WEBVTT 00:00:00.000 --> 00:00:02.420 align:middle line:90% [SQUEAKING] 00:00:02.420 --> 00:00:04.340 align:middle line:90% [RUSTLING] 00:00:04.340 --> 00:00:05.780 align:middle line:90% [CLICKING] 00:00:05.780 --> 00:00:10.237 align:middle line:90% 00:00:10.237 --> 00:00:12.320 align:middle line:84% IAN BALL: All right, so we're going to start today 00:00:12.320 --> 00:00:13.580 align:middle line:90% with another MobLab game. 00:00:13.580 --> 00:00:16.540 align:middle line:84% Hopefully everyone can get logged in. 00:00:16.540 --> 00:00:18.040 align:middle line:84% This is going to be-- the game we're 00:00:18.040 --> 00:00:20.380 align:middle line:84% going to play is called the Ultimatum Game. 00:00:20.380 --> 00:00:24.360 align:middle line:90% 00:00:24.360 --> 00:00:26.120 align:middle line:84% And this is probably the most famous game 00:00:26.120 --> 00:00:29.800 align:middle line:84% that's used in laboratory experiments about economics. 00:00:29.800 --> 00:00:31.880 align:middle line:84% So the way it's going to work is that you're 00:00:31.880 --> 00:00:34.140 align:middle line:84% going to be randomly assigned to two roles. 00:00:34.140 --> 00:00:36.520 align:middle line:84% One of you is going to be the proposer, and one of you 00:00:36.520 --> 00:00:38.480 align:middle line:90% is going to be the responder. 00:00:38.480 --> 00:00:40.400 align:middle line:90% And there's some pot of money. 00:00:40.400 --> 00:00:42.760 align:middle line:84% And the proposer just proposes how 00:00:42.760 --> 00:00:45.780 align:middle line:84% that money is split between them and the other person. 00:00:45.780 --> 00:00:47.280 align:middle line:84% And then the other person can choose 00:00:47.280 --> 00:00:51.240 align:middle line:84% to accept that, in which case you split the money in that way, 00:00:51.240 --> 00:00:54.400 align:middle line:84% or reject that, in which case no one gets any money. 00:00:54.400 --> 00:00:56.928 align:middle line:90% So player 1 makes an ultimatum. 00:00:56.928 --> 00:00:58.220 align:middle line:90% This is how much money you get. 00:00:58.220 --> 00:00:59.600 align:middle line:90% This is how much money I get. 00:00:59.600 --> 00:01:00.800 align:middle line:90% Take it or leave it. 00:01:00.800 --> 00:01:03.620 align:middle line:84% Player 2 says, I'll take it, or I'll leave it 00:01:03.620 --> 00:01:05.040 align:middle line:90% and then we both get nothing. 00:01:05.040 --> 00:01:06.020 align:middle line:90% So that's the game. 00:01:06.020 --> 00:01:09.040 align:middle line:84% I think in this version you have $100, maybe, to split. 00:01:09.040 --> 00:01:11.260 align:middle line:84% So let's play and see what happens. 00:01:11.260 --> 00:01:18.720 align:middle line:90% 00:01:18.720 --> 00:01:22.640 align:middle line:84% So here we have a table of the results. 00:01:22.640 --> 00:01:26.580 align:middle line:84% What we can see is the blue circle, 00:01:26.580 --> 00:01:28.700 align:middle line:84% these are offers that were accepted. 00:01:28.700 --> 00:01:30.720 align:middle line:84% So this is the amount that the proposer 00:01:30.720 --> 00:01:34.560 align:middle line:84% offered the responder could receive out of the 100. 00:01:34.560 --> 00:01:39.080 align:middle line:84% And then the orange circles are offers that were rejected. 00:01:39.080 --> 00:01:40.920 align:middle line:84% And then we see the averages here. 00:01:40.920 --> 00:01:43.480 align:middle line:84% So we definitely see that, on average, 00:01:43.480 --> 00:01:46.360 align:middle line:84% at least especially in these early rounds, 00:01:46.360 --> 00:01:50.160 align:middle line:84% the higher offers were more likely to be accepted, 00:01:50.160 --> 00:01:54.225 align:middle line:84% and the lower offers were more likely to be rejected. 00:01:54.225 --> 00:01:55.600 align:middle line:84% It looks like a lot of the offers 00:01:55.600 --> 00:01:58.830 align:middle line:84% were above zero, except maybe one down here. 00:01:58.830 --> 00:02:00.453 align:middle line:90% Someone offered zero. 00:02:00.453 --> 00:02:02.370 align:middle line:84% Anyone who played want to share their thoughts 00:02:02.370 --> 00:02:04.222 align:middle line:84% about how they approached this and how 00:02:04.222 --> 00:02:05.430 align:middle line:90% they thought about this game? 00:02:05.430 --> 00:02:10.948 align:middle line:90% 00:02:10.948 --> 00:02:12.990 align:middle line:84% So what about when you were the-- yeah, go ahead. 00:02:12.990 --> 00:02:15.670 align:middle line:84% AUDIENCE: I was the one who was receiving the offer. 00:02:15.670 --> 00:02:17.947 align:middle line:84% So as long as the amount was positive, 00:02:17.947 --> 00:02:19.530 align:middle line:84% I would receive it because otherwise I 00:02:19.530 --> 00:02:20.405 align:middle line:90% wouldn't [INAUDIBLE]. 00:02:20.405 --> 00:02:23.690 align:middle line:84% IAN BALL: OK, so that's the ultra-rational way to play. 00:02:23.690 --> 00:02:26.790 align:middle line:84% And that's what game theory would propose. 00:02:26.790 --> 00:02:29.013 align:middle line:84% Often in practice we don't see that happening. 00:02:29.013 --> 00:02:30.430 align:middle line:84% But that's a great way of playing. 00:02:30.430 --> 00:02:34.090 align:middle line:84% So other people who rejected positive offers, what 00:02:34.090 --> 00:02:35.410 align:middle line:90% was your thinking about that? 00:02:35.410 --> 00:02:35.910 align:middle line:90% Yeah? 00:02:35.910 --> 00:02:37.590 align:middle line:84% AUDIENCE: I guess I was less rational. 00:02:37.590 --> 00:02:40.990 align:middle line:84% If no one offered me 50, then I was like, well, 00:02:40.990 --> 00:02:42.110 align:middle line:90% I don't like you. 00:02:42.110 --> 00:02:43.910 align:middle line:90% So I'm not going to accept that. 00:02:43.910 --> 00:02:45.070 align:middle line:90% IAN BALL: OK, great. 00:02:45.070 --> 00:02:46.570 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:02:46.570 --> 00:02:48.290 align:middle line:84% IAN BALL: So a lot of people have 00:02:48.290 --> 00:02:50.150 align:middle line:90% this reaction about fairness. 00:02:50.150 --> 00:02:52.350 align:middle line:84% So this captures people's idea of fairness. 00:02:52.350 --> 00:02:55.810 align:middle line:84% Some people are willing to sacrifice financial gains 00:02:55.810 --> 00:02:58.730 align:middle line:84% in order to punish people who you might feel have morally 00:02:58.730 --> 00:03:01.530 align:middle line:84% wronged you or done something that's fundamentally unfair. 00:03:01.530 --> 00:03:04.570 align:middle line:84% Again, it's a pretty cheap way to punish someone when 00:03:04.570 --> 00:03:05.950 align:middle line:90% there's no money on the line. 00:03:05.950 --> 00:03:09.410 align:middle line:84% If it was $50,000 and they gave you 49,000, 00:03:09.410 --> 00:03:11.250 align:middle line:84% you'd probably still accept it, right? 00:03:11.250 --> 00:03:12.513 align:middle line:90% But yeah, that's useful. 00:03:12.513 --> 00:03:14.430 align:middle line:84% And then what about people who were proposing? 00:03:14.430 --> 00:03:17.830 align:middle line:90% What was your reasoning? 00:03:17.830 --> 00:03:20.410 align:middle line:90% 00:03:20.410 --> 00:03:21.950 align:middle line:90% Did you propose an even split? 00:03:21.950 --> 00:03:23.750 align:middle line:84% Did you propose more for yourself? 00:03:23.750 --> 00:03:24.470 align:middle line:90% What did you do? 00:03:24.470 --> 00:03:27.850 align:middle line:90% 00:03:27.850 --> 00:03:29.410 align:middle line:90% It's very quiet. 00:03:29.410 --> 00:03:31.210 align:middle line:90% Someone proposed. 00:03:31.210 --> 00:03:33.432 align:middle line:84% Some people proposed-- it looks like-- 00:03:33.432 --> 00:03:34.890 align:middle line:84% I guess, if we look at the game, it 00:03:34.890 --> 00:03:37.950 align:middle line:84% looks like no one gave more than 50 to the other person. 00:03:37.950 --> 00:03:40.370 align:middle line:84% So it looks like some people felt, 00:03:40.370 --> 00:03:42.630 align:middle line:90% let's be generous and give 50. 00:03:42.630 --> 00:03:44.850 align:middle line:84% And then other people wanted to keep a little more 00:03:44.850 --> 00:03:45.590 align:middle line:90% for themselves. 00:03:45.590 --> 00:03:47.850 align:middle line:84% Usually what happens in this game is people-- 00:03:47.850 --> 00:03:50.730 align:middle line:84% their reasoning is, well, I anticipate the responder might 00:03:50.730 --> 00:03:53.650 align:middle line:84% be a bit like you suggested and might not be happy with me 00:03:53.650 --> 00:03:56.020 align:middle line:84% if I give them nothing, but I don't have to give them 00:03:56.020 --> 00:03:57.000 align:middle line:90% all-- the full half. 00:03:57.000 --> 00:03:58.920 align:middle line:90% Maybe I'll give them 30 or 40. 00:03:58.920 --> 00:04:00.290 align:middle line:90% That's a pretty common response. 00:04:00.290 --> 00:04:02.540 align:middle line:84% And then the hope is that the person will accept this. 00:04:02.540 --> 00:04:04.540 align:middle line:84% Here, you actually played it in multiple rounds. 00:04:04.540 --> 00:04:07.860 align:middle line:84% Did anyone think about the dynamics of this game? 00:04:07.860 --> 00:04:10.280 align:middle line:84% So I think that's an important consideration in this game. 00:04:10.280 --> 00:04:12.363 align:middle line:84% Today, we're going to think about a static version 00:04:12.363 --> 00:04:13.460 align:middle line:90% of this game. 00:04:13.460 --> 00:04:15.500 align:middle line:84% But if you're playing this repeatedly, 00:04:15.500 --> 00:04:17.579 align:middle line:84% you might think, well, depending on what 00:04:17.579 --> 00:04:20.540 align:middle line:84% I do in this round, that might affect the way the person will 00:04:20.540 --> 00:04:22.160 align:middle line:90% play in future rounds. 00:04:22.160 --> 00:04:25.040 align:middle line:84% If they know that I'm someone who rejects really low offers, 00:04:25.040 --> 00:04:28.497 align:middle line:84% maybe they'll be forced or motivated to offer more to me 00:04:28.497 --> 00:04:29.080 align:middle line:90% in the future. 00:04:29.080 --> 00:04:30.872 align:middle line:84% And that's something we'll think about when 00:04:30.872 --> 00:04:32.380 align:middle line:90% we get into repeated games. 00:04:32.380 --> 00:04:44.460 align:middle line:84% So let's close this and move into more of a formal analysis 00:04:44.460 --> 00:04:47.460 align:middle line:90% of this game. 00:04:47.460 --> 00:04:50.060 align:middle line:84% So the topic for today is going to be what's 00:04:50.060 --> 00:04:51.280 align:middle line:90% called backward induction. 00:04:51.280 --> 00:04:52.980 align:middle line:84% But I think the best way to go about it is just 00:04:52.980 --> 00:04:55.480 align:middle line:84% to think through this example, and then, later in the class, 00:04:55.480 --> 00:04:57.100 align:middle line:84% I'll give a more formal definition. 00:04:57.100 --> 00:04:59.860 align:middle line:84% So let's look at an easier version of the ultimatum game. 00:04:59.860 --> 00:05:01.820 align:middle line:84% I think in the game you played, you 00:05:01.820 --> 00:05:05.313 align:middle line:84% could choose any number between maybe 0 and hundreds. 00:05:05.313 --> 00:05:06.480 align:middle line:90% Let's make it a bit simpler. 00:05:06.480 --> 00:05:12.140 align:middle line:84% So let's see the ultimatum game with maybe $3 at stake. 00:05:12.140 --> 00:05:15.060 align:middle line:84% And let's think about how this game works. 00:05:15.060 --> 00:05:22.140 align:middle line:84% So we have player 1, who's what we would call the proposer, 00:05:22.140 --> 00:05:25.000 align:middle line:84% and player 2, who's the responder. 00:05:25.000 --> 00:05:27.700 align:middle line:90% 00:05:27.700 --> 00:05:29.540 align:middle line:84% And the proposer is proposing how 00:05:29.540 --> 00:05:34.202 align:middle line:84% to split the $3 between themselves and the other person. 00:05:34.202 --> 00:05:35.660 align:middle line:84% But we're going to keep track of it 00:05:35.660 --> 00:05:39.400 align:middle line:84% as thinking about the proposer, P1, makes an offer. 00:05:39.400 --> 00:05:42.580 align:middle line:90% 00:05:42.580 --> 00:05:44.540 align:middle line:84% So we'll think of the offer as what 00:05:44.540 --> 00:05:46.860 align:middle line:90% they offer to the other person. 00:05:46.860 --> 00:05:49.480 align:middle line:84% And that means that they will keep the remaining amount. 00:05:49.480 --> 00:05:50.960 align:middle line:90% So there's two numbers going on. 00:05:50.960 --> 00:05:53.245 align:middle line:84% But because the numbers always sum to $3, 00:05:53.245 --> 00:05:55.370 align:middle line:84% we only have to keep track of one of those numbers. 00:05:55.370 --> 00:05:57.078 align:middle line:84% And we'll keep track of the number, which 00:05:57.078 --> 00:06:00.950 align:middle line:84% is the amount the responder would get. 00:06:00.950 --> 00:06:03.670 align:middle line:84% And then the responder, player 2, 00:06:03.670 --> 00:06:09.892 align:middle line:84% can either accept or reject this offer. 00:06:09.892 --> 00:06:11.350 align:middle line:84% And as usual, we're going to assume 00:06:11.350 --> 00:06:15.030 align:middle line:84% that people are risk neutral so that monetary payoffs coincide 00:06:15.030 --> 00:06:16.050 align:middle line:90% with utilities. 00:06:16.050 --> 00:06:20.710 align:middle line:90% 00:06:20.710 --> 00:06:24.310 align:middle line:84% So let's try to represent this game in the extensive form. 00:06:24.310 --> 00:06:26.790 align:middle line:90% So first, we have player 1. 00:06:26.790 --> 00:06:29.288 align:middle line:84% And how much can they offer to player 2? 00:06:29.288 --> 00:06:30.830 align:middle line:84% Well, we'll assume that they can only 00:06:30.830 --> 00:06:33.110 align:middle line:84% make offers that are in even dollar amounts just 00:06:33.110 --> 00:06:34.370 align:middle line:90% to make it a bit simpler. 00:06:34.370 --> 00:06:37.410 align:middle line:84% So they can either offer 0, 1, 2, or 3. 00:06:37.410 --> 00:06:44.750 align:middle line:90% 00:06:44.750 --> 00:06:48.770 align:middle line:84% And each of these offers, it's then player 2's turn to move. 00:06:48.770 --> 00:06:51.910 align:middle line:90% 00:06:51.910 --> 00:06:55.070 align:middle line:84% Of course, these are all in separate information sets. 00:06:55.070 --> 00:06:58.150 align:middle line:84% We have to remember that player 2 observes the offer that 00:06:58.150 --> 00:06:59.410 align:middle line:90% is made by player 1. 00:06:59.410 --> 00:07:03.070 align:middle line:84% The responder observes the offer made by the proposer. 00:07:03.070 --> 00:07:05.430 align:middle line:84% So they can distinguish these offers. 00:07:05.430 --> 00:07:17.310 align:middle line:84% And then they choose accept or reject, which I'll call A/R. 00:07:17.310 --> 00:07:19.390 align:middle line:84% And now let's make sure we get the payoffs right. 00:07:19.390 --> 00:07:23.030 align:middle line:84% So whenever the second player rejects the offer, 00:07:23.030 --> 00:07:25.803 align:middle line:84% no matter what the offer was, both players get 0. 00:07:25.803 --> 00:07:26.970 align:middle line:90% That's the rule of the game. 00:07:26.970 --> 00:07:29.190 align:middle line:84% If the offer is rejected, you both get 0. 00:07:29.190 --> 00:07:35.990 align:middle line:84% So we're going to have 0, 0; 0, 0; 0, 0; and 0, 0. 00:07:35.990 --> 00:07:39.150 align:middle line:84% Now, if the offer is accepted, then the number 00:07:39.150 --> 00:07:42.630 align:middle line:84% here is the offer that goes to player 2, which means player 00:07:42.630 --> 00:07:45.310 align:middle line:90% 1 gets the remainder of the $3. 00:07:45.310 --> 00:07:48.530 align:middle line:84% So up here, it's going to be 3, 0. 00:07:48.530 --> 00:07:51.600 align:middle line:84% The accepted offer is an offer of 0 to player 2, 00:07:51.600 --> 00:07:53.880 align:middle line:90% and that means player 1 keeps 3. 00:07:53.880 --> 00:07:56.440 align:middle line:90% Here, it's going to be 2, 1. 00:07:56.440 --> 00:07:59.180 align:middle line:84% The offer is 1 to player 2 is accepted. 00:07:59.180 --> 00:08:02.920 align:middle line:84% And that means the first player, the proposer, keeps 2. 00:08:02.920 --> 00:08:06.000 align:middle line:90% Then we have 1, 2, and 0, 3. 00:08:06.000 --> 00:08:10.680 align:middle line:90% 00:08:10.680 --> 00:08:13.280 align:middle line:90% So here's the game. 00:08:13.280 --> 00:08:22.280 align:middle line:84% And the first approach to analyzing this game is let's 00:08:22.280 --> 00:08:26.040 align:middle line:84% think about how we can represent this game in strategic form. 00:08:26.040 --> 00:08:27.940 align:middle line:90% So what is a strategy? 00:08:27.940 --> 00:08:30.940 align:middle line:84% Let's write down what the strategies are for the players. 00:08:30.940 --> 00:08:34.600 align:middle line:90% 00:08:34.600 --> 00:08:37.620 align:middle line:84% What about for player 1 and for player 2? 00:08:37.620 --> 00:08:42.600 align:middle line:84% So for player 1, what is a strategy in this game? 00:08:42.600 --> 00:08:44.780 align:middle line:90% Well, it's just an offer. 00:08:44.780 --> 00:08:49.940 align:middle line:84% So for player 1, a strategy is an element of the set 0, 1, 2, 00:08:49.940 --> 00:08:51.400 align:middle line:90% 3. 00:08:51.400 --> 00:08:53.880 align:middle line:84% They just choose, how much do they offer to player 2, 00:08:53.880 --> 00:08:55.400 align:middle line:90% to the responder? 00:08:55.400 --> 00:08:56.900 align:middle line:84% What about a strategy for player 2? 00:08:56.900 --> 00:08:58.960 align:middle line:90% This is a little trickier. 00:08:58.960 --> 00:09:00.960 align:middle line:84% How do we represent a strategy for player 2? 00:09:00.960 --> 00:09:04.540 align:middle line:84% Or what kind of mathematical object is it? 00:09:04.540 --> 00:09:05.040 align:middle line:90% Yeah? 00:09:05.040 --> 00:09:06.000 align:middle line:90% AUDIENCE: [INAUDIBLE] people? 00:09:06.000 --> 00:09:06.580 align:middle line:90% IAN BALL: Right. 00:09:06.580 --> 00:09:08.372 align:middle line:84% So it's a function of four things, which we 00:09:08.372 --> 00:09:09.860 align:middle line:90% can represent as a four-tuple. 00:09:09.860 --> 00:09:10.960 align:middle line:84% Let's first write it as a function, 00:09:10.960 --> 00:09:12.460 align:middle line:84% and then I'll represent it that way. 00:09:12.460 --> 00:09:15.920 align:middle line:84% So it's a function that says, as a function of what you offer me, 00:09:15.920 --> 00:09:18.600 align:middle line:84% I have to choose whether to accept that offer 00:09:18.600 --> 00:09:19.780 align:middle line:90% or reject that offer. 