WEBVTT 00:00:00.000 --> 00:00:01.500 align:middle line:90% [SQUEAKING] 00:00:01.500 --> 00:00:04.000 align:middle line:90% [RUSTLING] 00:00:04.000 --> 00:00:07.000 align:middle line:90% [CLICKING] 00:00:07.000 --> 00:00:10.300 align:middle line:90% 00:00:10.300 --> 00:00:13.720 align:middle line:84% IAN BELL: All right, let's get started. 00:00:13.720 --> 00:00:16.760 align:middle line:84% So today we're going to discuss subgame perfect 00:00:16.760 --> 00:00:17.900 align:middle line:90% Nash equilibrium. 00:00:17.900 --> 00:00:29.040 align:middle line:90% 00:00:29.040 --> 00:00:31.880 align:middle line:84% And I think the best way to motivate this 00:00:31.880 --> 00:00:34.760 align:middle line:90% is to start with an example. 00:00:34.760 --> 00:00:41.320 align:middle line:84% So let's start with a very simple example of a game. 00:00:41.320 --> 00:00:43.060 align:middle line:90% Let's draw it like this. 00:00:43.060 --> 00:00:48.280 align:middle line:90% 00:00:48.280 --> 00:00:50.640 align:middle line:84% The story is that we have two players. 00:00:50.640 --> 00:00:54.342 align:middle line:84% And the first player can choose whether to exit or enter, 00:00:54.342 --> 00:00:55.300 align:middle line:90% let's say, some market. 00:00:55.300 --> 00:00:56.760 align:middle line:90% We can think of them as firms. 00:00:56.760 --> 00:00:59.770 align:middle line:84% If they exit, then both players just get a payoff of 0. 00:00:59.770 --> 00:01:03.330 align:middle line:84% But if they enter, then the other firm, 00:01:03.330 --> 00:01:06.410 align:middle line:84% too, can choose whether to accommodate this new entrant 00:01:06.410 --> 00:01:07.130 align:middle line:90% or to fight. 00:01:07.130 --> 00:01:16.074 align:middle line:90% 00:01:16.074 --> 00:01:21.650 align:middle line:84% So if player 1 enters, player 2 chooses accommodate or fight. 00:01:21.650 --> 00:01:23.182 align:middle line:84% If they accommodate the new entrant, 00:01:23.182 --> 00:01:24.390 align:middle line:90% that's good for both players. 00:01:24.390 --> 00:01:25.530 align:middle line:90% They both get one. 00:01:25.530 --> 00:01:27.510 align:middle line:84% If they fight against the new entrant, 00:01:27.510 --> 00:01:28.790 align:middle line:90% that's bad for both players. 00:01:28.790 --> 00:01:31.850 align:middle line:90% They both get negative 1. 00:01:31.850 --> 00:01:37.030 align:middle line:84% So first, let's just represent this in strategic form. 00:01:37.030 --> 00:01:38.550 align:middle line:90% This is a bit of a review here. 00:01:38.550 --> 00:01:42.490 align:middle line:90% 00:01:42.490 --> 00:01:45.010 align:middle line:84% So if we represent this in strategic form, 00:01:45.010 --> 00:01:48.330 align:middle line:84% player 1 has two strategies, N and X. 00:01:48.330 --> 00:01:51.850 align:middle line:84% And player 2 has two strategies, A and F, 00:01:51.850 --> 00:01:53.170 align:middle line:90% and we can put them here. 00:01:53.170 --> 00:01:55.290 align:middle line:84% And then we just need to fill in the payoffs. 00:01:55.290 --> 00:01:58.600 align:middle line:84% Well, if player 1 does not enter, if they exit, 00:01:58.600 --> 00:02:05.140 align:middle line:84% then both players get a payoff of 0, so we have 0, 0 0, 0. 00:02:05.140 --> 00:02:08.820 align:middle line:84% If player 1 enters, then it depends what player 2 does. 00:02:08.820 --> 00:02:11.640 align:middle line:84% And if player 1 accommodates, we get 1,1. 00:02:11.640 --> 00:02:16.060 align:middle line:84% And if player 1 fights, we get negative 1, negative 1. 00:02:16.060 --> 00:02:19.140 align:middle line:84% So now as usual, let's try to solve for the Nash 00:02:19.140 --> 00:02:21.460 align:middle line:90% equilibria of this game. 00:02:21.460 --> 00:02:25.220 align:middle line:84% And I'll say, given, if player 2 accommodates, 00:02:25.220 --> 00:02:27.940 align:middle line:90% then I'm better off entering. 00:02:27.940 --> 00:02:31.780 align:middle line:84% If player 2 fights, then I'm better off exiting. 00:02:31.780 --> 00:02:38.000 align:middle line:84% And then if we do it the other way, we're going to get this. 00:02:38.000 --> 00:02:40.820 align:middle line:90% 00:02:40.820 --> 00:02:43.960 align:middle line:84% So here we see that we get two Nash equilibria. 00:02:43.960 --> 00:02:49.300 align:middle line:90% 00:02:49.300 --> 00:02:58.590 align:middle line:84% Namely, our two Nash equilibria are NA and XF. 00:02:58.590 --> 00:03:00.790 align:middle line:90% So what's the story here? 00:03:00.790 --> 00:03:04.670 align:middle line:84% One Nash equilibrium is player 1 enters, 00:03:04.670 --> 00:03:06.390 align:middle line:84% anticipating that player 2 is going 00:03:06.390 --> 00:03:09.230 align:middle line:90% to accommodate their entry. 00:03:09.230 --> 00:03:12.310 align:middle line:84% The other Nash equilibrium is XF. 00:03:12.310 --> 00:03:14.330 align:middle line:90% Player 1 exits. 00:03:14.330 --> 00:03:16.510 align:middle line:84% They don't enter because they anticipate 00:03:16.510 --> 00:03:18.830 align:middle line:90% that player 2 will fight. 00:03:18.830 --> 00:03:20.710 align:middle line:84% And both of these are Nash equilibria. 00:03:20.710 --> 00:03:22.830 align:middle line:84% Both players are playing optimally given the way 00:03:22.830 --> 00:03:25.070 align:middle line:90% the other player is behaving. 00:03:25.070 --> 00:03:29.310 align:middle line:84% But we argued last week, that in games like this, one of these 00:03:29.310 --> 00:03:31.250 align:middle line:84% is more compelling than the other. 00:03:31.250 --> 00:03:33.390 align:middle line:84% Do you find a problem with any-- does 00:03:33.390 --> 00:03:36.750 align:middle line:84% any of these Nash equilibria involve a non-credible threat 00:03:36.750 --> 00:03:39.350 align:middle line:84% that maybe makes it less compelling than the other Nash 00:03:39.350 --> 00:03:40.130 align:middle line:90% equilibrium? 00:03:40.130 --> 00:03:46.150 align:middle line:90% 00:03:46.150 --> 00:03:47.090 align:middle line:90% Yeah? 00:03:47.090 --> 00:03:50.630 align:middle line:84% AUDIENCE: Is the player 2 will fight [INAUDIBLE]? 00:03:50.630 --> 00:03:53.090 align:middle line:84% IAN BELL: Right, so this is a Nash equilibrium, 00:03:53.090 --> 00:03:55.070 align:middle line:84% but this involves a non-credible threat. 00:03:55.070 --> 00:04:00.650 align:middle line:90% 00:04:00.650 --> 00:04:01.530 align:middle line:90% Why? 00:04:01.530 --> 00:04:05.430 align:middle line:84% Well, in this equilibrium, player 1 is exiting. 00:04:05.430 --> 00:04:07.770 align:middle line:84% They're choosing not to enter because they anticipate 00:04:07.770 --> 00:04:09.535 align:middle line:90% that player 2 will fight them. 00:04:09.535 --> 00:04:10.910 align:middle line:84% But that's not a credible threat. 00:04:10.910 --> 00:04:13.410 align:middle line:84% If they actually were to enter, then player 2 00:04:13.410 --> 00:04:16.690 align:middle line:84% would find it optimal to accommodate, not to fight. 00:04:16.690 --> 00:04:21.170 align:middle line:84% So this second Nash equilibrium, it 00:04:21.170 --> 00:04:22.510 align:middle line:90% involves a non-credible threat. 00:04:22.510 --> 00:04:24.635 align:middle line:84% And if we want to be a little more formal about it, 00:04:24.635 --> 00:04:26.470 align:middle line:84% this is going to violate backward induction. 00:04:26.470 --> 00:04:34.570 align:middle line:90% 00:04:34.570 --> 00:04:37.410 align:middle line:84% We can easily apply backward induction to this game. 00:04:37.410 --> 00:04:39.170 align:middle line:90% We start with player 2. 00:04:39.170 --> 00:04:42.370 align:middle line:84% They'd rather accommodate than fight. 00:04:42.370 --> 00:04:44.470 align:middle line:84% And given that player 2 is going to accommodate, 00:04:44.470 --> 00:04:47.250 align:middle line:90% player 1 would rather enter. 00:04:47.250 --> 00:04:50.890 align:middle line:84% And indeed, backward induction gives us the other Nash 00:04:50.890 --> 00:04:54.020 align:middle line:90% equilibria of NA. 00:04:54.020 --> 00:04:56.580 align:middle line:84% So this is kind of a review of what we've done before. 00:04:56.580 --> 00:04:58.880 align:middle line:84% This is a game of perfect information. 00:04:58.880 --> 00:05:00.800 align:middle line:84% We were able to apply backward induction, 00:05:00.800 --> 00:05:03.340 align:middle line:84% and that was able to identify one Nash 00:05:03.340 --> 00:05:07.060 align:middle line:84% equilibrium as being compelling prediction in the game. 00:05:07.060 --> 00:05:08.740 align:middle line:84% And the other Nash equilibrium is 00:05:08.740 --> 00:05:11.680 align:middle line:84% being less compelling because it involved a non-credible threat, 00:05:11.680 --> 00:05:15.340 align:middle line:84% and more formally, it violated backward induction. 00:05:15.340 --> 00:05:18.080 align:middle line:84% But now let's modify this example slightly. 00:05:18.080 --> 00:05:23.220 align:middle line:90% 00:05:23.220 --> 00:05:28.780 align:middle line:84% Let's suppose that now player 1 chooses 00:05:28.780 --> 00:05:33.380 align:middle line:90% exit or enter as before, 0, 0. 00:05:33.380 --> 00:05:36.500 align:middle line:84% But now if player 1 chooses to enter, 00:05:36.500 --> 00:05:42.660 align:middle line:84% then player 1 and player 2 will play a simultaneous move game. 00:05:42.660 --> 00:05:45.620 align:middle line:84% And we can represent that by saying, well, player 2 is 00:05:45.620 --> 00:05:46.900 align:middle line:90% going to make a move. 00:05:46.900 --> 00:05:50.020 align:middle line:84% They're going to choose, say, accommodate or fight. 00:05:50.020 --> 00:05:53.910 align:middle line:84% But then player 1 is also making a move. 00:05:53.910 --> 00:05:58.270 align:middle line:84% So we'll put these in the same information set. 00:05:58.270 --> 00:06:02.810 align:middle line:84% And we'll say player 1 chooses, I think, top or bottom. 00:06:02.810 --> 00:06:08.150 align:middle line:90% 00:06:08.150 --> 00:06:12.070 align:middle line:84% So just so we understand what's going on here. 00:06:12.070 --> 00:06:16.670 align:middle line:84% Now, as before, player 1 first chooses whether to exit or enter 00:06:16.670 --> 00:06:17.950 align:middle line:90% the market. 00:06:17.950 --> 00:06:20.990 align:middle line:84% But if they enter the market, then player 1 and player 2 00:06:20.990 --> 00:06:23.510 align:middle line:84% play a simultaneous move game, where 00:06:23.510 --> 00:06:26.350 align:middle line:84% player 2 chooses between accommodate and fight, 00:06:26.350 --> 00:06:30.010 align:middle line:84% and player 1 chooses between the interpretation is so important, 00:06:30.010 --> 00:06:31.670 align:middle line:90% but T and B. 00:06:31.670 --> 00:06:34.030 align:middle line:84% And notice here that I've written this 00:06:34.030 --> 00:06:37.190 align:middle line:84% as if player 2 moves before player 1. 00:06:37.190 --> 00:06:40.550 align:middle line:84% But whenever we have simultaneous moves, 00:06:40.550 --> 00:06:43.090 align:middle line:84% and we want to capture that in the extensive form, 00:06:43.090 --> 00:06:46.950 align:middle line:84% the way that we do that is we can arbitrarily pick a sequence 00:06:46.950 --> 00:06:49.120 align:middle line:84% and then just assume that player 1 does not 00:06:49.120 --> 00:06:50.940 align:middle line:90% observe player 2's move. 00:06:50.940 --> 00:06:53.880 align:middle line:84% So if we had switched the order here, and said, 00:06:53.880 --> 00:06:56.580 align:middle line:84% player 1 moves, and then player 2 moves, 00:06:56.580 --> 00:06:58.460 align:middle line:84% not having observed what player 1 did, 00:06:58.460 --> 00:07:00.760 align:middle line:84% that would be equivalent to player 2 moving first 00:07:00.760 --> 00:07:02.520 align:middle line:84% and then player 1 moving not having 00:07:02.520 --> 00:07:04.840 align:middle line:90% observed what player 2 did. 00:07:04.840 --> 00:07:08.040 align:middle line:84% So now let me just fill in the payoffs here. 00:07:08.040 --> 00:07:12.400 align:middle line:90% Make sure I get this right. 00:07:12.400 --> 00:07:17.940 align:middle line:84% So we're going to have 2, 2, negative 1, negative 1. 00:07:17.940 --> 00:07:24.320 align:middle line:90% 00:07:24.320 --> 00:07:26.840 align:middle line:90% 2, 2 1, 1. 00:07:26.840 --> 00:07:28.440 align:middle line:90% Make sure I get this right. 00:07:28.440 --> 00:07:31.460 align:middle line:90% And then phi 2t. 00:07:31.460 --> 00:07:39.640 align:middle line:90% 00:07:39.640 --> 00:07:42.240 align:middle line:84% Great, so as before, accommodating 00:07:42.240 --> 00:07:43.560 align:middle line:90% is good for everyone. 00:07:43.560 --> 00:07:45.260 align:middle line:90% Fighting is bad for everyone. 00:07:45.260 --> 00:07:48.770 align:middle line:84% But now player 1'a choice between T and B can affect 00:07:48.770 --> 00:07:51.750 align:middle line:84% the intensity of that good or bad for the two players. 00:07:51.750 --> 00:07:55.850 align:middle line:90% 00:07:55.850 --> 00:08:00.390 align:middle line:84% So now let's go through our same analysis as before. 00:08:00.390 --> 00:08:03.070 align:middle line:84% Let's represent this in strategic form, 00:08:03.070 --> 00:08:06.090 align:middle line:84% and let's solve for the Nash equilibria of this game. 00:08:06.090 --> 00:08:09.430 align:middle line:84% So now player 1 has four strategies, right? 00:08:09.430 --> 00:08:14.210 align:middle line:84% Because player 1 chooses N or X and they also choose T or B. 00:08:14.210 --> 00:08:17.010 align:middle line:84% Notice player 1 here only has one information set. 00:08:17.010 --> 00:08:19.370 align:middle line:84% If these were in separate information sets, then 00:08:19.370 --> 00:08:21.830 align:middle line:84% player 1 would have to move at three information sets. 00:08:21.830 --> 00:08:24.290 align:middle line:84% But here we just have one information set 00:08:24.290 --> 00:08:25.890 align:middle line:90% and another information set. 00:08:25.890 --> 00:08:34.690 align:middle line:84% So for player 1, the strategies are NT, NB-- 00:08:34.690 --> 00:08:36.557 align:middle line:84% maybe I'll write it this way, yeah, sure. 00:08:36.557 --> 00:08:37.390 align:middle line:90% I'll do it this way. 00:08:37.390 --> 00:08:41.010 align:middle line:90% It doesn't matter, XT XB. 00:08:41.010 --> 00:08:50.060 align:middle line:84% And the strategies for player 2 are just A and F. 00:08:50.060 --> 00:08:53.160 align:middle line:84% And now we need to fill in the payoffs of this matrix. 00:08:53.160 --> 00:08:55.580 align:middle line:84% Well, let's do it systematically. 00:08:55.580 --> 00:08:58.860 align:middle line:84% If player 1 exits, then we know that 00:08:58.860 --> 00:09:00.740 align:middle line:84% no matter what player 2 or player 1 00:09:00.740 --> 00:09:06.060 align:middle line:84% subsequently does, we're going to get 0, 0. 00:09:06.060 --> 00:09:12.140 align:middle line:84% So here we're going to have 0, 0 0,0 0, o and 0, 0. 00:09:12.140 --> 00:09:14.180 align:middle line:84% Because in all of these strategies, 00:09:14.180 --> 00:09:16.040 align:middle line:84% player 1 first exits, and that's just 00:09:16.040 --> 00:09:17.540 align:middle line:84% ends the game with a payoff of 0, 0. 00:09:17.540 --> 00:09:20.180 align:middle line:90% 00:09:20.180 --> 00:09:21.940 align:middle line:84% Next, let's figure out what happens 00:09:21.940 --> 00:09:26.180 align:middle line:84% under the other two strategies or the other four strategies. 00:09:26.180 --> 00:09:30.620 align:middle line:90% So first player 1 enters. 00:09:30.620 --> 00:09:34.940 align:middle line:84% If player 2 accommodates, and player 1 plays T then 00:09:34.940 --> 00:09:36.240 align:middle line:90% we're going to get 2, 2. 00:09:36.240 --> 00:09:39.100 align:middle line:90% 00:09:39.100 --> 00:09:42.420 align:middle line:84% And if player 2 accommodates, and player 1 plays B, 00:09:42.420 --> 00:09:45.320 align:middle line:90% we're going to get 1, 1. 00:09:45.320 --> 00:09:48.460 align:middle line:84% And then if we do it over here, we're going to get-- 00:09:48.460 --> 00:09:56.998 align:middle line:90% 00:09:56.998 --> 00:09:57.540 align:middle line:90% I'll do that. 00:09:57.540 --> 00:10:03.280 align:middle line:84% Yeah, so all I've done is filled in the payoffs in this matrix. 00:10:03.280 --> 00:10:06.660 align:middle line:84% So now let's go through our standard procedure, 00:10:06.660 --> 00:10:09.500 align:middle line:84% and let's try to find the Nash equilibria of this game 00:10:09.500 --> 00:10:11.400 align:middle line:84% now that we have it in strategic form. 00:10:11.400 --> 00:10:14.920 align:middle line:84% We're going to say, if player 1 is going to accommodate, 00:10:14.920 --> 00:10:19.680 align:middle line:84% then player 2 finds it optimal to do this. 00:10:19.680 --> 00:10:23.920 align:middle line:84% If player 2 is going to fight, then player 1 00:10:23.920 --> 00:10:26.160 align:middle line:90% finds each of these optimal. 00:10:26.160 --> 00:10:28.080 align:middle line:84% And now we're going to go row by row. 00:10:28.080 --> 00:10:31.560 align:middle line:84% If player 1 is playing NT, then player 2 00:10:31.560 --> 00:10:33.920 align:middle line:84% would like to accommodate for an NB. 00:10:33.920 --> 00:10:35.860 align:middle line:90% Then we're here for here. 00:10:35.860 --> 00:10:42.770 align:middle line:90% 00:10:42.770 --> 00:10:45.670 align:middle line:84% So now we get three Nash equilibria. 