WEBVTT 00:00:00.000 --> 00:00:01.984 align:middle line:90% [SQUEAKING] 00:00:01.984 --> 00:00:03.472 align:middle line:90% [RUSTLING] 00:00:03.472 --> 00:00:04.960 align:middle line:90% [CLICKING] 00:00:04.960 --> 00:00:17.390 align:middle line:90% 00:00:17.390 --> 00:00:21.620 align:middle line:84% GLENN ELLISON: OK, so I'm going to start with something I also 00:00:21.620 --> 00:00:25.220 align:middle line:84% cover in [14.]122, which is collusion with perfect, 00:00:25.220 --> 00:00:27.360 align:middle line:84% simplest model of dynamic collusion. 00:00:27.360 --> 00:00:30.350 align:middle line:84% So we say the, I guess, reason for thinking 00:00:30.350 --> 00:00:32.210 align:middle line:84% about dynamic competition is, in part, 00:00:32.210 --> 00:00:35.540 align:middle line:84% because we think dynamic competition is a very good model 00:00:35.540 --> 00:00:38.150 align:middle line:90% for how many industries work. 00:00:38.150 --> 00:00:43.850 align:middle line:84% It's T-Mobile and Verizon and whatever 00:00:43.850 --> 00:00:46.340 align:middle line:84% competing against each other period after period 00:00:46.340 --> 00:00:47.073 align:middle line:90% after period. 00:00:47.073 --> 00:00:48.740 align:middle line:84% It's the same cable companies competing. 00:00:48.740 --> 00:00:51.390 align:middle line:84% It's the same companies competing, and many industries 00:00:51.390 --> 00:00:53.473 align:middle line:90% with a small number of firms. 00:00:53.473 --> 00:00:55.140 align:middle line:84% The small number of firms reflects that. 00:00:55.140 --> 00:00:57.182 align:middle line:84% It's hard to enter, and you have those same firms 00:00:57.182 --> 00:00:59.122 align:middle line:84% for over and over and over again. 00:00:59.122 --> 00:01:00.830 align:middle line:84% And then the second reason we're covering 00:01:00.830 --> 00:01:03.290 align:middle line:84% this is a lot of this course is about markups 00:01:03.290 --> 00:01:05.660 align:middle line:84% and thinking about where markups come from. 00:01:05.660 --> 00:01:07.130 align:middle line:84% I've talked about differentiation 00:01:07.130 --> 00:01:09.297 align:middle line:84% as a source of markups, and I've talked about search 00:01:09.297 --> 00:01:10.470 align:middle line:90% as a source of markups. 00:01:10.470 --> 00:01:14.300 align:middle line:84% I think we also think that dynamic collusion is 00:01:14.300 --> 00:01:16.050 align:middle line:84% another potential source of markups. 00:01:16.050 --> 00:01:17.700 align:middle line:90% And I did do the-- 00:01:17.700 --> 00:01:21.120 align:middle line:84% Ray Miller and Weinberg had that conclusion 00:01:21.120 --> 00:01:23.790 align:middle line:84% that it looks like, when you look at the Miller-Coors merger, 00:01:23.790 --> 00:01:26.580 align:middle line:84% part of the effect was Miller and Coors no longer 00:01:26.580 --> 00:01:30.060 align:middle line:84% pricing in static Nash form as they had before 00:01:30.060 --> 00:01:33.743 align:middle line:84% and AB InBev as they had before the merger. 00:01:33.743 --> 00:01:35.160 align:middle line:84% The theory of dynamic collusion is 00:01:35.160 --> 00:01:37.650 align:middle line:84% thinking about how it is that we can think about when firms are 00:01:37.650 --> 00:01:40.108 align:middle line:84% going to have markups or when they're going to do something 00:01:40.108 --> 00:01:41.520 align:middle line:90% other than static Nash. 00:01:41.520 --> 00:01:44.580 align:middle line:90% OK, so simplest model. 00:01:44.580 --> 00:01:46.420 align:middle line:90% I've got N identical firms. 00:01:46.420 --> 00:01:48.660 align:middle line:84% They have a constant marginal cost of c. 00:01:48.660 --> 00:01:51.450 align:middle line:84% They all have a discount factor of delta. 00:01:51.450 --> 00:01:54.240 align:middle line:84% We're going to have them play the simplest Bertrand pricing 00:01:54.240 --> 00:01:57.720 align:middle line:84% game where there's some market demand function, Q of p. 00:01:57.720 --> 00:01:59.970 align:middle line:84% So that means all of the demand goes to whichever 00:01:59.970 --> 00:02:02.100 align:middle line:90% firm charges the lowest price. 00:02:02.100 --> 00:02:04.570 align:middle line:90% The other firms sell 0. 00:02:04.570 --> 00:02:06.330 align:middle line:84% If firms charge equal prices, I'll 00:02:06.330 --> 00:02:09.419 align:middle line:90% assume they split the demand. 00:02:09.419 --> 00:02:12.940 align:middle line:84% Firms all observe prices at the end of every period. 00:02:12.940 --> 00:02:20.500 align:middle line:84% So at the history at time t is h t minus 1. 00:02:20.500 --> 00:02:24.210 align:middle line:84% It's all the prices charged by all the firms in every period, 00:02:24.210 --> 00:02:26.470 align:middle line:90% tau, less than t minus 1. 00:02:26.470 --> 00:02:29.340 align:middle line:84% And a strategy in a repeated game is a function. 00:02:29.340 --> 00:02:33.420 align:middle line:84% Firm i sets its time t price as a function of everything 00:02:33.420 --> 00:02:36.630 align:middle line:84% that it's observed in all previous periods. 00:02:36.630 --> 00:02:40.590 align:middle line:84% And when I teach this model in [14.]122, 00:02:40.590 --> 00:02:44.670 align:middle line:84% I note that this model has a subgame perfect equilibrium that 00:02:44.670 --> 00:02:47.430 align:middle line:84% looks like perfect competition, the static Nash equilibrium, 00:02:47.430 --> 00:02:50.580 align:middle line:84% which is just everybody plays price equal to cost in every 00:02:50.580 --> 00:02:52.440 align:middle line:90% period. 00:02:52.440 --> 00:02:55.950 align:middle line:84% But if delta is at least 1 minus 1 over N, 00:02:55.950 --> 00:02:58.200 align:middle line:84% this model also has a subgame perfect equilibrium that 00:02:58.200 --> 00:02:59.830 align:middle line:90% looks like perfect collusion. 00:02:59.830 --> 00:03:02.160 align:middle line:84% And what that is is the firms charge 00:03:02.160 --> 00:03:04.020 align:middle line:84% the monopoly price as long as everyone 00:03:04.020 --> 00:03:06.790 align:middle line:84% has charged the monopoly price in every previous period. 00:03:06.790 --> 00:03:08.670 align:middle line:84% And if anyone ever deviates, they 00:03:08.670 --> 00:03:12.045 align:middle line:84% set price equal to c in all future periods. 00:03:12.045 --> 00:03:14.940 align:middle line:84% That price war punishment, as we would call it-- 00:03:14.940 --> 00:03:16.530 align:middle line:84% that price war punishment is going 00:03:16.530 --> 00:03:18.690 align:middle line:84% to deter people from deviating and lead everyone 00:03:18.690 --> 00:03:21.630 align:middle line:84% to want to coordinate a monopoly price, if they're patient, 00:03:21.630 --> 00:03:23.585 align:middle line:84% and this is how patient they have to be. 00:03:23.585 --> 00:03:26.525 align:middle line:90% 00:03:26.525 --> 00:03:28.130 align:middle line:90% So some comments. 00:03:28.130 --> 00:03:30.290 align:middle line:84% This model identifies another factor 00:03:30.290 --> 00:03:32.210 align:middle line:90% that can determine markups. 00:03:32.210 --> 00:03:34.850 align:middle line:84% In particular, just repeated competition 00:03:34.850 --> 00:03:38.090 align:middle line:84% can lead to markups with no product differentiation, 00:03:38.090 --> 00:03:41.480 align:middle line:90% no search costs, nothing else. 00:03:41.480 --> 00:03:44.930 align:middle line:84% It has a lot of natural comparative statics. 00:03:44.930 --> 00:03:48.180 align:middle line:84% So it's easier to sustain collusion when delta is larger. 00:03:48.180 --> 00:03:51.170 align:middle line:84% So when delta is above this number, you can collude. 00:03:51.170 --> 00:03:53.990 align:middle line:84% When delta is below that number, you cannot. 00:03:53.990 --> 00:03:56.445 align:middle line:90% So what could delta mean? 00:03:56.445 --> 00:03:58.070 align:middle line:84% If you think about delta in this model, 00:03:58.070 --> 00:04:00.980 align:middle line:90% delta is the discount you apply. 00:04:00.980 --> 00:04:03.800 align:middle line:84% In between, you see someone cheat on the agreement, 00:04:03.800 --> 00:04:05.670 align:middle line:84% and then you start punishing them. 00:04:05.670 --> 00:04:08.770 align:middle line:84% And so delta is your time rate of discounting between those two 00:04:08.770 --> 00:04:09.270 align:middle line:90% events. 00:04:09.270 --> 00:04:11.810 align:middle line:84% Someone cheats, I start punishing. 00:04:11.810 --> 00:04:14.870 align:middle line:90% So what makes collusion harder? 00:04:14.870 --> 00:04:16.829 align:middle line:90% One would be detection lags. 00:04:16.829 --> 00:04:19.329 align:middle line:84% If people cheat on the agreement, 00:04:19.329 --> 00:04:21.690 align:middle line:84% you don't see them cheat for a while. 00:04:21.690 --> 00:04:24.930 align:middle line:90% That makes it harder to collude. 00:04:24.930 --> 00:04:26.900 align:middle line:90% Lumpy demands. 00:04:26.900 --> 00:04:29.358 align:middle line:84% In some industries, firms are selling goods 00:04:29.358 --> 00:04:31.400 align:middle line:84% over and over and over again and see their demand 00:04:31.400 --> 00:04:33.290 align:middle line:90% coming in real time. 00:04:33.290 --> 00:04:37.820 align:middle line:84% Other demand, like if you think about Raytheon competing 00:04:37.820 --> 00:04:41.220 align:middle line:84% with Lockheed, if they're still separate companies, to produce 00:04:41.220 --> 00:04:43.820 align:middle line:84% next-generation fighter jets for the US government, 00:04:43.820 --> 00:04:46.490 align:middle line:84% if those orders for new-generation fighter jet 00:04:46.490 --> 00:04:49.310 align:middle line:84% come along once every 15 years, if someone 00:04:49.310 --> 00:04:51.770 align:middle line:84% cheats on the agreement to collude on prices 00:04:51.770 --> 00:04:54.470 align:middle line:84% and you can't do anything about it for 15 years, 00:04:54.470 --> 00:04:56.130 align:middle line:90% delta is going to be very small. 00:04:56.130 --> 00:05:01.700 align:middle line:84% So whether demand is coming in continuously or lumpy matters. 00:05:01.700 --> 00:05:05.430 align:middle line:84% It's easier to collude in growing markets. 00:05:05.430 --> 00:05:07.250 align:middle line:84% It's easier to collude in growing markets 00:05:07.250 --> 00:05:10.940 align:middle line:84% because, again, the basic consideration here 00:05:10.940 --> 00:05:12.500 align:middle line:84% is how much do you hate-- how much do 00:05:12.500 --> 00:05:14.510 align:middle line:84% you dislike this punishment of switching 00:05:14.510 --> 00:05:17.610 align:middle line:84% to pricing at marginal cost and losing the collusive profits? 00:05:17.610 --> 00:05:19.980 align:middle line:84% And how much does that deter you from deviating? 00:05:19.980 --> 00:05:22.550 align:middle line:84% If the future is more important than the present, 00:05:22.550 --> 00:05:26.170 align:middle line:90% then that's like delta is large. 00:05:26.170 --> 00:05:28.230 align:middle line:90% That makes collusion easy. 00:05:28.230 --> 00:05:30.300 align:middle line:84% And on the other hand, technological change, 00:05:30.300 --> 00:05:31.883 align:middle line:84% if this is a market that's going to be 00:05:31.883 --> 00:05:35.140 align:middle line:84% replaced by some other product, then the future doesn't matter. 00:05:35.140 --> 00:05:37.495 align:middle line:84% If the future doesn't matter, then it's hard to collude. 00:05:37.495 --> 00:05:39.120 align:middle line:84% And so you might see collusion breaking 00:05:39.120 --> 00:05:42.150 align:middle line:84% down just because, when the market is on its way out, 00:05:42.150 --> 00:05:45.750 align:middle line:90% collusion could break down. 00:05:45.750 --> 00:05:47.850 align:middle line:84% Model also has a nice comparative static, 00:05:47.850 --> 00:05:51.330 align:middle line:84% that it's easier to sustain collusion when N is small. 00:05:51.330 --> 00:05:54.810 align:middle line:84% Because, again, when N is small, 1 minus 1 over N 00:05:54.810 --> 00:05:57.845 align:middle line:84% is a smaller number, so more markets where you could collude. 00:05:57.845 --> 00:06:00.810 align:middle line:90% 00:06:00.810 --> 00:06:04.590 align:middle line:84% Another common on this is that this theory also, in some ways, 00:06:04.590 --> 00:06:07.470 align:middle line:84% suggests that everything we've done in the entire semester 00:06:07.470 --> 00:06:08.820 align:middle line:90% is almost worthless. 00:06:08.820 --> 00:06:10.500 align:middle line:84% Because anytime you have a market 00:06:10.500 --> 00:06:13.560 align:middle line:84% with a small number of firms, how much differentiation 00:06:13.560 --> 00:06:20.130 align:middle line:84% there is, how much search costs there are, how much of whatever, 00:06:20.130 --> 00:06:23.490 align:middle line:84% how much multiproduct competition, all of that stuff, 00:06:23.490 --> 00:06:26.820 align:middle line:84% in some sense, is irrelevant because the firms could collude 00:06:26.820 --> 00:06:27.640 align:middle line:90% on anything. 00:06:27.640 --> 00:06:31.350 align:middle line:84% And so maybe one suggestion is that we either 00:06:31.350 --> 00:06:33.600 align:middle line:84% have to think about which equilibrium in this repeated 00:06:33.600 --> 00:06:35.340 align:middle line:84% game we think is going to happen, 00:06:35.340 --> 00:06:38.910 align:middle line:84% or we have to just accept that we can't always forecast what's 00:06:38.910 --> 00:06:40.920 align:middle line:84% going on and we're always-- we may always 00:06:40.920 --> 00:06:43.540 align:middle line:84% be surprised about how things are happening. 00:06:43.540 --> 00:06:45.540 align:middle line:84% The other thing I'd like to say about this model 00:06:45.540 --> 00:06:50.310 align:middle line:84% is that, numerically, it's just giving the wrong answer. 00:06:50.310 --> 00:06:56.220 align:middle line:84% It's saying that, with delta at least 1 minus 1 over N, 00:06:56.220 --> 00:06:57.595 align:middle line:90% we can collude. 00:06:57.595 --> 00:06:59.220 align:middle line:84% And if you think about it, if you think 00:06:59.220 --> 00:07:01.890 align:middle line:90% about a typical industry-- 00:07:01.890 --> 00:07:04.290 align:middle line:84% prices are posted on the internet or whatever. 00:07:04.290 --> 00:07:05.910 align:middle line:84% How long does it take you to observe 00:07:05.910 --> 00:07:08.808 align:middle line:90% that your rival has colluded? 00:07:08.808 --> 00:07:11.100 align:middle line:84% Like, if you're a T-Mobile and you see Verizon changing 00:07:11.100 --> 00:07:12.030 align:middle line:90% their prices-- 00:07:12.030 --> 00:07:14.070 align:middle line:84% they put a big TV ad campaign saying 00:07:14.070 --> 00:07:15.510 align:middle line:90% they've changed their prices-- 00:07:15.510 --> 00:07:19.230 align:middle line:84% within a week or within a day, you know what they've done. 00:07:19.230 --> 00:07:21.690 align:middle line:84% What's the discount factor over a day? 00:07:21.690 --> 00:07:26.060 align:middle line:84% Discount factor of a day is 0.000. 00:07:26.060 --> 00:07:30.860 align:middle line:84% Or delta should be 0.9999-- or five 9s or whatever. 00:07:30.860 --> 00:07:34.080 align:middle line:84% You could collude with thousands of firms. 00:07:34.080 --> 00:07:36.710 align:middle line:90% And so I think that, while-- 00:07:36.710 --> 00:07:41.134 align:middle line:84% even you put N equals 100, delta is 0.99, 00:07:41.134 --> 00:07:45.080 align:middle line:84% 0.99 is a good discount factor for a month or less than a 00:07:45.080 --> 00:07:45.860 align:middle line:90% month. 00:07:45.860 --> 00:07:47.685 align:middle line:84% So this would say, with 100 firms, 00:07:47.685 --> 00:07:49.310 align:middle line:84% even if it takes you a month to realize 00:07:49.310 --> 00:07:51.020 align:middle line:84% your rivals have changed their prices, 00:07:51.020 --> 00:07:52.525 align:middle line:90% you could collude perfectly. 00:07:52.525 --> 00:07:54.650 align:middle line:84% And I think we don't think that collusion typically 00:07:54.650 --> 00:07:57.770 align:middle line:84% happens with 100 firms when you have month-long delays. 00:07:57.770 --> 00:07:59.390 align:middle line:84% So there's something about collusion 00:07:59.390 --> 00:08:03.365 align:middle line:84% is just harder than this model suggests it should be. 00:08:03.365 --> 00:08:05.780 align:middle line:84% And so one thing you find in the IO literature 00:08:05.780 --> 00:08:11.300 align:middle line:84% is a bunch of factors that all may make collusion harder 00:08:11.300 --> 00:08:12.960 align:middle line:90% than it would be in this model. 00:08:12.960 --> 00:08:16.490 align:middle line:84% So I thought I would do is I'm just going to-- 00:08:16.490 --> 00:08:18.690 align:middle line:84% it's a fairly old literature, and it's all 00:08:18.690 --> 00:08:19.940 align:middle line:90% well-covered in Tirole's book. 00:08:19.940 --> 00:08:22.138 align:middle line:84% So I'll do a few of those models, 00:08:22.138 --> 00:08:24.430 align:middle line:84% but maybe I'll go through some of them kind of quickly. 00:08:24.430 --> 00:08:28.500 align:middle line:84% So the one I did want to do in more detail 00:08:28.500 --> 00:08:32.620 align:middle line:84% was this literature on collusion with imperfect monitoring. 00:08:32.620 --> 00:08:34.169 align:middle line:84% And I think this is, I think, what 00:08:34.169 --> 00:08:39.840 align:middle line:84% we think of as probably the most important reason why collusion 00:08:39.840 --> 00:08:44.340 align:middle line:84% is hard, which is that it's often hard to know 00:08:44.340 --> 00:08:45.590 align:middle line:90% when your rivals have cheated. 00:08:45.590 --> 00:08:47.173 align:middle line:84% It's easy to know when your rivals are 00:08:47.173 --> 00:08:49.770 align:middle line:84% cheated if your rivals are consumer-facing businesses that 00:08:49.770 --> 00:08:52.120 align:middle line:84% have to post prices that are visible. 00:08:52.120 --> 00:08:55.500 align:middle line:84% But in many industries where you're selling to businesses 00:08:55.500 --> 00:08:58.810 align:middle line:84% or selling elsewhere or selling with negotiated prices, 00:08:58.810 --> 00:09:00.810 align:middle line:84% it can be much harder to know if your rivals are 00:09:00.810 --> 00:09:01.852 align:middle line:90% cheating on an agreement. 00:09:01.852 --> 00:09:04.110 align:middle line:84% So I'm going to discuss this as-- 00:09:04.110 --> 00:09:05.653 align:middle line:90% ideas due to Green and Porter. 00:09:05.653 --> 00:09:07.320 align:middle line:84% This version of the model I'm giving you 00:09:07.320 --> 00:09:09.810 align:middle line:84% here is the one that's just out of Tirole's textbook, 00:09:09.810 --> 00:09:13.620 align:middle line:84% where Tirole reconfigured it to be an easier model 00:09:13.620 --> 00:09:16.660 align:middle line:84% to solve but, obviously, not as realistic. 00:09:16.660 --> 00:09:19.780 align:middle line:84% But so, in this story, I've got two firms. 00:09:19.780 --> 00:09:22.020 align:middle line:84% They have a constant marginal cost of c, 00:09:22.020 --> 00:09:25.240 align:middle line:84% and they're engaged in Bertrand competition. 00:09:25.240 --> 00:09:28.480 align:middle line:90% And market demand is noisy. 00:09:28.480 --> 00:09:31.870 align:middle line:84% So market demand is either Q of p, with some probability, 00:09:31.870 --> 00:09:35.840 align:middle line:84% 1 minus alpha, or it's 0 with probability alpha. 00:09:35.840 --> 00:09:38.380 align:middle line:84% So where that might make sense is 00:09:38.380 --> 00:09:40.900 align:middle line:84% you could have two firms that are 00:09:40.900 --> 00:09:43.330 align:middle line:84% competing to supply computer equipment 00:09:43.330 --> 00:09:45.010 align:middle line:90% to some large business. 00:09:45.010 --> 00:09:48.640 align:middle line:84% And many months, that business calls them up and says, 00:09:48.640 --> 00:09:53.170 align:middle line:84% I need 132 new computers, or I need 147 new desk chairs, 00:09:53.170 --> 00:09:54.220 align:middle line:90% or whatever. 