00:09:19.780 --> 00:09:21.780 align:middle line:84% And here we're looking at pure strategies. 00:09:21.780 --> 00:09:29.720 align:middle line:84% So it's a function from 0, 1, 2, 3 to either accept or reject. 00:09:29.720 --> 00:09:31.720 align:middle line:84% And how are we going to represent that function? 00:09:31.720 --> 00:09:35.520 align:middle line:84% As you suggested, the way we normally write down strategies 00:09:35.520 --> 00:09:39.680 align:middle line:84% is we put one blank space for each of the information 00:09:39.680 --> 00:09:41.640 align:middle line:90% sets of the player. 00:09:41.640 --> 00:09:44.480 align:middle line:84% In this case, the player has four information sets 00:09:44.480 --> 00:09:46.990 align:middle line:84% corresponding to the four possible offers. 00:09:46.990 --> 00:09:50.570 align:middle line:90% So we have 0, 1, 2, 3. 00:09:50.570 --> 00:09:54.490 align:middle line:84% And as an example, they might use a strategy like this. 00:09:54.490 --> 00:09:57.070 align:middle line:84% And this would be a very common strategy to use in this game. 00:09:57.070 --> 00:10:00.178 align:middle line:84% This would say, if you only offer me 0 or 1, 00:10:00.178 --> 00:10:01.470 align:middle line:90% I'm going to be upset with you. 00:10:01.470 --> 00:10:02.750 align:middle line:90% I'm going to find that unfair. 00:10:02.750 --> 00:10:05.510 align:middle line:84% So I'm going to reject that offer and punish both of us. 00:10:05.510 --> 00:10:09.570 align:middle line:84% If you give me 2 or 3, I'm very happy, 00:10:09.570 --> 00:10:11.080 align:middle line:84% and I'm going to accept that offer. 00:10:11.080 --> 00:10:11.830 align:middle line:90% I think it's fair. 00:10:11.830 --> 00:10:14.070 align:middle line:84% In fact, you've given me a majority of the pot, 00:10:14.070 --> 00:10:17.250 align:middle line:84% so I'm very happy, and I'll accept it. 00:10:17.250 --> 00:10:20.810 align:middle line:84% So now let's see if we can find a Nash equilibrium of this game 00:10:20.810 --> 00:10:22.350 align:middle line:90% involving this strategy. 00:10:22.350 --> 00:10:25.050 align:middle line:84% So remember, Nash equilibrium is a strategy profile. 00:10:25.050 --> 00:10:27.410 align:middle line:90% I've specified one strategy. 00:10:27.410 --> 00:10:30.090 align:middle line:84% Let's see-- can we find a strategy for player 1 00:10:30.090 --> 00:10:32.183 align:middle line:84% that makes this into a Nash equilibrium? 00:10:32.183 --> 00:10:33.350 align:middle line:90% So let's think this through. 00:10:33.350 --> 00:10:36.170 align:middle line:84% If you were player 1, if you were the proposer, 00:10:36.170 --> 00:10:39.410 align:middle line:84% and you expected or you believed that the responder was 00:10:39.410 --> 00:10:41.210 align:middle line:84% going to play in this way, how much 00:10:41.210 --> 00:10:45.490 align:middle line:90% would you offer the responder? 00:10:45.490 --> 00:10:46.190 align:middle line:90% Yeah, over here? 00:10:46.190 --> 00:10:46.690 align:middle line:90% AUDIENCE: 2. 00:10:46.690 --> 00:10:48.210 align:middle line:90% IAN BALL: 2, great. 00:10:48.210 --> 00:10:51.330 align:middle line:84% So we're going to have exactly S2 equals-- 00:10:51.330 --> 00:10:52.510 align:middle line:90% S1 equals 2. 00:10:52.510 --> 00:10:53.465 align:middle line:90% That's confusing. 00:10:53.465 --> 00:10:54.590 align:middle line:90% And explain your reasoning. 00:10:54.590 --> 00:10:55.507 align:middle line:90% Why would you offer 2? 00:10:55.507 --> 00:10:58.590 align:middle line:84% AUDIENCE: If you offer lower, both of you get 0. 00:10:58.590 --> 00:11:00.430 align:middle line:84% If you offer more, then you get nothing. 00:11:00.430 --> 00:11:02.590 align:middle line:90% So [INAUDIBLE]. 00:11:02.590 --> 00:11:03.690 align:middle line:90% IAN BALL: Great. 00:11:03.690 --> 00:11:05.770 align:middle line:84% So another way of saying that-- exactly right-- 00:11:05.770 --> 00:11:10.050 align:middle line:84% is, I'm making the lowest offer that will be accepted. 00:11:10.050 --> 00:11:12.370 align:middle line:84% I don't want to offer any less because there I'm 00:11:12.370 --> 00:11:14.570 align:middle line:90% clearly doing-- 00:11:14.570 --> 00:11:17.170 align:middle line:84% if I offer them less, it gets rejected. 00:11:17.170 --> 00:11:21.050 align:middle line:84% And if I offer them more, it would be-- 00:11:21.050 --> 00:11:24.090 align:middle line:84% it will be accepted, but it will be worse for me. 00:11:24.090 --> 00:11:28.290 align:middle line:84% So just to keep track, let's say these offers-- 00:11:28.290 --> 00:11:32.730 align:middle line:84% these offers are better for the proposer 00:11:32.730 --> 00:11:34.810 align:middle line:84% if they're accepted because I'm offering 00:11:34.810 --> 00:11:37.510 align:middle line:84% less money to the respondent, but they're not accepted. 00:11:37.510 --> 00:11:42.610 align:middle line:90% 00:11:42.610 --> 00:11:44.240 align:middle line:84% So when I say better for the proposer, 00:11:44.240 --> 00:11:46.440 align:middle line:84% I mean better for the proposer if they're accepted. 00:11:46.440 --> 00:11:49.020 align:middle line:84% The problem with these is they're not accepted. 00:11:49.020 --> 00:11:52.800 align:middle line:84% And then these higher offers are worse for the proposer. 00:11:52.800 --> 00:11:56.380 align:middle line:90% 00:11:56.380 --> 00:11:58.640 align:middle line:90% So I make the lowest offer. 00:11:58.640 --> 00:12:02.020 align:middle line:84% That is the offer that's best for me if accepted. 00:12:02.020 --> 00:12:04.860 align:middle line:84% And I find the lowest one that's actually going to be accepted. 00:12:04.860 --> 00:12:08.040 align:middle line:84% So this indeed is going to be a Nash equilibrium. 00:12:08.040 --> 00:12:10.860 align:middle line:90% 00:12:10.860 --> 00:12:12.400 align:middle line:90% And let's check this formally. 00:12:12.400 --> 00:12:17.420 align:middle line:84% Let's check that neither player has a profitable deviation. 00:12:17.420 --> 00:12:23.660 align:middle line:84% If player 1 is offering 2, let's check that player 2 cannot 00:12:23.660 --> 00:12:25.140 align:middle line:90% profitably deviate. 00:12:25.140 --> 00:12:29.500 align:middle line:84% Well, if you're offering me 2, there's only two possibilities. 00:12:29.500 --> 00:12:33.580 align:middle line:84% I accept the offer, and I get 2, or I reject the offer, 00:12:33.580 --> 00:12:34.682 align:middle line:90% and I get 0. 00:12:34.682 --> 00:12:36.140 align:middle line:84% So certainly I'm behaving optimally 00:12:36.140 --> 00:12:38.905 align:middle line:90% by accepting the offer. 00:12:38.905 --> 00:12:40.280 align:middle line:84% Now let's go the other direction. 00:12:40.280 --> 00:12:41.863 align:middle line:84% This is what we just reasoned through. 00:12:41.863 --> 00:12:44.820 align:middle line:84% Given that this is the way that player 2, the responder, 00:12:44.820 --> 00:12:49.120 align:middle line:84% is playing, the best that player 1 can possibly do is to offer 2, 00:12:49.120 --> 00:12:53.180 align:middle line:84% and that means keep 1, because the only way they could 00:12:53.180 --> 00:12:56.100 align:middle line:84% get more than 1 is if they made one of these offers 00:12:56.100 --> 00:12:57.200 align:middle line:90% and it was accepted. 00:12:57.200 --> 00:12:59.020 align:middle line:84% But that's impossible given the strategy 00:12:59.020 --> 00:13:01.340 align:middle line:90% that the responder is using. 00:13:01.340 --> 00:13:03.180 align:middle line:84% So we found one Nash equilibrium. 00:13:03.180 --> 00:13:05.218 align:middle line:84% There's a lot of Nash equilibrium in this game. 00:13:05.218 --> 00:13:07.260 align:middle line:84% But there's something about this Nash equilibrium 00:13:07.260 --> 00:13:09.900 align:middle line:90% that seems wrong. 00:13:09.900 --> 00:13:11.862 align:middle line:90% Or maybe it seems problematic. 00:13:11.862 --> 00:13:14.320 align:middle line:84% So maybe I'll say, what's wrong with this Nash equilibrium? 00:13:14.320 --> 00:13:21.460 align:middle line:90% 00:13:21.460 --> 00:13:24.020 align:middle line:84% And this is exactly what we want to capture today. 00:13:24.020 --> 00:13:26.960 align:middle line:84% So does anything feel off to you? 00:13:26.960 --> 00:13:30.360 align:middle line:84% Does it feel like there's a problem here? 00:13:30.360 --> 00:13:31.160 align:middle line:90% Yeah, in the front. 00:13:31.160 --> 00:13:33.700 align:middle line:84% AUDIENCE: It seems like one person is always 00:13:33.700 --> 00:13:34.520 align:middle line:90% beating the other. 00:13:34.520 --> 00:13:37.780 align:middle line:84% IAN BALL: OK, so that's true, but some games are unfair. 00:13:37.780 --> 00:13:39.035 align:middle line:90% So I think that's true. 00:13:39.035 --> 00:13:40.160 align:middle line:90% Maybe you're getting at it. 00:13:40.160 --> 00:13:42.130 align:middle line:84% But I would say let's add a little bit more to that. 00:13:42.130 --> 00:13:44.090 align:middle line:84% Yeah, so it is true that one player is always doing better, 00:13:44.090 --> 00:13:46.670 align:middle line:84% but I can certainly write down games where one player always 00:13:46.670 --> 00:13:47.290 align:middle line:90% loses. 00:13:47.290 --> 00:13:49.790 align:middle line:84% It doesn't necessarily mean they're playing irrationally. 00:13:49.790 --> 00:13:50.850 align:middle line:90% Any other thoughts in the front? 00:13:50.850 --> 00:13:51.190 align:middle line:90% Yeah? 00:13:51.190 --> 00:13:52.790 align:middle line:84% AUDIENCE: The responders should [INAUDIBLE] someone 00:13:52.790 --> 00:13:54.230 align:middle line:90% that's positive utility. 00:13:54.230 --> 00:13:55.310 align:middle line:90% IAN BALL: Great. 00:13:55.310 --> 00:13:59.330 align:middle line:84% So this information set seems a bit puzzling. 00:13:59.330 --> 00:14:03.990 align:middle line:84% So this says, if I'm offered 1, there are two choices for me. 00:14:03.990 --> 00:14:08.430 align:middle line:84% I either accept 1, and I get my $1, or I reject the offer, 00:14:08.430 --> 00:14:10.150 align:middle line:90% and I get 0. 00:14:10.150 --> 00:14:12.210 align:middle line:84% So it seems that in this contingency, 00:14:12.210 --> 00:14:15.110 align:middle line:84% accepting is strictly better than rejecting. 00:14:15.110 --> 00:14:19.250 align:middle line:84% And if you're not motivated by $1, let's call it $1 million. 00:14:19.250 --> 00:14:21.390 align:middle line:84% Then it makes it clear, I might be unhappy 00:14:21.390 --> 00:14:24.510 align:middle line:84% that you're only giving me 1 million out of the 3 million. 00:14:24.510 --> 00:14:28.910 align:middle line:84% But if my choice is take the 1 million or reject it and get 0, 00:14:28.910 --> 00:14:31.430 align:middle line:84% it seems like accepting is much better. 00:14:31.430 --> 00:14:34.090 align:middle line:84% But why is this still consistent with Nash equilibrium? 00:14:34.090 --> 00:14:35.110 align:middle line:90% So I agree with you. 00:14:35.110 --> 00:14:38.190 align:middle line:84% It seems like clearly this is a-- we might call this 00:14:38.190 --> 00:14:39.090 align:middle line:90% a "mistake." 00:14:39.090 --> 00:14:41.870 align:middle line:90% 00:14:41.870 --> 00:14:45.892 align:middle line:84% Or we might call this "suboptimal." 00:14:45.892 --> 00:14:47.350 align:middle line:84% But I thought Nash equilibrium said 00:14:47.350 --> 00:14:48.950 align:middle line:84% that players are playing optimal given the way 00:14:48.950 --> 00:14:50.117 align:middle line:90% their opponents are playing. 00:14:50.117 --> 00:14:51.130 align:middle line:90% So why is this allowed? 00:14:51.130 --> 00:14:54.410 align:middle line:84% It is a Nash equilibrium, but someone's playing suboptimally. 00:14:54.410 --> 00:14:57.610 align:middle line:90% So what's happening? 00:14:57.610 --> 00:15:02.310 align:middle line:90% 00:15:02.310 --> 00:15:02.870 align:middle line:90% Yeah? 00:15:02.870 --> 00:15:04.510 align:middle line:84% AUDIENCE: Because we're basing it off of the belief 00:15:04.510 --> 00:15:05.710 align:middle line:90% of how they're playing. 00:15:05.710 --> 00:15:06.510 align:middle line:90% IAN BALL: Exactly. 00:15:06.510 --> 00:15:11.870 align:middle line:84% So player 2 believes that player 1 is making an offer of 2. 00:15:11.870 --> 00:15:14.030 align:middle line:84% If I believe that player 1 is making an offer of 2, 00:15:14.030 --> 00:15:16.990 align:middle line:84% I'm never going to reach this contingency. 00:15:16.990 --> 00:15:19.470 align:middle line:84% And because I'm never going to reach this contingency, how 00:15:19.470 --> 00:15:22.710 align:middle line:84% I play this contingency doesn't affect my payoff given 00:15:22.710 --> 00:15:24.670 align:middle line:84% the strategy of the other player. 00:15:24.670 --> 00:15:28.910 align:middle line:84% So what we've identified is kind of a crucial issue with Nash 00:15:28.910 --> 00:15:33.070 align:middle line:84% equilibrium, maybe a limitation of Nash equilibrium, 00:15:33.070 --> 00:15:46.360 align:middle line:84% that Nash equilibrium does not require optimal play 00:15:46.360 --> 00:15:48.220 align:middle line:90% at unreached contingencies. 00:15:48.220 --> 00:15:50.880 align:middle line:90% 00:15:50.880 --> 00:15:53.520 align:middle line:84% We could be a little more precise-- 00:15:53.520 --> 00:15:55.040 align:middle line:84% contingencies that are not reached 00:15:55.040 --> 00:15:56.660 align:middle line:84% based on the play of the other player. 00:15:56.660 --> 00:15:58.320 align:middle line:84% But I'll be a little vague here-- 00:15:58.320 --> 00:16:01.060 align:middle line:90% "at unreached contingencies." 00:16:01.060 --> 00:16:08.160 align:middle line:90% 00:16:08.160 --> 00:16:12.160 align:middle line:84% And if we want to dig a little deeper, when we analyze the Nash 00:16:12.160 --> 00:16:14.880 align:middle line:84% equilibrium of this game, we converted this game 00:16:14.880 --> 00:16:17.320 align:middle line:90% into a strategic form. 00:16:17.320 --> 00:16:21.320 align:middle line:84% And we said, well, what player 2 does 00:16:21.320 --> 00:16:25.000 align:middle line:84% is player 2 chooses a complete contingent plan. 00:16:25.000 --> 00:16:28.400 align:middle line:84% They just say, if I'm offered 0, I'll reject. 00:16:28.400 --> 00:16:30.260 align:middle line:90% If I'm offered 1, I'll reject. 00:16:30.260 --> 00:16:31.540 align:middle line:90% If I'm offered 2, I'll accept. 00:16:31.540 --> 00:16:33.120 align:middle line:84% And if I'm offered 3, I'll accept. 00:16:33.120 --> 00:16:37.000 align:middle line:84% And then they form beliefs about what player 1 is going to do. 00:16:37.000 --> 00:16:46.480 align:middle line:84% So this reasoning of the complete contingent plan, 00:16:46.480 --> 00:16:48.600 align:middle line:84% in some sense it's missing something. 00:16:48.600 --> 00:16:51.480 align:middle line:84% If I think about the complete contingent plan approach, 00:16:51.480 --> 00:16:56.320 align:middle line:84% it's missing the fact that a deviation might be observable. 00:16:56.320 --> 00:17:08.740 align:middle line:84% So it doesn't capture the observability of deviations. 00:17:08.740 --> 00:17:12.640 align:middle line:90% 00:17:12.640 --> 00:17:13.819 align:middle line:90% What do I mean by that? 00:17:13.819 --> 00:17:16.319 align:middle line:84% Well, the logic of Nash equilibrium 00:17:16.319 --> 00:17:19.680 align:middle line:84% says, I have a belief that my opponent 00:17:19.680 --> 00:17:21.378 align:middle line:90% is playing this strategy. 00:17:21.378 --> 00:17:23.920 align:middle line:84% And given that belief, I'm going to choose my contingent plan 00:17:23.920 --> 00:17:24.597 align:middle line:90% optimally. 00:17:24.597 --> 00:17:25.180 align:middle line:90% And that's it. 00:17:25.180 --> 00:17:28.400 align:middle line:84% I have my contingent plan, and then it gets executed. 00:17:28.400 --> 00:17:31.320 align:middle line:84% But if player 1 is going to deviate, 00:17:31.320 --> 00:17:34.000 align:middle line:84% well, I'm actually going to get to observe that because if they 00:17:34.000 --> 00:17:36.770 align:middle line:90% offer me 1, I get to see that. 00:17:36.770 --> 00:17:39.970 align:middle line:84% So what this strategic form reduction of the dynamic game 00:17:39.970 --> 00:17:43.530 align:middle line:84% seems to be missing is the fact that if player 1 were 00:17:43.530 --> 00:17:46.450 align:middle line:84% to deviate and play 1, well, now I 00:17:46.450 --> 00:17:48.420 align:middle line:84% know that they're not playing 2 anymore. 00:17:48.420 --> 00:17:50.170 align:middle line:84% But the complete contingent plan reasoning 00:17:50.170 --> 00:17:53.590 align:middle line:84% tells me, well, my play, even at this contingency, 00:17:53.590 --> 00:17:57.770 align:middle line:84% is governed by my belief that player 1 is always offering 2. 00:17:57.770 --> 00:17:59.930 align:middle line:84% And it's ignoring the fact that if we actually 00:17:59.930 --> 00:18:03.010 align:middle line:84% reach this contingency, I learn that my opponent has 00:18:03.010 --> 00:18:05.370 align:middle line:84% deviated and is not playing according to my belief. 00:18:05.370 --> 00:18:07.930 align:middle line:84% So the fact that information arrives over time 00:18:07.930 --> 00:18:10.170 align:middle line:84% is totally missed by the strategic form 00:18:10.170 --> 00:18:14.130 align:middle line:84% because the whole tree and the extensive form game 00:18:14.130 --> 00:18:17.170 align:middle line:84% are missing from the strategic form representation. 00:18:17.170 --> 00:18:18.610 align:middle line:84% And that's going to be what we're 00:18:18.610 --> 00:18:22.050 align:middle line:84% going to try to study more formally and more rigorously 00:18:22.050 --> 00:18:22.850 align:middle line:90% today. 00:18:22.850 --> 00:18:26.690 align:middle line:84% So today, we're going to define something 00:18:26.690 --> 00:18:28.510 align:middle line:90% called backward induction. 00:18:28.510 --> 00:18:35.330 align:middle line:90% 00:18:35.330 --> 00:18:37.570 align:middle line:84% And the basic idea is that backward induction 00:18:37.570 --> 00:18:40.010 align:middle line:84% is going to require optimal play even 00:18:40.010 --> 00:18:41.590 align:middle line:90% at unreached contingencies-- 00:18:41.590 --> 00:18:54.955 align:middle line:90% 00:18:54.955 --> 00:18:56.830 align:middle line:84% obviously, at reached contingencies, as well. 00:18:56.830 --> 00:18:59.530 align:middle line:84% But what's new about it is that it requires optimal play 00:18:59.530 --> 00:19:01.190 align:middle line:90% at unreached contingencies. 