00:10:45.670 --> 00:10:49.010 align:middle line:90% 00:10:49.010 --> 00:10:50.190 align:middle line:90% Let's write these down. 00:10:50.190 --> 00:10:55.490 align:middle line:90% 00:10:55.490 --> 00:11:16.450 align:middle line:84% We get NT, A. We get XT, F. We get XB, F. 00:11:16.450 --> 00:11:21.770 align:middle line:84% All right, so here, player 1 enters 00:11:21.770 --> 00:11:24.530 align:middle line:84% the market anticipating that player 2 is 00:11:24.530 --> 00:11:26.130 align:middle line:90% going to accommodate. 00:11:26.130 --> 00:11:28.850 align:middle line:84% And then they follow this up by playing 00:11:28.850 --> 00:11:34.290 align:middle line:84% T. In both of these equilibria, player 1 exits the market 00:11:34.290 --> 00:11:37.090 align:middle line:84% because they anticipate that player 2 is going to fight. 00:11:37.090 --> 00:11:38.790 align:middle line:84% And then it doesn't actually matter. 00:11:38.790 --> 00:11:42.260 align:middle line:84% They could do T or B, and we have an equilibrium either way. 00:11:42.260 --> 00:11:44.380 align:middle line:84% But now let's ask the same question as before. 00:11:44.380 --> 00:11:50.020 align:middle line:84% Do any of these equilibria seem more compelling to you 00:11:50.020 --> 00:11:54.540 align:middle line:90% than any others? 00:11:54.540 --> 00:11:57.560 align:middle line:84% Do any of these involve non-credible threats? 00:11:57.560 --> 00:12:08.500 align:middle line:90% 00:12:08.500 --> 00:12:09.280 align:middle line:90% Any thoughts? 00:12:09.280 --> 00:12:14.300 align:middle line:90% 00:12:14.300 --> 00:12:14.860 align:middle line:90% Yes? 00:12:14.860 --> 00:12:16.320 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:12:16.320 --> 00:12:18.110 align:middle line:90% 00:12:18.110 --> 00:12:22.660 align:middle line:84% IAN BELL: Right, so these two seem to involve 00:12:22.660 --> 00:12:23.840 align:middle line:90% non-credible threats. 00:12:23.840 --> 00:12:24.440 align:middle line:90% Why? 00:12:24.440 --> 00:12:30.380 align:middle line:84% Well, player 2 is fighting when player 1 enters, 00:12:30.380 --> 00:12:33.820 align:middle line:84% but fighting just makes both players worse off. 00:12:33.820 --> 00:12:36.520 align:middle line:84% So we don't think that if player 1 actually enters, 00:12:36.520 --> 00:12:39.670 align:middle line:84% player 2 is going to be willing to follow through and fight. 00:12:39.670 --> 00:12:43.350 align:middle line:84% So intuitively, these involve non-credible threats. 00:12:43.350 --> 00:12:46.990 align:middle line:84% But formally, we can't apply backward induction to this game. 00:12:46.990 --> 00:12:49.270 align:middle line:84% Why can we not apply backward induction? 00:12:49.270 --> 00:12:50.310 align:middle line:90% Yeah? 00:12:50.310 --> 00:12:50.830 align:middle line:84% AUDIENCE: No perfect information. 00:12:50.830 --> 00:12:52.872 align:middle line:84% IAN BELL: It's not a game of perfect information. 00:12:52.872 --> 00:12:56.330 align:middle line:84% So the issue here is conceptually, 00:12:56.330 --> 00:12:58.880 align:middle line:84% we think that we'd like to select this equilibrium 00:12:58.880 --> 00:13:00.630 align:middle line:84% because we think that these equilibria are 00:13:00.630 --> 00:13:03.110 align:middle line:84% bad for the same reasons as in our first example 00:13:03.110 --> 00:13:05.830 align:middle line:84% up here because they involve a threat that will not actually 00:13:05.830 --> 00:13:08.270 align:middle line:84% be optimal for the player to carry out. 00:13:08.270 --> 00:13:11.270 align:middle line:84% However, we don't have the formal methodology 00:13:11.270 --> 00:13:15.110 align:middle line:84% to rule out these Nash equilibria because our existing 00:13:15.110 --> 00:13:17.270 align:middle line:84% techniques of backward induction don't apply 00:13:17.270 --> 00:13:19.870 align:middle line:84% to games of this form because this game does not 00:13:19.870 --> 00:13:21.490 align:middle line:90% have perfect information. 00:13:21.490 --> 00:13:29.750 align:middle line:84% So the problem here is that Backward Induction, BI, does not 00:13:29.750 --> 00:13:36.290 align:middle line:84% apply because this game does not have perfect information. 00:13:36.290 --> 00:13:39.080 align:middle line:90% 00:13:39.080 --> 00:13:42.140 align:middle line:84% Maybe we'll say, because this game has imperfect information. 00:13:42.140 --> 00:13:51.120 align:middle line:90% 00:13:51.120 --> 00:13:52.980 align:middle line:84% What is the imperfect information here? 00:13:52.980 --> 00:13:55.680 align:middle line:84% It's that when player 1 moves down here, 00:13:55.680 --> 00:13:59.360 align:middle line:84% player 1 has not observed what player 2's action is here. 00:13:59.360 --> 00:14:01.120 align:middle line:84% So they don't have perfect information 00:14:01.120 --> 00:14:03.160 align:middle line:90% about the previous moves. 00:14:03.160 --> 00:14:05.840 align:middle line:84% And indeed, that makes it difficult to apply backward 00:14:05.840 --> 00:14:10.040 align:middle line:84% induction because when we tried to apply backward induction, 00:14:10.040 --> 00:14:12.520 align:middle line:84% we wanted to say, well, how should a player move 00:14:12.520 --> 00:14:13.822 align:middle line:90% at each of these nodes? 00:14:13.822 --> 00:14:15.280 align:middle line:84% But if you're at an information set 00:14:15.280 --> 00:14:17.540 align:middle line:84% with multiple nodes in the information set, 00:14:17.540 --> 00:14:21.200 align:middle line:84% it's not clear how to carry out this procedure. 00:14:21.200 --> 00:14:27.640 align:middle line:84% So the plan for today, is to introduce a new solution 00:14:27.640 --> 00:14:30.720 align:middle line:84% concept of SPNE, which I wrote up here, 00:14:30.720 --> 00:14:33.120 align:middle line:84% Subgame Perfect Nash Equilibrium, 00:14:33.120 --> 00:14:36.710 align:middle line:84% that is going to capture-- so we'll say this in words. 00:14:36.710 --> 00:14:49.410 align:middle line:84% It will capture the idea of credibility, 00:14:49.410 --> 00:14:52.490 align:middle line:84% but it will apply to a larger class of games 00:14:52.490 --> 00:14:54.970 align:middle line:90% than backward induction did. 00:14:54.970 --> 00:15:02.230 align:middle line:84% But it-- to a larger class of games. 00:15:02.230 --> 00:15:07.050 align:middle line:90% 00:15:07.050 --> 00:15:16.770 align:middle line:84% Namely, it will apply to games that have imperfect information, 00:15:16.770 --> 00:15:24.250 align:middle line:84% and also, it will allow for infinite games, that is, games 00:15:24.250 --> 00:15:26.730 align:middle line:84% where we keep making moves over and over again, 00:15:26.730 --> 00:15:27.870 align:middle line:90% and there's no clear point. 00:15:27.870 --> 00:15:31.550 align:middle line:84% Remember, we argued last week, when we looked at negotiation, 00:15:31.550 --> 00:15:33.790 align:middle line:84% that there was this artificial aspect 00:15:33.790 --> 00:15:35.630 align:middle line:84% to our model of negotiation, that there 00:15:35.630 --> 00:15:37.370 align:middle line:84% was some time at which the world ended, 00:15:37.370 --> 00:15:39.703 align:middle line:84% and there couldn't be any negotiation beyond that point. 00:15:39.703 --> 00:15:41.570 align:middle line:84% We had to impose that artificial deadline 00:15:41.570 --> 00:15:44.070 align:middle line:84% because we didn't have the tools to analyze games that could 00:15:44.070 --> 00:15:45.910 align:middle line:90% potentially go on forever. 00:15:45.910 --> 00:15:48.790 align:middle line:84% And now with subgame perfection, we'll be able to do this. 00:15:48.790 --> 00:15:49.990 align:middle line:90% I want to be clear here. 00:15:49.990 --> 00:15:53.310 align:middle line:84% Of course, it also applies to games of perfect information 00:15:53.310 --> 00:15:54.650 align:middle line:90% and finite games. 00:15:54.650 --> 00:15:56.390 align:middle line:84% But in addition to those games, it 00:15:56.390 --> 00:15:58.970 align:middle line:84% applies to games of imperfect information and infinite games. 00:15:58.970 --> 00:16:01.470 align:middle line:90% So maybe I'll say, also. 00:16:01.470 --> 00:16:03.310 align:middle line:84% Of course, it also applies wherever 00:16:03.310 --> 00:16:04.610 align:middle line:90% backward induction applies. 00:16:04.610 --> 00:16:09.390 align:middle line:90% 00:16:09.390 --> 00:16:10.910 align:middle line:84% And maybe the last thing I'll say, 00:16:10.910 --> 00:16:16.610 align:middle line:84% is it extends backward induction. 00:16:16.610 --> 00:16:18.910 align:middle line:90% What do I mean by that? 00:16:18.910 --> 00:16:21.790 align:middle line:84% Whenever backward induction does apply, 00:16:21.790 --> 00:16:25.230 align:middle line:84% subgame perfect Nash equilibrium will give us the same answer. 00:16:25.230 --> 00:16:29.390 align:middle line:90% So i.e. 00:16:29.390 --> 00:16:33.720 align:middle line:84% it agrees with backward induction whenever 00:16:33.720 --> 00:16:35.020 align:middle line:90% backward induction applies. 00:16:35.020 --> 00:16:42.137 align:middle line:90% 00:16:42.137 --> 00:16:43.720 align:middle line:84% So we had this class of games where we 00:16:43.720 --> 00:16:45.180 align:middle line:90% could apply backward induction. 00:16:45.180 --> 00:16:46.640 align:middle line:84% Subgame perfect Nash equilibrium is 00:16:46.640 --> 00:16:49.820 align:middle line:84% going to give us the same answers on those nice games, 00:16:49.820 --> 00:16:52.560 align:middle line:84% but it's also going to give us a compelling answer in other games 00:16:52.560 --> 00:16:53.500 align:middle line:90% beyond that class. 00:16:53.500 --> 00:16:57.120 align:middle line:84% And that's what we're hoping to do today. 00:16:57.120 --> 00:17:02.640 align:middle line:84% OK, well, let's try to understand what went wrong, what 00:17:02.640 --> 00:17:06.599 align:middle line:90% was non-credible about this. 00:17:06.599 --> 00:17:09.060 align:middle line:84% And I'd encourage you one way to think of this-- 00:17:09.060 --> 00:17:12.240 align:middle line:90% 00:17:12.240 --> 00:17:15.480 align:middle line:84% well, we have the term subgame in the name subgame perfect Nash 00:17:15.480 --> 00:17:16.130 align:middle line:90% equilibrium. 00:17:16.130 --> 00:17:17.880 align:middle line:84% We're going to think of subgames, that is, 00:17:17.880 --> 00:17:20.880 align:middle line:84% smaller games that are part of the larger game. 00:17:20.880 --> 00:17:23.839 align:middle line:84% And notice that once we get to this node, 00:17:23.839 --> 00:17:28.140 align:middle line:84% we can think of the rest of this as a game in itself, 00:17:28.140 --> 00:17:33.450 align:middle line:84% as kind of a smaller subgame within the larger game. 00:17:33.450 --> 00:17:38.530 align:middle line:84% And if we think of this as a smaller subgame, 00:17:38.530 --> 00:17:43.050 align:middle line:84% the payoffs in this smaller subgame 00:17:43.050 --> 00:17:50.930 align:middle line:84% are actually going to look basically like these. 00:17:50.930 --> 00:17:52.730 align:middle line:84% So we can think of this smaller subgame 00:17:52.730 --> 00:17:57.090 align:middle line:84% as, well, player 2 is choosing between A and F, 00:17:57.090 --> 00:17:59.430 align:middle line:84% and player 1 is choosing between just T and B. 00:17:59.430 --> 00:18:04.170 align:middle line:84% So we can ignore the N. And we can compute what the payoffs are 00:18:04.170 --> 00:18:06.130 align:middle line:84% for each of those pairs of actions 00:18:06.130 --> 00:18:07.790 align:middle line:90% or strategies in this subgame. 00:18:07.790 --> 00:18:14.370 align:middle line:90% 00:18:14.370 --> 00:18:21.950 align:middle line:84% And we can see that only our combination of-- 00:18:21.950 --> 00:18:24.850 align:middle line:90% 00:18:24.850 --> 00:18:26.470 align:middle line:90% just look at this subgame. 00:18:26.470 --> 00:18:31.620 align:middle line:90% 00:18:31.620 --> 00:18:35.060 align:middle line:84% So in this equilibrium that we found compelling, 00:18:35.060 --> 00:18:40.100 align:middle line:84% in the subgame, player 1 plays T and player 2 plays A. 00:18:40.100 --> 00:18:42.125 align:middle line:84% And we can see that in this subgame, if we 00:18:42.125 --> 00:18:44.500 align:middle line:84% look at it as a single game, this will actually be a Nash 00:18:44.500 --> 00:18:46.700 align:middle line:90% equilibrium of the subgame. 00:18:46.700 --> 00:18:49.220 align:middle line:84% But if we look at how people play in this subgame in any 00:18:49.220 --> 00:18:54.820 align:middle line:84% of these strategies, TF or BF, these 00:18:54.820 --> 00:19:00.260 align:middle line:84% will not actually be Nash equilibrium of the subgame. 00:19:00.260 --> 00:19:03.100 align:middle line:84% And that's going to be kind of conceptually what 00:19:03.100 --> 00:19:05.760 align:middle line:84% violates our notion of subgame perfect Nash equilibrium. 00:19:05.760 --> 00:19:10.520 align:middle line:84% So let me try to be a little more precise about this. 00:19:10.520 --> 00:19:19.180 align:middle line:90% 00:19:19.180 --> 00:19:22.620 align:middle line:90% So what we can do-- 00:19:22.620 --> 00:19:27.730 align:middle line:84% so let's apply subgame perfect Nash equilibrium 00:19:27.730 --> 00:19:28.610 align:middle line:90% to this example. 00:19:28.610 --> 00:19:30.895 align:middle line:84% I said, we can represent this as its own subgame, 00:19:30.895 --> 00:19:32.770 align:middle line:84% but let's be a little more formal about this. 00:19:32.770 --> 00:19:35.870 align:middle line:84% We can think of this as we have player 1. 00:19:35.870 --> 00:19:39.590 align:middle line:84% They're choosing whether to exit or enter. 00:19:39.590 --> 00:19:42.190 align:middle line:84% And if they enter, we now have a subgame 00:19:42.190 --> 00:19:45.230 align:middle line:84% which is simultaneous move game, and we can just think of it 00:19:45.230 --> 00:19:47.090 align:middle line:90% as a block like this. 00:19:47.090 --> 00:19:55.790 align:middle line:90% 00:19:55.790 --> 00:20:01.090 align:middle line:84% And indeed, we get 2, 2 negative 2, negative 2. 00:20:01.090 --> 00:20:07.630 align:middle line:90% 00:20:07.630 --> 00:20:09.890 align:middle line:84% So this is not technically an extensive form game. 00:20:09.890 --> 00:20:11.570 align:middle line:84% This is kind of a heuristic derivation. 00:20:11.570 --> 00:20:14.270 align:middle line:84% But we can think of what's going on as after player 1 chooses 00:20:14.270 --> 00:20:16.770 align:middle line:84% whether to exit or enter the game, 00:20:16.770 --> 00:20:19.270 align:middle line:84% then we play this simultaneous move game. 00:20:19.270 --> 00:20:21.430 align:middle line:84% And what we would expect, and what we'd like, 00:20:21.430 --> 00:20:25.400 align:middle line:84% is that in this simultaneous move game, 00:20:25.400 --> 00:20:28.160 align:middle line:84% the players are actually playing a Nash equilibrium, 00:20:28.160 --> 00:20:30.760 align:middle line:84% so we can work within this simultaneous move game 00:20:30.760 --> 00:20:34.300 align:middle line:84% and figure out what are the Nash equilibria of this subgame. 00:20:34.300 --> 00:20:42.440 align:middle line:90% 00:20:42.440 --> 00:20:44.300 align:middle line:84% Well, we apply our standard procedure. 00:20:44.300 --> 00:20:45.660 align:middle line:90% We just underline things. 00:20:45.660 --> 00:20:50.040 align:middle line:90% 00:20:50.040 --> 00:20:58.240 align:middle line:84% And we see that only this is a Nash equilibrium. 00:20:58.240 --> 00:21:03.140 align:middle line:84% So we see that TA is the only one. 00:21:03.140 --> 00:21:06.080 align:middle line:90% 00:21:06.080 --> 00:21:10.080 align:middle line:84% So if we believe that once player 1 enters, 00:21:10.080 --> 00:21:13.680 align:middle line:84% from thereafter, the two players will play a Nash equilibrium 00:21:13.680 --> 00:21:16.080 align:middle line:84% in that subgame, then the only possibility 00:21:16.080 --> 00:21:18.400 align:middle line:90% is that thereafter they play TA. 00:21:18.400 --> 00:21:20.440 align:middle line:84% And anticipating that they're going to play TA, 00:21:20.440 --> 00:21:23.330 align:middle line:84% its optimal for player 1 to enter. 00:21:23.330 --> 00:21:26.930 align:middle line:84% And now we've exactly replicated our compelling equilibrium 00:21:26.930 --> 00:21:29.130 align:middle line:90% over here. 00:21:29.130 --> 00:21:32.215 align:middle line:84% So next I want to describe this procedure more formally. 00:21:32.215 --> 00:21:34.090 align:middle line:84% But are there any questions about the example 00:21:34.090 --> 00:21:39.730 align:middle line:84% and how we've done this so far in this reduction? 00:21:39.730 --> 00:21:42.130 align:middle line:90% No? 00:21:42.130 --> 00:21:42.670 align:middle line:90% OK. 00:21:42.670 --> 00:21:45.490 align:middle line:90% 00:21:45.490 --> 00:21:50.290 align:middle line:84% So what I argued is that in this big extensive form game, 00:21:50.290 --> 00:21:54.490 align:middle line:84% we could treat a subset of this game as a game in itself. 00:21:54.490 --> 00:21:56.340 align:middle line:90% And we call that a subgame. 00:21:56.340 --> 00:21:57.590 align:middle line:90% And that's what we want to do. 00:21:57.590 --> 00:22:00.450 align:middle line:84% So the first step to defining subgame perfect 00:22:00.450 --> 00:22:02.250 align:middle line:84% Nash equilibrium, is we first need 00:22:02.250 --> 00:22:14.010 align:middle line:84% to define subgames of an extensive form game G. 00:22:14.010 --> 00:22:17.310 align:middle line:84% And the intuition is-- intuitively it's clear. 