WEBVTT 00:00:00.000 --> 00:00:01.955 align:middle line:90% [SQUEAKING] 00:00:01.955 --> 00:00:03.895 align:middle line:90% [RUSTLING] 00:00:03.895 --> 00:00:05.835 align:middle line:90% [CLICKING] 00:00:05.835 --> 00:00:10.200 align:middle line:90% 00:00:10.200 --> 00:00:11.720 align:middle line:84% IAN BALL: OK, so today we're going 00:00:11.720 --> 00:00:16.040 align:middle line:84% to continue our study of subgame-perfect Nash equilibria. 00:00:16.040 --> 00:00:20.440 align:middle line:84% And specifically, we're going to look at multistage, 00:00:20.440 --> 00:00:24.080 align:middle line:84% something called multistage games. 00:00:24.080 --> 00:00:27.760 align:middle line:84% And the key technical tool that we're going to introduce today 00:00:27.760 --> 00:00:30.180 align:middle line:84% is something called the one-shot deviation principle. 00:00:30.180 --> 00:00:39.440 align:middle line:90% 00:00:39.440 --> 00:00:42.260 align:middle line:84% So I'll go into detail about what this is later in the class. 00:00:42.260 --> 00:00:45.400 align:middle line:84% But it's going to allow us to compute or to check 00:00:45.400 --> 00:00:48.280 align:middle line:84% that a strategy profile is, in fact, a subgame-perfect Nash 00:00:48.280 --> 00:00:51.560 align:middle line:84% equilibria, even in games that have an infinite horizon. 00:00:51.560 --> 00:00:53.240 align:middle line:84% And using this tool, we're then going 00:00:53.240 --> 00:00:55.080 align:middle line:84% to revisit one of the applications we looked 00:00:55.080 --> 00:00:57.760 align:middle line:84% at before, this price haggling game, where you have a buyer 00:00:57.760 --> 00:01:00.820 align:middle line:84% and a seller haggling over what the price is going to be 00:01:00.820 --> 00:01:02.120 align:middle line:90% that they exchange the good. 00:01:02.120 --> 00:01:04.480 align:middle line:84% And if you recall, when we first modeled that game, 00:01:04.480 --> 00:01:06.580 align:middle line:84% we had this artificial time at which 00:01:06.580 --> 00:01:09.180 align:middle line:84% the world ended because we didn't have the tools to analyze 00:01:09.180 --> 00:01:10.440 align:middle line:90% an infinite horizon game. 00:01:10.440 --> 00:01:12.340 align:middle line:84% But now we're going to apply these techniques to be 00:01:12.340 --> 00:01:14.090 align:middle line:84% able to fully analyze the infinite horizon 00:01:14.090 --> 00:01:16.060 align:middle line:90% version of bargaining. 00:01:16.060 --> 00:01:18.260 align:middle line:84% So I want to start by saying that 00:01:18.260 --> 00:01:21.660 align:middle line:84% what I did last class was true but was maybe 00:01:21.660 --> 00:01:22.560 align:middle line:90% a little more subtle. 00:01:22.560 --> 00:01:23.900 align:middle line:84% And I want to point out that there's kind of-- 00:01:23.900 --> 00:01:26.700 align:middle line:84% I want to give a cautionary note about some of the reasoning 00:01:26.700 --> 00:01:28.660 align:middle line:90% we did last class. 00:01:28.660 --> 00:01:31.560 align:middle line:84% So we looked at games maybe like this. 00:01:31.560 --> 00:01:32.980 align:middle line:90% Let's do this. 00:01:32.980 --> 00:01:35.300 align:middle line:84% Enter and exit game where we had first player 1 00:01:35.300 --> 00:01:37.320 align:middle line:90% chose whether to exit or enter. 00:01:37.320 --> 00:01:42.260 align:middle line:84% If they exited, let's say, each player got 1.5. 00:01:42.260 --> 00:01:45.000 align:middle line:84% And then if they entered-- let's say they played the Boston game, 00:01:45.000 --> 00:01:49.940 align:middle line:84% so player 2 moved and chose Celtics or Red Sox. 00:01:49.940 --> 00:01:57.480 align:middle line:84% And then player 1 moved and chose Celtics or Red Sox. 00:01:57.480 --> 00:02:02.840 align:middle line:84% And then the payoffs were, let's see, player 1, let's see, 00:02:02.840 --> 00:02:10.280 align:middle line:90% 2, 1, 1, 2, 0, 0, and 0, 0. 00:02:10.280 --> 00:02:14.480 align:middle line:84% And we argued that one equilibrium of this game 00:02:14.480 --> 00:02:22.560 align:middle line:84% is the equilibrium where we have XR, R. 00:02:22.560 --> 00:02:24.560 align:middle line:84% And we argue that this is a subgame-perfect Nash 00:02:24.560 --> 00:02:27.880 align:middle line:90% equilibrium. 00:02:27.880 --> 00:02:30.680 align:middle line:84% And let's see how we reasoned about that. 00:02:30.680 --> 00:02:32.720 align:middle line:84% Well, we said there's two subgames. 00:02:32.720 --> 00:02:34.240 align:middle line:84% There's the entire game, and there's 00:02:34.240 --> 00:02:35.540 align:middle line:90% the subgame that starts here. 00:02:35.540 --> 00:02:39.840 align:middle line:84% So the first step is always to identify our subgames. 00:02:39.840 --> 00:02:42.880 align:middle line:84% And then we said, let's find a Nash equilibrium 00:02:42.880 --> 00:02:45.180 align:middle line:84% within this subgame, within this smaller subgame. 00:02:45.180 --> 00:02:46.807 align:middle line:84% Remember, we always start at the end. 00:02:46.807 --> 00:02:49.140 align:middle line:84% We start with the smaller subgames that come at the end, 00:02:49.140 --> 00:02:51.480 align:middle line:90% and we work backwards. 00:02:51.480 --> 00:02:53.960 align:middle line:84% And we know that this subgame is the standard BOS 00:02:53.960 --> 00:02:55.460 align:middle line:90% game that we studied before. 00:02:55.460 --> 00:02:57.540 align:middle line:84% And we know that one equilibrium of this game 00:02:57.540 --> 00:02:59.900 align:middle line:84% is that both players go to the Red Sox game. 00:02:59.900 --> 00:03:01.480 align:middle line:84% So we filled in that equilibrium. 00:03:01.480 --> 00:03:06.232 align:middle line:90% 00:03:06.232 --> 00:03:07.440 align:middle line:90% And then we took a step back. 00:03:07.440 --> 00:03:09.440 align:middle line:84% We said, well, if this is the equilibrium that's 00:03:09.440 --> 00:03:14.300 align:middle line:84% being played in this subgame, then the payoff in this subgame 00:03:14.300 --> 00:03:16.060 align:middle line:90% is going to be 1, 2. 00:03:16.060 --> 00:03:17.820 align:middle line:84% We said if this subgame is reached, 00:03:17.820 --> 00:03:19.700 align:middle line:84% that's what the payoffs are going to be. 00:03:19.700 --> 00:03:21.580 align:middle line:84% And then we said if I'm player 1, well, 00:03:21.580 --> 00:03:25.700 align:middle line:84% I'm effectively choosing to exit and get a payoff of 1.5 00:03:25.700 --> 00:03:29.220 align:middle line:84% or to enter and get the payoff in the Nash equilibrium 00:03:29.220 --> 00:03:30.600 align:middle line:90% of this subgame and get 1. 00:03:30.600 --> 00:03:32.960 align:middle line:84% So I'm better off exiting, and we said great. 00:03:32.960 --> 00:03:34.700 align:middle line:90% This is an SPNE. 00:03:34.700 --> 00:03:35.480 align:middle line:90% So that's true. 00:03:35.480 --> 00:03:36.660 align:middle line:90% Everything is correct. 00:03:36.660 --> 00:03:40.740 align:middle line:84% But I want to argue that there's a little subtlety here. 00:03:40.740 --> 00:03:44.740 align:middle line:84% And the subtlety is that we of skipped a step. 00:03:44.740 --> 00:03:47.220 align:middle line:84% And what we didn't think about is-- 00:03:47.220 --> 00:04:00.800 align:middle line:84% and I'll say, what about complicated deviations 00:04:00.800 --> 00:04:01.700 align:middle line:90% by player 1? 00:04:01.700 --> 00:04:04.720 align:middle line:90% 00:04:04.720 --> 00:04:05.740 align:middle line:90% What do I mean by this? 00:04:05.740 --> 00:04:09.320 align:middle line:84% Well, we check that this strategy constitutes a Nash 00:04:09.320 --> 00:04:10.980 align:middle line:90% equilibrium in the subgame. 00:04:10.980 --> 00:04:12.337 align:middle line:90% That's pretty clear. 00:04:12.337 --> 00:04:14.920 align:middle line:84% But we also have to check that this constitutes an equilibrium 00:04:14.920 --> 00:04:17.360 align:middle line:84% in the full game because part of the definition 00:04:17.360 --> 00:04:19.279 align:middle line:84% of subgame-perfect Nash equilibrium 00:04:19.279 --> 00:04:21.519 align:middle line:84% is that the strategy profile must constitute a Nash 00:04:21.519 --> 00:04:23.540 align:middle line:84% equilibrium within every subgame, 00:04:23.540 --> 00:04:26.800 align:middle line:84% and the full game is one particular example of a subgame. 00:04:26.800 --> 00:04:30.720 align:middle line:84% So what we have to show is that player 1 does not 00:04:30.720 --> 00:04:34.080 align:middle line:84% have a profitable deviation in the full game. 00:04:34.080 --> 00:04:35.720 align:middle line:90% Now, what did we actually show? 00:04:35.720 --> 00:04:37.920 align:middle line:84% What we showed is, well, if I'm player 1, 00:04:37.920 --> 00:04:42.540 align:middle line:84% and I deviate to enter and then continue the strategy down here, 00:04:42.540 --> 00:04:44.480 align:middle line:90% that won't be profitable. 00:04:44.480 --> 00:04:46.400 align:middle line:84% But when we filled in this subgame 00:04:46.400 --> 00:04:48.840 align:middle line:84% and treated the payoff as 1, 2, we basically 00:04:48.840 --> 00:04:51.540 align:middle line:84% fixed the way that player 1 plays in this subgame. 00:04:51.540 --> 00:04:54.580 align:middle line:84% And in principle, player 1 could consider a more complicated 00:04:54.580 --> 00:04:55.208 align:middle line:90% deviation. 00:04:55.208 --> 00:04:57.000 align:middle line:84% What they could do is they could say, well, 00:04:57.000 --> 00:05:01.100 align:middle line:84% what if I enter the game and then deviate 00:05:01.100 --> 00:05:03.830 align:middle line:84% as well in this subgame and play differently in this subgame? 00:05:03.830 --> 00:05:06.080 align:middle line:84% How do we know that that's not going to be profitable? 00:05:06.080 --> 00:05:09.340 align:middle line:84% All we checked is that if they enter and then play 00:05:09.340 --> 00:05:12.002 align:middle line:84% as they're supposed to in the subgame, that's not profitable. 00:05:12.002 --> 00:05:13.460 align:middle line:84% But we haven't checked what happens 00:05:13.460 --> 00:05:15.900 align:middle line:84% if player 1 enters and then modifies 00:05:15.900 --> 00:05:17.540 align:middle line:90% their behavior in the subgame. 00:05:17.540 --> 00:05:19.380 align:middle line:84% So we didn't explicitly check that. 00:05:19.380 --> 00:05:20.440 align:middle line:90% I argue that that's OK. 00:05:20.440 --> 00:05:21.340 align:middle line:90% But why? 00:05:21.340 --> 00:05:24.220 align:middle line:84% Why do we not need to worry about these more complicated 00:05:24.220 --> 00:05:26.060 align:middle line:90% deviations? 00:05:26.060 --> 00:05:27.880 align:middle line:90% Does anyone see why? 00:05:27.880 --> 00:05:31.353 align:middle line:84% I think this is a subtle point, so if people are confused, 00:05:31.353 --> 00:05:32.520 align:middle line:90% feel free to ask a question. 00:05:32.520 --> 00:05:35.620 align:middle line:90% 00:05:35.620 --> 00:05:36.132 align:middle line:90% Yeah. 00:05:36.132 --> 00:05:38.340 align:middle line:84% AUDIENCE: Is it because we already performed backward 00:05:38.340 --> 00:05:40.220 align:middle line:90% induction on the subgame? 00:05:40.220 --> 00:05:41.280 align:middle line:90% IAN BALL: Exactly right. 00:05:41.280 --> 00:05:44.180 align:middle line:84% So what happens in one of these complicated deviations? 00:05:44.180 --> 00:05:48.060 align:middle line:84% We choose N here, but then instead of following the play 00:05:48.060 --> 00:05:50.990 align:middle line:84% that we're supposed to use in this subgame, 00:05:50.990 --> 00:05:53.130 align:middle line:84% we then deviate again in this subgame. 00:05:53.130 --> 00:05:55.370 align:middle line:84% But we already know that, in this subgame, 00:05:55.370 --> 00:05:57.570 align:middle line:84% our play constituted a Nash equilibrium. 00:05:57.570 --> 00:06:01.090 align:middle line:84% So this further deviation can't be any better 00:06:01.090 --> 00:06:05.010 align:middle line:84% because it involves suboptimal play within this subgame given 00:06:05.010 --> 00:06:07.010 align:middle line:84% the way the other player is playing. 00:06:07.010 --> 00:06:11.010 align:middle line:84% So it turns out what about these more complicated deviations? 00:06:11.010 --> 00:06:12.950 align:middle line:84% The answer is they can't be profitable. 00:06:12.950 --> 00:06:18.290 align:middle line:90% 00:06:18.290 --> 00:06:19.210 align:middle line:90% And why? 00:06:19.210 --> 00:06:23.010 align:middle line:84% Because we know that our strategy profile already 00:06:23.010 --> 00:06:25.175 align:middle line:84% constitutes a Nash equilibrium within this subgame. 00:06:25.175 --> 00:06:27.550 align:middle line:84% And therefore, given the way the other player is playing, 00:06:27.550 --> 00:06:29.730 align:middle line:84% we're already doing optimally in this subgame. 00:06:29.730 --> 00:06:31.730 align:middle line:84% So if I were to enter and then play suboptimally 00:06:31.730 --> 00:06:37.970 align:middle line:84% in the subgame, that can't make me better off 00:06:37.970 --> 00:06:39.990 align:middle line:84% because of backward induction, I'll say. 00:06:39.990 --> 00:06:45.413 align:middle line:90% 00:06:45.413 --> 00:06:47.330 align:middle line:84% And it's, quote, unquote, "backward induction" 00:06:47.330 --> 00:06:49.872 align:middle line:84% because we can't technically apply backward induction to this 00:06:49.872 --> 00:06:51.970 align:middle line:84% game, but it's the reasoning of working backwards. 00:06:51.970 --> 00:06:53.830 align:middle line:84% So maybe this is just so intuitive 00:06:53.830 --> 00:06:56.278 align:middle line:84% that everyone already intuited this. 00:06:56.278 --> 00:06:58.070 align:middle line:84% I think it is kind of intuitive, but I just 00:06:58.070 --> 00:07:01.270 align:middle line:84% want to highlight that there is kind of a subtlety going on. 00:07:01.270 --> 00:07:03.038 align:middle line:84% And in class today, because we're 00:07:03.038 --> 00:07:05.330 align:middle line:84% going to deal with games that have an infinite horizon, 00:07:05.330 --> 00:07:07.450 align:middle line:84% we have to be even more careful about the subtlety. 00:07:07.450 --> 00:07:07.950 align:middle line:90% Yes. 00:07:07.950 --> 00:07:10.565 align:middle line:84% AUDIENCE: So why isn't CC the Nash equilibrium? 00:07:10.565 --> 00:07:12.190 align:middle line:84% IAN BALL: I didn't say this is the only 00:07:12.190 --> 00:07:13.390 align:middle line:84% subgame-perfect Nash equilibrium. 00:07:13.390 --> 00:07:14.930 align:middle line:84% This is one subgame-perfect Nash equilibrium. 00:07:14.930 --> 00:07:16.888 align:middle line:84% There's also a subgame-perfect Nash equilibrium 00:07:16.888 --> 00:07:21.163 align:middle line:84% where the players play CC and the first player enters. 00:07:21.163 --> 00:07:22.830 align:middle line:84% That's a different subgame-perfect Nash. 00:07:22.830 --> 00:07:25.810 align:middle line:84% But here, I'm just questioning have we fully verified 00:07:25.810 --> 00:07:27.810 align:middle line:84% that this is a subgame-perfect Nash equilibrium? 00:07:27.810 --> 00:07:28.850 align:middle line:90% And I'm just arguing that it is. 00:07:28.850 --> 00:07:31.470 align:middle line:84% But the verification is a little more subtle than maybe you 00:07:31.470 --> 00:07:32.450 align:middle line:90% might think at first. 00:07:32.450 --> 00:07:32.950 align:middle line:90% Yeah. 00:07:32.950 --> 00:07:34.117 align:middle line:90% Any other questions on this? 00:07:34.117 --> 00:07:45.270 align:middle line:90% 00:07:45.270 --> 00:07:48.970 align:middle line:84% So it turns out that dealing with all these complicated 00:07:48.970 --> 00:07:52.030 align:middle line:84% deviations in really big games can be tricky. 00:07:52.030 --> 00:07:54.330 align:middle line:84% And we're now going to study a class of games 00:07:54.330 --> 00:07:56.230 align:middle line:84% where the analysis becomes pretty simple. 00:07:56.230 --> 00:07:57.605 align:middle line:84% And this is a class of games that 00:07:57.605 --> 00:07:59.670 align:middle line:84% covers a lot of applications of interest. 00:07:59.670 --> 00:08:02.070 align:middle line:84% And these are going to be called multistage games. 00:08:02.070 --> 00:08:06.370 align:middle line:90% 00:08:06.370 --> 00:08:08.170 align:middle line:84% And I want to emphasize that these possibly 00:08:08.170 --> 00:08:09.550 align:middle line:90% have an infinite horizon. 00:08:09.550 --> 00:08:21.530 align:middle line:90% 00:08:21.530 --> 00:08:22.910 align:middle line:90% So what is the definition? 00:08:22.910 --> 00:08:24.730 align:middle line:90% What is a multistage game? 00:08:24.730 --> 00:08:27.370 align:middle line:84% Well, it's a game that has multiple stages, so let's 00:08:27.370 --> 00:08:28.610 align:middle line:90% define what that means. 00:08:28.610 --> 00:08:34.556 align:middle line:84% So it's a game where we have stages, maybe 0, 1, 00:08:34.556 --> 00:08:37.429 align:middle line:84% 2, dot dot dot, and it could either be finite. 00:08:37.429 --> 00:08:42.549 align:middle line:84% It could either go to T, or it could go on forever. 00:08:42.549 --> 00:08:44.670 align:middle line:84% So I'm allowing both of those possibilities. 00:08:44.670 --> 00:08:49.130 align:middle line:84% So it could be finite or infinite. 00:08:49.130 --> 00:08:52.350 align:middle line:90% Both are allowed. 00:08:52.350 --> 00:08:53.770 align:middle line:84% But what happens in these stages? 00:08:53.770 --> 00:08:59.110 align:middle line:84% Well, there's two key properties of these stages. 00:08:59.110 --> 00:09:02.590 align:middle line:84% So in each stage, some subset of players 00:09:02.590 --> 00:09:04.210 align:middle line:84% are going to move simultaneously. 00:09:04.210 --> 00:09:06.990 align:middle line:90% 00:09:06.990 --> 00:09:15.450 align:middle line:84% So in each stage, some subset of players move simultaneously. 00:09:15.450 --> 00:09:28.830 align:middle line:90% 00:09:28.830 --> 00:09:30.830 align:middle line:84% And then the other key assumption 00:09:30.830 --> 00:09:33.350 align:middle line:84% is that at the beginning of each stage, 00:09:33.350 --> 00:09:35.430 align:middle line:84% everyone observes all the moves that 00:09:35.430 --> 00:09:37.410 align:middle line:90% were made prior to that stage. 