WEBVTT 00:03.420 --> 00:06.370 Prof: We're now at the stage where we're considering 00:06.369 --> 00:10.439 the implications of uncertainty, so I hope that the subtlety, 00:10.442 --> 00:13.392 and surprise element of the class, 00:13.390 --> 00:18.090 will gradually pick up without increasing the difficulty. 00:18.090 --> 00:20.470 The complexity will pick up a little bit, but not the 00:20.466 --> 00:21.056 difficulty. 00:21.060 --> 00:23.600 It's just you'll have to keep a few more things in your head, 00:23.599 --> 00:25.249 but the mathematics isn't any harder. 00:25.250 --> 00:31.000 So we ended last time talking about default and inferring 00:30.996 --> 00:36.126 default probabilities, and so I just want to finish 00:36.127 --> 00:38.587 off that discussion. 00:38.590 --> 00:42.050 So suppose that at any stage of the tree, you know, 00:42.051 --> 00:44.891 lots of things can happen in the world. 00:44.890 --> 00:48.130 We're always going to model the uncertainty in the future by a 00:48.127 --> 00:50.247 tree with different things happening, 00:50.250 --> 00:55.330 and at each of these nodes people are going to have a 00:55.327 --> 00:56.887 discount rate. 00:56.890 --> 01:02.640 So maybe it'll be r equals 20 percent, 01:02.640 --> 01:08.440 and here r could equal 15 percent, something like that, 01:08.438 --> 01:12.658 and we want to add to this the possibility that there's 01:12.658 --> 01:13.438 default. 01:13.438 --> 01:18.398 So if we add the possibility of default, and these things keep 01:18.397 --> 01:22.787 going and maybe there are payoffs at the end or payoffs 01:22.786 --> 01:24.246 along the way. 01:24.250 --> 01:27.430 If at any point in the tree like this one we add a new 01:27.430 --> 01:31.520 possibility, which is the default 01:31.522 --> 01:37.582 possibility, so this happens-- by the way, when do people 01:37.584 --> 01:37.974 default? 01:37.970 --> 01:40.300 They never default before they have to make a payment. 01:40.300 --> 01:42.830 So when do they default, exactly when they're supposed 01:42.833 --> 01:43.793 to make a payment. 01:43.790 --> 01:48.580 So suppose that this guy is going to default here when he's 01:48.575 --> 01:50.715 going to make a payment. 01:50.720 --> 01:53.530 At every possible scenario he would default there. 01:53.530 --> 01:56.980 So we've got a very simple model of default, 01:56.976 --> 02:01.696 so not a very realistic one where the guy defaults in all of 02:01.704 --> 02:04.114 these following scenarios. 02:04.109 --> 02:06.099 So something's just bad. 02:06.099 --> 02:08.899 Once he's gotten here you know that he's not going to make the 02:08.896 --> 02:10.086 payment the next period. 02:10.090 --> 02:13.410 We further assume that not only does he default there, 02:13.413 --> 02:16.113 but he defaults on everything thereafter. 02:16.110 --> 02:19.010 So the payoff is just going to be 0 here. 02:19.008 --> 02:20.848 So this is going to be--originally we had 02:20.852 --> 02:23.062 probabilities p_1, p_2, 02:23.055 --> 02:24.765 p_3, let's say for the 02:24.771 --> 02:28.941 probabilities, now we're going to have 02:28.938 --> 02:37.758 probabilities d for default and then 1 - d times all of these, 02:37.759 --> 02:38.679 right? 02:38.680 --> 02:40.800 So essentially what have we done? 02:40.800 --> 02:44.750 We've simply replaced in our calculation of payoffs and 02:44.745 --> 02:47.475 present values, we've simply replaced these 02:47.478 --> 02:49.788 possibilities with probability p_1, 02:49.788 --> 02:51.468 p_2, p_3. 02:51.470 --> 02:54.760 We added another possibility, but the payoffs are 0 here. 02:54.758 --> 02:57.378 Nothing's going to happen from then on except 0, 02:57.378 --> 02:59.738 and we said that happened with probability d, 02:59.740 --> 03:02.300 which means presumably, all of these have to be scaled 03:02.299 --> 03:04.279 down by that so they still add up to 1. 03:04.280 --> 03:18.110 So essentially the point I'm trying to make is that default 03:18.110 --> 03:29.080 that leads to 0 payoffs thereafter is just like 03:29.080 --> 03:34.090 discounting more. 03:34.090 --> 03:34.960 Why is that? 03:34.960 --> 03:38.660 Because whatever calculation you did for the value here of 03:38.662 --> 03:41.912 what the bond could possibly be worth there is-- 03:41.910 --> 03:45.150 it's all the same numbers as there were before except we've 03:45.147 --> 03:48.907 multiplied it by 1 - d, so it's the same thing. 03:48.910 --> 03:57.620 So instead of going 1 over 1 r times future payoffs, 03:57.622 --> 04:07.192 that's no default value, so that times future payoffs. 04:07.188 --> 04:13.678 Now we've got default value under this special kind of 04:13.679 --> 04:20.899 default is going to be 1 - d times 1 over 1 r times the same 04:20.903 --> 04:23.233 future payoffs. 04:23.230 --> 04:29.140 I could rewrite 1 - d as 1 over 1 s or something, 04:29.139 --> 04:36.649 and so then I really have just 1 over 1 r times 1 over 1 s. 04:36.649 --> 04:41.799 So that's just 1 over (1 r s r s). 04:41.800 --> 04:55.380 So that's going to equal 1 over (1 r s r s) times future values. 04:55.379 --> 04:59.599 So the effect of this special kind of default--we just get 0 04:59.600 --> 05:00.530 thereafter. 05:00.528 --> 05:02.908 The guy decides after this payment, "I'm not going to 05:02.913 --> 05:03.963 make any more payments. 05:03.959 --> 05:05.439 I'm defaulting from then on." 05:05.439 --> 05:10.079 That's the same thing when valuing the future payoffs, 05:10.079 --> 05:13.529 it's the same thing as instead of discounting by r, 05:13.528 --> 05:18.898 discounting by r s with a little bit r times s. 05:18.899 --> 05:20.899 If r and s are small numbers this [r times s] 05:20.899 --> 05:22.489 is probably quite a small number. 05:22.490 --> 05:30.300 So default probabilities get mapped into spreads they're 05:30.300 --> 05:31.580 called. 05:31.579 --> 05:36.429 They way to evaluate it is just you just multiply it by 1 - d, 05:36.430 --> 05:39.730 which is the same thing as discounting by a higher number, 05:39.730 --> 05:42.600 and that higher number is almost the same. 05:42.600 --> 05:46.100 It's very close to r d, as a matter of fact, 05:46.103 --> 05:48.553 because 1 - d is 1 over 1 s. 05:48.550 --> 06:01.680 If d is very small s is going to be very close to d as well. 06:01.680 --> 06:03.360 So what's the implication of this? 06:03.360 --> 06:10.750 The implication of this in a special case again where we just 06:10.752 --> 06:16.422 have no uncertainty, except we have default. 06:16.420 --> 06:19.280 So here there could be d_1, 06:19.278 --> 06:22.418 1 - d_1 and here there could be probability of 06:22.420 --> 06:27.230 default 2 and 1 - d_2, probability of default 3,1 - 06:27.232 --> 06:30.872 d_3, and we've got interest rates 06:30.870 --> 06:35.960 r_0-- so this will be i_0, 06:35.959 --> 06:40.939 i^(F)_1, i^(F)_2. 06:40.940 --> 06:44.070 So if you knew what the interest rate was going to be 06:44.067 --> 06:45.667 today, you knew what the interest rate 06:45.668 --> 06:47.738 was going to be tomorrow, you knew what the interest rate 06:47.735 --> 06:49.245 was going to be the day after tomorrow, 06:49.250 --> 06:51.120 there's no uncertainty about interest rates, 06:51.120 --> 06:52.890 they're perfectly anticipatable, 06:52.894 --> 06:55.984 but you know that there's a probability of default each 06:55.983 --> 06:56.503 time. 06:56.500 --> 06:59.960 So in stage 1 this guy might default before making his 06:59.956 --> 07:02.826 payment here, which case you're just going to 07:02.826 --> 07:03.476 get 0. 07:03.480 --> 07:07.030 In stage 2 he might default instead of making his payment, 07:07.033 --> 07:09.533 won't pay the coupon, he'll just default, 07:09.528 --> 07:11.648 or in year 3 he might default. 07:11.649 --> 07:17.349 So what's the implication of what we just said? 07:17.350 --> 07:23.090 You can evaluate this bond, the payoffs of the bond, 07:23.088 --> 07:28.038 so let's say it pays a coupon, C, C 100 C. 07:28.040 --> 07:31.730 All right, the way you would evaluate that without default is 07:31.728 --> 07:34.618 you would just take the value of the coupon, 07:34.620 --> 07:41.120 the present value would have been--you would have done it 07:41.124 --> 07:42.754 recursively. 07:42.750 --> 07:48.650 You would have gotten P_3 = 100 C. 07:48.649 --> 07:51.229 Then you would have said P_2, 07:51.230 --> 07:53.040 you would have gone in your computer-- 07:53.040 --> 08:00.480 you would have said P_2 is 1 over (1 08:00.475 --> 08:05.925 i^(F)_2) times 100 C. 08:05.930 --> 08:09.830 So the value here is the 100 C discounted by that forward rate, 08:09.829 --> 08:15.129 then P_1 would have been (C P_2) divided 08:15.132 --> 08:17.562 by (1 i^(F)_1). 08:17.560 --> 08:19.420 So you take the value here times that, 08:19.420 --> 08:21.060 so this is case there's no default, 08:21.060 --> 08:27.980 and P_0 would have been (C P_1) over (1 08:27.975 --> 08:30.475 i^(F)_0). 08:30.480 --> 08:33.140 So that's how you would have done it by backward induction. 08:33.139 --> 08:37.459 But now that you know there's a chance that there's default you 08:37.461 --> 08:40.671 have to not multiply by 1 i^(F)_2. 08:40.668 --> 08:44.758 You have to multiply all these things by the probability of 08:44.764 --> 08:45.474 default. 08:45.470 --> 08:51.620 So you'd have to multiply this by, if we change colors, 08:51.616 --> 08:52.866 by 1 - d. 08:52.870 --> 08:58.730 You have to multiply this by 1 - d_2, 08:58.730 --> 09:03.750 and you'd have to multiply this by (1 - d_1) times (1 09:03.750 --> 09:07.600 - d_2), and this by 1 -, 09:07.597 --> 09:13.027 sorry this would be 1 - d_3. 09:13.028 --> 09:17.068 This is (1 - d_3) times d_2, 09:17.072 --> 09:22.402 and this is (1 - d_1) times (1 - d_2) times 09:22.404 --> 09:24.474 (1 - d_3). 09:24.470 --> 09:27.990 So this value would be the old value you got here multiplied by 09:27.993 --> 09:30.613 1 - d_3, multiplied by this one minus 09:30.606 --> 09:31.626 d_3. 09:31.629 --> 09:34.239 This value is what you would have gotten here, 09:34.240 --> 09:37.370 but you've already scaled it down, so multiplied by 1 - 09:37.374 --> 09:40.164 d_2 and this is 1 - d_1. 09:40.159 --> 09:45.949 So what's the upshot? 09:45.950 --> 09:53.090 So that gives you the price with default. 09:53.090 --> 09:53.720 Student: Professor? 09:53.720 --> 09:54.480 Prof: Yep? 09:54.480 --> 09:57.800 Student: When you're doing the backwards induction 09:57.801 --> 10:01.361 should you be tacking on like that or should we just take one 10:01.359 --> 10:02.189 at a time. 10:02.190 --> 10:05.360 So P_0 should be multiplied by all 3, 10:05.361 --> 10:08.741 or isn't it that you're taking whatever you get as 10:08.740 --> 10:12.190 P_1 as given and just multiply by... 10:12.190 --> 10:13.740 Prof: Well, I could have done it two ways. 10:13.740 --> 10:16.980 I could have written--so what you're suggesting is a better 10:16.975 --> 10:19.425 way would have been to say P_2, 10:19.428 --> 10:25.368 the default bond P_2 is going to be (1 - 10:25.365 --> 10:29.965 d_3) times P_3, 10:29.970 --> 10:33.090 which is also equal to P_3 because there's no 10:33.086 --> 10:34.296 default after here. 10:34.299 --> 10:35.119 The world just ends. 10:35.120 --> 10:41.180 Then P_1 is equal to (1 - d_2) times 10:41.177 --> 10:45.717 P_2, but that P_2, 10:45.722 --> 10:50.952 remember, is already (1 - d_3) times 10:50.952 --> 10:52.932 P_3. 10:52.928 --> 10:59.318 And then P_0 is going to be (1 - d_1) times 10:59.317 --> 11:03.747 (1 - d_2) times P_1, 11:03.750 --> 11:07.900 but that's equal to (1 - d_1) times (1 - 11:07.899 --> 11:11.119 d_2( times P_1, 11:11.120 --> 11:18.700 which is (1 - d_3) times P_3, 11:18.700 --> 11:22.180 right? 11:22.178 --> 11:24.838 So all right, if you go from here to here the 11:24.841 --> 11:27.871 value here's 100 C, so if P_3 is just 100 11:27.865 --> 11:29.675 C let's leave it as 100 C. 11:29.679 --> 11:33.659 That's if the guy actually pays. 11:33.658 --> 11:40.078 So the present value would just be--oh, then you have to divide 11:40.081 --> 11:43.191 all this by 1 i_2. 11:43.190 --> 11:46.200 Sorry, this is 1 i^(F)_2. 11:46.200 --> 11:50.360 Oh, I'm making a mess of this. 11:50.360 --> 11:53.990 So usually you'll go back from here to here by discounting by 11:53.989 --> 11:56.739 the interest rate, but now we're going to have to 11:56.735 --> 11:59.635 also multiply by the probability that you default to go back 11:59.635 --> 12:00.025 here. 12:00.029 --> 12:01.279 So we get a lower number. 12:01.278 --> 12:04.788 P_2 is not just (100 C) divided by (1 i^(F)), 12:04.785 --> 12:08.095 that's the discounting, you also have to multiply by 12:08.096 --> 12:10.106 the probability of default. 