00:19:01.190 --> 00:19:06.715 align:middle line:90% 00:19:06.715 --> 00:19:08.090 align:middle line:84% And I want to make one more point 00:19:08.090 --> 00:19:09.770 align:middle line:84% about this, that what we're doing today 00:19:09.770 --> 00:19:12.310 align:middle line:84% is going to depend crucially on the extensive form. 00:19:12.310 --> 00:19:19.310 align:middle line:90% 00:19:19.310 --> 00:19:21.410 align:middle line:84% So, so far our approach in the class 00:19:21.410 --> 00:19:23.690 align:middle line:84% has been to start with the extensive form 00:19:23.690 --> 00:19:26.170 align:middle line:84% to try to capture the application we're studying, 00:19:26.170 --> 00:19:29.570 align:middle line:84% reduce it to the strategic form, and then do all our analysis 00:19:29.570 --> 00:19:32.100 align:middle line:90% within the strategic form. 00:19:32.100 --> 00:19:33.860 align:middle line:84% And so far what we looked at, Nash 00:19:33.860 --> 00:19:36.420 align:middle line:84% equilibrium, these various solution concepts, 00:19:36.420 --> 00:19:39.458 align:middle line:84% rationalizability-- only dependent on the strategic form. 00:19:39.458 --> 00:19:41.500 align:middle line:84% But today, we're going to take the extensive form 00:19:41.500 --> 00:19:42.247 align:middle line:90% more seriously. 00:19:42.247 --> 00:19:44.580 align:middle line:84% And we're going to argue that something might be missing 00:19:44.580 --> 00:19:48.940 align:middle line:84% when we reduce the extensive form to the strategic form. 00:19:48.940 --> 00:19:51.940 align:middle line:84% So let's see how this algorithm works by looking 00:19:51.940 --> 00:19:56.120 align:middle line:90% at another simple example. 00:19:56.120 --> 00:19:58.380 align:middle line:84% Let's look at the Boston game again. 00:19:58.380 --> 00:20:00.660 align:middle line:84% But in this case, the first player 00:20:00.660 --> 00:20:04.740 align:middle line:84% chooses what sporting event to go, and that move is observed. 00:20:04.740 --> 00:20:08.400 align:middle line:84% So the Boston game with observed moves. 00:20:08.400 --> 00:20:11.980 align:middle line:90% 00:20:11.980 --> 00:20:13.662 align:middle line:90% So what happens, we have-- 00:20:13.662 --> 00:20:15.620 align:middle line:84% maybe I'll draw it-- we can draw it either way. 00:20:15.620 --> 00:20:16.620 align:middle line:90% I'll draw it up here. 00:20:16.620 --> 00:20:20.300 align:middle line:84% We have player 1 first chooses to go to the Celtics 00:20:20.300 --> 00:20:23.580 align:middle line:90% game or the Red Sox game. 00:20:23.580 --> 00:20:28.300 align:middle line:84% And then player 2 observes whether player 1 has chosen 00:20:28.300 --> 00:20:29.680 align:middle line:90% the Celtics or the Red Sox. 00:20:29.680 --> 00:20:31.720 align:middle line:84% And then they choose Celtics or Red Sox. 00:20:31.720 --> 00:20:37.380 align:middle line:90% 00:20:37.380 --> 00:20:40.060 align:middle line:84% So the original version of this game that we studied 00:20:40.060 --> 00:20:42.620 align:middle line:84% was a simultaneous-move version of this game. 00:20:42.620 --> 00:20:44.560 align:middle line:84% And in the simultaneous-move version, 00:20:44.560 --> 00:20:47.140 align:middle line:84% we would represent that by putting these two nodes 00:20:47.140 --> 00:20:48.860 align:middle line:90% in the same information set. 00:20:48.860 --> 00:20:51.940 align:middle line:84% Now we're studying a different version, where player 1's move 00:20:51.940 --> 00:20:53.502 align:middle line:90% is observed by player 2. 00:20:53.502 --> 00:20:54.960 align:middle line:84% And now let's remember our payoffs. 00:20:54.960 --> 00:20:56.335 align:middle line:84% I think the convention we've used 00:20:56.335 --> 00:20:59.420 align:middle line:84% is that player 1 prefers Celtics to Red Sox. 00:20:59.420 --> 00:21:02.460 align:middle line:84% So if they both go to the Celtics game, it's 2 1. 00:21:02.460 --> 00:21:06.620 align:middle line:84% If they both go to Red Sox, it's 1 2, and then 0 0, and 0 0. 00:21:06.620 --> 00:21:09.860 align:middle line:84% So when I write extensive form games this way, 00:21:09.860 --> 00:21:11.340 align:middle line:84% I'll put the first player's payoff 00:21:11.340 --> 00:21:14.160 align:middle line:84% on top instead of to the left, but it has the same meaning. 00:21:14.160 --> 00:21:17.640 align:middle line:84% Instead of going 0, comma 0, or 0, comma 1, 00:21:17.640 --> 00:21:20.580 align:middle line:90% I'll just do 2 and then 1. 00:21:20.580 --> 00:21:25.440 align:middle line:84% So first let's represent this game in strategic form. 00:21:25.440 --> 00:21:31.070 align:middle line:90% 00:21:31.070 --> 00:21:33.110 align:middle line:84% So we first have to understand the strategies 00:21:33.110 --> 00:21:35.150 align:middle line:90% for the two players. 00:21:35.150 --> 00:21:38.390 align:middle line:84% So player 1's strategy set is pretty simple. 00:21:38.390 --> 00:21:40.530 align:middle line:84% They're just choosing Celtics or Red Sox. 00:21:40.530 --> 00:21:45.767 align:middle line:84% So we just have C and R. But player 2's strategy set 00:21:45.767 --> 00:21:46.850 align:middle line:90% is a bit more complicated. 00:21:46.850 --> 00:21:51.510 align:middle line:84% Just like we had before, player 2 has two information sets. 00:21:51.510 --> 00:21:53.110 align:middle line:84% So their strategy should be a vector 00:21:53.110 --> 00:21:56.390 align:middle line:84% with two components, one component saying 00:21:56.390 --> 00:21:58.630 align:middle line:84% what they do here and one component saying 00:21:58.630 --> 00:21:59.730 align:middle line:90% what they do here. 00:21:59.730 --> 00:22:01.590 align:middle line:84% So we can write player 2's strategies 00:22:01.590 --> 00:22:07.910 align:middle line:84% as CC, CR, RC, and RR, where the meaning 00:22:07.910 --> 00:22:11.070 align:middle line:84% is that we've agreed that we're going to order our information 00:22:11.070 --> 00:22:13.030 align:middle line:90% sets from left to right. 00:22:13.030 --> 00:22:15.950 align:middle line:84% So the first letter says what player 2 00:22:15.950 --> 00:22:18.910 align:middle line:90% does at this information set. 00:22:18.910 --> 00:22:21.190 align:middle line:84% And the second letter says what player 2 00:22:21.190 --> 00:22:24.430 align:middle line:90% does at this information set. 00:22:24.430 --> 00:22:26.150 align:middle line:84% So now we've defined the strategies. 00:22:26.150 --> 00:22:30.030 align:middle line:84% We now need to write out the extensive form-- 00:22:30.030 --> 00:22:31.950 align:middle line:90% sorry, the strategic form. 00:22:31.950 --> 00:22:33.750 align:middle line:90% So we have CR. 00:22:33.750 --> 00:22:39.410 align:middle line:90% And we have CC, CR, RC, RR. 00:22:39.410 --> 00:22:42.910 align:middle line:90% 00:22:42.910 --> 00:22:48.430 align:middle line:84% So now we want to fill in the payoffs here. 00:22:48.430 --> 00:22:53.150 align:middle line:84% So if player 1 goes to the Celtics game, 00:22:53.150 --> 00:22:55.010 align:middle line:84% when we're trying to compute payoffs, 00:22:55.010 --> 00:22:58.150 align:middle line:84% what matters is the first entry of player 2's 00:22:58.150 --> 00:23:00.950 align:middle line:84% strategy because the first entry of player 2's 00:23:00.950 --> 00:23:04.470 align:middle line:84% strategy precisely says what player 2 does 00:23:04.470 --> 00:23:08.310 align:middle line:84% after they observe that player 1 goes to the Celtics game. 00:23:08.310 --> 00:23:11.270 align:middle line:84% So under these two strategies, they go to the Celtics game, 00:23:11.270 --> 00:23:11.990 align:middle line:90% as well. 00:23:11.990 --> 00:23:17.110 align:middle line:84% And therefore the payoff is 2, 1; 2, 1. 00:23:17.110 --> 00:23:19.030 align:middle line:84% Under either of these strategies, 00:23:19.030 --> 00:23:20.990 align:middle line:84% player 1 goes to the Celtics game. 00:23:20.990 --> 00:23:24.115 align:middle line:84% But player 2 responds by going to the Red Sox game. 00:23:24.115 --> 00:23:25.490 align:middle line:84% They're not at the game together. 00:23:25.490 --> 00:23:26.940 align:middle line:90% And they both get a payoff of 0. 00:23:26.940 --> 00:23:32.560 align:middle line:90% 00:23:32.560 --> 00:23:35.920 align:middle line:84% Now let's look at what happens if player 1 goes to the Red Sox 00:23:35.920 --> 00:23:37.280 align:middle line:90% game. 00:23:37.280 --> 00:23:39.680 align:middle line:84% Now it's the second component of the strategy 00:23:39.680 --> 00:23:42.000 align:middle line:90% that determines the payoffs. 00:23:42.000 --> 00:23:44.960 align:middle line:84% So when do we both go to the Red Sox game? 00:23:44.960 --> 00:23:47.530 align:middle line:84% It's actually going to be under these two strategies 00:23:47.530 --> 00:23:49.280 align:middle line:84% because these are the two strategies where 00:23:49.280 --> 00:23:53.680 align:middle line:84% player 2 goes to the Red Sox game after player 1 does. 00:23:53.680 --> 00:23:58.680 align:middle line:84% So we're going to have 1, 2, and 1, 2. 00:23:58.680 --> 00:24:03.800 align:middle line:84% And then over here, we're going to have 0, 0 and 0, 0. 00:24:03.800 --> 00:24:08.060 align:middle line:84% So let's now try to compute the Nash equilibria of this game. 00:24:08.060 --> 00:24:11.000 align:middle line:84% We're going to use our standard trick-- 00:24:11.000 --> 00:24:13.000 align:middle line:90% fixing a strategy of player 2. 00:24:13.000 --> 00:24:14.038 align:middle line:90% Oh, yes? 00:24:14.038 --> 00:24:16.080 align:middle line:84% AUDIENCE: Could you clarify what the first letter 00:24:16.080 --> 00:24:17.340 align:middle line:90% means [? in ?] players too? 00:24:17.340 --> 00:24:18.465 align:middle line:90% IAN BALL: Yeah, absolutely. 00:24:18.465 --> 00:24:23.520 align:middle line:84% So this means-- so let me call these information sets 00:24:23.520 --> 00:24:31.080 align:middle line:90% maybe Ia and Ib. 00:24:31.080 --> 00:24:33.182 align:middle line:90% This is the first-- 00:24:33.182 --> 00:24:35.640 align:middle line:84% I'll use a so we don't get confused with the first player-- 00:24:35.640 --> 00:24:37.880 align:middle line:84% the first information set of player 2. 00:24:37.880 --> 00:24:40.600 align:middle line:84% And this is the second information set of player 2. 00:24:40.600 --> 00:24:43.680 align:middle line:90% And this is what they do at Ia. 00:24:43.680 --> 00:24:46.940 align:middle line:90% And this is what they do at Ib. 00:24:46.940 --> 00:24:49.060 align:middle line:84% So in words-- let me use this strategy. 00:24:49.060 --> 00:24:51.280 align:middle line:84% This is a better example since it's not constant. 00:24:51.280 --> 00:24:53.543 align:middle line:84% At the first information set, I play C. 00:24:53.543 --> 00:24:54.960 align:middle line:84% And at the second information set, 00:24:54.960 --> 00:24:59.360 align:middle line:84% I play R. That's my complete contingent plan. 00:24:59.360 --> 00:25:03.480 align:middle line:84% So now let's try to compute the Nash equilibria of this game. 00:25:03.480 --> 00:25:06.720 align:middle line:84% So as usual, we're going to fix the strategy of player 2 00:25:06.720 --> 00:25:08.640 align:middle line:84% and think about how well player 1 does 00:25:08.640 --> 00:25:10.300 align:middle line:90% with their different strategies. 00:25:10.300 --> 00:25:12.340 align:middle line:84% So given that player 2 does this, 00:25:12.340 --> 00:25:16.360 align:middle line:84% player 1 is better off choosing C because 2 is greater than 0. 00:25:16.360 --> 00:25:21.720 align:middle line:84% Given here, player 2 is better off here. 00:25:21.720 --> 00:25:23.490 align:middle line:90% Here, it doesn't matter. 00:25:23.490 --> 00:25:25.890 align:middle line:90% And here, we go and get this. 00:25:25.890 --> 00:25:27.690 align:middle line:84% So I'm just going column by column, 00:25:27.690 --> 00:25:30.690 align:middle line:84% fixing player 2's strategy and computing player 1's best 00:25:30.690 --> 00:25:32.808 align:middle line:90% response. 00:25:32.808 --> 00:25:34.350 align:middle line:84% Now I'm going to do it the other way. 00:25:34.350 --> 00:25:36.330 align:middle line:84% I'm going to fix player 1's strategy 00:25:36.330 --> 00:25:38.590 align:middle line:84% and compute player 2's best response. 00:25:38.590 --> 00:25:43.950 align:middle line:84% So I'm going to get this and this. 00:25:43.950 --> 00:25:47.930 align:middle line:90% 00:25:47.930 --> 00:25:52.450 align:middle line:84% So here, I get three Nash equilibria. 00:25:52.450 --> 00:25:55.890 align:middle line:84% Remember, Nash equilibrium can be visualized here 00:25:55.890 --> 00:25:59.530 align:middle line:84% by looking for a cell that has both numbers underlined. 00:25:59.530 --> 00:26:02.890 align:middle line:84% So we have three Nash equilibria. 00:26:02.890 --> 00:26:04.470 align:middle line:84% Player 1 goes to the Celtics game. 00:26:04.470 --> 00:26:07.250 align:middle line:84% And player 2 always goes to Celtics game. 00:26:07.250 --> 00:26:08.950 align:middle line:84% Player 2 goes to the Red Sox game, 00:26:08.950 --> 00:26:11.570 align:middle line:84% and player 2 always goes to the Red Sox game. 00:26:11.570 --> 00:26:14.230 align:middle line:84% And then this one, where player 2 goes to the Celtics game, 00:26:14.230 --> 00:26:16.430 align:middle line:84% and what player 2 does depends on what they observe. 00:26:16.430 --> 00:26:18.650 align:middle line:84% So they go to the Celtics game if player 1 goes to the Celtics 00:26:18.650 --> 00:26:20.430 align:middle line:84% game, but they go to the Red Sox game 00:26:20.430 --> 00:26:22.070 align:middle line:84% if player 2 goes to the Red Sox game. 00:26:22.070 --> 00:26:23.810 align:middle line:84% So we have three Nash equilibria. 00:26:23.810 --> 00:26:25.170 align:middle line:90% Which of these-- 00:26:25.170 --> 00:26:33.930 align:middle line:84% I'll write this-- which of these seems most compelling to you? 00:26:33.930 --> 00:26:37.458 align:middle line:84% Which of these seems the most reasonable? 00:26:37.458 --> 00:26:38.750 align:middle line:90% Or do they all seem reasonable? 00:26:38.750 --> 00:26:39.870 align:middle line:84% Or do none of them seem reasonable? 00:26:39.870 --> 00:26:40.990 align:middle line:90% I guess I gave it away. 00:26:40.990 --> 00:26:41.490 align:middle line:90% Yeah? 00:26:41.490 --> 00:26:44.450 align:middle line:90% AUDIENCE: [INAUDIBLE] C CR. 00:26:44.450 --> 00:26:45.850 align:middle line:90% IAN BALL: OK, this one. 00:26:45.850 --> 00:26:47.290 align:middle line:90% And tell me why. 00:26:47.290 --> 00:26:51.090 align:middle line:84% AUDIENCE: I mean, you're getting some sort of payoff either way 00:26:51.090 --> 00:26:53.310 align:middle line:84% or some positive payoff either way, 00:26:53.310 --> 00:26:57.370 align:middle line:84% whereas the other ones [INAUDIBLE] always [INAUDIBLE] 00:26:57.370 --> 00:27:00.003 align:middle line:84% the first player doesn't go to the Celtics game [INAUDIBLE]. 00:27:00.003 --> 00:27:00.670 align:middle line:90% IAN BALL: Right. 00:27:00.670 --> 00:27:02.212 align:middle line:84% So I think what you're thinking about 00:27:02.212 --> 00:27:05.570 align:middle line:84% is exactly, even at the contingency that's not reached, 00:27:05.570 --> 00:27:08.810 align:middle line:84% this second player is doing pretty well. 00:27:08.810 --> 00:27:13.690 align:middle line:84% So the issue, if we look at the first one or the last one, 00:27:13.690 --> 00:27:17.930 align:middle line:84% it is a Nash equilibrium, but it involves player 2 playing 00:27:17.930 --> 00:27:20.570 align:middle line:84% suboptimally at some contingency, 00:27:20.570 --> 00:27:23.850 align:middle line:84% namely in this Nash equilibrium, player 2 00:27:23.850 --> 00:27:25.410 align:middle line:84% goes to the Celtics game even when 00:27:25.410 --> 00:27:28.010 align:middle line:84% they see that player 1 has chosen the Red Sox game. 00:27:28.010 --> 00:27:30.310 align:middle line:84% That's suboptimal play because they'd 00:27:30.310 --> 00:27:31.810 align:middle line:84% rather join them at the Red Sox game 00:27:31.810 --> 00:27:35.210 align:middle line:84% than be all alone at the Celtics game. 00:27:35.210 --> 00:27:37.870 align:middle line:84% And similarly, this strategy profile, 00:27:37.870 --> 00:27:41.570 align:middle line:84% while it is a Nash equilibrium, involves suboptimal play 00:27:41.570 --> 00:27:44.930 align:middle line:84% at the first contingency because player 2 observes 00:27:44.930 --> 00:27:47.250 align:middle line:84% that player 1 has gone to the Celtics game, 00:27:47.250 --> 00:27:49.950 align:middle line:84% and they've responded not by going to the Celtics game, 00:27:49.950 --> 00:27:51.810 align:middle line:84% but by going to the Red Sox game. 00:27:51.810 --> 00:27:54.730 align:middle line:84% So each of these involves suboptimal play 00:27:54.730 --> 00:27:56.410 align:middle line:90% at unreached contingencies. 00:27:56.410 --> 00:27:58.490 align:middle line:84% And it's only this Nash equilibrium 00:27:58.490 --> 00:28:02.090 align:middle line:84% that seems to have this special property of optimal play 00:28:02.090 --> 00:28:03.850 align:middle line:90% at every contingency. 00:28:03.850 --> 00:28:08.090 align:middle line:84% So let's see how we could have tried to identify this. 00:28:08.090 --> 00:28:09.970 align:middle line:84% What algorithm could we have used? 00:28:09.970 --> 00:28:12.350 align:middle line:84% If I have colored chalk-- maybe not. 00:28:12.350 --> 00:28:14.370 align:middle line:90% So I'll just highlight it. 00:28:14.370 --> 00:28:16.590 align:middle line:84% Well, it's pretty intuitive to work backwards. 00:28:16.590 --> 00:28:18.660 align:middle line:84% And I think this is how most people approach 00:28:18.660 --> 00:28:20.860 align:middle line:90% problems like this. 00:28:20.860 --> 00:28:22.800 align:middle line:84% Let's analyze this working backwards. 00:28:22.800 --> 00:28:25.660 align:middle line:84% Let's start here at these last information sets 00:28:25.660 --> 00:28:28.980 align:middle line:84% and say, how should player 2 behave optimally 00:28:28.980 --> 00:28:30.797 align:middle line:84% if this information set is reached? 00:28:30.797 --> 00:28:31.880 align:middle line:90% So we'll look at player 2. 00:28:31.880 --> 00:28:34.860 align:middle line:84% We'll say, well, if we're here, we're certainly better off 00:28:34.860 --> 00:28:36.700 align:middle line:90% getting 1 than 0. 00:28:36.700 --> 00:28:41.540 align:middle line:84% So I'm going to choose this edge. 00:28:41.540 --> 00:28:43.780 align:middle line:84% And then we'll come over here and say, if I 00:28:43.780 --> 00:28:45.240 align:middle line:90% reach this information set-- 00:28:45.240 --> 00:28:46.560 align:middle line:90% I don't know if we'll reach it. 00:28:46.560 --> 00:28:48.240 align:middle line:84% I don't know how player 1 is playing. 