00:09:37.410 --> 00:09:40.190 align:middle line:90% 00:09:40.190 --> 00:09:49.690 align:middle line:84% So at the beginning of each stage 00:09:49.690 --> 00:09:51.310 align:middle line:90% all prior moves are observed. 00:09:51.310 --> 00:09:59.290 align:middle line:90% 00:09:59.290 --> 00:10:01.730 align:middle line:84% So sometimes these are called multistage games 00:10:01.730 --> 00:10:05.123 align:middle line:84% with observed actions to emphasize the second point. 00:10:05.123 --> 00:10:06.790 align:middle line:84% But I don't want to use that terminology 00:10:06.790 --> 00:10:09.330 align:middle line:84% because if we look within a stage, 00:10:09.330 --> 00:10:11.970 align:middle line:84% we have multiple players moving simultaneously. 00:10:11.970 --> 00:10:14.390 align:middle line:84% So at the stage where the player's moving simultaneously, 00:10:14.390 --> 00:10:16.432 align:middle line:84% they're not observing the other player's actions. 00:10:16.432 --> 00:10:19.170 align:middle line:84% So it's not actually a game of perfect information 00:10:19.170 --> 00:10:21.170 align:middle line:84% because we have these simultaneous moves 00:10:21.170 --> 00:10:23.355 align:middle line:90% within the game. 00:10:23.355 --> 00:10:24.730 align:middle line:84% Does anyone remember any examples 00:10:24.730 --> 00:10:27.530 align:middle line:84% of multistage games or any games that we've studied so far 00:10:27.530 --> 00:10:28.750 align:middle line:90% constituted multistage games? 00:10:28.750 --> 00:10:29.250 align:middle line:90% Yes. 00:10:29.250 --> 00:10:31.833 align:middle line:84% AUDIENCE: Could this be the one where firms have to pick entry 00:10:31.833 --> 00:10:34.510 align:middle line:84% and then they pick quantities after they observe? 00:10:34.510 --> 00:10:35.510 align:middle line:90% IAN BALL: Exactly right. 00:10:35.510 --> 00:10:37.790 align:middle line:84% So in that Cournot game with entry cost, 00:10:37.790 --> 00:10:40.650 align:middle line:84% that was exactly a multistage game that had two stages. 00:10:40.650 --> 00:10:44.350 align:middle line:84% Remember, in the first stage, the firms simultaneously 00:10:44.350 --> 00:10:46.990 align:middle line:84% chose whether to enter the market. 00:10:46.990 --> 00:10:49.390 align:middle line:84% That was stage-- I guess we can call it stage 0 00:10:49.390 --> 00:10:50.910 align:middle line:90% if you want to start at 0. 00:10:50.910 --> 00:10:53.030 align:middle line:84% And then, in the second stage, a subset 00:10:53.030 --> 00:10:55.670 align:middle line:84% of players, namely the firms who had chosen 00:10:55.670 --> 00:10:58.350 align:middle line:84% to enter in the first stage, then simultaneously 00:10:58.350 --> 00:11:00.130 align:middle line:84% chose what quantities to produce. 00:11:00.130 --> 00:11:02.510 align:middle line:84% And we made the critical assumption 00:11:02.510 --> 00:11:05.010 align:middle line:84% that, at the beginning of that second stage, 00:11:05.010 --> 00:11:07.430 align:middle line:84% everyone observed the moves in the first stage, that 00:11:07.430 --> 00:11:09.990 align:middle line:84% is, all the firms observed, which subset of firms 00:11:09.990 --> 00:11:11.438 align:middle line:90% had entered in the first stage. 00:11:11.438 --> 00:11:13.230 align:middle line:84% And maybe I should be a little more precise 00:11:13.230 --> 00:11:17.470 align:middle line:84% when I say some subset of players move simultaneously. 00:11:17.470 --> 00:11:20.590 align:middle line:84% It could be that the subset that moves simultaneously 00:11:20.590 --> 00:11:22.970 align:middle line:84% depends on what happened in previous stages. 00:11:22.970 --> 00:11:25.230 align:middle line:84% So if you look at that Cournot entry game, 00:11:25.230 --> 00:11:28.870 align:middle line:84% if firms 1 and 2 entered in the first stage, then firms 1 and 2 00:11:28.870 --> 00:11:30.890 align:middle line:84% play simultaneously in the second stage. 00:11:30.890 --> 00:11:34.390 align:middle line:84% If firms 3 and 4 entered in the first stage, then firms 3 and 4 00:11:34.390 --> 00:11:35.970 align:middle line:90% made choices simultaneously. 00:11:35.970 --> 00:11:38.690 align:middle line:84% So some subset, which maybe I'll say, 00:11:38.690 --> 00:11:48.370 align:middle line:84% it could be history dependent, where 00:11:48.370 --> 00:11:52.210 align:middle line:84% history means what happened in the previous stages. 00:11:52.210 --> 00:11:54.930 align:middle line:84% Another example, actually, is this game here. 00:11:54.930 --> 00:11:57.590 align:middle line:84% So how can we think of this game as a multistage game? 00:11:57.590 --> 00:12:00.530 align:middle line:90% 00:12:00.530 --> 00:12:02.355 align:middle line:84% Walk me through why we can think-- 00:12:02.355 --> 00:12:04.230 align:middle line:84% I mean, I model it as an extensive-form game, 00:12:04.230 --> 00:12:08.850 align:middle line:84% but how can we think of it as a multistage game? 00:12:08.850 --> 00:12:10.713 align:middle line:90% Well, in the first stage, or-- 00:12:10.713 --> 00:12:12.130 align:middle line:84% it's always awkward to say with 0. 00:12:12.130 --> 00:12:14.290 align:middle line:84% In the zero stage, in the initial stage, 00:12:14.290 --> 00:12:16.930 align:middle line:90% player 1 makes a move. 00:12:16.930 --> 00:12:18.757 align:middle line:90% Player 1 is a subset of players. 00:12:18.757 --> 00:12:20.590 align:middle line:84% It's a subset of players with just 1 player. 00:12:20.590 --> 00:12:22.850 align:middle line:84% That's the only player who moves. 00:12:22.850 --> 00:12:26.090 align:middle line:84% So maybe let's pull this down and look at this. 00:12:26.090 --> 00:12:28.450 align:middle line:84% So if we want to represent this as a multi-stage game, 00:12:28.450 --> 00:12:30.190 align:middle line:90% this is stage 0. 00:12:30.190 --> 00:12:33.770 align:middle line:90% 00:12:33.770 --> 00:12:37.510 align:middle line:84% And in stage 0, the only player who moves is player 1. 00:12:37.510 --> 00:12:42.970 align:middle line:84% Then we can think of this here as stage 1. 00:12:42.970 --> 00:12:43.470 align:middle line:90% Why? 00:12:43.470 --> 00:12:47.790 align:middle line:84% Well, remember, technically, we said player 2 moves first 00:12:47.790 --> 00:12:49.910 align:middle line:90% and then player 1 moves after. 00:12:49.910 --> 00:12:52.150 align:middle line:84% But because player 1 doesn't observe the move 00:12:52.150 --> 00:12:55.150 align:middle line:84% that player 2 makes, it's as if players 1 and 2 00:12:55.150 --> 00:12:57.050 align:middle line:90% are moving simultaneously. 00:12:57.050 --> 00:12:59.190 align:middle line:84% I could say player 1 moves and then 00:12:59.190 --> 00:13:01.490 align:middle line:84% player 2 moves not observing player 1's move. 00:13:01.490 --> 00:13:04.550 align:middle line:84% The order doesn't matter as long as the players don't observe 00:13:04.550 --> 00:13:05.870 align:middle line:90% what the other player's doing. 00:13:05.870 --> 00:13:07.670 align:middle line:84% So effectively what's happening in 00:13:07.670 --> 00:13:11.710 align:middle line:84% this game is the players are playing a simultaneous-move game 00:13:11.710 --> 00:13:14.270 align:middle line:84% within stage 1, and that simultaneous-move game 00:13:14.270 --> 00:13:16.710 align:middle line:84% is exactly the Boston game that we studied. 00:13:16.710 --> 00:13:18.390 align:middle line:84% Now we have to check one more thing. 00:13:18.390 --> 00:13:20.590 align:middle line:84% Is it true that, at the beginning of stage 1, 00:13:20.590 --> 00:13:23.990 align:middle line:84% the players know what happened in stage 0? 00:13:23.990 --> 00:13:24.830 align:middle line:90% Yes. 00:13:24.830 --> 00:13:26.830 align:middle line:84% If we look at this subgame, both players 00:13:26.830 --> 00:13:28.510 align:middle line:84% know that player 1 must've played 00:13:28.510 --> 00:13:32.730 align:middle line:84% enter because, when I'm player 1 here, 00:13:32.730 --> 00:13:35.327 align:middle line:84% maybe I don't know which node I'm at within this information 00:13:35.327 --> 00:13:37.910 align:middle line:84% set, but I certainly know that I'm on this branch of the tree, 00:13:37.910 --> 00:13:39.868 align:middle line:84% and therefore, I know what happened in stage 0. 00:13:39.868 --> 00:13:41.330 align:middle line:84% So this is a special case that we 00:13:41.330 --> 00:13:43.670 align:middle line:84% can think of as an example of a multistage game. 00:13:43.670 --> 00:13:47.930 align:middle line:90% 00:13:47.930 --> 00:13:51.890 align:middle line:84% So this is quite a flexible general class of games. 00:13:51.890 --> 00:13:55.170 align:middle line:84% And it turns out that it's very easy in this class of games 00:13:55.170 --> 00:13:57.450 align:middle line:84% to check whether a strategy profile is 00:13:57.450 --> 00:13:58.957 align:middle line:84% a subgame-perfect Nash equilibrium. 00:13:58.957 --> 00:14:00.790 align:middle line:84% And that's what we're going to go over next. 00:14:00.790 --> 00:14:13.530 align:middle line:90% 00:14:13.530 --> 00:14:16.090 align:middle line:84% So the next result is what's called the one-shot deviation 00:14:16.090 --> 00:14:16.830 align:middle line:90% principle. 00:14:16.830 --> 00:14:25.650 align:middle line:90% 00:14:25.650 --> 00:14:28.670 align:middle line:84% And maybe it's helpful to first, before I state the result, 00:14:28.670 --> 00:14:31.090 align:middle line:84% describe what I mean by a one-shot deviation. 00:14:31.090 --> 00:14:33.550 align:middle line:90% So what is a one-shot deviation? 00:14:33.550 --> 00:14:40.350 align:middle line:84% Well, let's suppose a player has some strategy s in the game. 00:14:40.350 --> 00:14:44.790 align:middle line:84% And I want to say, what does it mean for s prime-- 00:14:44.790 --> 00:14:48.350 align:middle line:84% maybe I'll call this i for s prime to be 00:14:48.350 --> 00:14:50.170 align:middle line:90% a one-shot deviation from s. 00:14:50.170 --> 00:15:01.830 align:middle line:90% 00:15:01.830 --> 00:15:05.530 align:middle line:84% Well, in principle, I could deviate from si 00:15:05.530 --> 00:15:08.325 align:middle line:90% in a very complicated way. 00:15:08.325 --> 00:15:10.450 align:middle line:84% There's many, many different histories in the game. 00:15:10.450 --> 00:15:13.710 align:middle line:90% The game's really complicated. 00:15:13.710 --> 00:15:15.910 align:middle line:84% I could make a lot of different deviations. 00:15:15.910 --> 00:15:19.050 align:middle line:84% A one-shot deviation means I only deviate once. 00:15:19.050 --> 00:15:22.470 align:middle line:84% Or more precisely, I only deviate at one history 00:15:22.470 --> 00:15:24.110 align:middle line:90% or one contingency. 00:15:24.110 --> 00:15:27.830 align:middle line:84% So in multistage games, we often refer 00:15:27.830 --> 00:15:32.050 align:middle line:84% to information sets as histories because, whenever 00:15:32.050 --> 00:15:35.970 align:middle line:84% a player is called upon to play in a multistage game, whatever 00:15:35.970 --> 00:15:38.810 align:middle line:84% stage it's in, what they know is exactly what happened 00:15:38.810 --> 00:15:40.290 align:middle line:90% in all the previous stages. 00:15:40.290 --> 00:15:42.330 align:middle line:84% So their information set is a history 00:15:42.330 --> 00:15:44.970 align:middle line:90% of play in all previous stages. 00:15:44.970 --> 00:15:48.730 align:middle line:84% So a one-shot deviation from si if-- 00:15:48.730 --> 00:15:50.610 align:middle line:90% what do we want to say? 00:15:50.610 --> 00:15:55.210 align:middle line:84% We want to say si prime of h is not 00:15:55.210 --> 00:16:02.110 align:middle line:84% equal to si of h at one history h. 00:16:02.110 --> 00:16:05.130 align:middle line:90% 00:16:05.130 --> 00:16:11.050 align:middle line:84% But si prime of h equals si of h prime prime 00:16:11.050 --> 00:16:20.490 align:middle line:90% at all other histories h prime. 00:16:20.490 --> 00:16:23.490 align:middle line:84% So remember, a strategy is a complete contingent plan. 00:16:23.490 --> 00:16:25.490 align:middle line:84% It says how player i is going to play 00:16:25.490 --> 00:16:27.810 align:middle line:84% at every history or every contingency at which they're 00:16:27.810 --> 00:16:29.550 align:middle line:90% asked to move. 00:16:29.550 --> 00:16:31.290 align:middle line:84% So when I compare two strategies, 00:16:31.290 --> 00:16:33.770 align:middle line:84% I can ask how different are these strategies? 00:16:33.770 --> 00:16:36.430 align:middle line:84% At how many histories do these strategies 00:16:36.430 --> 00:16:38.950 align:middle line:90% prescribe different behavior? 00:16:38.950 --> 00:16:42.230 align:middle line:84% And we say that si prime is a one-shot deviation from si 00:16:42.230 --> 00:16:44.950 align:middle line:84% if there's exactly one history at which they 00:16:44.950 --> 00:16:46.870 align:middle line:90% prescribe different behavior. 00:16:46.870 --> 00:16:48.630 align:middle line:84% And then, at all the other histories, 00:16:48.630 --> 00:16:50.450 align:middle line:84% they prescribe exactly the same behavior. 00:16:50.450 --> 00:16:55.450 align:middle line:90% 00:16:55.450 --> 00:16:58.910 align:middle line:84% But we have to be really, really careful about what 00:16:58.910 --> 00:17:02.070 align:middle line:90% this means at one history. 00:17:02.070 --> 00:17:06.990 align:middle line:84% So even if I modify my strategy at a single history, 00:17:06.990 --> 00:17:09.690 align:middle line:84% it could mean that play changes at many histories. 00:17:09.690 --> 00:17:12.470 align:middle line:84% So let's go back to our example of the Cournot game 00:17:12.470 --> 00:17:13.810 align:middle line:90% to understand what this means. 00:17:13.810 --> 00:17:23.349 align:middle line:90% 00:17:23.349 --> 00:17:26.150 align:middle line:84% Suppose we have a player who, in the first stage, 00:17:26.150 --> 00:17:28.303 align:middle line:84% was not supposed to enter the game, 00:17:28.303 --> 00:17:29.720 align:middle line:84% or they weren't entering the game. 00:17:29.720 --> 00:17:32.060 align:middle line:84% And now they contemplate a one-shot deviation. 00:17:32.060 --> 00:17:45.360 align:middle line:84% So let's say firm i deviates to enter at the initial history. 00:17:45.360 --> 00:17:48.420 align:middle line:90% 00:17:48.420 --> 00:17:52.812 align:middle line:84% So previously, the firm was not entering at this initial history 00:17:52.812 --> 00:17:54.020 align:middle line:90% at the beginning of the game. 00:17:54.020 --> 00:17:56.060 align:middle line:90% Now they enter. 00:17:56.060 --> 00:17:57.400 align:middle line:90% This is a one-shot deviation. 00:17:57.400 --> 00:18:00.660 align:middle line:84% They're only changing their strategy 00:18:00.660 --> 00:18:02.300 align:middle line:90% at that initial history. 00:18:02.300 --> 00:18:05.940 align:middle line:84% But now because they've entered at that initial history, 00:18:05.940 --> 00:18:09.140 align:middle line:84% now the way everyone plays in the second period 00:18:09.140 --> 00:18:10.660 align:middle line:90% becomes quite different. 00:18:10.660 --> 00:18:13.340 align:middle line:84% So even though if the firm is deviating at a single history, 00:18:13.340 --> 00:18:15.520 align:middle line:84% it changes play at all subsequent histories. 00:18:15.520 --> 00:18:25.320 align:middle line:90% 00:18:25.320 --> 00:18:26.000 align:middle line:90% Why? 00:18:26.000 --> 00:18:28.680 align:middle line:84% Well, now a different firm, a different number of firms 00:18:28.680 --> 00:18:30.720 align:middle line:84% have entered the game, so the quantities 00:18:30.720 --> 00:18:33.600 align:middle line:84% that every firm is going to produce in the second period 00:18:33.600 --> 00:18:35.560 align:middle line:84% has now changed because the quantities 00:18:35.560 --> 00:18:38.360 align:middle line:84% that they produce depend upon how many firms enter. 00:18:38.360 --> 00:18:42.800 align:middle line:84% So even if the deviation only changes behavior at one history, 00:18:42.800 --> 00:18:45.520 align:middle line:84% that then changes which future histories we reach 00:18:45.520 --> 00:18:47.820 align:middle line:84% and therefore can change a lot of things in the game. 00:18:47.820 --> 00:18:52.080 align:middle line:90% 00:18:52.080 --> 00:18:56.800 align:middle line:90% And we'll see that a bit later. 00:18:56.800 --> 00:18:57.420 align:middle line:90% All right. 00:18:57.420 --> 00:19:00.200 align:middle line:90% 00:19:00.200 --> 00:19:03.132 align:middle line:84% So now that we've defined what a one-shot deviation is, 00:19:03.132 --> 00:19:05.340 align:middle line:84% we can actually-- let me give a little more notation. 00:19:05.340 --> 00:19:08.240 align:middle line:84% We've said si prime is a one-shot deviation from si. 00:19:08.240 --> 00:19:09.880 align:middle line:90% We can be a little more precise. 00:19:09.880 --> 00:19:13.800 align:middle line:84% If si prime differs from si at precisely the history h, 00:19:13.800 --> 00:19:16.700 align:middle line:84% we can call this a one-shot deviation at h. 00:19:16.700 --> 00:19:20.360 align:middle line:84% It's a one-shot deviation where the strategy specifies 00:19:20.360 --> 00:19:22.260 align:middle line:90% different play precisely at h. 00:19:22.260 --> 00:19:24.860 align:middle line:84% So we'll maybe say a one-shot deviation from si, 00:19:24.860 --> 00:19:26.860 align:middle line:84% and maybe we'll add at h if we want 00:19:26.860 --> 00:19:29.740 align:middle line:84% to be more precise about which history player i is deviating 00:19:29.740 --> 00:19:30.240 align:middle line:90% at. 00:19:30.240 --> 00:19:36.940 align:middle line:90% 00:19:36.940 --> 00:19:38.440 align:middle line:90% OK. 00:19:38.440 --> 00:19:46.256 align:middle line:90% 00:19:46.256 --> 00:19:47.980 align:middle line:84% So now I think the final ingredient 00:19:47.980 --> 00:19:51.180 align:middle line:84% we need before we can state the one-shot deviation 00:19:51.180 --> 00:19:54.980 align:middle line:84% principle for multistage games is to keep track of, well, 00:19:54.980 --> 00:20:01.040 align:middle line:84% what are the subgames of a multistage game? 00:20:01.040 --> 00:20:09.995 align:middle line:90% 00:20:09.995 --> 00:20:11.620 align:middle line:84% Remember, that's always the first step. 00:20:11.620 --> 00:20:14.027 align:middle line:84% When you want to try to identify or solve 00:20:14.027 --> 00:20:16.360 align:middle line:84% for a subgame-perfect Nash equilibrium, you have to say, 00:20:16.360 --> 00:20:17.605 align:middle line:90% what's the set of subgames? 