12:10.110 --> 12:13.660 Then when you go back one period further you have to 12:13.657 --> 12:14.837 discount again. 12:14.840 --> 12:18.280 So I should have divided this by 1 i^(F)_1, 12:18.284 --> 12:21.804 you have to discount it, and also you have to multiply 12:21.797 --> 12:24.047 by the probability of default. 12:24.048 --> 12:26.858 But the thing you're bringing backwards is P_2, 12:26.857 --> 12:29.567 which has already taken into account the probability of 12:29.566 --> 12:30.866 default the next time. 12:30.870 --> 12:38.400 Then when you go back one step further you have to do the whole 12:38.402 --> 12:43.872 thing again divided by 1 i^(F)_0. 12:43.870 --> 12:47.360 Student: You shouldn't have a 1 - d_2 in 12:47.355 --> 12:49.525 there as well, shouldn't you just be 12:49.533 --> 12:52.213 discounting by 1 - d_1 and 0? 12:52.210 --> 12:56.660 Prof: This is P_1 not P_2. 12:56.658 --> 13:00.188 Student: You've already discounted by 1 - d_2. 13:00.190 --> 13:04.920 Prof: No, but I'm switching the Ps on 13:04.918 --> 13:05.578 you. 13:05.580 --> 13:12.030 When we go from here the value here we just calculated was 13:12.034 --> 13:16.804 going to be P_2, so here the value of 13:16.802 --> 13:20.362 P_1 taking into account default is (1 - 13:20.359 --> 13:23.249 d_2) times P_2, 13:23.250 --> 13:28.100 which already takes into account the default next time, 13:28.100 --> 13:31.690 times P_2 discounted by the interest rate here. 13:31.690 --> 13:33.970 So I've got the 1 - d_2 here. 13:33.970 --> 13:36.190 Now, when I discount back to here you're saying, 13:36.190 --> 13:38.300 "How come the d_2's showing up any 13:38.296 --> 13:40.136 more because I'm just at d_1, 13:40.139 --> 13:41.169 that's your question, right? 13:41.169 --> 13:42.719 So it doesn't show up. 13:42.720 --> 13:47.260 It's just P_0 is 1 - d_1-- 13:47.259 --> 13:48.569 oh, you're asking this, you're right, 13:48.570 --> 13:50.330 (1 - d_1)--you were right-- 13:50.330 --> 13:57.800 times P_1 divided by (1 i^(F)_0). 13:57.798 --> 13:59.518 That's right, but if I plug in for 13:59.522 --> 14:01.142 P_1, P_1 already 14:01.139 --> 14:02.809 had--that's where the d_2 came from, 14:02.808 --> 14:04.558 so P_1's got the d_2 in it. 14:04.558 --> 14:07.098 So it's (1 - d_1) times (1 - d_2) times 14:07.104 --> 14:08.954 P_2, and then P_2 had a 14:08.947 --> 14:09.997 P_3 in it. 14:10.000 --> 14:13.120 So I've got all the defaults in it. 14:13.120 --> 14:14.600 Are you with me now? 14:14.600 --> 14:16.300 So, sorry about that, so you were right, 14:16.302 --> 14:17.092 I said it wrong. 14:17.090 --> 14:19.280 So, but this isn't the point. 14:19.279 --> 14:21.149 This was supposed to be obvious. 14:21.149 --> 14:22.089 I didn't even think about it. 14:22.090 --> 14:27.640 The next step is the thing that's not obvious. 14:27.639 --> 14:29.419 Here are the potential cash flows. 14:29.418 --> 14:31.168 You're discounting them by the interest rate. 14:31.168 --> 14:33.538 You also have to discount it again by the fact that the guy 14:33.543 --> 14:34.733 might not actually pay you. 14:34.730 --> 14:36.470 So that gives you a lower present value. 14:36.470 --> 14:40.820 Yellow P_2 is less than the no default white 14:40.817 --> 14:42.127 P_2. 14:42.129 --> 14:44.479 When you discount again you're discounting the yellow 14:44.475 --> 14:47.225 P_2 by the interest rate here and also the fact that 14:47.226 --> 14:48.396 the guy might not pay. 14:48.399 --> 14:52.719 So you have to multiply by 1 - d_2 and also the fact 14:52.717 --> 14:55.477 that he might not pay, the forward rate, 14:55.479 --> 14:58.239 and you keep moving that backwards. 14:58.240 --> 15:03.930 So that was supposed to be obvious even though I made it 15:03.929 --> 15:05.999 sound complicated. 15:06.000 --> 15:23.960 What's slightly subtler is just saying the same thing backwards 15:23.957 --> 15:38.727 which is, suppose I knew all these forward rates. 15:38.730 --> 15:51.150 Suppose I had a bunch of bonds, suppose I had American bonds, 15:51.154 --> 15:54.264 coupon bonds. 15:54.259 --> 15:57.689 So the American coupon bonds are going to pay, 15:57.690 --> 16:02.040 you know, the 1-year pays a coupon C_1 and has a 16:02.035 --> 16:05.615 face of 100 and has a price Pi_1. 16:05.620 --> 16:08.620 The 2-year American bond has a coupon C_2, 16:08.616 --> 16:11.376 a face of 100, and a price of Pi_2. 16:11.379 --> 16:14.849 And let's say the 5-year has something, C_5, 16:14.846 --> 16:17.656 a face of 100 and a price Pi_5. 16:17.658 --> 16:23.118 Now from that we know that we can deduce what all the forwards 16:23.121 --> 16:23.661 are. 16:23.659 --> 16:30.459 We did that in the first class. 16:30.460 --> 16:37.290 So now suppose at the same time we have Argentina. 16:37.288 --> 16:41.228 Many Argentine sovereign bonds promise payments in dollars, 16:41.230 --> 16:45.240 by the way, they're trying to trade them internationally. 16:45.240 --> 16:49.550 So let's say we also have the Argentina bonds 16:49.552 --> 16:54.362 C-hat_1, 100, Pi-hat_1, 16:54.359 --> 16:58.399 that's the 1-year, down to the 5-year which is the 16:58.398 --> 17:03.468 Argentina C-hat_5, 100 and Pi-hat_5, 17:03.471 --> 17:04.851 its price. 17:04.848 --> 17:10.278 Now, let's suppose that Argentina could default whereas 17:10.279 --> 17:11.889 America can't. 17:11.890 --> 17:14.600 So it's quite likely that Pi-hat_1 will be less 17:14.596 --> 17:17.636 than the American Pi_1 and Pi-hat_5 is going 17:17.641 --> 17:20.011 to be less than the American Pi_5, 17:20.009 --> 17:22.019 because all these bonds might default. 17:22.019 --> 17:24.539 So if the coupons were the same, if C-hat_1 was 17:24.544 --> 17:27.204 the same as C_1 and C-hat_5 was the same 17:27.203 --> 17:29.593 as C_5 the fact that Argentina could default 17:29.593 --> 17:32.033 obviously would mean its bonds would trade less for the 17:32.028 --> 17:33.018 American ones. 17:33.019 --> 17:37.169 So the question is can you figure out the default 17:37.169 --> 17:42.439 probabilities very quickly in Argentina without having to do a 17:42.441 --> 17:47.241 lot of complicated calculations, and the answer is yes. 17:47.240 --> 17:48.060 And why is that? 17:48.058 --> 17:51.728 Because you could take this data and you could say--so we 17:51.729 --> 17:53.629 could just erase this here. 17:53.630 --> 18:01.000 We could say, assuming no default, 18:00.997 --> 18:10.817 we could explain these prices by finding, 18:10.818 --> 18:18.698 just like we did the America, the Argentine forwards, 18:18.700 --> 18:27.810 1 i-hat_0, 1 i-hat^(F)_1 and 1 18:27.807 --> 18:32.267 i-hat^(F)_4. 18:32.269 --> 18:34.029 So these are the Argentine forwards. 18:34.029 --> 18:37.209 Now, these forwards would be much bigger than the American 18:37.214 --> 18:37.834 forwards. 18:37.829 --> 18:38.829 Why is that? 18:38.828 --> 18:42.718 Because the prices in Argentina are so much lower. 18:42.720 --> 18:46.330 If you're assuming there is no default, assuming no default, 18:46.325 --> 18:49.255 contrary to fact, how could you explain all these 18:49.257 --> 18:50.477 very low prices? 18:50.480 --> 18:53.710 Well, you must think that in Argentina they've got very high 18:53.707 --> 18:55.947 interest rates and very high forwards, 18:55.950 --> 18:59.300 so they're discounting more and that's why they've got lower 18:59.298 --> 18:59.808 prices. 18:59.808 --> 19:02.478 And we know how to get those forwards assuming there was no 19:02.479 --> 19:02.939 default. 19:02.940 --> 19:05.660 So the trick, I'm merely pointing out now, 19:05.660 --> 19:10.170 is that if we now go back and say, ah ha, 19:10.170 --> 19:13.160 Argentina doesn't have different forwards because 19:13.163 --> 19:16.223 anyone in Argentina-- the bonds are denominated in 19:16.217 --> 19:19.737 dollars precisely so that people can be crossover investors. 19:19.740 --> 19:22.050 An American can put his money in Argentina. 19:22.048 --> 19:25.568 An Argentine can put his money in America, so you can move your 19:25.567 --> 19:26.927 money to either place. 19:26.930 --> 19:30.600 So it must be that the forward rates can't be different. 19:30.598 --> 19:33.368 If you knew for sure you were going to get paid in Argentina 19:33.374 --> 19:35.964 you'd have to have the same forward rates in America. 19:35.960 --> 19:39.200 So the reason these forward rates are higher is because 19:39.200 --> 19:41.000 there's a chance of default. 19:41.000 --> 19:44.550 So what is the chance of default? 19:44.548 --> 19:47.008 So I claim the chance of default is, and I was 19:47.008 --> 19:49.738 supposedly--you're supposed to realize this now. 19:49.740 --> 19:54.970 If I'd been clearer before you would see where I'm going. 19:54.970 --> 19:57.650 The chance of default is incredibly simple to find out. 19:57.650 --> 20:06.450 So it's 1 - d_t = what? 20:06.450 --> 20:08.330 Student: > 20:08.328 --> 20:11.998 Prof: But in terms of forwards is what? 20:12.000 --> 20:14.550 It's not Pi--this Pi, this isn't the 0 price. 20:14.548 --> 20:17.678 This is the big price of the bonds. 20:17.680 --> 20:22.650 So it's not Pi-hat_t over Pi_t. 20:22.650 --> 20:27.040 What is it though? 20:27.038 --> 20:29.228 Student: The ratio of forwards? 20:29.230 --> 20:38.830 Student: Pi_t - > 20:38.828 --> 20:45.628 Prof: This is going to be a bigger number than that, 20:45.631 --> 20:51.731 and in fact that ratio is the default probability. 20:51.730 --> 20:55.250 So this is assuming, remember, that if the Argentine 20:55.249 --> 20:59.179 bond defaults at this period, say, it's never going to pay 20:59.182 --> 21:00.842 anything after that. 21:00.838 --> 21:04.178 You're going to get 0 pay off and all the other Argentine 21:04.179 --> 21:05.729 bonds will also default. 21:05.730 --> 21:07.700 I claim that this [note: meaning, (1 - d_t = 21:07.703 --> 21:09.863 (1 i^(F)_t-1) over (1 i-hat^(F)_t-1)) ] 21:09.864 --> 21:12.364 is going to be the easy way of getting the default probability. 21:12.358 --> 21:15.928 And so the differences in the forwards is just explained by 21:15.929 --> 21:19.929 the default probabilities and so the extra Argentine interest, 21:19.930 --> 21:22.620 if this [the denominator] is a higher number than that 21:22.624 --> 21:26.674 [the numerator], 1 - d is approximately 1 over 21:26.670 --> 21:30.580 (1 d), and so it's basically if all 21:30.576 --> 21:37.016 the numbers i and d are small then i-hat^(F)_t - 1 21:37.016 --> 21:41.706 is approximately i^(F)_t - 1, 21:41.710 --> 21:45.260 the American one, plus this default rate in 21:45.263 --> 21:46.283 Argentina. 21:46.279 --> 21:50.679 So I should probably have a hat because I'm referring to 21:50.680 --> 21:52.680 Argentina with the hat. 21:52.680 --> 21:53.680 So why is that true. 21:53.680 --> 21:54.600 I just argued it. 21:54.599 --> 22:01.809 How could that possibly be true? 22:01.808 --> 22:06.108 So you see what I'm claiming that you have now a very simple 22:06.111 --> 22:09.761 algorithm for finding out, inferring what Argentina 22:09.756 --> 22:11.356 default rates are. 22:11.358 --> 22:14.348 Again, I'm making a special assumption that when Argentina 22:14.345 --> 22:15.755 defaults you get nothing. 22:15.759 --> 22:17.069 That really isn't the case. 22:17.068 --> 22:19.508 There's some huge convention that happens and all the 22:19.505 --> 22:21.515 countries get together they defaulted on. 22:21.519 --> 22:24.439 They've got some big meeting, and someone like Brady invents 22:24.443 --> 22:27.573 some idea where they'll owe less and they will be a writing down 22:27.567 --> 22:29.397 of principal, by the way. 22:29.400 --> 22:39.090 So whenever this happens there is recovery after a writing down 22:39.086 --> 22:41.426 of principal. 22:41.430 --> 22:43.480 So what all the countries do is they say, "Okay, 22:43.483 --> 22:45.423 we know that you can't pay all that you owe us. 22:45.420 --> 22:47.380 We'll settle for half of it. 22:47.380 --> 22:50.680 We'll write down the principal and we'll hold you to that half, 22:50.679 --> 22:51.849 or to a third of it. 22:51.848 --> 22:54.498 So this is one of the things we curiously haven't done in 22:54.497 --> 22:56.857 America where all these homeowners can't pay and we 22:56.859 --> 22:58.609 don't write down their principal. 22:58.608 --> 23:00.078 We just throw them out of their houses. 23:00.078 --> 23:04.248 But anyway, let's say you wrote the principal down to 0 in that 23:04.250 --> 23:05.260 special case. 23:05.259 --> 23:08.819 You could easily infer from the price of the Argentine bonds 23:08.818 --> 23:12.378 what the default probabilities were, and by this formula. 