00:09:54.220 --> 00:09:56.530 align:middle line:84% But then, some months, it doesn't call. 00:09:56.530 --> 00:09:59.510 align:middle line:84% So there's these two levels of demand. 00:09:59.510 --> 00:10:01.450 align:middle line:84% Either there's demand Q of p because the firm 00:10:01.450 --> 00:10:06.230 align:middle line:84% places a large order with one of you and it's 0. 00:10:06.230 --> 00:10:09.380 align:middle line:84% I'm going to assume that demand goes to the lower-priced firm 00:10:09.380 --> 00:10:12.410 align:middle line:84% or splits 50/50 if their prices are equal. 00:10:12.410 --> 00:10:14.750 align:middle line:84% And I'm going to assume there's imperfect observability 00:10:14.750 --> 00:10:17.660 align:middle line:84% in that firms see whether they've made 00:10:17.660 --> 00:10:20.090 align:middle line:90% a delivery to the customer. 00:10:20.090 --> 00:10:23.240 align:middle line:84% But they don't know what their rival's price is 00:10:23.240 --> 00:10:25.860 align:middle line:90% or what the rival's sales are. 00:10:25.860 --> 00:10:27.620 align:middle line:90% So you both have websites. 00:10:27.620 --> 00:10:29.975 align:middle line:84% The websites give your official price. 00:10:29.975 --> 00:10:32.600 align:middle line:84% But you don't know whether the firm called your rival and said, 00:10:32.600 --> 00:10:34.460 align:middle line:84% can you give me a 10% discount and I'll 00:10:34.460 --> 00:10:36.110 align:middle line:84% buy all my stuff from you this month? 00:10:36.110 --> 00:10:36.860 align:middle line:90% And then they did. 00:10:36.860 --> 00:10:38.260 align:middle line:84% And then they shipped it to them. 00:10:38.260 --> 00:10:40.400 align:middle line:84% All you can observe is your own demand. 00:10:40.400 --> 00:10:44.570 align:middle line:84% So the difficulty when that happens 00:10:44.570 --> 00:10:46.980 align:middle line:84% is we get this imperfect observability. 00:10:46.980 --> 00:10:50.810 align:middle line:84% Suppose there's some month where I get zero demand. 00:10:50.810 --> 00:10:52.150 align:middle line:90% One of two things happened. 00:10:52.150 --> 00:10:53.060 align:middle line:84% One thing that could have happened 00:10:53.060 --> 00:10:54.680 align:middle line:84% is we're just in a low-demand month 00:10:54.680 --> 00:10:56.730 align:middle line:84% and the firm didn't place any orders. 00:10:56.730 --> 00:10:58.397 align:middle line:84% The other thing that could have happened 00:10:58.397 --> 00:11:00.270 align:middle line:84% is my rival is cheating on the agreement, 00:11:00.270 --> 00:11:03.320 align:middle line:84% and the firm did make an order, and the entire order 00:11:03.320 --> 00:11:06.123 align:middle line:90% went to my rival. 00:11:06.123 --> 00:11:07.790 align:middle line:84% And obviously, you could call your rival 00:11:07.790 --> 00:11:09.257 align:middle line:90% and say, oh, I got zero demand. 00:11:09.257 --> 00:11:09.840 align:middle line:90% How about you? 00:11:09.840 --> 00:11:11.930 align:middle line:84% And then they can be like, yeah, zero demand, me too, 00:11:11.930 --> 00:11:12.450 align:middle line:90% or whatever. 00:11:12.450 --> 00:11:13.450 align:middle line:90% And you just don't know. 00:11:13.450 --> 00:11:17.420 align:middle line:84% Are they lying to you, or is this really what happened? 00:11:17.420 --> 00:11:20.570 align:middle line:84% So immediate conclusion-- no matter 00:11:20.570 --> 00:11:23.270 align:middle line:84% what the discount factor is, this model 00:11:23.270 --> 00:11:26.810 align:middle line:84% can't have an equilibrium in which both firms always 00:11:26.810 --> 00:11:29.865 align:middle line:84% charge the monopoly price on the equilibrium path. 00:11:29.865 --> 00:11:31.490 align:middle line:84% Because if you had an equilibrium where 00:11:31.490 --> 00:11:34.220 align:middle line:84% both firms were always charging the monopoly price, 00:11:34.220 --> 00:11:36.590 align:middle line:84% there would be some chance, given that these are i.i.d., 00:11:36.590 --> 00:11:39.470 align:middle line:84% that you'd have a million periods in a row 00:11:39.470 --> 00:11:41.360 align:middle line:90% where there's no demand. 00:11:41.360 --> 00:11:43.760 align:middle line:84% And if it was in every period on the equilibrium path, 00:11:43.760 --> 00:11:45.218 align:middle line:84% you always charge a monopoly price, 00:11:45.218 --> 00:11:48.080 align:middle line:84% then after that million periods of million zeros in a row, 00:11:48.080 --> 00:11:51.070 align:middle line:84% you still have to have both firms charge the monopoly price. 00:11:51.070 --> 00:11:53.470 align:middle line:84% But then if that was going to happen, 00:11:53.470 --> 00:11:55.970 align:middle line:84% obviously, it's better for you to cheat in the first million 00:11:55.970 --> 00:11:59.570 align:middle line:84% periods in a row and sell the entire market instead of half 00:11:59.570 --> 00:12:01.490 align:middle line:84% the market and leave your rival with nothing 00:12:01.490 --> 00:12:03.410 align:middle line:90% than to follow the equilibrium. 00:12:03.410 --> 00:12:07.490 align:middle line:84% So the fact that we've got that fundamental unobserved ability 00:12:07.490 --> 00:12:09.410 align:middle line:84% means there's no-- perfect collusion 00:12:09.410 --> 00:12:11.860 align:middle line:84% is just impossible, no matter how patient firms are. 00:12:11.860 --> 00:12:15.715 align:middle line:90% 00:12:15.715 --> 00:12:19.060 align:middle line:84% And then I think the key insight of Green and Porter 00:12:19.060 --> 00:12:22.870 align:middle line:84% is that while perfect collusion is impossible, 00:12:22.870 --> 00:12:25.640 align:middle line:84% you can get partial collusion in a model like this. 00:12:25.640 --> 00:12:31.150 align:middle line:84% So proposition, if alpha is less than a half, 00:12:31.150 --> 00:12:34.930 align:middle line:84% which you should think of the noise is not too large 00:12:34.930 --> 00:12:37.480 align:middle line:84% and delta sufficiently close to 1, 00:12:37.480 --> 00:12:40.420 align:middle line:84% the model has a partially collusive equilibrium. 00:12:40.420 --> 00:12:43.030 align:middle line:84% And what happens in this partially collusive equilibrium 00:12:43.030 --> 00:12:46.450 align:middle line:84% is the firms initially set the monopoly price. 00:12:46.450 --> 00:12:49.390 align:middle line:84% And then if they ever get zero demand, 00:12:49.390 --> 00:12:52.240 align:middle line:84% they switch to c for capital T periods 00:12:52.240 --> 00:12:54.800 align:middle line:84% and then they go back to the monopoly price. 00:12:54.800 --> 00:12:58.220 align:middle line:90% So what we would see-- 00:12:58.220 --> 00:12:59.930 align:middle line:84% what we would observe in this model 00:12:59.930 --> 00:13:04.130 align:middle line:84% is we would observe the firms going along, 00:13:04.130 --> 00:13:05.610 align:middle line:90% charging monopoly price. 00:13:05.610 --> 00:13:07.940 align:middle line:84% They would do that for some period of time. 00:13:07.940 --> 00:13:11.390 align:middle line:84% And then when there's one of these bad demand periods where 00:13:11.390 --> 00:13:14.540 align:middle line:84% the customer doesn't order from them, they would switch, 00:13:14.540 --> 00:13:19.370 align:middle line:84% and they would charge c for capital T periods. 00:13:19.370 --> 00:13:21.020 align:middle line:90% And then price would go back up. 00:13:21.020 --> 00:13:23.990 align:middle line:84% And then they would continue to collude for a while. 00:13:23.990 --> 00:13:26.035 align:middle line:90% Another price war. 00:13:26.035 --> 00:13:27.410 align:middle line:84% And so, in this equilibrium, we'd 00:13:27.410 --> 00:13:30.290 align:middle line:84% get this sort of funny price pattern that 00:13:30.290 --> 00:13:32.090 align:middle line:84% looks like that, where the firms go along, 00:13:32.090 --> 00:13:34.040 align:middle line:90% charging monopoly price. 00:13:34.040 --> 00:13:35.030 align:middle line:90% They jump down. 00:13:35.030 --> 00:13:36.830 align:middle line:90% They have a price war. 00:13:36.830 --> 00:13:39.175 align:middle line:84% And then they go on like that and so on. 00:13:39.175 --> 00:13:43.000 align:middle line:90% 00:13:43.000 --> 00:13:50.886 align:middle line:84% So then the theorem is that that is an equilibrium of this model. 00:13:50.886 --> 00:13:52.980 align:middle line:84% And intuitively, why don't you want 00:13:52.980 --> 00:13:56.700 align:middle line:84% to deviate from such an equilibrium? 00:13:56.700 --> 00:13:59.572 align:middle line:90% If you cheat in this model-- 00:13:59.572 --> 00:14:00.780 align:middle line:90% the other firm's charging Pm. 00:14:00.780 --> 00:14:04.430 align:middle line:84% You charge Pm minus epsilon to try to get all the demand. 00:14:04.430 --> 00:14:06.170 align:middle line:84% You do get some extra profit today 00:14:06.170 --> 00:14:09.840 align:middle line:84% from getting the whole market instead of half the market, 00:14:09.840 --> 00:14:14.900 align:middle line:84% but then you immediately trigger this T period price war. 00:14:14.900 --> 00:14:18.380 align:middle line:84% The money you lose in these T periods of a price war 00:14:18.380 --> 00:14:20.600 align:middle line:84% is going to offset the gain you get in the one 00:14:20.600 --> 00:14:22.852 align:middle line:90% period in which you succeeded. 00:14:22.852 --> 00:14:25.310 align:middle line:84% And T sometimes doesn't need to be bigger than two or three 00:14:25.310 --> 00:14:26.140 align:middle line:90% to make that true. 00:14:26.140 --> 00:14:30.280 align:middle line:90% 00:14:30.280 --> 00:14:30.780 align:middle line:90% OK. 00:14:30.780 --> 00:14:33.320 align:middle line:90% So how am I going to prove that? 00:14:33.320 --> 00:14:35.790 align:middle line:84% The way I'm going do it is I'm just going to do-- 00:14:35.790 --> 00:14:38.840 align:middle line:90% there's lights on the board. 00:14:38.840 --> 00:14:41.094 align:middle line:90% [VOCALIZING] 00:14:41.094 --> 00:14:41.687 align:middle line:90% 00:14:41.687 --> 00:14:43.270 align:middle line:84% OK, the way I'm going to do it is it's 00:14:43.270 --> 00:14:45.880 align:middle line:84% just going to be a simple dynamic programming 00:14:45.880 --> 00:14:47.460 align:middle line:90% kind of calculation. 00:14:47.460 --> 00:14:49.400 align:middle line:90% I make these two definitions. 00:14:49.400 --> 00:14:53.640 align:middle line:84% Vm is the present discounted value of payoffs at t equals 0. 00:14:53.640 --> 00:14:58.350 align:middle line:90% 00:14:58.350 --> 00:15:02.043 align:middle line:84% And Vp is the present discounted value of payoffs 00:15:02.043 --> 00:15:03.960 align:middle line:84% if we're at the start of the punishment phase. 00:15:03.960 --> 00:15:06.130 align:middle line:84% And the strategies are symmetric, 00:15:06.130 --> 00:15:09.690 align:middle line:84% so these things don't need an index for the players on them. 00:15:09.690 --> 00:15:13.647 align:middle line:84% And I just need to show that there's no profitable deviation. 00:15:13.647 --> 00:15:15.480 align:middle line:84% Showing that there's no profitable deviation 00:15:15.480 --> 00:15:17.240 align:middle line:84% during the punishment phase is easy. 00:15:17.240 --> 00:15:20.360 align:middle line:90% 00:15:20.360 --> 00:15:25.530 align:middle line:84% If we're down here and I decide that I'm supposed to charge c 00:15:25.530 --> 00:15:28.470 align:middle line:84% and I'm going to charge some other price, well, 00:15:28.470 --> 00:15:31.380 align:middle line:84% if I deviate and I charge a price that's less than c, 00:15:31.380 --> 00:15:33.330 align:middle line:84% all I do is sell at a loss, and I 00:15:33.330 --> 00:15:35.200 align:middle line:90% don't affect the future at all. 00:15:35.200 --> 00:15:38.250 align:middle line:84% And if I deviate and raise my price, 00:15:38.250 --> 00:15:40.518 align:middle line:84% I sell nothing, which is the same. 00:15:40.518 --> 00:15:42.060 align:middle line:84% And again, I don't affect the future, 00:15:42.060 --> 00:15:43.960 align:middle line:84% as I've described the strategies. 00:15:43.960 --> 00:15:47.128 align:middle line:84% So in the punishment phase, you're 00:15:47.128 --> 00:15:48.420 align:middle line:90% never going to want to deviate. 00:15:48.420 --> 00:15:50.920 align:middle line:84% You're never going to gain from deviating because all you do 00:15:50.920 --> 00:15:53.730 align:middle line:84% is lose or weakly lose in the current period 00:15:53.730 --> 00:15:55.950 align:middle line:84% and have no effect on the future. 00:15:55.950 --> 00:15:59.010 align:middle line:84% So it just remains to show that when you're at some point, 00:15:59.010 --> 00:16:01.500 align:middle line:84% like this, then you don't, also, want to deviate 00:16:01.500 --> 00:16:05.130 align:middle line:90% from the equilibrium. 00:16:05.130 --> 00:16:08.100 align:middle line:84% So to do that, what I need to show 00:16:08.100 --> 00:16:09.960 align:middle line:84% is, if I follow the equilibrium, I'm 00:16:09.960 --> 00:16:14.190 align:middle line:84% going to get this present discounted value Vm. 00:16:14.190 --> 00:16:18.927 align:middle line:90% If I do deviate, if there is-- 00:16:18.927 --> 00:16:20.760 align:middle line:84% we're in one of the positive demand periods. 00:16:20.760 --> 00:16:22.927 align:middle line:84% I'm going to get the full monopoly profit instead of 00:16:22.927 --> 00:16:25.590 align:middle line:90% one half the monopoly profit. 00:16:25.590 --> 00:16:27.443 align:middle line:84% But then, one period later, we're 00:16:27.443 --> 00:16:29.110 align:middle line:84% going to be starting a punishment phase. 00:16:29.110 --> 00:16:31.050 align:middle line:84% So I'm going to get delta times Vp. 00:16:31.050 --> 00:16:34.710 align:middle line:84% So I need to show that Vm is at least the right-hand side 00:16:34.710 --> 00:16:37.430 align:middle line:90% of that expression. 00:16:37.430 --> 00:16:43.040 align:middle line:84% So to show that, I just need to find Vm and Vp. 00:16:43.040 --> 00:16:47.270 align:middle line:84% And to do that, I just solve two equations and two unknowns. 00:16:47.270 --> 00:16:50.630 align:middle line:84% One recursive equation for Vm is the profit 00:16:50.630 --> 00:16:54.440 align:middle line:84% I get at the start of the game is with probability 1 minus 00:16:54.440 --> 00:16:55.610 align:middle line:90% alpha. 00:16:55.610 --> 00:16:57.710 align:middle line:90% I get pi m over 2. 00:16:57.710 --> 00:17:01.250 align:middle line:84% And then next period, we remain in the cooperative phase. 00:17:01.250 --> 00:17:04.880 align:middle line:84% And with probability alpha, one period later, we 00:17:04.880 --> 00:17:06.589 align:middle line:90% end up in the punishment phase. 00:17:06.589 --> 00:17:16.040 align:middle line:84% So this is one equation for Vm is 1 minus alpha times 00:17:16.040 --> 00:17:26.630 align:middle line:84% pi m over 2 plus delta Vm plus alpha delta Vp. 00:17:26.630 --> 00:17:31.680 align:middle line:84% And then Vp is delta to the T times Vm. 00:17:31.680 --> 00:17:35.376 align:middle line:90% 00:17:35.376 --> 00:17:39.660 align:middle line:84% So if you look at what's going on here, these are just-- 00:17:39.660 --> 00:17:45.000 align:middle line:84% you can think about bringing all of the Vm's and Vp's over 00:17:45.000 --> 00:17:46.330 align:middle line:90% to one side of the equation. 00:17:46.330 --> 00:17:50.490 align:middle line:84% So, for instance, this first one looks like Vm 00:17:50.490 --> 00:17:57.190 align:middle line:90% minus 1 minus alpha delta Vm. 00:17:57.190 --> 00:18:03.130 align:middle line:84% And then Vp is delta to the T Vm. 00:18:03.130 --> 00:18:07.120 align:middle line:84% So this is really minus alpha delta to the T 00:18:07.120 --> 00:18:16.420 align:middle line:84% plus 1 Vm equals 1 minus alpha times pi m over 2. 00:18:16.420 --> 00:18:19.810 align:middle line:84% So just by substituting in for this one, 00:18:19.810 --> 00:18:22.770 align:middle line:84% I just get an equation in one unknown. 00:18:22.770 --> 00:18:24.540 align:middle line:90% I can solve that for what Vm is. 00:18:24.540 --> 00:18:27.211 align:middle line:90% 00:18:27.211 --> 00:18:30.280 align:middle line:84% But it's just a two equation and two unknown problem. 00:18:30.280 --> 00:18:31.700 align:middle line:90% Anyway. 00:18:31.700 --> 00:18:32.960 align:middle line:90% So this is what I get. 00:18:32.960 --> 00:18:35.890 align:middle line:84% So this is the expression for what Vm is, and this is 00:18:35.890 --> 00:18:39.130 align:middle line:90% an expression for what Vp is. 00:18:39.130 --> 00:18:44.500 align:middle line:84% So then what we're trying to show is that this is true. 00:18:44.500 --> 00:18:46.630 align:middle line:90% OK? 00:18:46.630 --> 00:18:49.400 align:middle line:84% Again, I bring this term over to that side. 00:18:49.400 --> 00:19:00.230 align:middle line:84% So what I need to show is that Vm minus delta to the T plus 1 00:19:00.230 --> 00:19:07.585 align:middle line:84% Vm is at least pi m times 1 minus alpha. 00:19:07.585 --> 00:19:10.540 align:middle line:90% 00:19:10.540 --> 00:19:11.880 align:middle line:90% So that's what I wrote here. 00:19:11.880 --> 00:19:14.690 align:middle line:90% 00:19:14.690 --> 00:19:18.140 align:middle line:84% The Vm has a 1 minus alpha in it. 00:19:18.140 --> 00:19:21.440 align:middle line:90% I canceled the 1 minus alphas. 00:19:21.440 --> 00:19:25.770 align:middle line:84% This term also has a pi m in it, as does this term, 00:19:25.770 --> 00:19:27.550 align:middle line:90% so I cancel the pi m's. 00:19:27.550 --> 00:19:28.550 align:middle line:90% I multiply out. 00:19:28.550 --> 00:19:32.440 align:middle line:84% Anyway, I get-- a bunch of algebra. 00:19:32.440 --> 00:19:33.400 align:middle line:90% I get this expression. 00:19:33.400 --> 00:19:35.470 align:middle line:84% 2, 1 minus alpha, delta plus 2 alpha 00:19:35.470 --> 00:19:38.260 align:middle line:84% minus 1, delta t plus 1 is greater than 1. 00:19:38.260 --> 00:19:41.080 align:middle line:84% So if you look at this expression, 00:19:41.080 --> 00:19:45.220 align:middle line:84% if alpha is less than a half, 2 times 1 minus alpha 00:19:45.220 --> 00:19:46.550 align:middle line:90% is bigger than 1. 00:19:46.550 --> 00:19:50.980 align:middle line:84% So I have a number bigger than 1 times delta plus something 00:19:50.980 --> 00:19:54.310 align:middle line:84% with a delta to the T plus 1 has to be bigger than 1. 00:19:54.310 --> 00:19:56.950 align:middle line:84% So if 2, 1 minus alpha is bigger than 1 00:19:56.950 --> 00:20:01.310 align:middle line:84% and I choose delta close to 1, this is bigger than 1. 00:20:01.310 --> 00:20:03.670 align:middle line:84% And then I can choose capital T large enough 00:20:03.670 --> 00:20:05.870 align:middle line:84% to make this thing approximately 0. 00:20:05.870 --> 00:20:08.980 align:middle line:84% So I've got something bigger than 1, plus something 00:20:08.980 --> 00:20:12.200 align:middle line:84% that I can make as close to 0 as I want. 00:20:12.200 --> 00:20:14.380 align:middle line:90% That's going to be-- 00:20:14.380 --> 00:20:16.250 align:middle line:84% that is going to be greater than 1. 00:20:16.250 --> 00:20:19.990 align:middle line:84% So just the intuition is just make delta big. 00:20:19.990 --> 00:20:23.020 align:middle line:84% Make delta large so people care about the punishments 00:20:23.020 --> 00:20:25.663 align:middle line:84% about the future, and then make T large, 00:20:25.663 --> 00:20:27.580 align:middle line:84% which is just making the punishments as severe 00:20:27.580 --> 00:20:30.750 align:middle line:90% as you need it to be. 00:20:30.750 --> 00:20:35.225 align:middle line:84% So if delta is close to 1 and T is large, this is going to work. 00:20:35.225 --> 00:20:37.755 align:middle line:90% 00:20:37.755 --> 00:20:40.740 align:middle line:84% And I described that as T going to infinity. 