00:22:17.310 --> 00:22:20.530 align:middle line:84% What we want is we want to look at subsets of the tree that 00:22:20.530 --> 00:22:24.390 align:middle line:84% would constitute games by themselves if we thought of them 00:22:24.390 --> 00:22:26.670 align:middle line:90% separately, all by themselves. 00:22:26.670 --> 00:22:30.230 align:middle line:84% So what a subgame is going to consist of, 00:22:30.230 --> 00:22:41.350 align:middle line:84% is it's going to consist of a initial-- so a node and all 00:22:41.350 --> 00:22:43.190 align:middle line:90% its successors. 00:22:43.190 --> 00:22:44.290 align:middle line:90% So what does that mean? 00:22:44.290 --> 00:22:46.910 align:middle line:90% 00:22:46.910 --> 00:22:48.670 align:middle line:90% Let's take this as an example. 00:22:48.670 --> 00:22:52.110 align:middle line:84% I start here at a node, and I include every node 00:22:52.110 --> 00:22:53.470 align:middle line:90% that comes after this. 00:22:53.470 --> 00:22:56.150 align:middle line:84% So that's this node, this node, all the way to the terminal 00:22:56.150 --> 00:22:59.750 align:middle line:84% nodes, this node, this node, this node, this node. 00:22:59.750 --> 00:23:01.770 align:middle line:84% But we have to have certain conditions. 00:23:01.770 --> 00:23:05.270 align:middle line:90% 00:23:05.270 --> 00:23:14.590 align:middle line:84% The initial node has to be in its own information set. 00:23:14.590 --> 00:23:18.570 align:middle line:84% Maybe I'll emphasize-- so we have a single-- 00:23:18.570 --> 00:23:21.200 align:middle line:90% 00:23:21.200 --> 00:23:25.840 align:middle line:84% maybe I'll call this the initial node to give it context. 00:23:25.840 --> 00:23:28.460 align:middle line:84% So we pick some node, which will be our initial node, 00:23:28.460 --> 00:23:30.720 align:middle line:84% and then we include all successors, all nodes 00:23:30.720 --> 00:23:32.160 align:middle line:90% that come after that node. 00:23:32.160 --> 00:23:34.080 align:middle line:84% But then we have two requirements. 00:23:34.080 --> 00:23:38.920 align:middle line:84% Our first requirement is that the initial node is 00:23:38.920 --> 00:23:40.620 align:middle line:90% in a singleton information set. 00:23:40.620 --> 00:23:50.240 align:middle line:90% 00:23:50.240 --> 00:23:54.080 align:middle line:84% And this is just to ensure that this initial node and everything 00:23:54.080 --> 00:23:56.360 align:middle line:84% that comes after it would constitute 00:23:56.360 --> 00:23:58.020 align:middle line:90% a game in its own right. 00:23:58.020 --> 00:24:00.600 align:middle line:84% Remember, when we defined an extensive form game, 00:24:00.600 --> 00:24:02.720 align:middle line:84% one requirement of extensive form game 00:24:02.720 --> 00:24:04.700 align:middle line:84% is that it starts with an initial node, 00:24:04.700 --> 00:24:07.040 align:middle line:84% and that initial node is, by itself, 00:24:07.040 --> 00:24:08.800 align:middle line:90% in its own information set. 00:24:08.800 --> 00:24:11.880 align:middle line:84% So if we want this node to serve as the initial node 00:24:11.880 --> 00:24:15.100 align:middle line:84% of our subgame, it better be a singleton information set. 00:24:15.100 --> 00:24:18.930 align:middle line:90% 00:24:18.930 --> 00:24:22.970 align:middle line:84% And then the second requirement is that the subgame 00:24:22.970 --> 00:24:30.570 align:middle line:90% doesn't break information sets. 00:24:30.570 --> 00:24:32.390 align:middle line:84% Let's say what this means in words. 00:24:32.390 --> 00:24:37.910 align:middle line:90% 00:24:37.910 --> 00:24:39.770 align:middle line:84% What I mean by that is I take a node 00:24:39.770 --> 00:24:41.930 align:middle line:90% and all of its successor nodes. 00:24:41.930 --> 00:24:45.670 align:middle line:84% I draw a box or a curve around all these nodes, 00:24:45.670 --> 00:24:48.170 align:middle line:84% and I need to make sure that there's no information set that 00:24:48.170 --> 00:24:51.697 align:middle line:84% includes nodes in this game and also nodes outside the game. 00:24:51.697 --> 00:24:54.030 align:middle line:84% That would be what I mean, breaking the information set. 00:24:54.030 --> 00:24:59.730 align:middle line:90% 00:24:59.730 --> 00:25:03.890 align:middle line:84% And under these conditions, we indeed have a subgame. 00:25:03.890 --> 00:25:05.750 align:middle line:90% So let's note one observation. 00:25:05.750 --> 00:25:08.610 align:middle line:90% 00:25:08.610 --> 00:25:13.590 align:middle line:84% The entire game is a particular subgame. 00:25:13.590 --> 00:25:22.780 align:middle line:90% 00:25:22.780 --> 00:25:23.640 align:middle line:90% How can we see this? 00:25:23.640 --> 00:25:26.820 align:middle line:84% Why is the entire game necessarily a subgame? 00:25:26.820 --> 00:25:30.480 align:middle line:84% What do we choose as our initial node for this subgame? 00:25:30.480 --> 00:25:33.887 align:middle line:90% 00:25:33.887 --> 00:25:35.720 align:middle line:84% We just choose the initial node of the game. 00:25:35.720 --> 00:25:38.500 align:middle line:84% If we take the initial node of the game and then 00:25:38.500 --> 00:25:40.060 align:middle line:84% all its successors, that's just going 00:25:40.060 --> 00:25:41.520 align:middle line:90% to be every node in the game. 00:25:41.520 --> 00:25:43.940 align:middle line:84% And then we have to check are two conditions satisfied. 00:25:43.940 --> 00:25:46.620 align:middle line:84% Well, that initial node better be in a singleton information 00:25:46.620 --> 00:25:49.460 align:middle line:84% set because that's part of the definition of an extensive form 00:25:49.460 --> 00:25:50.260 align:middle line:90% game. 00:25:50.260 --> 00:25:51.945 align:middle line:84% And it can't break information sets 00:25:51.945 --> 00:25:53.320 align:middle line:84% because where else could they go? 00:25:53.320 --> 00:25:55.460 align:middle line:84% We're including every node in the game. 00:25:55.460 --> 00:26:01.180 align:middle line:84% So let's go back up here and see which nodes can start a subgame. 00:26:01.180 --> 00:26:03.260 align:middle line:90% So I'll start here. 00:26:03.260 --> 00:26:06.585 align:middle line:90% Can this node start a subgame? 00:26:06.585 --> 00:26:07.960 align:middle line:84% Yes, right, this is what we said. 00:26:07.960 --> 00:26:10.420 align:middle line:84% If we start here and look at everything after it, 00:26:10.420 --> 00:26:11.940 align:middle line:90% this gives us a subgame. 00:26:11.940 --> 00:26:15.350 align:middle line:84% And this is precisely the entire game. 00:26:15.350 --> 00:26:19.970 align:middle line:84% And maybe when I say initial node, 00:26:19.970 --> 00:26:23.510 align:middle line:84% let me say non-terminal node because I realize we'll get 00:26:23.510 --> 00:26:25.710 align:middle line:90% some-- 00:26:25.710 --> 00:26:27.390 align:middle line:90% so what about this node? 00:26:27.390 --> 00:26:30.530 align:middle line:84% Well, I guess we could think of this as a subgame if we wanted, 00:26:30.530 --> 00:26:32.270 align:middle line:84% but there's nothing to do once we're here because we're already 00:26:32.270 --> 00:26:34.530 align:middle line:84% at a terminal node, so we're not going to call that a subgame. 00:26:34.530 --> 00:26:35.470 align:middle line:84% And that's why we're going to say 00:26:35.470 --> 00:26:37.950 align:middle line:84% this is-- we have to start at a non-terminal node. 00:26:37.950 --> 00:26:40.935 align:middle line:90% 00:26:40.935 --> 00:26:42.630 align:middle line:90% What about this node? 00:26:42.630 --> 00:26:45.722 align:middle line:90% Can this node start a subgame? 00:26:45.722 --> 00:26:47.930 align:middle line:84% Yes, right, this is the subgame we already looked at. 00:26:47.930 --> 00:26:49.050 align:middle line:90% We started this node. 00:26:49.050 --> 00:26:51.390 align:middle line:84% We look at everything that comes after it, 00:26:51.390 --> 00:26:53.790 align:middle line:84% and we check that our conditions are satisfied. 00:26:53.790 --> 00:26:56.870 align:middle line:84% Is this initial node in a singleton information set? 00:26:56.870 --> 00:26:59.890 align:middle line:84% Yes, and do we break any information sets? 00:26:59.890 --> 00:27:01.590 align:middle line:84% Well, the only non-singleton information 00:27:01.590 --> 00:27:04.190 align:middle line:84% set we have to worry about breaking is this one, 00:27:04.190 --> 00:27:07.430 align:middle line:84% and we see that it's not broken, so we're good. 00:27:07.430 --> 00:27:09.010 align:middle line:90% What about here? 00:27:09.010 --> 00:27:12.310 align:middle line:84% Does this constitute a subgame starting at this node? 00:27:12.310 --> 00:27:13.760 align:middle line:90% And why not? 00:27:13.760 --> 00:27:15.173 align:middle line:90% AUDIENCE: It breaks infoset. 00:27:15.173 --> 00:27:16.340 align:middle line:90% IAN BELL: It breaks infoset. 00:27:16.340 --> 00:27:17.880 align:middle line:84% Actually, it violates both of our conditions. 00:27:17.880 --> 00:27:19.635 align:middle line:84% And generally, these can be intertwined. 00:27:19.635 --> 00:27:21.260 align:middle line:84% It's going to break an information set, 00:27:21.260 --> 00:27:23.343 align:middle line:84% and also, it's not in a singleton information set. 00:27:23.343 --> 00:27:25.168 align:middle line:90% So it actually violates both. 00:27:25.168 --> 00:27:26.460 align:middle line:90% And then the same is true here. 00:27:26.460 --> 00:27:28.800 align:middle line:84% If we tried to start a subgame here, 00:27:28.800 --> 00:27:31.760 align:middle line:84% we'd run into problems because this initial node is not 00:27:31.760 --> 00:27:33.820 align:middle line:90% in a singleton information set. 00:27:33.820 --> 00:27:37.840 align:middle line:84% And we would also break this information set. 00:27:37.840 --> 00:27:40.940 align:middle line:84% OK, so this is how we define subgames. 00:27:40.940 --> 00:27:44.520 align:middle line:84% And notice, if you're looking for subgames of a game, 00:27:44.520 --> 00:27:47.298 align:middle line:84% because every subgame starts with a non-terminal node, 00:27:47.298 --> 00:27:48.840 align:middle line:84% the way you can look for them is just 00:27:48.840 --> 00:27:50.860 align:middle line:84% go through each non-terminal node and say, 00:27:50.860 --> 00:27:53.800 align:middle line:84% can this non-terminal node start a subgame? 00:27:53.800 --> 00:27:55.080 align:middle line:90% So start there. 00:27:55.080 --> 00:27:56.760 align:middle line:84% If we wanted to start a subgame, we're 00:27:56.760 --> 00:27:59.140 align:middle line:84% going to have to include every node that comes after it. 00:27:59.140 --> 00:28:00.760 align:middle line:84% And then we just check if we do that, 00:28:00.760 --> 00:28:03.800 align:middle line:84% do we satisfy these two conditions? 00:28:03.800 --> 00:28:05.965 align:middle line:84% So this is how we identify subgames. 00:28:05.965 --> 00:28:07.840 align:middle line:84% And now that we understand what a subgame is, 00:28:07.840 --> 00:28:11.700 align:middle line:84% we can define what subgame perfect Nash equilibrium is. 00:28:11.700 --> 00:28:15.200 align:middle line:84% So here we'll give our definition. 00:28:15.200 --> 00:28:19.300 align:middle line:90% 00:28:19.300 --> 00:28:23.860 align:middle line:84% So here we're given an extensive-form game. 00:28:23.860 --> 00:28:34.420 align:middle line:84% So let's say we're given an extensive-form game G. Remember, 00:28:34.420 --> 00:28:36.100 align:middle line:84% Nash equilibrium could be applied 00:28:36.100 --> 00:28:37.980 align:middle line:90% to a strategic-form game. 00:28:37.980 --> 00:28:40.420 align:middle line:84% Subgame perfect Nash equilibrium requires 00:28:40.420 --> 00:28:42.700 align:middle line:84% us to look at the structure of the extensive form. 00:28:42.700 --> 00:28:45.660 align:middle line:84% So we have to be given an extensive-form game here, not 00:28:45.660 --> 00:28:47.380 align:middle line:90% a strategic form game. 00:28:47.380 --> 00:28:49.200 align:middle line:84% And we're given an extensive-form game. 00:28:49.200 --> 00:28:58.480 align:middle line:84% We say that a strategy profile, S star-- 00:28:58.480 --> 00:29:04.940 align:middle line:90% 00:29:04.940 --> 00:29:08.220 align:middle line:84% so what we're interested in is a strategy profile NG. 00:29:08.220 --> 00:29:10.750 align:middle line:84% Let's remember what a strategy profile is. 00:29:10.750 --> 00:29:13.590 align:middle line:84% It specifies a strategy for every player. 00:29:13.590 --> 00:29:15.870 align:middle line:84% And remember, in an extensive-form game, 00:29:15.870 --> 00:29:16.662 align:middle line:90% what is a strategy? 00:29:16.662 --> 00:29:18.078 align:middle line:84% Do you remember what is a strategy 00:29:18.078 --> 00:29:19.290 align:middle line:90% in an extensive-form game? 00:29:19.290 --> 00:29:22.930 align:middle line:90% 00:29:22.930 --> 00:29:23.430 align:middle line:90% Yeah? 00:29:23.430 --> 00:29:26.055 align:middle line:84% AUDIENCE: I believe you define it as a move for every info set? 00:29:26.055 --> 00:29:27.472 align:middle line:84% IAN BELL: Exactly, it's a function 00:29:27.472 --> 00:29:29.510 align:middle line:84% or a vector that says for every information 00:29:29.510 --> 00:29:32.330 align:middle line:84% set, what action or move do we take at that information set? 00:29:32.330 --> 00:29:34.455 align:middle line:84% So I'm not going to include that on the board here. 00:29:34.455 --> 00:29:36.830 align:middle line:84% But just keep in mind, that's what a strategy 00:29:36.830 --> 00:29:40.190 align:middle line:90% is in an extensive-form game. 00:29:40.190 --> 00:29:45.110 align:middle line:84% Is a, maybe I'll just write S, P, and E, 00:29:45.110 --> 00:29:50.470 align:middle line:84% so subgame perfect Nash equilibrium if-- 00:29:50.470 --> 00:29:53.430 align:middle line:84% well, intuitively what do we want? 00:29:53.430 --> 00:29:57.950 align:middle line:84% We want to look at every subgame of our game. 00:29:57.950 --> 00:29:59.950 align:middle line:84% And then we want to look at our strategy profile 00:29:59.950 --> 00:30:01.180 align:middle line:90% within that subgame. 00:30:01.180 --> 00:30:03.430 align:middle line:84% And we want to check that that would constitute a Nash 00:30:03.430 --> 00:30:06.350 align:middle line:90% equilibrium within that subgame. 00:30:06.350 --> 00:30:17.440 align:middle line:84% So let's say if for every subgame G prime of G-- 00:30:17.440 --> 00:30:19.900 align:middle line:84% so I'm going to impose a requirement on every subgame. 00:30:19.900 --> 00:30:21.317 align:middle line:84% And we've already gone through how 00:30:21.317 --> 00:30:24.760 align:middle line:84% we can identify all our subgames by looking at all the nodes. 00:30:24.760 --> 00:30:29.040 align:middle line:84% For every subgame G prime of G, well, we 00:30:29.040 --> 00:30:33.785 align:middle line:84% want to say that S star is a Nash equilibrium of the subgame. 00:30:33.785 --> 00:30:35.160 align:middle line:84% So at first, you might be tempted 00:30:35.160 --> 00:30:44.040 align:middle line:84% to say S star is a Nash equilibrium of g prime. 00:30:44.040 --> 00:30:46.300 align:middle line:84% And that's basically what we want to say. 00:30:46.300 --> 00:30:48.920 align:middle line:90% But there's an issue here. 00:30:48.920 --> 00:30:53.000 align:middle line:84% If S star is supposed to be a Nash equilibrium of G prime, 00:30:53.000 --> 00:30:55.480 align:middle line:84% well, the problem is that then that S star must 00:30:55.480 --> 00:30:58.560 align:middle line:90% be a strategy profile NG prime. 00:30:58.560 --> 00:31:01.440 align:middle line:84% But S is not a strategy profile necessarily NG prime. 00:31:01.440 --> 00:31:04.120 align:middle line:90% It's a strategy profile NG. 00:31:04.120 --> 00:31:07.970 align:middle line:84% So S star tells us what we do at information sets that 00:31:07.970 --> 00:31:09.418 align:middle line:90% may not be in the subgame. 00:31:09.418 --> 00:31:11.210 align:middle line:84% So we just have to be a little more careful 00:31:11.210 --> 00:31:13.610 align:middle line:90% and say, well, let's look here. 00:31:13.610 --> 00:31:15.610 align:middle line:84% I think we all know what we mean. 00:31:15.610 --> 00:31:19.250 align:middle line:84% We want the play specified by this strategy profile 00:31:19.250 --> 00:31:21.150 align:middle line:84% to be a Nash equilibrium down here. 00:31:21.150 --> 00:31:22.810 align:middle line:84% But of course, we can ignore what's 00:31:22.810 --> 00:31:25.370 align:middle line:84% done up here because this is not part of that subgame. 00:31:25.370 --> 00:31:31.730 align:middle line:84% So I'll use the notation, maybe, S star restricted to G prime. 00:31:31.730 --> 00:31:33.615 align:middle line:84% And this is just, I don't want to really go 00:31:33.615 --> 00:31:34.990 align:middle line:84% through formally the definitions, 00:31:34.990 --> 00:31:37.323 align:middle line:84% because I think it just makes it seem harder than it is. 00:31:37.323 --> 00:31:39.930 align:middle line:84% But we only look at the parts of the strategies that are 00:31:39.930 --> 00:31:41.850 align:middle line:90% relevant inside the subgame. 00:31:41.850 --> 00:31:50.130 align:middle line:84% So we might call this the restriction of S star 00:31:50.130 --> 00:31:51.170 align:middle line:90% to G prime. 