23:12.380 --> 23:17.670 And so the question is, why is that true? 23:17.670 --> 23:21.140 We know how to calculate the forwards in America given the 23:21.144 --> 23:22.124 American data. 23:22.118 --> 23:23.918 That's was one of the first things we did in class. 23:23.920 --> 23:27.180 We said that every American company in the whole country, 23:27.181 --> 23:29.221 financial company, is doing that. 23:29.220 --> 23:32.050 Everybody has those forwards calculated. 23:32.048 --> 23:34.318 Now, if you're given the Argentine data, 23:34.318 --> 23:36.908 which is, after all, just coupons and the prices of 23:36.906 --> 23:38.906 the bonds, you could find Argentine 23:38.911 --> 23:41.021 forwards assuming there's no default. 23:41.019 --> 23:44.529 But there is default, so it must be that they have 23:44.534 --> 23:48.054 access to the American interest rates in forwards, 23:48.048 --> 23:50.988 but the Argentine bond might default. 23:50.990 --> 23:53.520 Do you see what we did when we did this calculation? 23:53.519 --> 23:57.599 The difference between the backward induction in America 23:57.596 --> 24:01.526 from here to here was just discounting by the American 24:01.526 --> 24:04.356 forward-- to go to Argentina we had to 24:04.364 --> 24:08.184 discount by the American forward and multiply by 1 - d, 24:08.180 --> 24:09.770 so discounting it again. 24:09.769 --> 24:11.449 So all I'm saying is that in the U.S. 24:11.450 --> 24:16.290 when we went backwards we just discounted by this thing. 24:16.288 --> 24:27.338 In Argentina when you go backwards you have to discount 24:27.342 --> 24:36.352 by this thing, so those things have to be the 24:36.346 --> 24:38.186 same. 24:38.190 --> 24:43.330 So the Argentinian discount is like taking the-- 24:43.328 --> 24:45.288 I hope I haven't got the thing--so the American forwards 24:45.290 --> 24:47.110 are going to be less than the Argentine forwards. 24:47.109 --> 24:51.829 It's going to be like that. 24:51.829 --> 24:52.479 So that's it. 24:52.480 --> 24:55.850 There's nothing else to show except that whenever you're 24:55.852 --> 24:58.492 going backwards here you're discounting-- 24:58.490 --> 25:02.490 remember, you're discounting by the interest rate times the 25:02.491 --> 25:06.151 probability that you're actually going to pay off, 25:06.150 --> 25:13.830 so that's what it is in Argentina. 25:13.829 --> 25:16.659 Hang on. 25:16.660 --> 25:23.470 I hope I haven't inverted one of these, yeah, 25:23.471 --> 25:25.021 exactly. 25:25.019 --> 25:28.069 So if you write 1 - d times the American thing in the 25:28.069 --> 25:29.619 denominator, so as I said, 25:29.621 --> 25:32.311 to do the discounting in Argentina at every step from 25:32.308 --> 25:35.408 going back from here to here what did we do in Argentina? 25:35.410 --> 25:39.690 We simply took 1 - d-hat, that was the default 25:39.692 --> 25:44.262 rate--hat--in Argentina, and discounted it at the 25:44.260 --> 25:46.260 American forward. 25:46.259 --> 25:48.129 So that's what I did here. 25:48.130 --> 25:52.390 So if I take 1 - d-hat, this is multiplied by 1, 25:52.387 --> 25:57.007 so I take 1 - d-hat^( )multiplied by 1 over this. 25:57.009 --> 26:00.179 I just get one over the Argentine discount, 26:00.180 --> 26:02.970 and that's how we calculated--that's how we went 26:02.969 --> 26:06.649 backwards with our recursion just taking the interest rate, 26:06.650 --> 26:11.060 the discount 1 over (1 i) times (1 - d) and that's how we 26:11.064 --> 26:13.354 discounted going backwards. 26:13.348 --> 26:15.688 And so therefore in Argentina if you're forgetting that 26:15.690 --> 26:18.420 there's default and you're just thinking you have to discount at 26:18.421 --> 26:20.721 the right rate and you're getting this discount you're 26:20.719 --> 26:23.369 getting this number, but in reality you should have 26:23.371 --> 26:25.051 been taking this divided by that. 26:25.048 --> 26:28.888 So therefore figuring out this and knowing that tells you what 26:28.888 --> 26:29.958 this has to be. 26:29.960 --> 26:33.500 So it's extremely simple to deduce what the market thinks 26:33.497 --> 26:36.587 Argentinian default probabilities are year by year 26:36.593 --> 26:40.323 if you make the added assumption that once they default they 26:40.319 --> 26:41.899 default completely. 26:41.900 --> 26:44.590 And if you think you're only going to get a little bit back, 26:44.592 --> 26:46.922 well then the calculation won't change that much. 26:46.920 --> 26:48.180 Yeah? 26:48.180 --> 26:51.750 Student: Can we also do it by using the price of 0s? 26:51.750 --> 26:53.430 Prof: Yeah, so you could also do it by 26:53.426 --> 26:56.316 using the price of 0s, but to me the best thing is, 26:56.316 --> 26:59.216 the easiest thing is using the forwards, 26:59.220 --> 27:01.280 but you can also do it by 0s. 27:01.279 --> 27:05.019 So that's all I wanted to say. 27:05.019 --> 27:11.129 As I said, the one last thing to say is that if these numbers 27:11.132 --> 27:17.042 are all small then 1 over 1 - d--or, 1 - d is approximately 27:17.040 --> 27:19.690 equal to 1 over (1 d). 27:19.690 --> 27:23.810 If d is very small those are practically the same things. 27:23.808 --> 27:29.518 Then if you multiply 1 over (1 d) by 1 over (1 i) it's almost 1 27:29.520 --> 27:33.390 over (1 d i), so it's almost this thing. 27:33.390 --> 27:34.890 It's not quite true, literally true, 27:34.890 --> 27:38.330 but it's very close to say that the gap between Argentinian 27:38.327 --> 27:41.937 forwards and American forwards is just the default probability 27:41.942 --> 27:45.862 in Argentina, and that reason is why it's 27:45.855 --> 27:48.485 called a default spread. 27:48.490 --> 27:50.900 You just add some spread to the interest rate. 27:50.900 --> 27:54.730 You can guess by the spread what the probability of default 27:54.729 --> 27:55.059 is. 27:55.058 --> 27:59.208 If it's 8 percent interest there and 3 percent interest 27:59.213 --> 28:03.443 here somebody must think the probability of default is 5 28:03.444 --> 28:04.834 percent there. 28:04.829 --> 28:09.649 That's it. 28:09.650 --> 28:15.630 So let's now move to a tree where you have to make 28:15.625 --> 28:17.085 decisions. 28:17.088 --> 28:21.268 So I'm going to now describe the method of backward induction 28:21.271 --> 28:24.411 which occurs over and over and over again, 28:24.410 --> 28:26.180 and we've used it a couple of times, 28:26.180 --> 28:33.780 but not in its subtlest form, so backward induction. 28:33.779 --> 28:36.949 Now, who first invested the idea of backward induction? 28:36.950 --> 28:41.790 Well, the first person who spelled it out formally was 28:41.792 --> 28:45.092 Zermelo in 1910, I think, that's within a couple 28:45.086 --> 28:47.276 of years, a famous mathematician, 28:47.282 --> 28:49.072 Fraenkel-Zermelo Axioms. 28:49.068 --> 29:02.598 And he proved that chess--that there's an optimal strategy in 29:02.596 --> 29:09.356 chess by backward induction. 29:09.358 --> 29:12.218 So, for example, let's take a game. 29:12.220 --> 29:15.840 We always are on a tree, but now we're going to use a 29:15.836 --> 29:18.686 slightly extended definition of a tree. 29:18.690 --> 29:20.370 A tree is going to look like this. 29:20.368 --> 29:23.128 So it's a root, a finite number of branches 29:23.127 --> 29:26.607 from every--I don't want to formally define a tree. 29:26.608 --> 29:29.038 You know what it sort of looks like, and there's no reason why 29:29.040 --> 29:31.510 the number of branches has to be two or even has to be the same 29:31.511 --> 29:32.351 from every point. 29:32.348 --> 29:35.658 But the reason we're going to extend it is, 29:35.663 --> 29:39.533 the node is going to be described by who moves. 29:39.529 --> 29:43.829 So let's say white is moving here and black is moving here. 29:43.828 --> 29:50.108 Now, let's say the outcomes are a win for white, 29:50.106 --> 29:55.046 a win for black, a draw, or a draw. 29:55.048 --> 29:58.158 So the question is, so it's a two-move chess game, 29:58.160 --> 30:01.480 white moves first up or down, and then after white moves 30:01.480 --> 30:04.380 black moves up or down and then the game ends. 30:04.380 --> 30:07.630 And depending on where, position you reach either it's 30:07.634 --> 30:09.604 a win for white, a win for black, 30:09.599 --> 30:10.459 or a draw. 30:10.460 --> 30:12.420 So what should white do? 30:12.420 --> 30:20.520 Assuming that black is a smart player what should she do? 30:20.519 --> 30:22.009 So what did Zermelo do? 30:22.009 --> 30:24.239 He said not only there is an optimal strategy, 30:24.242 --> 30:27.422 but you know what the outcome should be with rational players. 30:27.420 --> 30:32.290 So Zermelo said if white goes up then black is clearly going 30:32.285 --> 30:34.755 to go down and win the game. 30:34.759 --> 30:37.539 So white ought to be thinking here, if I go up, 30:37.535 --> 30:40.545 the game, although it won't end for another period, 30:40.551 --> 30:41.881 it's already lost. 30:41.880 --> 30:45.100 So the value of the game is already 0. 30:45.098 --> 30:48.588 So this method of backward induction attaches the value. 30:48.588 --> 30:52.538 Here we have values at the end, and so to figure out what the 30:52.535 --> 30:56.675 right thing to do is by backward induction you can propagate the 30:56.678 --> 30:58.058 values backwards. 30:58.058 --> 31:02.458 If black makes the right choice here the payoff is black gets 31:02.463 --> 31:06.333 the negative of white, so the right choices here are 31:06.328 --> 31:09.408 black could get negative 1 or could get 0, 31:09.410 --> 31:11.230 so black clearly wants to get 0. 31:11.230 --> 31:14.560 So black could win the game by moving down, so black surely 31:14.563 --> 31:15.543 will move down. 31:15.538 --> 31:18.958 So I should think of the game as already lost here and pretend 31:18.960 --> 31:22.320 that I had a shorter tree with the final valuation of zero at 31:22.324 --> 31:23.114 this node. 31:23.108 --> 31:26.808 Similarly, if white goes down it doesn't matter what black 31:26.807 --> 31:29.207 does the game is going to be drawn. 31:29.210 --> 31:32.550 So white should think to himself the game is already a 31:32.548 --> 31:33.808 draw if I go down. 31:33.808 --> 31:36.618 And now white has an easy choice, do I want to move to a 31:36.621 --> 31:38.411 loss or do I want to move a draw. 31:38.410 --> 31:42.150 So I could just pick a move for black here. 31:42.150 --> 31:46.340 Clearly white is going to go down and therefore with correct 31:46.335 --> 31:48.175 play the game is a draw. 31:48.180 --> 31:51.150 So by backward induction you figure out the correct play. 31:51.150 --> 31:52.580 Now, why is this surprising? 31:52.578 --> 31:55.898 Because chess has an incredibly big tree, not an infinite tree, 31:55.898 --> 31:58.468 there are all these rules that keep it finite. 31:58.470 --> 32:01.660 If you reach to the same position three times it's 32:01.657 --> 32:02.957 considered a draw. 32:02.960 --> 32:06.140 If you make something like 50 moves in a row without a pawn 32:06.137 --> 32:07.287 moving it's a draw. 32:07.288 --> 32:10.948 Whatever those rules are--I used to play chess quite a bit. 32:10.950 --> 32:11.730 I've even forgotten. 32:11.730 --> 32:14.130 But whatever those rules are they're designed to make the 32:14.125 --> 32:15.705 game finite, so the tree is finite. 32:15.710 --> 32:18.560 And so it's impossible to see the whole tree, 32:18.560 --> 32:22.060 and how should you know what to do at the beginning? 32:22.058 --> 32:23.898 Well, you don't know what to do at the beginning until you know 32:23.897 --> 32:25.467 what black's going to do afterwards, and so what could 32:25.470 --> 32:26.330 happen later in the tree. 32:26.328 --> 32:29.028 But if you were fast enough to put the whole thing on a 32:29.025 --> 32:32.065 computer you could figure out what to do at the beginning, 32:32.068 --> 32:34.728 because your best move at the beginning depends on what you 32:34.726 --> 32:37.506 think black is going to do next, which depends on what he thinks 32:37.509 --> 32:40.109 you're going to do after that, which depends on what you think 32:40.106 --> 32:41.036 he'll do after that. 32:41.038 --> 32:43.728 But if the tree ends you can always go backward from the end 32:43.731 --> 32:46.151 to the beginning and figure out what to do at the very 32:46.151 --> 32:46.791 beginning. 32:46.788 --> 32:49.448 So this is a familiar argument to all of you, 32:49.445 --> 32:50.045 I think. 