00:20:40.740 --> 00:20:43.400 align:middle line:84% But if you think about the math of it, 00:20:43.400 --> 00:20:46.220 align:middle line:84% if I gain, I gain in one period, I 00:20:46.220 --> 00:20:48.440 align:middle line:90% gain half the monopoly profits. 00:20:48.440 --> 00:20:50.150 align:middle line:84% And then in every future period, I'm 00:20:50.150 --> 00:20:53.100 align:middle line:84% losing half the monopoly profits. 00:20:53.100 --> 00:20:55.220 align:middle line:84% So even losing half of the monopoly profits 00:20:55.220 --> 00:20:57.860 align:middle line:84% twice may be enough to make me not 00:20:57.860 --> 00:21:00.240 align:middle line:84% want to get half the monopoly profits once. 00:21:00.240 --> 00:21:03.650 align:middle line:84% So capital T equals 2 will often work for this. 00:21:03.650 --> 00:21:06.740 align:middle line:84% Capital T equals 3 will almost always work. 00:21:06.740 --> 00:21:09.920 align:middle line:84% There is problem with the 1 minus alphas in there, 00:21:09.920 --> 00:21:13.400 align:middle line:84% that you may not actually get punished here, because it 00:21:13.400 --> 00:21:15.390 align:middle line:90% may be that there is no demand. 00:21:15.390 --> 00:21:19.370 align:middle line:84% But if alpha is small so that the punishment really 00:21:19.370 --> 00:21:21.170 align:middle line:84% does occur, you just don't need T very big 00:21:21.170 --> 00:21:22.420 align:middle line:90% to get this thing to work. 00:21:22.420 --> 00:21:26.460 align:middle line:90% 00:21:26.460 --> 00:21:30.180 align:middle line:84% OK, so while it's a very artificial model, 00:21:30.180 --> 00:21:32.460 align:middle line:84% I think it's a very insightful one because it makes 00:21:32.460 --> 00:21:34.960 align:middle line:90% several important predictions. 00:21:34.960 --> 00:21:37.830 align:middle line:84% So first one is maybe unintuitive. 00:21:37.830 --> 00:21:41.903 align:middle line:84% Price wars should be observed in equilibrium. 00:21:41.903 --> 00:21:42.695 align:middle line:90% I think there was-- 00:21:42.695 --> 00:21:46.740 align:middle line:84% if you read the old literature, there's this very nice book 00:21:46.740 --> 00:21:54.150 align:middle line:84% by Tom Ulen about railroad cartels written in the 1970s 00:21:54.150 --> 00:21:56.610 align:middle line:84% that's thinking about these cartels tried to organize 00:21:56.610 --> 00:21:58.662 align:middle line:84% collusion, and the collusion kept breaking down. 00:21:58.662 --> 00:22:00.870 align:middle line:84% And they kept having price wars because they couldn't 00:22:00.870 --> 00:22:03.150 align:middle line:90% collude with each other. 00:22:03.150 --> 00:22:05.760 align:middle line:84% Green and Porter makes a point that, no, 00:22:05.760 --> 00:22:07.260 align:middle line:84% price wars are not something that's 00:22:07.260 --> 00:22:09.370 align:middle line:84% a sign that the cartel has broken down. 00:22:09.370 --> 00:22:11.220 align:middle line:84% If you have a well-designed cartel, 00:22:11.220 --> 00:22:12.862 align:middle line:84% it has to have these price wars in it. 00:22:12.862 --> 00:22:14.820 align:middle line:84% Because if it didn't have the price wars in it, 00:22:14.820 --> 00:22:16.270 align:middle line:90% people would be cheating. 00:22:16.270 --> 00:22:18.150 align:middle line:84% And so when we see price wars, we 00:22:18.150 --> 00:22:20.490 align:middle line:84% can think well-functioning cartel 00:22:20.490 --> 00:22:23.855 align:middle line:84% rather than poorly functioning cartel that keeps breaking down. 00:22:23.855 --> 00:22:26.640 align:middle line:90% 00:22:26.640 --> 00:22:29.680 align:middle line:84% The optimal collusion in this model can have T finite. 00:22:29.680 --> 00:22:31.170 align:middle line:90% Why is that? 00:22:31.170 --> 00:22:33.990 align:middle line:84% Certainly, making T infinite makes it the biggest punishment 00:22:33.990 --> 00:22:36.310 align:middle line:84% and makes people not want to deviate. 00:22:36.310 --> 00:22:39.180 align:middle line:84% But if you make T infinite and alpha's not super small, 00:22:39.180 --> 00:22:41.430 align:middle line:84% then just basically only collude at the very beginning 00:22:41.430 --> 00:22:42.460 align:middle line:90% of the game. 00:22:42.460 --> 00:22:45.270 align:middle line:84% And so what you want to do is have a punishment that 00:22:45.270 --> 00:22:48.337 align:middle line:84% keeps people from deviating but not so long 00:22:48.337 --> 00:22:50.920 align:middle line:84% that you lose a lot of revenue while you're in the punishment. 00:22:50.920 --> 00:22:53.963 align:middle line:84% So the best equilibrium at this point 00:22:53.963 --> 00:22:56.130 align:middle line:84% is going be the one that has the shortest T possible 00:22:56.130 --> 00:22:57.190 align:middle line:90% that deters collusion. 00:22:57.190 --> 00:22:59.930 align:middle line:84% So we should have these short, finite price wars. 00:22:59.930 --> 00:23:02.860 align:middle line:90% 00:23:02.860 --> 00:23:05.350 align:middle line:90% What causes price wars? 00:23:05.350 --> 00:23:07.420 align:middle line:84% In this model, the insight is price wars 00:23:07.420 --> 00:23:11.740 align:middle line:84% are caused by demand patterns that make the participants think 00:23:11.740 --> 00:23:12.970 align:middle line:90% someone has cheated. 00:23:12.970 --> 00:23:15.350 align:middle line:84% In equilibrium, no one ever cheats. 00:23:15.350 --> 00:23:17.750 align:middle line:84% It's not people cheat, price wars start. 00:23:17.750 --> 00:23:21.530 align:middle line:84% It's always suspicious demand patterns arise, 00:23:21.530 --> 00:23:24.820 align:middle line:84% people start price wars knowing that it was-- 00:23:24.820 --> 00:23:26.440 align:middle line:84% with probability 1 and equilibrium, 00:23:26.440 --> 00:23:28.340 align:middle line:84% it was a suspicious demand pattern. 00:23:28.340 --> 00:23:30.850 align:middle line:84% But it's suspicious demand patterns that 00:23:30.850 --> 00:23:33.588 align:middle line:90% trigger price wars among firms. 00:23:33.588 --> 00:23:35.380 align:middle line:84% And you might think, well, if the firms all 00:23:35.380 --> 00:23:37.755 align:middle line:84% know that it's a suspicious demand pattern that triggered 00:23:37.755 --> 00:23:40.600 align:middle line:84% it, can't they just renegotiate out of it and say, OK, look, 00:23:40.600 --> 00:23:42.820 align:middle line:84% let's go back and ignore that one 00:23:42.820 --> 00:23:46.270 align:middle line:84% because we all know the firm made no orders this week? 00:23:46.270 --> 00:23:47.692 align:middle line:84% Once you do that, it doesn't work. 00:23:47.692 --> 00:23:49.150 align:middle line:84% The firms have to commit that we're 00:23:49.150 --> 00:23:51.970 align:middle line:84% going to have these price wars when these suspicious demand 00:23:51.970 --> 00:23:55.000 align:middle line:84% patterns arise because it's the suspicious demand patterns that 00:23:55.000 --> 00:23:56.100 align:middle line:90% resemble someone cheating. 00:23:56.100 --> 00:23:59.360 align:middle line:90% 00:23:59.360 --> 00:24:00.080 align:middle line:90% Yes? 00:24:00.080 --> 00:24:01.455 align:middle line:84% AUDIENCE: Are there cases we know 00:24:01.455 --> 00:24:04.370 align:middle line:90% about where we know it was-- 00:24:04.370 --> 00:24:07.310 align:middle line:84% the price war was due to the cartel functioning and not 00:24:07.310 --> 00:24:09.027 align:middle line:90% just demand? 00:24:09.027 --> 00:24:09.860 align:middle line:90% GLENN ELLISON: Yeah. 00:24:09.860 --> 00:24:15.890 align:middle line:84% So I will-- in Wednesday's lecture, 00:24:15.890 --> 00:24:17.950 align:middle line:84% I'm going to talk about Rob Porter has this paper 00:24:17.950 --> 00:24:21.350 align:middle line:84% re-examining a railroad cartel from the 19th century. 00:24:21.350 --> 00:24:25.150 align:middle line:90% And there-- I don't know. 00:24:25.150 --> 00:24:27.160 align:middle line:84% I guess he at least-- he argues very much 00:24:27.160 --> 00:24:30.050 align:middle line:84% that this model looks like it's describing what's going on. 00:24:30.050 --> 00:24:35.800 align:middle line:84% I think it is an open question, in that example, 00:24:35.800 --> 00:24:39.700 align:middle line:84% whether it's cheating or whether it's suspicious demand shocks 00:24:39.700 --> 00:24:40.390 align:middle line:90% or whatever. 00:24:40.390 --> 00:24:43.120 align:middle line:84% There are plenty of cartels where 00:24:43.120 --> 00:24:45.580 align:middle line:84% we know that someone cheated and that someone cheated 00:24:45.580 --> 00:24:47.500 align:middle line:90% and that triggered price wars. 00:24:47.500 --> 00:24:49.990 align:middle line:84% The no one cheated is often harder 00:24:49.990 --> 00:24:53.890 align:middle line:84% to know empirically because the whole point is that these things 00:24:53.890 --> 00:24:54.860 align:middle line:90% are unobservable. 00:24:54.860 --> 00:24:58.000 align:middle line:84% And so how do you prove that no one did cheat and caused 00:24:58.000 --> 00:25:00.855 align:middle line:84% that shock that made people, everyone start the price wars? 00:25:00.855 --> 00:25:05.090 align:middle line:90% 00:25:05.090 --> 00:25:06.580 align:middle line:90% But actually, it's interesting. 00:25:06.580 --> 00:25:07.955 align:middle line:84% I will talk about this next time. 00:25:07.955 --> 00:25:12.420 align:middle line:84% But in the-- collusion wasn't illegal in the United States 00:25:12.420 --> 00:25:14.460 align:middle line:90% until 1887. 00:25:14.460 --> 00:25:18.040 align:middle line:84% And so in the 1880s, there were collusion consultants. 00:25:18.040 --> 00:25:21.630 align:middle line:84% So just like firms can go out and hire McKinsey today, 00:25:21.630 --> 00:25:24.540 align:middle line:84% in the 1880s, you can go and hire a collusion consultant 00:25:24.540 --> 00:25:26.460 align:middle line:90% and say, we have our industry. 00:25:26.460 --> 00:25:28.780 align:middle line:90% We're competing too much. 00:25:28.780 --> 00:25:31.538 align:middle line:84% We're not making-- this was the era of the Vanderbilts 00:25:31.538 --> 00:25:32.080 align:middle line:90% and whatever. 00:25:32.080 --> 00:25:34.320 align:middle line:90% We're not making enough money. 00:25:34.320 --> 00:25:38.550 align:middle line:84% My mansion in Newport only has 137 rooms. 00:25:38.550 --> 00:25:39.790 align:middle line:90% I need higher prices. 00:25:39.790 --> 00:25:40.680 align:middle line:90% And you would hire-- 00:25:40.680 --> 00:25:42.300 align:middle line:84% there were McKinsey-like firms that you would hire, 00:25:42.300 --> 00:25:44.160 align:middle line:84% and those firms would consult with you 00:25:44.160 --> 00:25:46.050 align:middle line:84% on how to set up your collusive agreement so 00:25:46.050 --> 00:25:50.210 align:middle line:84% that you would be able to keep prices up or whatever. 00:25:50.210 --> 00:25:54.192 align:middle line:90% 00:25:54.192 --> 00:25:58.980 align:middle line:84% This model is very stark, that you get these random-- 00:25:58.980 --> 00:26:01.890 align:middle line:84% no one ever gets away with cheating in this model. 00:26:01.890 --> 00:26:05.350 align:middle line:84% In a model with continuous demand shocks, 00:26:05.350 --> 00:26:07.035 align:middle line:84% you would get away with cheating. 00:26:07.035 --> 00:26:10.100 align:middle line:90% 00:26:10.100 --> 00:26:13.400 align:middle line:84% So let me say Green and Porter do a different version-- 00:26:13.400 --> 00:26:15.030 align:middle line:84% do a better version of this model. 00:26:15.030 --> 00:26:18.930 align:middle line:84% Green and Porter, they do a Cournot version 00:26:18.930 --> 00:26:28.320 align:middle line:84% of this model, where the firms choose Q1t and Q2t. 00:26:28.320 --> 00:26:36.390 align:middle line:84% And then you have price at time t equals P of Q1t plus Q2t 00:26:36.390 --> 00:26:38.130 align:middle line:90% plus epsilon i t. 00:26:38.130 --> 00:26:41.350 align:middle line:84% So you have something that looks like real demand shocks. 00:26:41.350 --> 00:26:43.380 align:middle line:90% The firms choose quantities. 00:26:43.380 --> 00:26:46.470 align:middle line:84% There's some, then, downward-sloping demand curve. 00:26:46.470 --> 00:26:49.150 align:middle line:84% But then demand is also just random in every period. 00:26:49.150 --> 00:26:51.010 align:middle line:84% The market-clearing price is random. 00:26:51.010 --> 00:26:54.960 align:middle line:84% If you think about this, there's some price 00:26:54.960 --> 00:26:57.180 align:middle line:84% they're trying to coordinate on the monopoly price 00:26:57.180 --> 00:26:58.500 align:middle line:90% as the outcome. 00:26:58.500 --> 00:27:02.070 align:middle line:84% But you're going to get this distribution of prices 00:27:02.070 --> 00:27:04.470 align:middle line:90% even in equilibrium. 00:27:04.470 --> 00:27:07.035 align:middle line:84% And so what would you do if, every period, 00:27:07.035 --> 00:27:08.410 align:middle line:84% you're going to get a price drawn 00:27:08.410 --> 00:27:11.140 align:middle line:84% from this normal distribution around Pm 00:27:11.140 --> 00:27:13.510 align:middle line:90% if everyone's behaving properly? 00:27:13.510 --> 00:27:16.000 align:middle line:84% You can't say, OK, if price isn't Pm, 00:27:16.000 --> 00:27:18.800 align:middle line:84% we start a price war because that would happen all the time. 00:27:18.800 --> 00:27:22.960 align:middle line:84% So what you do is you do some cutoff here, like p-hat. 00:27:22.960 --> 00:27:26.810 align:middle line:84% And if price is greater than p-hat, we keep colluding. 00:27:26.810 --> 00:27:28.720 align:middle line:84% If price is less than p-hat, we think 00:27:28.720 --> 00:27:30.460 align:middle line:84% it's likely enough that someone cheated 00:27:30.460 --> 00:27:32.380 align:middle line:90% that we start a price war. 00:27:32.380 --> 00:27:38.630 align:middle line:84% In a model like this, I can cheat and not get caught. 00:27:38.630 --> 00:27:43.150 align:middle line:84% If I cheat, I'm pushing the price distribution down. 00:27:43.150 --> 00:27:45.070 align:middle line:84% But pushing the price distribution down, 00:27:45.070 --> 00:27:46.570 align:middle line:84% there's still this large probability 00:27:46.570 --> 00:27:48.590 align:middle line:90% that we don't start a price war. 00:27:48.590 --> 00:27:52.910 align:middle line:84% But I do increase the probability of the price war. 00:27:52.910 --> 00:27:55.210 align:middle line:84% And you have to hope that the-- it's got to-- 00:27:55.210 --> 00:27:57.190 align:middle line:84% so the increased probability of a price war 00:27:57.190 --> 00:28:01.060 align:middle line:84% from a small deviation is enough to get you to not make 00:28:01.060 --> 00:28:02.226 align:middle line:90% that small deviation. 00:28:02.226 --> 00:28:05.260 align:middle line:90% 00:28:05.260 --> 00:28:10.440 align:middle line:84% In a more realistic model, firms will-- firms 00:28:10.440 --> 00:28:12.420 align:middle line:84% could cheat and get away with cheating 00:28:12.420 --> 00:28:14.200 align:middle line:84% and only get caught probabilistically. 00:28:14.200 --> 00:28:17.370 align:middle line:84% This is part of what also makes it harder for us to know, 00:28:17.370 --> 00:28:18.940 align:middle line:90% was there really cheating there? 00:28:18.940 --> 00:28:21.335 align:middle line:84% Because if you do epsilon cheating, there's only-- 00:28:21.335 --> 00:28:24.210 align:middle line:84% if you have an order epsilon cheat, 00:28:24.210 --> 00:28:25.650 align:middle line:84% you get an order epsilon increase 00:28:25.650 --> 00:28:27.420 align:middle line:84% in profit and an order epsilon increase 00:28:27.420 --> 00:28:28.900 align:middle line:90% in causing the price war. 00:28:28.900 --> 00:28:31.190 align:middle line:84% Hard to know if that's happening or not. 00:28:31.190 --> 00:28:34.735 align:middle line:90% 00:28:34.735 --> 00:28:35.460 align:middle line:90% Yeah? 00:28:35.460 --> 00:28:37.637 align:middle line:84% AUDIENCE: Sometimes, in this model, 00:28:37.637 --> 00:28:38.970 align:middle line:90% you will get away with cheating. 00:28:38.970 --> 00:28:43.530 align:middle line:84% Do you mean in the sense that there's 00:28:43.530 --> 00:28:45.600 align:middle line:84% a mixed strategy in which you cheat sometimes 00:28:45.600 --> 00:28:47.010 align:middle line:90% or in that the equilibrium-- 00:28:47.010 --> 00:28:47.760 align:middle line:90% GLENN ELLISON: No. 00:28:47.760 --> 00:28:49.470 align:middle line:84% So the equilibrium we're going to look for 00:28:49.470 --> 00:28:50.680 align:middle line:84% is an equilibrium with no cheating. 00:28:50.680 --> 00:28:52.263 align:middle line:84% It's an equilibrium where firms always 00:28:52.263 --> 00:28:54.340 align:middle line:90% charge exactly these prices. 00:28:54.340 --> 00:29:00.390 align:middle line:84% But if you did go to Q1t minus epsilon, 00:29:00.390 --> 00:29:03.060 align:middle line:84% then you'd just raise the probability of a price war 00:29:03.060 --> 00:29:04.000 align:middle line:90% by epsilon. 00:29:04.000 --> 00:29:08.670 align:middle line:84% So there's some probability, x star, 00:29:08.670 --> 00:29:10.030 align:middle line:90% that we should start price wars. 00:29:10.030 --> 00:29:12.030 align:middle line:84% And now price wars start with probability x star 00:29:12.030 --> 00:29:14.185 align:middle line:90% plus epsilon or something. 00:29:14.185 --> 00:29:14.685 align:middle line:90% Yeah. 00:29:14.685 --> 00:29:19.930 align:middle line:90% 00:29:19.930 --> 00:29:22.020 align:middle line:84% And sustaining collusion is more difficult 00:29:22.020 --> 00:29:23.650 align:middle line:90% when demands are noisier. 00:29:23.650 --> 00:29:28.920 align:middle line:84% Here in this model, when alpha gets large, you can't-- 00:29:28.920 --> 00:29:31.460 align:middle line:84% you need alpha less than a half to make this work. 00:29:31.460 --> 00:29:38.670 align:middle line:84% In this model, if the price distribution looks like this, 00:29:38.670 --> 00:29:41.040 align:middle line:84% it's relatively easy to collude because you just 00:29:41.040 --> 00:29:42.990 align:middle line:90% put a threshold here. 00:29:42.990 --> 00:29:46.000 align:middle line:84% And no one wants to shift demand down below that. 00:29:46.000 --> 00:29:50.920 align:middle line:84% But if prices look like this, it could be that, 00:29:50.920 --> 00:29:53.170 align:middle line:84% no matter where you draw the cut-off, 00:29:53.170 --> 00:29:56.320 align:middle line:84% the increase in probability of a price war from cheating 00:29:56.320 --> 00:29:59.650 align:middle line:84% is so small that you're not pushing much mass 00:29:59.650 --> 00:30:01.100 align:middle line:90% beyond any one of these points. 00:30:01.100 --> 00:30:02.840 align:middle line:90% It's hard to stop collusion. 00:30:02.840 --> 00:30:03.770 align:middle line:90% It's hard to collude. 00:30:03.770 --> 00:30:08.860 align:middle line:84% So the closer we get to that first model example of you 00:30:08.860 --> 00:30:10.655 align:middle line:84% see someone cheat, you know they cheated, 00:30:10.655 --> 00:30:12.280 align:middle line:84% the easier it is to sustain collusion-- 00:30:12.280 --> 00:30:15.060 align:middle line:84% and the more there's noise in that, the harder it is. 00:30:15.060 --> 00:30:19.260 align:middle line:90% 00:30:19.260 --> 00:30:21.500 align:middle line:90% Questions? 00:30:21.500 --> 00:30:26.810 align:middle line:84% So obviously, this was a very beautiful 1984 paper 00:30:26.810 --> 00:30:28.910 align:middle line:90% that first did this. 00:30:28.910 --> 00:30:32.340 align:middle line:84% As I said, this is how Green and Porter actually did the model. 00:30:32.340 --> 00:30:36.303 align:middle line:84% It's harder to solve but more economically sensible. 00:30:36.303 --> 00:30:40.080 align:middle line:90% 00:30:40.080 --> 00:30:42.400 align:middle line:84% There's a big subsequent game theory literature. 00:30:42.400 --> 00:30:46.950 align:middle line:84% If you take [14.]126, they cover many of these models. 