00:31:51.170 --> 00:31:53.890 align:middle line:84% And remember, since we said a strategy specifies 00:31:53.890 --> 00:31:56.228 align:middle line:84% a move at every information set, here we're 00:31:56.228 --> 00:31:57.770 align:middle line:84% only going to keep track of the moves 00:31:57.770 --> 00:32:00.670 align:middle line:84% that information sets that are contained NG prime. 00:32:00.670 --> 00:32:04.380 align:middle line:90% 00:32:04.380 --> 00:32:06.800 align:middle line:84% Again, this is a graduate game theory class. 00:32:06.800 --> 00:32:08.680 align:middle line:84% We get really formal about introducing 00:32:08.680 --> 00:32:10.680 align:middle line:84% notation for all these things and what it means, 00:32:10.680 --> 00:32:13.740 align:middle line:84% but I think it's pretty clear conceptually what 00:32:13.740 --> 00:32:14.600 align:middle line:90% these things mean. 00:32:14.600 --> 00:32:18.180 align:middle line:90% 00:32:18.180 --> 00:32:26.420 align:middle line:84% So first question if we have a subgame perfect Nash 00:32:26.420 --> 00:32:29.740 align:middle line:84% equilibrium, do we know it's a Nash equilibrium? 00:32:29.740 --> 00:32:33.960 align:middle line:84% So if we have a subgame perfect Nash equilibrium, 00:32:33.960 --> 00:32:36.240 align:middle line:84% does that imply that something is a Nash equilibrium? 00:32:36.240 --> 00:32:38.020 align:middle line:90% So give me a strategy profile. 00:32:38.020 --> 00:32:41.060 align:middle line:84% Suppose I know it's a subgame perfect Nash equilibrium. 00:32:41.060 --> 00:32:43.795 align:middle line:84% Do I know that it's necessarily a Nash equilibrium. 00:32:43.795 --> 00:32:45.920 align:middle line:84% And maybe a better way, instead of the implication, 00:32:45.920 --> 00:32:47.780 align:middle line:90% let's just do this graphically. 00:32:47.780 --> 00:32:52.300 align:middle line:84% So what we want to say is it the case that the set of SPNE 00:32:52.300 --> 00:32:55.160 align:middle line:84% is a subset of the set of Nash equilibrium? 00:32:55.160 --> 00:32:57.860 align:middle line:90% 00:32:57.860 --> 00:33:00.420 align:middle line:90% So we'd like that to be true. 00:33:00.420 --> 00:33:02.600 align:middle line:84% Just like when we came over here, we said, 00:33:02.600 --> 00:33:04.470 align:middle line:84% there's a lot of Nash equilibria, 00:33:04.470 --> 00:33:06.870 align:middle line:84% and we wanted the subgame perfect Nash equilibrium 00:33:06.870 --> 00:33:09.050 align:middle line:84% to be a special one of those Nash equilibria. 00:33:09.050 --> 00:33:12.150 align:middle line:84% So of course, they could agree, but we 00:33:12.150 --> 00:33:14.350 align:middle line:84% want to make sure that every subgame perfect Nash 00:33:14.350 --> 00:33:16.933 align:middle line:84% equilibrium, the point is, we want it to be a Nash equilibrium 00:33:16.933 --> 00:33:19.430 align:middle line:84% and satisfy these stronger assumptions. 00:33:19.430 --> 00:33:21.673 align:middle line:84% So it should be a Nash equilibrium. 00:33:21.673 --> 00:33:23.590 align:middle line:84% How can we see that this is a Nash equilibrium 00:33:23.590 --> 00:33:26.790 align:middle line:90% from the definition? 00:33:26.790 --> 00:33:29.178 align:middle line:90% So this doesn't require-- 00:33:29.178 --> 00:33:29.970 align:middle line:90% you have to see it. 00:33:29.970 --> 00:33:31.330 align:middle line:90% But there's not a long argument. 00:33:31.330 --> 00:33:32.122 align:middle line:90% It's pretty simple. 00:33:32.122 --> 00:33:32.630 align:middle line:90% Yeah? 00:33:32.630 --> 00:33:34.070 align:middle line:90% AUDIENCE: G is a subgame of G. 00:33:34.070 --> 00:33:37.190 align:middle line:84% IAN BELL: Right, so why is this true? 00:33:37.190 --> 00:33:41.690 align:middle line:90% Well, G prime G is a subgame. 00:33:41.690 --> 00:33:44.670 align:middle line:90% 00:33:44.670 --> 00:33:46.750 align:middle line:90% So let's look at our definition. 00:33:46.750 --> 00:33:50.990 align:middle line:84% Our definition says for every subgame, this statement holds. 00:33:50.990 --> 00:33:53.510 align:middle line:84% So in particular, it holds for the special subgame G 00:33:53.510 --> 00:33:55.510 align:middle line:90% prime equals G. 00:33:55.510 --> 00:34:00.030 align:middle line:84% So if we plug that in, it says the restriction of S star to G 00:34:00.030 --> 00:34:03.840 align:middle line:84% is a Nash equilibrium of G. But what is the restriction 00:34:03.840 --> 00:34:05.600 align:middle line:90% of the strategy profile to G? 00:34:05.600 --> 00:34:07.520 align:middle line:84% It's just the strategy profile itself 00:34:07.520 --> 00:34:09.800 align:middle line:90% because G is the entire game. 00:34:09.800 --> 00:34:12.080 align:middle line:84% So this is stronger than Nash equilibrium, 00:34:12.080 --> 00:34:16.360 align:middle line:84% but it implies Nash equilibrium, OK. 00:34:16.360 --> 00:34:24.719 align:middle line:90% 00:34:24.719 --> 00:34:27.360 align:middle line:84% Let's maybe go through some examples 00:34:27.360 --> 00:34:30.040 align:middle line:84% to see how we would approach this. 00:34:30.040 --> 00:34:32.580 align:middle line:90% Let me come down here. 00:34:32.580 --> 00:34:50.360 align:middle line:90% 00:34:50.360 --> 00:34:52.699 align:middle line:84% Are there any questions before I go to the examples? 00:34:52.699 --> 00:34:55.400 align:middle line:90% 00:34:55.400 --> 00:34:57.780 align:middle line:90% So let's do one more example. 00:34:57.780 --> 00:35:10.420 align:middle line:84% And this is going to be a variant of our Boston game, 00:35:10.420 --> 00:35:15.320 align:middle line:84% where first, we're going to have player 1. 00:35:15.320 --> 00:35:21.180 align:middle line:90% 00:35:21.180 --> 00:35:22.980 align:middle line:84% And they can choose maybe again, I'll 00:35:22.980 --> 00:35:25.100 align:middle line:90% call it entering and exiting. 00:35:25.100 --> 00:35:34.660 align:middle line:84% And if they exit, we'll say both players get 1.5. 00:35:34.660 --> 00:35:37.113 align:middle line:84% So we're going to think of exiting. 00:35:37.113 --> 00:35:39.280 align:middle line:84% Well, this is just really to illustrate the concept. 00:35:39.280 --> 00:35:41.090 align:middle line:84% But this means we're not really going 00:35:41.090 --> 00:35:43.260 align:middle line:84% to-- we're going to do some alternative activity, 00:35:43.260 --> 00:35:45.838 align:middle line:84% rather than the Red Sox or the Celtics game. 00:35:45.838 --> 00:35:47.880 align:middle line:84% But then if they enter, then we're going to play. 00:35:47.880 --> 00:35:50.940 align:middle line:84% This is like they say they're sick, and they just stay home. 00:35:50.940 --> 00:35:54.260 align:middle line:84% So now we're going to have, let's say, player 2 00:35:54.260 --> 00:35:56.420 align:middle line:90% is going to move-- 00:35:56.420 --> 00:35:59.390 align:middle line:90% choose Celtics or Red Sox. 00:35:59.390 --> 00:36:04.670 align:middle line:84% And then player 1, not observing what player 2 does, 00:36:04.670 --> 00:36:08.258 align:middle line:84% is going to choose Celtics or Red Sox. 00:36:08.258 --> 00:36:10.550 align:middle line:84% And we're going to follow our convention that, I think, 00:36:10.550 --> 00:36:13.310 align:middle line:84% player one prefers Celtics to Red Sox, 00:36:13.310 --> 00:36:21.790 align:middle line:84% so we're going to have 2, 1 1,2 0, 0. 00:36:21.790 --> 00:36:24.030 align:middle line:90% Zero. 00:36:24.030 --> 00:36:29.510 align:middle line:84% So now let's try to compute the SPNE of this game. 00:36:29.510 --> 00:36:38.070 align:middle line:84% So here's kind of a recipe for solving SPNE. 00:36:38.070 --> 00:36:40.290 align:middle line:84% Step one is always identify the subgames. 00:36:40.290 --> 00:36:50.390 align:middle line:90% 00:36:50.390 --> 00:36:55.610 align:middle line:84% So how many subgames does this game have? 00:36:55.610 --> 00:36:58.280 align:middle line:90% 00:36:58.280 --> 00:36:59.540 align:middle line:90% Can we identify the subgames? 00:36:59.540 --> 00:37:02.280 align:middle line:90% Well, let's go through. 00:37:02.280 --> 00:37:04.080 align:middle line:84% Starting here is definitely a subgame 00:37:04.080 --> 00:37:06.760 align:middle line:84% because the entire game is always a subgame. 00:37:06.760 --> 00:37:09.380 align:middle line:84% This doesn't start a subgame because this is a terminal node. 00:37:09.380 --> 00:37:11.640 align:middle line:90% So we can't start here. 00:37:11.640 --> 00:37:14.280 align:middle line:84% This node will start a subgame because if we 00:37:14.280 --> 00:37:17.200 align:middle line:84% go here and everything below it, then we 00:37:17.200 --> 00:37:19.600 align:middle line:84% have a single node by itself, and we're not 00:37:19.600 --> 00:37:24.460 align:middle line:84% going to break any information sets if we go like that, 00:37:24.460 --> 00:37:26.280 align:middle line:90% so that looks good. 00:37:26.280 --> 00:37:27.380 align:middle line:90% What about this node here? 00:37:27.380 --> 00:37:29.200 align:middle line:90% Does this start a subgame? 00:37:29.200 --> 00:37:31.090 align:middle line:84% No, because this is not a singleton. 00:37:31.090 --> 00:37:33.340 align:middle line:84% And then similarly down here, this is not a singleton. 00:37:33.340 --> 00:37:35.200 align:middle line:84% So it actually is exactly the same structure 00:37:35.200 --> 00:37:36.720 align:middle line:90% as this game over here. 00:37:36.720 --> 00:37:38.940 align:middle line:84% So in this case, we're going to have two subgames. 00:37:38.940 --> 00:37:43.640 align:middle line:90% 00:37:43.640 --> 00:37:47.560 align:middle line:84% And then a good approach, step two 00:37:47.560 --> 00:37:50.660 align:middle line:84% is to work backwards, just like we did with backward induction. 00:37:50.660 --> 00:37:52.960 align:middle line:84% So we want to start with the smallest subgames 00:37:52.960 --> 00:37:56.530 align:middle line:84% and try to solve for an equilibrium within that subgame 00:37:56.530 --> 00:37:58.110 align:middle line:84% and then kind of build up backwards. 00:37:58.110 --> 00:38:00.510 align:middle line:84% So step two is going to be, let's work backwards. 00:38:00.510 --> 00:38:07.730 align:middle line:90% 00:38:07.730 --> 00:38:09.570 align:middle line:90% So let's see how that works. 00:38:09.570 --> 00:38:13.370 align:middle line:84% Well, here, working backwards is pretty easy 00:38:13.370 --> 00:38:14.810 align:middle line:84% because there's only two subgames. 00:38:14.810 --> 00:38:16.310 align:middle line:84% So we certainly know where to start. 00:38:16.310 --> 00:38:19.330 align:middle line:84% If we want to work backwards, we want to start with this subgame. 00:38:19.330 --> 00:38:23.010 align:middle line:84% So let's look at this subgame, and let's represent 00:38:23.010 --> 00:38:27.810 align:middle line:90% this in strategic form. 00:38:27.810 --> 00:38:33.050 align:middle line:84% So we have Celtics, Red Sox, Celtics, Red Sox. 00:38:33.050 --> 00:38:35.930 align:middle line:84% Player 1 is choosing between Celtics and Red Sox. 00:38:35.930 --> 00:38:38.030 align:middle line:84% Player 2 is choosing between Celtics and Red Sox. 00:38:38.030 --> 00:38:40.490 align:middle line:84% And we know the payoffs are just like the Boston game 00:38:40.490 --> 00:38:41.510 align:middle line:90% we studied before. 00:38:41.510 --> 00:38:47.010 align:middle line:90% 00:38:47.010 --> 00:38:51.810 align:middle line:84% So this subgame of the larger game 00:38:51.810 --> 00:38:54.957 align:middle line:84% reduces to a very simple, simultaneous move game 00:38:54.957 --> 00:38:56.540 align:middle line:84% that we can think of in strategic form 00:38:56.540 --> 00:38:58.340 align:middle line:90% that we've already analyzed. 00:38:58.340 --> 00:39:02.680 align:middle line:84% So what are the Nash equilibria of this game? 00:39:02.680 --> 00:39:03.560 align:middle line:90% Let's look at pure. 00:39:03.560 --> 00:39:05.310 align:middle line:84% We could also talk about mixed strategies, 00:39:05.310 --> 00:39:07.663 align:middle line:84% but let's focus on pure strategies for now. 00:39:07.663 --> 00:39:09.580 align:middle line:84% This is the game we've already analyzed, yeah? 00:39:09.580 --> 00:39:10.660 align:middle line:90% AUDIENCE: CC and RR. 00:39:10.660 --> 00:39:12.940 align:middle line:84% IAN BELL: Great, so we have two Nash equilibria. 00:39:12.940 --> 00:39:16.780 align:middle line:90% We have CC and RR, right? 00:39:16.780 --> 00:39:19.540 align:middle line:84% So notice this is actually a pretty important difference 00:39:19.540 --> 00:39:23.412 align:middle line:84% with backward induction applied to perfect information games. 00:39:23.412 --> 00:39:25.620 align:middle line:84% When we did backward induction to perfect information 00:39:25.620 --> 00:39:27.880 align:middle line:84% games, most of the time, we didn't have ties. 00:39:27.880 --> 00:39:31.300 align:middle line:84% We didn't have multiple solutions because at each point, 00:39:31.300 --> 00:39:32.958 align:middle line:84% a single player was making a move. 00:39:32.958 --> 00:39:35.000 align:middle line:84% And generally one move was better than the other, 00:39:35.000 --> 00:39:36.220 align:middle line:90% and we just selected that. 00:39:36.220 --> 00:39:38.320 align:middle line:84% But here, when we're working backwards, 00:39:38.320 --> 00:39:40.580 align:middle line:84% we're working at the level of an entire subgame. 00:39:40.580 --> 00:39:43.700 align:middle line:84% And we know that generally games can have many equilibria. 00:39:43.700 --> 00:39:45.840 align:middle line:84% So when we work backwards, what we're going to do, 00:39:45.840 --> 00:39:48.900 align:middle line:84% if we wanted to find all of the Nash equilibria 00:39:48.900 --> 00:39:51.550 align:middle line:84% that are subgame perfect, we'd actually 00:39:51.550 --> 00:39:53.070 align:middle line:90% need to make a choice here. 00:39:53.070 --> 00:39:56.150 align:middle line:84% We need to first try this and then work backwards and see 00:39:56.150 --> 00:39:56.953 align:middle line:90% what happens. 00:39:56.953 --> 00:39:59.370 align:middle line:84% And then try this and work backwards and see what happens. 00:39:59.370 --> 00:40:02.110 align:middle line:84% And in the end, we may find multiple subgame perfect Nash 00:40:02.110 --> 00:40:02.890 align:middle line:90% equilibrium. 00:40:02.890 --> 00:40:07.070 align:middle line:84% So by working backwards, we start at our smallest subgame. 00:40:07.070 --> 00:40:10.530 align:middle line:84% We choose a Nash equilibrium of that subgame 00:40:10.530 --> 00:40:11.890 align:middle line:90% and then we work backwards. 00:40:11.890 --> 00:40:12.830 align:middle line:84% And then we have to do it all over 00:40:12.830 --> 00:40:14.090 align:middle line:90% again with a different choice. 00:40:14.090 --> 00:40:16.632 align:middle line:84% So you can see, eventually this could get pretty complicated. 00:40:16.632 --> 00:40:19.230 align:middle line:90% So let's first make choice one. 00:40:19.230 --> 00:40:20.930 align:middle line:84% So here, we have two choices, right? 00:40:20.930 --> 00:40:24.710 align:middle line:90% 00:40:24.710 --> 00:40:29.830 align:middle line:84% So the first choice would be CC, and the second choice 00:40:29.830 --> 00:40:32.010 align:middle line:90% would be RR, in this subgame. 00:40:32.010 --> 00:40:36.750 align:middle line:90% 00:40:36.750 --> 00:40:39.870 align:middle line:84% Well now, let's start with our first choice, CC. 00:40:39.870 --> 00:40:41.950 align:middle line:84% And we'll do what we did in backward induction is 00:40:41.950 --> 00:40:45.550 align:middle line:84% we'll treat this subgame as if once we reach this subgame, 00:40:45.550 --> 00:40:46.830 align:middle line:90% this is what happens. 00:40:46.830 --> 00:40:51.890 align:middle line:84% So let's highlight CC is going to take us down here. 00:40:51.890 --> 00:40:53.850 align:middle line:84% So now let's move a step back and look 00:40:53.850 --> 00:40:55.970 align:middle line:90% at the choice for player 1. 00:40:55.970 --> 00:41:00.850 align:middle line:84% So player 1, anticipating that if we get to this subgame, 00:41:00.850 --> 00:41:03.330 align:middle line:84% CC is going to be played, and therefore, the payoff 00:41:03.330 --> 00:41:08.423 align:middle line:84% is going to be 2, 1, does player 1 want to exit or enter? 00:41:08.423 --> 00:41:09.170 align:middle line:90% AUDIENCE: Enter. 00:41:09.170 --> 00:41:12.410 align:middle line:84% IAN BELL: They want to enter, right? 00:41:12.410 --> 00:41:15.250 align:middle line:84% So now when you make choice one, and we work backwards, 00:41:15.250 --> 00:41:20.370 align:middle line:84% we see that player 1 wants to enter. 00:41:20.370 --> 00:41:23.810 align:middle line:84% And now we have one subgame perfect Nash equilibrium. 00:41:23.810 --> 00:41:25.910 align:middle line:84% But we have to be careful to write it down. 00:41:25.910 --> 00:41:26.410 align:middle line:90% Let's see. 00:41:26.410 --> 00:41:28.370 align:middle line:84% What is player 1's full strategy? 00:41:28.370 --> 00:41:31.050 align:middle line:84% Well, player 1 is going to enter here, 00:41:31.050 --> 00:41:33.710 align:middle line:84% and they're going to choose C down here. 00:41:33.710 --> 00:41:36.250 align:middle line:84% So remember at the end, we want a strategy profile. 00:41:36.250 --> 00:41:40.390 align:middle line:84% So our strategy profile is going to be this. 00:41:40.390 --> 00:41:42.250 align:middle line:84% We have to list how player 1 is going 00:41:42.250 --> 00:41:43.930 align:middle line:90% to behave at every contingency. 00:41:43.930 --> 00:41:46.690 align:middle line:84% At this contingency they're going to enter. 