32:50.048 --> 32:55.478 It was a beautiful argument in chess in 1910 and then it was 32:55.482 --> 32:59.722 anticipated--I mean, in mathematics in 1910. 32:59.720 --> 33:01.720 The chess players, of course, all knew about it, 33:01.720 --> 33:08.560 so Steinitz who was a world champion from when to when, 33:08.558 --> 33:14.828 something like 1870 or so to, or 1880 let's say to let's see, 33:14.829 --> 33:18.109 19,21, to 1894. 33:18.108 --> 33:19.598 I think he was world champion from then to there. 33:19.599 --> 33:22.139 Lasker became the champion then. 33:22.140 --> 33:26.270 So he wrote a bunch of books and stuff in which he said 33:26.271 --> 33:29.871 there's a backward induction value to chess, 33:29.868 --> 33:33.318 but since we can't figure that out on general principles you 33:33.324 --> 33:36.724 can tell by looking at the configuration of pieces what the 33:36.720 --> 33:44.440 right possible move is, and so you can have positional 33:44.435 --> 33:47.435 values, so I'll do that, 33:47.440 --> 33:51.350 and then you can have the backward induction values. 33:51.348 --> 33:56.138 So for instance a positional value might tell you that having 33:56.143 --> 34:00.083 doubled pawns is a bad thing, having control of the center's 34:00.076 --> 34:02.966 a good thing and you add up all those pluses and minuses and you 34:02.967 --> 34:04.387 get these positional values. 34:04.390 --> 34:11.040 And so he said if you've got the right positional algorithm, 34:11.039 --> 34:14.229 right positional understanding, your positional sense of what 34:14.231 --> 34:16.661 to do, you only need to analyze one 34:16.655 --> 34:17.415 move deep. 34:17.420 --> 34:19.920 You can figure out what the best position's going to be and 34:19.916 --> 34:21.806 move that way, and if you really understand 34:21.807 --> 34:23.667 the game properly that positional thinking, 34:23.670 --> 34:26.910 that strategic thinking, so it's called strategic 34:26.909 --> 34:30.079 thinking, is going to give you the same 34:30.083 --> 34:33.753 decision as the exhaustive analysis of all the 34:33.751 --> 34:37.501 possibilities which was tactical thinking, 34:37.500 --> 34:39.590 so the two should amount to the same thing. 34:39.590 --> 34:41.250 Now, in fact, people can't do the full 34:41.251 --> 34:43.811 tactical thing and also they don't have the full strategic 34:43.811 --> 34:46.461 understanding either, so they kind of mix strategy 34:46.456 --> 34:49.406 and tactics and that's what makes the game interesting. 34:49.409 --> 34:52.339 So no one has ever written this, but I'm sure there's an 34:52.344 --> 34:55.554 interesting study to be made about what games are interesting 34:55.547 --> 34:58.587 and they must be the kinds of games where there's always a 34:58.590 --> 35:00.620 mixture of strategy and tactics. 35:00.619 --> 35:02.929 In game theory, as we describe it in economics, 35:02.925 --> 35:04.725 there's no such thing as strategy. 35:04.730 --> 35:05.730 All this is out. 35:05.730 --> 35:08.070 It's all just backward induction, which is what I'm 35:08.068 --> 35:08.768 teaching you. 35:08.768 --> 35:12.098 So the way computers play chess, incidentally, 35:12.099 --> 35:17.739 is--and the guy who invented this is Shannon, 35:17.739 --> 35:26.309 so he's a famous professor of information systems, 35:26.309 --> 35:27.609 so an engineering professor. 35:27.610 --> 35:30.320 So he said, well, you can't look at the whole 35:30.318 --> 35:34.068 tree which is too long in chess, so what you should do--so all 35:34.074 --> 35:35.864 this extends way further. 35:35.860 --> 35:37.860 Maybe you can only look two moves deep. 35:37.860 --> 35:41.730 So what Shannon recommended is look as far as you can put in 35:41.733 --> 35:45.203 your computer, apply some positional thinking 35:45.199 --> 35:49.819 to evaluate these positions at the pseudo end of the tree. 35:49.820 --> 35:51.550 So it's really not a win for white. 35:51.550 --> 35:54.480 Let's just pretend White's so far ahead that we'll call it a 35:54.476 --> 35:56.556 win, and a loss, and a draw, and a draw. 35:56.559 --> 35:58.679 That's just by looking at positional values. 35:58.679 --> 36:01.629 And then having assigned those terminal nodes values, 36:01.630 --> 36:04.900 now by backward induction you can figure out what the value is 36:04.903 --> 36:07.913 here and exactly what the right first move to make is. 36:07.909 --> 36:10.359 And after white's move black will come here, 36:10.360 --> 36:13.670 black can now look two moves deep, so black's going to look 36:13.666 --> 36:15.716 from here all the way down here. 36:15.719 --> 36:18.749 He's going to do his positional evaluator to these nodes and try 36:18.753 --> 36:21.303 and figure out what they're worth and then do backward 36:21.304 --> 36:23.764 induction to figure out what his right move is. 36:23.760 --> 36:26.630 Anyway, that's basically the idea of all chess algorithms, 36:26.628 --> 36:29.448 and then they've gotten refined by saying--wait a minute, 36:29.447 --> 36:31.157 there are lots of refinements. 36:31.159 --> 36:32.439 So I used to very interested in this. 36:32.440 --> 36:34.610 I don't think I'll talk more about it unless anyone wants to 36:34.610 --> 36:35.310 ask me something. 36:35.309 --> 36:37.869 So there's the origin of backward induction, 36:37.869 --> 36:41.829 Zermelo's proof and it's obviously a big deal in chess 36:41.827 --> 36:46.157 and the chess players all knew about it before Zermelo, 36:46.159 --> 36:48.609 but they didn't write anything down as formal as Zermelo. 36:48.610 --> 36:52.890 So how does this apply to everything we do in economics? 36:52.889 --> 36:57.139 Well, I want to give a series of examples culminating in 36:57.135 --> 37:00.605 market examples, but starting off a little far 37:00.608 --> 37:01.688 from life. 37:01.690 --> 37:08.420 So the first one I want to give is the red and the black. 37:08.420 --> 37:09.610 So these are just two games. 37:09.610 --> 37:12.590 This is the first one I invented ten years ago, 37:12.594 --> 37:14.734 but I don't think they're--anyway, 37:14.733 --> 37:17.203 I think they aren't that original. 37:17.199 --> 37:18.549 I thought they were when I invented them. 37:18.550 --> 37:21.410 But anyway, so the red and the black works like this. 37:21.409 --> 37:27.479 There's a deck of cards, 52 cards, a deck of 52 cards, 37:27.476 --> 37:32.966 so half of them are--26 red, right, and 26 black, 37:32.971 --> 37:37.551 which is all I care about the cards. 37:37.550 --> 37:39.620 And someone offers you a game and they say, 37:39.619 --> 37:43.549 okay, the deck is upside down, they've been shuffled, 37:43.550 --> 37:47.260 you can turn over a card and if it's black I'll pay you 1 37:47.262 --> 37:47.862 dollar. 37:47.860 --> 37:50.330 If it's red you have to pay me 1 dollar. 37:50.329 --> 37:55.049 So I'm offering you this chance to play this game and of course 37:55.054 --> 37:57.804 you can quit whenever you want to. 37:57.800 --> 38:01.610 I can't keep forcing you to play, so anytime you want to you 38:01.608 --> 38:02.318 can quit. 38:02.320 --> 38:11.920 So that's the game, can stop whenever you want. 38:11.920 --> 38:14.330 And once you draw the card you throw it away. 38:14.329 --> 38:18.659 And so all these examples are going to be examples of stopping 38:18.655 --> 38:22.405 games and you'll see in economics that when you prepay 38:22.414 --> 38:27.314 on a mortgage or when you call a bond you're stopping the thing, 38:27.309 --> 38:29.049 the contract's ending. 38:29.050 --> 38:31.360 Life is going on, but that contract is ending, 38:31.355 --> 38:33.655 so want to know, when's the right time to take 38:33.659 --> 38:34.889 an action like that? 38:34.889 --> 38:38.219 So red and black is a simple game like that where you turn 38:38.221 --> 38:39.041 over a card. 38:39.039 --> 38:41.719 If it's black, you're in the black you win 1 38:41.719 --> 38:42.279 dollar. 38:42.280 --> 38:46.360 If it's red you lose 1 dollar and you can stop whenever you 38:46.356 --> 38:46.846 want. 38:46.849 --> 38:51.159 So you have an option, so call this an option, 38:51.163 --> 38:55.863 and most people totally underestimate the value of 38:55.858 --> 38:57.008 options. 38:57.010 --> 39:00.870 So let's just figure out how to figure out the optimal thing to 39:00.867 --> 39:01.177 do. 39:01.179 --> 39:02.529 What would you do in this game? 39:02.530 --> 39:05.060 Would you play if I gave you the chance to play? 39:05.059 --> 39:07.499 I think I did this on the very first day. 39:07.500 --> 39:10.210 Yes, you're about to say something? 39:10.210 --> 39:12.610 Your hand twitched. 39:12.610 --> 39:15.540 Student: I was going to say we have deceasing marginal 39:15.536 --> 39:17.236 utility as well, so assuming we have 39:17.244 --> 39:18.664 > 39:18.659 --> 39:21.729 you wouldn't play the game because you would derive less 39:21.726 --> 39:24.116 utility from winning 1 dollar than you would 39:24.123 --> 39:25.743 > 39:25.739 --> 39:30.599 in magnitude that the loss > 39:30.599 --> 39:33.409 Prof: So I'm going to now disagree with what you said, 39:33.414 --> 39:35.344 but it's very interesting what he said. 39:35.340 --> 39:37.570 He said, look, if you draw a card at the 39:37.570 --> 39:40.890 beginning it's 50/50 whether you're going to win or not. 39:40.889 --> 39:42.309 If you win you get 1 dollar. 39:42.309 --> 39:45.809 If you lose you lose 1 dollar, 50/50 chance. 39:45.809 --> 39:50.539 If you're a little bit afraid, if a 1 dollar loss is more 39:50.543 --> 39:55.703 important to you than a 1 dollar gain right away it's not very 39:55.701 --> 39:58.461 good odds, I mean, it's barely even, 39:58.461 --> 40:02.161 and if you're a little bit risk averse and it's barely even you 40:02.164 --> 40:03.244 shouldn't play. 40:03.239 --> 40:07.409 But now, is it really barely even, this game? 40:07.409 --> 40:08.199 Yep. 40:08.199 --> 40:10.969 Student: Well, I mean, I think you should play 40:10.965 --> 40:14.045 because even if you get the first 26 red at that point just 40:14.050 --> 40:17.140 go to the end and you haven't lost anything except the time 40:17.135 --> 40:19.045 you've spent playing the game. 40:19.050 --> 40:21.650 You might as well play on the off chance that you'll get some 40:21.652 --> 40:22.522 blacks ones first. 40:22.518 --> 40:24.618 Prof: Right, so you can't possibly lose if 40:24.617 --> 40:25.577 you play this right. 40:25.579 --> 40:27.379 You can always go to the very end of the deck. 40:27.380 --> 40:29.620 We're ignoring a good point. 40:29.619 --> 40:32.499 We're ignoring your utility of time, so you can always go to 40:32.496 --> 40:34.786 the end of the deck and assure yourself of 0. 40:34.789 --> 40:39.559 So this is actually a pretty valuable option to be able to 40:39.563 --> 40:41.243 stop like he says. 40:41.239 --> 40:43.979 If the first one's black you could stop and then you've won 1 40:43.978 --> 40:44.388 dollar. 40:44.389 --> 40:46.369 If a whole bunch of them are red and you lose, 40:46.369 --> 40:49.099 well, you can always go to the end of the deck and get zero. 40:49.099 --> 40:50.819 So you're never going to lose and you have a chance of 40:50.824 --> 40:51.154 winning. 40:51.150 --> 40:52.320 So obviously you should play. 40:52.320 --> 40:56.190 Even if you are risk averse you should play, but now the 40:56.188 --> 40:58.578 question is can we tell exactly. 40:58.579 --> 41:00.179 Suppose you're risk neutral? 41:00.179 --> 41:10.039 How many dollars do you expect to win would you guess? 41:10.039 --> 41:11.409 Yep? 41:11.409 --> 41:12.109 Student: 0. 41:12.110 --> 41:14.450 Prof: You'd expect to get 0. 41:14.449 --> 41:17.809 Now he just made an argument that you should expect more than 41:17.811 --> 41:20.781 0, because for instance he said take this strategy. 41:20.780 --> 41:22.000 Pick a card. 41:22.000 --> 41:24.870 If it's black you win 1 dollar, quit. 41:24.869 --> 41:25.889 You're 1 dollar ahead. 41:25.889 --> 41:29.719 So at 50 percent of the time you're plus 1 dollar. 41:29.719 --> 41:32.769 If it's red the first time just close your eyes and play to the 41:32.768 --> 41:35.918 end of the game and you're going to get 0 because you're going to 41:35.918 --> 41:37.638 win 26 times and lose 26 times. 41:37.639 --> 41:40.739 So that's equal to .5. 41:40.739 --> 41:44.339 So there's one strategy that gets you 50 cents on average. 41:44.340 --> 41:47.250 You can't lose and half the time you'll get 1 dollar, 41:47.250 --> 41:49.490 but that may not be the best strategy. 41:49.489 --> 41:54.969 Student: You can play a bunch of times and at worst 41:54.967 --> 42:00.637 you'll break even and at best you could get all 26 black. 