00:30:46.950 --> 00:30:50.340 align:middle line:90% One very nice paper on this-- 00:30:50.340 --> 00:30:53.760 align:middle line:84% Abreu, Pearce, Stachetti, Econometrica, 1990-- 00:30:53.760 --> 00:31:00.150 align:middle line:84% discusses more general strongly symmetric models like this. 00:31:00.150 --> 00:31:03.710 align:middle line:84% What Abreu, Pearce, and Stachetti 00:31:03.710 --> 00:31:06.440 align:middle line:84% show is that in the more general version, like Green 00:31:06.440 --> 00:31:10.700 align:middle line:84% and Porter's, you actually want to have-- 00:31:10.700 --> 00:31:13.280 align:middle line:84% they give this dynamic programming 00:31:13.280 --> 00:31:16.385 align:middle line:84% thing where it is basically always a two-- 00:31:16.385 --> 00:31:23.600 align:middle line:90% 00:31:23.600 --> 00:31:25.920 align:middle line:90% always a two-phase thing. 00:31:25.920 --> 00:31:28.530 align:middle line:84% But you actually-- what you end up doing is you do punishment. 00:31:28.530 --> 00:31:30.650 align:middle line:90% You do incentives on both sides. 00:31:30.650 --> 00:31:34.070 align:middle line:84% Here, firms don't want to cheat from the high price 00:31:34.070 --> 00:31:36.660 align:middle line:84% because if they cheat, we start a price war. 00:31:36.660 --> 00:31:38.720 align:middle line:84% But on the price war phase, what gets firms 00:31:38.720 --> 00:31:40.250 align:middle line:84% to charge these really low prices 00:31:40.250 --> 00:31:43.470 align:middle line:84% is the lower is the resulting price. 00:31:43.470 --> 00:31:44.760 align:middle line:90% You only switch back-- 00:31:44.760 --> 00:31:48.560 align:middle line:84% you switch back to the top not when-- 00:31:48.560 --> 00:31:52.460 align:middle line:84% not after T periods, but when price ends up low enough. 00:31:52.460 --> 00:31:56.300 align:middle line:84% And so it's like the firms are trying to drive price low enough 00:31:56.300 --> 00:32:00.840 align:middle line:84% to get back to a price war-- get back to colluding again. 00:32:00.840 --> 00:32:03.630 align:middle line:84% And so, in some sense, you use the incentives on both sides. 00:32:03.630 --> 00:32:06.930 align:middle line:84% Here, they have an incentive not to cheat because it causes this. 00:32:06.930 --> 00:32:08.910 align:middle line:84% Here, they have an incentive to price low 00:32:08.910 --> 00:32:11.190 align:middle line:84% because the lower they price, the more likely 00:32:11.190 --> 00:32:12.790 align:middle line:90% it is to jump back up. 00:32:12.790 --> 00:32:17.640 align:middle line:84% And so you still get this two-phase thing going on, 00:32:17.640 --> 00:32:22.110 align:middle line:84% but you get this two-phase thing where you have incentives. 00:32:22.110 --> 00:32:23.640 align:middle line:84% Rare events cause us to jump down, 00:32:23.640 --> 00:32:25.600 align:middle line:84% and then rare events cause us to jump back up, 00:32:25.600 --> 00:32:28.530 align:middle line:84% and that's the optimal collusive agreement. 00:32:28.530 --> 00:32:29.160 align:middle line:90% Yeah? 00:32:29.160 --> 00:32:31.577 align:middle line:84% AUDIENCE: Is the reason why they're pushing the price even 00:32:31.577 --> 00:32:33.960 align:middle line:84% below cost just to make the punishment painful enough 00:32:33.960 --> 00:32:35.085 align:middle line:90% that everyone goes back up? 00:32:35.085 --> 00:32:37.777 align:middle line:90% Or why would that happen? 00:32:37.777 --> 00:32:38.610 align:middle line:90% GLENN ELLISON: Yeah. 00:32:38.610 --> 00:32:45.030 align:middle line:84% Well, if you think about it, punishments are wasteful, right? 00:32:45.030 --> 00:32:46.810 align:middle line:90% You're in a punishment phase. 00:32:46.810 --> 00:32:48.540 align:middle line:84% You had a period where there was demand, 00:32:48.540 --> 00:32:50.550 align:middle line:90% and you served at a profit zero. 00:32:50.550 --> 00:32:54.473 align:middle line:84% So wouldn't it be better to lose a little bit more money today? 00:32:54.473 --> 00:32:57.230 align:middle line:90% 00:32:57.230 --> 00:33:02.670 align:middle line:84% Because your D pi D P evaluated at c is small or whatever. 00:33:02.670 --> 00:33:06.020 align:middle line:84% So you lose more money today, and then you 00:33:06.020 --> 00:33:08.970 align:middle line:84% can get more money back in the future or something like that. 00:33:08.970 --> 00:33:14.880 align:middle line:84% So, just in general, you want to give firms as much incentive 00:33:14.880 --> 00:33:16.260 align:middle line:90% as you can to make them low. 00:33:16.260 --> 00:33:17.980 align:middle line:84% And a good incentive to price low 00:33:17.980 --> 00:33:20.265 align:middle line:84% is that things will get better in the future if you-- 00:33:20.265 --> 00:33:22.140 align:middle line:84% the lower your price today, the more likely 00:33:22.140 --> 00:33:23.490 align:middle line:84% it is things will get better in the future 00:33:23.490 --> 00:33:24.865 align:middle line:84% is an extra incentive to get them 00:33:24.865 --> 00:33:28.550 align:middle line:90% to carry out the pricing low. 00:33:28.550 --> 00:33:29.050 align:middle line:90% Yeah. 00:33:29.050 --> 00:33:32.940 align:middle line:90% 00:33:32.940 --> 00:33:36.630 align:middle line:84% I'm not sure if we need it in that p equals c example 00:33:36.630 --> 00:33:37.980 align:middle line:90% with pure Bertrand competition. 00:33:37.980 --> 00:33:42.300 align:middle line:84% But in general, you want the incentives on both sides. 00:33:42.300 --> 00:33:44.400 align:middle line:90% i guess a thing about-- 00:33:44.400 --> 00:33:47.590 align:middle line:84% Abreu, Pearce, Stachetti also writes that, in some sense-- 00:33:47.590 --> 00:33:49.350 align:middle line:84% what they'll show you is that it's wrong-- 00:33:49.350 --> 00:33:50.850 align:middle line:84% if you're trying to think about what 00:33:50.850 --> 00:33:52.830 align:middle line:84% are all the possible strategies you can use, 00:33:52.830 --> 00:33:56.520 align:middle line:84% it's generally a bad idea to focus on strategy space 00:33:56.520 --> 00:33:59.560 align:middle line:84% because strategy space is very, very complicated. 00:33:59.560 --> 00:34:02.130 align:middle line:84% What you want to do is focus on payoff space, where 00:34:02.130 --> 00:34:05.880 align:middle line:84% these things are just numbers, and think about what's 00:34:05.880 --> 00:34:07.395 align:middle line:90% the highest Vm we can use. 00:34:07.395 --> 00:34:09.699 align:middle line:84% And to support the highest Vm possible, 00:34:09.699 --> 00:34:12.210 align:middle line:84% we want to use the lowest Vp possible. 00:34:12.210 --> 00:34:14.100 align:middle line:84% And to just think about what constraints 00:34:14.100 --> 00:34:17.100 align:middle line:84% the Vm and Vp have to solve and think 00:34:17.100 --> 00:34:19.389 align:middle line:84% about that from a dynamic programming perspective, 00:34:19.389 --> 00:34:21.900 align:middle line:84% just what's the highest Vm, what's the lowest Vp. 00:34:21.900 --> 00:34:24.159 align:middle line:84% What constraint would Vp have to satisfy? 00:34:24.159 --> 00:34:26.159 align:middle line:84% What constraint would Vm have to satisfy 00:34:26.159 --> 00:34:27.719 align:middle line:90% from not wanting to deviate? 00:34:27.719 --> 00:34:29.550 align:middle line:90% And then use that to-- 00:34:29.550 --> 00:34:30.804 align:middle line:90% I don't know. 00:34:30.804 --> 00:34:33.429 align:middle line:84% I would say that's a big insight from Abreu, Pearce, Stachetti, 00:34:33.429 --> 00:34:35.050 align:middle line:84% is we want to focus on this payoff space 00:34:35.050 --> 00:34:36.310 align:middle line:90% rather than on strategy space. 00:34:36.310 --> 00:34:39.730 align:middle line:84% Because payoff space can be just two dimensional, 00:34:39.730 --> 00:34:43.330 align:middle line:84% whereas strategy space is infinite dimensional. 00:34:43.330 --> 00:34:44.320 align:middle line:90% OK, then. 00:34:44.320 --> 00:34:47.050 align:middle line:84% Fudenberg, Levine, Maskin, a few years later, 00:34:47.050 --> 00:34:51.000 align:middle line:84% have this paper showing that, in many models, 00:34:51.000 --> 00:34:53.400 align:middle line:84% we can avoid the inefficiency that's 00:34:53.400 --> 00:34:56.489 align:middle line:84% in Green-Porter and Abreu, Pearce, Stachetti 00:34:56.489 --> 00:34:59.580 align:middle line:90% by using asymmetric strategies. 00:34:59.580 --> 00:35:04.350 align:middle line:84% The idea here is that let's suppose that you actually 00:35:04.350 --> 00:35:06.870 align:middle line:84% get a signal not only of that someone cheated, 00:35:06.870 --> 00:35:08.200 align:middle line:90% but who cheated. 00:35:08.200 --> 00:35:11.190 align:middle line:84% Now, the Green-Porter model, you can't tell who cheated. 00:35:11.190 --> 00:35:14.490 align:middle line:84% If this is firms are supposed to be choosing these quantities, 00:35:14.490 --> 00:35:18.010 align:middle line:84% price is too low, there's no common signal of who cheated. 00:35:18.010 --> 00:35:20.850 align:middle line:84% There's just that someone cheated. 00:35:20.850 --> 00:35:24.740 align:middle line:84% If the demand was at all asymmetric with regard 00:35:24.740 --> 00:35:27.415 align:middle line:84% to Firm One's price and Firm Two's price or Firm 00:35:27.415 --> 00:35:29.040 align:middle line:84% One's quantity and Firm Two's quantity, 00:35:29.040 --> 00:35:31.350 align:middle line:84% then you would get a signal about who cheated. 00:35:31.350 --> 00:35:33.230 align:middle line:84% And so what Fudenberg, Levine, Maskin say 00:35:33.230 --> 00:35:35.522 align:middle line:84% is that, well, if you actually had a signal about who's 00:35:35.522 --> 00:35:37.550 align:middle line:84% cheated, you should get rid of the price wars, 00:35:37.550 --> 00:35:40.490 align:middle line:84% and you should just do, say, a market share adjustment. 00:35:40.490 --> 00:35:42.890 align:middle line:90% If Firm One prices-- 00:35:42.890 --> 00:35:46.670 align:middle line:84% if it looks like the demand makes me think Firm One cheated, 00:35:46.670 --> 00:35:51.385 align:middle line:84% then I should have, instead of punishing Firm-- 00:35:51.385 --> 00:35:52.760 align:middle line:84% instead of having a price war, we 00:35:52.760 --> 00:35:55.862 align:middle line:84% should just have a market share reallocation, 00:35:55.862 --> 00:35:57.320 align:middle line:84% where, in the next period, Firm One 00:35:57.320 --> 00:35:58.987 align:middle line:84% gets a smaller market share and Firm Two 00:35:58.987 --> 00:36:01.010 align:middle line:90% gets a larger market share. 00:36:01.010 --> 00:36:03.650 align:middle line:84% And by doing market share reallocations, 00:36:03.650 --> 00:36:05.720 align:middle line:84% we avoid all the inefficiency of the price war, 00:36:05.720 --> 00:36:08.270 align:middle line:90% and we still punish people. 00:36:08.270 --> 00:36:12.650 align:middle line:84% So, for instance, if you assumed there were these two firms, one 00:36:12.650 --> 00:36:14.990 align:middle line:84% of them had larger demand on the Eastern US 00:36:14.990 --> 00:36:17.630 align:middle line:84% and one had larger demand on the Western US, 00:36:17.630 --> 00:36:20.172 align:middle line:84% then if you saw prices go down more in the East 00:36:20.172 --> 00:36:22.880 align:middle line:84% than in the West, you'd be like, OK, the East Coast firm cheated. 00:36:22.880 --> 00:36:25.370 align:middle line:84% Therefore, next period, East Coast firm 00:36:25.370 --> 00:36:29.535 align:middle line:84% has to charge monopoly price plus something. 00:36:29.535 --> 00:36:32.160 align:middle line:84% And the West Coast firms charge monopoly price minus something, 00:36:32.160 --> 00:36:34.430 align:middle line:84% and we reallocate market shares so 00:36:34.430 --> 00:36:36.600 align:middle line:84% that the East Coast firm is relatively disadvantaged 00:36:36.600 --> 00:36:40.670 align:middle line:84% but where there's no first-order loss in profits. 00:36:40.670 --> 00:36:43.500 align:middle line:84% So in general, that's the Fudenberg, Levine, Maskin ideas. 00:36:43.500 --> 00:36:46.040 align:middle line:90% If there is an asymmetry-- 00:36:46.040 --> 00:36:48.780 align:middle line:84% Green-Porter is unusual in not having any asymmetry. 00:36:48.780 --> 00:36:50.690 align:middle line:84% But if there is an asymmetry, you just 00:36:50.690 --> 00:36:53.150 align:middle line:84% want to reallocate market shares between the firms 00:36:53.150 --> 00:36:56.990 align:middle line:90% rather than doing that. 00:36:56.990 --> 00:36:57.590 align:middle line:90% Anyway. 00:36:57.590 --> 00:37:00.680 align:middle line:84% So there's one place here where there's 00:37:00.680 --> 00:37:05.465 align:middle line:84% recent literature, is the private monitoring, where-- 00:37:05.465 --> 00:37:10.340 align:middle line:90% 00:37:10.340 --> 00:37:12.710 align:middle line:84% all these papers are focusing on this public monitoring 00:37:12.710 --> 00:37:15.980 align:middle line:84% where everyone observes the same signal. 00:37:15.980 --> 00:37:19.310 align:middle line:84% Everyone observes the same signal of what happened. 00:37:19.310 --> 00:37:23.450 align:middle line:84% If you imagine-- if you had a model where the firm said P1t-- 00:37:23.450 --> 00:37:26.720 align:middle line:90% so we have these firms set. 00:37:26.720 --> 00:37:27.650 align:middle line:90% P1t. 00:37:27.650 --> 00:37:34.310 align:middle line:84% And then you get demands Q1t equals some function Q of P1t, 00:37:34.310 --> 00:37:48.170 align:middle line:84% P2t plus epsilon one t and Q2t, Q2 of P1t, P2t plus epsilon 2t. 00:37:48.170 --> 00:37:50.480 align:middle line:84% This would be what we would call a private monitoring 00:37:50.480 --> 00:37:54.920 align:middle line:84% model, where each firm gets a different signal. 00:37:54.920 --> 00:37:56.840 align:middle line:90% So Firm One observes its demand. 00:37:56.840 --> 00:37:58.790 align:middle line:90% Firm Two observes its demands. 00:37:58.790 --> 00:38:01.310 align:middle line:84% And they have different signals because they 00:38:01.310 --> 00:38:02.880 align:middle line:90% have different epsilons. 00:38:02.880 --> 00:38:04.760 align:middle line:84% This would be a private monitoring model 00:38:04.760 --> 00:38:08.620 align:middle line:84% where I don't know what signal you got 00:38:08.620 --> 00:38:10.850 align:middle line:84% and you don't know what signal I got. 00:38:10.850 --> 00:38:14.580 align:middle line:84% Anyway, it's harder to solve models 00:38:14.580 --> 00:38:16.710 align:middle line:84% once you have that private uncertainty instead 00:38:16.710 --> 00:38:17.980 align:middle line:90% of the public uncertainty. 00:38:17.980 --> 00:38:19.890 align:middle line:90% There's a literature on that. 00:38:19.890 --> 00:38:23.040 align:middle line:84% Actually, Alex has a very nice, recent paper arguing that 00:38:23.040 --> 00:38:25.270 align:middle line:84% sometimes, actually, private monitoring 00:38:25.270 --> 00:38:26.520 align:middle line:90% will make it more complicated. 00:38:26.520 --> 00:38:29.160 align:middle line:84% Could even make collusion easier. 00:38:29.160 --> 00:38:31.652 align:middle line:84% Although, obviously, sometimes, the normal case would be it 00:38:31.652 --> 00:38:32.610 align:middle line:90% makes collusion harder. 00:38:32.610 --> 00:38:35.240 align:middle line:84% But some cases where that can even make collusion easier. 00:38:35.240 --> 00:38:39.615 align:middle line:90% 00:38:39.615 --> 00:38:43.040 align:middle line:90% Any questions? 00:38:43.040 --> 00:38:46.135 align:middle line:84% OK, the next set of things I wanted to just-- 00:38:46.135 --> 00:38:48.260 align:middle line:84% I'm going to go through and highlight them quickly. 00:38:48.260 --> 00:38:49.927 align:middle line:84% Maybe I'll come back to them at the end. 00:38:49.927 --> 00:38:52.730 align:middle line:84% But I wanted to get to more recent material at the end. 00:38:52.730 --> 00:38:55.110 align:middle line:90% So just some classics. 00:38:55.110 --> 00:39:02.270 align:middle line:84% Rotemberg and Saloner, 1986, argues that variable demand 00:39:02.270 --> 00:39:05.030 align:middle line:90% can make collusion harder. 00:39:05.030 --> 00:39:08.270 align:middle line:84% Let me just say, the basic argument here is-- as I said, 00:39:08.270 --> 00:39:11.660 align:middle line:84% it's harder to collude when the present is 00:39:11.660 --> 00:39:13.940 align:middle line:90% more important than the future. 00:39:13.940 --> 00:39:17.160 align:middle line:84% And when the future is more important, it's easy to collude. 00:39:17.160 --> 00:39:20.160 align:middle line:84% When the future is less important, it's hard to collude. 00:39:20.160 --> 00:39:22.670 align:middle line:84% And so the basic intuition for Rotemberg and Saloner 00:39:22.670 --> 00:39:28.510 align:middle line:84% was that imagine we just have firms competing, 00:39:28.510 --> 00:39:31.240 align:middle line:84% and they're just facing a macro economy 00:39:31.240 --> 00:39:33.400 align:middle line:90% that has ups and downs to it. 00:39:33.400 --> 00:39:39.010 align:middle line:84% So if you just have business cycles like this, where this 00:39:39.010 --> 00:39:42.710 align:middle line:90% is t and this is Qt of p star-- 00:39:42.710 --> 00:39:45.460 align:middle line:84% so imagine you have firms that are trying to collude 00:39:45.460 --> 00:39:46.675 align:middle line:90% with their business cycles. 00:39:46.675 --> 00:39:48.550 align:middle line:84% And so there are times when they're selling-- 00:39:48.550 --> 00:39:50.860 align:middle line:84% 1955, they're selling a lot of cars. 00:39:50.860 --> 00:39:53.445 align:middle line:84% 1957, they're not selling as many cars. 00:39:53.445 --> 00:39:57.570 align:middle line:90% 00:39:57.570 --> 00:39:59.310 align:middle line:84% You know, Rotemberg and Saloner note 00:39:59.310 --> 00:40:01.920 align:middle line:84% that, in an economy like this, it's 00:40:01.920 --> 00:40:05.880 align:middle line:84% harder to collude when demand is variable because here 00:40:05.880 --> 00:40:08.220 align:middle line:84% the present matters a lot relative to the future, 00:40:08.220 --> 00:40:11.440 align:middle line:84% whereas here the present future would matter more. 00:40:11.440 --> 00:40:14.110 align:middle line:84% So here, it's relatively easy to collude. 00:40:14.110 --> 00:40:16.970 align:middle line:90% Here, it's hard to collude. 00:40:16.970 --> 00:40:20.480 align:middle line:84% And so, then, what Rotemberg and Saloner ask is, well, what would 00:40:20.480 --> 00:40:24.680 align:middle line:84% firms do if they were colluding in a market like this? 00:40:24.680 --> 00:40:29.930 align:middle line:84% And imagine the delta is such that you 00:40:29.930 --> 00:40:32.780 align:middle line:84% can't collude on the monopoly price here, 00:40:32.780 --> 00:40:35.610 align:middle line:84% but you can collude on the monopoly price here. 00:40:35.610 --> 00:40:40.520 align:middle line:84% So what do you do in these periods? 00:40:40.520 --> 00:40:41.680 align:middle line:90% So if there was some-- 00:40:41.680 --> 00:40:45.610 align:middle line:90% 00:40:45.610 --> 00:40:47.900 align:middle line:84% imagine this is just a multiplicative shock to demand. 00:40:47.900 --> 00:40:49.900 align:middle line:84% So what I'd like to do is have the same monopoly 00:40:49.900 --> 00:40:51.400 align:middle line:90% price in every period. 00:40:51.400 --> 00:40:54.320 align:middle line:84% I'd like to have the same monopoly price in every period, 00:40:54.320 --> 00:40:55.960 align:middle line:84% but what I'm going to need to do is, 00:40:55.960 --> 00:40:58.690 align:middle line:84% when we're in these value of the high demand shocks, 00:40:58.690 --> 00:41:02.080 align:middle line:84% I'm going to have to drag down my price 00:41:02.080 --> 00:41:05.523 align:middle line:84% and price it less than the monopoly price. 00:41:05.523 --> 00:41:07.940 align:middle line:84% And I'm going to have to price it less than monopoly price 00:41:07.940 --> 00:41:09.500 align:middle line:84% because I just pulled my price down 00:41:09.500 --> 00:41:11.180 align:middle line:90% to reduce the incentive to-- 00:41:11.180 --> 00:41:13.310 align:middle line:90% temptation to deviate. 