00:41:46.690 --> 00:41:49.140 align:middle line:84% At this contingency, they're going to play C. 00:41:49.140 --> 00:41:51.980 align:middle line:84% And actually, I should have highlighted this as well. 00:41:51.980 --> 00:41:53.640 align:middle line:84% Even though it doesn't affect payoffs, 00:41:53.640 --> 00:41:58.420 align:middle line:84% remember, if player 1 is choosing C, 00:41:58.420 --> 00:42:01.118 align:middle line:84% then they have to make the same choice at every node 00:42:01.118 --> 00:42:02.160 align:middle line:90% of their information set. 00:42:02.160 --> 00:42:03.577 align:middle line:84% So I should have highlighted this. 00:42:03.577 --> 00:42:06.500 align:middle line:90% Let me make that clear. 00:42:06.500 --> 00:42:08.900 align:middle line:90% So player 1 chooses N here. 00:42:08.900 --> 00:42:11.260 align:middle line:84% They choose C at this information set. 00:42:11.260 --> 00:42:13.760 align:middle line:84% So we get NC as the strategy for player 1. 00:42:13.760 --> 00:42:16.380 align:middle line:84% And then for player 2, we just get C. 00:42:16.380 --> 00:42:18.760 align:middle line:84% Now let's go through the same procedure, 00:42:18.760 --> 00:42:22.460 align:middle line:84% making a different choice of our equilibrium in this subgame. 00:42:22.460 --> 00:42:25.180 align:middle line:90% So now our choice is RR. 00:42:25.180 --> 00:42:26.640 align:middle line:90% I don't want to mess up. 00:42:26.640 --> 00:42:28.190 align:middle line:84% I'd have to redraw the graph, so it's 00:42:28.190 --> 00:42:29.440 align:middle line:90% going to be too hard to erase. 00:42:29.440 --> 00:42:30.815 align:middle line:84% So let's just do it in our heads. 00:42:30.815 --> 00:42:36.380 align:middle line:84% If we choose RR, then upon reaching this subgame, we go RR, 00:42:36.380 --> 00:42:38.740 align:middle line:90% The payoff is going to be 1, 2. 00:42:38.740 --> 00:42:41.960 align:middle line:84% So now when we take a step back, and we look at player 1, 00:42:41.960 --> 00:42:44.680 align:middle line:90% what is optimal for player 1? 00:42:44.680 --> 00:42:45.180 align:middle line:90% Yeah? 00:42:45.180 --> 00:42:45.805 align:middle line:90% AUDIENCE: Exit. 00:42:45.805 --> 00:42:48.110 align:middle line:84% IAN BELL: They want to exit because 1.5 00:42:48.110 --> 00:42:50.150 align:middle line:90% is a higher payoff than 1, 2. 00:42:50.150 --> 00:42:52.630 align:middle line:84% So now when I go back, I'm going to get exit. 00:42:52.630 --> 00:42:54.310 align:middle line:84% And now if I put it all together, 00:42:54.310 --> 00:42:57.470 align:middle line:84% what is the complete contingent plan for player 1? 00:42:57.470 --> 00:43:02.970 align:middle line:84% It's exit at this node and play R at this point. 00:43:02.970 --> 00:43:14.130 align:middle line:90% 00:43:14.130 --> 00:43:18.870 align:middle line:84% Now, so great, in this game, we found two subgame perfect Nash 00:43:18.870 --> 00:43:20.830 align:middle line:90% equilibria. 00:43:20.830 --> 00:43:24.310 align:middle line:84% But there could be some other Nash equilibria 00:43:24.310 --> 00:43:25.730 align:middle line:90% that aren't subgame perfect. 00:43:25.730 --> 00:43:28.670 align:middle line:90% 00:43:28.670 --> 00:43:33.425 align:middle line:84% Can anyone see an example of a Nash equilibrium 00:43:33.425 --> 00:43:35.050 align:middle line:84% that's not going to be subgame perfect? 00:43:35.050 --> 00:43:37.010 align:middle line:90% Maybe, it's a bit tricky. 00:43:37.010 --> 00:43:38.410 align:middle line:84% But let's just look at this here. 00:43:38.410 --> 00:43:43.310 align:middle line:84% Let's focus on this subgame perfect Nash equilibrium. 00:43:43.310 --> 00:43:51.920 align:middle line:84% We know that if player 1 exits, we 00:43:51.920 --> 00:43:53.760 align:middle line:84% might say, well, if they're going to exit, 00:43:53.760 --> 00:43:56.920 align:middle line:84% it doesn't really matter what they do down here. 00:43:56.920 --> 00:43:58.720 align:middle line:84% So you might be tempted to say, let's look 00:43:58.720 --> 00:44:02.440 align:middle line:84% at a different strategy profile, say, 00:44:02.440 --> 00:44:11.543 align:middle line:84% X C, R. And let's check if this is going to be a Nash 00:44:11.543 --> 00:44:13.960 align:middle line:84% equilibrium, or if it's going to be a subgame perfect Nash 00:44:13.960 --> 00:44:14.505 align:middle line:90% equilibrium? 00:44:14.505 --> 00:44:16.880 align:middle line:84% So we don't think it's going to be a subgame perfect Nash 00:44:16.880 --> 00:44:19.900 align:middle line:84% equilibrium because it didn't get spit out of this procedure. 00:44:19.900 --> 00:44:22.400 align:middle line:84% But let's see if it's going to be a Nash equilibrium. 00:44:22.400 --> 00:44:27.480 align:middle line:84% Given that player 1 is playing XC as their strategy, 00:44:27.480 --> 00:44:32.080 align:middle line:84% can player 2 do any better than choosing R? 00:44:32.080 --> 00:44:34.120 align:middle line:84% So does player 2 have a profitable deviation 00:44:34.120 --> 00:44:35.180 align:middle line:90% from this strategy? 00:44:35.180 --> 00:44:38.320 align:middle line:90% 00:44:38.320 --> 00:44:39.220 align:middle line:90% Why, why not? 00:44:39.220 --> 00:44:43.287 align:middle line:90% 00:44:43.287 --> 00:44:43.870 align:middle line:90% Thoughts here? 00:44:43.870 --> 00:44:44.370 align:middle line:90% Yeah? 00:44:44.370 --> 00:44:46.247 align:middle line:84% AUDIENCE: Player 2 doesn't have a choice. 00:44:46.247 --> 00:44:48.830 align:middle line:84% IAN BELL: Right, I mean, they have a complete contingent plan, 00:44:48.830 --> 00:44:52.050 align:middle line:84% but if player 1 is playing X, then we're 00:44:52.050 --> 00:44:54.570 align:middle line:84% definitely going to end up here whatever player 2 does. 00:44:54.570 --> 00:44:56.690 align:middle line:84% So if player 2 were to deviate, the payoff 00:44:56.690 --> 00:44:58.450 align:middle line:90% would be exactly the same. 00:44:58.450 --> 00:45:01.570 align:middle line:84% And that means that any deviation by player 2, 00:45:01.570 --> 00:45:03.210 align:middle line:90% will not be profitable. 00:45:03.210 --> 00:45:05.570 align:middle line:84% No deviation is profitable, so player 2 does not 00:45:05.570 --> 00:45:08.610 align:middle line:90% have a profitable deviation. 00:45:08.610 --> 00:45:09.950 align:middle line:90% What about player 1? 00:45:09.950 --> 00:45:11.910 align:middle line:84% Does player 1 have a profitable deviation? 00:45:11.910 --> 00:45:28.130 align:middle line:90% 00:45:28.130 --> 00:45:29.250 align:middle line:90% Any thoughts, yeah? 00:45:29.250 --> 00:45:31.630 align:middle line:84% AUDIENCE: Player 2 has to make R [INAUDIBLE]. 00:45:31.630 --> 00:45:33.370 align:middle line:84% IAN BELL: And let's see why that is. 00:45:33.370 --> 00:45:35.130 align:middle line:84% So we have to be pretty careful here. 00:45:35.130 --> 00:45:38.130 align:middle line:84% If player 2 is playing R, well by exiting, 00:45:38.130 --> 00:45:40.660 align:middle line:90% I'm getting a payoff of 1.5. 00:45:40.660 --> 00:45:42.180 align:middle line:90% Could I ever do better? 00:45:42.180 --> 00:45:44.700 align:middle line:84% Well, I could enter and then play C. 00:45:44.700 --> 00:45:49.420 align:middle line:84% So if I enter, player 2 plays R and then I play C, I get 0, 0. 00:45:49.420 --> 00:45:52.460 align:middle line:84% That's definitely worse, but I think I have a better deviation. 00:45:52.460 --> 00:45:56.080 align:middle line:84% What is my better deviation going to be? 00:45:56.080 --> 00:45:56.580 align:middle line:90% Yeah? 00:45:56.580 --> 00:45:57.380 align:middle line:90% AUDIENCE: 3R. 00:45:57.380 --> 00:45:59.380 align:middle line:84% IAN BELL: Yeah, so in fact, instead of XC, 00:45:59.380 --> 00:46:02.300 align:middle line:84% I think my best deviation is going to be NR. 00:46:02.300 --> 00:46:05.428 align:middle line:84% So instead of doing X, I'm going to enter. 00:46:05.428 --> 00:46:07.220 align:middle line:84% And then because I believe that player 2 is 00:46:07.220 --> 00:46:08.845 align:middle line:84% going to go to the Red Sox game, I also 00:46:08.845 --> 00:46:12.260 align:middle line:84% want to go to the Red Sox game, and that gives me a payoff of 1. 00:46:12.260 --> 00:46:15.000 align:middle line:84% But that's still worse than my payoff of 1.5. 00:46:15.000 --> 00:46:20.380 align:middle line:84% So indeed, this is a Nash equilibrium, but not 00:46:20.380 --> 00:46:22.120 align:middle line:84% a subgame perfect Nash equilibrium. 00:46:22.120 --> 00:46:26.460 align:middle line:90% 00:46:26.460 --> 00:46:28.500 align:middle line:84% So if we go to our circle over here, 00:46:28.500 --> 00:46:30.780 align:middle line:90% this lies somewhere in here. 00:46:30.780 --> 00:46:33.380 align:middle line:84% It's a Nash equilibrium, but it's not a subgame perfect Nash 00:46:33.380 --> 00:46:33.960 align:middle line:90% equilibrium. 00:46:33.960 --> 00:46:36.460 align:middle line:84% We already checked that it's a Nash equilibrium, or at least 00:46:36.460 --> 00:46:37.160 align:middle line:90% verbally. 00:46:37.160 --> 00:46:39.320 align:middle line:84% We said neither player can profitably 00:46:39.320 --> 00:46:41.700 align:middle line:84% deviate from this strategy profile, 00:46:41.700 --> 00:46:44.000 align:middle line:84% and that implies that it's a Nash equilibrium. 00:46:44.000 --> 00:46:46.040 align:middle line:84% We argued it shouldn't be a subgame perfect Nash 00:46:46.040 --> 00:46:48.660 align:middle line:84% equilibrium because it didn't come out of our procedure. 00:46:48.660 --> 00:46:49.760 align:middle line:90% But let's formally check. 00:46:49.760 --> 00:46:53.420 align:middle line:84% What's violated about this strategy profile? 00:46:53.420 --> 00:46:56.680 align:middle line:84% Why does this strategy profile not satisfy our definition 00:46:56.680 --> 00:46:58.860 align:middle line:84% of subgame perfect Nash equilibrium? 00:46:58.860 --> 00:47:03.760 align:middle line:90% 00:47:03.760 --> 00:47:04.660 align:middle line:90% So what's wrong? 00:47:04.660 --> 00:47:07.440 align:middle line:84% It is a Nash equilibrium, but it's not subgame perfect, 00:47:07.440 --> 00:47:08.460 align:middle line:90% and why is that? 00:47:08.460 --> 00:47:13.200 align:middle line:90% 00:47:13.200 --> 00:47:16.040 align:middle line:90% Thoughts? 00:47:16.040 --> 00:47:19.420 align:middle line:90% Well, let's look at our subgame. 00:47:19.420 --> 00:47:22.520 align:middle line:84% In our subgame, player 1 is playing C, 00:47:22.520 --> 00:47:26.160 align:middle line:84% and player 2 is playing R in our subgame here. 00:47:26.160 --> 00:47:28.200 align:middle line:84% And that means player 1 is playing 00:47:28.200 --> 00:47:32.760 align:middle line:84% C. Player 2 is playing R. We're at 0, 0 over here. 00:47:32.760 --> 00:47:35.280 align:middle line:84% This is not a Nash equilibrium of the subgame. 00:47:35.280 --> 00:47:38.450 align:middle line:84% Because in this subgame, in fact, either player 00:47:38.450 --> 00:47:39.370 align:middle line:90% could deviate. 00:47:39.370 --> 00:47:44.130 align:middle line:84% If player 2 changes to R, they would get a higher payoff 00:47:44.130 --> 00:47:46.730 align:middle line:84% because then player 1 would get 1. 00:47:46.730 --> 00:47:49.010 align:middle line:84% And similarly, player 2 could deviate to C, 00:47:49.010 --> 00:47:50.927 align:middle line:84% and they would get a higher payoff. 00:47:50.927 --> 00:47:52.510 align:middle line:84% But we have to be really careful here. 00:47:52.510 --> 00:47:55.370 align:middle line:84% I just said that it is a Nash equilibrium 00:47:55.370 --> 00:47:57.530 align:middle line:84% and neither player can profitably deviate. 00:47:57.530 --> 00:47:59.410 align:middle line:84% And now I'm saying in the subgame, 00:47:59.410 --> 00:48:00.830 align:middle line:90% a player can profitably deviate. 00:48:00.830 --> 00:48:03.650 align:middle line:90% So what's going on here? 00:48:03.650 --> 00:48:05.190 align:middle line:90% So it is a Nash equilibrium. 00:48:05.190 --> 00:48:07.930 align:middle line:84% No one has a profitable deviation in the entire game. 00:48:07.930 --> 00:48:10.970 align:middle line:84% But I'm saying, in this subgame down here, 00:48:10.970 --> 00:48:13.150 align:middle line:84% there are profitable deviations the players have. 00:48:13.150 --> 00:48:14.858 align:middle line:84% And that's why it violates the definition 00:48:14.858 --> 00:48:16.810 align:middle line:84% of subgame perfect Nash equilibrium. 00:48:16.810 --> 00:48:17.830 align:middle line:90% So what's going on? 00:48:17.830 --> 00:48:25.482 align:middle line:90% 00:48:25.482 --> 00:48:27.690 align:middle line:84% So what's going on is there are profitable deviations 00:48:27.690 --> 00:48:30.490 align:middle line:84% in subgames that are not reached. 00:48:30.490 --> 00:48:32.190 align:middle line:90% So why? 00:48:32.190 --> 00:48:35.210 align:middle line:90% 00:48:35.210 --> 00:48:37.980 align:middle line:84% Maybe I'll make the more general point. 00:48:37.980 --> 00:48:39.960 align:middle line:84% When we're at a point here, what's going on? 00:48:39.960 --> 00:48:42.620 align:middle line:90% 00:48:42.620 --> 00:48:44.420 align:middle line:84% How can we have a strategy profile that's 00:48:44.420 --> 00:48:46.780 align:middle line:84% a Nash equilibrium, but not a subgame perfect Nash 00:48:46.780 --> 00:48:47.600 align:middle line:90% equilibrium? 00:48:47.600 --> 00:48:49.420 align:middle line:84% What must be the case is that there 00:48:49.420 --> 00:48:57.340 align:middle line:84% must be a profitable deviation, and we 00:48:57.340 --> 00:48:59.520 align:middle line:84% have to be precise about what that means. 00:48:59.520 --> 00:49:09.960 align:middle line:84% Profitable deviation within the subgame that is not reached. 00:49:09.960 --> 00:49:18.460 align:middle line:90% 00:49:18.460 --> 00:49:20.920 align:middle line:84% And this is exactly the idea of a non-credible threat. 00:49:20.920 --> 00:49:24.400 align:middle line:84% It says, if we were to get to this subgame, 00:49:24.400 --> 00:49:27.500 align:middle line:84% if we were to get to this node, some player 00:49:27.500 --> 00:49:29.580 align:middle line:90% could do strictly better. 00:49:29.580 --> 00:49:33.220 align:middle line:84% That's why this strategy profile violates subgame perfect 00:49:33.220 --> 00:49:35.070 align:middle line:90% Nash equilibrium. 00:49:35.070 --> 00:49:37.290 align:middle line:84% But it doesn't violate Nash equilibrium 00:49:37.290 --> 00:49:39.570 align:middle line:84% because under this strategy profile, 00:49:39.570 --> 00:49:41.618 align:middle line:84% we don't actually get to this subgame. 00:49:41.618 --> 00:49:43.910 align:middle line:84% So the fact that I could do better if this subgame were 00:49:43.910 --> 00:49:46.090 align:middle line:84% reached, doesn't mean I could do better 00:49:46.090 --> 00:49:47.815 align:middle line:84% because when I change my play here, 00:49:47.815 --> 00:49:50.190 align:middle line:84% I'm not actually affecting our final payoff because we're 00:49:50.190 --> 00:49:53.110 align:middle line:90% going to get stuck over here. 00:49:53.110 --> 00:49:56.090 align:middle line:84% So it's exactly what we talked about with backward induction. 00:49:56.090 --> 00:49:58.230 align:middle line:84% Remember, we said Nash equilibrium does not 00:49:58.230 --> 00:50:02.630 align:middle line:84% require optimal play at unreached contingencies. 00:50:02.630 --> 00:50:05.430 align:middle line:84% Similarly, Nash equilibrium does not necessarily 00:50:05.430 --> 00:50:10.070 align:middle line:84% require Nash equilibrium play in unreached subgames. 00:50:10.070 --> 00:50:13.670 align:middle line:84% So we're just kind of replacing contingencies with subgames. 00:50:13.670 --> 00:50:16.270 align:middle line:84% And indeed, there may be a profitable deviation 00:50:16.270 --> 00:50:18.150 align:middle line:84% at a subgame that is not reached, 00:50:18.150 --> 00:50:20.550 align:middle line:84% but that doesn't mean there's a profitable deviation 00:50:20.550 --> 00:50:22.067 align:middle line:90% within the entire game. 00:50:22.067 --> 00:50:23.650 align:middle line:84% I think that's kind of a subtle point. 00:50:23.650 --> 00:50:26.110 align:middle line:84% And that's the heart of subgame perfection. 00:50:26.110 --> 00:50:27.590 align:middle line:90% Any questions about that? 00:50:27.590 --> 00:50:30.010 align:middle line:84% And if not, we'll move to a more substantive example. 00:50:30.010 --> 00:50:32.560 align:middle line:90% 00:50:32.560 --> 00:50:38.900 align:middle line:84% All right, so now let's go over here. 00:50:38.900 --> 00:50:53.353 align:middle line:90% 00:50:53.353 --> 00:50:55.520 align:middle line:84% And now we're going to consider an application where 00:50:55.520 --> 00:50:57.680 align:middle line:84% we can apply subgame perfection to try 00:50:57.680 --> 00:51:00.600 align:middle line:84% to make a prediction about behavior in this game. 00:51:00.600 --> 00:51:10.820 align:middle line:84% So this is going to be Cournot competition with entry costs. 00:51:10.820 --> 00:51:15.530 align:middle line:90% 00:51:15.530 --> 00:51:17.780 align:middle line:84% So we're going to look at classic Cournot competition. 00:51:17.780 --> 00:51:20.780 align:middle line:84% Remember, Cournot competition is quantity competition. 00:51:20.780 --> 00:51:24.245 align:middle line:84% So each firm chooses how much quantity to bring to market. 