42:00.639 --> 42:01.529 Prof: Right. 42:01.530 --> 42:04.320 So he's saying this isn't ambitious enough. 42:04.320 --> 42:06.450 This surely gets a half a dollar, but you could do much 42:06.445 --> 42:06.795 better. 42:06.800 --> 42:08.060 Like, let's just wait. 42:08.059 --> 42:11.489 The first time suppose you get 1 dollar. 42:11.489 --> 42:14.859 Suppose you get black the first time, so that gives you 1 42:14.858 --> 42:15.398 dollar. 42:15.400 --> 42:19.870 Now, the trouble is the deck is starting to turn against you. 42:19.869 --> 42:23.489 Now it's 25 blacks and 26 reds. 42:23.489 --> 42:26.399 So what would you do then? 42:26.400 --> 42:28.070 Student: I'd stop. 42:28.070 --> 42:30.800 Prof: You'd stop or keep going? 42:30.800 --> 42:36.480 Well, the deck is against you, so now your very next draw is 42:36.481 --> 42:37.831 unfavorable. 42:37.829 --> 42:40.459 And, by the way, playing to the end of the deck 42:40.456 --> 42:43.816 is going to lose you 1 dollar because there are 25 black and 42:43.824 --> 42:45.634 26 red, so this argument that if you 42:45.630 --> 42:48.150 just play to the end of the deck you'll break even it's not true 42:48.150 --> 42:50.030 after you've already taken a black one out, 42:50.030 --> 42:50.860 so you could lose it. 42:50.860 --> 42:54.060 From then on you're starting to run a little bit of a risk. 42:54.059 --> 42:55.989 So we're ignoring risk aversion. 42:55.989 --> 42:57.609 We're just caring about expected dollars. 42:57.610 --> 43:01.640 The fact is the deck is against you, so should you play or not? 43:01.639 --> 43:05.889 And so first reaction is hell no, the deck's against me. 43:05.889 --> 43:07.679 Why should I draw another card? 43:07.679 --> 43:10.629 But you still have the option of going to the end of the deck, 43:10.630 --> 43:13.390 so the most you could lose is 1 dollar if you went all the way 43:13.391 --> 43:15.291 to the end of the deck, and who knows, 43:15.289 --> 43:18.479 maybe you'll get a run of more black cards in the beginning and 43:18.481 --> 43:20.181 make a lot more than 1 dollar. 43:20.179 --> 43:24.349 So you should choose another card. 43:24.349 --> 43:26.929 In which case if you get black again-- 43:26.929 --> 43:29.199 if you got red on the next card you'd be breaking even, 43:29.199 --> 43:32.779 but now it's 25/25 and by the previous argument it's obvious 43:32.784 --> 43:36.314 you should pick another one because the worst you can do is 43:36.309 --> 43:37.829 break even from then. 43:37.829 --> 43:40.319 But what if you got two blacks in a row? 43:40.320 --> 43:43.750 Well, now the deck is way against you. 43:43.750 --> 43:48.770 It's 26 red and 24 black, so now you've only got a 48 43:48.773 --> 43:53.993 percent chance of drawing a black one the next time. 43:53.989 --> 43:56.649 The deck is going further against you. 43:56.650 --> 44:00.330 Should you really draw another card? 44:00.329 --> 44:02.279 It's more likely to be red. 44:02.280 --> 44:05.310 Well, the answer is yes. 44:05.309 --> 44:11.399 And suppose you got a black one again, meaning you're three up, 44:11.398 --> 44:14.148 and now the deck is 26/23. 44:14.150 --> 44:16.890 Sorry, I went the other way, 26 red, 23 black. 44:16.889 --> 44:19.129 It's getting further and further against you. 44:19.130 --> 44:21.250 Should you draw another card? 44:21.250 --> 44:25.930 Well, what you've got is you've got a bad deck working against 44:25.929 --> 44:30.379 you, but you've got this option working in favor of you. 44:30.380 --> 44:33.180 So the question is just how valuable is the option. 44:33.179 --> 44:36.009 And like I said, people always underestimate the 44:36.014 --> 44:37.164 value of options. 44:37.159 --> 44:40.109 And so--okay, go ahead. 44:40.110 --> 44:42.940 Student: Don't you want to play until your 44:42.938 --> 44:46.178 lose-condition is either balanced out or worse than your 44:46.179 --> 44:47.239 win condition? 44:47.239 --> 44:49.629 Prof: Yes, but what is that condition? 44:49.630 --> 44:52.960 Student: Thirteen blacks > 44:52.960 --> 44:56.180 Prof: If you got ten blacks in a row you would keep 44:56.181 --> 44:58.841 drawing blacks, is that what you were saying? 44:58.840 --> 45:00.340 Student: Well, I mean, at that point if you 45:00.344 --> 45:01.884 assume that that's all > 45:01.880 --> 45:05.160 , assuming that you pick another red and then play out 45:05.155 --> 45:08.365 the end of the game you could lose, it would be... 45:08.369 --> 45:09.389 Prof: Thirteen more. 45:09.389 --> 45:13.059 So you're saying you want to keep drawing blacks until if you 45:13.063 --> 45:16.863 play to the end of the deck you would lose as much as you'd won 45:16.860 --> 45:18.330 up until that point. 45:18.329 --> 45:22.659 So you want to never run the risk of losing more. 45:22.659 --> 45:24.599 But you see, that strategy would get you to 45:24.601 --> 45:25.851 quit after the first one. 45:25.849 --> 45:29.859 After the first black if you ran to the end of the--your 45:29.855 --> 45:32.035 strategy doesn't make sense. 45:32.039 --> 45:34.989 By going to the end of the deck you're always going to undo 45:34.992 --> 45:37.132 everything you've won until that point, 45:37.130 --> 45:41.050 because you'll be zero no matter what if you go to the end 45:41.054 --> 45:42.024 of the deck. 45:42.019 --> 45:43.819 Yep? 45:43.820 --> 45:47.020 Student: Shouldn't you quit whenever you have 1 dollar, 45:47.018 --> 45:51.568 because say if you have more black cards than red cards then 45:51.570 --> 45:54.810 the number of black cards left in the... 45:54.809 --> 45:56.769 Prof: Deck is less... 45:56.768 --> 45:59.318 Student: Is lower than the number of red cards left in 45:59.322 --> 46:01.792 the deck, so you should quit whenever you have 1 dollar. 46:01.789 --> 46:02.769 Prof: No. 46:02.768 --> 46:07.318 So he's saying just the commonsensical thing. 46:07.320 --> 46:11.320 After you draw one black card he would quit because now the 46:11.322 --> 46:12.842 deck is against you. 46:12.840 --> 46:14.790 It's 25 red [correction: 26 red] 46:14.791 --> 46:16.051 and only 25 black. 46:16.050 --> 46:17.240 The deck's against you. 46:17.239 --> 46:20.109 Why go on and play against an unfavorable deck? 46:20.110 --> 46:23.210 You've got your dollar, be satisfied and quit. 46:23.210 --> 46:26.690 That's his recommendation, but that's wrong and it's 46:26.686 --> 46:29.616 because you're doing what everybody does. 46:29.619 --> 46:31.859 You underestimate the option. 46:31.860 --> 46:35.500 The option is incredibly valuable here and now I just 46:35.503 --> 46:39.503 want to show how to compute what your optimal strategy is, 46:39.496 --> 46:42.016 and I think you'll be surprised. 46:42.018 --> 46:45.338 You should keep drawing, not three times--if you got a 46:45.342 --> 46:47.982 fourth black card, so you've already made 4 46:47.976 --> 46:48.726 dollars. 46:48.730 --> 46:49.650 It's a sunk cost. 46:49.650 --> 46:50.990 You've got this horrible deck. 46:50.989 --> 46:52.769 It's 26,22. 46:52.769 --> 46:53.989 Should you keep playing? 46:53.989 --> 46:55.189 Yes you should. 46:55.190 --> 46:56.220 Yes you can. 46:56.219 --> 47:01.829 If you get a fifth black card in a row you're up 5 dollars. 47:01.829 --> 47:03.929 The deck is horribly against you. 47:03.929 --> 47:05.479 Should you keep playing? 47:05.480 --> 47:06.790 Yes you should. 47:06.789 --> 47:10.239 So anyway, it's really shocking, I think. 47:10.239 --> 47:13.009 So now let's just see how to compute this out so that we 47:13.014 --> 47:14.584 don't have to argue about it. 47:14.579 --> 47:18.659 It's just a little bit of mathematics and you just see how 47:18.664 --> 47:21.034 surprising this calculation is. 47:21.030 --> 47:22.130 So how would you do it? 47:22.130 --> 47:32.820 47:32.820 --> 47:37.360 Well, the key is to figure out how to put it into a tree. 47:37.360 --> 47:39.910 So I'm not going to draw the picture because it gets too 47:39.913 --> 47:43.023 complicated, but basically what you want to 47:43.021 --> 47:46.171 know is how valuable-- remember the tree in backward 47:46.166 --> 47:46.656 induction? 47:46.659 --> 47:50.849 It was, take the thing at the end and then figure out by 47:50.851 --> 47:53.521 propagating the values backwards. 47:53.518 --> 47:59.658 So if I have black and red here, and I've got 1 black card 47:59.659 --> 48:04.289 and no red cards, the value to me of that is 48:04.289 --> 48:05.259 what? 48:05.260 --> 48:06.760 I'm going to win 1 dollar for sure. 48:06.760 --> 48:09.250 If there's only 1 black card left in the deck and no red 48:09.251 --> 48:12.061 cards I know I'm going to play to the end and get 1 dollar, 48:12.059 --> 48:15.689 and obviously the value of 2 black cards and no red cards is 48:15.688 --> 48:16.978 2 dollars etcetera. 48:16.980 --> 48:22.910 And I also know, what's the value to me of no 48:22.911 --> 48:24.801 black cards? 48:24.800 --> 48:27.460 You know at every stage what's left in the deck because you've 48:27.460 --> 48:28.640 seen what came up before. 48:28.639 --> 48:32.719 So if you're at the very end of the deck with no black cards and 48:32.721 --> 48:36.221 1 red card what's the value of that position to you? 48:36.219 --> 48:36.789 Student: 0. 48:36.789 --> 48:39.719 Prof: 0, not minus 1,0, 48:39.722 --> 48:41.142 why is that? 48:41.139 --> 48:44.719 Because you're going to quit, you're not going to play and 48:44.724 --> 48:47.874 that's the critical step, seeing that this is 0. 48:47.869 --> 48:49.459 Someone said minus 1. 48:49.460 --> 48:52.150 That difference between 0 and minus 1, that's the whole heart 48:52.148 --> 48:52.818 of the thing. 48:52.820 --> 48:55.890 So 0 and 2, the value of that is also 0. 48:55.889 --> 48:57.749 You're just going to quit. 48:57.750 --> 49:01.820 So what in general is the value? 49:01.820 --> 49:06.710 What is the value of V if there are B black cards and R red 49:06.710 --> 49:09.410 cards what's the value to you? 49:09.409 --> 49:16.419 Well, a crucial step is that you can choose to quit by not 49:16.420 --> 49:17.650 playing. 49:17.650 --> 49:19.860 So this is the value from then on to you. 49:19.860 --> 49:23.350 So B black cards left, R red cards left. 49:23.349 --> 49:28.179 You could get 0 by quitting, or you could draw a card. 49:28.179 --> 49:31.729 Now, what happens to you if you draw a card? 49:31.730 --> 49:34.410 What happens to you if you draw a card? 49:34.409 --> 49:42.209 Well, with probability B over (B R) you win 1 dollar, 49:42.210 --> 49:49.110 right, but then you move on to the new deck. 49:49.110 --> 50:00.010 So what do I write V here, V of B - 1 and R, 50:00.010 --> 50:09.290 but with probability R over (B R), you drew a red card, 50:09.289 --> 50:16.309 so that's minus 1, but then you move onto a deck 50:16.307 --> 50:22.337 that has 1 less red card, and that's it. 50:22.340 --> 50:26.490 You either decide to stop or if you've decided to draw a card 50:26.487 --> 50:30.357 you know what the chances of getting a black card are. 50:30.360 --> 50:35.060 You look at the black cards, 26. 50:35.059 --> 50:35.889 I'm down there. 50:35.889 --> 50:37.609 It's 21 out of 47. 50:37.610 --> 50:40.220 It sounds horrible, 21 out of 47, 50:40.215 --> 50:41.595 I win 1 dollar. 50:41.599 --> 50:44.369 26 out of 47 I lose 1 dollar. 50:44.369 --> 50:48.819 So the immediate draw is terrible, but if I get a black 50:48.820 --> 50:53.770 card I move to this situation, and if I get a red card I move 50:53.766 --> 50:55.576 to this situation. 50:55.579 --> 51:04.919 So do you agree with me that that's what the value's going to 51:04.920 --> 51:05.700 be? 51:05.699 --> 51:07.709 Are there any questions? 51:07.710 --> 51:09.820 This is a critical formula, a critical spot. 51:09.820 --> 51:15.500 Does everyone--Sophia you're now in trouble. 51:15.500 --> 51:18.990 Somebody came and said hello to me after class and now I know a 51:18.989 --> 51:21.239 name, so does this formula make sense? 51:21.239 --> 51:23.039 Student: Yes. 51:23.039 --> 51:24.839 Prof: Kathleen, yes? 51:24.840 --> 51:26.030 Is it Katherine or Kathleen? 51:26.030 --> 51:27.080 Student: Katherine. 51:27.079 --> 51:27.819 Prof: Katherine, okay. 51:27.820 --> 51:30.050 Katherine, so you agree with this formula, 51:30.054 --> 51:30.494 right? 51:30.489 --> 51:32.349 But this formula's the key. 51:32.349 --> 51:34.049 It's just like our tree. 51:34.050 --> 51:37.360 Once you know what the values are down here you can always go 51:37.360 --> 51:39.680 backwards and figure out the value here. 51:39.679 --> 51:44.949 So what is the tree? 51:44.949 --> 51:47.799 The tree is going to have--well, I'm going I'm going 51:47.795 --> 51:49.185 to do it on a computer. 