00:41:13.310 --> 00:41:17.440 align:middle line:84% And so what Rotemberg and Saloner 00:41:17.440 --> 00:41:20.980 align:middle line:84% argue is that, if you have a cartel, 00:41:20.980 --> 00:41:22.885 align:middle line:84% it's harder to collude during the booms. 00:41:22.885 --> 00:41:24.760 align:middle line:84% That doesn't mean you should have price wars. 00:41:24.760 --> 00:41:26.135 align:middle line:84% That inclusion should break down. 00:41:26.135 --> 00:41:29.140 align:middle line:84% But what you should have is this countercyclical pricing, 00:41:29.140 --> 00:41:33.220 align:middle line:84% where, because demand is high, we drop our price down. 00:41:33.220 --> 00:41:39.053 align:middle line:84% And we could actually see lower prices here than we see here. 00:41:39.053 --> 00:41:40.720 align:middle line:84% Now, obviously, depending on what demand 00:41:40.720 --> 00:41:43.750 align:middle line:84% specification you make, if I do linear demands-- 00:41:43.750 --> 00:41:46.880 align:middle line:84% in linear demands, price goes up when demand goes up. 00:41:46.880 --> 00:41:50.080 align:middle line:84% So it could be that the monopoly price isn't a flat line. 00:41:50.080 --> 00:41:53.560 align:middle line:84% It could be the monopoly price would also be something 00:41:53.560 --> 00:41:55.510 align:middle line:90% that mirrors the demand. 00:41:55.510 --> 00:41:57.730 align:middle line:84% Then what Rotemberg and Saloner say is, 00:41:57.730 --> 00:42:00.220 align:middle line:84% OK, you're going to pull down your price here. 00:42:00.220 --> 00:42:05.020 align:middle line:84% But pulling down your price here, it could still be the-- 00:42:05.020 --> 00:42:08.440 align:middle line:84% it could still be higher than the price here because it's-- 00:42:08.440 --> 00:42:10.990 align:middle line:84% I've pulled down my price, but I pulled it 00:42:10.990 --> 00:42:14.230 align:middle line:84% down from something that was starting even higher. 00:42:14.230 --> 00:42:17.020 align:middle line:84% But anyway, the basic idea of Rotemberg and Saloner 00:42:17.020 --> 00:42:20.560 align:middle line:84% is that you can get this countercyclical pricing that 00:42:20.560 --> 00:42:23.140 align:middle line:84% comes from a countercyclical incentive 00:42:23.140 --> 00:42:25.960 align:middle line:84% to deviate on a cartel agreement. 00:42:25.960 --> 00:42:28.660 align:middle line:84% Anyway, on the slides, I've worked out 00:42:28.660 --> 00:42:32.470 align:middle line:84% an example in a two-state model where demand is just i.i.d. 00:42:32.470 --> 00:42:34.870 align:middle line:90% high or low. 00:42:34.870 --> 00:42:38.170 align:middle line:84% And when demand is i.i.d. high or low, the algebra is easy, 00:42:38.170 --> 00:42:40.988 align:middle line:84% and you can work out what you do. 00:42:40.988 --> 00:42:43.155 align:middle line:84% But you can look at that on the slides, if you like. 00:42:43.155 --> 00:42:46.860 align:middle line:90% 00:42:46.860 --> 00:42:47.360 align:middle line:90% Let's see. 00:42:47.360 --> 00:42:49.140 align:middle line:84% Which of these things do I want to say? 00:42:49.140 --> 00:42:49.640 align:middle line:90% Yeah. 00:42:49.640 --> 00:42:53.740 align:middle line:84% So for intermediate, markups are countercyclical. 00:42:53.740 --> 00:42:58.690 align:middle line:84% Yeah, If you add imperfect observation to this, 00:42:58.690 --> 00:43:00.790 align:middle line:84% intuition is still that collusion is harder 00:43:00.790 --> 00:43:04.030 align:middle line:84% when demand is high than when demand is low. 00:43:04.030 --> 00:43:07.060 align:middle line:84% And so what are you going to have to do? 00:43:07.060 --> 00:43:08.515 align:middle line:90% One of two things has to happen. 00:43:08.515 --> 00:43:12.040 align:middle line:90% 00:43:12.040 --> 00:43:13.870 align:middle line:84% Either you have to reduce the prices 00:43:13.870 --> 00:43:17.560 align:middle line:84% and reduce the markups, or breakdowns of collusion become 00:43:17.560 --> 00:43:19.550 align:middle line:90% more likely, or both. 00:43:19.550 --> 00:43:22.930 align:middle line:84% But in general, you could adjust either of two ways. 00:43:22.930 --> 00:43:25.540 align:middle line:84% You could either drop the prices, 00:43:25.540 --> 00:43:27.400 align:middle line:84% or you could keep the prices up and just 00:43:27.400 --> 00:43:29.960 align:middle line:84% have more frequent breakdowns of collusion. 00:43:29.960 --> 00:43:33.035 align:middle line:84% Or you could do a combination of the two of them. 00:43:33.035 --> 00:43:34.660 align:middle line:84% Or you could do one, but not the other, 00:43:34.660 --> 00:43:36.650 align:middle line:84% possibly, depending on the specification. 00:43:36.650 --> 00:43:38.710 align:middle line:84% But it's just the general idea should 00:43:38.710 --> 00:43:41.865 align:middle line:84% be that collusion is harder and something has to work worse. 00:43:41.865 --> 00:43:50.630 align:middle line:90% 00:43:50.630 --> 00:43:52.850 align:middle line:90% Second idea I'll do quickly. 00:43:52.850 --> 00:43:55.880 align:middle line:90% Private cost shocks. 00:43:55.880 --> 00:44:00.950 align:middle line:84% So Athey and Bagwell note that efficient collusion 00:44:00.950 --> 00:44:03.715 align:middle line:84% you can think of as having two components to it. 00:44:03.715 --> 00:44:08.680 align:middle line:90% 00:44:08.680 --> 00:44:12.010 align:middle line:84% One is that firms want to charge the monopoly price. 00:44:12.010 --> 00:44:13.720 align:middle line:84% The other is that the colluding firms 00:44:13.720 --> 00:44:15.880 align:middle line:90% want to produce efficiently. 00:44:15.880 --> 00:44:17.710 align:middle line:84% And in some cases, the firms are going 00:44:17.710 --> 00:44:19.990 align:middle line:84% to have cost shocks that make it efficient for one 00:44:19.990 --> 00:44:22.100 align:middle line:90% of the two of them to serve. 00:44:22.100 --> 00:44:25.990 align:middle line:84% So, for instance, for this one, for an example, think about-- 00:44:25.990 --> 00:44:29.110 align:middle line:84% let's imagine-- and this may not be so unrealistic-- 00:44:29.110 --> 00:44:33.070 align:middle line:84% that in a city like Cambridge, if there's a giant paving job 00:44:33.070 --> 00:44:34.060 align:middle line:90% that needs to be done-- 00:44:34.060 --> 00:44:36.580 align:middle line:84% you need to repave Memorial Drive or whatever-- 00:44:36.580 --> 00:44:40.620 align:middle line:84% there's a very small number of firms that could do that job. 00:44:40.620 --> 00:44:42.780 align:middle line:84% And so what the city of Cambridge is going to do 00:44:42.780 --> 00:44:43.830 align:middle line:90% is put out for bids. 00:44:43.830 --> 00:44:46.990 align:middle line:84% Like, we need someone to repave two miles of Memorial Drive. 00:44:46.990 --> 00:44:50.970 align:middle line:84% There's a very limited number of firms who could do that job. 00:44:50.970 --> 00:44:55.140 align:middle line:84% Those two firms are going to have different costs of doing 00:44:55.140 --> 00:44:57.960 align:middle line:84% the job in each period because their equipment may 00:44:57.960 --> 00:44:59.350 align:middle line:90% be in use elsewhere. 00:44:59.350 --> 00:45:01.710 align:middle line:84% It may be that I've got a job in Revere 00:45:01.710 --> 00:45:04.320 align:middle line:84% where I'm paving some big road, and it wouldn't be costly 00:45:04.320 --> 00:45:07.367 align:middle line:84% for me to also get extra manpower and equipment 00:45:07.367 --> 00:45:08.200 align:middle line:90% to get to Cambridge. 00:45:08.200 --> 00:45:10.200 align:middle line:84% Or it could be that I don't have any other jobs, 00:45:10.200 --> 00:45:12.870 align:middle line:84% and so this would be a good time for me to serve Cambridge. 00:45:12.870 --> 00:45:16.020 align:middle line:90% So what they assume is that-- 00:45:16.020 --> 00:45:17.640 align:middle line:90% we have these two firms. 00:45:17.640 --> 00:45:19.620 align:middle line:90% They're serving some demand. 00:45:19.620 --> 00:45:22.722 align:middle line:84% And the firms have random cost shocks. 00:45:22.722 --> 00:45:24.180 align:middle line:84% Again, to make it simple, they just 00:45:24.180 --> 00:45:27.280 align:middle line:84% go with two different levels of cost, C L or C H. 00:45:27.280 --> 00:45:29.040 align:middle line:84% And what the firms would like to do 00:45:29.040 --> 00:45:34.080 align:middle line:84% is collude on charging monopoly price and collude on whichever 00:45:34.080 --> 00:45:39.390 align:middle line:84% of us has low cost wins the job and whoever has high cost 00:45:39.390 --> 00:45:40.852 align:middle line:90% doesn't win the job. 00:45:40.852 --> 00:45:43.060 align:middle line:84% So in some sense, what we should do every period is-- 00:45:43.060 --> 00:45:50.850 align:middle line:90% 00:45:50.850 --> 00:45:53.250 align:middle line:84% so the city of Cambridge just repeatedly issuing 00:45:53.250 --> 00:45:55.440 align:middle line:90% these requests for bids. 00:45:55.440 --> 00:45:59.160 align:middle line:84% If our costs are C L, C H, what we should do 00:45:59.160 --> 00:46:03.210 align:middle line:84% is I bid the monopoly price and you bid the monopoly price 00:46:03.210 --> 00:46:05.650 align:middle line:90% plus epsilon. 00:46:05.650 --> 00:46:11.110 align:middle line:84% And if the costs are C H, C L, then we 00:46:11.110 --> 00:46:18.050 align:middle line:84% can bid monopoly price plus epsilon monopoly price. 00:46:18.050 --> 00:46:22.370 align:middle line:84% And if the costs are C H, C H, we 00:46:22.370 --> 00:46:25.800 align:middle line:84% can just go ahead and bid monopoly price, monopoly price. 00:46:25.800 --> 00:46:29.630 align:middle line:84% And just one of us is chosen at random to win that auction. 00:46:29.630 --> 00:46:31.345 align:middle line:84% So this is what we would like to do. 00:46:31.345 --> 00:46:36.880 align:middle line:90% 00:46:36.880 --> 00:46:37.760 align:middle line:90% How do you do that? 00:46:37.760 --> 00:46:39.670 align:middle line:84% Well, the firms have this other problem now. 00:46:39.670 --> 00:46:42.070 align:middle line:84% The problem is this communication between them. 00:46:42.070 --> 00:46:44.280 align:middle line:84% The question is, who gets to bid the monopoly price, 00:46:44.280 --> 00:46:46.030 align:middle line:84% and who's supposed to bid the higher price 00:46:46.030 --> 00:46:47.627 align:middle line:90% and let the other firm thing? 00:46:47.627 --> 00:46:49.210 align:middle line:84% In some sense, every period, the firms 00:46:49.210 --> 00:46:52.220 align:middle line:84% can get together and be like, OK, who wins this job? 00:46:52.220 --> 00:46:53.300 align:middle line:90% Who has the C L cost? 00:46:53.300 --> 00:46:54.220 align:middle line:84% And everyone's going to raise their hand. 00:46:54.220 --> 00:46:54.720 align:middle line:90% Oh, yeah. 00:46:54.720 --> 00:46:56.450 align:middle line:90% My cost is C L this period. 00:46:56.450 --> 00:46:59.050 align:middle line:90% I should get this job. 00:46:59.050 --> 00:47:00.910 align:middle line:84% Bad luck for you, but I guess you bid 00:47:00.910 --> 00:47:02.687 align:middle line:90% price plus epsilon or whatever. 00:47:02.687 --> 00:47:05.020 align:middle line:84% And then, again, you have this temptation, over and over 00:47:05.020 --> 00:47:05.980 align:middle line:84% and over again, the firm would raise their hands 00:47:05.980 --> 00:47:08.105 align:middle line:84% and be like, oh, yep, me, just happened to have all 00:47:08.105 --> 00:47:11.383 align:middle line:84% my equipment, a bunch of spare labor. 00:47:11.383 --> 00:47:13.050 align:middle line:84% I need to win this job again this month. 00:47:13.050 --> 00:47:15.880 align:middle line:90% 00:47:15.880 --> 00:47:19.300 align:middle line:84% What Athey-Bagwell shows, that's an additional obstacle 00:47:19.300 --> 00:47:21.880 align:middle line:90% to collusion, this desire to-- 00:47:21.880 --> 00:47:25.900 align:middle line:84% that having the need to shift the market shares across firms 00:47:25.900 --> 00:47:28.030 align:middle line:90% is an extra problem. 00:47:28.030 --> 00:47:32.530 align:middle line:84% One thing they show is, if delta is close to one, 00:47:32.530 --> 00:47:34.970 align:middle line:84% the firms can still achieve full collusion. 00:47:34.970 --> 00:47:36.970 align:middle line:90% And this isn't a problem. 00:47:36.970 --> 00:47:40.630 align:middle line:84% The way they can do this is with this dynamic allocation 00:47:40.630 --> 00:47:42.040 align:middle line:90% of market share. 00:47:42.040 --> 00:47:46.780 align:middle line:84% So the rule could be, if my costs are really low 00:47:46.780 --> 00:47:51.220 align:middle line:84% and your costs are high, I go ahead and bid Pm and I win. 00:47:51.220 --> 00:47:58.720 align:middle line:84% But then the next time C L, C L happens, 00:47:58.720 --> 00:48:01.300 align:middle line:84% if you said your cost was low and you 00:48:01.300 --> 00:48:03.640 align:middle line:84% wanted to be the winner of this auction, 00:48:03.640 --> 00:48:05.410 align:middle line:84% then the next time an auction comes around 00:48:05.410 --> 00:48:10.795 align:middle line:84% where we both raise our hands, you have to set the high price, 00:48:10.795 --> 00:48:12.670 align:middle line:84% and the other firm gets to set the low price. 00:48:12.670 --> 00:48:14.850 align:middle line:84% So it's like you dynamically make up 00:48:14.850 --> 00:48:17.550 align:middle line:84% for the market share gains that firms got from claiming 00:48:17.550 --> 00:48:19.470 align:middle line:84% their costs was low by saying, OK, 00:48:19.470 --> 00:48:22.050 align:middle line:84% if you claim your cost is low today, then 00:48:22.050 --> 00:48:25.740 align:middle line:84% the next time we both claim our costs are low, I get the demand, 00:48:25.740 --> 00:48:27.750 align:middle line:90% and you don't get the demand. 00:48:27.750 --> 00:48:31.830 align:middle line:84% And so the idea is that, by dynamically 00:48:31.830 --> 00:48:40.180 align:middle line:84% reallocating market share, we can sometimes collude optimally. 00:48:40.180 --> 00:48:43.570 align:middle line:84% When delta is not as close to one, however, 00:48:43.570 --> 00:48:45.078 align:middle line:90% this stops working. 00:48:45.078 --> 00:48:47.620 align:middle line:84% And the problem is, you can get this same problem I discussed 00:48:47.620 --> 00:48:49.537 align:middle line:84% before of just some firm, over and over again, 00:48:49.537 --> 00:48:51.940 align:middle line:84% says, yeah, my costs are low, my costs are low. 00:48:51.940 --> 00:48:53.950 align:middle line:90% My costs are low. 00:48:53.950 --> 00:48:57.080 align:middle line:84% You can't keep promising, OK, next time your costs-- 00:48:57.080 --> 00:48:58.930 align:middle line:90% next time both costs are low-- 00:48:58.930 --> 00:49:00.700 align:middle line:84% in some sense, you're accumulating debts 00:49:00.700 --> 00:49:01.480 align:middle line:90% to the other firm. 00:49:01.480 --> 00:49:03.105 align:middle line:84% And you can't accumulate too many debts 00:49:03.105 --> 00:49:04.850 align:middle line:90% before this thing stops working. 00:49:04.850 --> 00:49:07.930 align:middle line:84% So at some point, you would have to get to an agreement where, 00:49:07.930 --> 00:49:11.030 align:middle line:84% OK, if you've done too many times in a row, 00:49:11.030 --> 00:49:13.190 align:middle line:84% even if your costs are low and my costs are high, 00:49:13.190 --> 00:49:15.650 align:middle line:84% you still have to back out and do this. 00:49:15.650 --> 00:49:18.172 align:middle line:84% Or we can stop you from deviating-- stop you 00:49:18.172 --> 00:49:20.880 align:middle line:84% from doing that by saying you've done it too many times in a row. 00:49:20.880 --> 00:49:22.490 align:middle line:84% If you claim your costs are low, you 00:49:22.490 --> 00:49:24.920 align:middle line:84% have to cut your price below the monopoly price 00:49:24.920 --> 00:49:27.200 align:middle line:84% instead of charging the monopoly price. 00:49:27.200 --> 00:49:29.480 align:middle line:84% Because we need some extra disincentive. 00:49:29.480 --> 00:49:32.330 align:middle line:84% Once you're already saying, in the year 2045, 00:49:32.330 --> 00:49:34.530 align:middle line:84% I'll give you market share, that doesn't work. 00:49:34.530 --> 00:49:35.315 align:middle line:90% So you have to-- 00:49:35.315 --> 00:49:37.460 align:middle line:84% for the firm, if you're going to say your costs are low, 00:49:37.460 --> 00:49:39.085 align:middle line:84% we're going to make you cut your price. 00:49:39.085 --> 00:49:41.720 align:middle line:90% 00:49:41.720 --> 00:49:42.590 align:middle line:90% So anyway. 00:49:42.590 --> 00:49:45.830 align:middle line:84% I think the message of Athey and Bagwell is that it's the cost 00:49:45.830 --> 00:49:47.810 align:middle line:84% shocks are a problem, but they're-- 00:49:47.810 --> 00:49:51.110 align:middle line:84% if firms are clever and can do these dynamic reallocations, 00:49:51.110 --> 00:49:53.510 align:middle line:84% because it's frequent things, that may not 00:49:53.510 --> 00:49:54.995 align:middle line:90% be such an obstacle after all. 00:49:54.995 --> 00:49:58.820 align:middle line:90% 00:49:58.820 --> 00:50:01.470 align:middle line:84% Harrington has this paper about antitrust authorities. 00:50:01.470 --> 00:50:05.150 align:middle line:84% Obviously, in practice, we do think that one of the reasons 00:50:05.150 --> 00:50:07.200 align:middle line:84% firms don't collude is because it's illegal 00:50:07.200 --> 00:50:08.870 align:middle line:90% and you can go to jail. 00:50:08.870 --> 00:50:12.540 align:middle line:84% And business executives who make millions of dollars 00:50:12.540 --> 00:50:15.280 align:middle line:84% don't like spending time in federal prison. 00:50:15.280 --> 00:50:16.606 align:middle line:90% It's unpleasant. 00:50:16.606 --> 00:50:21.120 align:middle line:90% 00:50:21.120 --> 00:50:24.600 align:middle line:84% Obviously, so that is a constraint on antitrust, 00:50:24.600 --> 00:50:28.680 align:middle line:84% on collusion is that people can go to jail. 00:50:28.680 --> 00:50:33.090 align:middle line:84% And firms can pay very large fines. 00:50:33.090 --> 00:50:35.730 align:middle line:84% One thing that Harrington notes is that that is true. 00:50:35.730 --> 00:50:39.450 align:middle line:84% But you can imagine reasons that doesn't work so well. 00:50:39.450 --> 00:50:44.820 align:middle line:84% So, for instance, once you have those penalties there, 00:50:44.820 --> 00:50:47.620 align:middle line:84% firms really don't want the penalties to get triggered. 00:50:47.620 --> 00:50:49.470 align:middle line:84% And it may be that something that 00:50:49.470 --> 00:50:52.890 align:middle line:84% triggers a penalty is a breakdown of the cartel 00:50:52.890 --> 00:50:53.440 align:middle line:90% agreement. 00:50:53.440 --> 00:50:57.810 align:middle line:84% So if we have a price war and the price drops from 100 to 60 00:50:57.810 --> 00:51:01.278 align:middle line:84% and then goes back up again, that's 00:51:01.278 --> 00:51:02.820 align:middle line:84% a clear sign to antitrust authorities 00:51:02.820 --> 00:51:04.560 align:middle line:84% that maybe we should look into what's going on here 00:51:04.560 --> 00:51:06.347 align:middle line:84% and are they colluding with each other. 00:51:06.347 --> 00:51:08.430 align:middle line:84% Therefore, you may not want to cut price and cheat 00:51:08.430 --> 00:51:10.120 align:middle line:84% from the agreement in the first place. 