00:28:48.240 --> 00:28:51.460 align:middle line:84% But let's suppose that we get to this contingency. 00:28:51.460 --> 00:28:54.420 align:middle line:90% Well, if I choose C, I get 0. 00:28:54.420 --> 00:28:56.060 align:middle line:90% If I choose R, I get 2. 00:28:56.060 --> 00:28:59.020 align:middle line:90% So R is better. 00:28:59.020 --> 00:29:01.780 align:middle line:84% And then we can work our way up a step further. 00:29:01.780 --> 00:29:04.020 align:middle line:84% And now we can say, well, suppose I'm player 1 00:29:04.020 --> 00:29:05.700 align:middle line:90% and I'm choosing how to play. 00:29:05.700 --> 00:29:09.060 align:middle line:84% I'm anticipating how player 2 is going to play 00:29:09.060 --> 00:29:10.900 align:middle line:90% at each of these contingencies. 00:29:10.900 --> 00:29:14.100 align:middle line:84% So if I choose C, I anticipate we'll go down here, 00:29:14.100 --> 00:29:15.980 align:middle line:90% and I'll get a payoff of 2. 00:29:15.980 --> 00:29:19.300 align:middle line:84% And if I choose R, I anticipate player 2 will do this, 00:29:19.300 --> 00:29:21.300 align:middle line:90% and I'll get a payoff of 1. 00:29:21.300 --> 00:29:25.220 align:middle line:84% 2 is higher than 1, so I'll go here. 00:29:25.220 --> 00:29:28.540 align:middle line:84% And sure enough, we've exactly identified this special Nash 00:29:28.540 --> 00:29:30.140 align:middle line:90% equilibrium. 00:29:30.140 --> 00:29:31.620 align:middle line:84% When you apply this procedure, it's 00:29:31.620 --> 00:29:33.980 align:middle line:84% very tempting to then-- you follow the path down 00:29:33.980 --> 00:29:36.740 align:middle line:84% and you say, ah, 2 1 is the answer. 00:29:36.740 --> 00:29:38.820 align:middle line:90% But no, 2 1 is just the payoff. 00:29:38.820 --> 00:29:41.440 align:middle line:84% The answer that you're looking for is a strategy profile. 00:29:41.440 --> 00:29:44.620 align:middle line:84% The strategy profile is player 1 plays C, 00:29:44.620 --> 00:29:47.340 align:middle line:84% and player 2 makes a contingent plan 00:29:47.340 --> 00:29:50.500 align:middle line:84% to play C at this information set and R at this information 00:29:50.500 --> 00:29:51.000 align:middle line:90% set. 00:29:51.000 --> 00:30:02.900 align:middle line:90% 00:30:02.900 --> 00:30:05.440 align:middle line:84% So now let's formalize this procedure. 00:30:05.440 --> 00:30:05.940 align:middle line:90% Yes? 00:30:05.940 --> 00:30:08.460 align:middle line:84% AUDIENCE: Since this is an [INAUDIBLE] version 00:30:08.460 --> 00:30:11.300 align:middle line:84% of the [INAUDIBLE] game, then would players here 00:30:11.300 --> 00:30:16.150 align:middle line:84% have no reason to choose the 0 0 payoffs in order to convince 00:30:16.150 --> 00:30:18.690 align:middle line:90% player 1 to switch strategies? 00:30:18.690 --> 00:30:22.630 align:middle line:84% Or is that only in the game where they don't 00:30:22.630 --> 00:30:24.210 align:middle line:90% know each other's choices? 00:30:24.210 --> 00:30:27.430 align:middle line:84% IAN BALL: Wait, so maybe what you're getting at 00:30:27.430 --> 00:30:31.390 align:middle line:84% is, let's look at player 1's payoffs in these equilibria. 00:30:31.390 --> 00:30:34.190 align:middle line:84% So we have this equilibrium, 2, 1, 00:30:34.190 --> 00:30:37.710 align:middle line:84% that gives-- it's an equilibrium that gives the payoffs 2 and 1. 00:30:37.710 --> 00:30:39.550 align:middle line:84% And we have another Nash equilibrium 00:30:39.550 --> 00:30:41.550 align:middle line:90% that gives the payoffs 1, 2. 00:30:41.550 --> 00:30:43.910 align:middle line:84% So you might say, if you're player 2, 00:30:43.910 --> 00:30:46.350 align:middle line:84% you do better in this equilibrium 00:30:46.350 --> 00:30:48.190 align:middle line:90% than this equilibrium. 00:30:48.190 --> 00:30:51.430 align:middle line:84% And indeed, this gets at a central issue, 00:30:51.430 --> 00:30:56.750 align:middle line:84% another way of thinking about backward induction, that how-- 00:30:56.750 --> 00:31:00.383 align:middle line:84% let's suppose I'm player 2 and I want to get to this equilibrium. 00:31:00.383 --> 00:31:01.050 align:middle line:90% What would I do? 00:31:01.050 --> 00:31:03.350 align:middle line:84% Maybe I would say something like, look, 00:31:03.350 --> 00:31:05.210 align:middle line:84% I'm always going to go to the Red Sox game. 00:31:05.210 --> 00:31:06.390 align:middle line:90% That's just who I am. 00:31:06.390 --> 00:31:09.550 align:middle line:84% Whatever I see you do, I'm going to go to the Red Sox game. 00:31:09.550 --> 00:31:12.590 align:middle line:84% And it's almost like a threat because the way I punish you 00:31:12.590 --> 00:31:14.610 align:middle line:84% is by just saying, I'm going to the Red Sox game 00:31:14.610 --> 00:31:16.590 align:middle line:90% whatever you do. 00:31:16.590 --> 00:31:18.830 align:middle line:84% The issue is that another way of saying it 00:31:18.830 --> 00:31:21.370 align:middle line:84% is that this involves a non-credible threat. 00:31:21.370 --> 00:31:25.670 align:middle line:90% 00:31:25.670 --> 00:31:27.790 align:middle line:84% So let's understand, why might player 2 00:31:27.790 --> 00:31:28.950 align:middle line:90% want to make this threat? 00:31:28.950 --> 00:31:32.470 align:middle line:84% It would be great for player 2 if player 1 believes 00:31:32.470 --> 00:31:34.910 align:middle line:84% that player 2 is always going to go to the Red Sox game 00:31:34.910 --> 00:31:36.565 align:middle line:84% because player 2 loves the Red Sox. 00:31:36.565 --> 00:31:39.190 align:middle line:84% They love player one to believe that because then player 1 will 00:31:39.190 --> 00:31:40.710 align:middle line:90% choose the Red Sox themselves. 00:31:40.710 --> 00:31:42.750 align:middle line:84% The problem is if player 1 thinks it through, 00:31:42.750 --> 00:31:44.250 align:middle line:84% they're going to say, wait a second, 00:31:44.250 --> 00:31:47.390 align:middle line:84% you're threatening to go to the Red Sox game whatever I do. 00:31:47.390 --> 00:31:49.710 align:middle line:84% But if I actually go to the Celtics game, 00:31:49.710 --> 00:31:52.510 align:middle line:84% are you really going to find it optimal to follow through 00:31:52.510 --> 00:31:53.372 align:middle line:90% on that threat? 00:31:53.372 --> 00:31:54.830 align:middle line:84% And the answer is no, because given 00:31:54.830 --> 00:31:56.288 align:middle line:84% that I'm going to the Celtics game, 00:31:56.288 --> 00:31:58.510 align:middle line:84% you'd rather go to the Celtics than go alone. 00:31:58.510 --> 00:32:01.430 align:middle line:84% So one other way of thinking about backward induction 00:32:01.430 --> 00:32:03.910 align:middle line:84% is that it's a principled way of ruling out 00:32:03.910 --> 00:32:06.750 align:middle line:84% equilibria that involve non-credible threats 00:32:06.750 --> 00:32:08.530 align:middle line:84% at contingencies that are not reached. 00:32:08.530 --> 00:32:09.290 align:middle line:90% Great question. 00:32:09.290 --> 00:32:11.450 align:middle line:84% Yeah, any other questions on this? 00:32:11.450 --> 00:32:11.950 align:middle line:90% Yes? 00:32:11.950 --> 00:32:14.440 align:middle line:84% AUDIENCE: Then would you say offering 1 00:32:14.440 --> 00:32:17.100 align:middle line:84% to a person then rejecting it is a non-credible threat? 00:32:17.100 --> 00:32:18.560 align:middle line:90% IAN BALL: Yes. 00:32:18.560 --> 00:32:22.280 align:middle line:84% So similarly, this was a Nash equilibrium that 00:32:22.280 --> 00:32:24.600 align:middle line:84% involves a non-credible threat, and therefore it's 00:32:24.600 --> 00:32:28.660 align:middle line:84% a Nash equilibrium, but it will not satisfy backward induction. 00:32:28.660 --> 00:32:31.420 align:middle line:84% So in both of these cases-- these are very symmetric. 00:32:31.420 --> 00:32:34.360 align:middle line:84% In both cases, we have many Nash equilibria. 00:32:34.360 --> 00:32:37.240 align:middle line:84% But some of the Nash equilibria involve non-credible threats. 00:32:37.240 --> 00:32:39.240 align:middle line:84% And we're going to introduce an algorithm called 00:32:39.240 --> 00:32:42.605 align:middle line:84% backward induction that will rule out those equilibria. 00:32:42.605 --> 00:32:44.480 align:middle line:84% But I think the point that could be confusing 00:32:44.480 --> 00:32:47.120 align:middle line:84% is even though I'm saying this is not credible, 00:32:47.120 --> 00:32:48.800 align:middle line:90% it's still an equilibrium. 00:32:48.800 --> 00:32:50.720 align:middle line:84% And similarly over here, this was exactly 00:32:50.720 --> 00:32:52.360 align:middle line:84% a non-credible threat, but it was still 00:32:52.360 --> 00:32:53.760 align:middle line:90% part of an equilibrium. 00:32:53.760 --> 00:32:56.160 align:middle line:90% Yeah. 00:32:56.160 --> 00:32:57.020 align:middle line:90% Any other questions? 00:32:57.020 --> 00:32:58.800 align:middle line:90% This is great. 00:32:58.800 --> 00:33:02.640 align:middle line:84% OK, so now let's be a bit more formal 00:33:02.640 --> 00:33:06.240 align:middle line:84% or just say what our algorithm is. 00:33:06.240 --> 00:33:09.340 align:middle line:84% So this is-- we'll call this the backward induction algorithm. 00:33:09.340 --> 00:33:19.160 align:middle line:90% 00:33:19.160 --> 00:33:20.680 align:middle line:90% So we already did it here. 00:33:20.680 --> 00:33:22.180 align:middle line:84% The algorithm is you work backwards. 00:33:22.180 --> 00:33:23.940 align:middle line:90% But let's say what that means. 00:33:23.940 --> 00:33:27.520 align:middle line:84% Well, first, what's the input to the algorithm? 00:33:27.520 --> 00:33:30.680 align:middle line:84% And formally, the input to the algorithm 00:33:30.680 --> 00:33:34.860 align:middle line:90% is a finite extensive-form game. 00:33:34.860 --> 00:33:42.440 align:middle line:90% 00:33:42.440 --> 00:33:45.120 align:middle line:84% Really, the important aspect of finiteness 00:33:45.120 --> 00:33:49.320 align:middle line:84% is that the players alternate moves only finitely many times. 00:33:49.320 --> 00:33:51.720 align:middle line:84% If one player at some node can choose 00:33:51.720 --> 00:33:55.480 align:middle line:84% among infinitely many moves, like choosing 00:33:55.480 --> 00:33:58.220 align:middle line:84% a number between 0 and 100, it's still going to be OK. 00:33:58.220 --> 00:34:00.360 align:middle line:84% But to be extra careful, we'll only-- 00:34:00.360 --> 00:34:03.460 align:middle line:84% we'll formally define the algorithm for finite games. 00:34:03.460 --> 00:34:05.160 align:middle line:84% But we will be a bit informal and then 00:34:05.160 --> 00:34:08.320 align:middle line:84% apply it to some infinite games where it won't create problems. 00:34:08.320 --> 00:34:11.969 align:middle line:84% The other important aspect is it's an extensive-form game. 00:34:11.969 --> 00:34:15.330 align:middle line:84% The whole point of this is that this algorithm is faithfully 00:34:15.330 --> 00:34:17.042 align:middle line:84% capturing the dynamics of the game, 00:34:17.042 --> 00:34:19.250 align:middle line:84% and we wouldn't be able to observe that if we started 00:34:19.250 --> 00:34:20.690 align:middle line:90% with a strategic-form game. 00:34:20.690 --> 00:34:23.810 align:middle line:84% So notice this is fundamentally different from Nash equilibrium. 00:34:23.810 --> 00:34:26.449 align:middle line:84% When we define Nash equilibrium, our starting point 00:34:26.449 --> 00:34:29.050 align:middle line:90% was a given strategic-form game. 00:34:29.050 --> 00:34:32.250 align:middle line:84% Here, our starting point is an extensive-form game. 00:34:32.250 --> 00:34:33.929 align:middle line:84% And the other final thing that we need 00:34:33.929 --> 00:34:36.830 align:middle line:84% is this is a game with perfect information. 00:34:36.830 --> 00:34:43.650 align:middle line:90% 00:34:43.650 --> 00:34:45.770 align:middle line:84% What do I mean by perfect information? 00:34:45.770 --> 00:34:48.949 align:middle line:84% I mean every information set is a singleton. 00:34:48.949 --> 00:35:01.970 align:middle line:90% 00:35:01.970 --> 00:35:04.510 align:middle line:84% And as long as information sets are singletons, 00:35:04.510 --> 00:35:07.170 align:middle line:84% you could even have moves by nature, that's OK, 00:35:07.170 --> 00:35:09.770 align:middle line:84% but only moves by nature that everyone gets to see. 00:35:09.770 --> 00:35:11.938 align:middle line:84% So poker, where you have a private hand, 00:35:11.938 --> 00:35:13.730 align:middle line:84% that's not going to be perfect information. 00:35:13.730 --> 00:35:16.850 align:middle line:84% But if nature flipped a coin and everyone saw that, 00:35:16.850 --> 00:35:20.050 align:middle line:90% that would be OK. 00:35:20.050 --> 00:35:22.530 align:middle line:90% Can you see what would go wrong? 00:35:22.530 --> 00:35:26.230 align:middle line:84% Let's say these two nodes were in the same information set 00:35:26.230 --> 00:35:28.570 align:middle line:84% and we tried to perform the same procedure. 00:35:28.570 --> 00:35:29.670 align:middle line:90% What would go wrong? 00:35:29.670 --> 00:35:33.530 align:middle line:90% 00:35:33.530 --> 00:35:35.810 align:middle line:84% Well, if we were at this information set 00:35:35.810 --> 00:35:37.650 align:middle line:84% and we tried to say how should player 00:35:37.650 --> 00:35:39.310 align:middle line:84% 2 play at this information set, we 00:35:39.310 --> 00:35:40.810 align:middle line:84% wouldn't know how to play because it 00:35:40.810 --> 00:35:43.018 align:middle line:84% would depend on which of the nodes in the information 00:35:43.018 --> 00:35:43.710 align:middle line:90% set we were at. 00:35:43.710 --> 00:35:45.210 align:middle line:84% And later in the course, we'll think 00:35:45.210 --> 00:35:46.905 align:middle line:90% about how to approach that. 00:35:46.905 --> 00:35:49.030 align:middle line:84% But for now, our algorithm-- "later in the course," 00:35:49.030 --> 00:35:50.570 align:middle line:90% just in two days-- 00:35:50.570 --> 00:35:52.930 align:middle line:84% but for now, we're only going to apply this 00:35:52.930 --> 00:35:56.088 align:middle line:84% to games of perfect information so that, at any node we get to, 00:35:56.088 --> 00:35:58.130 align:middle line:84% we know that we're at that node, and therefore we 00:35:58.130 --> 00:35:59.590 align:middle line:90% can compute the optimal play. 00:35:59.590 --> 00:36:01.788 align:middle line:84% If we had a non-singleton information set 00:36:01.788 --> 00:36:04.330 align:middle line:84% and we don't know which node we're at within that information 00:36:04.330 --> 00:36:08.500 align:middle line:84% set, it's not clear what optimal play is. 00:36:08.500 --> 00:36:10.700 align:middle line:90% So this is our input. 00:36:10.700 --> 00:36:12.000 align:middle line:90% And what is our output? 00:36:12.000 --> 00:36:14.700 align:middle line:90% 00:36:14.700 --> 00:36:16.860 align:middle line:84% We have to keep in mind that the output is always 00:36:16.860 --> 00:36:18.120 align:middle line:90% a strategy profile. 00:36:18.120 --> 00:36:22.940 align:middle line:90% 00:36:22.940 --> 00:36:24.280 align:middle line:90% It's not a pair of payoffs. 00:36:24.280 --> 00:36:25.640 align:middle line:90% It's the full strategy profile. 00:36:25.640 --> 00:36:27.500 align:middle line:84% And this is the most common mistake I see. 00:36:27.500 --> 00:36:29.800 align:middle line:84% People just list the payoffs they get at the end. 00:36:29.800 --> 00:36:31.920 align:middle line:84% They follow the paths down, and they say 2 1. 00:36:31.920 --> 00:36:36.100 align:middle line:84% But you really have to specify the full strategy profile. 00:36:36.100 --> 00:36:38.720 align:middle line:84% Now, I could formally define what the algorithm is, 00:36:38.720 --> 00:36:40.270 align:middle line:84% but I think you see how it works. 00:36:40.270 --> 00:36:42.020 align:middle line:84% I can introduce all this notation and say, 00:36:42.020 --> 00:36:43.900 align:middle line:84% you start at the end, and you play optimally, 00:36:43.900 --> 00:36:45.233 align:middle line:90% and you work your way backwards. 00:36:45.233 --> 00:36:47.560 align:middle line:84% But I think it's just easier to say, you see how it is. 00:36:47.560 --> 00:36:50.520 align:middle line:84% But let me talk about a few properties of this algorithm. 00:36:50.520 --> 00:36:54.460 align:middle line:84% So the output is not just any strategy profile. 00:36:54.460 --> 00:36:57.700 align:middle line:84% A first observation is that it is a pure strategy 00:36:57.700 --> 00:37:04.880 align:middle line:84% profile because, at each point, we just choose one action. 00:37:04.880 --> 00:37:06.540 align:middle line:90% We don't need any mixing. 00:37:06.540 --> 00:37:10.000 align:middle line:84% Now, if the player were indifferent at some point, 00:37:10.000 --> 00:37:11.280 align:middle line:90% you might have two choices. 00:37:11.280 --> 00:37:13.420 align:middle line:84% There might be two ways they could play optimally. 00:37:13.420 --> 00:37:15.980 align:middle line:84% And in that case, just choose one arbitrarily. 00:37:15.980 --> 00:37:18.800 align:middle line:84% So always break ties yourself because, at the end, 00:37:18.800 --> 00:37:21.100 align:middle line:84% we always want to get a pure strategy 00:37:21.100 --> 00:37:23.220 align:middle line:90% equilibrium when we apply this. 00:37:23.220 --> 00:37:27.060 align:middle line:84% And that kind of indicates another property of this, that, 00:37:27.060 --> 00:37:32.620 align:middle line:84% in fact, the output is going to be unique 00:37:32.620 --> 00:37:34.080 align:middle line:90% as long as there are no ties. 00:37:34.080 --> 00:37:39.220 align:middle line:90% 00:37:39.220 --> 00:37:42.700 align:middle line:84% So as long as we never get to a point where there's a tie, 00:37:42.700 --> 00:37:44.960 align:middle line:84% we never have any flexibility in what we do. 00:37:44.960 --> 00:37:46.700 align:middle line:84% So when we work backwards, we're always 00:37:46.700 --> 00:37:49.648 align:middle line:84% going to have a unique edge to choose. 00:37:49.648 --> 00:37:51.440 align:middle line:84% There's never going to be multiple choices. 00:37:51.440 --> 00:37:54.620 align:middle line:90% 00:37:54.620 --> 00:37:57.200 align:middle line:84% And this isn't just going to be any strategy profile. 00:37:57.200 --> 00:37:59.180 align:middle line:84% This is also going to be a Nash equilibrium. 