00:20:17.605 --> 00:20:18.480 align:middle line:90% Any thoughts on this? 00:20:18.480 --> 00:20:20.380 align:middle line:84% What are the subgames going to be? 00:20:20.380 --> 00:20:22.120 align:middle line:84% Remember, to find subgames, you just 00:20:22.120 --> 00:20:25.480 align:middle line:84% have to find nodes in the game that can start a subgame. 00:20:25.480 --> 00:20:27.920 align:middle line:84% So what are the set of possible subgames 00:20:27.920 --> 00:20:29.297 align:middle line:90% in this multistage game? 00:20:29.297 --> 00:20:31.880 align:middle line:84% And it may be helpful to think we know we studied this Cournot 00:20:31.880 --> 00:20:32.860 align:middle line:90% entry game on Tuesday. 00:20:32.860 --> 00:20:35.480 align:middle line:84% That was a multistage game, and we identified the subgames 00:20:35.480 --> 00:20:37.080 align:middle line:84% of that game, so that might guide you 00:20:37.080 --> 00:20:39.060 align:middle line:84% about what the subgames are in this game. 00:20:39.060 --> 00:20:42.440 align:middle line:90% 00:20:42.440 --> 00:20:44.600 align:middle line:84% Well, let's go through step by step. 00:20:44.600 --> 00:20:47.120 align:middle line:90% What about in stage 0? 00:20:47.120 --> 00:20:49.580 align:middle line:84% How many subgames are there that start in stage 0? 00:20:49.580 --> 00:20:53.280 align:middle line:90% 00:20:53.280 --> 00:20:55.440 align:middle line:84% There's just one-- the entire game. 00:20:55.440 --> 00:20:56.660 align:middle line:90% We start in stage 0. 00:20:56.660 --> 00:20:58.320 align:middle line:90% Nothing has happened so far. 00:20:58.320 --> 00:21:00.380 align:middle line:90% So this is just the whole game. 00:21:00.380 --> 00:21:04.000 align:middle line:90% 00:21:04.000 --> 00:21:05.300 align:middle line:90% What about in stage 1? 00:21:05.300 --> 00:21:10.880 align:middle line:90% 00:21:10.880 --> 00:21:13.740 align:middle line:84% How many subgames start in stage 1? 00:21:13.740 --> 00:21:19.780 align:middle line:90% 00:21:19.780 --> 00:21:21.620 align:middle line:84% Well, remember, in the Cournot game, 00:21:21.620 --> 00:21:25.300 align:middle line:84% we had a subgame associated with every subset of firms that could 00:21:25.300 --> 00:21:27.180 align:middle line:90% enter in the previous stage. 00:21:27.180 --> 00:21:30.780 align:middle line:84% And it's going to be the same idea here that, in stage 1, 00:21:30.780 --> 00:21:33.620 align:middle line:84% every history of play in stage 0 is 00:21:33.620 --> 00:21:36.860 align:middle line:84% going to initiate a new subgame starting in stage 1. 00:21:36.860 --> 00:21:45.600 align:middle line:84% So stage 1, we have a subgame for each stage 0 history. 00:21:45.600 --> 00:21:48.700 align:middle line:90% 00:21:48.700 --> 00:21:51.500 align:middle line:84% That is, however people played in stage 0, 00:21:51.500 --> 00:21:54.160 align:middle line:84% we all know that's how people played because we observe it. 00:21:54.160 --> 00:21:56.580 align:middle line:84% And now we have a subgame starting at that history of play 00:21:56.580 --> 00:21:59.260 align:middle line:90% from here on from here onwards. 00:21:59.260 --> 00:22:00.560 align:middle line:90% What about in stage 2? 00:22:00.560 --> 00:22:07.780 align:middle line:90% 00:22:07.780 --> 00:22:13.180 align:middle line:84% Maybe I'll call this h0, where h0 says 00:22:13.180 --> 00:22:15.540 align:middle line:90% how people played in stage 0. 00:22:15.540 --> 00:22:16.993 align:middle line:90% What about in stage 2? 00:22:16.993 --> 00:22:18.660 align:middle line:84% How many subgames do we have in stage 2? 00:22:18.660 --> 00:22:21.280 align:middle line:84% Now it's getting a little more complicated. 00:22:21.280 --> 00:22:23.960 align:middle line:84% Well, what do we know in stage 2? 00:22:23.960 --> 00:22:26.640 align:middle line:84% We know how people played in stage 0. 00:22:26.640 --> 00:22:28.800 align:middle line:84% And we know how people played in stage 1. 00:22:28.800 --> 00:22:32.360 align:middle line:84% So any history of play in stage 0 and in stage 1 00:22:32.360 --> 00:22:35.120 align:middle line:84% is going to start a subgame in stage 2. 00:22:35.120 --> 00:22:39.080 align:middle line:84% So now we have this for each, maybe I'll say, history. 00:22:39.080 --> 00:22:42.900 align:middle line:84% Maybe I'll call it h0, h1 that says, 00:22:42.900 --> 00:22:45.220 align:middle line:84% this is how we played in stage 0. 00:22:45.220 --> 00:22:46.940 align:middle line:84% This is how we played in stage 1, 00:22:46.940 --> 00:22:49.440 align:middle line:84% and now we're going to move on to stage 2. 00:22:49.440 --> 00:22:54.160 align:middle line:84% And in general, we're going to see that, in stage t, 00:22:54.160 --> 00:22:59.480 align:middle line:84% we're going to have a much more complicated history h0, h1, up 00:22:59.480 --> 00:23:02.280 align:middle line:90% to ht minus 1. 00:23:02.280 --> 00:23:05.760 align:middle line:84% So at stage t, we know how everyone played up 00:23:05.760 --> 00:23:08.120 align:middle line:90% to stage t minus 1. 00:23:08.120 --> 00:23:10.120 align:middle line:84% And since we know that, that's going 00:23:10.120 --> 00:23:14.240 align:middle line:84% to start a new subgame from stage t onward. 00:23:14.240 --> 00:23:17.660 align:middle line:84% So we see there's going to be a lot of histories. 00:23:17.660 --> 00:23:20.160 align:middle line:90% And it's going to be helpful. 00:23:20.160 --> 00:23:21.900 align:middle line:84% We'll just use a generic term history 00:23:21.900 --> 00:23:23.480 align:middle line:90% to mean a history at any point. 00:23:23.480 --> 00:23:25.180 align:middle line:84% So it could be a history here that 00:23:25.180 --> 00:23:26.737 align:middle line:90% just says how we played in h0. 00:23:26.737 --> 00:23:29.320 align:middle line:84% It could be a history here that says how we played in 0 and 1. 00:23:29.320 --> 00:23:31.480 align:middle line:84% It could be a history here, how we played up in here. 00:23:31.480 --> 00:23:32.460 align:middle line:84% But we'll look at all those together, 00:23:32.460 --> 00:23:33.877 align:middle line:84% and we'll just look at arbitrary-- 00:23:33.877 --> 00:23:36.600 align:middle line:84% we'll call something a history if it's a history at any stage. 00:23:36.600 --> 00:23:39.900 align:middle line:90% 00:23:39.900 --> 00:23:42.460 align:middle line:84% So what are the subgames of a multistage game? 00:23:42.460 --> 00:23:47.940 align:middle line:84% They are described by the histories of the game. 00:23:47.940 --> 00:23:49.000 align:middle line:90% So we have the-- 00:23:49.000 --> 00:23:52.460 align:middle line:90% 00:23:52.460 --> 00:23:54.060 align:middle line:90% maybe I'll call it the-- 00:23:54.060 --> 00:24:01.760 align:middle line:84% we have an h subgame for every history h. 00:24:01.760 --> 00:24:06.100 align:middle line:90% 00:24:06.100 --> 00:24:08.340 align:middle line:90% So give me a history h. 00:24:08.340 --> 00:24:10.150 align:middle line:90% It could be a history like this. 00:24:10.150 --> 00:24:11.400 align:middle line:90% It could be history like this. 00:24:11.400 --> 00:24:12.650 align:middle line:90% It could be history like this. 00:24:12.650 --> 00:24:15.080 align:middle line:84% That's going to start a subgame from here on out. 00:24:15.080 --> 00:24:19.020 align:middle line:84% And conversely, any subgame is exactly going to have this form. 00:24:19.020 --> 00:24:21.640 align:middle line:84% So we have an association between the subgames 00:24:21.640 --> 00:24:25.760 align:middle line:84% of a multistage game and the histories of play in the game. 00:24:25.760 --> 00:24:34.420 align:middle line:84% Just so we can make this concrete, in this example, 00:24:34.420 --> 00:24:37.520 align:middle line:84% we had one subgame starting in stage 0, which 00:24:37.520 --> 00:24:39.680 align:middle line:84% was the subgame following the null history where 00:24:39.680 --> 00:24:41.320 align:middle line:90% nothing had happened. 00:24:41.320 --> 00:24:43.040 align:middle line:84% And then we actually had one subgame 00:24:43.040 --> 00:24:46.720 align:middle line:84% starting in stage 1 following the single history n. 00:24:46.720 --> 00:24:49.160 align:middle line:84% And that was the only possible history at which we 00:24:49.160 --> 00:24:50.880 align:middle line:90% were called to play in stage 1. 00:24:50.880 --> 00:24:53.620 align:middle line:84% So that's how we map it into this example over here. 00:24:53.620 --> 00:24:58.520 align:middle line:90% 00:24:58.520 --> 00:25:02.520 align:middle line:84% OK, so now that we've stated that we can now 00:25:02.520 --> 00:25:04.980 align:middle line:84% give our theorem on the one-shot deviation principle. 00:25:04.980 --> 00:25:08.160 align:middle line:90% 00:25:08.160 --> 00:25:11.660 align:middle line:84% So let's consider a multistage game. 00:25:11.660 --> 00:25:19.660 align:middle line:90% 00:25:19.660 --> 00:25:30.580 align:middle line:84% You say, for any strategy profile s star, 00:25:30.580 --> 00:25:31.840 align:middle line:90% the following are equivalent. 00:25:31.840 --> 00:25:41.100 align:middle line:90% 00:25:41.100 --> 00:25:43.740 align:middle line:84% So we're going to list some statements about a strategy 00:25:43.740 --> 00:25:45.900 align:middle line:84% profile s star that are equivalent. 00:25:45.900 --> 00:25:49.580 align:middle line:84% So all the statements hold, or none of the statements hold. 00:25:49.580 --> 00:25:52.620 align:middle line:84% So the first statement is what we're interested in. 00:25:52.620 --> 00:25:56.700 align:middle line:84% It's that s star is a subgame-perfect Nash 00:25:56.700 --> 00:25:58.940 align:middle line:90% equilibrium. 00:25:58.940 --> 00:26:00.440 align:middle line:84% So this is what we're interested in. 00:26:00.440 --> 00:26:03.340 align:middle line:84% You give me a strategy s star, a strategy profile s star, 00:26:03.340 --> 00:26:05.820 align:middle line:84% we want to understand, is this strategy profile 00:26:05.820 --> 00:26:08.580 align:middle line:84% a subgame-perfect Nash equilibrium? 00:26:08.580 --> 00:26:10.140 align:middle line:84% Now, before we really do anything, 00:26:10.140 --> 00:26:12.120 align:middle line:84% let's just understand the definition 00:26:12.120 --> 00:26:13.860 align:middle line:84% of a subgame-perfect Nash equilibrium. 00:26:13.860 --> 00:26:16.360 align:middle line:84% So the first step is going to be pretty easy. 00:26:16.360 --> 00:26:18.760 align:middle line:84% I just need to apply the definition 00:26:18.760 --> 00:26:21.560 align:middle line:84% of subgame-perfect Nash equilibrium 00:26:21.560 --> 00:26:24.383 align:middle line:84% to this specific multistage game. 00:26:24.383 --> 00:26:25.800 align:middle line:84% So we know that, what does it mean 00:26:25.800 --> 00:26:27.440 align:middle line:84% for s star to be a subgame-perfect Nash 00:26:27.440 --> 00:26:28.280 align:middle line:90% equilibrium? 00:26:28.280 --> 00:26:31.800 align:middle line:84% It means, in any subgame, the restriction of the strategy 00:26:31.800 --> 00:26:35.000 align:middle line:84% profile to that subgame constitutes a Nash equilibrium 00:26:35.000 --> 00:26:36.160 align:middle line:90% in that subgame. 00:26:36.160 --> 00:26:38.080 align:middle line:84% But now let's use our observation 00:26:38.080 --> 00:26:40.560 align:middle line:84% over here about the structure of the subgames. 00:26:40.560 --> 00:26:42.900 align:middle line:84% So instead of making a statement about every subgame, 00:26:42.900 --> 00:26:44.840 align:middle line:84% we can make a statement about every history 00:26:44.840 --> 00:26:49.380 align:middle line:84% because a subgame is basically equivalent to a history. 00:26:49.380 --> 00:26:52.900 align:middle line:84% So we'll say if and only if, for every history h-- 00:26:52.900 --> 00:26:58.080 align:middle line:90% 00:26:58.080 --> 00:27:01.000 align:middle line:84% well, what we want to say is that the restriction 00:27:01.000 --> 00:27:06.040 align:middle line:84% of s star to the h subgame is a Nash equilibrium of the h 00:27:06.040 --> 00:27:06.840 align:middle line:90% subgame. 00:27:06.840 --> 00:27:09.240 align:middle line:84% So we'll say, for every history h, 00:27:09.240 --> 00:27:20.580 align:middle line:84% s star h is a Nash equilibrium of the h subgame, 00:27:20.580 --> 00:27:22.380 align:middle line:90% where what is the h subgame? 00:27:22.380 --> 00:27:23.800 align:middle line:84% This is what I described up here. 00:27:23.800 --> 00:27:26.980 align:middle line:84% It's the subgame that starts at history h, where 00:27:26.980 --> 00:27:30.180 align:middle line:84% the players have observed the path of play along this history 00:27:30.180 --> 00:27:31.720 align:middle line:90% h up to the current stage. 00:27:31.720 --> 00:27:34.540 align:middle line:84% And then they can choose how to behave from here onward. 00:27:34.540 --> 00:27:36.020 align:middle line:90% What is s star h? 00:27:36.020 --> 00:27:42.900 align:middle line:84% This is notation for the restriction of s star 00:27:42.900 --> 00:27:44.040 align:middle line:90% to the h subgame. 00:27:44.040 --> 00:27:47.380 align:middle line:90% 00:27:47.380 --> 00:27:50.620 align:middle line:84% Remember, we can't say that s star is a Nash equilibrium 00:27:50.620 --> 00:27:53.620 align:middle line:84% of the h subgame because s star is not a strategy 00:27:53.620 --> 00:27:55.160 align:middle line:90% profile in the h subgame. 00:27:55.160 --> 00:27:57.300 align:middle line:84% It specifies play at all these histories 00:27:57.300 --> 00:27:58.940 align:middle line:90% that may not be in that subgame. 00:27:58.940 --> 00:28:02.260 align:middle line:84% So here we just focus on the part of s star 00:28:02.260 --> 00:28:04.340 align:middle line:84% or the part of the strategy in s star 00:28:04.340 --> 00:28:09.010 align:middle line:84% that specify behavior within this particular subgame. 00:28:09.010 --> 00:28:11.070 align:middle line:84% So so far, we haven't really done anything. 00:28:11.070 --> 00:28:13.690 align:middle line:84% So far, we've just stated the definition 00:28:13.690 --> 00:28:15.530 align:middle line:84% of subgame-perfect Nash equilibrium 00:28:15.530 --> 00:28:18.770 align:middle line:84% in this particular context where the class of subgames 00:28:18.770 --> 00:28:20.730 align:middle line:90% has a certain structure. 00:28:20.730 --> 00:28:21.422 align:middle line:90% Any questions? 00:28:21.422 --> 00:28:23.130 align:middle line:84% I think this lecture can be a little more 00:28:23.130 --> 00:28:25.710 align:middle line:84% abstract than some of the other ones, so I want to check in. 00:28:25.710 --> 00:28:31.108 align:middle line:84% Any questions about this, about what any of the words mean? 00:28:31.108 --> 00:28:33.150 align:middle line:84% It's great if people come after class and ask me, 00:28:33.150 --> 00:28:36.947 align:middle line:84% but if you ask me in class, you can help your classmates as well 00:28:36.947 --> 00:28:38.030 align:middle line:90% who might be confused too. 00:28:38.030 --> 00:28:38.370 align:middle line:90% Yeah. 00:28:38.370 --> 00:28:40.953 align:middle line:84% AUDIENCE: What is the difference between these two statements? 00:28:40.953 --> 00:28:43.270 align:middle line:84% IAN BALL: Oh, we're getting to the real results later. 00:28:43.270 --> 00:28:44.630 align:middle line:90% So this is just preparation. 00:28:44.630 --> 00:28:48.170 align:middle line:84% So, yeah, if you think this is obvious, that's fine. 00:28:48.170 --> 00:28:50.870 align:middle line:84% It's basically, once you understand the definitions, 00:28:50.870 --> 00:28:53.490 align:middle line:84% yeah, we haven't gone much-- so the heart of the theorem 00:28:53.490 --> 00:28:54.470 align:middle line:90% is going to be 3. 00:28:54.470 --> 00:29:04.750 align:middle line:90% 00:29:04.750 --> 00:29:07.652 align:middle line:84% So we're, again, not going to do much in 0.3. 00:29:07.652 --> 00:29:09.110 align:middle line:84% We're just going to break down what 00:29:09.110 --> 00:29:14.670 align:middle line:84% it means for s star h be a Nash equilibrium of the h subgame. 00:29:14.670 --> 00:29:17.950 align:middle line:84% So this means that, for every history h-- 00:29:17.950 --> 00:29:19.250 align:middle line:90% I'll just use this-- 00:29:19.250 --> 00:29:21.870 align:middle line:90% 00:29:21.870 --> 00:29:38.670 align:middle line:84% no player has a profitable unilateral deviation 00:29:38.670 --> 00:29:42.370 align:middle line:90% from s star h in the h subgame. 00:29:42.370 --> 00:29:54.950 align:middle line:90% 00:29:54.950 --> 00:29:57.150 align:middle line:84% So here, I still haven't really done anything. 00:29:57.150 --> 00:30:00.890 align:middle line:84% I've just written out the definition of Nash equilibrium. 00:30:00.890 --> 00:30:03.130 align:middle line:84% What does it mean for s star h to be a Nash 00:30:03.130 --> 00:30:04.530 align:middle line:90% equilibrium of the h subgame? 00:30:04.530 --> 00:30:06.610 align:middle line:84% It just means that in the h subgame, 00:30:06.610 --> 00:30:09.330 align:middle line:84% no player can profitably deviate unilaterally 00:30:09.330 --> 00:30:10.793 align:middle line:90% from the strategy profile. 00:30:10.793 --> 00:30:12.210 align:middle line:84% If a player starts at the strategy 00:30:12.210 --> 00:30:13.770 align:middle line:84% they're supposed to play, and they 00:30:13.770 --> 00:30:15.530 align:middle line:84% deviate to a different strategy, that 00:30:15.530 --> 00:30:18.010 align:middle line:84% can't be profitable within the subgame, just the definition 00:30:18.010 --> 00:30:19.298 align:middle line:90% of Nash equilibrium. 00:30:19.298 --> 00:30:21.090 align:middle line:84% So what makes this a theorem and what makes 00:30:21.090 --> 00:30:22.470 align:middle line:90% this hard is the final step. 00:30:22.470 --> 00:30:25.070 align:middle line:84% But I just want to make clear the structure of this. 00:30:25.070 --> 00:30:29.770 align:middle line:90% So now comes the actual result. 