00:51:10.120 --> 00:51:11.745 align:middle line:84% And therefore, you may-- in some sense, 00:51:11.745 --> 00:51:14.088 align:middle line:84% you can drift into collusion and get even more collusion 00:51:14.088 --> 00:51:15.630 align:middle line:84% with the antitrust authority than you 00:51:15.630 --> 00:51:19.020 align:middle line:84% can get without because no one wants to risk starting the price 00:51:19.020 --> 00:51:22.020 align:middle line:84% war that then makes people notice that there was a cartel 00:51:22.020 --> 00:51:24.465 align:middle line:90% agreement going on. 00:51:24.465 --> 00:51:29.332 align:middle line:84% I think while I do think the main version of that 00:51:29.332 --> 00:51:31.290 align:middle line:84% is certainly true, that collusion being illegal 00:51:31.290 --> 00:51:34.770 align:middle line:90% is a deterrent, it can also be-- 00:51:34.770 --> 00:51:37.170 align:middle line:84% not well-designed collusion policies 00:51:37.170 --> 00:51:40.560 align:middle line:84% can also, then, in some sense, lock people into cartels. 00:51:40.560 --> 00:51:42.120 align:middle line:84% Something you often see in practice 00:51:42.120 --> 00:51:45.540 align:middle line:84% reflecting this is that, sometimes, these things have 00:51:45.540 --> 00:51:48.455 align:middle line:84% whistleblower protections, that if you go to the government 00:51:48.455 --> 00:51:49.830 align:middle line:84% and say, hey, I've been colluding 00:51:49.830 --> 00:51:52.147 align:middle line:84% with these guys for the past five years, 00:51:52.147 --> 00:51:54.480 align:middle line:84% you get off with a much lighter punishment than everyone 00:51:54.480 --> 00:51:55.810 align:middle line:90% that you were colluding with. 00:51:55.810 --> 00:51:58.385 align:middle line:84% And that can be a way to try to get rid of this problem. 00:51:58.385 --> 00:52:03.370 align:middle line:90% 00:52:03.370 --> 00:52:05.010 align:middle line:90% Multimarket contact. 00:52:05.010 --> 00:52:05.920 align:middle line:90% Limited time. 00:52:05.920 --> 00:52:08.100 align:middle line:84% I will leave that for you to read, also. 00:52:08.100 --> 00:52:09.275 align:middle line:90% So things I wanted to do. 00:52:09.275 --> 00:52:13.610 align:middle line:90% 00:52:13.610 --> 00:52:16.670 align:middle line:84% How do people in IO get away from this indeterminacy 00:52:16.670 --> 00:52:18.488 align:middle line:84% and not just always have this conclusion 00:52:18.488 --> 00:52:20.030 align:middle line:84% that we can't do empirical work or we 00:52:20.030 --> 00:52:23.620 align:middle line:84% can't do other things because we can't tell what's going on? 00:52:23.620 --> 00:52:26.950 align:middle line:90% 00:52:26.950 --> 00:52:31.140 align:middle line:84% One thing we do is we adopt this static Nash framework of just, 00:52:31.140 --> 00:52:32.890 align:middle line:84% OK, we're just going to assume static Nash 00:52:32.890 --> 00:52:34.600 align:middle line:90% described what goes on. 00:52:34.600 --> 00:52:36.430 align:middle line:84% In a lot of models, you can't just 00:52:36.430 --> 00:52:38.650 align:middle line:84% assume static Nash is what goes on 00:52:38.650 --> 00:52:43.700 align:middle line:84% because there is some variable that's changing over time. 00:52:43.700 --> 00:52:46.630 align:middle line:84% Like, firms are investing in their equality 00:52:46.630 --> 00:52:48.130 align:middle line:90% or in reducing costs. 00:52:48.130 --> 00:52:50.755 align:middle line:84% Or firms are setting prices, and it's 00:52:50.755 --> 00:52:53.380 align:middle line:84% costly to change their prices, so whatever price they set today 00:52:53.380 --> 00:52:55.270 align:middle line:90% are locked in for the future. 00:52:55.270 --> 00:52:58.968 align:middle line:90% We need some model where firms-- 00:52:58.968 --> 00:53:01.510 align:middle line:84% there needs to be some trade-off between today and the future 00:53:01.510 --> 00:53:04.090 align:middle line:84% to think about staggered price-setting 00:53:04.090 --> 00:53:08.200 align:middle line:84% or to think about cost, quality investments, or whatever. 00:53:08.200 --> 00:53:12.400 align:middle line:84% What people often do is to use this concept called 00:53:12.400 --> 00:53:15.100 align:middle line:90% Markov perfect equilibrium. 00:53:15.100 --> 00:53:18.460 align:middle line:84% In a Markov perfect equilibrium, players' strategies 00:53:18.460 --> 00:53:21.380 align:middle line:84% are assumed to depend only on payoff-relevant variables. 00:53:21.380 --> 00:53:25.850 align:middle line:84% So, for instance, you can price on-- 00:53:25.850 --> 00:53:32.660 align:middle line:84% my price at time t is a function of my cost and your costs. 00:53:32.660 --> 00:53:36.000 align:middle line:84% These things are relevant to the payoffs in the game. 00:53:36.000 --> 00:53:38.570 align:middle line:84% And therefore, if these things are changing the payoffs 00:53:38.570 --> 00:53:42.500 align:middle line:84% in the game, I'm allowed to price based on this. 00:53:42.500 --> 00:53:46.250 align:middle line:84% But I can't be doing I'm pricing today 00:53:46.250 --> 00:53:49.220 align:middle line:84% based on the price I set at t minus 1 00:53:49.220 --> 00:53:51.650 align:middle line:84% and the price you set at t minus 1 00:53:51.650 --> 00:53:55.020 align:middle line:84% because those prices are completely irrelevant today. 00:53:55.020 --> 00:53:57.440 align:middle line:84% So in Markov perfect equilibrium, what we want to do 00:53:57.440 --> 00:54:01.100 align:middle line:84% is assume that firms can set their strategies based 00:54:01.100 --> 00:54:03.890 align:middle line:84% on payoff-relevant variables but assume 00:54:03.890 --> 00:54:05.630 align:middle line:84% that they can't set their price based 00:54:05.630 --> 00:54:09.270 align:middle line:90% on payoff-irrelevant variables. 00:54:09.270 --> 00:54:10.750 align:middle line:90% So saying that can't happen. 00:54:10.750 --> 00:54:15.290 align:middle line:84% This is what rules out those grim-trigger punishments 00:54:15.290 --> 00:54:17.490 align:middle line:90% in the repeated Bertrand game. 00:54:17.490 --> 00:54:21.410 align:middle line:84% You can't say, if you said price Pm minus epsilon yesterday 00:54:21.410 --> 00:54:25.140 align:middle line:84% instead of Pm yesterday, then we go to a price war forever. 00:54:25.140 --> 00:54:27.630 align:middle line:84% I'm basically trying to recreate-- 00:54:27.630 --> 00:54:29.730 align:middle line:84% get as close as I can to static Nash by saying, 00:54:29.730 --> 00:54:32.460 align:middle line:84% you can only price based on things that are payoff relevant, 00:54:32.460 --> 00:54:36.770 align:middle line:84% not based on things that are payoff irrelevant. 00:54:36.770 --> 00:54:38.870 align:middle line:84% And people often think of this as a way 00:54:38.870 --> 00:54:45.330 align:middle line:84% to get back to obtaining unique equilibrium. 00:54:45.330 --> 00:54:46.830 align:middle line:84% We see an interesting observation 00:54:46.830 --> 00:54:49.410 align:middle line:84% is that people often think of Markov perfect equilibrium 00:54:49.410 --> 00:54:50.380 align:middle line:90% as doing that. 00:54:50.380 --> 00:54:53.520 align:middle line:84% But in practice, it doesn't always do that. 00:54:53.520 --> 00:54:54.630 align:middle line:90% And so there's this-- 00:54:54.630 --> 00:54:56.910 align:middle line:84% I talk about this interesting, beautiful paper 00:54:56.910 --> 00:54:59.760 align:middle line:84% by Maskin and Tirole that just give examples 00:54:59.760 --> 00:55:02.250 align:middle line:84% of what kind of Markov perfect equilibrium 00:55:02.250 --> 00:55:05.970 align:middle line:84% could we have in an alternating-move price game. 00:55:05.970 --> 00:55:10.365 align:middle line:84% So in Maskin and Tirole's game, it's just this-- 00:55:10.365 --> 00:55:12.870 align:middle line:90% 00:55:12.870 --> 00:55:14.770 align:middle line:90% it's this alternating-move game. 00:55:14.770 --> 00:55:16.290 align:middle line:90% We have two players. 00:55:16.290 --> 00:55:20.190 align:middle line:90% Let's say Player 1 and Player 2. 00:55:20.190 --> 00:55:25.830 align:middle line:84% And Player 1 picks some price at, say, 00:55:25.830 --> 00:55:29.220 align:middle line:84% t equals 1 and sets some price that's 00:55:29.220 --> 00:55:32.700 align:middle line:84% going to have to hold for two periods. 00:55:32.700 --> 00:55:38.280 align:middle line:84% And then Player 2 gets to pick a price in period t equals 2, 00:55:38.280 --> 00:55:41.010 align:middle line:84% and that has to be there for two periods. 00:55:41.010 --> 00:55:44.280 align:middle line:84% And then Player 1 chooses a new price at three 00:55:44.280 --> 00:55:47.810 align:middle line:90% that holds for 3 and 4. 00:55:47.810 --> 00:55:50.163 align:middle line:84% And Player 2 chooses a price that holds for 4 and 5. 00:55:50.163 --> 00:55:51.830 align:middle line:84% But the idea that we're trying to get at 00:55:51.830 --> 00:55:55.458 align:middle line:84% is, in reality, price changing is asynchronous. 00:55:55.458 --> 00:55:57.500 align:middle line:84% If you look at two gas stations across the street 00:55:57.500 --> 00:56:01.370 align:middle line:84% from each other, it's not like the guys climb on the signs 00:56:01.370 --> 00:56:04.760 align:middle line:84% and change the price listings at exactly the same time 00:56:04.760 --> 00:56:07.400 align:middle line:84% and change the pumps at exactly the same time. 00:56:07.400 --> 00:56:09.198 align:middle line:84% It's like somebody changes their price, 00:56:09.198 --> 00:56:11.240 align:middle line:84% and then the other guy sees the first one's price 00:56:11.240 --> 00:56:12.530 align:middle line:90% and changes his price. 00:56:12.530 --> 00:56:15.650 align:middle line:84% And they will tend to go back and forth like that. 00:56:15.650 --> 00:56:16.730 align:middle line:90% Each one observe it-- 00:56:16.730 --> 00:56:18.740 align:middle line:84% needing some time to observe one's price, 00:56:18.740 --> 00:56:21.420 align:middle line:84% do something [? rather, ?] and then change their price. 00:56:21.420 --> 00:56:23.540 align:middle line:84% So they consider this stylized game 00:56:23.540 --> 00:56:28.910 align:middle line:84% where Firm One chooses prices at t equals 1, t equals 3, 00:56:28.910 --> 00:56:30.140 align:middle line:90% t equals 5. 00:56:30.140 --> 00:56:33.140 align:middle line:84% And Firm Two chooses prices at t equals 2, t equals 4, 00:56:33.140 --> 00:56:34.030 align:middle line:90% t equals 6. 00:56:34.030 --> 00:56:36.890 align:middle line:90% 00:56:36.890 --> 00:56:40.050 align:middle line:84% And then, besides that, they do repeated Bertrand competition. 00:56:40.050 --> 00:56:42.300 align:middle line:84% So just repeated Bertrand competition. 00:56:42.300 --> 00:56:44.480 align:middle line:90% D of p is 1 minus p. 00:56:44.480 --> 00:56:47.210 align:middle line:84% And to make a simple example, they restrict prices 00:56:47.210 --> 00:56:49.230 align:middle line:90% to this simple grid-- 00:56:49.230 --> 00:56:53.450 align:middle line:90% 0, 1/6, 2/6, 3/6, 4/6, 5/6. 00:56:53.450 --> 00:56:56.630 align:middle line:84% One observation is that, in this model, 00:56:56.630 --> 00:57:00.650 align:middle line:84% there's no Markov equilibrium in which the firms price at cost. 00:57:00.650 --> 00:57:04.760 align:middle line:84% So the Bertrand equilibrium of just pricing at c goes away. 00:57:04.760 --> 00:57:06.240 align:middle line:90% And why is that? 00:57:06.240 --> 00:57:08.180 align:middle line:84% Well, because imagine you had an equilibrium 00:57:08.180 --> 00:57:11.960 align:middle line:84% where the firms were supposed to price at cost in all periods. 00:57:11.960 --> 00:57:14.750 align:middle line:84% What Firm One could do is just, in the first period, instead of 00:57:14.750 --> 00:57:18.650 align:middle line:84% pricing at c, I raise my price and I charge. 00:57:18.650 --> 00:57:20.510 align:middle line:90% So suppose c was 0. 00:57:20.510 --> 00:57:25.540 align:middle line:84% Instead of price 0, I raise my price to 4/6. 00:57:25.540 --> 00:57:27.630 align:middle line:90% I set a really high price. 00:57:27.630 --> 00:57:30.440 align:middle line:84% Then in period two, is Firm Two going 00:57:30.440 --> 00:57:32.320 align:middle line:90% to continue pricing at cost? 00:57:32.320 --> 00:57:36.990 align:middle line:84% Well, if it continues pricing at 0, it earns zero profits. 00:57:36.990 --> 00:57:38.410 align:middle line:84% You've given it this opportunity. 00:57:38.410 --> 00:57:40.577 align:middle line:84% It knows you're locked in at this really high price. 00:57:40.577 --> 00:57:44.820 align:middle line:84% It could raise its price to 3/6 and charge them at-- which is 00:57:44.820 --> 00:57:47.670 align:middle line:84% the monopoly price in this model and earn the monopoly price 00:57:47.670 --> 00:57:49.040 align:middle line:90% for two periods. 00:57:49.040 --> 00:57:53.670 align:middle line:84% If it does that, then, here, I can price at 2/6 and earn some 00:57:53.670 --> 00:57:56.610 align:middle line:84% money, and I'm earning 0 in equilibrium. 00:57:56.610 --> 00:57:59.340 align:middle line:84% So we can't have a zero-profit equilibrium 00:57:59.340 --> 00:58:02.078 align:middle line:84% here because firms could always raise their prices, 00:58:02.078 --> 00:58:03.870 align:middle line:84% tempt the other firm to raise their prices, 00:58:03.870 --> 00:58:06.840 align:middle line:84% then undercut that other firm and earn something positive. 00:58:06.840 --> 00:58:10.440 align:middle line:84% So zero-profit equilibria go away in this model. 00:58:10.440 --> 00:58:13.260 align:middle line:90% They don't exist. 00:58:13.260 --> 00:58:15.630 align:middle line:90% What does exist? 00:58:15.630 --> 00:58:20.070 align:middle line:84% One thing that still exists here is collusive agreement, 00:58:20.070 --> 00:58:21.630 align:middle line:84% collusive equilibrium, like the ones 00:58:21.630 --> 00:58:23.380 align:middle line:84% that Markov equilibrium is, in some sense, 00:58:23.380 --> 00:58:24.930 align:middle line:90% designed to get rid of. 00:58:24.930 --> 00:58:27.030 align:middle line:84% So consider this collusive equilibrium where, 00:58:27.030 --> 00:58:30.990 align:middle line:84% basically, we always, whatever the other firm 00:58:30.990 --> 00:58:34.140 align:middle line:84% is charging, if it's high, we charge the monopoly price 00:58:34.140 --> 00:58:35.730 align:middle line:90% of a half. 00:58:35.730 --> 00:58:40.560 align:middle line:84% And then if someone undercuts the other firm and goes to 2/6, 00:58:40.560 --> 00:58:43.390 align:middle line:84% then you undercut them and go to 1/6. 00:58:43.390 --> 00:58:46.080 align:middle line:84% And then if the other firm is charging 1/6, 00:58:46.080 --> 00:58:50.670 align:middle line:84% you either understick at 1/6 or you go back to the monopoly 00:58:50.670 --> 00:58:51.602 align:middle line:90% price. 00:58:51.602 --> 00:58:53.310 align:middle line:84% So this is like a-- you can think of this 00:58:53.310 --> 00:58:56.910 align:middle line:90% as a probabilistic price war. 00:58:56.910 --> 00:58:58.450 align:middle line:90% We're going along. 00:58:58.450 --> 00:59:00.435 align:middle line:84% We're supposed to be charging this-- 00:59:00.435 --> 00:59:04.550 align:middle line:84% be Pt-- we're supposed to be charging this price of one 00:59:04.550 --> 00:59:05.675 align:middle line:90% half in every period. 00:59:05.675 --> 00:59:08.640 align:middle line:90% 00:59:08.640 --> 00:59:13.770 align:middle line:84% If someone deviates and instead goes here, then the next firm, 00:59:13.770 --> 00:59:15.280 align:middle line:90% the firm goes there. 00:59:15.280 --> 00:59:18.420 align:middle line:84% And then they have this randomization 00:59:18.420 --> 00:59:19.480 align:middle line:90% between if they do that. 00:59:19.480 --> 00:59:22.620 align:middle line:84% And then somebody gives up and goes back to one half, 00:59:22.620 --> 00:59:24.070 align:middle line:90% and then we continue. 00:59:24.070 --> 00:59:27.450 align:middle line:84% So what they basically do in this Markov framework is, 00:59:27.450 --> 00:59:30.720 align:middle line:84% because the other firm's previous period price is payoff 00:59:30.720 --> 00:59:33.570 align:middle line:84% relevant, you have to include it in the Markov state space 00:59:33.570 --> 00:59:34.840 align:middle line:90% that you price on. 00:59:34.840 --> 00:59:38.260 align:middle line:84% And so we can get these price, probabilistic price where 00:59:38.260 --> 00:59:40.760 align:middle line:84% we're looking just like Green and Porter, then jump back up. 00:59:40.760 --> 00:59:43.780 align:middle line:90% 00:59:43.780 --> 00:59:47.608 align:middle line:84% But this is also a paper-- like a folk theorem paper, 00:59:47.608 --> 00:59:49.400 align:middle line:84% there's also equilibrium multiplicity here. 00:59:49.400 --> 00:59:51.510 align:middle line:84% And we get these other equilibria too. 00:59:51.510 --> 00:59:53.560 align:middle line:84% So one interesting one they point out 00:59:53.560 --> 00:59:55.760 align:middle line:84% are these things called Edgeworth cycles. 00:59:55.760 --> 00:59:59.530 align:middle line:84% Edgeworth cycles, we never settle down on pricing. 00:59:59.530 --> 01:00:04.240 align:middle line:84% And what the firms just do is forever undercut each other. 01:00:04.240 --> 01:00:07.120 align:middle line:90% So the firms just do this. 01:00:07.120 --> 01:00:13.800 align:middle line:84% They start out at a high price of 4/6, 01:00:13.800 --> 01:00:17.100 align:middle line:84% and then they just undercut each other. 01:00:17.100 --> 01:00:18.307 align:middle line:90% They get down to the bottom. 01:00:18.307 --> 01:00:19.140 align:middle line:90% There's some mixing. 01:00:19.140 --> 01:00:21.360 align:middle line:90% And then somebody wants to end. 01:00:21.360 --> 01:00:24.910 align:middle line:84% Again, this is like incentive to end-- 01:00:24.910 --> 01:00:26.940 align:middle line:84% you have this incentive to end the punishment, 01:00:26.940 --> 01:00:28.410 align:middle line:84% where it's like each one wants the other guy 01:00:28.410 --> 01:00:29.327 align:middle line:90% to end the punishment. 01:00:29.327 --> 01:00:32.550 align:middle line:84% But then someone jumps up to 5/6 and lets the other one undercut 01:00:32.550 --> 01:00:34.600 align:middle line:90% him in the next period to 4/6. 01:00:34.600 --> 01:00:36.515 align:middle line:84% And then we just continue like this. 01:00:36.515 --> 01:00:39.210 align:middle line:90% 01:00:39.210 --> 01:00:43.170 align:middle line:84% So we can get these Edgeworth-cycle equilibria 01:00:43.170 --> 01:00:45.630 align:middle line:84% where the firms just continually undercut each other 01:00:45.630 --> 01:00:47.290 align:middle line:90% over and over and over again. 01:00:47.290 --> 01:00:50.580 align:middle line:84% And then, at the bottom of the cycle, 01:00:50.580 --> 01:00:53.740 align:middle line:84% you've driven the price all the way to 0. 01:00:53.740 --> 01:00:54.490 align:middle line:90% What do you do? 01:00:54.490 --> 01:00:56.070 align:middle line:84% At some point, you give up and say, OK, we've 01:00:56.070 --> 01:00:57.153 align:middle line:90% been at 0 for long enough. 01:00:57.153 --> 01:00:59.940 align:middle line:84% I'm going to go ahead and price at 5/6, also earning nothing, 01:00:59.940 --> 01:01:02.130 align:middle line:84% but then let you undercut me to get to 4/6. 01:01:02.130 --> 01:01:04.830 align:middle line:84% And then I'll get to 3/6 two periods from now. 01:01:04.830 --> 01:01:09.540 align:middle line:84% So we also have these Edgeworth-cycle equilibria. 01:01:09.540 --> 01:01:11.790 align:middle line:84% And so I think the lesson here with this paper 01:01:11.790 --> 01:01:14.940 align:middle line:84% was that we may think of Markov equilibrium 01:01:14.940 --> 01:01:21.120 align:middle line:84% as eliminating multiplicity, but if-- 01:01:21.120 --> 01:01:24.930 align:middle line:84% here, it's because the previous period's price 01:01:24.930 --> 01:01:26.610 align:middle line:84% is payoff relevant, and that would 01:01:26.610 --> 01:01:29.070 align:middle line:84% happen in any staggered price-setting game. 01:01:29.070 --> 01:01:31.560 align:middle line:84% Or you could also do this based on other things that 01:01:31.560 --> 01:01:32.470 align:middle line:90% are payoff relevant. 01:01:32.470 --> 01:01:36.070 align:middle line:84% Like, if costs are there, we can also bootstrap on the costs. 