51:49.190 --> 52:01.590 So now we can just do this on a computer. 52:01.590 --> 52:06.960 I hope I don't have to do that. 52:06.960 --> 52:13.310 So you have this, by the way, it's on the web. 52:13.309 --> 52:17.469 Oh no! 52:17.469 --> 52:19.439 Why are all these, oh okay, there aren't question 52:19.440 --> 52:19.770 marks. 52:19.769 --> 52:20.829 So here it is. 52:20.829 --> 52:25.059 So you can see that on this, I can do it with this, 52:25.059 --> 52:30.059 on this thing it's--by the way, I did my own spreadsheet and an 52:30.063 --> 52:34.913 undergraduate last year thought it was so messy that she just 52:34.905 --> 52:36.515 redid it for me. 52:36.519 --> 52:38.399 So this is her doing. 52:38.400 --> 52:40.080 It looks much better than I did. 52:40.079 --> 52:42.009 So anyway, here are the number of cards. 52:42.010 --> 52:45.210 This is what the number of black cards left. 52:45.210 --> 52:47.670 This is the number of red cards left, 52:47.670 --> 52:51.400 and then when you go to the corresponding coordinate like 52:51.400 --> 52:54.410 this one, this is the value of the game 52:54.407 --> 52:57.677 when you have 1 black card and 1 red card. 52:57.679 --> 53:00.879 Even though there's an even deck it's a favorable game to 53:00.876 --> 53:01.216 you. 53:01.219 --> 53:02.129 Why is that? 53:02.130 --> 53:04.540 Because you draw the first card, if it's black, 53:04.536 --> 53:07.776 which happens with probability 50 percent, you win 1 dollar. 53:07.780 --> 53:08.960 So you've got half a dollar. 53:08.960 --> 53:11.440 If it's red, which happens with 50 percent 53:11.443 --> 53:15.203 probability, you draw the next one and so you end up with 0. 53:15.199 --> 53:16.829 And by the way, if you got a black the first 53:16.827 --> 53:17.847 time, obviously you stop. 53:17.849 --> 53:20.269 So you get 50 percent chance of 1 dollar and stopping or 50 53:20.268 --> 53:22.688 percent chance of going to the end and getting nothing, 53:22.690 --> 53:25.270 so it's value of 50 percent, but that's a bad way of 53:25.273 --> 53:26.593 calculating that number. 53:26.590 --> 53:29.160 She's got a much better way of calculating it. 53:29.159 --> 53:33.419 So what she said is if you had no red cards and only black 53:33.422 --> 53:37.842 cards the value is to go to the end of the deck and just win 53:37.836 --> 53:38.806 them all. 53:38.809 --> 53:42.439 So if you've got only red cards in the deck and the top line is 53:42.443 --> 53:45.613 no black cards obviously you should quit right away. 53:45.610 --> 53:49.110 So we've got this first thing trivially done. 53:49.110 --> 53:51.020 Now, how do you figure out this thing? 53:51.018 --> 53:54.198 Well, if you look at the formula up there it's just the 53:54.204 --> 53:56.924 formula you wrote, we wrote, which is you could 53:56.916 --> 53:58.446 quit if you wanted to. 53:58.449 --> 54:01.929 So you have to take the max of 0, but if you go on and draw, 54:01.929 --> 54:05.699 I can't read what's written there, if you go on and draw 54:05.702 --> 54:08.862 there's going to be the probability of red, 54:08.860 --> 54:12.420 this is the probability of getting a red over the total 54:12.422 --> 54:15.132 number of cards times losing 1 dollar, 54:15.130 --> 54:18.760 plus what happens is you then move. 54:18.760 --> 54:29.970 If you drew red then you go to--what did she do? 54:29.969 --> 54:32.769 Student: > 54:32.768 --> 54:33.608 the value. 54:33.610 --> 54:36.640 Prof: C 4, oh C 4 is here, 54:36.641 --> 54:37.401 right. 54:37.400 --> 54:39.260 So sorry, I was getting confused with the cards. 54:39.260 --> 54:40.970 So C 4 is this squared. 54:40.969 --> 54:42.419 That's the value in here. 54:42.420 --> 54:45.270 So C 4 says if you draw a red card the first time, 54:45.268 --> 54:47.678 it happens with this probability, the number of reds 54:47.675 --> 54:51.085 over the total number of cards, you lose 1 dollar and then you 54:51.090 --> 54:53.960 move to the position C 4, which is 1 back here, 54:53.963 --> 54:57.583 the one where you've got 1 less red card and just a black card 54:57.576 --> 54:58.046 left. 54:58.050 --> 55:00.280 On the other hand, you could have drawn the first 55:00.277 --> 55:02.217 time, not the red one but the black 55:02.215 --> 55:04.535 one divided by the total number of cards, 55:04.539 --> 55:06.799 and you would have won 1 dollar, but then you would have 55:06.802 --> 55:09.152 moved to the position where you had 1 less black card, 55:09.150 --> 55:11.850 which is D 3, which is up here. 55:11.849 --> 55:14.869 So instead of doing the whole game, she says, 55:14.873 --> 55:18.453 half the time you win, but then you move over to here 55:18.447 --> 55:20.027 and get that value. 55:20.030 --> 55:23.280 Half the time you lose, and then you lose 1 dollar, 55:23.275 --> 55:25.415 and then you move over to here. 55:25.420 --> 55:26.160 So she's done that. 55:26.159 --> 55:29.559 That same formula appears in every box, so all you had to do 55:29.556 --> 55:30.646 was just copy it. 55:30.650 --> 55:33.140 It's max of 0, and then the chance that you're 55:33.139 --> 55:36.109 going to lose 1 dollar, which is the number here of 55:36.112 --> 55:39.742 reds over the totals times minus 1 dollar and then moving over to 55:39.744 --> 55:40.204 here. 55:40.199 --> 55:42.939 Or you could get the probability of winning 1 dollar 55:42.936 --> 55:44.166 with the black cards. 55:44.170 --> 55:48.170 So you win 1 dollar, but then you move up to here. 55:48.170 --> 55:53.160 So it's very simple. 55:53.159 --> 55:54.859 So she's done it. 55:54.860 --> 55:57.810 And notice that although the deck is even at 1 card each, 55:57.807 --> 56:00.017 so it sounds like a fair game, it's not. 56:00.018 --> 56:02.238 It's a favorable game because you have an option, 56:02.235 --> 56:04.865 so you all understood that, but the thing is the option is 56:04.865 --> 56:06.615 much more valuable than you think. 56:06.619 --> 56:09.949 So let's see what the value of the game is. 56:09.949 --> 56:15.539 It's when you had 26 black cards and 26 red cards, 56:15.541 --> 56:19.651 so we have to go way over to here. 56:19.650 --> 56:22.210 Sorry. 56:22.210 --> 56:23.640 Where am I going? 56:23.639 --> 56:24.569 That's not the right answer. 56:24.570 --> 56:27.720 Here it is, 26,26, the value of the game is 2.6 56:27.721 --> 56:30.051 dollars, not just half a dollar. 56:30.050 --> 56:32.350 You wanted to quit, wherever you are, 56:32.353 --> 56:33.573 at half a dollar. 56:33.570 --> 56:35.150 He's not looking up anymore. 56:35.150 --> 56:41.780 So he wanted to quit after the first draw, but it's much better 56:41.784 --> 56:43.074 than that. 56:43.070 --> 56:47.560 So now what the shocking thing, though, is, so this means with 56:47.561 --> 56:51.541 26,26 you have a favorable game and you should draw. 56:51.539 --> 56:53.059 If you didn't draw it'd be worth 0. 56:53.059 --> 56:54.519 So obviously you're supposed to draw here. 56:54.518 --> 56:56.828 If you get a black card you're going to go here. 56:56.829 --> 56:58.249 So here you're down. 56:58.250 --> 57:01.290 You've gotten a card and you've got one more, 57:01.289 --> 57:03.969 you know, it wasn't that likely you were going to get black, 57:03.969 --> 57:06.649 but if you did--50 percent chance you go here. 57:06.650 --> 57:09.820 So you win 1 dollar and now you're at this position. 57:09.820 --> 57:14.640 Now, if you're supposed to stop at that point, 57:14.637 --> 57:17.527 what would you have done? 57:17.530 --> 57:20.430 If you were supposed to stop at that point you would have had 57:20.427 --> 57:20.907 value 0. 57:20.909 --> 57:23.529 So the fact that that number is positive is telling you even 57:23.534 --> 57:26.834 when the deck is against you, 25, you can't see it, 57:26.831 --> 57:31.651 it's 25 blacks and still 26 reds it's still a favorable 57:31.646 --> 57:32.356 game. 57:32.360 --> 57:35.560 You should draw a card and if by some miracle you won 1 dollar 57:35.561 --> 57:37.191 you would have moved to here. 57:37.190 --> 57:41.000 So now you're 26 red and 24 black, but the game is still 57:40.998 --> 57:41.828 favorable. 57:41.829 --> 57:47.009 You should draw another card, and if you win again you're 57:47.012 --> 57:47.662 here. 57:47.659 --> 57:49.889 Now you've gone one, two, three times you've drawn 57:49.889 --> 57:50.299 blacks. 57:50.300 --> 57:51.710 You should still draw another one. 57:51.710 --> 57:54.830 Four times getting blacks you should still draw another one. 57:54.829 --> 57:59.959 Five consecutive black draws the deck is now 26 against you 57:59.963 --> 58:02.003 and 21 in your favor. 58:02.000 --> 58:03.680 You should draw again. 58:03.679 --> 58:06.729 The game is still slightly favorable, and it just seems 58:06.733 --> 58:10.303 shocking that could be the case, but this is the proof that it's 58:10.295 --> 58:11.025 the case. 58:11.030 --> 58:15.240 So anyway, that illustrates the power of the option of being 58:15.242 --> 58:16.602 able to continue. 58:16.599 --> 58:19.469 And if you work, you're going to work out low 58:19.472 --> 58:22.962 numbers 2 and 3 in the homework, and then you'll see very 58:22.963 --> 58:25.773 clearly why it is that this option is just so powerful. 58:25.769 --> 58:27.369 It's uncannily strong. 58:27.369 --> 58:29.169 So are there any questions about this? 58:29.170 --> 58:30.020 Yes? 58:30.018 --> 58:32.108 Student: I'm still a little bit confused. 58:32.110 --> 58:35.430 I know that the option value is positive, but the probability 58:35.425 --> 58:37.575 actually is a little bit against you. 58:37.579 --> 58:44.139 So especially after five wins why do you want another draw, 58:44.139 --> 58:46.629 because yes, the option value is a little 58:46.625 --> 58:49.985 bit positive, but the probability is still 58:49.989 --> 58:51.089 against you? 58:51.090 --> 58:52.170 Prof: This isn't the option value. 58:52.170 --> 58:54.370 This is the value of playing. 58:54.369 --> 58:58.099 So it says the option value is more important than the fact 58:58.103 --> 59:00.233 that the cards are against you. 59:00.230 --> 59:03.010 So he's asking, my TA is asking, 59:03.005 --> 59:08.195 how could it possibly be that the deck is now 26 red and 21 59:08.199 --> 59:13.389 black totally against you and according to this calculation 59:13.391 --> 59:15.901 you should still draw? 59:15.900 --> 59:19.040 He can't see the advantage in drawing because the odds are 59:19.039 --> 59:22.069 pretty high you're going to get a red card next time. 59:22.070 --> 59:23.420 Well, that's true. 59:23.420 --> 59:28.110 You're going to get a red card next time, but the thing is your 59:28.106 --> 59:29.766 downside is limited. 59:29.768 --> 59:30.848 Here's a way of thinking about it. 59:30.849 --> 59:34.369 You're up 5 cards. 59:34.369 --> 59:37.279 You can never lose more than 5 dollars from that point on 59:37.282 --> 59:40.092 because you can always play to the end of the deck, 59:40.090 --> 59:44.870 right, which means you lose the five back that you already won. 59:44.869 --> 59:48.129 So there's a downside--the downside of losing is limited 59:48.125 --> 59:48.535 here. 59:48.539 --> 59:51.549 On the other hand there's a big upside to you. 59:51.550 --> 59:56.840 You might by some miracle draw 10 consecutive black ones at 59:56.842 --> 1:00:00.222 that point and then you could quit. 1:00:00.219 --> 1:00:02.399 So your upside is much bigger than your downside. 1:00:02.400 --> 1:00:04.860 Now, the upside is less probable than the downside so 1:00:04.862 --> 1:00:07.232 it's not so obvious which is going to be bigger. 1:00:07.230 --> 1:00:09.580 Is the option value more important or is the fact that 1:00:09.581 --> 1:00:11.401 the deck is against you more important? 1:00:11.400 --> 1:00:13.470 It would be impossible to intuit the answer, 1:00:13.465 --> 1:00:15.045 but we don't have to intuit it. 1:00:15.050 --> 1:00:16.350 We just proved it. 1:00:16.349 --> 1:00:21.629 We solved for the optimal strategy. 1:00:21.630 --> 1:00:23.220 Are there any other questions? 1:00:23.219 --> 1:00:27.269 It's quite amazing, right, this--I'm going to pause 1:00:27.273 --> 1:00:28.493 for a second. 1:00:28.489 --> 1:00:29.429 Yes? 1:00:29.429 --> 1:00:32.289 Student: How do you calculate the profit at a given 1:00:32.288 --> 1:00:32.838 position? 1:00:32.840 --> 1:00:34.670 Prof: That's what we just did. 1:00:34.670 --> 1:00:40.490 So let's try it again. 1:00:40.489 --> 1:00:46.129 Student: It may not be a total profit. 1:00:46.130 --> 1:00:50.410 Prof: So what we did, this number V is the expected 1:00:50.409 --> 1:00:55.629 profit you're going to make if you start with B black cards and 1:00:55.626 --> 1:00:56.886 R red cards. 