00:37:59.180 --> 00:38:08.030 align:middle line:90% 00:38:08.030 --> 00:38:11.910 align:middle line:90% So what does this algorithm do? 00:38:11.910 --> 00:38:16.790 align:middle line:84% It gives us a pure-strategy Nash equilibrium. 00:38:16.790 --> 00:38:19.550 align:middle line:84% And unless there are ties, it always 00:38:19.550 --> 00:38:22.030 align:middle line:84% gives us the same pure-strategy Nash equilibrium. 00:38:22.030 --> 00:38:25.210 align:middle line:84% But let's keep in mind a warning. 00:38:25.210 --> 00:38:27.830 align:middle line:90% 00:38:27.830 --> 00:38:33.170 align:middle line:84% As we already saw, there may be other Nash equilibria. 00:38:33.170 --> 00:38:38.750 align:middle line:90% 00:38:38.750 --> 00:38:40.990 align:middle line:84% So it's true that the output of this algorithm 00:38:40.990 --> 00:38:44.490 align:middle line:84% is a Nash equilibrium, but it's a very special Nash equilibrium. 00:38:44.490 --> 00:38:47.510 align:middle line:84% And it's possible that there are other Nash equilibria that would 00:38:47.510 --> 00:38:50.390 align:middle line:84% not-- that cannot be outputs of this algorithm. 00:38:50.390 --> 00:38:52.390 align:middle line:84% And those would be the Nash equilibria 00:38:52.390 --> 00:38:54.470 align:middle line:84% that involve non-credible threats, 00:38:54.470 --> 00:38:56.970 align:middle line:90% as we've identified previously. 00:38:56.970 --> 00:39:08.030 align:middle line:90% 00:39:08.030 --> 00:39:11.630 align:middle line:84% Now, I think one nice implication 00:39:11.630 --> 00:39:18.010 align:middle line:84% of this that we shouldn't forget is that, recall-- 00:39:18.010 --> 00:39:20.790 align:middle line:90% 00:39:20.790 --> 00:39:26.550 align:middle line:84% recall that some of the strategic-form games 00:39:26.550 --> 00:39:30.610 align:middle line:84% we looked at before did not have pure-strategy Nash equilibria. 00:39:30.610 --> 00:39:48.830 align:middle line:90% 00:39:48.830 --> 00:39:52.190 align:middle line:84% Can you remember an example of a game that 00:39:52.190 --> 00:39:54.190 align:middle line:84% doesn't have-- a finite game that 00:39:54.190 --> 00:39:57.290 align:middle line:84% doesn't have a pure-strategy Nash equilibrium? 00:39:57.290 --> 00:40:05.020 align:middle line:90% 00:40:05.020 --> 00:40:05.520 align:middle line:90% Yeah? 00:40:05.520 --> 00:40:06.360 align:middle line:90% AUDIENCE: Rock, paper, scissors. 00:40:06.360 --> 00:40:07.700 align:middle line:84% IAN BALL: Rock, paper, scissors is a great example. 00:40:07.700 --> 00:40:08.620 align:middle line:90% So it's unique. 00:40:08.620 --> 00:40:12.760 align:middle line:84% Nash equilibrium is we each mix one third, one third, one third. 00:40:12.760 --> 00:40:14.680 align:middle line:90% So wait a second. 00:40:14.680 --> 00:40:17.680 align:middle line:84% Rock, paper, scissors doesn't have a pure-strategy Nash 00:40:17.680 --> 00:40:18.223 align:middle line:90% equilibrium. 00:40:18.223 --> 00:40:19.640 align:middle line:84% But here, I have an algorithm that 00:40:19.640 --> 00:40:21.740 align:middle line:84% always gives us a pure-strategy Nash equilibrium. 00:40:21.740 --> 00:40:23.180 align:middle line:90% So what's going on here? 00:40:23.180 --> 00:40:27.440 align:middle line:90% 00:40:27.440 --> 00:40:31.040 align:middle line:84% Well, the input to our algorithm is a finite extensive-form game 00:40:31.040 --> 00:40:33.000 align:middle line:90% with perfect information. 00:40:33.000 --> 00:40:36.680 align:middle line:84% So notice, rock, paper, scissors is a simultaneous move game. 00:40:36.680 --> 00:40:38.400 align:middle line:84% If we wanted to represent rock, paper, 00:40:38.400 --> 00:40:42.920 align:middle line:84% scissors in extensive form, we'd have an information set 00:40:42.920 --> 00:40:45.200 align:middle line:84% with multiple nodes, and therefore this 00:40:45.200 --> 00:40:47.640 align:middle line:84% would not be a game of perfect information, 00:40:47.640 --> 00:40:49.920 align:middle line:84% and therefore algorithm would not apply. 00:40:49.920 --> 00:40:51.903 align:middle line:84% So one implication of this algorithm-- 00:40:51.903 --> 00:40:53.320 align:middle line:84% you can think of it as a theorem-- 00:40:53.320 --> 00:40:56.960 align:middle line:84% is that every extensive-form game with perfect information 00:40:56.960 --> 00:40:59.160 align:middle line:84% has a pure-strategy Nash equilibrium. 00:40:59.160 --> 00:41:01.840 align:middle line:84% So Nash's theorem told us that any game 00:41:01.840 --> 00:41:03.940 align:middle line:84% has a mixed-strategy Nash equilibrium. 00:41:03.940 --> 00:41:07.060 align:middle line:84% Some games only have mixed-strategy Nash equilibria. 00:41:07.060 --> 00:41:10.140 align:middle line:84% But for this special class of games with perfect information, 00:41:10.140 --> 00:41:12.860 align:middle line:84% we always have a pure-strategy Nash equilibrium. 00:41:12.860 --> 00:41:16.040 align:middle line:90% 00:41:16.040 --> 00:41:18.360 align:middle line:84% So maybe I'll say-- what's the answer here? 00:41:18.360 --> 00:41:20.660 align:middle line:84% Perfect-information games are special. 00:41:20.660 --> 00:41:28.426 align:middle line:90% 00:41:28.426 --> 00:41:30.426 align:middle line:84% And in fact, what's nice about this algorithm is 00:41:30.426 --> 00:41:32.760 align:middle line:84% it doesn't just show that a pure-strategy Nash 00:41:32.760 --> 00:41:36.280 align:middle line:84% equilibrium exists, it gives us an algorithm for finding it, 00:41:36.280 --> 00:41:39.720 align:middle line:84% whereas computing Nash equilibria in arbitrary games 00:41:39.720 --> 00:41:42.840 align:middle line:90% is generally much harder. 00:41:42.840 --> 00:41:46.080 align:middle line:84% And this actually formally confirms a claim 00:41:46.080 --> 00:41:47.700 align:middle line:90% I made in a previous class. 00:41:47.700 --> 00:41:52.760 align:middle line:84% If you recall, I said that the value of chess 00:41:52.760 --> 00:41:55.640 align:middle line:90% is plus 1, minus 1, or 0. 00:41:55.640 --> 00:41:58.730 align:middle line:84% Can anyone see-- I didn't actually check that claim. 00:41:58.730 --> 00:42:03.890 align:middle line:84% Can anyone see why this follows from what we've said so far? 00:42:03.890 --> 00:42:05.930 align:middle line:84% So we know chess has a value because chess 00:42:05.930 --> 00:42:07.890 align:middle line:90% is a zero-sum game. 00:42:07.890 --> 00:42:09.370 align:middle line:84% But in general, the value of a game 00:42:09.370 --> 00:42:14.170 align:middle line:84% could be something like 0.3 or negative 0.7. 00:42:14.170 --> 00:42:16.050 align:middle line:84% So we know chess is some value that's going 00:42:16.050 --> 00:42:19.055 align:middle line:90% to be between negative 1 and 1. 00:42:19.055 --> 00:42:20.930 align:middle line:84% But how do we know that it has a value that's 00:42:20.930 --> 00:42:26.470 align:middle line:84% either 0, negative 1, or 1, and not, say, 0.7? 00:42:26.470 --> 00:42:26.970 align:middle line:90% Yeah? 00:42:26.970 --> 00:42:28.370 align:middle line:84% AUDIENCE: Win, draw, [? or loss. ?] 00:42:28.370 --> 00:42:29.828 align:middle line:84% IAN BALL: So those are the outcomes 00:42:29.828 --> 00:42:31.230 align:middle line:90% of the game, that's true. 00:42:31.230 --> 00:42:34.630 align:middle line:84% But I can give you examples of games-- let's take rock, paper-- 00:42:34.630 --> 00:42:36.330 align:middle line:90% well, let's take-- 00:42:36.330 --> 00:42:39.730 align:middle line:84% rock, paper, scissors is not a good example. 00:42:39.730 --> 00:42:42.635 align:middle line:84% The-- actually, hide and seek is also not a good example. 00:42:42.635 --> 00:42:44.010 align:middle line:84% There are games that can give you 00:42:44.010 --> 00:42:46.850 align:middle line:84% with 1's and 2's and only whole-number payoffs, where 00:42:46.850 --> 00:42:49.610 align:middle line:84% the value of the game is not a whole number because it involves 00:42:49.610 --> 00:42:50.110 align:middle line:90% mixing. 00:42:50.110 --> 00:42:54.270 align:middle line:84% So rock, paper, scissors, if I slightly change things, 00:42:54.270 --> 00:42:55.690 align:middle line:90% I'd be able to get that. 00:42:55.690 --> 00:43:00.210 align:middle line:84% So that shows that, indeed, the value can't be below negative 1 00:43:00.210 --> 00:43:01.250 align:middle line:90% or above 1. 00:43:01.250 --> 00:43:04.150 align:middle line:84% But in principle, players could play mixed strategies. 00:43:04.150 --> 00:43:06.890 align:middle line:84% And maybe the value-- we often saw in zero-sum games 00:43:06.890 --> 00:43:09.410 align:middle line:84% that the security strategies involved mixing. 00:43:09.410 --> 00:43:12.710 align:middle line:84% And that meant that the value was a fraction. 00:43:12.710 --> 00:43:13.510 align:middle line:90% Yeah? 00:43:13.510 --> 00:43:15.510 align:middle line:84% AUDIENCE: I mean, I don't have the exact answer, 00:43:15.510 --> 00:43:17.427 align:middle line:84% but does it have something to do with the fact 00:43:17.427 --> 00:43:21.390 align:middle line:84% that one player has one step's worth of perfect information, 00:43:21.390 --> 00:43:25.105 align:middle line:84% [? that ?] you always know what happens if you [INAUDIBLE]. 00:43:25.105 --> 00:43:27.230 align:middle line:84% IAN BALL: Yeah, I think it's even easier than that. 00:43:27.230 --> 00:43:28.730 align:middle line:84% They do have perfect information. 00:43:28.730 --> 00:43:30.950 align:middle line:84% So chess is a game of perfect information. 00:43:30.950 --> 00:43:32.950 align:middle line:84% Whenever I move, I see all the moves-- 00:43:32.950 --> 00:43:35.290 align:middle line:84% I see the board, and I know exactly how 00:43:35.290 --> 00:43:36.650 align:middle line:90% people have moved before. 00:43:36.650 --> 00:43:38.030 align:middle line:90% So let's understand this. 00:43:38.030 --> 00:43:40.890 align:middle line:84% Chess is a game of perfect information, 00:43:40.890 --> 00:43:44.950 align:middle line:84% PI, which means I can apply backward induction to chess, 00:43:44.950 --> 00:43:45.515 align:middle line:90% in principle. 00:43:45.515 --> 00:43:46.890 align:middle line:84% Of course, I can't actually do it 00:43:46.890 --> 00:43:49.450 align:middle line:90% because the game tree explodes. 00:43:49.450 --> 00:43:53.570 align:middle line:84% No computer-- all the NVIDIA chips in the world 00:43:53.570 --> 00:43:56.700 align:middle line:84% can't solve chess, but because it's 00:43:56.700 --> 00:43:59.300 align:middle line:84% a game of perfect information, I know 00:43:59.300 --> 00:44:04.980 align:middle line:84% it has a pure-strategy Nash equilibrium from backward 00:44:04.980 --> 00:44:07.220 align:middle line:90% induction. 00:44:07.220 --> 00:44:09.600 align:middle line:84% And because it has a pure-strategy Nash equilibrium-- 00:44:09.600 --> 00:44:11.740 align:middle line:84% well, because it's a zero-sum game, 00:44:11.740 --> 00:44:14.760 align:middle line:84% I know that in that pure-strategy Nash equilibrium, 00:44:14.760 --> 00:44:18.220 align:middle line:84% each player is actually playing a security strategy. 00:44:18.220 --> 00:44:20.980 align:middle line:84% So it must consist-- because it's 00:44:20.980 --> 00:44:27.980 align:middle line:84% zero-sum, that consists of security strategies. 00:44:27.980 --> 00:44:30.780 align:middle line:84% And the payoff in that pure-strategy Nash equilibrium 00:44:30.780 --> 00:44:32.500 align:middle line:84% consisting of security strategies 00:44:32.500 --> 00:44:34.080 align:middle line:84% is exactly the value of the game. 00:44:34.080 --> 00:44:41.180 align:middle line:90% 00:44:41.180 --> 00:44:42.940 align:middle line:84% So in general, the value of the game 00:44:42.940 --> 00:44:45.700 align:middle line:84% is always the payoff when each player plays their security 00:44:45.700 --> 00:44:46.660 align:middle line:90% strategy. 00:44:46.660 --> 00:44:49.400 align:middle line:84% But often, those security strategies involve mixing. 00:44:49.400 --> 00:44:50.800 align:middle line:90% Those are mixed strategies. 00:44:50.800 --> 00:44:52.820 align:middle line:84% And therefore the payoff isn't necessarily 00:44:52.820 --> 00:44:54.280 align:middle line:90% plus 1, minus 1, or 0. 00:44:54.280 --> 00:44:56.060 align:middle line:90% It could be something like 1/2. 00:44:56.060 --> 00:44:58.380 align:middle line:84% But this tells us that, in fact, the value of the game 00:44:58.380 --> 00:45:00.220 align:middle line:90% is plus 1, minus 1, and 0. 00:45:00.220 --> 00:45:02.740 align:middle line:84% And even more amazingly, it tells us 00:45:02.740 --> 00:45:06.580 align:middle line:84% that the security strategies are pure. 00:45:06.580 --> 00:45:09.740 align:middle line:84% So if I say, what's the optimal way to play chess, it's not, 00:45:09.740 --> 00:45:11.760 align:middle line:84% oh, I'm going to randomize, like it is in poker. 00:45:11.760 --> 00:45:12.840 align:middle line:90% There's literally a book. 00:45:12.840 --> 00:45:15.173 align:middle line:84% It would be bigger than the size of the universe, 00:45:15.173 --> 00:45:17.840 align:middle line:84% but there is a book I could, in principle, write down that says, 00:45:17.840 --> 00:45:20.500 align:middle line:84% given this move on the board, this is how you should play. 00:45:20.500 --> 00:45:21.880 align:middle line:84% Given this position on the board, 00:45:21.880 --> 00:45:23.060 align:middle line:90% this is how you should play. 00:45:23.060 --> 00:45:26.060 align:middle line:84% And that would ensure either that I always win, 00:45:26.060 --> 00:45:28.940 align:middle line:84% depending on if the value is 1 and I'm the first player, 00:45:28.940 --> 00:45:31.087 align:middle line:84% maybe I always draw, or maybe the other player 00:45:31.087 --> 00:45:31.920 align:middle line:90% has such a strategy. 00:45:31.920 --> 00:45:35.140 align:middle line:84% We just don't know in chess who has the advantage. 00:45:35.140 --> 00:45:36.387 align:middle line:90% In checkers, we do know. 00:45:36.387 --> 00:45:37.720 align:middle line:90% So what's the value of checkers? 00:45:37.720 --> 00:45:38.440 align:middle line:90% Does anyone know? 00:45:38.440 --> 00:45:42.020 align:middle line:90% 00:45:42.020 --> 00:45:42.980 align:middle line:90% Yeah? 00:45:42.980 --> 00:45:44.537 align:middle line:84% AUDIENCE: The first player has won. 00:45:44.537 --> 00:45:46.120 align:middle line:84% IAN BALL: No, I think it's actually 0. 00:45:46.120 --> 00:45:47.953 align:middle line:84% I think, checkers, if people play optimally, 00:45:47.953 --> 00:45:49.260 align:middle line:90% I think it ends in a tie. 00:45:49.260 --> 00:45:50.880 align:middle line:84% Though, there may be different variants of checkers, 00:45:50.880 --> 00:45:51.970 align:middle line:90% so I have to check that. 00:45:51.970 --> 00:45:53.390 align:middle line:84% But I think, in standard checkers, 00:45:53.390 --> 00:45:54.530 align:middle line:90% it always ends in a tie. 00:45:54.530 --> 00:45:55.150 align:middle line:90% Yeah. 00:45:55.150 --> 00:45:55.730 align:middle line:90% Yeah? 00:45:55.730 --> 00:45:57.970 align:middle line:84% AUDIENCE: When you say a game has a value, 00:45:57.970 --> 00:45:59.053 align:middle line:90% what are you referring to? 00:45:59.053 --> 00:46:01.730 align:middle line:84% IAN BALL: So this is something specific to zero-sum games. 00:46:01.730 --> 00:46:04.370 align:middle line:84% That's a lecture we did, I guess, on Thursday. 00:46:04.370 --> 00:46:06.690 align:middle line:84% Yeah, so it's referencing a previous lecture. 00:46:06.690 --> 00:46:08.170 align:middle line:84% Yeah, I'm happy to talk more after. 00:46:08.170 --> 00:46:10.830 align:middle line:90% 00:46:10.830 --> 00:46:13.790 align:middle line:90% Any other questions on this? 00:46:13.790 --> 00:46:21.990 align:middle line:84% So now let's do one final example of backward induction. 00:46:21.990 --> 00:46:28.290 align:middle line:84% And that will be another quantity competition game. 00:46:28.290 --> 00:46:35.173 align:middle line:90% 00:46:35.173 --> 00:46:37.990 align:middle line:84% So, so far we looked at Cournot competition. 00:46:37.990 --> 00:46:40.747 align:middle line:84% Here we'll look at something called Stackelberg competition. 00:46:40.747 --> 00:46:42.330 align:middle line:84% I mean, the names don't really matter. 00:46:42.330 --> 00:46:48.390 align:middle line:90% 00:46:48.390 --> 00:46:50.450 align:middle line:84% So in Cournot competition, remember, 00:46:50.450 --> 00:46:54.710 align:middle line:84% we had two firms that were each choosing how much to produce 00:46:54.710 --> 00:46:56.590 align:middle line:90% and how much to bring to market. 00:46:56.590 --> 00:46:59.150 align:middle line:84% But the assumption in Cournot was that the firms 00:46:59.150 --> 00:47:00.742 align:middle line:90% move simultaneously. 00:47:00.742 --> 00:47:02.950 align:middle line:84% And in Stackelberg, we're going to imagine that there 00:47:02.950 --> 00:47:04.270 align:middle line:90% may be a first mover. 00:47:04.270 --> 00:47:06.710 align:middle line:90% Maybe I harvest my crops first. 00:47:06.710 --> 00:47:08.890 align:middle line:84% I get to choose to bring them to market. 00:47:08.890 --> 00:47:11.350 align:middle line:84% And then the next farmer sees how much I bring to market 00:47:11.350 --> 00:47:13.950 align:middle line:84% and then chooses how much to bring themselves. 00:47:13.950 --> 00:47:17.590 align:middle line:84% So we have-- it's still quantity competition. 00:47:17.590 --> 00:47:19.310 align:middle line:90% I'll emphasize this. 00:47:19.310 --> 00:47:25.390 align:middle line:84% But it's-- sequential quantity competition would maybe be-- 00:47:25.390 --> 00:47:26.835 align:middle line:90% we often name these with people. 00:47:26.835 --> 00:47:28.210 align:middle line:84% But if you don't know the person, 00:47:28.210 --> 00:47:29.543 align:middle line:90% it's not a very meaningful name. 00:47:29.543 --> 00:47:32.590 align:middle line:84% So we might call this sequential quantity competition 00:47:32.590 --> 00:47:34.170 align:middle line:84% if we want a more descriptive title. 00:47:34.170 --> 00:47:37.430 align:middle line:90% 00:47:37.430 --> 00:47:38.950 align:middle line:90% So let's see how this works. 00:47:38.950 --> 00:47:44.990 align:middle line:84% We still have two firms, 1 and 2. 00:47:44.990 --> 00:47:49.560 align:middle line:84% And they are choosing quantities as before. 00:47:49.560 --> 00:47:52.880 align:middle line:90% And we have market demand-- 00:47:52.880 --> 00:48:01.040 align:middle line:84% maybe I'll say inverse demand P of Q equals 00:48:01.040 --> 00:48:06.280 align:middle line:84% to the maximum of 1 minus Q and 0. 