00:51:24.245 --> 00:51:26.120 align:middle line:84% And then the total quantity brought to market 00:51:26.120 --> 00:51:28.520 align:middle line:84% determines the market price, and then each firm 00:51:28.520 --> 00:51:32.450 align:middle line:84% sells their quantity at that market price. 00:51:32.450 --> 00:51:36.950 align:middle line:84% And we're going to start with everything exactly as before. 00:51:36.950 --> 00:51:39.530 align:middle line:90% So we have n firms. 00:51:39.530 --> 00:51:41.630 align:middle line:84% We have constant marginal cost, c. 00:51:41.630 --> 00:51:47.330 align:middle line:90% 00:51:47.330 --> 00:51:50.570 align:middle line:84% Remember this is the constant marginal cost of production. 00:51:50.570 --> 00:51:53.290 align:middle line:84% And then we're going to specify the same convenient demand 00:51:53.290 --> 00:51:58.690 align:middle line:84% that we usually specify, which is P of Q. 00:51:58.690 --> 00:52:00.830 align:middle line:84% If quantity Q is brought to market, 00:52:00.830 --> 00:52:02.570 align:middle line:84% what is the market price going to be? 00:52:02.570 --> 00:52:04.450 align:middle line:84% Well, intuitively, the market price 00:52:04.450 --> 00:52:07.690 align:middle line:84% should get smaller the more that's brought to market. 00:52:07.690 --> 00:52:12.530 align:middle line:84% And we're going to use the same specification the maximum of 1 00:52:12.530 --> 00:52:14.130 align:middle line:90% minus Q and 0. 00:52:14.130 --> 00:52:15.810 align:middle line:90% Exactly what we said before. 00:52:15.810 --> 00:52:18.490 align:middle line:84% So if more than one unit is brought to market, 00:52:18.490 --> 00:52:21.250 align:middle line:90% the price is just 0. 00:52:21.250 --> 00:52:24.950 align:middle line:84% And if nothing is brought to market, the price is 1, 00:52:24.950 --> 00:52:27.870 align:middle line:84% and then we interpolate between that linearly. 00:52:27.870 --> 00:52:30.820 align:middle line:90% 00:52:30.820 --> 00:52:35.540 align:middle line:84% But now the key difference is that the firms can't just 00:52:35.540 --> 00:52:39.040 align:middle line:90% produce immediately. 00:52:39.040 --> 00:52:42.260 align:middle line:84% They have to choose whether they want to enter this industry. 00:52:42.260 --> 00:52:47.320 align:middle line:84% So I can't just produce corn at my farm today. 00:52:47.320 --> 00:52:49.765 align:middle line:84% I have to first buy a farm and start entering the market 00:52:49.765 --> 00:52:51.640 align:middle line:84% and decide that I'm going to become a farmer. 00:52:51.640 --> 00:52:54.700 align:middle line:84% So in this example, the firms, each firm 00:52:54.700 --> 00:52:57.848 align:middle line:84% has to first make a choice about whether they want to participate 00:52:57.848 --> 00:52:59.640 align:middle line:84% in this industry, whether they want to say, 00:52:59.640 --> 00:53:02.080 align:middle line:84% pay the upfront costs that would allow them to produce. 00:53:02.080 --> 00:53:03.580 align:middle line:84% So in your Econ course, you probably 00:53:03.580 --> 00:53:06.500 align:middle line:84% heard the difference between fixed costs and marginal costs. 00:53:06.500 --> 00:53:09.600 align:middle line:84% Here, c is playing the role of marginal costs. 00:53:09.600 --> 00:53:11.847 align:middle line:84% This is a per unit cost I pay to produce. 00:53:11.847 --> 00:53:13.680 align:middle line:84% But I also have something like a fixed cost. 00:53:13.680 --> 00:53:15.460 align:middle line:84% I have to buy a factory to start producing 00:53:15.460 --> 00:53:17.220 align:middle line:84% or buy a farm to start producing. 00:53:17.220 --> 00:53:19.820 align:middle line:84% | this game is going to proceed in two stages. 00:53:19.820 --> 00:53:24.220 align:middle line:84% In stage 1, each firm simultaneously 00:53:24.220 --> 00:53:27.000 align:middle line:84% chooses whether to enter the market. 00:53:27.000 --> 00:53:28.960 align:middle line:84% In other words, pay the up-front cost 00:53:28.960 --> 00:53:30.600 align:middle line:90% to be a player in the market. 00:53:30.600 --> 00:53:42.380 align:middle line:84% So each firm simultaneously chooses whether to enter. 00:53:42.380 --> 00:53:46.080 align:middle line:90% 00:53:46.080 --> 00:53:49.120 align:middle line:90% And entering has a cost. 00:53:49.120 --> 00:53:52.400 align:middle line:84% I'll say, at cost-- well, we'd like to say c, 00:53:52.400 --> 00:53:54.400 align:middle line:84% but we already have c here, so I'll 00:53:54.400 --> 00:53:57.120 align:middle line:84% say it cost F. You could think of this as a fee 00:53:57.120 --> 00:53:58.840 align:middle line:90% or as a fixed cost. 00:53:58.840 --> 00:54:02.720 align:middle line:84% This is building a factory, buying the farm. 00:54:02.720 --> 00:54:08.040 align:middle line:84% And then stage 2, the firms who've 00:54:08.040 --> 00:54:10.300 align:middle line:84% entered each choose how much to produce. 00:54:10.300 --> 00:54:14.880 align:middle line:90% 00:54:14.880 --> 00:54:16.800 align:middle line:84% Maybe I'll say, the present firms, 00:54:16.800 --> 00:54:20.060 align:middle line:84% that is the firms who've entered, choose quantities. 00:54:20.060 --> 00:54:25.160 align:middle line:90% 00:54:25.160 --> 00:54:26.750 align:middle line:90% And again, this is simultaneous. 00:54:26.750 --> 00:54:33.090 align:middle line:90% 00:54:33.090 --> 00:54:37.730 align:middle line:84% So first, each firm chooses whether to enter at some cost F. 00:54:37.730 --> 00:54:40.370 align:middle line:84% Some subset of the firms has entered. 00:54:40.370 --> 00:54:42.610 align:middle line:84% And then the present-- this is not great grammar, 00:54:42.610 --> 00:54:44.402 align:middle line:84% but I just mean the firms that are present, 00:54:44.402 --> 00:54:47.210 align:middle line:84% the entered firms choose simultaneously 00:54:47.210 --> 00:54:48.510 align:middle line:90% the quantities to produce. 00:54:48.510 --> 00:54:51.570 align:middle line:84% And then we just play the Cournot game. 00:54:51.570 --> 00:54:53.925 align:middle line:84% I would say here, though, we have to be really careful. 00:54:53.925 --> 00:54:55.550 align:middle line:84% There's one detail I haven't specified. 00:54:55.550 --> 00:54:57.970 align:middle line:84% There's something I need to specify as part of the game. 00:54:57.970 --> 00:55:01.770 align:middle line:84% What have I not told you that's important? 00:55:01.770 --> 00:55:04.150 align:middle line:84% So we have the players, we have the actions, 00:55:04.150 --> 00:55:07.810 align:middle line:84% we have the payoffs, but information. 00:55:07.810 --> 00:55:11.250 align:middle line:84% So we have to be clear here, what do these firms in stage 2 00:55:11.250 --> 00:55:12.450 align:middle line:90% know? 00:55:12.450 --> 00:55:14.395 align:middle line:84% Do they know who else has entered? 00:55:14.395 --> 00:55:15.770 align:middle line:84% That's an important consideration 00:55:15.770 --> 00:55:17.090 align:middle line:90% in the way we model the game. 00:55:17.090 --> 00:55:19.490 align:middle line:84% We're going to assume that they do observe this. 00:55:19.490 --> 00:55:28.860 align:middle line:84% So I'll say, the set of firms that enter is observed, 00:55:28.860 --> 00:55:30.880 align:middle line:84% or maybe I'll just say, entering is observed. 00:55:30.880 --> 00:55:40.660 align:middle line:90% 00:55:40.660 --> 00:55:43.487 align:middle line:84% So if you want to start a factory, you want to buy a farm, 00:55:43.487 --> 00:55:44.320 align:middle line:90% people observe that. 00:55:44.320 --> 00:55:45.680 align:middle line:84% People see that you're now in the market. 00:55:45.680 --> 00:55:47.805 align:middle line:84% And that's potentially going to affect the way they 00:55:47.805 --> 00:55:49.220 align:middle line:90% behave in the second stage. 00:55:49.220 --> 00:55:50.120 align:middle line:90% And this is crucial. 00:55:50.120 --> 00:55:53.110 align:middle line:84% If you were a firm, and you were deciding how much to produce, 00:55:53.110 --> 00:55:54.860 align:middle line:84% part of your market research would exactly 00:55:54.860 --> 00:55:57.580 align:middle line:84% be to see what other firms are present in the market? 00:55:57.580 --> 00:55:59.340 align:middle line:84% And is there a new entrant that may change 00:55:59.340 --> 00:56:00.560 align:middle line:90% how much you want to produce? 00:56:00.560 --> 00:56:02.810 align:middle line:84% So we're going to assume that this market research has 00:56:02.810 --> 00:56:04.860 align:middle line:84% been done, and that this entrance decision is 00:56:04.860 --> 00:56:07.380 align:middle line:90% publicly observable. 00:56:07.380 --> 00:56:16.260 align:middle line:84% So now, we'd like to solve for a subgame perfect Nash 00:56:16.260 --> 00:56:18.820 align:middle line:90% equilibrium of this game. 00:56:18.820 --> 00:56:24.150 align:middle line:90% So let's solve for SPNE. 00:56:24.150 --> 00:56:27.030 align:middle line:84% And let's follow our procedure, step 1, 00:56:27.030 --> 00:56:28.530 align:middle line:90% identify all the subgames. 00:56:28.530 --> 00:56:32.110 align:middle line:90% 00:56:32.110 --> 00:56:33.290 align:middle line:90% So any thoughts? 00:56:33.290 --> 00:56:40.750 align:middle line:84% What are going to be the subgames of this game? 00:56:40.750 --> 00:56:44.150 align:middle line:84% Well, certainly, the entire game is a subgame. 00:56:44.150 --> 00:56:48.070 align:middle line:84% So first, let's find the subgames. 00:56:48.070 --> 00:56:49.850 align:middle line:90% Well, we have the entire game. 00:56:49.850 --> 00:56:52.510 align:middle line:90% 00:56:52.510 --> 00:56:53.370 align:middle line:90% That's always easy. 00:56:53.370 --> 00:56:55.910 align:middle line:84% That's what happens at the beginning of stage one. 00:56:55.910 --> 00:57:00.510 align:middle line:84% But then there's going to be a lot of subgames in stage 2. 00:57:00.510 --> 00:57:06.448 align:middle line:84% So in stage 2, what's going to start a subgame? 00:57:06.448 --> 00:57:07.990 align:middle line:84% A subgame can only start at a history 00:57:07.990 --> 00:57:09.693 align:middle line:90% that's observed by the players. 00:57:09.693 --> 00:57:11.110 align:middle line:84% That's another way of saying, when 00:57:11.110 --> 00:57:13.750 align:middle line:84% we say that the initial node must be in its own information 00:57:13.750 --> 00:57:16.835 align:middle line:84% set, think of that as saying a subgame can start at the history 00:57:16.835 --> 00:57:19.210 align:middle line:84% if everyone knows we're at that history when we're at it. 00:57:19.210 --> 00:57:21.880 align:middle line:84% There can't be any uncertainty about which history we're at. 00:57:21.880 --> 00:57:23.780 align:middle line:90% So what is observed? 00:57:23.780 --> 00:57:26.320 align:middle line:84% Well, here, what is the history at stage 2? 00:57:26.320 --> 00:57:29.400 align:middle line:84% Well, what we know at stage 2, is which firms have entered 00:57:29.400 --> 00:57:31.640 align:middle line:90% and which firms haven't. 00:57:31.640 --> 00:57:33.440 align:middle line:84% And in fact, each of those histories 00:57:33.440 --> 00:57:35.720 align:middle line:90% will start a different subgame. 00:57:35.720 --> 00:57:37.560 align:middle line:84% So in fact, the subgames are going 00:57:37.560 --> 00:57:41.280 align:middle line:84% to be parametrized by the set of firms who entered. 00:57:41.280 --> 00:57:43.960 align:middle line:84% I don't have to specify who didn't enter because that's just 00:57:43.960 --> 00:57:44.698 align:middle line:90% everyone else. 00:57:44.698 --> 00:57:46.740 align:middle line:84% Once I know who entered, I know who didn't enter. 00:57:46.740 --> 00:57:49.380 align:middle line:90% 00:57:49.380 --> 00:57:53.240 align:middle line:90% And this is in stage 1. 00:57:53.240 --> 00:57:57.080 align:middle line:84% And then we have a subgame parametrized by-- 00:57:57.080 --> 00:58:01.120 align:middle line:90% going to say parametrized. 00:58:01.120 --> 00:58:05.580 align:middle line:84% That just means we can discard the subgame by this quantity. 00:58:05.580 --> 00:58:11.100 align:middle line:84% Maybe I'll call it I subset of 1 through n. 00:58:11.100 --> 00:58:14.240 align:middle line:90% 00:58:14.240 --> 00:58:16.080 align:middle line:84% And the interpretation of this is 00:58:16.080 --> 00:58:17.500 align:middle line:90% the set of players who entered. 00:58:17.500 --> 00:58:30.770 align:middle line:90% 00:58:30.770 --> 00:58:36.490 align:middle line:84% And I guess, maybe, if we want to be really formal, 00:58:36.490 --> 00:58:38.770 align:middle line:84% let's say it's not an empty subset. 00:58:38.770 --> 00:58:40.490 align:middle line:84% Because if no one enters, then we're 00:58:40.490 --> 00:58:42.610 align:middle line:84% actually at a terminal node because no one has any decisions 00:58:42.610 --> 00:58:43.110 align:middle line:90% to make. 00:58:43.110 --> 00:58:45.730 align:middle line:84% So that's not going to start a subgame. 00:58:45.730 --> 00:58:49.130 align:middle line:84% So maybe I'll just write it like this. 00:58:49.130 --> 00:58:50.470 align:middle line:90% Sorry, this is math notation. 00:58:50.470 --> 00:58:54.210 align:middle line:84% All it means is we have some non-empty subset of firms 00:58:54.210 --> 00:58:54.990 align:middle line:90% have entered. 00:58:54.990 --> 00:58:56.330 align:middle line:90% We all observe that. 00:58:56.330 --> 00:58:59.510 align:middle line:84% And then based upon that, we all decide how to play. 00:58:59.510 --> 00:59:02.732 align:middle line:84% Of course, remember, if this is the set of firms who've entered, 00:59:02.732 --> 00:59:04.690 align:middle line:84% this is also the set of firms who have a choice 00:59:04.690 --> 00:59:06.010 align:middle line:90% to make in the next period. 00:59:06.010 --> 00:59:07.510 align:middle line:84% The other firms aren't even present. 00:59:07.510 --> 00:59:09.135 align:middle line:84% They don't even matter in this subgame. 00:59:09.135 --> 00:59:13.130 align:middle line:84% So sometimes a subgame might involve 00:59:13.130 --> 00:59:15.290 align:middle line:90% only a subset of players. 00:59:15.290 --> 00:59:16.630 align:middle line:90% And that's happening here. 00:59:16.630 --> 00:59:19.870 align:middle line:84% Only these players are going to be involved in the subgame. 00:59:19.870 --> 00:59:23.110 align:middle line:84% Because if you don't enter, you have no choice to make. 00:59:23.110 --> 00:59:29.830 align:middle line:84% OK, so now let's try to work backwards and try to solve this. 00:59:29.830 --> 00:59:31.730 align:middle line:90% So step 2 is working backwards. 00:59:31.730 --> 00:59:42.070 align:middle line:90% 00:59:42.070 --> 00:59:44.350 align:middle line:90% So I've achieved step 1. 00:59:44.350 --> 00:59:46.310 align:middle line:90% I've identified the subgames. 00:59:46.310 --> 00:59:48.405 align:middle line:90% Now I want to work backwards. 00:59:48.405 --> 00:59:50.030 align:middle line:84% So I'm going to start with the subgames 00:59:50.030 --> 00:59:53.110 align:middle line:84% here and try to find a Nash equilibrium of each 00:59:53.110 --> 00:59:55.030 align:middle line:90% of these subgames. 00:59:55.030 --> 01:00:01.230 align:middle line:84% So suppose we're in stage 2, and suppose 01:00:01.230 --> 01:00:04.390 align:middle line:84% that the set of firms that has entered is I. 01:00:04.390 --> 01:00:10.770 align:middle line:84% So let's say we're given non-empty subset. 01:00:10.770 --> 01:00:20.360 align:middle line:90% 01:00:20.360 --> 01:00:23.200 align:middle line:84% So I'm saying, suppose we're at the contingency 01:00:23.200 --> 01:00:25.760 align:middle line:84% where this is the set of firms who've entered. 01:00:25.760 --> 01:00:27.720 align:middle line:90% That defines a subgame. 01:00:27.720 --> 01:00:29.960 align:middle line:84% And now I need to try to solve for a Nash 01:00:29.960 --> 01:00:31.560 align:middle line:90% equilibrium of this subgame. 01:00:31.560 --> 01:00:36.640 align:middle line:84% So I need to think, well, what are the substrategies 01:00:36.640 --> 01:00:37.340 align:middle line:90% in this subgame? 01:00:37.340 --> 01:00:38.840 align:middle line:84% Remember, we just need to think what 01:00:38.840 --> 01:00:41.680 align:middle line:84% are the decisions that need to be made within this subgame? 01:00:41.680 --> 01:00:48.288 align:middle line:84% Well, what we need is each player firm i in I-- 01:00:48.288 --> 01:00:50.580 align:middle line:84% the other firms don't have any decision to make, right? 01:00:50.580 --> 01:00:52.120 align:middle line:84% It's only the firms who've entered. 01:00:52.120 --> 01:00:53.680 align:middle line:84% Each firm i and I is going to choose 01:00:53.680 --> 01:00:55.700 align:middle line:90% a quantity q I to produce. 01:00:55.700 --> 01:01:00.700 align:middle line:90% 01:01:00.700 --> 01:01:04.460 align:middle line:84% SO these are the strategies within this subgame. 01:01:04.460 --> 01:01:07.760 align:middle line:84% Once I'm in this subgame, each player i who's involved just 01:01:07.760 --> 01:01:09.172 align:middle line:90% needs to choose some quantity. 01:01:09.172 --> 01:01:11.880 align:middle line:84% And I think we just said this is just some non-negative quantity. 01:01:11.880 --> 01:01:15.770 align:middle line:90% 01:01:15.770 --> 01:01:22.305 align:middle line:84% So now we'd like to solve for a Nash equilibrium of the subgame. 01:01:22.305 --> 01:01:23.430 align:middle line:90% Do we know what to do here? 01:01:23.430 --> 01:01:25.870 align:middle line:84% Does this fit in with something we've studied previously? 