00:30:29.770 --> 00:30:32.290 align:middle line:84% And what it says is, at every history, 00:30:32.290 --> 00:30:54.530 align:middle line:84% no player has a profitable unilateral one-shot deviation 00:30:54.530 --> 00:30:55.550 align:middle line:90% at history h. 00:30:55.550 --> 00:30:58.250 align:middle line:90% 00:30:58.250 --> 00:31:04.630 align:middle line:84% One-shot deviation, maybe I'll say, from s star h 00:31:04.630 --> 00:31:06.910 align:middle line:90% at history h in the h subgame. 00:31:06.910 --> 00:31:12.470 align:middle line:90% 00:31:12.470 --> 00:31:14.630 align:middle line:84% So this is where the theorem comes in. 00:31:14.630 --> 00:31:16.830 align:middle line:84% Up here, we're just saying no player 00:31:16.830 --> 00:31:19.770 align:middle line:84% has a profitable deviation of any kind. 00:31:19.770 --> 00:31:21.710 align:middle line:84% It could be a really complicated deviation 00:31:21.710 --> 00:31:24.910 align:middle line:84% where they deviate at many, many histories. 00:31:24.910 --> 00:31:27.950 align:middle line:84% Statement four says, in fact, no player 00:31:27.950 --> 00:31:31.790 align:middle line:84% has a profitable one-shot deviation at history h. 00:31:31.790 --> 00:31:33.870 align:middle line:84% And that means they're going to deviate-- 00:31:33.870 --> 00:31:35.430 align:middle line:84% we're only considering deviations 00:31:35.430 --> 00:31:38.810 align:middle line:84% to strategies that differ from their equilibrium strategy 00:31:38.810 --> 00:31:40.430 align:middle line:90% exactly at history h. 00:31:40.430 --> 00:31:45.710 align:middle line:84% And at every other history, it specifies the same behavior. 00:31:45.710 --> 00:31:52.390 align:middle line:84% So let's just understand one direction of this is easy. 00:31:52.390 --> 00:31:57.210 align:middle line:84% So I think the equivalence of 1 through 3-- 00:31:57.210 --> 00:31:59.890 align:middle line:90% 00:31:59.890 --> 00:32:02.230 align:middle line:84% 1, 2, and 3 are equivalent by definition. 00:32:02.230 --> 00:32:08.770 align:middle line:90% 00:32:08.770 --> 00:32:11.583 align:middle line:84% The equivalence of 1 and 2 follows 00:32:11.583 --> 00:32:13.250 align:middle line:84% is just a statement about the definition 00:32:13.250 --> 00:32:15.390 align:middle line:84% of subgame-perfect Nash equilibrium 00:32:15.390 --> 00:32:18.330 align:middle line:84% and the structure of the subgames. 00:32:18.330 --> 00:32:20.970 align:middle line:84% The equivalence of 2 and 3 is just 00:32:20.970 --> 00:32:23.243 align:middle line:84% the statement of the definition of Nash equilibrium. 00:32:23.243 --> 00:32:25.410 align:middle line:84% We're just breaking down what it means for something 00:32:25.410 --> 00:32:26.830 align:middle line:90% to be a Nash equilibrium. 00:32:26.830 --> 00:32:30.290 align:middle line:84% It means no player has a profitable unilateral deviation. 00:32:30.290 --> 00:32:36.030 align:middle line:84% And then the implication from 3 to 4 is also immediate. 00:32:36.030 --> 00:32:37.530 align:middle line:90% Why? 00:32:37.530 --> 00:32:40.236 align:middle line:90% 00:32:40.236 --> 00:32:44.290 align:middle line:90% Why is this immediate? 00:32:44.290 --> 00:32:48.730 align:middle line:90% 00:32:48.730 --> 00:32:49.702 align:middle line:90% Yeah. 00:32:49.702 --> 00:32:51.410 align:middle line:84% AUDIENCE: Because a one-shot deviation is 00:32:51.410 --> 00:32:52.950 align:middle line:90% a type of unilateral deviation. 00:32:52.950 --> 00:32:55.867 align:middle line:84% IAN BALL: Exactly, if no deviation is profitable, 00:32:55.867 --> 00:32:58.450 align:middle line:84% if I already know that there's no deviation that's profitable, 00:32:58.450 --> 00:33:01.075 align:middle line:84% well, certainly, there can't be a profitable one-shot deviation 00:33:01.075 --> 00:33:03.390 align:middle line:84% because that's just a special kind of deviation. 00:33:03.390 --> 00:33:06.095 align:middle line:84% So again, not a trick question, just really easy. 00:33:06.095 --> 00:33:07.470 align:middle line:84% If we know nothing is profitable, 00:33:07.470 --> 00:33:09.178 align:middle line:84% then certainly this special kind of thing 00:33:09.178 --> 00:33:11.650 align:middle line:90% can't be profitable, just logic. 00:33:11.650 --> 00:33:14.630 align:middle line:84% So the heart of the theorem, which we're not even 00:33:14.630 --> 00:33:17.690 align:middle line:84% going to prove because it's actually quite tricky is 4 to 3. 00:33:17.690 --> 00:33:23.250 align:middle line:84% This is I don't know, hard, important, substantive I'll say. 00:33:23.250 --> 00:33:26.750 align:middle line:90% 00:33:26.750 --> 00:33:30.450 align:middle line:84% Because 3 to 4 is clear-- if no deviation is profitable, 00:33:30.450 --> 00:33:32.950 align:middle line:84% then clearly a one-shot deviation can't be profitable. 00:33:32.950 --> 00:33:35.410 align:middle line:84% But the other direction is not clear. 00:33:35.410 --> 00:33:38.150 align:middle line:84% Just because I can't profit with a one-shot deviation, 00:33:38.150 --> 00:33:39.670 align:middle line:84% how do I know that I can't profit 00:33:39.670 --> 00:33:42.030 align:middle line:84% with a much more complicated deviation? 00:33:42.030 --> 00:33:44.910 align:middle line:84% So the idea is if a simple deviation is not profitable, 00:33:44.910 --> 00:33:47.190 align:middle line:84% where I'm using simple colloquially, 00:33:47.190 --> 00:33:50.230 align:middle line:84% then a complicated deviation also can't be profitable. 00:33:50.230 --> 00:33:52.750 align:middle line:84% Notice that was exactly the issue 00:33:52.750 --> 00:33:56.030 align:middle line:84% when we talked here about complicated deviations. 00:33:56.030 --> 00:33:59.530 align:middle line:84% We observed that, in the last class, we actually only checked, 00:33:59.530 --> 00:34:01.077 align:middle line:84% quote, unquote, "simple deviations." 00:34:01.077 --> 00:34:03.410 align:middle line:84% We didn't check that more complicated deviations weren't 00:34:03.410 --> 00:34:04.250 align:middle line:90% profitable. 00:34:04.250 --> 00:34:08.067 align:middle line:84% But I argued that, intuitively, if the simple deviations aren't 00:34:08.067 --> 00:34:10.650 align:middle line:84% profitable, then we know that the more complicated ones aren't 00:34:10.650 --> 00:34:11.389 align:middle line:90% profitable. 00:34:11.389 --> 00:34:13.190 align:middle line:84% And that's exactly the same logic here, 00:34:13.190 --> 00:34:15.815 align:middle line:84% but it's just mathematically a lot more tricky because the game 00:34:15.815 --> 00:34:17.110 align:middle line:90% can possibly go on forever. 00:34:17.110 --> 00:34:21.094 align:middle line:84% But the substantive idea is still there. 00:34:21.094 --> 00:34:22.469 align:middle line:84% So we're not going to prove this. 00:34:22.469 --> 00:34:25.610 align:middle line:90% 00:34:25.610 --> 00:34:28.530 align:middle line:90% But it's true. 00:34:28.530 --> 00:34:32.978 align:middle line:84% And what it means is that it becomes much easier to check 00:34:32.978 --> 00:34:34.770 align:middle line:84% whether something is a subgame-perfect Nash 00:34:34.770 --> 00:34:37.130 align:middle line:84% equilibrium, so given this theorem, 00:34:37.130 --> 00:34:38.710 align:middle line:84% if I give you a strategy profile, 00:34:38.710 --> 00:34:40.393 align:middle line:84% say in a quiz or an exam or a game, 00:34:40.393 --> 00:34:42.810 align:middle line:84% and you want to understand, is this a subgame-perfect Nash 00:34:42.810 --> 00:34:45.929 align:middle line:84% equilibrium, you don't have to check that there's-- that no 00:34:45.929 --> 00:34:47.310 align:middle line:90% deviation is profitable. 00:34:47.310 --> 00:34:50.070 align:middle line:84% You only have to worry about the one-shot deviations. 00:34:50.070 --> 00:34:52.929 align:middle line:84% So it allows you-- so what's the upshot? 00:34:52.929 --> 00:34:56.949 align:middle line:90% 00:34:56.949 --> 00:35:00.510 align:middle line:84% The upshot is that, to verify a subgame-perfect Nash 00:35:00.510 --> 00:35:01.370 align:middle line:90% equilibrium. 00:35:01.370 --> 00:35:03.990 align:middle line:90% 00:35:03.990 --> 00:35:06.710 align:middle line:84% To verify that a strategy profile is an SPNE, 00:35:06.710 --> 00:35:08.970 align:middle line:84% it's enough to check the one-shot deviations. 00:35:08.970 --> 00:35:17.390 align:middle line:90% 00:35:17.390 --> 00:35:21.330 align:middle line:84% That one-shot deviations are not profitable. 00:35:21.330 --> 00:35:27.222 align:middle line:90% 00:35:27.222 --> 00:35:29.430 align:middle line:84% Because if I know that the one-shot deviations aren't 00:35:29.430 --> 00:35:32.110 align:middle line:84% profitable, the theorem tells me that no deviations are 00:35:32.110 --> 00:35:35.710 align:middle line:90% profitable, and then I'm done. 00:35:35.710 --> 00:35:38.030 align:middle line:90% Now, that sounds nice. 00:35:38.030 --> 00:35:40.390 align:middle line:84% We only have to check the one-shot deviations, 00:35:40.390 --> 00:35:42.590 align:middle line:84% but this is actually still quite complicated 00:35:42.590 --> 00:35:45.230 align:middle line:84% because we have to check the one-shot deviations still 00:35:45.230 --> 00:35:47.350 align:middle line:90% in every subgame. 00:35:47.350 --> 00:35:50.230 align:middle line:84% So checking one-shot deviations may not be so hard, but notice, 00:35:50.230 --> 00:35:52.770 align:middle line:84% we still have the quantifier for every history. 00:35:52.770 --> 00:35:54.895 align:middle line:84% So, yes, we only have to check one-shot deviations, 00:35:54.895 --> 00:35:56.937 align:middle line:84% but we have to check them at every single history 00:35:56.937 --> 00:35:57.650 align:middle line:90% of the game. 00:35:57.650 --> 00:36:00.910 align:middle line:84% And in general, doing this may be complicated. 00:36:00.910 --> 00:36:04.650 align:middle line:90% 00:36:04.650 --> 00:36:10.730 align:middle line:84% I should maybe be a little more precise. 00:36:10.730 --> 00:36:15.570 align:middle line:84% I don't think this is a big issue in this class. 00:36:15.570 --> 00:36:18.750 align:middle line:84% And maybe this demonstrates that this step 4 to 3 is not obvious. 00:36:18.750 --> 00:36:20.130 align:middle line:84% You need one technical assumption 00:36:20.130 --> 00:36:21.530 align:middle line:90% for this theorem to be true. 00:36:21.530 --> 00:36:24.410 align:middle line:84% So I said in any multistage game. 00:36:24.410 --> 00:36:28.290 align:middle line:84% I should really say in any multistage game-- 00:36:28.290 --> 00:36:34.350 align:middle line:84% I should really say in any continuous multistage game. 00:36:34.350 --> 00:36:36.850 align:middle line:84% We're not really going to deal with discontinuous multistage 00:36:36.850 --> 00:36:37.750 align:middle line:90% games in this class. 00:36:37.750 --> 00:36:39.047 align:middle line:90% But you can look at the notes. 00:36:39.047 --> 00:36:41.130 align:middle line:84% I think they say a little bit about the definition 00:36:41.130 --> 00:36:42.150 align:middle line:90% of continuous. 00:36:42.150 --> 00:36:46.290 align:middle line:84% Continuous just means if the game is infinite, 00:36:46.290 --> 00:36:49.310 align:middle line:84% if everyone plays exactly the same way a billion periods 00:36:49.310 --> 00:36:52.830 align:middle line:84% into the game, then how they play beyond a billion periods 00:36:52.830 --> 00:36:55.030 align:middle line:84% shouldn't have a big effect on their payoffs? 00:36:55.030 --> 00:36:59.830 align:middle line:84% So basically, if we want to understand 00:36:59.830 --> 00:37:01.422 align:middle line:84% what people's payoffs are in the game, 00:37:01.422 --> 00:37:03.630 align:middle line:84% we just have to look at how they play, say, a billion 00:37:03.630 --> 00:37:04.630 align:middle line:84% periods into the game, and that'll 00:37:04.630 --> 00:37:05.950 align:middle line:84% give us a pretty good approximation 00:37:05.950 --> 00:37:07.242 align:middle line:90% for what their payoffs will be. 00:37:07.242 --> 00:37:10.630 align:middle line:84% It can't be that things, 2 trillion periods into the future 00:37:10.630 --> 00:37:11.850 align:middle line:90% determine everything. 00:37:11.850 --> 00:37:14.510 align:middle line:84% So a classic example of what will satisfy this 00:37:14.510 --> 00:37:17.470 align:middle line:84% is if we just sum up the payoffs in each period with a discount 00:37:17.470 --> 00:37:20.350 align:middle line:84% factor because then things that happen really far in the future 00:37:20.350 --> 00:37:21.333 align:middle line:90% get discounted a lot. 00:37:21.333 --> 00:37:23.250 align:middle line:84% And therefore, they don't play much of a role. 00:37:23.250 --> 00:37:25.962 align:middle line:84% But this is kind of a technical thing that won't be covered in-- 00:37:25.962 --> 00:37:28.170 align:middle line:84% everything we study in this class will be continuous. 00:37:28.170 --> 00:37:29.628 align:middle line:84% You don't have to worry about this, 00:37:29.628 --> 00:37:33.670 align:middle line:84% but I'm happy to talk after if you have questions about this. 00:37:33.670 --> 00:37:35.170 align:middle line:90% OK. 00:37:35.170 --> 00:37:38.710 align:middle line:90% 00:37:38.710 --> 00:37:40.710 align:middle line:84% If you like math, I think this is a nice theorem 00:37:40.710 --> 00:37:42.330 align:middle line:84% to think about and think about how you would prove it, 00:37:42.330 --> 00:37:43.830 align:middle line:84% but that's not going to be something 00:37:43.830 --> 00:37:45.670 align:middle line:90% we'll cover in this class. 00:37:45.670 --> 00:37:48.210 align:middle line:84% OK, so now the question is, how do we use this theorem? 00:37:48.210 --> 00:37:51.450 align:middle line:84% And we're going to use this theorem 00:37:51.450 --> 00:37:54.370 align:middle line:84% to analyze a classical game, just like we talked about 00:37:54.370 --> 00:37:57.210 align:middle line:84% before of price haggling but now with a potentially 00:37:57.210 --> 00:37:58.230 align:middle line:90% infinite horizon. 00:37:58.230 --> 00:38:17.250 align:middle line:90% 00:38:17.250 --> 00:38:19.010 align:middle line:84% So now we're going to look at maybe 00:38:19.010 --> 00:38:28.490 align:middle line:84% what I'll call infinite-horizon alternating-offer bargaining. 00:38:28.490 --> 00:38:41.170 align:middle line:90% 00:38:41.170 --> 00:38:43.650 align:middle line:84% So let's just go through the words. 00:38:43.650 --> 00:38:45.330 align:middle line:84% Let's start with alternating offer. 00:38:45.330 --> 00:38:47.038 align:middle line:84% This is just like we talked about before. 00:38:47.038 --> 00:38:49.410 align:middle line:84% We have two parties there negotiating or bargaining-- 00:38:49.410 --> 00:38:51.350 align:middle line:84% I'll use these terms interchangeably-- 00:38:51.350 --> 00:38:53.083 align:middle line:90% over something. 00:38:53.083 --> 00:38:54.250 align:middle line:90% One side comes to the table. 00:38:54.250 --> 00:38:55.750 align:middle line:90% They say, here's my proposal. 00:38:55.750 --> 00:38:58.477 align:middle line:84% The other side says either, great, we're done. 00:38:58.477 --> 00:39:00.810 align:middle line:84% Or the other side says, no, I'm rejecting your proposal. 00:39:00.810 --> 00:39:03.110 align:middle line:84% I'm going to come back with my proposal. 00:39:03.110 --> 00:39:05.015 align:middle line:84% And then the other side makes a proposal. 00:39:05.015 --> 00:39:06.890 align:middle line:84% Now, that can either be accepted or rejected, 00:39:06.890 --> 00:39:07.730 align:middle line:90% and then we go again. 00:39:07.730 --> 00:39:09.090 align:middle line:84% And we keep alternating our offers. 00:39:09.090 --> 00:39:11.340 align:middle line:84% You might see this if you're at a used car dealership. 00:39:11.340 --> 00:39:14.170 align:middle line:84% You might say, I'll pay $10,000 for the car. 00:39:14.170 --> 00:39:15.725 align:middle line:84% The dealer says, no, it's 15,000. 00:39:15.725 --> 00:39:16.850 align:middle line:90% You say, what about 12,000? 00:39:16.850 --> 00:39:18.290 align:middle line:90% He says, oh, what about 14,000? 00:39:18.290 --> 00:39:20.950 align:middle line:90% And you converge. 00:39:20.950 --> 00:39:23.310 align:middle line:84% Infinite horizon means, well, we don't literally 00:39:23.310 --> 00:39:25.570 align:middle line:84% think that negotiations will go on forever, 00:39:25.570 --> 00:39:27.410 align:middle line:84% but there's no clear last period. 00:39:27.410 --> 00:39:29.470 align:middle line:84% Whenever we're negotiating, it's always possible 00:39:29.470 --> 00:39:31.790 align:middle line:84% that the offer can be rejected, and our negotiations 00:39:31.790 --> 00:39:33.735 align:middle line:84% or bargaining could continue one more period. 00:39:33.735 --> 00:39:35.110 align:middle line:84% So the infinite horizon allows us 00:39:35.110 --> 00:39:38.810 align:middle line:84% to rule out these kind of weird last-period effects that we saw, 00:39:38.810 --> 00:39:40.990 align:middle line:84% where we know that this is our very last opportunity 00:39:40.990 --> 00:39:43.470 align:middle line:84% to negotiate, and whoever makes an offer in that period 00:39:43.470 --> 00:39:44.720 align:middle line:90% has a lot of bargaining power. 00:39:44.720 --> 00:39:47.700 align:middle line:84% So we want to shut down that weird last-period effect 00:39:47.700 --> 00:39:49.920 align:middle line:84% by saying, in principle, there's no end date. 00:39:49.920 --> 00:39:53.100 align:middle line:84% The negotiations could just be ongoing. 00:39:53.100 --> 00:39:55.020 align:middle line:84% And we want a very abstract model 00:39:55.020 --> 00:39:56.900 align:middle line:84% of this that can capture not just 00:39:56.900 --> 00:39:59.640 align:middle line:84% haggling over the price of something, but also, 00:39:59.640 --> 00:40:00.980 align:middle line:90% say, in international relations. 