00:48:06.280 --> 00:48:08.160 align:middle line:84% So this is exactly what we had before. 00:48:08.160 --> 00:48:11.760 align:middle line:84% The more quantity that's brought to market, the lower the price 00:48:11.760 --> 00:48:12.520 align:middle line:90% is. 00:48:12.520 --> 00:48:16.200 align:middle line:84% If quantity 1 is brought to market, the price is 0. 00:48:16.200 --> 00:48:19.280 align:middle line:84% If quantity 0 is brought to market, the price is 1. 00:48:19.280 --> 00:48:20.940 align:middle line:84% And the price can never go negative. 00:48:20.940 --> 00:48:23.680 align:middle line:84% So if tons is brought to market, we just say the price is 0. 00:48:23.680 --> 00:48:26.360 align:middle line:90% The market price is 0. 00:48:26.360 --> 00:48:34.860 align:middle line:84% And let's also assume marginal cost C. And as usual, 00:48:34.860 --> 00:48:38.640 align:middle line:84% we'll assume C is between 0 and 1. 00:48:38.640 --> 00:48:40.960 align:middle line:84% So remember, marginal cost C means 00:48:40.960 --> 00:48:43.360 align:middle line:84% whatever quantity Q of the good I produce, 00:48:43.360 --> 00:48:46.100 align:middle line:84% it costs me C times Q to produce that amount. 00:48:46.100 --> 00:48:50.120 align:middle line:84% So it's a linear cost-- constant marginal cost. 00:48:50.120 --> 00:48:54.320 align:middle line:84% But the key difference relative to Cournot competition 00:48:54.320 --> 00:48:56.540 align:middle line:90% is that firm 1 moves first. 00:48:56.540 --> 00:49:03.520 align:middle line:90% 00:49:03.520 --> 00:49:10.660 align:middle line:84% And then firm 2 observes firm 1's choice. 00:49:10.660 --> 00:49:16.440 align:middle line:90% 00:49:16.440 --> 00:49:17.340 align:middle line:90% So we can draw this. 00:49:17.340 --> 00:49:19.100 align:middle line:84% We can try to draw this in extensive form. 00:49:19.100 --> 00:49:20.600 align:middle line:84% It's a little tricky because they're 00:49:20.600 --> 00:49:23.540 align:middle line:84% making choices among infinitely many quantities. 00:49:23.540 --> 00:49:27.720 align:middle line:84% But if we wanted to draw it, we would say, here's firm 1. 00:49:27.720 --> 00:49:31.360 align:middle line:84% They're making an infinite choice over how much to produce. 00:49:31.360 --> 00:49:35.120 align:middle line:90% Maybe we'll call this Q1. 00:49:35.120 --> 00:49:39.600 align:middle line:84% Maybe I'll take a particular point here. 00:49:39.600 --> 00:49:43.920 align:middle line:84% And then we have firm 2, who's making 00:49:43.920 --> 00:49:46.850 align:middle line:90% an infinite choice over Q2. 00:49:46.850 --> 00:49:49.410 align:middle line:84% So it's a little hard to formally represent 00:49:49.410 --> 00:49:52.090 align:middle line:84% this in extensive form when we have infinitely many things. 00:49:52.090 --> 00:49:55.530 align:middle line:84% But what I mean here is player 1 chooses 00:49:55.530 --> 00:49:59.930 align:middle line:84% a quantity Q1 to bring to market, any non-negative number. 00:49:59.930 --> 00:50:02.542 align:middle line:84% Whatever quantity that is, player 2 observes it. 00:50:02.542 --> 00:50:04.250 align:middle line:84% I'll warn you that it's particularly hard 00:50:04.250 --> 00:50:07.950 align:middle line:84% to represent information sets with infinitely many players. 00:50:07.950 --> 00:50:10.930 align:middle line:84% So it's crucial here sometimes to write it in words 00:50:10.930 --> 00:50:13.343 align:middle line:84% that player 2 observes the quantity that's 00:50:13.343 --> 00:50:15.010 align:middle line:84% produced by player 1 because it's really 00:50:15.010 --> 00:50:16.730 align:middle line:90% hard to draw information sets. 00:50:16.730 --> 00:50:18.790 align:middle line:84% And then player 2 chooses the quantity Q2. 00:50:18.790 --> 00:50:26.690 align:middle line:90% 00:50:26.690 --> 00:50:29.090 align:middle line:90% So now let's write down payoffs. 00:50:29.090 --> 00:50:32.090 align:middle line:84% The payoffs are actually going to look exactly the same 00:50:32.090 --> 00:50:34.250 align:middle line:84% as they did in Cournot competition 00:50:34.250 --> 00:50:37.650 align:middle line:84% because the payoffs just depend on how much each firm brings 00:50:37.650 --> 00:50:38.350 align:middle line:90% to market. 00:50:38.350 --> 00:50:40.570 align:middle line:84% So this is kind of like our BoS game, 00:50:40.570 --> 00:50:44.490 align:middle line:84% where the payoffs are the same whether player 2 observes 00:50:44.490 --> 00:50:47.023 align:middle line:84% player 1's move or not, but the equilibria of the game 00:50:47.023 --> 00:50:48.690 align:middle line:84% are going to be very different depending 00:50:48.690 --> 00:50:50.450 align:middle line:84% on whether that move is observed. 00:50:50.450 --> 00:50:51.910 align:middle line:90% So what are our payoffs? 00:50:51.910 --> 00:50:53.590 align:middle line:90% Let's look at, say, firm 1. 00:50:53.590 --> 00:50:56.570 align:middle line:90% 00:50:56.570 --> 00:51:05.730 align:middle line:84% Well, as before, we want to say, what is firm 1's profit, pi 1, 00:51:05.730 --> 00:51:08.370 align:middle line:84% if they bring quantity Q1 to market 00:51:08.370 --> 00:51:11.170 align:middle line:84% and firm 2 brings quantity Q2 to market? 00:51:11.170 --> 00:51:13.770 align:middle line:84% So we always have to think of profit in two components. 00:51:13.770 --> 00:51:17.570 align:middle line:84% It's the profit per unit times the number of units you have. 00:51:17.570 --> 00:51:23.370 align:middle line:84% So it's going to be Q1 times what? 00:51:23.370 --> 00:51:30.250 align:middle line:84% Well, maybe I'll write it like this. 00:51:30.250 --> 00:51:32.730 align:middle line:84% Q1 is the amount I bring to market. 00:51:32.730 --> 00:51:35.770 align:middle line:84% I sell it at a price of P of Q1 plus Q2 00:51:35.770 --> 00:51:39.410 align:middle line:84% because Q1 plus Q2 is the total quantity brought to market. 00:51:39.410 --> 00:51:41.290 align:middle line:84% That's what determines the price. 00:51:41.290 --> 00:51:44.250 align:middle line:84% And then, of course, I also have to pay my constant marginal cost 00:51:44.250 --> 00:51:45.170 align:middle line:90% per unit. 00:51:45.170 --> 00:51:48.450 align:middle line:84% But let me plug in a formula here, again with my little cheat 00:51:48.450 --> 00:51:50.955 align:middle line:84% that I'm going to assume Q is not so big that this 00:51:50.955 --> 00:51:51.830 align:middle line:90% is a negative number. 00:51:51.830 --> 00:51:56.490 align:middle line:84% So I'm just going to write this as Q1 times 1 minus Q1 00:51:56.490 --> 00:52:01.050 align:middle line:84% minus Q2 minus C, where technically this formula only 00:52:01.050 --> 00:52:04.930 align:middle line:84% applies if Q1 plus Q2 is between 0 and 1, but in the end 00:52:04.930 --> 00:52:05.478 align:middle line:90% it will be. 00:52:05.478 --> 00:52:07.270 align:middle line:84% You might say, why are there two negatives? 00:52:07.270 --> 00:52:10.770 align:middle line:84% Well, remember, it's minus, in parentheses, Q1 plus Q2. 00:52:10.770 --> 00:52:14.330 align:middle line:84% And that's why I get two minuses. 00:52:14.330 --> 00:52:18.010 align:middle line:84% So now we've specified the payoffs. 00:52:18.010 --> 00:52:21.290 align:middle line:84% We still have to specify strategies, though. 00:52:21.290 --> 00:52:26.850 align:middle line:84% And this, I think, is where it gets a bit tricky. 00:52:26.850 --> 00:52:27.430 align:middle line:90% Any ideas? 00:52:27.430 --> 00:52:29.430 align:middle line:84% Let's start with player 1, firm 1. 00:52:29.430 --> 00:52:32.210 align:middle line:84% Maybe I'll call them F1 and F2 for firm. 00:52:32.210 --> 00:52:34.190 align:middle line:90% What is a strategy for firm 1? 00:52:34.190 --> 00:52:39.890 align:middle line:90% 00:52:39.890 --> 00:52:40.410 align:middle line:90% Yeah? 00:52:40.410 --> 00:52:42.140 align:middle line:84% AUDIENCE: Whatever value of Q1 would 00:52:42.140 --> 00:52:44.560 align:middle line:84% lead to the derivative of Q1 times [INAUDIBLE]. 00:52:44.560 --> 00:52:46.000 align:middle line:84% IAN BALL: Well, now you're talking about optimal. 00:52:46.000 --> 00:52:48.417 align:middle line:84% Sorry, I'm just saying, what is the set of all strategies, 00:52:48.417 --> 00:52:49.980 align:middle line:90% before we talk about optimum. 00:52:49.980 --> 00:52:51.140 align:middle line:90% So a strategy for-- 00:52:51.140 --> 00:52:52.980 align:middle line:84% I'm just saying, what is a strategy? 00:52:52.980 --> 00:52:56.283 align:middle line:84% AUDIENCE: [INAUDIBLE] for any non-negative value. 00:52:56.283 --> 00:52:57.700 align:middle line:84% IAN BALL: Yeah, so it's just going 00:52:57.700 --> 00:53:01.380 align:middle line:84% to be Q1, which we have to choose exactly how much they 00:53:01.380 --> 00:53:02.280 align:middle line:90% can produce. 00:53:02.280 --> 00:53:04.658 align:middle line:84% Let's say it's a number between, say, 0 and infinity, 00:53:04.658 --> 00:53:05.700 align:middle line:90% so a non-negative number. 00:53:05.700 --> 00:53:06.540 align:middle line:90% Let's say that. 00:53:06.540 --> 00:53:09.300 align:middle line:90% The details don't really matter. 00:53:09.300 --> 00:53:11.238 align:middle line:84% Whether it's 0 to 1 or 0 to infinity, 00:53:11.238 --> 00:53:12.280 align:middle line:90% it doesn't really matter. 00:53:12.280 --> 00:53:13.280 align:middle line:90% But what about firm 2? 00:53:13.280 --> 00:53:15.060 align:middle line:84% This is where it gets quite subtle. 00:53:15.060 --> 00:53:17.740 align:middle line:90% What is a strategy for firm 2? 00:53:17.740 --> 00:53:21.220 align:middle line:84% Well, if we took our previous approach, 00:53:21.220 --> 00:53:22.760 align:middle line:90% we'd have to get a vector. 00:53:22.760 --> 00:53:25.020 align:middle line:84% And we'd have to have a different spot in the vector 00:53:25.020 --> 00:53:26.540 align:middle line:90% for every contingency. 00:53:26.540 --> 00:53:28.520 align:middle line:84% But now there's infinitely many contingencies. 00:53:28.520 --> 00:53:30.420 align:middle line:84% So that's going to become a bit messy. 00:53:30.420 --> 00:53:33.780 align:middle line:90% So we really need a function. 00:53:33.780 --> 00:53:39.100 align:middle line:84% What kind of function is this going to be? 00:53:39.100 --> 00:53:43.620 align:middle line:84% Well, firm 2 observes the quantity chosen by firm 1, 00:53:43.620 --> 00:53:45.860 align:middle line:84% and then they choose a quantity themselves. 00:53:45.860 --> 00:53:55.880 align:middle line:84% So sometimes it might be easier to do 0, 1, 00:53:55.880 --> 00:53:58.740 align:middle line:90% but I'll stick with it this way. 00:53:58.740 --> 00:53:59.800 align:middle line:90% So what does this say? 00:53:59.800 --> 00:54:00.883 align:middle line:90% What's the interpretation? 00:54:00.883 --> 00:54:01.440 align:middle line:90% Yeah? 00:54:01.440 --> 00:54:03.380 align:middle line:84% AUDIENCE: Is C going to be involved [? in this one? ?] 00:54:03.380 --> 00:54:05.600 align:middle line:84% Is C going to be considered an input to the function? 00:54:05.600 --> 00:54:06.980 align:middle line:90% Or will C be implied? 00:54:06.980 --> 00:54:11.200 align:middle line:84% IAN BALL: No, so C is a commonly known parameter-- is known. 00:54:11.200 --> 00:54:13.500 align:middle line:84% AUDIENCE: So it is not considered an input 00:54:13.500 --> 00:54:14.560 align:middle line:90% to the function? 00:54:14.560 --> 00:54:16.600 align:middle line:84% IAN BALL: No, because C is like 2 here. 00:54:16.600 --> 00:54:18.700 align:middle line:90% It doesn't change-- 00:54:18.700 --> 00:54:22.800 align:middle line:84% C is a known parameter that affects the payoff functions, 00:54:22.800 --> 00:54:25.360 align:middle line:84% but it doesn't affect the definition of the strategies. 00:54:25.360 --> 00:54:27.380 align:middle line:84% So you think of it, in a game, there's 00:54:27.380 --> 00:54:30.080 align:middle line:84% a fundamental distinction between the set of strategies, 00:54:30.080 --> 00:54:32.600 align:middle line:84% what people can do, and the utility functions, 00:54:32.600 --> 00:54:35.500 align:middle line:84% which is how well they do when they choose those strategies. 00:54:35.500 --> 00:54:38.080 align:middle line:84% C is just-- we could call it a half if we wanted. 00:54:38.080 --> 00:54:41.050 align:middle line:84% It's just a known number that enters the utility functions 00:54:41.050 --> 00:54:43.010 align:middle line:84% but has no role in the strategies. 00:54:43.010 --> 00:54:46.670 align:middle line:84% One way of seeing this is if in this game, I made this a 3, 00:54:46.670 --> 00:54:48.450 align:middle line:84% it wouldn't change the set of strategies. 00:54:48.450 --> 00:54:50.075 align:middle line:84% It would just change the payoffs people 00:54:50.075 --> 00:54:51.710 align:middle line:90% get from those strategies. 00:54:51.710 --> 00:54:54.655 align:middle line:84% And I could call this alpha, and the same would be true. 00:54:54.655 --> 00:54:55.530 align:middle line:90% But that's important. 00:54:55.530 --> 00:54:57.890 align:middle line:84% If C was private information, then it would get tricky. 00:54:57.890 --> 00:54:59.890 align:middle line:84% But we're going to assume it's known and public. 00:54:59.890 --> 00:55:02.150 align:middle line:84% Everyone knows it, just like everyone 00:55:02.150 --> 00:55:05.630 align:middle line:84% knows this market function, big P. 00:55:05.630 --> 00:55:08.030 align:middle line:84% Great, so what's the interpretation of this? 00:55:08.030 --> 00:55:14.310 align:middle line:84% This says, for any Q1 that I observe, what is the amount 00:55:14.310 --> 00:55:18.030 align:middle line:90% Q2 of Q1 that I produce? 00:55:18.030 --> 00:55:21.150 align:middle line:84% So Q1 is the quantity that I observe 00:55:21.150 --> 00:55:23.548 align:middle line:90% for 1 producing hypothetically. 00:55:23.548 --> 00:55:25.590 align:middle line:84% It's not how much they actually produce, but it's 00:55:25.590 --> 00:55:28.590 align:middle line:84% if I were to observe them producing Q1, 00:55:28.590 --> 00:55:30.430 align:middle line:84% how much would I produce as firm 2? 00:55:30.430 --> 00:55:33.390 align:middle line:90% I would produce Q2 of Q1. 00:55:33.390 --> 00:55:36.510 align:middle line:90% OK, now, a big warning here. 00:55:36.510 --> 00:55:39.390 align:middle line:84% I think this is maybe the fault of sometimes 00:55:39.390 --> 00:55:40.570 align:middle line:90% the way we teach math. 00:55:40.570 --> 00:55:43.710 align:middle line:90% 00:55:43.710 --> 00:55:46.950 align:middle line:90% Q2 is a function. 00:55:46.950 --> 00:55:49.532 align:middle line:90% Q2 of Q1 is a number. 00:55:49.532 --> 00:55:50.990 align:middle line:84% And this creates a lot of confusion 00:55:50.990 --> 00:55:55.470 align:middle line:84% because, in math class, we often write f of x to mean a function. 00:55:55.470 --> 00:55:57.170 align:middle line:90% But really, f of x is a number. 00:55:57.170 --> 00:55:59.167 align:middle line:84% It's the value of the function at x. 00:55:59.167 --> 00:56:00.750 align:middle line:84% So it's really important in this class 00:56:00.750 --> 00:56:05.550 align:middle line:84% that we understand f by itself is a function, f 00:56:05.550 --> 00:56:07.110 align:middle line:90% of x is a number-- 00:56:07.110 --> 00:56:09.950 align:middle line:84% the value of the function evaluated at x. 00:56:09.950 --> 00:56:12.470 align:middle line:84% So in problem sets, it's really important 00:56:12.470 --> 00:56:15.470 align:middle line:84% that we understand that the strategy for firm 2 00:56:15.470 --> 00:56:17.510 align:middle line:90% is the function Q2. 00:56:17.510 --> 00:56:21.150 align:middle line:84% People often say the strategy is Q2 of Q1, but no. 00:56:21.150 --> 00:56:23.930 align:middle line:90% Q2 of Q1 is a number. 00:56:23.930 --> 00:56:27.890 align:middle line:84% It's the quantity produced at a particular contingency, Q1. 00:56:27.890 --> 00:56:29.390 align:middle line:84% So it's really important that you 00:56:29.390 --> 00:56:31.670 align:middle line:90% understand function notation. 00:56:31.670 --> 00:56:34.450 align:middle line:84% If in doubt, you need to, on problem sets or on quizzes, 00:56:34.450 --> 00:56:36.420 align:middle line:84% be very clear about what you mean by things 00:56:36.420 --> 00:56:38.253 align:middle line:84% because if you say the strategy is Q2 of Q1, 00:56:38.253 --> 00:56:39.795 align:middle line:84% we're not going to really know if you 00:56:39.795 --> 00:56:42.398 align:middle line:84% mean a number or a function because of this ambiguity 00:56:42.398 --> 00:56:42.940 align:middle line:90% that we have. 00:56:42.940 --> 00:56:46.080 align:middle line:90% So try to be precise. 00:56:46.080 --> 00:56:47.360 align:middle line:90% Great. 00:56:47.360 --> 00:56:50.660 align:middle line:84% So now let's try to actually analyze this. 00:56:50.660 --> 00:57:07.920 align:middle line:90% 00:57:07.920 --> 00:57:10.400 align:middle line:84% So now what we'd like to do is apply backward induction 00:57:10.400 --> 00:57:11.160 align:middle line:90% to this game. 00:57:11.160 --> 00:57:13.240 align:middle line:84% Now, I know technically I said it only applies 00:57:13.240 --> 00:57:14.700 align:middle line:90% to finite extensive form games. 00:57:14.700 --> 00:57:16.280 align:middle line:84% But we are going to be able to apply it here, 00:57:16.280 --> 00:57:17.480 align:middle line:90% and it won't be a problem. 00:57:17.480 --> 00:57:19.300 align:middle line:84% So let's apply backward induction. 00:57:19.300 --> 00:57:25.600 align:middle line:90% 00:57:25.600 --> 00:57:26.800 align:middle line:90% And what are we going to do? 00:57:26.800 --> 00:57:30.160 align:middle line:84% Well, we're going to work backwards, like we always do. 00:57:30.160 --> 00:57:31.600 align:middle line:90% So let's start here. 00:57:31.600 --> 00:57:35.760 align:middle line:84% And we have to say, as a function of the amount Q1 00:57:35.760 --> 00:57:40.040 align:middle line:84% that firm 1 has produced, what should firm 2 do? 00:57:40.040 --> 00:57:49.760 align:middle line:84% So let's consider maybe what I'll call contingency or infoset 00:57:49.760 --> 00:57:50.840 align:middle line:90% Q1. 00:57:50.840 --> 00:57:53.800 align:middle line:84% So this says, put yourself in the position of firm 2. 00:57:53.800 --> 00:57:56.778 align:middle line:84% You've observed how much firm 1 has brought to market. 00:57:56.778 --> 00:57:59.320 align:middle line:84% And you've observed that the amount they've brought to market 00:57:59.320 --> 00:58:00.360 align:middle line:90% is Q1. 