1:00:56.889 --> 1:01:01.859 And now the intuitive mind figures that if B is less than R 1:01:01.862 --> 1:01:06.582 you've got an unfavorable deck and you should just quit, 1:01:06.579 --> 1:01:08.979 but that's not the case. 1:01:08.980 --> 1:01:11.300 You can figure out what the profit is, how, 1:01:11.295 --> 1:01:12.945 by doing backward induction. 1:01:12.949 --> 1:01:15.559 You couldn't tell what the value of this bond is here with 1:01:15.557 --> 1:01:18.077 all these defaults and stuff until you started computing 1:01:18.076 --> 1:01:21.126 backwards until you got to here, so the same way here. 1:01:21.130 --> 1:01:24.990 We know at the edges it's very obvious when all the cards are 1:01:24.989 --> 1:01:27.949 black or all the cards are red, that's up here, 1:01:27.947 --> 1:01:30.197 it's obvious what the value is. 1:01:30.199 --> 1:01:33.569 But if you have a position here you can figure out what the 1:01:33.565 --> 1:01:36.345 value is of being in that position of 1 and 1. 1:01:36.349 --> 1:01:38.209 You could quit and be 0, or you could say, 1:01:38.210 --> 1:01:40.750 what are my chances of getting a black card and winning 1 1:01:40.750 --> 1:01:41.250 dollar? 1:01:41.250 --> 1:01:43.610 If I get a black card then I move to this position, 1:01:43.610 --> 1:01:46.350 but I already figured out this position's value because I'm 1:01:46.349 --> 1:01:48.049 doing backward induction, right? 1:01:48.050 --> 1:01:51.820 That's got one less--sorry, if I draw a black card I go 1:01:51.820 --> 1:01:52.590 this way. 1:01:52.590 --> 1:01:55.470 It's got 1 less black card and we already know that value of 1:01:55.465 --> 1:01:56.485 that position is 0. 1:01:56.489 --> 1:01:59.259 So to figure out the value of this position I know the chance 1:01:59.262 --> 1:02:01.622 of getting a black card, then I'm going to end up in 1:02:01.621 --> 1:02:03.241 that position which is value 0. 1:02:03.239 --> 1:02:05.609 I won't draw any more. 1:02:05.610 --> 1:02:09.050 Or I'm going to get a red card and then I'm going to move to 1:02:09.054 --> 1:02:12.564 this position over here whose value I've already computed. 1:02:12.559 --> 1:02:14.459 So that gives me the value here. 1:02:14.460 --> 1:02:16.450 How do I figure out the value here? 1:02:16.449 --> 1:02:19.629 Well, it's now 2 reds and 1 black. 1:02:19.630 --> 1:02:21.000 So this looks really bad. 1:02:21.000 --> 1:02:24.910 Actually this position, the value of this I happen to 1:02:24.907 --> 1:02:25.807 know is 0. 1:02:25.809 --> 1:02:27.989 How could it be, with 2 red cards and only 1 1:02:27.985 --> 1:02:30.665 black card actually the value of the position is 0? 1:02:30.670 --> 1:02:31.910 Well, what do I do? 1:02:31.909 --> 1:02:36.019 The chances of getting a black card the first time are 1 third. 1:02:36.018 --> 1:02:40.348 So V of (1,2) is going to be 1 third of getting a black card 1:02:40.349 --> 1:02:44.969 plus then I go to here which is no black cards left and just red 1:02:44.972 --> 1:02:48.652 cards, which obviously is 0, 1:02:48.650 --> 1:02:50.700 plus 2 thirds. 1:02:50.699 --> 1:02:53.799 2 thirds of the time I get a red card and lose 1 dollar, 1:02:53.800 --> 1:02:56.840 but then I'm going to move to here with 1 red card where I 1:02:56.842 --> 1:02:59.182 have V of (1 and 1), which I've already figured out 1:02:59.182 --> 1:03:00.002 the answer to, right? 1:03:00.000 --> 1:03:04.990 This was V of (1 and 1) has value a half. 1:03:04.989 --> 1:03:09.619 So therefore V of (1 and 2) is going to be 2 thirds times minus 1:03:09.617 --> 1:03:11.257 1 V of (1,2), right? 1:03:11.260 --> 1:03:15.020 I just drew a black card so it's no longer--no, 1:03:15.021 --> 1:03:18.211 I drew a red card, so it's (1 and 1). 1:03:18.210 --> 1:03:20.260 So I started with 1 black and 2 reds, 1:03:20.260 --> 1:03:22.100 1 third of the time I get a black card, 1:03:22.099 --> 1:03:24.509 2 thirds of the times I get a red card, 1:03:24.510 --> 1:03:28.140 but after getting the red card the position is now 1 black and 1:03:28.143 --> 1:03:28.623 1 red. 1:03:28.619 --> 1:03:30.109 The red card disappeared. 1:03:30.110 --> 1:03:31.190 That's over here. 1:03:31.190 --> 1:03:33.490 So I get a black card, I move to here. 1:03:33.489 --> 1:03:35.909 If I get a red card I move to there, 1:03:35.909 --> 1:03:42.839 but V of (1,1) is worth a half, so that's equal to 1 third 1:03:42.842 --> 1:03:48.562 times (1 0) 2 thirds times (minus 1 1 half), 1:03:48.559 --> 1:03:54.719 which equals 1 third - 1 third which equals 0. 1:03:54.719 --> 1:03:58.489 So starting at this point you've got 1 red card and 2 1:03:58.492 --> 1:03:59.512 black cards. 1:03:59.510 --> 1:04:02.160 It looks horrible to pick a card, 2 thirds of the time 1:04:02.159 --> 1:04:04.059 you're going to get the wrong card, 1:04:04.059 --> 1:04:07.779 but you still have a position that's actually equal in value, 1:04:07.780 --> 1:04:12.880 because if you get that black card which wins you stop. 1:04:12.880 --> 1:04:15.730 If you get a red card you're now in a position that's equal 1:04:15.731 --> 1:04:17.551 deck and that's favorable for you, 1:04:17.550 --> 1:04:20.000 because if you get another black card you stop, 1:04:20.000 --> 1:04:26.890 and if you get a red card you keep playing until the end. 1:04:26.889 --> 1:04:27.669 So that's it. 1:04:27.670 --> 1:04:29.780 So how can you do this by backward induction? 1:04:29.780 --> 1:04:33.630 You have the stuff on the edges and then you solve for all these 1:04:33.626 --> 1:04:36.856 things along the side here, and having done that now I can 1:04:36.858 --> 1:04:39.468 solve for this one because I've got up and to the left. 1:04:39.469 --> 1:04:42.539 Now I do that row and then I can do this whole row, 1:04:42.539 --> 1:04:45.839 and then I can just by backward induction do the whole thing, 1:04:45.840 --> 1:04:47.720 and the computer does that instantly. 1:04:47.719 --> 1:04:50.809 So it figures out the value of every single node and it's 1:04:50.813 --> 1:04:52.473 shocking what the answer is. 1:04:52.469 --> 1:05:00.589 So are there any other questions about this? 1:05:00.590 --> 1:05:03.250 Ben, do you-- so... 1:05:03.250 --> 1:05:04.790 Student: Yeah, I figured it out. 1:05:04.789 --> 1:05:07.549 I tell you initially I'm thinking about it. 1:05:07.550 --> 1:05:13.090 Well, maybe this value is actually an option value. 1:05:13.090 --> 1:05:17.000 For example if you choose to play this game and after you win 1:05:17.003 --> 1:05:21.053 or you lose then you'll get an option to continue the game, 1:05:21.050 --> 1:05:24.930 and so I separated that option value to this. 1:05:24.929 --> 1:05:25.909 Prof: Well, that is your option. 1:05:25.909 --> 1:05:30.599 The option is always to keep playing or to stop, 1:05:30.599 --> 1:05:33.809 but the value I wrote down is the value of the game to you, 1:05:33.809 --> 1:05:38.729 of being able to play the whole game however you want. 1:05:38.730 --> 1:05:42.460 So now let's do another example. 1:05:42.460 --> 1:05:43.340 Yes? 1:05:43.340 --> 1:05:45.540 Student: Would you call somebody rational who doesn't 1:05:45.536 --> 1:05:47.506 play the game up to that point that you showed me, 1:05:47.510 --> 1:05:50.420 the point .05 > 1:05:50.420 --> 1:05:52.220 Prof: Right, so if you got... 1:05:52.219 --> 1:05:55.309 Student: My question is, is it preference to risk or is 1:05:55.313 --> 1:05:57.093 it actually the rationality of... 1:05:57.090 --> 1:05:59.250 Prof: Right, so this is going to become very 1:05:59.250 --> 1:06:00.330 important very shortly. 1:06:00.329 --> 1:06:03.669 So his question is I just proved if you can call that a 1:06:03.670 --> 1:06:06.770 proof by computer, the computer proved that even 1:06:06.773 --> 1:06:10.823 if you got 5 blacks in a row you should still draw another card. 1:06:10.820 --> 1:06:14.220 Of course things are quite risky now because there's a very 1:06:14.224 --> 1:06:17.634 good chance you're going to lose on that very next card. 1:06:17.630 --> 1:06:20.920 So he's saying if you're risk averse maybe you would stop 1:06:20.920 --> 1:06:22.930 there, and how can you distinguish 1:06:22.931 --> 1:06:26.171 somebody who's risk averse from somebody who's just dumb and 1:06:26.172 --> 1:06:27.822 can't make the calculation. 1:06:27.820 --> 1:06:31.200 So that's going to be a question we're going to take up 1:06:31.202 --> 1:06:32.772 in the very next class. 1:06:32.768 --> 1:06:36.268 But I would say that it's usually because people are dumb 1:06:36.273 --> 1:06:38.343 and can't make the calculation. 1:06:38.340 --> 1:06:41.660 So they just don't realize how favorable the situation is 1:06:41.657 --> 1:06:45.567 they're in by having this option to be able to play to the end, 1:06:45.570 --> 1:06:47.330 to stop when they want to stop. 1:06:47.329 --> 1:06:50.079 So let's just do one more example. 1:06:50.079 --> 1:06:57.709 Suppose that you are undergraduates and you want to 1:06:57.711 --> 1:06:59.851 get married. 1:06:59.849 --> 1:07:01.779 You've been told that's a good idea, 1:07:01.780 --> 1:07:05.820 and you--it's going to be a very sexist thing, 1:07:05.820 --> 1:07:10.790 but anyway this is also a game I invented which turned out not 1:07:10.789 --> 1:07:13.479 to be as original as I thought. 1:07:13.480 --> 1:07:16.930 So I call this the optimal marriage problem. 1:07:16.929 --> 1:07:23.039 So let's say you knew you were going to meet 1,000 women. 1:07:23.039 --> 1:07:27.489 You're telling it from the guy's point of view. 1:07:27.489 --> 1:07:33.029 You're going to meet 1,000 women and each woman you meet 1:07:33.034 --> 1:07:39.084 her suitability you can't tell until you meet her and talk to 1:07:39.081 --> 1:07:44.811 her, and after you meet her each 1:07:44.809 --> 1:07:54.629 woman's suitability is uniformly distributed on (0,1). 1:07:54.630 --> 1:07:56.080 So what do I mean by that? 1:07:56.079 --> 1:07:58.419 I mean you meet her, you talk to her, 1:07:58.418 --> 1:08:01.208 you get to know her, and before you met her, 1:08:01.210 --> 1:08:04.590 you have no idea how suitable she's going to be. 1:08:04.590 --> 1:08:06.730 After you've talked to her you understand how suitable. 1:08:06.730 --> 1:08:07.960 The best, it's 1. 1:08:07.960 --> 1:08:11.070 The worst, it's 0, and it could be a draw anywhere 1:08:11.068 --> 1:08:12.208 between 0 and 1. 1:08:12.210 --> 1:08:13.720 Before you meet her you have no idea. 1:08:13.719 --> 1:08:17.079 After you meet her you know exactly how suitable she is, 1:08:17.081 --> 1:08:20.871 and there are going to be 1,000 of them that you could meet. 1:08:20.868 --> 1:08:28.438 The problem is that after you've talked to a woman you can 1:08:28.444 --> 1:08:32.964 marry her then, or you can move on, 1:08:32.962 --> 1:08:40.672 but once you've moved on you can never go back to her. 1:08:40.670 --> 1:08:42.700 So you understand the problem. 1:08:42.698 --> 1:08:46.808 The problem is that let's say the first woman is .95 or .90. 1:08:46.810 --> 1:08:48.700 You think, gosh how suitable. 1:08:48.698 --> 1:08:52.278 This is a great match, but I've got 999 more women to 1:08:52.284 --> 1:08:52.634 go. 1:08:52.630 --> 1:08:56.890 Maybe I'll do better, and then you get 0s from then 1:08:56.887 --> 1:08:59.167 on, and so you've missed your .90, 1:08:59.168 --> 1:09:02.468 and so you're going to end up marrying the last one who's 1:09:02.470 --> 1:09:03.710 maybe a 0 for you. 1:09:03.710 --> 1:09:10.040 That doesn't mean she's a 0, just for you a 0. 1:09:10.039 --> 1:09:10.829 I'm trying. 1:09:10.828 --> 1:09:16.348 Anyway, so what should your optimal strategy be and are you 1:09:16.349 --> 1:09:19.299 playing the optimal strategy. 1:09:19.300 --> 1:09:21.730 So what do you think, just intuitively, 1:09:21.725 --> 1:09:23.635 what's the optimal strategy? 1:09:23.640 --> 1:09:26.070 Of course we're going to do it by backward induction, 1:09:26.073 --> 1:09:28.183 but what do you think it's going to look like, 1:09:28.179 --> 1:09:29.349 the optimal strategy? 1:09:29.350 --> 1:09:29.740 Yes? 1:09:29.738 --> 1:09:31.928 Student: The further you get into the game the less 1:09:31.930 --> 1:09:32.930 selective you should be. 1:09:32.930 --> 1:09:36.190 Prof: Right, that's what's going to happen. 1:09:36.189 --> 1:09:39.669 We're going to prove this, but he's exactly picked--he 1:09:39.672 --> 1:09:42.962 said, you set a threshold here at the beginning. 1:09:42.960 --> 1:09:45.900 You'll marry her if she's above some number. 1:09:45.899 --> 1:09:48.069 You keep to that threshold for a while. 