00:40:00.980 --> 00:40:02.140 align:middle line:84% Right now, there's some negotiations 00:40:02.140 --> 00:40:03.500 align:middle line:90% going on in the Middle East. 00:40:03.500 --> 00:40:05.420 align:middle line:84% How can we have a very general model that 00:40:05.420 --> 00:40:07.500 align:middle line:90% can capture these negotiations? 00:40:07.500 --> 00:40:09.460 align:middle line:90% So what we want to do. 00:40:09.460 --> 00:40:12.500 align:middle line:84% Well, that's hard because what we're negotiating over 00:40:12.500 --> 00:40:13.600 align:middle line:90% may be very complicated. 00:40:13.600 --> 00:40:20.480 align:middle line:84% So maybe I'll say the terms of dispute may be complicated. 00:40:20.480 --> 00:40:25.340 align:middle line:90% 00:40:25.340 --> 00:40:27.000 align:middle line:90% So how can we represent this? 00:40:27.000 --> 00:40:29.580 align:middle line:84% Well, what's relevant to the parties 00:40:29.580 --> 00:40:32.440 align:middle line:84% is not the details that we're negotiating over, 00:40:32.440 --> 00:40:35.140 align:middle line:84% but the utilities we get from those details. 00:40:35.140 --> 00:40:39.260 align:middle line:84% So we're going to reduce these complicated terms to utilities. 00:40:39.260 --> 00:40:44.685 align:middle line:84% We're going to reduce to maybe utility space. 00:40:44.685 --> 00:40:47.060 align:middle line:84% Now, of course, in reality, this is not how we negotiate. 00:40:47.060 --> 00:40:50.320 align:middle line:84% But we might say, instead of thinking of terms of negotiation 00:40:50.320 --> 00:40:54.580 align:middle line:84% as a 20-point plan, about the terms of some deal, 00:40:54.580 --> 00:40:57.000 align:middle line:84% we might say, well, let's represent that 20-point plan 00:40:57.000 --> 00:40:59.800 align:middle line:84% by the utility it gives to party 1 and the utility it gives 00:40:59.800 --> 00:41:00.463 align:middle line:90% to party 2. 00:41:00.463 --> 00:41:02.380 align:middle line:84% Now, of course, in reality, we don't say this. 00:41:02.380 --> 00:41:03.860 align:middle line:90% We don't say, here's my offer. 00:41:03.860 --> 00:41:05.927 align:middle line:84% You get utility 7, and I get utility 3. 00:41:05.927 --> 00:41:07.260 align:middle line:90% No one actually speaks that way. 00:41:07.260 --> 00:41:10.480 align:middle line:84% But what we're saying is the strategically relevant aspect 00:41:10.480 --> 00:41:13.720 align:middle line:84% of the terms of some offer is simply how much utility 00:41:13.720 --> 00:41:15.040 align:middle line:90% each party gets. 00:41:15.040 --> 00:41:18.180 align:middle line:84% And we can just treat the offers-- in utility space, 00:41:18.180 --> 00:41:20.080 align:middle line:84% we can represent them by the utilities 00:41:20.080 --> 00:41:22.160 align:middle line:90% they give to each party. 00:41:22.160 --> 00:41:24.560 align:middle line:84% So what we're going to say is we're going 00:41:24.560 --> 00:41:28.280 align:middle line:90% to think of utility space. 00:41:28.280 --> 00:41:35.160 align:middle line:84% We have player 1's utility and player 2's utility. 00:41:35.160 --> 00:41:39.260 align:middle line:84% And the set of all possible agreements we can reach we're 00:41:39.260 --> 00:41:41.900 align:middle line:84% simply going to represent by the set of all possible utility 00:41:41.900 --> 00:41:43.980 align:middle line:90% pairs we could get. 00:41:43.980 --> 00:41:46.720 align:middle line:84% So we're going to have some set x. 00:41:46.720 --> 00:41:48.903 align:middle line:84% Let me draw a very simple example of a set, 00:41:48.903 --> 00:41:50.320 align:middle line:84% but we could be much more general. 00:41:50.320 --> 00:41:56.220 align:middle line:90% 00:41:56.220 --> 00:42:02.180 align:middle line:84% So maybe this is our set x and this represents 00:42:02.180 --> 00:42:05.800 align:middle line:90% feasible utility pairs. 00:42:05.800 --> 00:42:09.460 align:middle line:90% 00:42:09.460 --> 00:42:10.820 align:middle line:90% So what do I mean by that? 00:42:10.820 --> 00:42:13.360 align:middle line:84% If you give me some point in this set x, 00:42:13.360 --> 00:42:14.860 align:middle line:84% there exists some agreement we could 00:42:14.860 --> 00:42:20.100 align:middle line:84% reach that would generate these utilities for the two sides. 00:42:20.100 --> 00:42:22.600 align:middle line:84% What that agreement looks like I'm not taking a stand on. 00:42:22.600 --> 00:42:24.220 align:middle line:84% We're being very abstract, but we just 00:42:24.220 --> 00:42:26.393 align:middle line:84% want to say it's possible to achieve this. 00:42:26.393 --> 00:42:28.060 align:middle line:84% And then if you give me a point out here 00:42:28.060 --> 00:42:30.020 align:middle line:84% that's not an x, what I'm saying is, 00:42:30.020 --> 00:42:31.660 align:middle line:84% there is no agreement that could give 00:42:31.660 --> 00:42:33.305 align:middle line:90% both sides this much utility. 00:42:33.305 --> 00:42:35.180 align:middle line:84% It would be great if there was some deal that 00:42:35.180 --> 00:42:36.800 align:middle line:84% makes us both really, really happy, 00:42:36.800 --> 00:42:39.160 align:middle line:84% but there's just nothing feasible that 00:42:39.160 --> 00:42:41.200 align:middle line:90% would give us this utility. 00:42:41.200 --> 00:42:43.480 align:middle line:84% And then the one more thing we have to specify, 00:42:43.480 --> 00:42:46.120 align:middle line:84% well, negotiations can always break down. 00:42:46.120 --> 00:42:49.000 align:middle line:84% So we have to say what happens if negotiations break down, 00:42:49.000 --> 00:42:50.640 align:middle line:90% and we don't reach an agreement. 00:42:50.640 --> 00:42:52.000 align:middle line:84% If we don't reach an agreement, maybe we 00:42:52.000 --> 00:42:53.140 align:middle line:90% revert to the status quo. 00:42:53.140 --> 00:42:55.320 align:middle line:84% We have some disagreement outcome. 00:42:55.320 --> 00:42:58.080 align:middle line:84% But again, we want to reduce that disagreement outcome also 00:42:58.080 --> 00:42:59.480 align:middle line:90% to utility space. 00:42:59.480 --> 00:43:01.610 align:middle line:84% So we often normalize it to the origin. 00:43:01.610 --> 00:43:03.860 align:middle line:84% And this is going to be called the disagreement point. 00:43:03.860 --> 00:43:07.992 align:middle line:90% 00:43:07.992 --> 00:43:10.200 align:middle line:84% And what this says is if we don't reach an agreement, 00:43:10.200 --> 00:43:11.283 align:middle line:90% something will happen. 00:43:11.283 --> 00:43:12.700 align:middle line:84% I don't know exactly what that is, 00:43:12.700 --> 00:43:15.955 align:middle line:84% but whatever it is, we can assign utility to it. 00:43:15.955 --> 00:43:18.080 align:middle line:84% And this is going to represent the utilities we get 00:43:18.080 --> 00:43:19.420 align:middle line:90% if we don't reach an agreement. 00:43:19.420 --> 00:43:23.000 align:middle line:90% 00:43:23.000 --> 00:43:26.400 align:middle line:84% It's very standard to write our feasible set x to lie 00:43:26.400 --> 00:43:29.120 align:middle line:90% above the disagreement point. 00:43:29.120 --> 00:43:29.780 align:middle line:90% Why is that? 00:43:29.780 --> 00:43:31.640 align:middle line:84% What if there was some agreement we 00:43:31.640 --> 00:43:34.233 align:middle line:84% could make that would give utilities over here? 00:43:34.233 --> 00:43:35.900 align:middle line:84% I'm arguing that this is not going to be 00:43:35.900 --> 00:43:37.680 align:middle line:90% so relevant to our negotiation. 00:43:37.680 --> 00:43:40.140 align:middle line:84% Let me put one down here as well. 00:43:40.140 --> 00:43:42.393 align:middle line:84% Why is something like this, some point like this 00:43:42.393 --> 00:43:44.560 align:middle line:84% not really going to be relevant to our negotiations? 00:43:44.560 --> 00:43:45.300 align:middle line:90% Yeah, in the front. 00:43:45.300 --> 00:43:47.842 align:middle line:84% AUDIENCE: Why bargain at all if you can just disagree and get 00:43:47.842 --> 00:43:48.770 align:middle line:90% better utility? 00:43:48.770 --> 00:43:49.520 align:middle line:90% IAN BALL: I agree. 00:43:49.520 --> 00:43:50.500 align:middle line:90% But for whom? 00:43:50.500 --> 00:43:51.520 align:middle line:90% Let's be more specific. 00:43:51.520 --> 00:43:55.820 align:middle line:84% So if we disagree, we don't both get better utility. 00:43:55.820 --> 00:43:58.500 align:middle line:84% Who gets better utility in this example? 00:43:58.500 --> 00:44:01.580 align:middle line:90% AUDIENCE: So it would be u2. 00:44:01.580 --> 00:44:05.100 align:middle line:84% IAN BALL: Yes, so what I mean here is if we both disagree-- 00:44:05.100 --> 00:44:07.260 align:middle line:84% so disagreement-- and maybe I should be more clear. 00:44:07.260 --> 00:44:09.260 align:middle line:84% If we disagree and we don't reach a deal, 00:44:09.260 --> 00:44:11.980 align:middle line:84% this specifies the utilities that both of us get. 00:44:11.980 --> 00:44:15.060 align:middle line:84% So let's understand-- this deal is great for player 2. 00:44:15.060 --> 00:44:18.020 align:middle line:84% It's a lot better than the disagreement point for player 2. 00:44:18.020 --> 00:44:20.300 align:middle line:84% The problem is that it's worse for player 1. 00:44:20.300 --> 00:44:24.340 align:middle line:84% So the idea is player 1 would never accept something like this 00:44:24.340 --> 00:44:27.180 align:middle line:84% because player 1 can just walk away from the negotiating table, 00:44:27.180 --> 00:44:30.385 align:middle line:84% force the disagreement point, and get this point here. 00:44:30.385 --> 00:44:32.760 align:middle line:84% So I should say, when we talk about a disagreement point, 00:44:32.760 --> 00:44:34.343 align:middle line:84% there's kind of an implicit assumption 00:44:34.343 --> 00:44:37.360 align:middle line:84% here that each player has unilateral power to reach 00:44:37.360 --> 00:44:38.500 align:middle line:90% the disagreement point. 00:44:38.500 --> 00:44:40.208 align:middle line:84% And this is usually true in negotiations. 00:44:40.208 --> 00:44:42.140 align:middle line:90% Any side can just say, I'm done. 00:44:42.140 --> 00:44:43.820 align:middle line:90% I'm walking away from the table. 00:44:43.820 --> 00:44:45.280 align:middle line:90% And we go down here. 00:44:45.280 --> 00:44:46.380 align:middle line:90% OK, what about this point? 00:44:46.380 --> 00:44:49.480 align:middle line:84% Why could we never agree on a point like this? 00:44:49.480 --> 00:44:50.900 align:middle line:90% Who would veto that? 00:44:50.900 --> 00:44:51.448 align:middle line:90% Yeah. 00:44:51.448 --> 00:44:52.240 align:middle line:90% AUDIENCE: Player 2. 00:44:52.240 --> 00:44:54.532 align:middle line:84% IAN BALL: Player 2 would veto that because player 2 can 00:44:54.532 --> 00:44:57.380 align:middle line:84% get this much utility by walking away from the table. 00:44:57.380 --> 00:44:59.500 align:middle line:84% And down here, they get a strictly lower utility. 00:44:59.500 --> 00:45:01.660 align:middle line:84% So maybe there's some feasible things down here. 00:45:01.660 --> 00:45:02.720 align:middle line:84% But we usually don't even include them 00:45:02.720 --> 00:45:04.428 align:middle line:84% because they're not relevant to the game. 00:45:04.428 --> 00:45:08.378 align:middle line:84% So what's relevant are the set of feasible utilities. 00:45:08.378 --> 00:45:09.420 align:middle line:90% They have to be feasible. 00:45:09.420 --> 00:45:11.170 align:middle line:84% Otherwise, they're certainly not relevant. 00:45:11.170 --> 00:45:13.120 align:middle line:84% And they deliver utility that's weakly 00:45:13.120 --> 00:45:15.980 align:middle line:84% better than the disagreement point for both players. 00:45:15.980 --> 00:45:17.720 align:middle line:84% So neither party would unilaterally 00:45:17.720 --> 00:45:19.880 align:middle line:90% reject this agreement. 00:45:19.880 --> 00:45:23.280 align:middle line:84% But the question is, where are we going to be within here? 00:45:23.280 --> 00:45:26.800 align:middle line:84% That's the goal of the analysis to understand 00:45:26.800 --> 00:45:29.027 align:middle line:90% what's going to happen. 00:45:29.027 --> 00:45:30.860 align:middle line:84% In this game, I think it's pretty intuitive. 00:45:30.860 --> 00:45:34.440 align:middle line:84% You might think that we'll be somewhere along this line here. 00:45:34.440 --> 00:45:37.480 align:middle line:84% It seems intuitive that we don't want to throw away utility. 00:45:37.480 --> 00:45:37.980 align:middle line:90% But where? 00:45:37.980 --> 00:45:40.820 align:middle line:84% I mean, player 2 would love to be up here. 00:45:40.820 --> 00:45:42.760 align:middle line:84% Player 1 would love to be down here. 00:45:42.760 --> 00:45:45.720 align:middle line:84% And it's not clear where along this frontier we're going to be. 00:45:45.720 --> 00:45:48.580 align:middle line:90% 00:45:48.580 --> 00:45:51.030 align:middle line:84% So today, we're going to analyze this special case. 00:45:51.030 --> 00:45:52.780 align:middle line:84% We could look at an arbitrary disagreement 00:45:52.780 --> 00:45:56.580 align:middle line:84% point and an arbitrary feasible set x and analyze that game. 00:45:56.580 --> 00:45:59.500 align:middle line:84% We're going to look at the special case described here. 00:45:59.500 --> 00:46:03.740 align:middle line:84% Now you might say, what does this represent? 00:46:03.740 --> 00:46:08.700 align:middle line:84% It turns out, this is exactly price haggling, this picture. 00:46:08.700 --> 00:46:16.260 align:middle line:84% This is exactly price haggling between a seller and a buyer 00:46:16.260 --> 00:46:18.200 align:middle line:84% where the buyer values the good at 1. 00:46:18.200 --> 00:46:26.020 align:middle line:90% 00:46:26.020 --> 00:46:28.120 align:middle line:84% Let's see why this represents that? 00:46:28.120 --> 00:46:32.960 align:middle line:84% What is our disagreement point in price haggling? 00:46:32.960 --> 00:46:35.640 align:middle line:84% What can happen if we're a buyer and a seller come together, 00:46:35.640 --> 00:46:37.600 align:middle line:90% and they haggle over the price? 00:46:37.600 --> 00:46:40.480 align:middle line:84% Well, they can always just walk away and not exchange. 00:46:40.480 --> 00:46:43.808 align:middle line:84% So if there's no exchange, we each get utility 0. 00:46:43.808 --> 00:46:45.100 align:middle line:90% The buyer doesn't get the good. 00:46:45.100 --> 00:46:46.200 align:middle line:90% They don't pay anything. 00:46:46.200 --> 00:46:47.780 align:middle line:84% The seller doesn't get any revenue. 00:46:47.780 --> 00:46:48.940 align:middle line:90% So we get utility 0. 00:46:48.940 --> 00:46:52.300 align:middle line:84% And now I need to decide who's the seller and the buyer. 00:46:52.300 --> 00:46:56.600 align:middle line:84% Let's say this is the seller, and this is the buyer. 00:46:56.600 --> 00:46:58.180 align:middle line:90% What about this point over here? 00:46:58.180 --> 00:47:03.560 align:middle line:90% 00:47:03.560 --> 00:47:07.400 align:middle line:84% So at this point, the seller gets utility 1, 00:47:07.400 --> 00:47:09.060 align:middle line:90% and the buyer gets utility 0. 00:47:09.060 --> 00:47:12.000 align:middle line:90% What does this represent? 00:47:12.000 --> 00:47:14.960 align:middle line:84% How could, in our simple price haggling game, 00:47:14.960 --> 00:47:17.760 align:middle line:84% where we have value and we have risk-neutral agents 00:47:17.760 --> 00:47:19.232 align:middle line:90% get to this point? 00:47:19.232 --> 00:47:20.940 align:middle line:84% So this is the best thing for the seller? 00:47:20.940 --> 00:47:21.920 align:middle line:90% Yeah. 00:47:21.920 --> 00:47:24.322 align:middle line:84% AUDIENCE: The buyer put the max they're willing to pay. 00:47:24.322 --> 00:47:26.280 align:middle line:84% IAN BALL: And how much are they willing to pay? 00:47:26.280 --> 00:47:27.107 align:middle line:90% AUDIENCE: 1. 00:47:27.107 --> 00:47:28.940 align:middle line:84% IAN BALL: 1, so this would be the case where 00:47:28.940 --> 00:47:31.740 align:middle line:90% the buyer buys at a price of 1. 00:47:31.740 --> 00:47:35.500 align:middle line:84% So if the buyer buys at a price of 1, the seller's revenue is 1. 00:47:35.500 --> 00:47:37.380 align:middle line:90% So the seller's utility is 1. 00:47:37.380 --> 00:47:40.680 align:middle line:84% The buyer's utility is 0 because they value the good at 1, 00:47:40.680 --> 00:47:41.757 align:middle line:90% but they paid 1 for it. 00:47:41.757 --> 00:47:43.840 align:middle line:84% So what does it mean to value a good at something? 00:47:43.840 --> 00:47:45.240 align:middle line:84% It means if you pay that much for it, 00:47:45.240 --> 00:47:46.320 align:middle line:90% it's as if you don't have it. 00:47:46.320 --> 00:47:47.720 align:middle line:84% Your utility is exactly the same, 00:47:47.720 --> 00:47:49.700 align:middle line:90% so the buyer's utility is 0. 00:47:49.700 --> 00:47:51.233 align:middle line:90% What about up here? 00:47:51.233 --> 00:47:52.400 align:middle line:90% How do we get to this point? 00:47:52.400 --> 00:47:52.860 align:middle line:90% Yeah. 00:47:52.860 --> 00:47:53.760 align:middle line:84% AUDIENCE: The buyer got it for free. 00:47:53.760 --> 00:47:55.302 align:middle line:84% IAN BALL: The buyer gets it for free. 00:47:55.302 --> 00:47:57.580 align:middle line:84% So here, they exchange the good at a price of 0. 00:47:57.580 --> 00:48:00.100 align:middle line:84% The buyer's utility is 1 because they get the good for value 00:48:00.100 --> 00:48:02.380 align:middle line:90% of 1, but they pay nothing. 00:48:02.380 --> 00:48:04.420 align:middle line:84% And the seller's utility is 0 because they 00:48:04.420 --> 00:48:07.100 align:middle line:90% don't raise any revenue. 00:48:07.100 --> 00:48:10.340 align:middle line:84% Now, you can see we can then move along this line 00:48:10.340 --> 00:48:13.060 align:middle line:84% as we vary the price between 0 and 1. 00:48:13.060 --> 00:48:15.160 align:middle line:84% As we vary the price between 0 and 1, 00:48:15.160 --> 00:48:18.357 align:middle line:84% we're going to move along this line here. 00:48:18.357 --> 00:48:19.940 align:middle line:84% Now, I guess, technically, we can also 00:48:19.940 --> 00:48:21.023 align:middle line:90% get in this shaded region. 