00:58:00.360 --> 00:58:03.440 align:middle line:84% And now you're choosing Q2, how much you should 00:58:03.440 --> 00:58:04.920 align:middle line:90% bring to market yourself. 00:58:04.920 --> 00:58:06.200 align:middle line:90% So what are you choosing? 00:58:06.200 --> 00:58:10.840 align:middle line:84% Well, you're going to maximize over Q2-- 00:58:10.840 --> 00:58:13.800 align:middle line:90% Q2 is what you're choosing-- 00:58:13.800 --> 00:58:19.040 align:middle line:90% pi 2 of Q1, Q2. 00:58:19.040 --> 00:58:21.160 align:middle line:84% So this is essentially just what we 00:58:21.160 --> 00:58:24.100 align:middle line:84% did before when we talked about computing a best response. 00:58:24.100 --> 00:58:27.180 align:middle line:84% Given that Q1 is the amount produced, 00:58:27.180 --> 00:58:31.330 align:middle line:84% I want to choose Q2 to maximize my payoff where Q1 appears 00:58:31.330 --> 00:58:32.580 align:middle line:90% in here because this is fixed. 00:58:32.580 --> 00:58:33.830 align:middle line:90% I can't control this. 00:58:33.830 --> 00:58:34.890 align:middle line:90% This is already chosen. 00:58:34.890 --> 00:58:37.130 align:middle line:90% And this is what I can control. 00:58:37.130 --> 00:58:37.950 align:middle line:90% And what do we get? 00:58:37.950 --> 00:58:39.810 align:middle line:90% Well, this is Q2. 00:58:39.810 --> 00:58:41.570 align:middle line:84% I'm just going to copy this formula 00:58:41.570 --> 00:58:46.750 align:middle line:84% but switch it because we're now focusing on firm 2. 00:58:46.750 --> 00:58:57.690 align:middle line:84% 1 minus Q1 minus Q2 minus C. And if we use our trick, 00:58:57.690 --> 00:59:00.810 align:middle line:84% this is something times a constant minus that something. 00:59:00.810 --> 00:59:03.090 align:middle line:84% It's always optimal to just do the something 00:59:03.090 --> 00:59:07.490 align:middle line:84% the constant-- sorry, this is Q2 times the constant minus Q2. 00:59:07.490 --> 00:59:11.170 align:middle line:84% It's always optimal to set Q2 to be the constant divided by 2. 00:59:11.170 --> 00:59:17.930 align:middle line:84% So what we're going to get is Q2, maybe I'll say star of Q1 00:59:17.930 --> 00:59:18.910 align:middle line:90% is equal to-- 00:59:18.910 --> 00:59:22.890 align:middle line:90% 00:59:22.890 --> 00:59:24.690 align:middle line:90% so we're almost there. 00:59:24.690 --> 00:59:27.150 align:middle line:84% I just mechanically took a derivative. 00:59:27.150 --> 00:59:30.390 align:middle line:84% But there is one issue I have to be a little careful about. 00:59:30.390 --> 00:59:34.450 align:middle line:84% What if Q1 is really, really big? 00:59:34.450 --> 00:59:36.280 align:middle line:84% So now my assumption that, oh, we 00:59:36.280 --> 00:59:38.030 align:middle line:84% don't have to worry about negative prices, 00:59:38.030 --> 00:59:39.870 align:middle line:84% we don't have to worry about max or min, 00:59:39.870 --> 00:59:41.970 align:middle line:84% turns out we have to be a little bit careful. 00:59:41.970 --> 00:59:43.670 align:middle line:84% If Q1-- let's just think it through. 00:59:43.670 --> 00:59:49.450 align:middle line:84% If Q1 is a billion, what should firm 2 produce? 00:59:49.450 --> 00:59:52.110 align:middle line:84% Nothing, because if Q1 is a billion, 00:59:52.110 --> 00:59:53.930 align:middle line:90% then the market price is 0. 00:59:53.930 --> 00:59:56.770 align:middle line:84% My cost is C. So I definitely don't want to produce anything. 00:59:56.770 --> 00:59:58.970 align:middle line:84% And unfortunately, it turns out that if we just 00:59:58.970 --> 01:00:01.065 align:middle line:84% compute this number and check if it's positive, 01:00:01.065 --> 01:00:02.190 align:middle line:90% we'll get the right answer. 01:00:02.190 --> 01:00:03.690 align:middle line:90% So if it's positive, we're good. 01:00:03.690 --> 01:00:06.410 align:middle line:84% If it's negative, we set it to 0. 01:00:06.410 --> 01:00:09.850 align:middle line:84% And we often-- the notation I'll use for that is I'll 01:00:09.850 --> 01:00:13.090 align:middle line:84% put a little plus at the bottom of it. 01:00:13.090 --> 01:00:15.088 align:middle line:90% Or-- let me just write-- 01:00:15.088 --> 01:00:17.130 align:middle line:84% I like that notation, but I think that might just 01:00:17.130 --> 01:00:17.910 align:middle line:90% cause confusion. 01:00:17.910 --> 01:00:23.698 align:middle line:84% So let's just write max of this is 0. 01:00:23.698 --> 01:00:25.930 align:middle line:90% Need a little more space. 01:00:25.930 --> 01:00:28.370 align:middle line:90% Getting too cute here. 01:00:28.370 --> 01:00:34.500 align:middle line:84% So it's the-- we have 1 minus Q1 minus C over 2. 01:00:34.500 --> 01:00:37.300 align:middle line:84% But if this is negative, I just want to produce 0. 01:00:37.300 --> 01:00:42.020 align:middle line:84% So I'm going to write the maximum of this and 0. 01:00:42.020 --> 01:00:44.440 align:middle line:84% I could use the plus, but I think this is clearer. 01:00:44.440 --> 01:00:49.220 align:middle line:90% 01:00:49.220 --> 01:00:51.260 align:middle line:84% And I wrote Q2 star just to indicate 01:00:51.260 --> 01:00:55.020 align:middle line:84% that this is going to be part of my special backward induction 01:00:55.020 --> 01:00:57.660 align:middle line:90% strategy profile. 01:00:57.660 --> 01:01:03.180 align:middle line:84% And let's understand, Q2 star at a particular Q1 01:01:03.180 --> 01:01:04.860 align:middle line:90% is exactly this number. 01:01:04.860 --> 01:01:06.640 align:middle line:84% But by going through this procedure, 01:01:06.640 --> 01:01:09.140 align:middle line:84% I'm going to obtain a function Q2 star. 01:01:09.140 --> 01:01:15.760 align:middle line:84% So Q2 star-- the function is firm 2's strategy. 01:01:15.760 --> 01:01:20.780 align:middle line:90% 01:01:20.780 --> 01:01:23.800 align:middle line:84% It's the function where, whatever Q1 you plug into it, 01:01:23.800 --> 01:01:24.880 align:middle line:90% this is what you get out. 01:01:24.880 --> 01:01:29.020 align:middle line:90% 01:01:29.020 --> 01:01:31.800 align:middle line:84% OK, so now we go one step further. 01:01:31.800 --> 01:01:35.700 align:middle line:90% 01:01:35.700 --> 01:01:36.640 align:middle line:90% So we work backwards. 01:01:36.640 --> 01:01:40.900 align:middle line:84% We figured out what happens at the second stage with firm 2. 01:01:40.900 --> 01:01:44.040 align:middle line:84% And now we have to figure out how firm 1 should behave. 01:01:44.040 --> 01:01:57.180 align:middle line:90% 01:01:57.180 --> 01:02:02.900 align:middle line:90% So now we're at firm 1. 01:02:02.900 --> 01:02:05.160 align:middle line:84% Technically, we're at the initial contingency. 01:02:05.160 --> 01:02:07.340 align:middle line:84% Nothing has really happened so far in the game. 01:02:07.340 --> 01:02:09.980 align:middle line:90% And what is firm 1's problem? 01:02:09.980 --> 01:02:13.340 align:middle line:90% They're maximizing over Q1 what? 01:02:13.340 --> 01:02:17.340 align:middle line:84% They're choosing Q1 to maximize what? 01:02:17.340 --> 01:02:20.140 align:middle line:90% Any thoughts here? 01:02:20.140 --> 01:02:23.560 align:middle line:84% Well, I think the first step is it's going to be a profit. 01:02:23.560 --> 01:02:26.020 align:middle line:90% So let's put pi 1 here. 01:02:26.020 --> 01:02:28.590 align:middle line:84% And if they choose Q1, well, definitely Q1 01:02:28.590 --> 01:02:30.927 align:middle line:84% should be the first component in here. 01:02:30.927 --> 01:02:33.510 align:middle line:84% But the hard thing is, what do we put in the second component? 01:02:33.510 --> 01:02:36.850 align:middle line:84% If I choose Q1, what will firm 2 do? 01:02:36.850 --> 01:02:37.750 align:middle line:90% Yes? 01:02:37.750 --> 01:02:39.110 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:02:39.110 --> 01:02:39.950 align:middle line:90% IAN BALL: Exactly. 01:02:39.950 --> 01:02:46.710 align:middle line:84% So what I'll put in here is Q2 star of Q1. 01:02:46.710 --> 01:02:48.670 align:middle line:84% So notice what I'm doing here as firm 1. 01:02:48.670 --> 01:02:50.990 align:middle line:84% I'm choosing how much quantity to produce, 01:02:50.990 --> 01:02:54.170 align:middle line:84% recognizing that when I choose Q1, 01:02:54.170 --> 01:02:56.150 align:middle line:84% that's going to directly affect my payoff. 01:02:56.150 --> 01:02:58.150 align:middle line:84% But the Q1 I choose is also going 01:02:58.150 --> 01:03:01.430 align:middle line:84% to affect how much the other firm brings to market. 01:03:01.430 --> 01:03:03.270 align:middle line:84% And specifically, we can see that this 01:03:03.270 --> 01:03:05.510 align:middle line:90% is a decreasing function of Q1. 01:03:05.510 --> 01:03:07.850 align:middle line:84% So the more that I bring to market, 01:03:07.850 --> 01:03:10.590 align:middle line:84% the more the next farmer sees, wow, this person 01:03:10.590 --> 01:03:11.850 align:middle line:90% brought a lot to market. 01:03:11.850 --> 01:03:14.030 align:middle line:84% That's really going to bring down the market price. 01:03:14.030 --> 01:03:15.930 align:middle line:84% I'm going to bring less to market. 01:03:15.930 --> 01:03:19.510 align:middle line:84% So I can influence how much firm 2 brings to market 01:03:19.510 --> 01:03:22.815 align:middle line:90% through my choice of Q1. 01:03:22.815 --> 01:03:24.190 align:middle line:84% And now we're just going to write 01:03:24.190 --> 01:03:25.990 align:middle line:90% this, substitute everything in. 01:03:25.990 --> 01:03:29.710 align:middle line:84% And we'll use the trick that let's hope that we never 01:03:29.710 --> 01:03:30.448 align:middle line:90% deal with the 0. 01:03:30.448 --> 01:03:31.990 align:middle line:84% So let's assume everything's positive 01:03:31.990 --> 01:03:33.610 align:middle line:84% and solve it and see what happens. 01:03:33.610 --> 01:03:38.670 align:middle line:84% So we're going to get Q1 is the amount that I produce. 01:03:38.670 --> 01:03:40.510 align:middle line:90% Then we're going to get-- 01:03:40.510 --> 01:03:42.190 align:middle line:84% maybe I'll write it really carefully-- 01:03:42.190 --> 01:03:50.850 align:middle line:84% Q1 plus Q2 star of Q1 minus C because for each unit I produce, 01:03:50.850 --> 01:03:52.510 align:middle line:84% I have to pay a cost of C. And I'm 01:03:52.510 --> 01:03:54.010 align:middle line:90% going to get the market price. 01:03:54.010 --> 01:03:55.010 align:middle line:90% What's the market price? 01:03:55.010 --> 01:03:57.610 align:middle line:84% Well, it depends on the market production level, 01:03:57.610 --> 01:03:59.150 align:middle line:84% which is the sum of what I produce 01:03:59.150 --> 01:04:02.590 align:middle line:84% and what I'm going to induce firm 2 to produce. 01:04:02.590 --> 01:04:14.190 align:middle line:84% So this is Q1 times 1 minus Q1 minus Q2 star of Q1 minus C. 01:04:14.190 --> 01:04:16.270 align:middle line:84% And here, there's kind of a trick. 01:04:16.270 --> 01:04:22.590 align:middle line:84% We have 1 minus Q1 minus C minus, again, 1 minus Q1 minus C 01:04:22.590 --> 01:04:23.510 align:middle line:90% but over 2. 01:04:23.510 --> 01:04:26.500 align:middle line:84% So we have 1 minus Q1 minus C minus half of it. 01:04:26.500 --> 01:04:27.620 align:middle line:90% So we have half left. 01:04:27.620 --> 01:04:34.100 align:middle line:84% So we're going to get Q1 times 1 minus Q1 minus C over 2. 01:04:34.100 --> 01:04:36.760 align:middle line:90% 01:04:36.760 --> 01:04:39.840 align:middle line:84% The Q2 star is gone because I substituted in the expression 01:04:39.840 --> 01:04:41.340 align:middle line:90% for Q2 star above. 01:04:41.340 --> 01:04:47.240 align:middle line:90% 01:04:47.240 --> 01:04:49.452 align:middle line:84% And it turns out the half here doesn't actually 01:04:49.452 --> 01:04:51.160 align:middle line:84% affect the optimum, because we could just 01:04:51.160 --> 01:04:52.480 align:middle line:90% pull out the half to the front. 01:04:52.480 --> 01:04:57.600 align:middle line:84% And we're going to get Q1 star equals 1 minus C over 2 01:04:57.600 --> 01:04:59.640 align:middle line:90% if we do the algebra. 01:04:59.640 --> 01:05:01.660 align:middle line:84% It can be tricky to see this half here. 01:05:01.660 --> 01:05:04.143 align:middle line:84% But remember, if we just put the half in the front, 01:05:04.143 --> 01:05:06.560 align:middle line:84% you can see-- if we multiply the objective by a half, that 01:05:06.560 --> 01:05:07.757 align:middle line:90% doesn't change the optimum. 01:05:07.757 --> 01:05:09.840 align:middle line:84% It changes the optimal value, but not the optimum. 01:05:09.840 --> 01:05:13.600 align:middle line:90% So we'll just get this. 01:05:13.600 --> 01:05:16.440 align:middle line:84% Now let's be really clear about this. 01:05:16.440 --> 01:05:19.740 align:middle line:84% The algorithm has told us a strategy profile. 01:05:19.740 --> 01:05:22.880 align:middle line:84% It said firm 1 is producing this amount, 01:05:22.880 --> 01:05:26.200 align:middle line:84% and firm 2 is using this complete contingent plan 01:05:26.200 --> 01:05:29.297 align:middle line:84% that specifies what they produce as a function of every Q1. 01:05:29.297 --> 01:05:30.880 align:middle line:84% But now we'd like to understand, well, 01:05:30.880 --> 01:05:32.680 align:middle line:90% how much is actually produced? 01:05:32.680 --> 01:05:36.900 align:middle line:84% Because all we know is firm 2's complete contingent plan. 01:05:36.900 --> 01:05:39.680 align:middle line:84% Let's figure out the actual production level for firm 2. 01:05:39.680 --> 01:05:40.780 align:middle line:90% What is that going to be? 01:05:40.780 --> 01:05:44.920 align:middle line:90% 01:05:44.920 --> 01:05:47.040 align:middle line:90% Well, we need to plug in-- 01:05:47.040 --> 01:05:49.800 align:middle line:84% what I want to look at is Q2 star 01:05:49.800 --> 01:05:53.600 align:middle line:84% is the strategy that backward induction gave us for firm 2. 01:05:53.600 --> 01:05:56.680 align:middle line:84% But now I'm going to plug into that function the strategy 01:05:56.680 --> 01:05:58.880 align:middle line:84% that backward induction gave us for firm 1, which 01:05:58.880 --> 01:06:02.240 align:middle line:90% is just a number, the quantity. 01:06:02.240 --> 01:06:04.400 align:middle line:84% And this is going to tell me how much is actually 01:06:04.400 --> 01:06:09.080 align:middle line:84% produced as opposed to firm 2's complete contingent plan. 01:06:09.080 --> 01:06:14.540 align:middle line:84% And I have to-- little careful here. 01:06:14.540 --> 01:06:18.860 align:middle line:84% It's 1 minus C over 2 minus quarter. 01:06:18.860 --> 01:06:21.670 align:middle line:84% And I think it should be 1 minus C over 4. 01:06:21.670 --> 01:06:25.490 align:middle line:90% 01:06:25.490 --> 01:06:30.330 align:middle line:90% So now we've analyzed the game. 01:06:30.330 --> 01:06:32.910 align:middle line:84% But now we can ask-- we've analyzed it mathematically. 01:06:32.910 --> 01:06:36.090 align:middle line:84% Let's ask some econ questions about it. 01:06:36.090 --> 01:06:39.930 align:middle line:84% And in particular, we've looked at how much each firm produces 01:06:39.930 --> 01:06:41.290 align:middle line:90% in the equilibrium. 01:06:41.290 --> 01:06:45.090 align:middle line:84% Does anyone recall, if we did Cournot with two firms, 01:06:45.090 --> 01:06:48.196 align:middle line:90% how much each firm produced? 01:06:48.196 --> 01:06:49.427 align:middle line:90% Anyone remember? 01:06:49.427 --> 01:06:51.010 align:middle line:84% We had the same-- were the same demand 01:06:51.010 --> 01:06:52.350 align:middle line:90% function, and the same costs. 01:06:52.350 --> 01:06:56.210 align:middle line:90% 01:06:56.210 --> 01:06:56.862 align:middle line:90% No? 01:06:56.862 --> 01:06:57.570 align:middle line:90% I'll just say it. 01:06:57.570 --> 01:06:59.030 align:middle line:90% It's 1 minus C over 3. 01:06:59.030 --> 01:07:03.062 align:middle line:90% 01:07:03.062 --> 01:07:06.510 align:middle line:84% So can anyone comment on the differences here? 01:07:06.510 --> 01:07:13.550 align:middle line:84% So in Cournot, Q1 star equals Q2 star equals 1 minus C over 3. 01:07:13.550 --> 01:07:15.610 align:middle line:84% So when the firms move simultaneously, 01:07:15.610 --> 01:07:18.290 align:middle line:84% they both produced 1 minus C over 3. 01:07:18.290 --> 01:07:20.330 align:middle line:84% When the firms move sequentially, 01:07:20.330 --> 01:07:22.250 align:middle line:84% firm 1 produced 1 minus C over 2, 01:07:22.250 --> 01:07:27.490 align:middle line:84% and firm 2 produced 1 minus C over 4. 01:07:27.490 --> 01:07:32.378 align:middle line:84% Any intuition for why we get this? 01:07:32.378 --> 01:07:33.670 align:middle line:90% What's the key difference here? 01:07:33.670 --> 01:07:36.010 align:middle line:84% Let's think of it from firm 1's perspective. 01:07:36.010 --> 01:07:40.850 align:middle line:84% What's the key difference in firm 1's optimization problem 01:07:40.850 --> 01:07:43.622 align:middle line:84% between the simultaneous-move Cournot game 01:07:43.622 --> 01:07:44.830 align:middle line:90% and the sequential-move game? 01:07:44.830 --> 01:07:46.622 align:middle line:84% Remember, we already saw in the Boston game 01:07:46.622 --> 01:07:48.330 align:middle line:84% that simultaneity versus sequential play 01:07:48.330 --> 01:07:50.570 align:middle line:90% can have a big difference. 01:07:50.570 --> 01:07:51.570 align:middle line:90% Yeah? 01:07:51.570 --> 01:07:53.430 align:middle line:84% AUDIENCE: Maybe in the Cournot game, 01:07:53.430 --> 01:07:56.230 align:middle line:84% they both know that they're choosing the same quantity. 01:07:56.230 --> 01:07:59.930 align:middle line:84% So they have to keep in mind that the other person could 01:07:59.930 --> 01:08:03.110 align:middle line:84% be doing the same thing while the other [INAUDIBLE]. 01:08:03.110 --> 01:08:05.710 align:middle line:84% IAN BALL: Stackelberg or sequential quantity competition? 01:08:05.710 --> 01:08:06.510 align:middle line:90% Yeah. 01:08:06.510 --> 01:08:09.130 align:middle line:84% AUDIENCE: [INAUDIBLE] player 1 knows [INAUDIBLE] 01:08:09.130 --> 01:08:10.730 align:middle line:90% player 2's going to know. 01:08:10.730 --> 01:08:13.928 align:middle line:84% Player 2's obviously not going to [INAUDIBLE]. 01:08:13.928 --> 01:08:16.470 align:middle line:84% IAN BALL: So I think the one word you said there was crucial. 01:08:16.470 --> 01:08:18.138 align:middle line:84% You said firm 1 can influence firm 2, 01:08:18.138 --> 01:08:19.680 align:middle line:84% and I think that's the crucial thing. 01:08:19.680 --> 01:08:21.580 align:middle line:84% So let's think about the Nash equilibrium 01:08:21.580 --> 01:08:22.939 align:middle line:90% reasoning in Cournot. 