1:09:48.069 --> 1:09:51.049 Then you haven't married anyone and you'd say, 1:09:51.045 --> 1:09:54.875 oh my god, I'm running out of women and then your standards 1:09:54.878 --> 1:09:56.068 just collapse. 1:09:56.069 --> 1:09:59.129 Desperation sets in. 1:09:59.130 --> 1:10:02.370 So that's absolutely right, but the only interesting thing 1:10:02.369 --> 1:10:05.269 is to figure out how high the standard should be. 1:10:05.270 --> 1:10:09.260 So how high do you think it is at the beginning? 1:10:09.260 --> 1:10:13.470 What would you say the number is at the beginning? 1:10:13.470 --> 1:10:16.180 Now, let me give you a hint. 1:10:16.180 --> 1:10:22.210 If you divide up--here's 1 and there's 1,000 women, 1:10:22.211 --> 1:10:23.901 so here's 0. 1:10:23.899 --> 1:10:25.429 So they're randomly picked. 1:10:25.430 --> 1:10:30.140 So if you could look at all the women, and pick out the most 1:10:30.137 --> 1:10:33.807 suitable one, what would her suitability be? 1:10:33.810 --> 1:10:41.340 Well, her suitability would be--so top, the top on average 1:10:41.340 --> 1:10:46.890 will be something like 1,000 over 1,001. 1:10:46.890 --> 1:10:48.280 This is a famous problem. 1:10:48.279 --> 1:10:52.709 If you take N people randomly, you take N numbers you pick 1:10:52.712 --> 1:10:57.182 randomly uniformly on (0, 1) the top one on average is 1:10:57.176 --> 1:11:00.056 going to be, if there's 1,000 women it's 1:11:00.060 --> 1:11:01.220 1,000 over 1,001. 1:11:01.220 --> 1:11:06.730 Second top is going to be 999 over 1,001. 1:11:06.729 --> 1:11:11.019 So this very standard statistical result actually was 1:11:11.018 --> 1:11:15.388 derived by a former Yale professor in World War II. 1:11:15.390 --> 1:11:18.540 The Americans captured German tanks which had all their serial 1:11:18.539 --> 1:11:19.469 numbers on them. 1:11:19.470 --> 1:11:24.150 The first tank was number 1, the second tank was number 2, 1:11:24.149 --> 1:11:26.139 the third tank they made was number 3, 1:11:26.140 --> 1:11:28.750 so we captured a bunch of them and then we had to guess, 1:11:28.750 --> 1:11:30.460 how many tanks did they make? 1:11:30.460 --> 1:11:34.380 Anyhow, so it's related to this idea that if they're uniformly 1:11:34.380 --> 1:11:37.390 distributed on (0, 1) the top one's going to be on 1:11:37.391 --> 1:11:42.641 average 1,000 over 1,001, 999 over 1,001 etcetera. 1:11:42.640 --> 1:11:47.080 So what standard would you set for the first one? 1:11:47.078 --> 1:11:47.408 Right? 1:11:47.408 --> 1:11:49.438 You have to set some threshold here. 1:11:49.439 --> 1:11:52.399 Here's one and here's 0. 1:11:52.399 --> 1:11:55.879 By the end you'll take the last woman you've got--you're going 1:11:55.877 --> 1:11:57.527 to take her no matter what. 1:11:57.529 --> 1:12:01.219 So what should the threshold be? 1:12:01.220 --> 1:12:02.330 Well, it's hard to tell. 1:12:02.328 --> 1:12:04.198 We can't do it except by backward induction. 1:12:04.199 --> 1:12:06.449 So what would you guess? 1:12:06.448 --> 1:12:07.418 Student: 1 > 1:12:07.420 --> 1:12:09.760 Prof: 1, well then you'll never take her 1:12:09.762 --> 1:12:12.462 if it's 1 because the odds of getting exactly 1 are 0, 1:12:12.462 --> 1:12:13.892 so what would you guess? 1:12:13.890 --> 1:12:15.650 Student: > 1:12:15.649 --> 1:12:19.519 Prof: So you'd set the threshold this high, 1:12:19.519 --> 1:12:21.019 1,000 over 1,001. 1:12:21.020 --> 1:12:24.060 So that means you're expecting to get as good a woman as if you 1:12:24.056 --> 1:12:26.406 could go to the very end and look at all of them, 1:12:26.408 --> 1:12:27.828 you never make a mistake. 1:12:27.828 --> 1:12:29.888 I told you, there's a chance you'll make a mistake. 1:12:29.890 --> 1:12:31.030 The first one's the best. 1:12:31.029 --> 1:12:33.699 It doesn't quite come to your threshold and all the rest are 1:12:33.695 --> 1:12:35.725 worse then you end up with a disaster here. 1:12:35.729 --> 1:12:40.069 So you have a chance of not doing that well, 1:12:40.069 --> 1:12:41.819 so you're setting too high a standard here, 1:12:41.819 --> 1:12:44.189 because you have a very good chance of saying no to all these 1:12:44.194 --> 1:12:45.784 women and then ending up with what's-- 1:12:45.779 --> 1:12:46.489 right? 1:12:46.488 --> 1:12:49.278 However, you're on the right track. 1:12:49.279 --> 1:12:52.799 So amazingly, this is the answer, 1:12:52.800 --> 1:12:55.440 this is the threshold. 1:12:55.439 --> 1:13:00.249 So you should set the threshold at where you expect the second 1:13:00.253 --> 1:13:03.693 highest woman to be, the second highest match to be, 1:13:03.686 --> 1:13:06.756 and that is why there's so many novels about the other woman 1:13:06.764 --> 1:13:09.634 because if you're playing optimally you should be ending 1:13:09.632 --> 1:13:12.662 up with the second best woman and there should be one other 1:13:12.658 --> 1:13:15.788 woman that at the end of your life you regret that you didn't 1:13:15.787 --> 1:13:18.587 wait for, but only one other woman. 1:13:18.590 --> 1:13:23.470 So anyway, I just want to prove this to you in the same way we 1:13:23.465 --> 1:13:28.415 proved it before just by solving for backward induction for the 1:13:28.420 --> 1:13:30.600 optimal-- so we can just do this by 1:13:30.604 --> 1:13:31.524 backward induction. 1:13:31.520 --> 1:13:34.540 We know exactly what to do. 1:13:34.538 --> 1:13:36.408 It's one thing to say what you should do. 1:13:36.408 --> 1:13:38.498 It's another thing to prove that's what you should do and 1:13:38.496 --> 1:13:39.536 I'm going to prove it now. 1:13:39.538 --> 1:13:46.478 So you can just see by backward induction how easy it is to do 1:13:46.481 --> 1:13:48.191 these things. 1:13:48.189 --> 1:13:49.299 I'm going to have to take four minutes. 1:13:49.300 --> 1:13:51.870 If you can hang out for four minutes we'll get this. 1:13:51.868 --> 1:13:55.128 So what happens with 2 women left? 1:13:55.130 --> 1:13:58.310 What's V of 2? 1:13:58.310 --> 1:13:59.770 What should you do? 1:13:59.770 --> 1:14:05.230 What should your threshold be for 2 women? 1:14:05.229 --> 1:14:09.799 So here's the threshold. 1:14:09.800 --> 1:14:15.520 Threshold for 1 woman is 0. 1:14:15.520 --> 1:14:19.830 If it's the last woman there whatever she is that's it. 1:14:19.829 --> 1:14:21.059 You might as well marry her. 1:14:21.060 --> 1:14:23.070 It can't be negative. 1:14:23.069 --> 1:14:27.769 And then the value--unlike Herodotus-- 1:14:27.770 --> 1:14:30.820 the value of 1 is going to be a half, 1:14:30.819 --> 1:14:32.849 right, because if there's 1 woman left you're going to take 1:14:32.850 --> 1:14:34.740 her no matter what and on average you'll get a half. 1:14:34.738 --> 1:14:38.168 So the question is now, what's the threshold when there 1:14:38.170 --> 1:14:41.410 are 2 women left and what's your expected payoff? 1:14:41.408 --> 1:14:44.248 So what's the threshold if there are 2 women left? 1:14:44.250 --> 1:14:46.890 You see the second to last woman, you talk to her, 1:14:46.893 --> 1:14:48.893 you find out how good the match is. 1:14:48.890 --> 1:14:52.650 You should take her if the match is above what? 1:14:52.649 --> 1:14:54.459 Student: 1 over 1,001? 1:14:54.460 --> 1:14:58.030 Prof: No, there's only 1 woman left after 1:14:58.029 --> 1:14:58.789 her, so. 1:14:58.788 --> 1:15:00.818 So a half, right, because if you don't take her 1:15:00.823 --> 1:15:03.483 you're going to go to the last woman and on an average you're 1:15:03.475 --> 1:15:05.875 going to get a half, so there's no point in taking 1:15:05.878 --> 1:15:08.658 someone whose match is less than a half when the very next step 1:15:08.657 --> 1:15:10.557 you're, on average, you're going to get 1:15:10.555 --> 1:15:10.915 a half. 1:15:10.920 --> 1:15:12.590 So your threshold is a half. 1:15:12.590 --> 1:15:15.260 So what's your expected quality of the match? 1:15:15.260 --> 1:15:22.660 Well, with probability a half she's going to be above a half, 1:15:22.658 --> 1:15:26.108 and if she's above a half she's going to be half way between 1 1:15:26.108 --> 1:15:28.708 and a half, so it's 3 quarters. 1:15:28.710 --> 1:15:32.260 And with probability, sorry, and with probability a 1:15:32.255 --> 1:15:36.645 half she's going to be below a half and you're going to pass on 1:15:36.652 --> 1:15:41.052 her and go to the last one and it'll be on average a half. 1:15:41.050 --> 1:15:45.600 So this is 3 eighths plus, no. 1:15:45.600 --> 1:15:48.850 Probability a half, she's going to be 3 quarters 1:15:48.854 --> 1:15:52.254 and probability a half, she'll be a half so it's 3 1:15:52.246 --> 1:15:54.666 eighths 1 quarter, is 5 eights. 1:15:54.670 --> 1:15:58.820 Now, what if there are 3 women, what should the threshold be 1:15:58.823 --> 1:16:01.643 for 3 women and what's the value of 3? 1:16:01.640 --> 1:16:02.680 Student: 5 eighths. 1:16:02.680 --> 1:16:06.060 Prof: So the threshold should be V of 2, 1:16:06.055 --> 1:16:09.205 which is 5 eighths, and what's the value? 1:16:09.210 --> 1:16:12.150 The value's going to be, what's the chance that you take 1:16:12.154 --> 1:16:13.444 the one you just meet? 1:16:13.439 --> 1:16:15.629 Well, the odds that she's above a half [correction: 1:16:15.625 --> 1:16:16.845 above V(2)] is 3 eighths, 1:16:16.850 --> 1:16:22.770 so it's 3 eighths times the average of 1, 1:16:22.770 --> 1:16:25.000 you know, half way between, so he's above 5 eights, 1:16:25.000 --> 1:16:27.210 so she's somewhere between 1 and 5 eighths. 1:16:27.210 --> 1:16:31.230 So that's going to be (1 5 eighths) over 2, 1:16:31.225 --> 1:16:36.955 plus 5 eighths of the time you pass on her and then you get 5 1:16:36.962 --> 1:16:38.112 eighths. 1:16:38.109 --> 1:16:46.149 So that equals, if we just do that a little bit 1:16:46.152 --> 1:16:54.022 more generally, it's 1 - V (2)--So that's (1 - 1:16:54.019 --> 1:17:04.159 V (2)) times (1 V (2)) divided by 2 V (2) times V (2). 1:17:04.159 --> 1:17:06.459 So that's the formula. 1:17:06.460 --> 1:17:11.800 So in general V (t), you set the threshold at V 1:17:11.804 --> 1:17:12.854 -(t-1). 1:17:12.850 --> 1:17:19.290 So with probability V (t-1) you're going to get (1 V (t-1)) 1:17:19.288 --> 1:17:23.948 divided by 2, and with probability 1 minus, 1:17:23.952 --> 1:17:24.732 no. 1:17:24.729 --> 1:17:28.279 This is 1 - V (t-1), with probability of V (t-1) 1:17:28.282 --> 1:17:32.592 you're going to get--you're going to pass on her and go to 1:17:32.590 --> 1:17:35.840 the next thing so you get just V (t-1). 1:17:35.840 --> 1:17:37.420 So that's just a formula. 1:17:37.420 --> 1:17:40.730 V(t) equals some function of V(t-1). 1:17:40.729 --> 1:17:43.519 So you can program that into a computer. 1:17:43.520 --> 1:17:46.440 So I'm ending now with this one picture. 1:17:46.439 --> 1:17:47.469 That's the end of it. 1:17:47.470 --> 1:17:47.980 Sorry. 1:17:47.979 --> 1:17:49.979 I know I've gone over, but this is the last picture. 1:17:49.979 --> 1:17:51.199 It'll only take a second. 1:17:51.198 --> 1:17:57.188 So 0, yield curve optimal marriage, so here it is, 1:17:57.194 --> 1:18:02.584 with one woman your value--oh shit, sorry. 1:18:02.578 --> 1:18:17.898 With 1 woman the value is a half, 1 match to go. 1:18:17.899 --> 1:18:22.139 So the number on the left is with 1 match to go the value's a 1:18:22.135 --> 1:18:26.295 half, with 2 it's 5 eighths, with 3 we computed that too. 1:18:26.300 --> 1:18:28.620 So you can tell for however many women you want. 1:18:28.618 --> 1:18:33.838 Now, what I've done on the right number, 1:18:33.836 --> 1:18:37.176 is this N - 1 over N 1. 1:18:37.180 --> 1:18:40.730 So that's the second best woman how good she'd be on average for 1:18:40.734 --> 1:18:42.534 you, and as you go down further and 1:18:42.527 --> 1:18:45.227 further you see these numbers are getting to be the same. 1:18:45.229 --> 1:18:48.369 So this number and this number's practically the same 1:18:48.367 --> 1:18:51.687 and if you go down to the very bottom you'll see they're 1:18:51.685 --> 1:18:52.525 identical. 1:18:52.529 --> 1:18:57.199 Up to an incredible number of decimal places these two numbers 1:18:57.201 --> 1:18:58.351 are the same. 1:18:58.350 --> 1:19:00.670 So if there are enough women you're going to get exactly the 1:19:00.671 --> 1:19:02.921 second best, and it's going to be the problem of the other 1:19:02.916 --> 1:19:03.306 woman. 1:19:03.310 --> 1:19:06.890 But anyway, the point of all this was to just illustrate how 1:19:06.885 --> 1:19:08.395 powerful the option is. 1:19:08.399 --> 1:19:11.189 It's as if you could go to the very end and pick out the second 1:19:11.194 --> 1:19:13.584 best one even though you have to do them sequentially, 1:19:13.583 --> 1:19:13.903 so. 1:19:13.899 --> 1:19:18.999