00:48:21.023 --> 00:48:22.840 align:middle line:84% How would we get in this shaded region? 00:48:22.840 --> 00:48:24.090 align:middle line:90% I mean, this is a bit strange. 00:48:24.090 --> 00:48:27.800 align:middle line:84% No one would really do this, but what could we do? 00:48:27.800 --> 00:48:31.960 align:middle line:84% Well, technically I could, as the buyer, I could, say, 00:48:31.960 --> 00:48:35.880 align:middle line:84% pay you $0.80 for the good and then burn $0.10 and just throw 00:48:35.880 --> 00:48:36.450 align:middle line:90% it away. 00:48:36.450 --> 00:48:37.700 align:middle line:90% We probably wouldn't get that. 00:48:37.700 --> 00:48:41.075 align:middle line:84% But often, we want to think of this as a convex set. 00:48:41.075 --> 00:48:43.200 align:middle line:84% So technically, we could always burn money and just 00:48:43.200 --> 00:48:44.580 align:middle line:84% throw things away and go down here. 00:48:44.580 --> 00:48:45.955 align:middle line:84% We don't expect this will happen. 00:48:45.955 --> 00:48:50.340 align:middle line:84% But in the monetary context, this maybe seems crazy, 00:48:50.340 --> 00:48:52.300 align:middle line:84% but you certainly see in negotiations, 00:48:52.300 --> 00:48:54.760 align:middle line:84% people are sometimes willing to put forward 00:48:54.760 --> 00:48:57.560 align:middle line:84% proposals that would harm both parties as a threat. 00:48:57.560 --> 00:48:59.960 align:middle line:84% So in the monetary context, maybe it 00:48:59.960 --> 00:49:01.640 align:middle line:84% doesn't seem reasonable, but it's 00:49:01.640 --> 00:49:04.503 align:middle line:84% important to think about the possibility of threats 00:49:04.503 --> 00:49:05.920 align:middle line:84% that hurt both parties, and that's 00:49:05.920 --> 00:49:07.840 align:middle line:90% why we want to fill this in. 00:49:07.840 --> 00:49:12.520 align:middle line:84% OK, so we've seen how this fits into our example. 00:49:12.520 --> 00:49:15.600 align:middle line:84% And now we just need to be a little more formal. 00:49:15.600 --> 00:49:27.060 align:middle line:84% So if we get an agreement, x1, x2, in X-- 00:49:27.060 --> 00:49:29.020 align:middle line:90% so let's go back to the terms. 00:49:29.020 --> 00:49:33.035 align:middle line:84% Maybe player 1-- going to be a little more abstract. 00:49:33.035 --> 00:49:35.160 align:middle line:84% We can think of this in the price haggling context. 00:49:35.160 --> 00:49:36.660 align:middle line:84% But we'll be abstract and just think 00:49:36.660 --> 00:49:38.140 align:middle line:90% about players 1 and player 2. 00:49:38.140 --> 00:49:41.860 align:middle line:84% If we reach an agreement x1, x2 in X, then, 00:49:41.860 --> 00:49:44.100 align:middle line:84% well, we think that player 1 will get utility x1, 00:49:44.100 --> 00:49:45.900 align:middle line:84% and player 2 will get utility x2. 00:49:45.900 --> 00:49:48.660 align:middle line:84% But we also want to represent the cost of delay. 00:49:48.660 --> 00:49:49.940 align:middle line:90% Delay is costly. 00:49:49.940 --> 00:49:51.820 align:middle line:84% So suppose we reach an agreement here 00:49:51.820 --> 00:50:03.860 align:middle line:84% at time t, where t is going to go 0, 1, 2 on forever. 00:50:03.860 --> 00:50:05.560 align:middle line:84% Then the utilities are going to be, 00:50:05.560 --> 00:50:09.260 align:middle line:84% well, player 1 gets utility delta to the t x1 00:50:09.260 --> 00:50:13.020 align:middle line:84% and player 2 gets utility delta to the t x2. 00:50:13.020 --> 00:50:20.460 align:middle line:84% And here, where delta 0 is less than delta is less than 1. 00:50:20.460 --> 00:50:22.060 align:middle line:84% So this is just like we had before. 00:50:22.060 --> 00:50:24.920 align:middle line:90% We have a discount factor delta. 00:50:24.920 --> 00:50:27.040 align:middle line:84% The later on that we reach an agreement, 00:50:27.040 --> 00:50:28.980 align:middle line:84% the less utility each of us gets. 00:50:28.980 --> 00:50:32.040 align:middle line:90% 00:50:32.040 --> 00:50:36.380 align:middle line:84% And as time passes, things get worse and worse for both of us. 00:50:36.380 --> 00:50:39.620 align:middle line:84% If we make an agreement in period 0-- remember, 00:50:39.620 --> 00:50:42.200 align:middle line:84% any nonzero number to the 0 is just 1. 00:50:42.200 --> 00:50:46.280 align:middle line:84% So if we agree immediately in period 0, we just get x1 and x2. 00:50:46.280 --> 00:50:49.660 align:middle line:84% If we agree in period 1, we get delta x1, delta x2. 00:50:49.660 --> 00:50:51.560 align:middle line:84% If we agree in period 10, we get delta 00:50:51.560 --> 00:50:54.200 align:middle line:84% to the 10 x1, delta to the 10 x2. 00:50:54.200 --> 00:50:56.980 align:middle line:84% And again, delta is going to parameterize our patients. 00:50:56.980 --> 00:50:59.718 align:middle line:84% So what happens if delta is very close to 1? 00:50:59.718 --> 00:51:00.760 align:middle line:90% What does that represent? 00:51:00.760 --> 00:51:03.840 align:middle line:90% Is that patience are impatience? 00:51:03.840 --> 00:51:05.088 align:middle line:90% That's-- yeah. 00:51:05.088 --> 00:51:05.880 align:middle line:90% AUDIENCE: Patience. 00:51:05.880 --> 00:51:06.740 align:middle line:90% IAN BALL: Patience, right. 00:51:06.740 --> 00:51:08.920 align:middle line:84% If delta's close to 1, even if t is really large, 00:51:08.920 --> 00:51:10.440 align:middle line:90% it doesn't hurt us very much. 00:51:10.440 --> 00:51:13.540 align:middle line:84% If delta's close to 0, we're very impatient. 00:51:13.540 --> 00:51:15.620 align:middle line:84% And in the extreme case, if delta is 0, 00:51:15.620 --> 00:51:17.420 align:middle line:84% then, basically, the world ends tomorrow, 00:51:17.420 --> 00:51:18.920 align:middle line:84% and the only way we can reach a deal 00:51:18.920 --> 00:51:22.350 align:middle line:84% is if we reach a deal in the first period. 00:51:22.350 --> 00:51:24.350 align:middle line:90% OK, so this is our model. 00:51:24.350 --> 00:51:27.190 align:middle line:84% Now we have to specify the sequence 00:51:27.190 --> 00:51:30.190 align:middle line:90% in which we make offers. 00:51:30.190 --> 00:51:32.310 align:middle line:84% And I think the standard way we'll 00:51:32.310 --> 00:51:39.070 align:middle line:84% do it is we have time 0, 1, 2, 3, and so on. 00:51:39.070 --> 00:51:41.150 align:middle line:84% And it's alternating offers, so let's say 00:51:41.150 --> 00:51:45.970 align:middle line:84% that, in the even periods, player 1 makes an offer-- 00:51:45.970 --> 00:51:48.870 align:middle line:90% 00:51:48.870 --> 00:51:50.450 align:middle line:84% so maybe I'll say makes an offer. 00:51:50.450 --> 00:51:55.310 align:middle line:90% 00:51:55.310 --> 00:51:57.890 align:middle line:84% And then player two can either accept or reject that offer. 00:51:57.890 --> 00:52:07.030 align:middle line:90% 00:52:07.030 --> 00:52:09.930 align:middle line:84% And then, in the odd periods, player 2 is the one who moves. 00:52:09.930 --> 00:52:23.810 align:middle line:90% 00:52:23.810 --> 00:52:27.090 align:middle line:84% OK, so just to be clear, player 1 makes an offer. 00:52:27.090 --> 00:52:30.110 align:middle line:84% If player 2 accepts that offer, that's it. 00:52:30.110 --> 00:52:33.410 align:middle line:84% We implement the offer, and we stop. 00:52:33.410 --> 00:52:35.570 align:middle line:84% If player 2 rejects that offer, then player 2 00:52:35.570 --> 00:52:37.330 align:middle line:84% comes back and makes their offer. 00:52:37.330 --> 00:52:38.890 align:middle line:84% Player 1 can either accept or reject. 00:52:38.890 --> 00:52:41.730 align:middle line:84% If they accept it, we make that offer, and we accept that offer, 00:52:41.730 --> 00:52:42.448 align:middle line:90% and we stop. 00:52:42.448 --> 00:52:44.490 align:middle line:84% If they reject it, we move on to the next period, 00:52:44.490 --> 00:52:46.010 align:middle line:90% and we go on like this. 00:52:46.010 --> 00:52:47.670 align:middle line:90% Now, what is an offer, though? 00:52:47.670 --> 00:52:50.470 align:middle line:84% Remember, in our price haggling game, an offer was a price. 00:52:50.470 --> 00:52:52.290 align:middle line:90% We said what price is offered? 00:52:52.290 --> 00:52:56.738 align:middle line:84% But more generally, an offer is simply a point in x. 00:52:56.738 --> 00:52:59.030 align:middle line:84% Remember, that's how we're going to represent an offer. 00:52:59.030 --> 00:53:01.070 align:middle line:84% An offer is to say, let's stop now. 00:53:01.070 --> 00:53:03.470 align:middle line:84% Let's reach this agreement, and let's get that utility. 00:53:03.470 --> 00:53:12.370 align:middle line:84% So let's be clear that an offer is always a point in x. 00:53:12.370 --> 00:53:14.570 align:middle line:90% I've specified the payoffs. 00:53:14.570 --> 00:53:16.990 align:middle line:84% If we reach an agreement x in time 00:53:16.990 --> 00:53:19.730 align:middle line:84% t, and that's generally how the game will end. 00:53:19.730 --> 00:53:21.820 align:middle line:84% At some time t, the offer will be accepted. 00:53:21.820 --> 00:53:24.070 align:middle line:84% Some offer will be accepted, and now we can write down 00:53:24.070 --> 00:53:25.230 align:middle line:90% what the payoffs are. 00:53:25.230 --> 00:53:26.990 align:middle line:84% But I just have to be more careful here. 00:53:26.990 --> 00:53:30.470 align:middle line:84% What happens if we just go on forever and nothing is accepted? 00:53:30.470 --> 00:53:32.430 align:middle line:84% That's one thing that could happen in the game? 00:53:32.430 --> 00:53:35.270 align:middle line:84% So I need to specify what the utilities are in that case? 00:53:35.270 --> 00:53:41.350 align:middle line:84% And if it's never accepted, well, then we just 00:53:41.350 --> 00:53:43.670 align:middle line:84% move to our disagreement point, which, in this case, 00:53:43.670 --> 00:53:46.110 align:middle line:90% is going to be 0, 0. 00:53:46.110 --> 00:53:47.550 align:middle line:90% It's never accepted. 00:53:47.550 --> 00:53:49.170 align:middle line:90% We go to our disagreement point. 00:53:49.170 --> 00:53:53.070 align:middle line:90% 00:53:53.070 --> 00:53:56.902 align:middle line:84% And in this case, we'll say that utility is 0, 0. 00:53:56.902 --> 00:53:59.110 align:middle line:84% We have to specify that because that's one thing that 00:53:59.110 --> 00:53:59.930 align:middle line:90% could happen in the game. 00:53:59.930 --> 00:54:00.470 align:middle line:90% Yes. 00:54:00.470 --> 00:54:05.030 align:middle line:84% AUDIENCE: So does this represent that because of time decay, 00:54:05.030 --> 00:54:07.850 align:middle line:84% we're heading towards 0, 0 asymptotically? 00:54:07.850 --> 00:54:10.527 align:middle line:84% Or do we actually hit the point 0, 0? 00:54:10.527 --> 00:54:12.110 align:middle line:84% Is it just like, both people are like, 00:54:12.110 --> 00:54:14.070 align:middle line:90% OK, let's give up, disagree. 00:54:14.070 --> 00:54:15.570 align:middle line:84% IAN BALL: Yeah, I think interpreting 00:54:15.570 --> 00:54:18.410 align:middle line:84% what disagreement means in the infinite horizon 00:54:18.410 --> 00:54:22.850 align:middle line:84% is a little tricky because when does it actually happen? 00:54:22.850 --> 00:54:29.670 align:middle line:84% Yeah so I think there's a few different interpretations. 00:54:29.670 --> 00:54:35.810 align:middle line:84% I would maybe interpret the disagreement point-- 00:54:35.810 --> 00:54:44.030 align:middle line:84% so one interpretation-- yeah, I don't want to confuse things. 00:54:44.030 --> 00:54:46.250 align:middle line:84% One interpretation could be, in each period, 00:54:46.250 --> 00:54:49.010 align:middle line:84% until we reach a deal, we just get a flow utility of 0 00:54:49.010 --> 00:54:50.290 align:middle line:90% in that period. 00:54:50.290 --> 00:54:52.070 align:middle line:84% And then, once we reach an agreement, 00:54:52.070 --> 00:54:54.722 align:middle line:84% we get the agreement utility from that period onward. 00:54:54.722 --> 00:54:56.430 align:middle line:84% And then we discount in some R utilities. 00:54:56.430 --> 00:54:58.890 align:middle line:84% That would be one interpretation you could give. 00:54:58.890 --> 00:55:02.270 align:middle line:84% So it's, really, our payoffs would be like 0, 0, 0, 0, 00:55:02.270 --> 00:55:06.030 align:middle line:84% and then, once we reach an agreement of x here, 00:55:06.030 --> 00:55:08.670 align:middle line:84% then we get that agreement utility x, x, x, x forever. 00:55:08.670 --> 00:55:11.030 align:middle line:84% And then we do the average discounted payoff from that. 00:55:11.030 --> 00:55:12.590 align:middle line:84% I don't want to create complications, 00:55:12.590 --> 00:55:14.715 align:middle line:84% but if you did that, it would reduce to this model. 00:55:14.715 --> 00:55:17.175 align:middle line:84% So that would maybe be one way of thinking about it. 00:55:17.175 --> 00:55:18.050 align:middle line:90% Because you're right. 00:55:18.050 --> 00:55:20.313 align:middle line:84% If things go on forever, there's not a point 00:55:20.313 --> 00:55:21.730 align:middle line:84% where we say, oh, we've disagreed. 00:55:21.730 --> 00:55:23.290 align:middle line:84% It just says, well, we didn't reach an agreement today, 00:55:23.290 --> 00:55:24.540 align:middle line:90% and that just happens forever. 00:55:24.540 --> 00:55:27.330 align:middle line:84% So maybe it's better to think of it in terms of flow payoffs. 00:55:27.330 --> 00:55:30.110 align:middle line:90% Good question. 00:55:30.110 --> 00:55:36.035 align:middle line:84% OK, so now we'd like to analyze this game. 00:55:36.035 --> 00:55:37.910 align:middle line:84% We did the whole one-shot deviation principle 00:55:37.910 --> 00:55:41.070 align:middle line:84% to help us study games like this. 00:55:41.070 --> 00:55:44.870 align:middle line:84% But it's not even clear that this is a multistage game 00:55:44.870 --> 00:55:47.550 align:middle line:84% because players are not moving simultaneously. 00:55:47.550 --> 00:55:48.930 align:middle line:90% Players are moving sequentially. 00:55:48.930 --> 00:55:51.410 align:middle line:84% Player 1 makes an offer, and then player 2 observes this. 00:55:51.410 --> 00:55:52.970 align:middle line:84% So why is this a multistage game. 00:55:52.970 --> 00:56:06.470 align:middle line:90% 00:56:06.470 --> 00:56:08.490 align:middle line:84% Well, it turns out, it is a multistage game, 00:56:08.490 --> 00:56:11.010 align:middle line:84% but it's just the stages are not given by time. 00:56:11.010 --> 00:56:13.670 align:middle line:90% So technically, this is stage 0. 00:56:13.670 --> 00:56:15.610 align:middle line:84% Player 1 is the only player who moves. 00:56:15.610 --> 00:56:16.970 align:middle line:90% This is stage 1. 00:56:16.970 --> 00:56:18.030 align:middle line:90% This is stage 2. 00:56:18.030 --> 00:56:21.110 align:middle line:84% This is stage 3, 4, 5, 6, 7, 8, 9. 00:56:21.110 --> 00:56:22.490 align:middle line:90% So it is a multistage game. 00:56:22.490 --> 00:56:23.930 align:middle line:84% The multistage game just requires 00:56:23.930 --> 00:56:26.210 align:middle line:84% that, within each stage, a subset of players 00:56:26.210 --> 00:56:27.343 align:middle line:90% move simultaneously. 00:56:27.343 --> 00:56:29.010 align:middle line:84% Well, in this game, the subset who moves 00:56:29.010 --> 00:56:32.650 align:middle line:84% is just a single player, and the history of past play 00:56:32.650 --> 00:56:35.170 align:middle line:84% is always observed, so it's a multistage game. 00:56:35.170 --> 00:56:38.330 align:middle line:84% So we can now apply the one-shot deviation principle 00:56:38.330 --> 00:56:41.330 align:middle line:84% to try to understand subgame-perfect Nash equilibria 00:56:41.330 --> 00:56:43.530 align:middle line:90% of this game. 00:56:43.530 --> 00:56:45.830 align:middle line:84% Now, just like with our finite case, 00:56:45.830 --> 00:56:48.330 align:middle line:84% the strategies in this game are really complicated. 00:56:48.330 --> 00:56:50.730 align:middle line:90% What is a strategy for player 2? 00:56:50.730 --> 00:56:54.870 align:middle line:84% It has to say, whatever offer I've given in period 0, 00:56:54.870 --> 00:56:56.490 align:middle line:90% will I accept or reject that? 00:56:56.490 --> 00:56:58.250 align:middle line:84% Then, based on that history, what 00:56:58.250 --> 00:57:00.690 align:middle line:84% offer will I make in period 1 for every possible history 00:57:00.690 --> 00:57:01.515 align:middle line:90% of play there? 00:57:01.515 --> 00:57:02.390 align:middle line:90% And on and on and on. 00:57:02.390 --> 00:57:05.650 align:middle line:84% So it's going to have to specify a complete contingent plan that 00:57:05.650 --> 00:57:07.970 align:middle line:84% says what I will either accept or offer 00:57:07.970 --> 00:57:09.373 align:middle line:90% at every history of past play. 00:57:09.373 --> 00:57:11.290 align:middle line:84% That's going to be really, really complicated, 00:57:11.290 --> 00:57:13.415 align:middle line:84% but we're going to, hopefully, be able to solve it. 00:57:13.415 --> 00:57:15.790 align:middle line:84% And then make a prediction about what's going to happen? 00:57:15.790 --> 00:57:18.830 align:middle line:84% Any guesses, I guess, before we really get into it, 00:57:18.830 --> 00:57:22.270 align:middle line:84% if we look at our agreement space here, 00:57:22.270 --> 00:57:25.670 align:middle line:84% any intuition about where we might end up 00:57:25.670 --> 00:57:26.850 align:middle line:90% in terms of payoffs? 00:57:26.850 --> 00:57:30.250 align:middle line:84% So any predictions about what might happen? 00:57:30.250 --> 00:57:31.310 align:middle line:90% Yeah. 00:57:31.310 --> 00:57:33.990 align:middle line:84% AUDIENCE: I mean, just looking at the graph over time, 00:57:33.990 --> 00:57:35.910 align:middle line:90% the line will move to the left. 00:57:35.910 --> 00:57:37.190 align:middle line:90% IAN BALL: Exactly. 00:57:37.190 --> 00:57:37.930 align:middle line:90% That's right. 