01:08:22.939 --> 01:08:26.020 align:middle line:84% In Cournot, I'm taking as given the amount 01:08:26.020 --> 01:08:27.427 align:middle line:90% that my opponent is producing. 01:08:27.427 --> 01:08:29.760 align:middle line:84% And I'm saying, given the amount that they're producing, 01:08:29.760 --> 01:08:31.500 align:middle line:84% the amount that I'm producing is optimal, 01:08:31.500 --> 01:08:33.899 align:middle line:84% taking as fixed the amount they produce. 01:08:33.899 --> 01:08:35.420 align:middle line:84% Here, it's fundamentally different 01:08:35.420 --> 01:08:39.380 align:middle line:84% because, because I move first, I know that my choice of quantity 01:08:39.380 --> 01:08:41.120 align:middle line:90% affects how much they produce. 01:08:41.120 --> 01:08:43.700 align:middle line:84% So I know if I increase my production, 01:08:43.700 --> 01:08:46.200 align:middle line:84% I can cause them to decrease their production. 01:08:46.200 --> 01:08:49.000 align:middle line:84% So that makes more production for firm 1 more attractive. 01:08:49.000 --> 01:08:51.143 align:middle line:84% In general, I don't want the other person 01:08:51.143 --> 01:08:53.060 align:middle line:84% to produce because that brings down the price. 01:08:53.060 --> 01:08:55.880 align:middle line:84% So if I know that by raising my production, 01:08:55.880 --> 01:08:57.779 align:middle line:84% I can decrease their production, that 01:08:57.779 --> 01:09:02.500 align:middle line:84% gives me a further incentive for me to raise my production level. 01:09:02.500 --> 01:09:05.220 align:middle line:84% So that's why firm 1 produces more. 01:09:05.220 --> 01:09:07.760 align:middle line:84% Firm 2's choices actually looks very similar. 01:09:07.760 --> 01:09:10.240 align:middle line:84% Firm 2 takes firm 1's quantity as given. 01:09:10.240 --> 01:09:12.620 align:middle line:84% So firm 1's incentives are essentially the same-- sorry, 01:09:12.620 --> 01:09:15.319 align:middle line:84% firm 2's incentives are essentially the same. 01:09:15.319 --> 01:09:17.740 align:middle line:84% But the issue is firm 1 is producing so much more. 01:09:17.740 --> 01:09:21.240 align:middle line:84% It's optimal for them to produce a lot less than in Cournot. 01:09:21.240 --> 01:09:24.779 align:middle line:90% 01:09:24.779 --> 01:09:25.880 align:middle line:90% So what's key? 01:09:25.880 --> 01:09:28.160 align:middle line:84% The key word you used is "influence." 01:09:28.160 --> 01:09:30.700 align:middle line:90% 01:09:30.700 --> 01:09:36.540 align:middle line:84% Can my quantity influence the quantity that you choose? 01:09:36.540 --> 01:09:38.180 align:middle line:84% All right, so now we've found-- we've 01:09:38.180 --> 01:09:39.720 align:middle line:90% applied backward induction. 01:09:39.720 --> 01:09:42.740 align:middle line:84% We found a particular Nash equilibrium 01:09:42.740 --> 01:09:44.000 align:middle line:90% that seems reasonable to us. 01:09:44.000 --> 01:09:46.083 align:middle line:84% It doesn't involve any credible-- any non-credible 01:09:46.083 --> 01:09:46.620 align:middle line:90% threats. 01:09:46.620 --> 01:09:48.120 align:middle line:84% But it's still good to keep in mind, 01:09:48.120 --> 01:09:51.260 align:middle line:84% could we find any other Nash equilibria that 01:09:51.260 --> 01:09:52.760 align:middle line:90% do involve credible threats? 01:09:52.760 --> 01:10:03.700 align:middle line:90% 01:10:03.700 --> 01:10:09.145 align:middle line:84% So could we find other Nash equilibria 01:10:09.145 --> 01:10:11.020 align:middle line:84% that maybe we don't find compelling, but just 01:10:11.020 --> 01:10:13.640 align:middle line:84% to understand what's going on here? 01:10:13.640 --> 01:10:17.095 align:middle line:84% What would be a threat that firm 2 might be tempted to make? 01:10:17.095 --> 01:10:18.470 align:middle line:84% If we think back to our reasoning 01:10:18.470 --> 01:10:20.630 align:middle line:84% about-- we gave the example of, I'll 01:10:20.630 --> 01:10:23.710 align:middle line:84% threaten to always go to the Red Sox game whatever you do. 01:10:23.710 --> 01:10:25.590 align:middle line:84% What might be a threat that firm 2 might 01:10:25.590 --> 01:10:28.730 align:middle line:84% want to impose to try to benefit themselves, 01:10:28.730 --> 01:10:30.090 align:middle line:90% if we had a very smart firm 2? 01:10:30.090 --> 01:10:30.350 align:middle line:90% Yeah? 01:10:30.350 --> 01:10:31.767 align:middle line:84% AUDIENCE: Make a threat to produce 01:10:31.767 --> 01:10:34.910 align:middle line:84% a lot to drive the price down so that firm 1 won't 01:10:34.910 --> 01:10:37.528 align:middle line:84% make any money, and then, yeah, they're both just worse off. 01:10:37.528 --> 01:10:38.570 align:middle line:90% IAN BALL: Great, exactly. 01:10:38.570 --> 01:10:41.990 align:middle line:90% So firm 2 can make a threat. 01:10:41.990 --> 01:10:45.310 align:middle line:84% Firm 2 might threaten-- there's even a term for this-- 01:10:45.310 --> 01:10:47.390 align:middle line:84% to flood the market, meaning produce 01:10:47.390 --> 01:10:50.130 align:middle line:84% a huge amount of quantity to pull down the market price. 01:10:50.130 --> 01:11:02.550 align:middle line:90% 01:11:02.550 --> 01:11:03.730 align:middle line:90% But let's say a little more. 01:11:03.730 --> 01:11:05.130 align:middle line:84% When would they want to impose this? 01:11:05.130 --> 01:11:06.210 align:middle line:84% When would they flood the market? 01:11:06.210 --> 01:11:08.250 align:middle line:84% Under what conditions or what circumstances? 01:11:08.250 --> 01:11:13.350 align:middle line:90% 01:11:13.350 --> 01:11:16.750 align:middle line:84% What does firm 2 want firm 1 to do? 01:11:16.750 --> 01:11:19.420 align:middle line:90% Produce a lot or a little? 01:11:19.420 --> 01:11:21.170 align:middle line:84% You can answer or someone else can answer. 01:11:21.170 --> 01:11:21.510 align:middle line:90% Yeah? 01:11:21.510 --> 01:11:24.070 align:middle line:84% AUDIENCE: Well, I guess firm 2 would want firm 1 to produce 01:11:24.070 --> 01:11:25.410 align:middle line:90% 1 minus C over 3. 01:11:25.410 --> 01:11:25.910 align:middle line:90% But-- 01:11:25.910 --> 01:11:27.150 align:middle line:84% IAN BALL: Or in general, they actually 01:11:27.150 --> 01:11:28.370 align:middle line:84% might want them to produce either even less. 01:11:28.370 --> 01:11:30.370 align:middle line:84% But in general, they want them to produce not very much. 01:11:30.370 --> 01:11:32.310 align:middle line:84% So when would they threaten-- when would they 01:11:32.310 --> 01:11:33.018 align:middle line:90% flood the market? 01:11:33.018 --> 01:11:33.830 align:middle line:90% Under what-- 01:11:33.830 --> 01:11:34.670 align:middle line:84% AUDIENCE: When they produce too much. 01:11:34.670 --> 01:11:36.212 align:middle line:84% IAN BALL: When they produce too much. 01:11:36.212 --> 01:11:38.430 align:middle line:84% So what firm 2 might say is, look, you better 01:11:38.430 --> 01:11:43.130 align:middle line:84% keep your production level low because if you produce too much, 01:11:43.130 --> 01:11:44.490 align:middle line:90% I'm going to flood the market. 01:11:44.490 --> 01:11:46.510 align:middle line:84% So firm 2 might threaten to flood the market 01:11:46.510 --> 01:11:50.650 align:middle line:90% if firm 1 produces too much. 01:11:50.650 --> 01:11:55.350 align:middle line:90% 01:11:55.350 --> 01:11:58.790 align:middle line:84% And this actually gets into dynamics that we'll study later. 01:11:58.790 --> 01:12:01.290 align:middle line:84% Oil cartels essentially function exactly in this way. 01:12:01.290 --> 01:12:03.450 align:middle line:84% I might say, you need to keep production low, 01:12:03.450 --> 01:12:04.750 align:middle line:84% and if you don't, I'm going to punish you 01:12:04.750 --> 01:12:05.880 align:middle line:90% by flooding the market Yes? 01:12:05.880 --> 01:12:08.130 align:middle line:84% AUDIENCE: How would this be a credible threat, though? 01:12:08.130 --> 01:12:10.580 align:middle line:84% Because firms 2 loses money by flooding. 01:12:10.580 --> 01:12:12.580 align:middle line:84% IAN BALL: So this will not be a credible threat. 01:12:12.580 --> 01:12:15.403 align:middle line:84% That's why it didn't survive our backward induction procedure. 01:12:15.403 --> 01:12:16.820 align:middle line:84% But it will be a Nash equilibrium. 01:12:16.820 --> 01:12:19.400 align:middle line:84% So what I want to illustrate is how a Nash equilibrium can 01:12:19.400 --> 01:12:21.280 align:middle line:90% involve non-credible threats. 01:12:21.280 --> 01:12:24.740 align:middle line:84% Backward induction rules out those non-credible threats, 01:12:24.740 --> 01:12:26.360 align:middle line:84% but Nash equilibrium can allow them 01:12:26.360 --> 01:12:28.400 align:middle line:90% if we design them carefully. 01:12:28.400 --> 01:12:29.100 align:middle line:90% Other question? 01:12:29.100 --> 01:12:29.600 align:middle line:90% Yeah? 01:12:29.600 --> 01:12:32.440 align:middle line:84% AUDIENCE: If you remove some of the simplifying assumptions 01:12:32.440 --> 01:12:34.920 align:middle line:84% we make for this, can you make it a credible threat 01:12:34.920 --> 01:12:38.080 align:middle line:84% with things like infrastructure and capital investments 01:12:38.080 --> 01:12:39.397 align:middle line:90% in production capacity? 01:12:39.397 --> 01:12:41.480 align:middle line:84% IAN BALL: I think the best way to make it credible 01:12:41.480 --> 01:12:42.820 align:middle line:84% would be to say it's a repeated game, 01:12:42.820 --> 01:12:44.080 align:middle line:84% and that's what we'll study later on. 01:12:44.080 --> 01:12:45.640 align:middle line:84% We're assuming it's a one-off thing. 01:12:45.640 --> 01:12:48.900 align:middle line:84% But really what might happen is, every day we come to market, 01:12:48.900 --> 01:12:51.400 align:middle line:84% and I might say, well, if day after day you keep doing this, 01:12:51.400 --> 01:12:52.483 align:middle line:90% then I'm going to do this. 01:12:52.483 --> 01:12:54.400 align:middle line:84% And then we can get these complicated rewards 01:12:54.400 --> 01:12:55.500 align:middle line:90% and punishments over time. 01:12:55.500 --> 01:12:57.080 align:middle line:84% And that's going to be a big focus of the second part 01:12:57.080 --> 01:12:58.000 align:middle line:90% of our course. 01:12:58.000 --> 01:12:59.340 align:middle line:84% So that's a good way to make it credible. 01:12:59.340 --> 01:12:59.640 align:middle line:90% Yes? 01:12:59.640 --> 01:13:01.420 align:middle line:84% AUDIENCE: If it were a Nash equilibrium-- 01:13:01.420 --> 01:13:02.860 align:middle line:90% so this is a Nash equilibrium. 01:13:02.860 --> 01:13:04.920 align:middle line:84% Is it the idea that firm 1, knowing 01:13:04.920 --> 01:13:07.220 align:middle line:84% firm 2 can threaten the market, will deviate? 01:13:07.220 --> 01:13:08.720 align:middle line:84% IAN BALL: Exactly, so let's actually 01:13:08.720 --> 01:13:10.000 align:middle line:84% construct our Nash equilibrium because we 01:13:10.000 --> 01:13:11.220 align:middle line:90% haven't constructed one yet. 01:13:11.220 --> 01:13:18.760 align:middle line:84% So let's fix some number, Q1, which is, let's say, 01:13:18.760 --> 01:13:20.080 align:middle line:90% really small. 01:13:20.080 --> 01:13:26.980 align:middle line:84% So let's say-- it doesn't really matter exactly, but we do-- 01:13:26.980 --> 01:13:32.440 align:middle line:90% 01:13:32.440 --> 01:13:34.560 align:middle line:84% yeah, let's say 1 minus C over 4. 01:13:34.560 --> 01:13:36.640 align:middle line:90% It doesn't matter. 01:13:36.640 --> 01:13:41.060 align:middle line:84% And now what is Q2 star going to be? 01:13:41.060 --> 01:13:50.080 align:middle line:90% 01:13:50.080 --> 01:13:53.883 align:middle line:84% Well, let's use our idea that-- there's 01:13:53.883 --> 01:13:55.300 align:middle line:84% a few different ways we can do it, 01:13:55.300 --> 01:13:56.280 align:middle line:84% but let's use an idea that they're 01:13:56.280 --> 01:13:58.405 align:middle line:84% going to threaten to flood the market if production 01:13:58.405 --> 01:14:00.280 align:middle line:90% is too high. 01:14:00.280 --> 01:14:02.280 align:middle line:90% So let's look at two cases. 01:14:02.280 --> 01:14:07.200 align:middle line:84% If Q1 is less than-- let me call this Q1 bar. 01:14:07.200 --> 01:14:12.330 align:middle line:84% If Q1 is less than Q1 bar and if Q1 is greater than or-- 01:14:12.330 --> 01:14:16.010 align:middle line:84% less than or equal to and greater than Q1. 01:14:16.010 --> 01:14:18.890 align:middle line:84% So if Q1 is greater than Q1 bar, firm 2 01:14:18.890 --> 01:14:20.170 align:middle line:90% is going to flood the market. 01:14:20.170 --> 01:14:21.910 align:middle line:84% Again, the exact amount doesn't matter. 01:14:21.910 --> 01:14:23.030 align:middle line:90% Let's say 100. 01:14:23.030 --> 01:14:23.790 align:middle line:90% It doesn't matter. 01:14:23.790 --> 01:14:27.210 align:middle line:90% 01:14:27.210 --> 01:14:29.625 align:middle line:84% What if Q1 is less than or equal to Q1 bar? 01:14:29.625 --> 01:14:31.250 align:middle line:84% We actually have to be a little careful 01:14:31.250 --> 01:14:33.042 align:middle line:84% to make sure it's still a Nash equilibrium. 01:14:33.042 --> 01:14:40.610 align:middle line:84% So let's follow our idea here, and let's make this 1 minus Q1 01:14:40.610 --> 01:14:45.490 align:middle line:90% minus C over 2. 01:14:45.490 --> 01:14:46.420 align:middle line:90% There's many examples. 01:14:46.420 --> 01:14:48.670 align:middle line:84% And we could figure out exactly how big Q1 bar can go. 01:14:48.670 --> 01:14:49.990 align:middle line:84% But I think this one should work. 01:14:49.990 --> 01:14:51.948 align:middle line:84% And let's just see that we understand that this 01:14:51.948 --> 01:14:53.890 align:middle line:90% is a Nash equilibrium. 01:14:53.890 --> 01:14:58.910 align:middle line:84% So given Q2's strategy, why is this the optimal choice of Q1? 01:14:58.910 --> 01:15:00.810 align:middle line:90% Can anyone explain in words? 01:15:00.810 --> 01:15:03.150 align:middle line:84% I mean, we see the math, but can we explain it in words? 01:15:03.150 --> 01:15:06.930 align:middle line:90% 01:15:06.930 --> 01:15:09.490 align:middle line:84% Well, why don't I want to produce more than this? 01:15:09.490 --> 01:15:10.070 align:middle line:90% Yeah? 01:15:10.070 --> 01:15:11.170 align:middle line:84% AUDIENCE: That's the most you can 01:15:11.170 --> 01:15:12.910 align:middle line:84% produce without the market getting flooded. 01:15:12.910 --> 01:15:13.660 align:middle line:90% IAN BALL: Exactly. 01:15:13.660 --> 01:15:16.630 align:middle line:84% So if I produce-- it's the most I can produce without flooding. 01:15:16.630 --> 01:15:19.090 align:middle line:84% And you can check that, when the market's not flooded, 01:15:19.090 --> 01:15:21.010 align:middle line:84% my payoff is going to be increasing 01:15:21.010 --> 01:15:22.230 align:middle line:90% at least until this point. 01:15:22.230 --> 01:15:25.550 align:middle line:84% So if I produce less than this, the market doesn't get flooded. 01:15:25.550 --> 01:15:27.770 align:middle line:84% But I'm producing so little, I'm not doing very well. 01:15:27.770 --> 01:15:29.272 align:middle line:84% If I produce more than this, that 01:15:29.272 --> 01:15:31.230 align:middle line:84% would be good if the market didn't get flooded. 01:15:31.230 --> 01:15:33.750 align:middle line:84% But if I produce more than this, the market gets flooded, 01:15:33.750 --> 01:15:35.030 align:middle line:90% so I don't want to do it. 01:15:35.030 --> 01:15:39.770 align:middle line:84% So player 1 is behaving optimal given firm 2's strategy. 01:15:39.770 --> 01:15:42.450 align:middle line:84% How can we see that firm 2 is behaving optimally given firm 01:15:42.450 --> 01:15:43.070 align:middle line:90% 1's strategy? 01:15:43.070 --> 01:15:45.090 align:middle line:84% What's the one-- the only one aspect 01:15:45.090 --> 01:15:46.570 align:middle line:84% of this strategy that's important 01:15:46.570 --> 01:15:50.890 align:middle line:84% if we want to check that it's a Nash equilibrium? 01:15:50.890 --> 01:15:53.730 align:middle line:84% What's important is, when firm 1 chooses 01:15:53.730 --> 01:16:00.010 align:middle line:84% Q1 bar, which is in this case-- firm 2 responds optimally 01:16:00.010 --> 01:16:02.537 align:middle line:84% because that's the only thing that affects firm 2's payoff, 01:16:02.537 --> 01:16:04.870 align:middle line:84% what they do in the contingency that's actually reached. 01:16:04.870 --> 01:16:07.620 align:middle line:84% So you can check that, given that firm 1 is producing 01:16:07.620 --> 01:16:11.260 align:middle line:84% this amount, firm 2 is going to produce 1 minus Q1 bar minus C 01:16:11.260 --> 01:16:11.880 align:middle line:90% over 2. 01:16:11.880 --> 01:16:14.180 align:middle line:84% And it turns out that's optimal for firm 2 given firm 01:16:14.180 --> 01:16:15.120 align:middle line:90% 1's behavior. 01:16:15.120 --> 01:16:16.820 align:middle line:84% So this will be a Nash equilibrium. 01:16:16.820 --> 01:16:20.840 align:middle line:84% But it doesn't survive backward induction. 01:16:20.840 --> 01:16:24.720 align:middle line:84% What is a contingency at which firm 2 is behaving suboptimally? 01:16:24.720 --> 01:16:29.220 align:middle line:90% 01:16:29.220 --> 01:16:31.540 align:middle line:84% Well, really anything up here-- actually, 01:16:31.540 --> 01:16:32.960 align:middle line:84% not if Q1 is really, really high. 01:16:32.960 --> 01:16:39.140 align:middle line:84% But if Q1 is anywhere between Q1 bar and 1 minus C, 01:16:39.140 --> 01:16:42.660 align:middle line:84% firm 2 could respond and make positive profits, 01:16:42.660 --> 01:16:44.600 align:middle line:84% but they're instead flooding the market, 01:16:44.600 --> 01:16:47.620 align:middle line:84% causing the price to drop to 0, and making profits of 0. 01:16:47.620 --> 01:16:51.360 align:middle line:84% So at unreached contingencies, firm 2 is behaving suboptimally. 01:16:51.360 --> 01:16:53.560 align:middle line:84% That's why this doesn't survive backward induction. 01:16:53.560 --> 01:16:56.318 align:middle line:84% But at the reached contingency, they are behaving optimally. 01:16:56.318 --> 01:16:58.360 align:middle line:84% And that's why it constitutes a Nash equilibrium. 01:16:58.360 --> 01:17:00.527 align:middle line:84% And let me stop there, and I'll see you on Thursday. 01:17:00.527 --> 01:17:02.330 align:middle line:90% Thank you. 01:17:02.330 --> 01:17:05.000 align:middle line:90%