01:01:25.870 --> 01:01:29.050 align:middle line:90% 01:01:29.050 --> 01:01:31.010 align:middle line:90% Any thoughts? 01:01:31.010 --> 01:01:34.130 align:middle line:84% Well, this subgame is just the classical Cournot competition 01:01:34.130 --> 01:01:38.530 align:middle line:84% game we studied but with a possibly smaller set of firms. 01:01:38.530 --> 01:01:41.450 align:middle line:84% So we already know how to analyze the subgame 01:01:41.450 --> 01:01:44.410 align:middle line:84% because the subgame reduces to something we've already studied. 01:01:44.410 --> 01:01:53.170 align:middle line:84% So this is just standard Cournot with-- 01:01:53.170 --> 01:01:56.830 align:middle line:84% well, effectively, we only have as many firms as have entered. 01:01:56.830 --> 01:01:58.650 align:middle line:84% So we need some notation for that. 01:01:58.650 --> 01:02:03.890 align:middle line:84% I'll say what this means is, if I have a set I, 01:02:03.890 --> 01:02:05.870 align:middle line:84% this means the number of elements of that set. 01:02:05.870 --> 01:02:08.290 align:middle line:84% So this is just counting how many firms have entered, 01:02:08.290 --> 01:02:09.590 align:middle line:90% so with this many firms. 01:02:09.590 --> 01:02:14.540 align:middle line:90% 01:02:14.540 --> 01:02:18.020 align:middle line:90% And we know how to solve that. 01:02:18.020 --> 01:02:21.620 align:middle line:84% Notice one thing that all that matters for the Cournot outcome 01:02:21.620 --> 01:02:23.180 align:middle line:90% is the number of firms. 01:02:23.180 --> 01:02:25.760 align:middle line:84% So even if a different set of firms is entered, 01:02:25.760 --> 01:02:27.980 align:middle line:84% each entering firm is going to produce-- the amount 01:02:27.980 --> 01:02:30.260 align:middle line:84% that each firm who's entered will produce depends only 01:02:30.260 --> 01:02:32.240 align:middle line:84% on the number of firms who have entered, 01:02:32.240 --> 01:02:33.900 align:middle line:90% not the identities of the firms. 01:02:33.900 --> 01:02:35.460 align:middle line:84% But that's because of our assumption 01:02:35.460 --> 01:02:37.220 align:middle line:84% that the firms are all symmetric. 01:02:37.220 --> 01:02:39.340 align:middle line:84% If one firm could produce at a lower marginal cost 01:02:39.340 --> 01:02:40.620 align:middle line:84% than another firm, then we'd have 01:02:40.620 --> 01:02:42.520 align:middle line:84% to keep track of the identities of the firms. 01:02:42.520 --> 01:02:45.080 align:middle line:84% But since all the firms are ex-ante equivalent, 01:02:45.080 --> 01:02:47.560 align:middle line:84% ex-ante identical, we don't really care who's entered. 01:02:47.560 --> 01:02:50.420 align:middle line:84% We just care how many people have entered. 01:02:50.420 --> 01:02:52.240 align:middle line:84% And what is the solution of this? 01:02:52.240 --> 01:02:58.580 align:middle line:84% Does anyone recall what is qi star going 01:02:58.580 --> 01:03:01.640 align:middle line:90% to be, anyone remember? 01:03:01.640 --> 01:03:06.060 align:middle line:84% This is, I think for tests, a good kind of thing to know. 01:03:06.060 --> 01:03:09.180 align:middle line:84% Well, we have 1 minus c on the top because the more costly 01:03:09.180 --> 01:03:12.310 align:middle line:84% it is to produce, the less each firm produces. 01:03:12.310 --> 01:03:15.210 align:middle line:84% And on the bottom, well, the more firms there are, 01:03:15.210 --> 01:03:16.710 align:middle line:90% the less we produce. 01:03:16.710 --> 01:03:19.870 align:middle line:84% We know a monopolist produces 1 minus c over 2. 01:03:19.870 --> 01:03:24.130 align:middle line:84% So if we expand that formula, or that pattern, 01:03:24.130 --> 01:03:26.870 align:middle line:84% we're going to get the number of firms plus 1. 01:03:26.870 --> 01:03:32.230 align:middle line:84% So if I am the only firm, so the cardinality of I is 1, 01:03:32.230 --> 01:03:34.290 align:middle line:84% then I is the only firm and monopolist. 01:03:34.290 --> 01:03:36.110 align:middle line:90% And I choose 1 minus c over 2. 01:03:36.110 --> 01:03:38.937 align:middle line:84% If there's two firms, we each produce 1 minus c over 3. 01:03:38.937 --> 01:03:41.270 align:middle line:84% If there's three firms, we each produce 1 minus c over 4 01:03:41.270 --> 01:03:41.970 align:middle line:90% and so on. 01:03:41.970 --> 01:03:44.590 align:middle line:90% 01:03:44.590 --> 01:03:48.350 align:middle line:84% Now let's think back to how we approach this problem before. 01:03:48.350 --> 01:03:50.190 align:middle line:90% We looked at the subgame. 01:03:50.190 --> 01:03:52.610 align:middle line:84% We found a Nash equilibrium of the subgame. 01:03:52.610 --> 01:03:55.230 align:middle line:84% But then the key step was to compute the payoffs in that Nash 01:03:55.230 --> 01:03:57.708 align:middle line:84% equilibrium because the payoffs in that Nash equilibrium 01:03:57.708 --> 01:03:59.750 align:middle line:84% determined the choices that the players are going 01:03:59.750 --> 01:04:01.470 align:middle line:90% to make higher up in the tree. 01:04:01.470 --> 01:04:04.470 align:middle line:84% So this is what will actually be done, but let's compute what 01:04:04.470 --> 01:04:07.510 align:middle line:90% the payoffs will actually be. 01:04:07.510 --> 01:04:10.780 align:middle line:90% So how do we compute payoffs? 01:04:10.780 --> 01:04:12.760 align:middle line:84% Well, my payoff depends on how much I produce 01:04:12.760 --> 01:04:14.780 align:middle line:84% and also what the market price is. 01:04:14.780 --> 01:04:17.800 align:middle line:84% So let's now compute the market price. 01:04:17.800 --> 01:04:21.360 align:middle line:84% So the market price is a function 01:04:21.360 --> 01:04:24.720 align:middle line:90% of the total amount produced. 01:04:24.720 --> 01:04:26.800 align:middle line:84% Each of the firms is producing this amount. 01:04:26.800 --> 01:04:31.900 align:middle line:90% But there's I of those firms. 01:04:31.900 --> 01:04:36.120 align:middle line:84% So the total amount produced is going to be I 01:04:36.120 --> 01:04:43.200 align:middle line:90% over I plus 1 times 1 minus c. 01:04:43.200 --> 01:04:45.120 align:middle line:90% Do you see we got this, right? 01:04:45.120 --> 01:04:49.295 align:middle line:84% Each firm is producing 1 minus c over I plus 1. 01:04:49.295 --> 01:04:51.920 align:middle line:84% But there's I firms altogether, so when we add it all up, This. 01:04:51.920 --> 01:04:55.280 align:middle line:90% Is the total amount produced. 01:04:55.280 --> 01:04:57.140 align:middle line:84% And now what is the formula for P-- 01:04:57.140 --> 01:05:00.320 align:middle line:90% 01:05:00.320 --> 01:05:03.600 align:middle line:84% it's actually maybe a cleaner way 01:05:03.600 --> 01:05:06.020 align:middle line:84% of doing it is we don't really just care about p. 01:05:06.020 --> 01:05:08.090 align:middle line:90% We care about p minus c. 01:05:08.090 --> 01:05:12.010 align:middle line:84% Because this is going to be the profit per unit 01:05:12.010 --> 01:05:13.250 align:middle line:90% that each firm gets. 01:05:13.250 --> 01:05:15.530 align:middle line:84% Because each firm is producing this amount. 01:05:15.530 --> 01:05:17.770 align:middle line:84% This is going to be the market price. 01:05:17.770 --> 01:05:19.970 align:middle line:84% This is the cost per unit, so the difference 01:05:19.970 --> 01:05:23.170 align:middle line:84% is going to be the profit per unit. 01:05:23.170 --> 01:05:25.630 align:middle line:84% And here, we get a nice algebra simplification 01:05:25.630 --> 01:05:29.930 align:middle line:90% because we get p is 1 minus q. 01:05:29.930 --> 01:05:34.810 align:middle line:84% So we're going to get 1 minus this minus c. 01:05:34.810 --> 01:05:38.650 align:middle line:84% But that's 1 minus c minus some fraction of 1 minus c. 01:05:38.650 --> 01:05:41.650 align:middle line:84% So we're just going to get the residual fraction of 1 minus c. 01:05:41.650 --> 01:05:49.970 align:middle line:84% And we're exactly going to get 1 minus c over this. 01:05:49.970 --> 01:05:51.070 align:middle line:90% You see how this happens? 01:05:51.070 --> 01:05:55.330 align:middle line:84% We get all of 1 minus c minus the fraction i over I 01:05:55.330 --> 01:05:57.730 align:middle line:90% plus 1 of 1 minus c. 01:05:57.730 --> 01:05:59.750 align:middle line:84% So what's 1 minus i over I plus 1? 01:05:59.750 --> 01:06:01.280 align:middle line:90% It's just one over I plus 1. 01:06:01.280 --> 01:06:03.030 align:middle line:84% If that's tricky, we can go over it again. 01:06:03.030 --> 01:06:05.910 align:middle line:90% But any questions on that? 01:06:05.910 --> 01:06:07.070 align:middle line:90% No? 01:06:07.070 --> 01:06:09.730 align:middle line:90% OK, and this also makes sense. 01:06:09.730 --> 01:06:14.030 align:middle line:84% My profit per unit is decreasing in my cost 01:06:14.030 --> 01:06:18.070 align:middle line:84% and also decreasing the number of firms who are entered. 01:06:18.070 --> 01:06:23.510 align:middle line:84% So now what we can write down is-- 01:06:23.510 --> 01:06:25.790 align:middle line:84% we can actually compute-- so this is a good, 01:06:25.790 --> 01:06:27.050 align:middle line:90% I think, thing to remember. 01:06:27.050 --> 01:06:36.110 align:middle line:84% What we're computing is the per firm Cournot profit. 01:06:36.110 --> 01:06:38.870 align:middle line:84% And it's a bit messy to keep writing cardinality of I, 01:06:38.870 --> 01:06:40.730 align:middle line:90% so let's say with k firms. 01:06:40.730 --> 01:06:43.535 align:middle line:90% 01:06:43.535 --> 01:06:45.910 align:middle line:84% So what I want to say is if we're playing a Cournot game, 01:06:45.910 --> 01:06:48.910 align:middle line:84% and there's k firms all together, 01:06:48.910 --> 01:06:52.270 align:middle line:84% what is each firm going to get in equilibrium? 01:06:52.270 --> 01:06:55.310 align:middle line:90% Well, it's going to be-- 01:06:55.310 --> 01:06:59.803 align:middle line:90% so I'll call this pi k star. 01:06:59.803 --> 01:07:01.470 align:middle line:84% And it's going to be the amount produced 01:07:01.470 --> 01:07:05.920 align:middle line:84% by each firm times the profit per unit. 01:07:05.920 --> 01:07:08.240 align:middle line:90% So this is going to be-- 01:07:08.240 --> 01:07:12.532 align:middle line:84% well, I'll just use this formula --1 minus c over k plus 1. 01:07:12.532 --> 01:07:13.740 align:middle line:90% Notice it's exactly the same. 01:07:13.740 --> 01:07:15.490 align:middle line:84% I've just changed notation because I don't 01:07:15.490 --> 01:07:17.822 align:middle line:90% want to keep writing this. 01:07:17.822 --> 01:07:19.280 align:middle line:84% And then I'm going to multiply that 01:07:19.280 --> 01:07:22.500 align:middle line:84% by the profit per unit, which is exactly this. 01:07:22.500 --> 01:07:23.660 align:middle line:90% Again, I'll use my k. 01:07:23.660 --> 01:07:27.360 align:middle line:90% 01:07:27.360 --> 01:07:30.400 align:middle line:84% And it actually is a pretty simple formula that-- 01:07:30.400 --> 01:07:33.202 align:middle line:84% I think, this is kind of a good formula to memorize for exams. 01:07:33.202 --> 01:07:34.660 align:middle line:84% I mean, you can always rederive it, 01:07:34.660 --> 01:07:37.320 align:middle line:84% but I think that's kind of a nice thing to keep in mind. 01:07:37.320 --> 01:07:39.200 align:middle line:84% Let's make sure we see what we're doing here. 01:07:39.200 --> 01:07:43.280 align:middle line:90% The first term is the quantity. 01:07:43.280 --> 01:07:48.400 align:middle line:84% The second term is the profit per unit. 01:07:48.400 --> 01:07:51.503 align:middle line:84% And it turns out that these are exactly the same. 01:07:51.503 --> 01:07:52.420 align:middle line:90% Why are they the same? 01:07:52.420 --> 01:07:55.117 align:middle line:84% Well, this goes back to you're maximizing area of a rectangle 01:07:55.117 --> 01:07:57.700 align:middle line:84% subject to a perimeter, and you always want it to be a square. 01:07:57.700 --> 01:07:58.992 align:middle line:90% But that doesn't really matter. 01:07:58.992 --> 01:08:01.490 align:middle line:90% This is what you get. 01:08:01.490 --> 01:08:08.530 align:middle line:84% OK, so now we can understand if k firms enter, 01:08:08.530 --> 01:08:09.730 align:middle line:90% what's going to happen? 01:08:09.730 --> 01:08:13.170 align:middle line:84% If k firms enter, and we play a Nash equilibrium 01:08:13.170 --> 01:08:16.170 align:middle line:84% of that subgame, then the k entering firms 01:08:16.170 --> 01:08:18.490 align:middle line:90% will each produce this amount. 01:08:18.490 --> 01:08:20.370 align:middle line:84% They'll each get this profit per unit, 01:08:20.370 --> 01:08:22.770 align:middle line:84% and they'll each get this total profit. 01:08:22.770 --> 01:08:27.850 align:middle line:90% And now we can move to stage 1. 01:08:27.850 --> 01:08:31.810 align:middle line:90% So let's go here. 01:08:31.810 --> 01:08:35.370 align:middle line:84% So now let's move back to stage 1. 01:08:35.370 --> 01:08:37.430 align:middle line:84% So we've actually simultaneously solved. 01:08:37.430 --> 01:08:38.850 align:middle line:84% At first, this seemed really hard. 01:08:38.850 --> 01:08:41.550 align:middle line:84% We had a huge array of subgames to deal with. 01:08:41.550 --> 01:08:43.729 align:middle line:84% But we actually gave one general formula 01:08:43.729 --> 01:08:45.830 align:middle line:90% that works for any subgame. 01:08:45.830 --> 01:08:48.210 align:middle line:84% Whatever subgame you give me, I just compute 01:08:48.210 --> 01:08:49.810 align:middle line:90% how many firms have entered. 01:08:49.810 --> 01:08:51.330 align:middle line:90% That becomes my k. 01:08:51.330 --> 01:08:54.510 align:middle line:84% And now I've computed the Nash equilibrium of that subgame. 01:08:54.510 --> 01:08:56.890 align:middle line:84% So I've actually covered every subgame at once. 01:08:56.890 --> 01:09:00.899 align:middle line:84% And now we need to move to stage 1. 01:09:00.899 --> 01:09:05.640 align:middle line:84% So the question at stage 1 is an entry decision. 01:09:05.640 --> 01:09:16.899 align:middle line:84% So now each firm chooses maybe enter N/X, enter or exit. 01:09:16.899 --> 01:09:18.819 align:middle line:84% A better term would be enter or not enter. 01:09:18.819 --> 01:09:24.760 align:middle line:84% But yeah, enter and exit is cleaner to say. 01:09:24.760 --> 01:09:30.620 align:middle line:84% Well, now, this becomes a bit tricky because let's look 01:09:30.620 --> 01:09:33.859 align:middle line:90% at each firm's decision problem. 01:09:33.859 --> 01:09:36.680 align:middle line:84% If they exit, that is if they don't enter at all, 01:09:36.680 --> 01:09:39.420 align:middle line:84% what is their payoff going to be? 01:09:39.420 --> 01:09:40.939 align:middle line:90% Just 0, right? 01:09:40.939 --> 01:09:43.660 align:middle line:90% 01:09:43.660 --> 01:09:46.220 align:middle line:84% If they enter, and then after entering, 01:09:46.220 --> 01:09:48.180 align:middle line:84% everyone plays a Nash equilibrium, 01:09:48.180 --> 01:09:50.740 align:middle line:84% what is their payoff going to be? 01:09:50.740 --> 01:09:52.500 align:middle line:84% Well, this is tricky because it depends 01:09:52.500 --> 01:09:54.737 align:middle line:90% on how many other firms enter. 01:09:54.737 --> 01:09:56.320 align:middle line:84% So we have a formula for their payoff, 01:09:56.320 --> 01:09:59.670 align:middle line:84% but it's going to depend on the total number of firms who enter. 01:09:59.670 --> 01:10:08.070 align:middle line:84% So it's going to be pi k star, which equals 1 minus c over k 01:10:08.070 --> 01:10:13.290 align:middle line:84% plus 1 squared if the total number of firms who enter is k. 01:10:13.290 --> 01:10:18.150 align:middle line:90% 01:10:18.150 --> 01:10:21.490 align:middle line:84% Which means if k minus 1 other firms also enter. 01:10:21.490 --> 01:10:31.390 align:middle line:90% 01:10:31.390 --> 01:10:34.470 align:middle line:84% So if k minus 1 firms also enter, well, when I enter, 01:10:34.470 --> 01:10:38.770 align:middle line:84% that makes k, and therefore, this is going to be my payoff. 01:10:38.770 --> 01:10:45.413 align:middle line:90% 01:10:45.413 --> 01:10:46.830 align:middle line:84% I think if you just stare at this, 01:10:46.830 --> 01:10:48.730 align:middle line:84% it starts getting hard because you've got to figure out the k. 01:10:48.730 --> 01:10:49.890 align:middle line:90% The k determines the enter. 01:10:49.890 --> 01:10:50.750 align:middle line:90% It gets hard. 01:10:50.750 --> 01:10:52.930 align:middle line:84% So I think the best way to do it is to say, 01:10:52.930 --> 01:10:57.440 align:middle line:84% can we find an equilibrium where k firms enter? 01:10:57.440 --> 01:11:02.780 align:middle line:84% So let's look for an equilibrium. 01:11:02.780 --> 01:11:13.237 align:middle line:90% 01:11:13.237 --> 01:11:15.320 align:middle line:84% And if there's an equilibrium where k firms enter, 01:11:15.320 --> 01:11:16.820 align:middle line:84% it's not actually going to matter which k. 01:11:16.820 --> 01:11:18.195 align:middle line:84% Then there will be an equilibrium 01:11:18.195 --> 01:11:21.300 align:middle line:84% for any subset of size k because all the firms are the same. 01:11:21.300 --> 01:11:22.740 align:middle line:90% But this may not exist. 