00:57:37.930 --> 00:57:40.530 align:middle line:84% So if we made an agreement in the initial period, 00:57:40.530 --> 00:57:42.930 align:middle line:84% we could-- that's the only way we can get on this line. 00:57:42.930 --> 00:57:46.270 align:middle line:84% And any intuition about where we might end up? 00:57:46.270 --> 00:57:49.047 align:middle line:84% I mean, we'll analyze it, but what do you think 00:57:49.047 --> 00:57:50.130 align:middle line:90% might happen in this game? 00:57:50.130 --> 00:57:50.930 align:middle line:90% Who's going to do better? 00:57:50.930 --> 00:57:51.430 align:middle line:90% Yeah. 00:57:51.430 --> 00:57:53.668 align:middle line:90% 00:57:53.668 --> 00:57:54.210 align:middle line:90% Make a guess. 00:57:54.210 --> 00:57:54.570 align:middle line:90% It's fine. 00:57:54.570 --> 00:57:54.750 align:middle line:90% Yeah. 00:57:54.750 --> 00:57:56.667 align:middle line:84% AUDIENCE: Well, my thought was, last time, you 00:57:56.667 --> 00:57:59.390 align:middle line:84% talked about how it's dependent on the fact 00:57:59.390 --> 00:58:01.050 align:middle line:90% that the negotiation was finite. 00:58:01.050 --> 00:58:04.710 align:middle line:84% So it would get sped up because both knew that they had 00:58:04.710 --> 00:58:06.130 align:middle line:90% to reach an agreement quickly. 00:58:06.130 --> 00:58:08.750 align:middle line:84% So I feel like it's going to devolve a little bit more. 00:58:08.750 --> 00:58:11.850 align:middle line:84% And maybe if it's infinite, then maybe the first person 00:58:11.850 --> 00:58:13.833 align:middle line:84% who has the offer has the advantage. 00:58:13.833 --> 00:58:14.750 align:middle line:90% IAN BALL: That's true. 00:58:14.750 --> 00:58:16.530 align:middle line:90% So I agree. 00:58:16.530 --> 00:58:20.110 align:middle line:84% We still have complete information here. 00:58:20.110 --> 00:58:22.272 align:middle line:84% So we talked about, last time, one reason 00:58:22.272 --> 00:58:24.730 align:middle line:84% that we're not going to see actually delay and disagreement 00:58:24.730 --> 00:58:27.370 align:middle line:84% is that we all agree, for each offer, exactly what utilities 00:58:27.370 --> 00:58:28.190 align:middle line:90% that gives us. 00:58:28.190 --> 00:58:31.057 align:middle line:84% In reality, if you make a peace proposal in a negotiating plan, 00:58:31.057 --> 00:58:32.890 align:middle line:84% you don't know how much utility that's going 00:58:32.890 --> 00:58:33.973 align:middle line:90% to give to the other side. 00:58:33.973 --> 00:58:35.910 align:middle line:84% So there's been a big assumption here, 00:58:35.910 --> 00:58:37.290 align:middle line:84% and that's actually going to be a reason why we're going 00:58:37.290 --> 00:58:39.010 align:middle line:90% to see an agreement initially. 00:58:39.010 --> 00:58:41.660 align:middle line:84% And indeed, the first mover, player 1, 00:58:41.660 --> 00:58:43.410 align:middle line:84% is going to have an advantage in this game 00:58:43.410 --> 00:58:45.708 align:middle line:84% because they have the power of moving first. 00:58:45.708 --> 00:58:47.250 align:middle line:84% And indeed, there's going to be a bit 00:58:47.250 --> 00:58:48.810 align:middle line:84% of a first-mover advantage, so we're 00:58:48.810 --> 00:58:52.650 align:middle line:84% going to end up along this line but a little closer to player 1. 00:58:52.650 --> 00:58:54.910 align:middle line:84% And the more impatient the players are, 00:58:54.910 --> 00:58:56.730 align:middle line:84% the closer we're going to be to player 1 00:58:56.730 --> 00:58:58.530 align:middle line:84% because the more impatient the players are-- let's 00:58:58.530 --> 00:59:01.155 align:middle line:84% think of the extreme case just before we get into the analysis. 00:59:01.155 --> 00:59:03.730 align:middle line:84% I think this is good, whenever you approach a problem, to try 00:59:03.730 --> 00:59:05.980 align:middle line:84% to think through what might happen before you formally 00:59:05.980 --> 00:59:06.530 align:middle line:90% analyze it. 00:59:06.530 --> 00:59:08.130 align:middle line:90% Is that a question? 00:59:08.130 --> 00:59:09.310 align:middle line:90% Yeah, yeah, go ahead. 00:59:09.310 --> 00:59:11.750 align:middle line:84% AUDIENCE: Does this only apply if we assume that they have 00:59:11.750 --> 00:59:13.430 align:middle line:90% the same level of patience? 00:59:13.430 --> 00:59:16.045 align:middle line:84% IAN BALL: Good, so I'm assuming that they have 00:59:16.045 --> 00:59:17.170 align:middle line:90% the same level of patience. 00:59:17.170 --> 00:59:18.670 align:middle line:84% A very natural extension of the model 00:59:18.670 --> 00:59:20.450 align:middle line:84% would be to say that they have different discount factors. 00:59:20.450 --> 00:59:21.770 align:middle line:84% And indeed, that would change the results. 00:59:21.770 --> 00:59:23.430 align:middle line:84% And there may be a problem set on that 00:59:23.430 --> 00:59:25.805 align:middle line:84% or the problem is the algebra gets quite messy, 00:59:25.805 --> 00:59:26.930 align:middle line:90% so I often don't assign it. 00:59:26.930 --> 00:59:28.370 align:middle line:90% But that's a classic extension. 00:59:28.370 --> 00:59:30.430 align:middle line:90% Yes, exactly. 00:59:30.430 --> 00:59:31.390 align:middle line:90% Yes. 00:59:31.390 --> 00:59:33.310 align:middle line:84% AUDIENCE: Yeah, I mean if delta were 0, 00:59:33.310 --> 00:59:36.143 align:middle line:84% then they would just make whatever deal you want. 00:59:36.143 --> 00:59:37.810 align:middle line:84% IAN BALL: Exactly, so let's get to that. 00:59:37.810 --> 00:59:39.430 align:middle line:90% So let's say delta is 0. 00:59:39.430 --> 00:59:41.230 align:middle line:84% So the world ends after tomorrow. 00:59:41.230 --> 00:59:42.533 align:middle line:90% Player 1 is making a proposal. 00:59:42.533 --> 00:59:44.450 align:middle line:84% What do you think player 1's going to propose? 00:59:44.450 --> 00:59:48.350 align:middle line:84% And where are we going to be on this triangle? 00:59:48.350 --> 00:59:49.030 align:middle line:90% Yeah. 00:59:49.030 --> 00:59:50.720 align:middle line:84% AUDIENCE: I mean, I think that we 00:59:50.720 --> 00:59:54.598 align:middle line:84% could make this 0.1 for player 2 but as long as it's above 0. 00:59:54.598 --> 00:59:56.390 align:middle line:84% IAN BALL: Right, so we're basically-- maybe 00:59:56.390 --> 00:59:58.068 align:middle line:84% there's an issue with 0.1, 0.001, 00:59:58.068 --> 00:59:59.610 align:middle line:84% but we're basically going to be here. 00:59:59.610 --> 01:00:02.590 align:middle line:84% So player 1's going to have all the power. 01:00:02.590 --> 01:00:05.210 align:middle line:84% If the world ends tomorrow, then you as player 2 01:00:05.210 --> 01:00:06.750 align:middle line:84% have to accept whatever I give you. 01:00:06.750 --> 01:00:08.710 align:middle line:84% So I'm going to make an offer way down here. 01:00:08.710 --> 01:00:10.377 align:middle line:84% I'm going to give you almost no utility, 01:00:10.377 --> 01:00:11.530 align:middle line:90% and we're going to be here. 01:00:11.530 --> 01:00:13.570 align:middle line:84% And then, intuitively, as the players 01:00:13.570 --> 01:00:15.370 align:middle line:90% become very, very patient. 01:00:15.370 --> 01:00:17.730 align:middle line:84% Well, now the first-mover advantage that player 1 gets 01:00:17.730 --> 01:00:19.183 align:middle line:90% is going to shrink. 01:00:19.183 --> 01:00:20.850 align:middle line:84% If we're very patient, the fact that you 01:00:20.850 --> 01:00:23.090 align:middle line:84% might have to wait till tomorrow isn't a big deal. 01:00:23.090 --> 01:00:25.770 align:middle line:84% And as we get more and more patient, we're going to move, 01:00:25.770 --> 01:00:28.250 align:middle line:84% and we're going to converge to the midpoint of here 01:00:28.250 --> 01:00:31.690 align:middle line:84% because, as delta goes to 1, the fact that player 1 moves first 01:00:31.690 --> 01:00:33.350 align:middle line:84% is not really an advantage at all. 01:00:33.350 --> 01:00:35.450 align:middle line:84% And therefore, we expect the symmetric solution 01:00:35.450 --> 01:00:36.438 align:middle line:90% where we end up here. 01:00:36.438 --> 01:00:38.230 align:middle line:84% And that's exactly what we're going to see. 01:00:38.230 --> 01:00:40.110 align:middle line:90% But that's just a preview. 01:00:40.110 --> 01:00:40.610 align:middle line:90% Yes. 01:00:40.610 --> 01:00:42.110 align:middle line:84% AUDIENCE: Does that mean we're never 01:00:42.110 --> 01:00:44.570 align:middle line:84% going to be on the left side of the midpoint where player 01:00:44.570 --> 01:00:46.268 align:middle line:90% 2 isn't getting [INAUDIBLE]? 01:00:46.268 --> 01:00:47.310 align:middle line:90% IAN BALL: That's correct. 01:00:47.310 --> 01:00:49.430 align:middle line:84% We are not because player 1 moves first. 01:00:49.430 --> 01:00:51.610 align:middle line:84% So if player 2 moves first, we would have the mirror 01:00:51.610 --> 01:00:53.730 align:middle line:84% image, where player 2 moves first 01:00:53.730 --> 01:00:55.930 align:middle line:84% and the discount factor was 0, we'd be up here, 01:00:55.930 --> 01:00:57.770 align:middle line:84% and then we would converge down to here. 01:00:57.770 --> 01:00:58.970 align:middle line:84% AUDIENCE: So when we talk about converge, 01:00:58.970 --> 01:01:00.190 align:middle line:90% we actually hit the midpoint? 01:01:00.190 --> 01:01:02.282 align:middle line:90% Or do we just get-- 01:01:02.282 --> 01:01:04.490 align:middle line:84% IAN BALL: In this case, we will not literally hit it. 01:01:04.490 --> 01:01:05.830 align:middle line:90% We will just converge. 01:01:05.830 --> 01:01:09.888 align:middle line:84% We will get arbitrarily close as delta goes to S yeah. 01:01:09.888 --> 01:01:11.430 align:middle line:84% Because what would happen, you'd want 01:01:11.430 --> 01:01:13.430 align:middle line:84% to say we hit it when delta equals 1. 01:01:13.430 --> 01:01:17.120 align:middle line:84% Actually, if delta equals 1, we then 01:01:17.120 --> 01:01:19.370 align:middle line:84% have to be a little more careful about delta equals 1. 01:01:19.370 --> 01:01:21.210 align:middle line:84% So there could be multiple equilibria. 01:01:21.210 --> 01:01:23.150 align:middle line:84% So let's not-- we're not going to deal with the exact delta 01:01:23.150 --> 01:01:23.930 align:middle line:90% equals 1 case. 01:01:23.930 --> 01:01:24.990 align:middle line:90% Yeah. 01:01:24.990 --> 01:01:26.150 align:middle line:90% OK. 01:01:26.150 --> 01:01:31.310 align:middle line:84% Great, so let's try to analyze this. 01:01:31.310 --> 01:01:32.250 align:middle line:90% Let's go down here. 01:01:32.250 --> 01:01:46.710 align:middle line:90% 01:01:46.710 --> 01:01:48.800 align:middle line:84% So as we said, specifying a strategy in this game 01:01:48.800 --> 01:01:50.050 align:middle line:90% is really, really complicated. 01:01:50.050 --> 01:01:51.130 align:middle line:90% There's so many histories. 01:01:51.130 --> 01:01:53.797 align:middle line:84% But what we're going to look for is a very, very simple strategy 01:01:53.797 --> 01:01:55.590 align:middle line:84% where people's behavior doesn't really 01:01:55.590 --> 01:01:57.830 align:middle line:84% depend on the details of the past, that, basically, 01:01:57.830 --> 01:02:00.420 align:middle line:84% when it's your turn to propose, you basically 01:02:00.420 --> 01:02:02.600 align:middle line:84% propose the same thing no matter what else happened. 01:02:02.600 --> 01:02:04.808 align:middle line:84% And we're going to check that this is an equilibrium. 01:02:04.808 --> 01:02:11.020 align:middle line:84% So here's our conjectured, maybe our candidate, SPNE. 01:02:11.020 --> 01:02:13.940 align:middle line:84% So we'll check that this is actually an SPNE. 01:02:13.940 --> 01:02:16.820 align:middle line:84% Well, as we said, we think that whoever proposes, 01:02:16.820 --> 01:02:19.780 align:middle line:84% whatever history we're at, has some advantage. 01:02:19.780 --> 01:02:28.220 align:middle line:84% And what's going to happen is, at any history, 01:02:28.220 --> 01:02:30.060 align:middle line:90% well, someone's the proposer. 01:02:30.060 --> 01:02:31.520 align:middle line:84% It's either player 1 or player 2, 01:02:31.520 --> 01:02:33.800 align:middle line:84% depending on whether it's an even history or an odd history. 01:02:33.800 --> 01:02:35.092 align:middle line:90% But let's not worry about that. 01:02:35.092 --> 01:02:44.060 align:middle line:84% And any history, the proposer proposes-- 01:02:44.060 --> 01:02:47.340 align:middle line:84% well, I'd like to use a vector in x, 01:02:47.340 --> 01:02:49.100 align:middle line:84% but the problem is a vector in x tells me 01:02:49.100 --> 01:02:51.020 align:middle line:84% the payoffs for player 1 and player 2, 01:02:51.020 --> 01:02:53.480 align:middle line:84% and the proposer could be player 1 or player 2. 01:02:53.480 --> 01:02:55.940 align:middle line:84% So I really want to think in terms of whoever's proposing 01:02:55.940 --> 01:02:58.120 align:middle line:84% and who's responding rather than the actual labels 01:02:58.120 --> 01:02:58.960 align:middle line:90% of the players. 01:02:58.960 --> 01:03:01.560 align:middle line:84% So let's say the proposer proposes 01:03:01.560 --> 01:03:04.400 align:middle line:84% some utility for herself and some utility 01:03:04.400 --> 01:03:06.180 align:middle line:84% for the other player, for the respondent. 01:03:06.180 --> 01:03:07.638 align:middle line:84% So they're going to propose-- well, 01:03:07.638 --> 01:03:09.680 align:middle line:84% let's think they're the proposer. 01:03:09.680 --> 01:03:11.780 align:middle line:90% So they have a bit more power. 01:03:11.780 --> 01:03:15.560 align:middle line:84% So we think they're going to propose more than 1/2 01:03:15.560 --> 01:03:19.080 align:middle line:84% to themselves and less than 1/2 the other player. 01:03:19.080 --> 01:03:20.860 align:middle line:84% And in it's going to be exactly-- 01:03:20.860 --> 01:03:24.800 align:middle line:84% this is a formula we saw in class a few weeks ago-- 01:03:24.800 --> 01:03:27.622 align:middle line:90% 1 over 1 plus delta. 01:03:27.622 --> 01:03:28.580 align:middle line:90% I'll write it this way. 01:03:28.580 --> 01:03:31.400 align:middle line:90% 01:03:31.400 --> 01:03:35.360 align:middle line:90% Maybe I'll say for herself. 01:03:35.360 --> 01:03:39.740 align:middle line:84% And then delta over 1 plus delta for opponent. 01:03:39.740 --> 01:03:43.120 align:middle line:90% 01:03:43.120 --> 01:03:44.640 align:middle line:90% So I'm proposing. 01:03:44.640 --> 01:03:46.220 align:middle line:90% I'm proposing a split. 01:03:46.220 --> 01:03:51.660 align:middle line:84% Remember, if we look at this line, along this line, 01:03:51.660 --> 01:03:54.020 align:middle line:84% the sum of the components of this vector is always 1. 01:03:54.020 --> 01:03:55.853 align:middle line:84% So we're basically thinking, how do we split 01:03:55.853 --> 01:03:57.380 align:middle line:90% the 1 unit of utility we have. 01:03:57.380 --> 01:04:00.220 align:middle line:84% And I'm going to propose to keep 1 over 1 01:04:00.220 --> 01:04:03.900 align:middle line:84% plus delta for myself as the proposer, and delta over 1 01:04:03.900 --> 01:04:06.118 align:middle line:90% plus delta for the opponent. 01:04:06.118 --> 01:04:08.660 align:middle line:84% And let's just understand how this fits in with our intuition 01:04:08.660 --> 01:04:12.380 align:middle line:84% that we said before, if delta gets very close to 1, 01:04:12.380 --> 01:04:14.980 align:middle line:84% then I'm basically proposing an even split. 01:04:14.980 --> 01:04:17.380 align:middle line:90% It's very close to 1/2, 1/2. 01:04:17.380 --> 01:04:19.980 align:middle line:84% If delta gets very close to 0, then I'm 01:04:19.980 --> 01:04:22.340 align:middle line:84% basically taking everything for myself. 01:04:22.340 --> 01:04:26.460 align:middle line:84% So this comports with the intuition that we presented. 01:04:26.460 --> 01:04:29.780 align:middle line:84% So in any history, the proposal proposes this. 01:04:29.780 --> 01:04:34.040 align:middle line:84% And after any proposal, what does the respondent do? 01:04:34.040 --> 01:04:37.580 align:middle line:90% 01:04:37.580 --> 01:04:41.140 align:middle line:84% Remember, we can't just say what the respondent does in response 01:04:41.140 --> 01:04:44.420 align:middle line:84% to this proposal because we have to specify the respondent 01:04:44.420 --> 01:04:47.780 align:middle line:84% strategy at every history, including some proposals 01:04:47.780 --> 01:04:50.100 align:middle line:84% that we don't expect to be actually be made. 01:04:50.100 --> 01:04:56.440 align:middle line:84% So let's say the respondent faces 01:04:56.440 --> 01:05:01.840 align:middle line:84% some proposal, any proposal, maybe 01:05:01.840 --> 01:05:03.340 align:middle line:90% I'll say an arbitrary proposal. 01:05:03.340 --> 01:05:06.320 align:middle line:90% 01:05:06.320 --> 01:05:10.080 align:middle line:84% Any intuition about, well, certainly the respondent 01:05:10.080 --> 01:05:15.260 align:middle line:84% should accept if they're offered enough and reject otherwise. 01:05:15.260 --> 01:05:19.720 align:middle line:90% 01:05:19.720 --> 01:05:21.800 align:middle line:90% So what is enough? 01:05:21.800 --> 01:05:25.060 align:middle line:84% Accept if-- it's always tricky with the notation, 01:05:25.060 --> 01:05:27.440 align:middle line:84% so what I mean is, accept if what's 01:05:27.440 --> 01:05:29.760 align:middle line:84% offered to the respondent-- the respondents component 01:05:29.760 --> 01:05:32.260 align:middle line:90% of that vector is large enough. 01:05:32.260 --> 01:05:32.760 align:middle line:90% Maybe 01:05:32.760 --> 01:05:39.000 align:middle line:84% I'll say, accept if she gets at least what? 01:05:39.000 --> 01:05:41.260 align:middle line:84% So meaning there's some offer that's made. 01:05:41.260 --> 01:05:43.743 align:middle line:84% The offer specifies a utility for both players. 01:05:43.743 --> 01:05:45.160 align:middle line:84% What the respondent cares about is 01:05:45.160 --> 01:05:48.280 align:middle line:84% the utility that she's offered, and she gets at least what? 01:05:48.280 --> 01:05:51.540 align:middle line:84% What is the critical value she's going to need to get? 01:05:51.540 --> 01:05:56.700 align:middle line:90% 01:05:56.700 --> 01:05:58.820 align:middle line:90% Any thoughts? 