1:12:09.789 --> 1:12:14.129 Should you shoot earlier than d* when you're playing against 1:12:14.127 --> 1:12:17.357 the Harvard guy, or later than d* when you're 1:12:17.363 --> 1:12:20.013 playing against the Harvard guy? 1:12:20.010 --> 1:12:22.020 Let's try our Yale guys and see what they think, 1:12:22.018 --> 1:12:23.128 what do you think Chevy? 1:12:23.130 --> 1:12:24.450 Student: Definitely not earlier. 1:12:24.449 --> 1:12:25.969 Professor Ben Polak: Definitely not earlier, 1:12:25.968 --> 1:12:26.848 that's the key thing right. 1:12:26.850 --> 1:12:28.380 Now why? Why definitely not earlier? 1:12:28.380 --> 1:12:33.410 Student: Because if you miss, the other person has a 1:12:33.406 --> 1:12:37.216 probability of 1, you have a higher chance of 1:12:37.220 --> 1:12:40.240 missing. Professor Ben Polak: All 1:12:40.236 --> 1:12:42.336 right, so I claim Chevy's right. 1:12:42.340 --> 1:12:44.280 That's good because I've just claimed that Yale football 1:12:44.281 --> 1:12:45.271 players are sophisticated. 1:12:45.270 --> 1:12:47.380 Chevy's right, that even if you're playing 1:12:47.381 --> 1:12:49.961 against a Harvard guy you shouldn't shoot before d* 1:12:49.956 --> 1:12:53.046 because it was a dominant strategy not to shoot before 1:12:53.046 --> 1:12:54.916 d*. It doesn't matter whether you 1:12:54.919 --> 1:12:57.929 think the Harvard guy is going to be dumb enough to shoot early 1:12:57.933 --> 1:12:59.513 or not. If he is dumb enough to shoot 1:12:59.507 --> 1:13:00.517 early, so much the better. 1:13:00.520 --> 1:13:02.470 You should wait until D*. 1:13:02.470 --> 1:13:04.720 Notice that argument doesn't depend on you playing against 1:13:04.724 --> 1:13:07.554 somebody who is sophisticated, or someone who's less 1:13:07.546 --> 1:13:10.876 sophisticated, like a Harvard football player, 1:13:10.880 --> 1:13:13.770 or somebody who's basically a chair, 1:13:13.770 --> 1:13:16.420 like a Harvard football player. 1:13:16.420 --> 1:13:19.460 You shouldn't shoot before d* because it's a dominant strategy 1:13:19.455 --> 1:13:20.695 not to shoot before d*. 1:13:20.699 --> 1:13:22.789 Now, you might want to wait a little to see if they're not 1:13:22.794 --> 1:13:24.844 going to shoot early, to see if he's not going to 1:13:24.838 --> 1:13:26.968 shoot, but you certainly shouldn't shoot early. 1:13:26.970 --> 1:13:28.800 Let me finish with one other thing. 1:13:28.800 --> 1:13:32.540 Every time when we've played this game in class, 1:13:32.541 --> 1:13:37.081 whether it's here or up in SOM, people shoot too early. 1:13:37.080 --> 1:13:41.160 They miss. You can do the econometrics on 1:13:41.162 --> 1:13:43.122 this, you could figure out that, on average--average 1:13:43.123 --> 1:13:45.163 abilities--I'm sometimes getting the better shots, 1:13:45.159 --> 1:13:48.759 sometimes I'm getting the worse shots--on average I should see 1:13:48.764 --> 1:13:52.314 people hitting about half of the time over a large sample. 1:13:52.310 --> 1:13:55.090 But here I tend to see people miss, as we did today, 1:13:55.090 --> 1:13:56.290 almost all the time. 1:13:56.290 --> 1:14:03.360 Why do we see so many misses? 1:14:03.359 --> 1:14:06.539 So one problem may be that people are just overconfident. 1:14:06.539 --> 1:14:09.279 They're overconfident on their ability to throw. 1:14:09.279 --> 1:14:12.349 And there's a large literature in Economics about how people 1:14:12.353 --> 1:14:13.763 tend to be overconfident. 1:14:13.760 --> 1:14:16.200 But there's another possible explanation, and let me just 1:14:16.201 --> 1:14:18.251 push it past you as the last thing for today. 1:14:18.250 --> 1:14:21.660 I think Americans--I think this doesn't go for the 1:14:21.661 --> 1:14:25.561 Brits--Americans have what I call a "pro-active bias." 1:14:25.560 --> 1:14:28.660 You guys are brought up since you're in kindergarten--maybe 1:14:28.655 --> 1:14:31.425 before--and you're told you have to be pro-active. 1:14:31.430 --> 1:14:33.840 You have "to make the world come to you." 1:14:33.840 --> 1:14:37.970 And my evidence for this is based on sophisticated empirical 1:14:37.970 --> 1:14:40.070 work watching Sports Center. 1:14:40.069 --> 1:14:44.619 So on Sports Center when they interview these sweaty athletes 1:14:44.622 --> 1:14:47.582 after the game, the sweaty athletes say, 1:14:47.581 --> 1:14:50.921 it's great I now control my own destiny. 1:14:50.920 --> 1:14:52.560 Well, I'm a Brit. 1:14:52.560 --> 1:14:55.090 I think controlling my own destiny sounds kind of scary to 1:14:55.088 --> 1:14:56.308 me; it doesn't sound like a good 1:14:56.310 --> 1:15:00.350 thing at all. In fact, if I wanted to control 1:15:00.350 --> 1:15:05.490 my own destiny, I wouldn't have got married. 1:15:05.489 --> 1:15:10.039 That's going to be edited off the film, but the point I want 1:15:10.041 --> 1:15:12.831 to make is this. Every time I play this game, 1:15:12.829 --> 1:15:15.479 when I ask people why they shoot early I hear the same 1:15:15.479 --> 1:15:18.029 thing, and it's evidence to this proactive bias. 1:15:18.029 --> 1:15:23.199 People say, well at least I went down swinging and the 1:15:23.204 --> 1:15:28.484 problem is: the aim in life is not to go down swinging, 1:15:28.476 --> 1:15:30.816 it's not to go down. 1:15:30.819 --> 1:15:36.989 So one lesson to get from this lecture is, sometimes waiting is 1:15:36.988 --> 1:15:40.308 a good strategy. Alright, and we'll come back to 1:15:40.309 --> 1:15:40.999 it on Monday.