01:01:36.070 --> 01:01:39.340 align:middle line:84% But so what we can get is multiplicity, 01:01:39.340 --> 01:01:41.870 align:middle line:84% and we can actually get multiple interesting patterns. 01:01:41.870 --> 01:01:44.210 align:middle line:84% We can get the collusive pattern we've seen before. 01:01:44.210 --> 01:01:47.110 align:middle line:84% You could also get this kind of pattern as an equilibrium. 01:01:47.110 --> 01:01:49.360 align:middle line:84% And the one thing you can't get in that model, 01:01:49.360 --> 01:01:51.890 align:middle line:84% interestingly enough, is competitive pricing. 01:01:51.890 --> 01:01:53.290 align:middle line:90% So competitive doesn't work. 01:01:53.290 --> 01:01:55.120 align:middle line:90% But this is possible. 01:01:55.120 --> 01:01:58.420 align:middle line:84% Sometimes, it's funny that perpetual cycling of prices 01:01:58.420 --> 01:02:02.740 align:middle line:84% makes more economic sense than just static Nash pricing 01:02:02.740 --> 01:02:03.420 align:middle line:90% in this model. 01:02:03.420 --> 01:02:07.750 align:middle line:90% 01:02:07.750 --> 01:02:10.340 align:middle line:90% Any questions on that? 01:02:10.340 --> 01:02:12.800 align:middle line:84% Obviously, this is a simple numerical example. 01:02:12.800 --> 01:02:16.603 align:middle line:84% But if things happen in this numerical example, 01:02:16.603 --> 01:02:18.020 align:middle line:84% similar things are going to happen 01:02:18.020 --> 01:02:19.700 align:middle line:84% in lots of more complicated models, 01:02:19.700 --> 01:02:23.390 align:middle line:84% that you can get these multiple types of Markov equilibria. 01:02:23.390 --> 01:02:26.738 align:middle line:90% 01:02:26.738 --> 01:02:29.030 align:middle line:84% So then the final thing I was going to talk about today 01:02:29.030 --> 01:02:34.100 align:middle line:84% is a more recent paper, Calvano, Calzolari, Denicolo, Pastorello. 01:02:34.100 --> 01:02:37.070 align:middle line:84% This is a 2020 AER paper motivated 01:02:37.070 --> 01:02:39.530 align:middle line:84% by thinking about online businesses. 01:02:39.530 --> 01:02:43.580 align:middle line:84% And they note that, in many online businesses, 01:02:43.580 --> 01:02:45.565 align:middle line:84% firms are setting prices repeatedly. 01:02:45.565 --> 01:02:46.940 align:middle line:84% And we really are in a case where 01:02:46.940 --> 01:02:51.140 align:middle line:90% delta is very, very close to 1. 01:02:51.140 --> 01:02:54.050 align:middle line:84% If you think about Hertz competing with Avis, competing 01:02:54.050 --> 01:02:58.590 align:middle line:84% with Budget on selling rental cars through Orbitz, 01:02:58.590 --> 01:03:01.270 align:middle line:84% they can change their price every second. 01:03:01.270 --> 01:03:03.840 align:middle line:84% They have a computer monitoring the other prices and changing 01:03:03.840 --> 01:03:04.870 align:middle line:90% prices. 01:03:04.870 --> 01:03:08.110 align:middle line:84% You think about competing sellers selling through Amazon. 01:03:08.110 --> 01:03:10.630 align:middle line:84% Many firms are trying to sell umbrellas through Amazon. 01:03:10.630 --> 01:03:11.850 align:middle line:84% They're competing with all the other firms trying 01:03:11.850 --> 01:03:14.700 align:middle line:84% to sell exactly the same umbrella or the same Nike shoes 01:03:14.700 --> 01:03:16.030 align:middle line:90% through Amazon. 01:03:16.030 --> 01:03:19.090 align:middle line:84% You can also think about Amazon competing with Target. 01:03:19.090 --> 01:03:21.690 align:middle line:84% There, the prices are not appearing side by side. 01:03:21.690 --> 01:03:24.420 align:middle line:84% But it's still Amazon choosing the price at which it's 01:03:24.420 --> 01:03:26.250 align:middle line:84% going to sell something, Target and Walmart 01:03:26.250 --> 01:03:28.458 align:middle line:84% choosing the price at which they're going to sell it, 01:03:28.458 --> 01:03:30.580 align:middle line:84% and the demand is a function of all those prices. 01:03:30.580 --> 01:03:33.480 align:middle line:84% And again, Amazon is not having a guy 01:03:33.480 --> 01:03:38.160 align:middle line:84% decide what's the price at which I'm going to sell this pen. 01:03:38.160 --> 01:03:41.965 align:middle line:84% Amazon has got a computer that's monitoring the internet 01:03:41.965 --> 01:03:44.340 align:middle line:84% and changing prices continuously and trying to figure out 01:03:44.340 --> 01:03:47.040 align:middle line:90% what a good price is. 01:03:47.040 --> 01:03:52.190 align:middle line:84% And so what the paper's asking is, what do we 01:03:52.190 --> 01:03:53.940 align:middle line:84% think is going to happen in that situation 01:03:53.940 --> 01:03:56.490 align:middle line:84% where you have computers setting prices, 01:03:56.490 --> 01:03:58.270 align:middle line:84% competing against other computers 01:03:58.270 --> 01:03:59.545 align:middle line:90% in a repeated-game setting? 01:03:59.545 --> 01:04:02.160 align:middle line:90% 01:04:02.160 --> 01:04:05.100 align:middle line:84% We know that there should be many, many equilibria. 01:04:05.100 --> 01:04:08.730 align:middle line:84% In some sense, the question is, which equilibrium 01:04:08.730 --> 01:04:12.150 align:middle line:84% will the computers find of the many repeated-game equilibria 01:04:12.150 --> 01:04:13.500 align:middle line:90% that exist? 01:04:13.500 --> 01:04:16.170 align:middle line:84% And they notice that we should-- when 01:04:16.170 --> 01:04:18.540 align:middle line:84% we think about what the computers are going to be doing, 01:04:18.540 --> 01:04:21.570 align:middle line:84% the computers are going to be trying 01:04:21.570 --> 01:04:23.340 align:middle line:90% to learn what's happening. 01:04:23.340 --> 01:04:25.800 align:middle line:84% The computers are not born knowing, 01:04:25.800 --> 01:04:30.460 align:middle line:84% what's my demand when my price is 39 and your price is 38? 01:04:30.460 --> 01:04:32.530 align:middle line:84% The computers have to figure that out. 01:04:32.530 --> 01:04:35.610 align:middle line:84% And the computers also have to figure out what's your strategy. 01:04:35.610 --> 01:04:39.360 align:middle line:84% If you are at 39 and I undercut to 38, do you undercut to 37? 01:04:39.360 --> 01:04:41.430 align:middle line:90% Or do you stick with 38? 01:04:41.430 --> 01:04:43.350 align:middle line:84% The computers, in some sense, have 01:04:43.350 --> 01:04:46.480 align:middle line:84% to figure out what should we be doing to profit maximize. 01:04:46.480 --> 01:04:48.450 align:middle line:84% And so they try to think about what 01:04:48.450 --> 01:04:51.810 align:middle line:84% would happen when you have agents who are trying to learn 01:04:51.810 --> 01:04:53.170 align:middle line:90% competing with each other. 01:04:53.170 --> 01:04:55.750 align:middle line:84% And in some sense, this is resurrecting-- 01:04:55.750 --> 01:04:59.280 align:middle line:84% there was a large theory literature on this in the 1990s. 01:04:59.280 --> 01:05:03.130 align:middle line:84% Drew has this very nice book with David Levine from 1998 01:05:03.130 --> 01:05:04.990 align:middle line:84% discussing lots of theoretical results 01:05:04.990 --> 01:05:07.460 align:middle line:84% about what happens if people are naively trying to learn. 01:05:07.460 --> 01:05:09.350 align:middle line:84% And at the time, the theorists were 01:05:09.350 --> 01:05:11.350 align:middle line:84% arguing that this is a very interesting question 01:05:11.350 --> 01:05:14.170 align:middle line:84% because if people are naively trying to learn, 01:05:14.170 --> 01:05:16.270 align:middle line:84% do they learn an equilibrium or not? 01:05:16.270 --> 01:05:17.710 align:middle line:84% To believe in equilibrium, we need 01:05:17.710 --> 01:05:19.850 align:middle line:84% to think that that process converges. 01:05:19.850 --> 01:05:22.060 align:middle line:84% And if that process does converge, 01:05:22.060 --> 01:05:23.620 align:middle line:84% then the question is, well, if we 01:05:23.620 --> 01:05:25.217 align:middle line:84% have a game with multiple equilibria, 01:05:25.217 --> 01:05:27.550 align:middle line:84% are there some equilibria that are more likely to emerge 01:05:27.550 --> 01:05:29.500 align:middle line:90% from that learning than others? 01:05:29.500 --> 01:05:32.440 align:middle line:84% So, in some sense, that's what Calvano et al. 01:05:32.440 --> 01:05:34.790 align:middle line:90% are trying to do here. 01:05:34.790 --> 01:05:39.770 align:middle line:84% So they bring in a more complicated learning model than 01:05:39.770 --> 01:05:42.860 align:middle line:84% existed in the 1990s theory literature. 01:05:42.860 --> 01:05:45.890 align:middle line:84% But it's a learning literature, thinking 01:05:45.890 --> 01:05:48.920 align:middle line:84% of how operations, research, or computer scientists 01:05:48.920 --> 01:05:50.720 align:middle line:84% would say you should design a pricing 01:05:50.720 --> 01:05:52.490 align:middle line:90% algorithm for your website. 01:05:52.490 --> 01:05:57.920 align:middle line:84% So we have a situation where profits pi depend on s-- 01:05:57.920 --> 01:06:00.170 align:middle line:84% a state of the world that's observable to the firms 01:06:00.170 --> 01:06:01.520 align:middle line:90% when they're setting prices-- 01:06:01.520 --> 01:06:03.560 align:middle line:84% and a-- an action that they take which you 01:06:03.560 --> 01:06:06.470 align:middle line:90% can think of as their price. 01:06:06.470 --> 01:06:10.160 align:middle line:84% And the profits depend on this observable state, this action, 01:06:10.160 --> 01:06:12.480 align:middle line:84% but it also depends on a random shock. 01:06:12.480 --> 01:06:15.770 align:middle line:84% So I set my rental car price at $39. 01:06:15.770 --> 01:06:18.430 align:middle line:90% Someone else sets it at $38. 01:06:18.430 --> 01:06:21.560 align:middle line:84% And then look at demand over the next day, 01:06:21.560 --> 01:06:24.400 align:middle line:84% there's some random number of people who reserve a car at LAX 01:06:24.400 --> 01:06:27.640 align:middle line:90% on that date given those prices. 01:06:27.640 --> 01:06:30.190 align:middle line:84% And then there's some state transition process, 01:06:30.190 --> 01:06:33.100 align:middle line:84% probability that the state goes to s prime given s and a. 01:06:33.100 --> 01:06:34.758 align:middle line:90% That's also unknown. 01:06:34.758 --> 01:06:36.550 align:middle line:84% So again, the staggered price-setting game, 01:06:36.550 --> 01:06:38.508 align:middle line:84% you can think of the s, the state of the world, 01:06:38.508 --> 01:06:41.320 align:middle line:84% as everybody's price on that rental car website. 01:06:41.320 --> 01:06:46.017 align:middle line:84% The current prices today out of LAX are $41, $40, $40. 01:06:46.017 --> 01:06:46.850 align:middle line:90% Something like that. 01:06:46.850 --> 01:06:47.990 align:middle line:90% That's today's state. 01:06:47.990 --> 01:06:50.330 align:middle line:84% I observe the prices are $41, $40, $40, where 01:06:50.330 --> 01:06:51.580 align:middle line:90% I'm the third of those prices. 01:06:51.580 --> 01:06:54.260 align:middle line:90% I choose what price do I charge. 01:06:54.260 --> 01:06:57.920 align:middle line:84% And then I'm going to get some profit over the course 01:06:57.920 --> 01:06:58.740 align:middle line:90% of the next day. 01:06:58.740 --> 01:06:59.990 align:middle line:84% And then, the next day, when I go back 01:06:59.990 --> 01:07:01.820 align:middle line:84% to check the website again, all the prices 01:07:01.820 --> 01:07:02.903 align:middle line:90% are going to be different. 01:07:02.903 --> 01:07:05.570 align:middle line:90% It may be $35, $38, $39. 01:07:05.570 --> 01:07:08.420 align:middle line:90% And then I'm going to try-- 01:07:08.420 --> 01:07:11.275 align:middle line:84% sometimes, I don't know what that state transition 01:07:11.275 --> 01:07:13.650 align:middle line:84% process is because I don't know what my rivals are doing. 01:07:13.650 --> 01:07:15.775 align:middle line:84% And I also don't know what my profits are because I 01:07:15.775 --> 01:07:18.470 align:middle line:90% don't know the demand curve. 01:07:18.470 --> 01:07:19.910 align:middle line:90% So what would you do? 01:07:19.910 --> 01:07:23.450 align:middle line:84% Well, dynamic programming suggests that a good thing to do 01:07:23.450 --> 01:07:26.900 align:middle line:84% is to try to learn this function, Q of s, a. 01:07:26.900 --> 01:07:29.660 align:middle line:84% Often, we talk about this value function, the value 01:07:29.660 --> 01:07:32.450 align:middle line:90% that you get being in state. 01:07:32.450 --> 01:07:33.740 align:middle line:90% I'm going to find v of s, a. 01:07:33.740 --> 01:07:38.720 align:middle line:84% It's going to be the max over a, Q of s, a. 01:07:38.720 --> 01:07:40.350 align:middle line:90% Sorry, that's behind the board. 01:07:40.350 --> 01:07:43.100 align:middle line:84% So Q of s, comma, a is the profit 01:07:43.100 --> 01:07:47.450 align:middle line:84% you get if you're in state s and you play a today 01:07:47.450 --> 01:07:50.540 align:middle line:84% and then you play optimally in all future periods. 01:07:50.540 --> 01:07:55.550 align:middle line:84% So Q of s, a, the recursive definition of it is Q of s, 01:07:55.550 --> 01:07:58.400 align:middle line:84% a is the expected profit you get from taking action a 01:07:58.400 --> 01:08:03.950 align:middle line:84% when you observe state s plus delta times the expectation 01:08:03.950 --> 01:08:07.280 align:middle line:84% over what tomorrow's state is going to be given that state was 01:08:07.280 --> 01:08:11.270 align:middle line:84% s and you took a of the maximum over all actions 01:08:11.270 --> 01:08:14.690 align:middle line:84% you're going to take next period, Q of s prime, a prime. 01:08:14.690 --> 01:08:17.094 align:middle line:90% So I'm going to take action a. 01:08:17.094 --> 01:08:20.250 align:middle line:84% And then, next period, I'm going to see some new state s prime. 01:08:20.250 --> 01:08:22.740 align:middle line:84% And therefore, I'm going to choose some new action a prime. 01:08:22.740 --> 01:08:25.950 align:middle line:84% And then I'm going to get this discounted 01:08:25.950 --> 01:08:30.750 align:middle line:84% present value of profits, Q of s prime, a prime. 01:08:30.750 --> 01:08:34.080 align:middle line:84% So if I knew this object, Q of s, a, 01:08:34.080 --> 01:08:35.640 align:middle line:84% then I can behave optimally, right? 01:08:35.640 --> 01:08:37.200 align:middle line:90% I just do the argmax over all-- 01:08:37.200 --> 01:08:38.080 align:middle line:90% I observe a state. 01:08:38.080 --> 01:08:41.880 align:middle line:84% I just choose the a that maximizes Q of s, a. 01:08:41.880 --> 01:08:45.390 align:middle line:84% So if you think about it, Q is just a matrix of numbers 01:08:45.390 --> 01:08:46.294 align:middle line:90% that I need to know. 01:08:46.294 --> 01:08:51.939 align:middle line:90% 01:08:51.939 --> 01:08:53.800 align:middle line:90% I have all the possible states-- 01:08:53.800 --> 01:08:57.467 align:middle line:84% s, s prime, s double prime, s triple prime. 01:08:57.467 --> 01:08:59.050 align:middle line:84% And there are all the possible actions 01:08:59.050 --> 01:09:02.689 align:middle line:84% I can take-- a, a prime, a double prime, a triple prime. 01:09:02.689 --> 01:09:06.310 align:middle line:84% I just need to know what's my discounted present value in each 01:09:06.310 --> 01:09:08.260 align:middle line:90% of these-- 01:09:08.260 --> 01:09:09.687 align:middle line:90% in all of these things. 01:09:09.687 --> 01:09:11.979 align:middle line:84% If I know the numbers in this matrix, I just-- whatever 01:09:11.979 --> 01:09:14.470 align:middle line:84% state I'm in, I just maximize the numbers in that column, 01:09:14.470 --> 01:09:17.120 align:middle line:90% and that's how I behave. 01:09:17.120 --> 01:09:20.270 align:middle line:84% So here's a standard learning algorithm 01:09:20.270 --> 01:09:21.770 align:middle line:84% for how you might program a computer 01:09:21.770 --> 01:09:23.979 align:middle line:90% to behave in this situation. 01:09:23.979 --> 01:09:27.380 align:middle line:84% You conjecture some initial matrix Q. 01:09:27.380 --> 01:09:31.130 align:middle line:84% So maybe you have some initial conjecture demand curve, 01:09:31.130 --> 01:09:34.040 align:middle line:84% and maybe you assume just some flat conjecture 01:09:34.040 --> 01:09:37.220 align:middle line:84% about how your rivals are going to price and figure out, 01:09:37.220 --> 01:09:40.979 align:middle line:84% what profit do I think I'm going to get if there's no dynamics? 01:09:40.979 --> 01:09:42.930 align:middle line:84% I just guess what the demand curve is. 01:09:42.930 --> 01:09:45.680 align:middle line:90% I just initialize Q to some-- 01:09:45.680 --> 01:09:47.470 align:middle line:90% I initialize Q to something. 01:09:47.470 --> 01:09:50.069 align:middle line:84% So I assume that this is the profit-maximizing thing. 01:09:50.069 --> 01:09:53.250 align:middle line:90% So my profit is 2, 2, 2, 2, 2. 01:09:53.250 --> 01:09:57.426 align:middle line:90% 1.9, 1.9, 1.9, 1.9. 01:09:57.426 --> 01:09:58.890 align:middle line:90% 1.8. 01:09:58.890 --> 01:10:01.130 align:middle line:84% I just initialize this matrix to something, 01:10:01.130 --> 01:10:02.870 align:middle line:84% guessing that I think this is the best 01:10:02.870 --> 01:10:05.220 align:middle line:84% and I think my profits stayed independent. 01:10:05.220 --> 01:10:08.530 align:middle line:90% 01:10:08.530 --> 01:10:12.160 align:middle line:84% Then, at every point t, I figure out, OK, 01:10:12.160 --> 01:10:16.880 align:middle line:84% if this is the state at time 0, that's my optimal action. 01:10:16.880 --> 01:10:20.510 align:middle line:90% So I take it. 01:10:20.510 --> 01:10:22.510 align:middle line:84% So I choose the profit-maximizing action 01:10:22.510 --> 01:10:24.260 align:middle line:84% given my current knowledge of the Q matrix 01:10:24.260 --> 01:10:26.650 align:middle line:84% and my current observation of the state. 01:10:26.650 --> 01:10:28.670 align:middle line:84% Well, I choose that with large probability. 01:10:28.670 --> 01:10:32.480 align:middle line:84% But with epsilon probability, I experiment because I don't know. 01:10:32.480 --> 01:10:34.520 align:middle line:84% It may be that this price is not good. 01:10:34.520 --> 01:10:35.827 align:middle line:90% So I usually choose this. 01:10:35.827 --> 01:10:37.910 align:middle line:84% But with some probability, I experiment and choose 01:10:37.910 --> 01:10:40.400 align:middle line:90% a different action instead. 01:10:40.400 --> 01:10:43.580 align:middle line:84% And then when I choose this action, 01:10:43.580 --> 01:10:46.940 align:middle line:84% I update the number in this cell by saying 01:10:46.940 --> 01:10:49.520 align:middle line:84% I'm going to take a weighted average of what 01:10:49.520 --> 01:10:51.000 align:middle line:90% I thought the number was-- 01:10:51.000 --> 01:10:52.410 align:middle line:90% so I thought the number was 2. 01:10:52.410 --> 01:10:54.500 align:middle line:84% So I take a weighted average of 2. 01:10:54.500 --> 01:10:57.710 align:middle line:90% And what I now-- 01:10:57.710 --> 01:10:59.450 align:middle line:84% what my current period experience 01:10:59.450 --> 01:11:04.430 align:middle line:84% is, which is I got profit pi t, and then I got profit pi t 01:11:04.430 --> 01:11:06.873 align:middle line:84% and then I end up in this state, and I think, 01:11:06.873 --> 01:11:08.790 align:middle line:84% what's the best I'm going to do in this state? 01:11:08.790 --> 01:11:09.890 align:middle line:90% And so I do-- 01:11:09.890 --> 01:11:12.620 align:middle line:84% I know that I got profit pi t, and I 01:11:12.620 --> 01:11:15.300 align:middle line:84% know that I ended up in state s t plus 1. 01:11:15.300 --> 01:11:19.980 align:middle line:84% So I do this plus what I expect from optimal play in that state, 01:11:19.980 --> 01:11:22.170 align:middle line:84% still using my Q matrix that I started with. 01:11:22.170 --> 01:11:24.350 align:middle line:84% So here, actually, I think I'm still going to get 2, 01:11:24.350 --> 01:11:25.767 align:middle line:84% even though the state has changed. 