01:11:22.740 --> 01:11:24.440 align:middle line:84% So let's try to think, if there's 01:11:24.440 --> 01:11:28.240 align:middle line:84% an equilibrium in which k firms enter, what must be true? 01:11:28.240 --> 01:11:31.160 align:middle line:84% It means each of the firms that entered, 01:11:31.160 --> 01:11:33.600 align:middle line:84% finds it optimal to enter, and each of the firms 01:11:33.600 --> 01:11:37.600 align:middle line:84% that doesn't enter, finds it optimal not to enter. 01:11:37.600 --> 01:11:40.520 align:middle line:84% So I guess the question is, does any firm 01:11:40.520 --> 01:11:45.273 align:middle line:84% have a profitable deviation at stage 1? 01:11:45.273 --> 01:11:46.940 align:middle line:84% So let's look for profitable deviations. 01:11:46.940 --> 01:11:53.840 align:middle line:84% Let's consider a firm that entered. 01:11:53.840 --> 01:11:54.500 align:middle line:90% What did I use? 01:11:54.500 --> 01:11:59.800 align:middle line:84% N. So let's consider this deviation. 01:11:59.800 --> 01:12:03.540 align:middle line:84% Let's look at a single firm that did enter. 01:12:03.540 --> 01:12:07.420 align:middle line:84% So here k firms, let's assume k is at least 1. 01:12:07.420 --> 01:12:09.060 align:middle line:84% Let's look at a firm that entered 01:12:09.060 --> 01:12:14.220 align:middle line:84% and consider what would happen if they deviated and exited? 01:12:14.220 --> 01:12:18.140 align:middle line:90% Well, this firm is entering. 01:12:18.140 --> 01:12:21.600 align:middle line:84% So what is their payoff from entering? 01:12:21.600 --> 01:12:23.540 align:middle line:84% This is their payoff in equilibrium. 01:12:23.540 --> 01:12:26.300 align:middle line:90% Their payoff is-- well, what? 01:12:26.300 --> 01:12:28.980 align:middle line:84% So let's take the perspective of one of the k firms 01:12:28.980 --> 01:12:30.100 align:middle line:90% who has entered. 01:12:30.100 --> 01:12:34.220 align:middle line:84% And they're saying, what is my payoff going to be in this game? 01:12:34.220 --> 01:12:38.380 align:middle line:84% Well k firms in total enter, so my payoff 01:12:38.380 --> 01:12:41.000 align:middle line:90% is pi k star from entering. 01:12:41.000 --> 01:12:45.820 align:middle line:84% But I have to pay this cost, this fee, F. And, I'm sorry. 01:12:45.820 --> 01:12:51.820 align:middle line:84% I missed that part, minus F. Because if I enter, 01:12:51.820 --> 01:12:53.470 align:middle line:84% I get this in that subgame, but I also 01:12:53.470 --> 01:12:54.845 align:middle line:84% have to pay the fee for entering. 01:12:54.845 --> 01:12:57.590 align:middle line:84% Otherwise, this would be a pretty easy problem, OK. 01:12:57.590 --> 01:13:02.430 align:middle line:84% So this is what I get in equilibrium. 01:13:02.430 --> 01:13:03.650 align:middle line:90% What if I exit? 01:13:03.650 --> 01:13:06.350 align:middle line:90% 01:13:06.350 --> 01:13:08.090 align:middle line:90% Well, then I get 0, right? 01:13:08.090 --> 01:13:10.950 align:middle line:90% 01:13:10.950 --> 01:13:14.790 align:middle line:84% So what we need is each of these firms that 01:13:14.790 --> 01:13:17.270 align:middle line:84% entered is willing to enter because this is 01:13:17.270 --> 01:13:20.782 align:middle line:90% greater than or equal to this. 01:13:20.782 --> 01:13:22.710 align:middle line:84% But now you might say, wait a second. 01:13:22.710 --> 01:13:25.125 align:middle line:84% If each of these firms found it optimal to enter, 01:13:25.125 --> 01:13:27.750 align:middle line:84% doesn't that mean we're going to have a deviation because isn't 01:13:27.750 --> 01:13:29.550 align:middle line:84% one of the firms that didn't enter also 01:13:29.550 --> 01:13:31.295 align:middle line:90% going to want to enter? 01:13:31.295 --> 01:13:32.670 align:middle line:84% So now it seems like we're stuck. 01:13:32.670 --> 01:13:35.045 align:middle line:84% We have to make sure that the firms that entered actually 01:13:35.045 --> 01:13:36.430 align:middle line:90% find it optimal to enter. 01:13:36.430 --> 01:13:37.972 align:middle line:84% But if they find it optimal to enter, 01:13:37.972 --> 01:13:40.138 align:middle line:84% why doesn't one of the other firms that didn't enter 01:13:40.138 --> 01:13:41.250 align:middle line:90% deviate and enter as well? 01:13:41.250 --> 01:13:43.708 align:middle line:90% 01:13:43.708 --> 01:13:45.250 align:middle line:84% Well, let's go through their problem. 01:13:45.250 --> 01:13:49.270 align:middle line:84% Let's look at a firm that didn't enter and consider 01:13:49.270 --> 01:13:51.080 align:middle line:90% their deviation. 01:13:51.080 --> 01:13:55.640 align:middle line:84% So if they didn't enter, they're getting a payoff of 0. 01:13:55.640 --> 01:13:57.880 align:middle line:90% What if one of these firms-- 01:13:57.880 --> 01:14:00.440 align:middle line:84% the arrows may not-- hopefully, what I'm saying verbally 01:14:00.440 --> 01:14:02.023 align:middle line:84% makes it clear what these arrows mean. 01:14:02.023 --> 01:14:03.760 align:middle line:84% You might want to take notes on this. 01:14:03.760 --> 01:14:07.200 align:middle line:84% So if I'm one of the firms that didn't enter, and I deviate now, 01:14:07.200 --> 01:14:09.380 align:middle line:84% and I enter, what is my payoff going to be? 01:14:09.380 --> 01:14:12.880 align:middle line:90% 01:14:12.880 --> 01:14:13.800 align:middle line:90% Yeah? 01:14:13.800 --> 01:14:16.600 align:middle line:84% AUDIENCE: Is it pi k plus 1 star minus F? 01:14:16.600 --> 01:14:19.040 align:middle line:84% IAN BELL: Exactly, and this is the key difference. 01:14:19.040 --> 01:14:21.280 align:middle line:84% If I'm one of the firms who's entered, 01:14:21.280 --> 01:14:24.280 align:middle line:84% there's only k minus 1 other firms who've entered. 01:14:24.280 --> 01:14:26.260 align:middle line:84% But if I'm one of the firms who hadn't entered, 01:14:26.260 --> 01:14:28.680 align:middle line:84% now I face a different problem because I face 01:14:28.680 --> 01:14:30.480 align:middle line:90% k other firms who've entered. 01:14:30.480 --> 01:14:33.480 align:middle line:84% And this means that if I don't enter-- sorry, if I enter, 01:14:33.480 --> 01:14:41.215 align:middle line:84% I'll get pi k plus 1 star minus F. 01:14:41.215 --> 01:14:43.340 align:middle line:84% Because there's already k the firms who've entered. 01:14:43.340 --> 01:14:45.400 align:middle line:84% If I deviate and enter as well, now there's k plus 1 of us 01:14:45.400 --> 01:14:46.360 align:middle line:90% who've entered. 01:14:46.360 --> 01:14:51.010 align:middle line:84% And I need this also to be unprofitable. 01:14:51.010 --> 01:14:51.910 align:middle line:90% So I need this. 01:14:51.910 --> 01:14:55.450 align:middle line:90% 01:14:55.450 --> 01:14:58.030 align:middle line:84% So what is my condition for this equilibrium? 01:14:58.030 --> 01:15:00.930 align:middle line:90% 01:15:00.930 --> 01:15:04.370 align:middle line:84% Well, F has to be small enough so that the firms that enter 01:15:04.370 --> 01:15:07.650 align:middle line:84% find it optimal to do so, but large enough 01:15:07.650 --> 01:15:09.250 align:middle line:84% so that the firms that don't enter 01:15:09.250 --> 01:15:12.290 align:middle line:90% find it optimal to not enter. 01:15:12.290 --> 01:15:20.650 align:middle line:84% And what we need is pi k plus 1 star is less than or equal to F, 01:15:20.650 --> 01:15:23.070 align:middle line:84% is less than or equal to pi k star. 01:15:23.070 --> 01:15:28.850 align:middle line:90% 01:15:28.850 --> 01:15:33.570 align:middle line:84% And indeed, these pi stars are decreasing in k. 01:15:33.570 --> 01:15:35.930 align:middle line:90% Let's look at our formula. 01:15:35.930 --> 01:15:38.890 align:middle line:84% This is definitely decreasing in k. 01:15:38.890 --> 01:15:40.930 align:middle line:90% So this makes sense. 01:15:40.930 --> 01:15:43.930 align:middle line:84% This inequality says, if I'm one of the firms that did enter, 01:15:43.930 --> 01:15:46.250 align:middle line:84% and I'm getting pi k star, it was worth it for me 01:15:46.250 --> 01:15:49.500 align:middle line:84% to enter because what I'm getting is better than the fee 01:15:49.500 --> 01:15:50.610 align:middle line:90% I had to pay to enter. 01:15:50.610 --> 01:15:52.860 align:middle line:84% But this is saying, well, if I'm one of the firms that 01:15:52.860 --> 01:15:55.712 align:middle line:84% didn't enter, if I contemplate deviating and entering, 01:15:55.712 --> 01:15:57.420 align:middle line:84% I'm only going to get this much, and that 01:15:57.420 --> 01:16:00.820 align:middle line:84% can't be worth it for me because F is higher. 01:16:00.820 --> 01:16:03.360 align:middle line:84% So let's just draw a little graph to finish. 01:16:03.360 --> 01:16:18.100 align:middle line:90% 01:16:18.100 --> 01:16:20.020 align:middle line:84% So what I want to do is, I want to plot 01:16:20.020 --> 01:16:23.000 align:middle line:90% all these pi k's on this chart. 01:16:23.000 --> 01:16:36.592 align:middle line:84% So we're going to get something like pi 1 star, pi 2 star. 01:16:36.592 --> 01:16:38.300 align:middle line:84% It's quadratic, so they're going to start 01:16:38.300 --> 01:16:47.690 align:middle line:84% getting closer together pi 3 star, pi 4 star, pi n star. 01:16:47.690 --> 01:16:50.710 align:middle line:90% 01:16:50.710 --> 01:16:52.970 align:middle line:84% So we can plot all these points In the graph. 01:16:52.970 --> 01:16:57.030 align:middle line:90% 01:16:57.030 --> 01:16:59.750 align:middle line:84% And we can try to see what's going to happen based on where 01:16:59.750 --> 01:17:01.830 align:middle line:90% F is relative to these points. 01:17:01.830 --> 01:17:04.230 align:middle line:84% So given the demand parameters, I 01:17:04.230 --> 01:17:07.910 align:middle line:84% can graph each of these numbers all the way up to n 01:17:07.910 --> 01:17:09.910 align:middle line:84% because n is the total number of firms. 01:17:09.910 --> 01:17:11.508 align:middle line:84% And let's first say what happens-- 01:17:11.508 --> 01:17:14.050 align:middle line:84% we have to be a little careful if F is exactly at a boundary. 01:17:14.050 --> 01:17:15.175 align:middle line:90% Let's not worry about that. 01:17:15.175 --> 01:17:18.230 align:middle line:84% Let's say what happens if F is here. 01:17:18.230 --> 01:17:19.450 align:middle line:90% How many firms enter? 01:17:19.450 --> 01:17:23.110 align:middle line:90% 01:17:23.110 --> 01:17:25.670 align:middle line:84% Here, no firms enter because this says, 01:17:25.670 --> 01:17:29.150 align:middle line:84% the entry cost is higher than monopoly profits. 01:17:29.150 --> 01:17:32.710 align:middle line:84% So the only equilibrium is no one enters. 01:17:32.710 --> 01:17:35.310 align:middle line:84% That's an equilibrium because if I were to enter and deviate, 01:17:35.310 --> 01:17:36.970 align:middle line:84% then I would get monopoly profits. 01:17:36.970 --> 01:17:40.070 align:middle line:84% But that's not enough to offset my fixed cost. 01:17:40.070 --> 01:17:44.270 align:middle line:84% So if F is here, we see 0 firms enter. 01:17:44.270 --> 01:17:46.270 align:middle line:84% So here, I want to keep track of how many enter. 01:17:46.270 --> 01:17:50.850 align:middle line:90% 01:17:50.850 --> 01:17:54.450 align:middle line:90% What if the cost is here? 01:17:54.450 --> 01:17:57.370 align:middle line:90% Now how many are going to enter? 01:17:57.370 --> 01:17:58.190 align:middle line:90% One, right? 01:17:58.190 --> 01:18:00.910 align:middle line:84% This satisfies my equation here with k equals 1. 01:18:00.910 --> 01:18:03.570 align:middle line:84% There's an equilibrium with one firm entering. 01:18:03.570 --> 01:18:06.550 align:middle line:84% If f is between pi 2 star and pi 1, 01:18:06.550 --> 01:18:09.970 align:middle line:84% that's exactly my formula here, so we get that one enter. 01:18:09.970 --> 01:18:13.450 align:middle line:84% And now we can keep going down to here. 01:18:13.450 --> 01:18:14.590 align:middle line:90% Two are going to enter. 01:18:14.590 --> 01:18:16.610 align:middle line:90% We're going to keep going. 01:18:16.610 --> 01:18:23.050 align:middle line:84% And I guess the tricky thing is what happens if F is here? 01:18:23.050 --> 01:18:27.530 align:middle line:84% If F is smaller than pi n star, then actually 01:18:27.530 --> 01:18:29.170 align:middle line:90% all n are going to enter. 01:18:29.170 --> 01:18:31.497 align:middle line:84% And even if F gets all the way down to 0, 01:18:31.497 --> 01:18:33.330 align:middle line:84% there's no more firms than n that can enter. 01:18:33.330 --> 01:18:35.247 align:middle line:84% So we can't continue this pattern indefinitely 01:18:35.247 --> 01:18:36.510 align:middle line:90% because there's only n firms. 01:18:36.510 --> 01:18:38.302 align:middle line:84% The most that can happen is they all enter. 01:18:38.302 --> 01:18:42.940 align:middle line:90% 01:18:42.940 --> 01:18:46.340 align:middle line:84% So here we've used subgame perfect Nash equilibrium 01:18:46.340 --> 01:18:47.700 align:middle line:90% as a solution concept. 01:18:47.700 --> 01:18:50.220 align:middle line:84% We've gotten a unique prediction in the game. 01:18:50.220 --> 01:18:52.540 align:middle line:84% And we've actually I think, delivered 01:18:52.540 --> 01:18:56.240 align:middle line:84% kind of a nice qualitative, industrial organization insight, 01:18:56.240 --> 01:18:59.460 align:middle line:84% which says industries that have large fixed costs, 01:18:59.460 --> 01:19:01.740 align:middle line:84% will tend to have fewer firms in them. 01:19:01.740 --> 01:19:03.980 align:middle line:84% And industries that have very small fixed costs, 01:19:03.980 --> 01:19:05.960 align:middle line:84% will tend to have more firms in them. 01:19:05.960 --> 01:19:07.500 align:middle line:90% And this is exactly what we see. 01:19:07.500 --> 01:19:09.860 align:middle line:84% We see we tend to see monopolies in industries 01:19:09.860 --> 01:19:11.813 align:middle line:84% like cell phone networks and utilities, 01:19:11.813 --> 01:19:13.980 align:middle line:84% where you have to pay this huge fixed cost to create 01:19:13.980 --> 01:19:14.973 align:middle line:90% your network. 01:19:14.973 --> 01:19:16.640 align:middle line:84% But once you've created that fixed cost, 01:19:16.640 --> 01:19:18.220 align:middle line:90% the marginal cost is pretty low. 01:19:18.220 --> 01:19:19.460 align:middle line:90% What's going on here? 01:19:19.460 --> 01:19:21.780 align:middle line:84% Well, if F is up here, it's worth 01:19:21.780 --> 01:19:23.460 align:middle line:84% it for, say, the monopoly utility 01:19:23.460 --> 01:19:24.918 align:middle line:90% provider to enter the market. 01:19:24.918 --> 01:19:26.460 align:middle line:84% But it's not worth it for anyone else 01:19:26.460 --> 01:19:29.700 align:middle line:84% to enter because if they were to enter, they would only-- 01:19:29.700 --> 01:19:31.660 align:middle line:90% sorry, I should be here. 01:19:31.660 --> 01:19:33.280 align:middle line:90% Let's say fixed costs are here. 01:19:33.280 --> 01:19:35.500 align:middle line:84% It was worth it for the first utility company 01:19:35.500 --> 01:19:38.420 align:middle line:84% to enter because they're getting monopoly profits that's 01:19:38.420 --> 01:19:39.940 align:middle line:90% higher than the fixed cost. 01:19:39.940 --> 01:19:42.392 align:middle line:84% But if a second firm contemplates entering, 01:19:42.392 --> 01:19:44.350 align:middle line:84% they're going to have to pay this fixed cost F, 01:19:44.350 --> 01:19:45.750 align:middle line:84% but they're only going to get these profits 01:19:45.750 --> 01:19:47.310 align:middle line:84% pi 2 star because they're going to have 01:19:47.310 --> 01:19:48.930 align:middle line:84% duopoly competition with the other firm, 01:19:48.930 --> 01:19:50.090 align:middle line:84% and it won't be worth it for them. 01:19:50.090 --> 01:19:51.930 align:middle line:84% So the simple model delivered, I think, 01:19:51.930 --> 01:19:55.292 align:middle line:84% a pretty robust prediction about the organization of industries. 01:19:55.292 --> 01:19:56.750 align:middle line:84% I guess one question I'll leave you 01:19:56.750 --> 01:20:00.110 align:middle line:84% with, though, is suppose we have n firms, 01:20:00.110 --> 01:20:03.110 align:middle line:84% and our equilibrium says k is strictly less than n, 01:20:03.110 --> 01:20:04.870 align:middle line:90% should enter. 01:20:04.870 --> 01:20:05.950 align:middle line:90% That's our equilibrium. 01:20:05.950 --> 01:20:06.790 align:middle line:90% K of them enter. 01:20:06.790 --> 01:20:08.590 align:middle line:90% N minus k don't. 01:20:08.590 --> 01:20:09.910 align:middle line:90% Well, which k? 01:20:09.910 --> 01:20:13.390 align:middle line:84% How do they coordinate on which k are the mentor? 01:20:13.390 --> 01:20:15.750 align:middle line:84% And I think that's kind of a puzzle 01:20:15.750 --> 01:20:19.110 align:middle line:84% that maybe this model can't really explain. 01:20:19.110 --> 01:20:22.610 align:middle line:84% I'd like to be one of the firms that enters. 01:20:22.610 --> 01:20:24.910 align:middle line:84% But if too many of us enter, it's 01:20:24.910 --> 01:20:27.250 align:middle line:84% not an equilibrium, so how do we coordinate on this? 01:20:27.250 --> 01:20:28.018 align:middle line:90% Do we mix? 01:20:28.018 --> 01:20:29.810 align:middle line:84% Do we flip a coin, decide whether to enter? 01:20:29.810 --> 01:20:31.560 align:middle line:84% Is there something missing from the model? 01:20:31.560 --> 01:20:35.300 align:middle line:84% Things to think about, but I'll see you everyone on Thursday. 01:20:35.300 --> 01:20:40.000 align:middle line:90%