01:11:25.767 --> 01:11:28.900 align:middle line:90% 01:11:28.900 --> 01:11:29.400 align:middle line:90% Anyway. 01:11:29.400 --> 01:11:34.140 align:middle line:84% So theorem from people who study these kinds of learning 01:11:34.140 --> 01:11:36.240 align:middle line:84% algorithms is that, if I'm playing 01:11:36.240 --> 01:11:37.830 align:middle line:84% against a stationary environment, 01:11:37.830 --> 01:11:40.050 align:middle line:84% there really is just some exogenous process 01:11:40.050 --> 01:11:43.810 align:middle line:84% by which states transition, which can depend on my action. 01:11:43.810 --> 01:11:45.310 align:middle line:84% And it depends on the current state. 01:11:45.310 --> 01:11:48.900 align:middle line:84% But if there's some Markov state transitions like this 01:11:48.900 --> 01:11:50.790 align:middle line:84% and there's some profit function-- 01:11:50.790 --> 01:11:52.330 align:middle line:90% I don't even know what it is-- 01:11:52.330 --> 01:11:54.660 align:middle line:84% this Q-learning algorithm finds the best thing 01:11:54.660 --> 01:11:57.370 align:middle line:90% to do in the long run. 01:11:57.370 --> 01:12:00.760 align:middle line:84% Again, you need some of ability of full support of the Markov 01:12:00.760 --> 01:12:02.085 align:middle line:90% process so you don't get stuck. 01:12:02.085 --> 01:12:03.710 align:middle line:84% Like, there was some great thing to do, 01:12:03.710 --> 01:12:06.293 align:middle line:84% but once you leave that, you can never get back to that state. 01:12:06.293 --> 01:12:10.810 align:middle line:84% But if there's-- if the Markov process is going to cover all 01:12:10.810 --> 01:12:13.390 align:middle line:84% states, this thing will eventually find it. 01:12:13.390 --> 01:12:16.210 align:middle line:90% Because you're just updating-- 01:12:16.210 --> 01:12:18.910 align:middle line:84% every period, you're just updating one element in this 01:12:18.910 --> 01:12:20.800 align:middle line:84% to reflect the profit you get in this state 01:12:20.800 --> 01:12:22.220 align:middle line:84% and then the profit you get in the next state. 01:12:22.220 --> 01:12:23.050 align:middle line:84% And then if you're in this state, 01:12:23.050 --> 01:12:26.110 align:middle line:84% you update the profit here to reflect where you end up there. 01:12:26.110 --> 01:12:27.970 align:middle line:84% You just keep updating the profits, 01:12:27.970 --> 01:12:30.640 align:middle line:84% taking in a weighted average of what you thought they were 01:12:30.640 --> 01:12:33.950 align:middle line:90% and the current thing. 01:12:33.950 --> 01:12:36.940 align:middle line:84% So at some point, the current, the actual experience 01:12:36.940 --> 01:12:39.800 align:middle line:84% starts to overwhelm the past experience. 01:12:39.800 --> 01:12:41.086 align:middle line:90% This thing converges. 01:12:41.086 --> 01:12:44.790 align:middle line:90% 01:12:44.790 --> 01:12:47.432 align:middle line:90% And so Calvano et al. 01:12:47.432 --> 01:12:49.890 align:middle line:84% conjectured that this would be a reasonable thing for firms 01:12:49.890 --> 01:12:52.980 align:middle line:84% to do if they're pricing against another firm on a travel 01:12:52.980 --> 01:12:53.790 align:middle line:90% website. 01:12:53.790 --> 01:12:56.430 align:middle line:84% If I were to assign you-- like, here, I 01:12:56.430 --> 01:13:01.097 align:middle line:84% want you to design a pricing algorithm for Hertz to use, 01:13:01.097 --> 01:13:02.680 align:middle line:84% that you might have come up with this. 01:13:02.680 --> 01:13:06.000 align:middle line:90% 01:13:06.000 --> 01:13:08.160 align:middle line:84% OK, so what they do is then they study 01:13:08.160 --> 01:13:10.750 align:middle line:84% what would happen here in a repeated competition game. 01:13:10.750 --> 01:13:13.610 align:middle line:84% So they go to things that look very much like we're used to. 01:13:13.610 --> 01:13:17.370 align:middle line:84% So they go to a logit price competition game 01:13:17.370 --> 01:13:21.810 align:middle line:84% where consumers have these v minus p plus epsilon 01:13:21.810 --> 01:13:22.560 align:middle line:90% preferences. 01:13:22.560 --> 01:13:26.700 align:middle line:84% And the epsilons are logit distributed errors. 01:13:26.700 --> 01:13:30.680 align:middle line:84% Their base model has two firms in it. 01:13:30.680 --> 01:13:33.920 align:middle line:84% The two firms have identical costs. 01:13:33.920 --> 01:13:37.080 align:middle line:84% They have a grid that has 15 possible prices. 01:13:37.080 --> 01:13:42.930 align:middle line:84% They have a discount factor of 0.95. 01:13:42.930 --> 01:13:45.460 align:middle line:84% They don't know what parameters firms would use. 01:13:45.460 --> 01:13:48.820 align:middle line:84% Alpha, if you remember, was the weight on current profits, 01:13:48.820 --> 01:13:51.640 align:middle line:84% how fast you update the numbers in the Q matrix 01:13:51.640 --> 01:13:53.770 align:middle line:84% when you get profits in each period. 01:13:53.770 --> 01:13:56.953 align:middle line:84% And again, here, profits are random because the customer-- 01:13:56.953 --> 01:13:58.620 align:middle line:84% because the epsilons might buy from you, 01:13:58.620 --> 01:14:00.690 align:middle line:90% might not buy from you. 01:14:00.690 --> 01:14:04.470 align:middle line:84% And again, you don't know the logit parameter, 01:14:04.470 --> 01:14:07.170 align:middle line:84% but you just know that higher price less likely, 01:14:07.170 --> 01:14:08.730 align:middle line:90% but I don't know the elasticity. 01:14:08.730 --> 01:14:10.470 align:middle line:84% And then the second parameter beta, 01:14:10.470 --> 01:14:14.970 align:middle line:84% which governs how fast the experimentation rate goes to 0. 01:14:14.970 --> 01:14:17.740 align:middle line:84% As I said, you're going to want the-- early in the game, 01:14:17.740 --> 01:14:19.860 align:middle line:84% you're going to want to experiment a lot. 01:14:19.860 --> 01:14:21.820 align:middle line:84% After you've been playing this thing for a million periods, 01:14:21.820 --> 01:14:23.445 align:middle line:84% you want to stop experimenting and just 01:14:23.445 --> 01:14:24.730 align:middle line:90% start picking the best thing. 01:14:24.730 --> 01:14:27.090 align:middle line:84% And so these two parameters, alpha 01:14:27.090 --> 01:14:29.310 align:middle line:84% is how fast you adjust your Q matrix, 01:14:29.310 --> 01:14:32.220 align:middle line:84% and beta is how quickly you abandon-- 01:14:32.220 --> 01:14:33.930 align:middle line:84% how long you abandon-- how slowly 01:14:33.930 --> 01:14:37.510 align:middle line:90% you abandon experimenting. 01:14:37.510 --> 01:14:38.050 align:middle line:90% Anyway. 01:14:38.050 --> 01:14:40.960 align:middle line:84% They declare learning to have converged when optimal play is 01:14:40.960 --> 01:14:44.440 align:middle line:84% unchanged in their model for 100,000 periods in a row. 01:14:44.440 --> 01:14:46.060 align:middle line:84% Obviously, they had a big computer. 01:14:46.060 --> 01:14:49.150 align:middle line:84% And they also, actually, consider 10,000 different 01:14:49.150 --> 01:14:51.130 align:middle line:84% parameter combinations, alpha and beta, 01:14:51.130 --> 01:14:54.730 align:middle line:84% to be able to talk about what happens here based on what alpha 01:14:54.730 --> 01:14:57.820 align:middle line:90% and beta do firms use. 01:14:57.820 --> 01:15:00.430 align:middle line:84% Main results they have are that, really, it turns out 01:15:00.430 --> 01:15:02.500 align:middle line:84% that, for a wide range of alpha and beta, 01:15:02.500 --> 01:15:04.280 align:middle line:84% pretty much the same things happen. 01:15:04.280 --> 01:15:05.590 align:middle line:90% What happens? 01:15:05.590 --> 01:15:09.820 align:middle line:84% First, profits are usually 70% to 90% of the way 01:15:09.820 --> 01:15:12.720 align:middle line:90% from static Nash to monopoly. 01:15:12.720 --> 01:15:14.770 align:middle line:84% So what we find the computers learning 01:15:14.770 --> 01:15:17.310 align:middle line:90% is to collude with each other. 01:15:17.310 --> 01:15:19.620 align:middle line:84% And so they end up with these collusive pricing 01:15:19.620 --> 01:15:24.605 align:middle line:84% that's pretty far towards pure monopoly from static Nash. 01:15:24.605 --> 01:15:27.780 align:middle line:90% 01:15:27.780 --> 01:15:29.890 align:middle line:90% Where do the computers end up? 01:15:29.890 --> 01:15:31.560 align:middle line:84% The computers end up playing things that 01:15:31.560 --> 01:15:33.280 align:middle line:90% are epsilon Nash equilibrium. 01:15:33.280 --> 01:15:36.150 align:middle line:84% In some sense, the computers invent punishments 01:15:36.150 --> 01:15:37.980 align:middle line:90% with finite price wars. 01:15:37.980 --> 01:15:40.890 align:middle line:84% So the computers develop these beliefs 01:15:40.890 --> 01:15:42.400 align:middle line:90% about what's going to happen. 01:15:42.400 --> 01:15:45.880 align:middle line:84% That means that we all charge this monopoly price. 01:15:45.880 --> 01:15:49.320 align:middle line:84% And then if-- so let's suppose this is the state where 01:15:49.320 --> 01:15:51.540 align:middle line:90% we're both charging price two. 01:15:51.540 --> 01:15:53.400 align:middle line:84% And then what the computers think 01:15:53.400 --> 01:15:58.155 align:middle line:84% is that, if they deviate to 1.92 and we're in this state 01:15:58.155 --> 01:16:05.850 align:middle line:84% s prime, then in the state 1.92, profits are lower. 01:16:05.850 --> 01:16:09.840 align:middle line:84% And profits are lower in a way that's going to make the state-- 01:16:09.840 --> 01:16:11.410 align:middle line:90% the other firm undercut. 01:16:11.410 --> 01:16:13.890 align:middle line:84% And we're going to transition and follow 01:16:13.890 --> 01:16:15.450 align:middle line:90% this arrow in the matrix. 01:16:15.450 --> 01:16:19.470 align:middle line:84% In some sense, the computers learn that, from here, 01:16:19.470 --> 01:16:23.760 align:middle line:84% profits are going to be such that I want to undercut 01:16:23.760 --> 01:16:25.552 align:middle line:90% and charge an even lower price. 01:16:25.552 --> 01:16:27.010 align:middle line:84% And then, the next time, he's going 01:16:27.010 --> 01:16:29.090 align:middle line:84% to want to undercut and charge an even lower price. 01:16:29.090 --> 01:16:30.465 align:middle line:84% And the computers, in some sense, 01:16:30.465 --> 01:16:32.680 align:middle line:84% learn that the payoffs are such that you 01:16:32.680 --> 01:16:35.290 align:middle line:84% want to do something not so different from one 01:16:35.290 --> 01:16:38.835 align:middle line:84% of these mass [? interal ?] cycles off the equilibrium path. 01:16:38.835 --> 01:16:42.350 align:middle line:90% 01:16:42.350 --> 01:16:47.240 align:middle line:84% So even when you set delta approximately equal to 0, 01:16:47.240 --> 01:16:50.420 align:middle line:84% the computers still learn that they shouldn't-- 01:16:50.420 --> 01:16:51.723 align:middle line:90% delta is positive but nonzero. 01:16:51.723 --> 01:16:53.640 align:middle line:84% The computers still learn they should collude. 01:16:53.640 --> 01:16:55.490 align:middle line:84% They don't collude much when delta is small. 01:16:55.490 --> 01:16:58.980 align:middle line:84% They collude with 20% of the weight of the monopoly price. 01:16:58.980 --> 01:17:00.710 align:middle line:84% But it seems that, whatever you pick, 01:17:00.710 --> 01:17:02.930 align:middle line:84% the computers are colluding that way. 01:17:02.930 --> 01:17:05.750 align:middle line:84% And then, as you increase the number of firms, 01:17:05.750 --> 01:17:08.550 align:middle line:84% it does decrease the collusive level of prices. 01:17:08.550 --> 01:17:11.990 align:middle line:84% So if we want to ask what causes prices to go down 01:17:11.990 --> 01:17:16.610 align:middle line:84% as delta increases, like I said, the pure theory version 01:17:16.610 --> 01:17:19.310 align:middle line:84% has a very hard time with that because 1 minus 1 over a quarter 01:17:19.310 --> 01:17:20.580 align:middle line:90% is not a big number. 01:17:20.580 --> 01:17:22.078 align:middle line:84% But this says that prices do seem 01:17:22.078 --> 01:17:24.620 align:middle line:84% to decline the number of firms, and they learn less collusive 01:17:24.620 --> 01:17:26.810 align:middle line:84% equilibria when the number of firms is larger. 01:17:26.810 --> 01:17:29.040 align:middle line:90% But still, high with four firms. 01:17:29.040 --> 01:17:32.600 align:middle line:84% And let me say, it's hard to write papers 01:17:32.600 --> 01:17:35.150 align:middle line:84% like this because you're used to-- in a theory paper, 01:17:35.150 --> 01:17:36.680 align:middle line:90% people want to see theorems. 01:17:36.680 --> 01:17:38.780 align:middle line:84% And you can read this as an example 01:17:38.780 --> 01:17:42.270 align:middle line:84% for, how do I present results from a paper that 01:17:42.270 --> 01:17:44.683 align:middle line:84% has lots of simulations instead of theories? 01:17:44.683 --> 01:17:46.850 align:middle line:84% And they have a lot of nice, pretty graphs in there, 01:17:46.850 --> 01:17:47.850 align:middle line:90% and I think that helps. 01:17:47.850 --> 01:17:52.080 align:middle line:84% So this is their profit gain as a function of the two 01:17:52.080 --> 01:17:55.530 align:middle line:84% parameters, alpha and beta, where this color here 01:17:55.530 --> 01:17:57.280 align:middle line:90% is 90% of the way there. 01:17:57.280 --> 01:17:59.973 align:middle line:84% This color here is 60% or 70% of the way there. 01:17:59.973 --> 01:18:01.890 align:middle line:84% But they're showing you that, for many things, 01:18:01.890 --> 01:18:05.440 align:middle line:84% it looks like 70% to 90% of the way there. 01:18:05.440 --> 01:18:09.100 align:middle line:84% Here's their graph of profits versus the discount factor. 01:18:09.100 --> 01:18:13.020 align:middle line:84% So when discount factors are anything below 0.4 or 0.5, 01:18:13.020 --> 01:18:15.930 align:middle line:84% we're roughly getting 20% of the way to the monopoly profits, 01:18:15.930 --> 01:18:17.130 align:middle line:90% on average. 01:18:17.130 --> 01:18:20.460 align:middle line:84% But then, as people become more patient, the computers-- 01:18:20.460 --> 01:18:22.825 align:middle line:84% the more patient computers learn more and more 01:18:22.825 --> 01:18:23.950 align:middle line:90% to collude with each other. 01:18:23.950 --> 01:18:25.800 align:middle line:84% And when the computers are very patient, 01:18:25.800 --> 01:18:27.840 align:middle line:84% they're learning 90-something percent of the way 01:18:27.840 --> 01:18:28.880 align:middle line:90% to the monopoly price. 01:18:28.880 --> 01:18:32.220 align:middle line:90% 01:18:32.220 --> 01:18:34.930 align:middle line:84% They have illustrations of what the price wars look like. 01:18:34.930 --> 01:18:36.210 align:middle line:90% So this was just an example. 01:18:36.210 --> 01:18:38.100 align:middle line:90% Take a final strategy. 01:18:38.100 --> 01:18:39.390 align:middle line:90% Someone undercuts. 01:18:39.390 --> 01:18:41.220 align:middle line:84% The other firm, then, does undercut them, 01:18:41.220 --> 01:18:43.870 align:middle line:84% and it takes them a while to get back to the monopoly price. 01:18:43.870 --> 01:18:47.310 align:middle line:84% So the computers have learned to follow some pattern like this, 01:18:47.310 --> 01:18:50.460 align:middle line:90% that has finite price wars. 01:18:50.460 --> 01:18:53.795 align:middle line:84% And this is their illustration of what one of the equilibrium 01:18:53.795 --> 01:18:55.170 align:middle line:84% strategies looks like, where this 01:18:55.170 --> 01:18:58.120 align:middle line:84% is the absorbing state where we charge the monopoly price. 01:18:58.120 --> 01:19:01.500 align:middle line:84% And if you deviate, you follow one of these paths, like this, 01:19:01.500 --> 01:19:04.640 align:middle line:84% and then eventually come back, six or eight or whatever periods 01:19:04.640 --> 01:19:05.140 align:middle line:90% later. 01:19:05.140 --> 01:19:06.515 align:middle line:84% But the computers, in some sense, 01:19:06.515 --> 01:19:08.910 align:middle line:84% created this network of ways that you traverse 01:19:08.910 --> 01:19:13.530 align:middle line:84% this thing that eventually gets you back to the collusive price. 01:19:13.530 --> 01:19:15.660 align:middle line:84% And just, yeah, a cute illustration of what 01:19:15.660 --> 01:19:16.810 align:middle line:90% happens in these things. 01:19:16.810 --> 01:19:19.647 align:middle line:84% You do see a lot of stories on this of this happens on Amazon, 01:19:19.647 --> 01:19:21.730 align:middle line:84% that firms set these automatic pricing algorithms. 01:19:21.730 --> 01:19:23.490 align:middle line:84% Then you go to buy something on Amazon, 01:19:23.490 --> 01:19:25.120 align:middle line:90% you see these crazy prices. 01:19:25.120 --> 01:19:28.500 align:middle line:84% This is just some illustration I found of this pillow was being 01:19:28.500 --> 01:19:32.060 align:middle line:90% sold on Amazon for $10,235. 01:19:32.060 --> 01:19:34.270 align:middle line:84% And clearly, it was-- there were two firms selling 01:19:34.270 --> 01:19:39.130 align:middle line:84% this particular Downright Eiderdown 434 Thread Count Silk 01:19:39.130 --> 01:19:41.547 align:middle line:84% Pillow, and the computers were playing against each other. 01:19:41.547 --> 01:19:43.672 align:middle line:84% And somehow, they went off into a part of the space 01:19:43.672 --> 01:19:46.438 align:middle line:84% where each one thought, if I'm a little more expensive than him, 01:19:46.438 --> 01:19:47.980 align:middle line:84% I'm a little more expensive than him, 01:19:47.980 --> 01:19:50.000 align:middle line:84% something great is going to happen. 01:19:50.000 --> 01:19:51.190 align:middle line:90% And they ended up with-- 01:19:51.190 --> 01:19:54.590 align:middle line:84% they competed theirselves off to $10,000 pillow. 01:19:54.590 --> 01:19:56.590 align:middle line:90% And maybe the pillow doesn't-- 01:19:56.590 --> 01:19:59.180 align:middle line:84% once you get out here, you try undercutting from $10,000 01:19:59.180 --> 01:20:01.550 align:middle line:84% to $9,000, it makes no difference. 01:20:01.550 --> 01:20:03.520 align:middle line:84% You don't notice your way back to a pillow 01:20:03.520 --> 01:20:05.085 align:middle line:90% should be $30 or whatever. 01:20:05.085 --> 01:20:07.643 align:middle line:90% 01:20:07.643 --> 01:20:10.060 align:middle line:84% But you can get lots of things that make it clear evidence 01:20:10.060 --> 01:20:11.935 align:middle line:84% that it does seem like, in a lot of products, 01:20:11.935 --> 01:20:15.290 align:middle line:84% there are algorithms competing, and the algorithms work 01:20:15.290 --> 01:20:16.700 align:middle line:90% sometimes and not others. 01:20:16.700 --> 01:20:20.070 align:middle line:90% 01:20:20.070 --> 01:20:21.000 align:middle line:90% Any questions? 01:20:21.000 --> 01:20:22.370 align:middle line:90% So anyway, if that's it-- 01:20:22.370 --> 01:20:26.780 align:middle line:84% so next class, I'm doing empirical stuff. 01:20:26.780 --> 01:20:29.737 align:middle line:90% So it's Rob Porter's paper. 01:20:29.737 --> 01:20:31.320 align:middle line:84% If you want to pick one thing to read, 01:20:31.320 --> 01:20:34.440 align:middle line:84% Rob Porter's paper on railroad cartels. 01:20:34.440 --> 01:20:37.370 align:middle line:84% I have a paper building on Rob's paper 01:20:37.370 --> 01:20:40.602 align:middle line:90% and then, also, a paper about-- 01:20:40.602 --> 01:20:42.560 align:middle line:84% talk about a paper about Edgeworth price cycles 01:20:42.560 --> 01:20:44.310 align:middle line:90% in practice. 01:20:44.310 --> 01:20:57.000 align:middle line:90%