01:05:58.820 --> 01:06:01.140 align:middle line:84% Well, in this equilibrium, the proposer 01:06:01.140 --> 01:06:03.020 align:middle line:84% is giving this much to the other player. 01:06:03.020 --> 01:06:06.220 align:middle line:84% So why are they giving the other player this much? 01:06:06.220 --> 01:06:09.500 align:middle line:84% That's the least they can give them that they'll still accept. 01:06:09.500 --> 01:06:12.960 align:middle line:84% So it's exactly going to be delta over 1 plus delta. 01:06:12.960 --> 01:06:16.100 align:middle line:90% 01:06:16.100 --> 01:06:17.760 align:middle line:84% So let's just understand the structure. 01:06:17.760 --> 01:06:20.880 align:middle line:84% When I'm proposer, I anticipate that the other player, 01:06:20.880 --> 01:06:24.180 align:middle line:84% the respondent, will only accept my offer if I offer to give them 01:06:24.180 --> 01:06:26.060 align:middle line:90% at least this much utility. 01:06:26.060 --> 01:06:27.200 align:middle line:90% So what am I going to do? 01:06:27.200 --> 01:06:29.237 align:middle line:84% Well, I could give them more utility than this, 01:06:29.237 --> 01:06:31.320 align:middle line:84% and they'd accept it, but I don't want to do that. 01:06:31.320 --> 01:06:33.900 align:middle line:90% That leaves less utility for me. 01:06:33.900 --> 01:06:37.500 align:middle line:84% So I'm going to give them the smallest amount of utility 01:06:37.500 --> 01:06:40.780 align:middle line:90% that they will actually accept. 01:06:40.780 --> 01:06:42.680 align:middle line:84% And this is going to be our strategy. 01:06:42.680 --> 01:06:44.480 align:middle line:84% So remember, this specifies a lot. 01:06:44.480 --> 01:06:47.260 align:middle line:84% If we're at some really complicated history, whatever's 01:06:47.260 --> 01:06:49.180 align:middle line:84% happened, whoever's the proposer is going 01:06:49.180 --> 01:06:51.760 align:middle line:90% to propose exactly this split. 01:06:51.760 --> 01:06:54.100 align:middle line:84% And if we're down here, whatever's offered, 01:06:54.100 --> 01:06:55.600 align:middle line:84% the respondent is going to follow up 01:06:55.600 --> 01:06:56.740 align:middle line:90% with exactly this strategy. 01:06:56.740 --> 01:06:58.780 align:middle line:84% So this is a complete contingent plan in the game. 01:06:58.780 --> 01:07:00.613 align:middle line:84% I didn't write it out really mathematically, 01:07:00.613 --> 01:07:02.880 align:middle line:84% but it's capturing a lot of complexity here. 01:07:02.880 --> 01:07:03.440 align:middle line:90% Yes. 01:07:03.440 --> 01:07:06.160 align:middle line:84% AUDIENCE: I think as long as the proposer proposes [INAUDIBLE] 01:07:06.160 --> 01:07:07.625 align:middle line:90% the first step, the game ends. 01:07:07.625 --> 01:07:09.000 align:middle line:84% IAN BALL: Exactly right, which is 01:07:09.000 --> 01:07:12.120 align:middle line:84% exactly what we found in the finite horizon game as well. 01:07:12.120 --> 01:07:14.520 align:middle line:84% We said there's no disagreement, there's no delay. 01:07:14.520 --> 01:07:17.460 align:middle line:84% We immediately end in the first period, 01:07:17.460 --> 01:07:19.880 align:middle line:84% and we immediately end with these splits. 01:07:19.880 --> 01:07:21.855 align:middle line:84% And these splits exactly correspond 01:07:21.855 --> 01:07:23.980 align:middle line:84% to the points along the line that I was describing. 01:07:23.980 --> 01:07:26.720 align:middle line:84% So if these strategies are followed, then, 01:07:26.720 --> 01:07:30.400 align:middle line:84% in the first period, player 1 proposes this split. 01:07:30.400 --> 01:07:33.020 align:middle line:84% And therefore, player 1 gets 1 over 1 plus delta. 01:07:33.020 --> 01:07:34.840 align:middle line:84% And as we said, if delta is close to 0, 01:07:34.840 --> 01:07:36.180 align:middle line:90% this converges to 1/2. 01:07:36.180 --> 01:07:37.520 align:middle line:90% If delta is close to-- 01:07:37.520 --> 01:07:40.520 align:middle line:84% sorry, if delta's close to 0, this converges to 1. 01:07:40.520 --> 01:07:43.760 align:middle line:84% And if delta converges to 1, this converges to 1/2. 01:07:43.760 --> 01:07:49.580 align:middle line:84% So it's exactly what we talked about down here. 01:07:49.580 --> 01:07:53.060 align:middle line:84% OK, so now we want to verify that this 01:07:53.060 --> 01:07:57.040 align:middle line:84% is a subgame-perfect Nash equilibrium. 01:07:57.040 --> 01:08:16.700 align:middle line:90% 01:08:16.700 --> 01:08:18.420 align:middle line:90% So what do we need to do? 01:08:18.420 --> 01:08:20.740 align:middle line:84% We need to check that the proposer has 01:08:20.740 --> 01:08:23.020 align:middle line:84% no profitable one-shot deviation. 01:08:23.020 --> 01:08:26.140 align:middle line:84% And the responder has no profitable one-shot deviation. 01:08:26.140 --> 01:08:29.540 align:middle line:90% So we need to check. 01:08:29.540 --> 01:08:36.160 align:middle line:84% No profitable one-shot deviations. 01:08:36.160 --> 01:08:40.380 align:middle line:90% 01:08:40.380 --> 01:08:43.979 align:middle line:90% So let's start with proposer. 01:08:43.979 --> 01:08:47.800 align:middle line:84% There's a few things they could do. 01:08:47.800 --> 01:08:54.080 align:middle line:84% They could, let's say, offer the responder more than delta 01:08:54.080 --> 01:08:55.520 align:middle line:90% over 1 plus delta. 01:08:55.520 --> 01:08:59.160 align:middle line:84% Or they could offer the responder less than delta 01:08:59.160 --> 01:09:00.149 align:middle line:90% over 1 plus delta. 01:09:00.149 --> 01:09:02.399 align:middle line:84% There's a lot of things they could do, but let me just 01:09:02.399 --> 01:09:05.080 align:middle line:90% organize them in this way. 01:09:05.080 --> 01:09:08.479 align:middle line:84% So suppose I'm the proposer, and I deviate, at some history, 01:09:08.479 --> 01:09:13.460 align:middle line:84% by offering the respondent less than delta over 1 plus delta. 01:09:13.460 --> 01:09:14.460 align:middle line:90% Well, then what happens? 01:09:14.460 --> 01:09:18.439 align:middle line:90% 01:09:18.439 --> 01:09:23.200 align:middle line:84% Well, according to the respondent's strategy, 01:09:23.200 --> 01:09:24.700 align:middle line:84% this offer is going to be rejected. 01:09:24.700 --> 01:09:31.800 align:middle line:90% 01:09:31.800 --> 01:09:35.600 align:middle line:84% So if this offer is rejected, what payoff 01:09:35.600 --> 01:09:37.840 align:middle line:84% am I going to get as the proposer? 01:09:37.840 --> 01:09:39.840 align:middle line:90% Well, today, I'm the proposer. 01:09:39.840 --> 01:09:42.880 align:middle line:84% I've proposed an offer that's been rejected. 01:09:42.880 --> 01:09:45.340 align:middle line:84% That means, tomorrow, the person who rejected my offer 01:09:45.340 --> 01:09:47.939 align:middle line:84% is going to propose something tomorrow. 01:09:47.939 --> 01:09:50.500 align:middle line:84% And then I have to decide what to do. 01:09:50.500 --> 01:09:53.540 align:middle line:84% But we're only looking at one-shot deviations. 01:09:53.540 --> 01:09:56.980 align:middle line:84% So if I'm the proposer, I'm only deviating here. 01:09:56.980 --> 01:09:58.860 align:middle line:84% So once we get to tomorrow, I'm going 01:09:58.860 --> 01:10:01.220 align:middle line:84% to follow the strategy I'm supposed to follow. 01:10:01.220 --> 01:10:03.800 align:middle line:84% So if I follow the strategy I'm supposed to follow tomorrow, 01:10:03.800 --> 01:10:06.260 align:middle line:84% because that's what a one-shot deviation means, then, 01:10:06.260 --> 01:10:10.940 align:middle line:84% tomorrow, well, I'm going to be offered delta over 1 plus delta. 01:10:10.940 --> 01:10:12.780 align:middle line:84% And I'm going to accept it because this 01:10:12.780 --> 01:10:14.460 align:middle line:84% is-- if I didn't accept it, that would 01:10:14.460 --> 01:10:17.540 align:middle line:84% be a multi-shot deviation, which we don't have to worry about. 01:10:17.540 --> 01:10:19.780 align:middle line:90% So it's going to be rejected. 01:10:19.780 --> 01:10:28.740 align:middle line:84% So tomorrow, I'm going to be offered delta over 1 plus delta. 01:10:28.740 --> 01:10:32.020 align:middle line:90% And I'm going to accept. 01:10:32.020 --> 01:10:34.780 align:middle line:90% Notice the trickiness. 01:10:34.780 --> 01:10:38.140 align:middle line:84% Whether I want to offer less than delta over 1 01:10:38.140 --> 01:10:39.740 align:middle line:84% plus delta to the other player today 01:10:39.740 --> 01:10:42.960 align:middle line:84% depends on how I'm going to play in the rest of the game. 01:10:42.960 --> 01:10:44.720 align:middle line:84% The one-shot deviation principle allows 01:10:44.720 --> 01:10:46.400 align:middle line:84% us to take as fixed how I'm going 01:10:46.400 --> 01:10:48.560 align:middle line:84% to play in the rest of the game and focus on a very 01:10:48.560 --> 01:10:50.220 align:middle line:90% specific kind of deviation. 01:10:50.220 --> 01:10:52.840 align:middle line:84% And it's easier to check that this very specific deviation's 01:10:52.840 --> 01:10:53.687 align:middle line:90% not profitable. 01:10:53.687 --> 01:10:55.520 align:middle line:84% There might be really complicated deviations 01:10:55.520 --> 01:10:57.793 align:middle line:90% where I offer too little today. 01:10:57.793 --> 01:10:59.460 align:middle line:84% Then I also do something weird tomorrow. 01:10:59.460 --> 01:11:00.820 align:middle line:84% Then I do something weird the next day. 01:11:00.820 --> 01:11:03.487 align:middle line:84% But we don't have to worry about these by the one-shot deviation 01:11:03.487 --> 01:11:04.040 align:middle line:90% principle. 01:11:04.040 --> 01:11:06.980 align:middle line:84% So what happens if tomorrow I'm offered delta over 1 plus delta, 01:11:06.980 --> 01:11:07.940 align:middle line:90% and I accept? 01:11:07.940 --> 01:11:10.380 align:middle line:84% What is my discounted payoff going to be? 01:11:10.380 --> 01:11:14.680 align:middle line:90% 01:11:14.680 --> 01:11:19.920 align:middle line:84% Discounted into today's units, what will my payoff be? 01:11:19.920 --> 01:11:22.480 align:middle line:84% Well, I'm getting delta over 1 plus delta tomorrow, 01:11:22.480 --> 01:11:24.600 align:middle line:84% but I have to discount that back to today. 01:11:24.600 --> 01:11:26.700 align:middle line:84% So today it's going to be delta times that. 01:11:26.700 --> 01:11:30.780 align:middle line:84% So my payoff is going to be delta squared over 1 plus delta. 01:11:30.780 --> 01:11:33.840 align:middle line:90% 01:11:33.840 --> 01:11:37.300 align:middle line:84% But if I don't deviate, I get 1 over 1 plus delta. 01:11:37.300 --> 01:11:38.640 align:middle line:90% So that's certainly better. 01:11:38.640 --> 01:11:40.500 align:middle line:84% So this deviation is not profitable. 01:11:40.500 --> 01:11:41.800 align:middle line:90% It's strictly unprofitable. 01:11:41.800 --> 01:11:47.940 align:middle line:90% 01:11:47.940 --> 01:11:50.788 align:middle line:84% What if I offer more than delta over 1 01:11:50.788 --> 01:11:52.080 align:middle line:90% plus delta to the other player? 01:11:52.080 --> 01:11:52.920 align:middle line:90% Can this be profitable? 01:11:52.920 --> 01:11:53.878 align:middle line:90% What's going to happen? 01:11:53.878 --> 01:11:58.340 align:middle line:90% 01:11:58.340 --> 01:11:58.847 align:middle line:90% Yeah. 01:11:58.847 --> 01:12:00.180 align:middle line:90% AUDIENCE: [INAUDIBLE] to accept. 01:12:00.180 --> 01:12:01.440 align:middle line:84% IAN BALL: Right, so it'll be accepted. 01:12:01.440 --> 01:12:02.160 align:middle line:90% And then what? 01:12:02.160 --> 01:12:03.700 align:middle line:90% But is that good for me? 01:12:03.700 --> 01:12:07.740 align:middle line:84% AUDIENCE: No, because now you're only getting like-- 01:12:07.740 --> 01:12:11.880 align:middle line:84% now you're losing R minus delta over 1 plus delta. 01:12:11.880 --> 01:12:15.393 align:middle line:84% IAN BALL: Exactly, so if I give the responder more than delta 01:12:15.393 --> 01:12:18.580 align:middle line:84% over 1 plus delta, I can't be getting more than 1 01:12:18.580 --> 01:12:20.940 align:middle line:84% over 1 plus delta, so this must be worse 01:12:20.940 --> 01:12:25.940 align:middle line:90% for me and unprofitable. 01:12:25.940 --> 01:12:28.360 align:middle line:84% So you can write out all the math really carefully, 01:12:28.360 --> 01:12:31.180 align:middle line:84% but I think it's better to just think through the story. 01:12:31.180 --> 01:12:34.560 align:middle line:84% So what are the two kinds of deviations we're considering? 01:12:34.560 --> 01:12:36.700 align:middle line:84% We're considering one-shot deviations where 01:12:36.700 --> 01:12:40.080 align:middle line:84% the proposer proposed a different offer today but then 01:12:40.080 --> 01:12:41.600 align:middle line:84% follows their equilibrium strategy, 01:12:41.600 --> 01:12:45.240 align:middle line:84% or their candidate equilibrium strategy, forever after. 01:12:45.240 --> 01:12:48.120 align:middle line:84% One kind of deviation is a deviation 01:12:48.120 --> 01:12:50.360 align:middle line:84% where they are very generous, and they 01:12:50.360 --> 01:12:53.000 align:middle line:84% offer to give more to the other player. 01:12:53.000 --> 01:12:55.060 align:middle line:84% The other player is going to accept this offer, 01:12:55.060 --> 01:12:57.120 align:middle line:84% but I'm hurting myself as the proposer by being 01:12:57.120 --> 01:12:59.260 align:middle line:84% too generous, more generous than I need to be, 01:12:59.260 --> 01:13:01.400 align:middle line:84% so this deviation won't be profitable. 01:13:01.400 --> 01:13:03.120 align:middle line:84% Another thing I could do is I could 01:13:03.120 --> 01:13:06.260 align:middle line:84% offer them less than what I'm supposed to in equilibrium. 01:13:06.260 --> 01:13:07.800 align:middle line:90% I could be too stingy. 01:13:07.800 --> 01:13:10.760 align:middle line:84% If I'm too stingy, the offer will be rejected. 01:13:10.760 --> 01:13:12.800 align:middle line:84% That means, tomorrow, the other player 01:13:12.800 --> 01:13:14.760 align:middle line:90% will make a counteroffer to me. 01:13:14.760 --> 01:13:19.200 align:middle line:84% The counteroffer to me is going to be delta over 1 plus delta. 01:13:19.200 --> 01:13:21.120 align:middle line:84% And I'm going to accept it precisely 01:13:21.120 --> 01:13:23.100 align:middle line:84% because this is a one-shot deviation. 01:13:23.100 --> 01:13:25.780 align:middle line:84% And not accepting it would be a multi-shot deviation. 01:13:25.780 --> 01:13:28.168 align:middle line:84% But if I accept it, I actually get a lower payoff, 01:13:28.168 --> 01:13:29.960 align:middle line:84% and it's farther in the future, which means 01:13:29.960 --> 01:13:31.480 align:middle line:90% it's definitely worse for me. 01:13:31.480 --> 01:13:33.680 align:middle line:84% So we've checked that the proposer does not 01:13:33.680 --> 01:13:35.780 align:middle line:84% have any profitable one-shot deviations. 01:13:35.780 --> 01:13:38.550 align:middle line:84% We have not directly checked multi-shot deviations 01:13:38.550 --> 01:13:41.230 align:middle line:84% where the proposer is too stingy today 01:13:41.230 --> 01:13:43.130 align:middle line:84% and then also rejects the offer tomorrow 01:13:43.130 --> 01:13:44.810 align:middle line:84% and then does more complicated things. 01:13:44.810 --> 01:13:46.310 align:middle line:84% But the one-shot deviation principle 01:13:46.310 --> 01:13:48.710 align:middle line:84% says we don't have to worry about that. 01:13:48.710 --> 01:13:51.750 align:middle line:90% Any questions on that? 01:13:51.750 --> 01:13:56.350 align:middle line:84% OK, now let's look at the other side. 01:13:56.350 --> 01:13:57.650 align:middle line:90% Let's look at the responder. 01:13:57.650 --> 01:14:16.070 align:middle line:90% 01:14:16.070 --> 01:14:20.010 align:middle line:84% The responder is offered some amount. 01:14:20.010 --> 01:14:22.790 align:middle line:90% 01:14:22.790 --> 01:14:26.150 align:middle line:90% And there's really two cases. 01:14:26.150 --> 01:14:29.830 align:middle line:84% So let's say the responder is offered 01:14:29.830 --> 01:14:32.530 align:middle line:84% at least-- so we're going to split the histories in two. 01:14:32.530 --> 01:14:40.490 align:middle line:90% 01:14:40.490 --> 01:14:43.210 align:middle line:84% So there's some class of histories where a lot of things 01:14:43.210 --> 01:14:43.710 align:middle line:90% happen. 01:14:43.710 --> 01:14:45.390 align:middle line:84% We don't worry about it, but then, at the end of it, 01:14:45.390 --> 01:14:47.090 align:middle line:84% I'm the responder, and I'm offered 01:14:47.090 --> 01:14:50.290 align:middle line:84% some amount that's weakly greater than delta 01:14:50.290 --> 01:14:51.890 align:middle line:90% over 1 plus delta. 01:14:51.890 --> 01:14:54.110 align:middle line:84% And what I'm supposed to do is accept this. 01:14:54.110 --> 01:15:00.827 align:middle line:90% 01:15:00.827 --> 01:15:02.410 align:middle line:84% And then there's other histories where 01:15:02.410 --> 01:15:04.890 align:middle line:84% I'm offered strictly less than delta over 1 plus delta. 01:15:04.890 --> 01:15:07.630 align:middle line:84% And again, I can choose whether to accept or reject. 01:15:07.630 --> 01:15:11.530 align:middle line:84% But I'm supposed to reject these. 01:15:11.530 --> 01:15:15.690 align:middle line:84% And we need to check that I don't 01:15:15.690 --> 01:15:18.450 align:middle line:84% have a profitable deviation, a profitable one-shot deviation 01:15:18.450 --> 01:15:19.430 align:middle line:90% as the responder. 01:15:19.430 --> 01:15:23.010 align:middle line:90% 01:15:23.010 --> 01:15:25.610 align:middle line:90% So when I'm offered-- 01:15:25.610 --> 01:15:29.510 align:middle line:84% it's easy to compute what I get when I accept these offers. 01:15:29.510 --> 01:15:30.578 align:middle line:90% So let's write that down. 01:15:30.578 --> 01:15:32.370 align:middle line:84% If I'm offered more than this and I accept, 01:15:32.370 --> 01:15:34.570 align:middle line:84% then I'm going to get whatever this amount is 01:15:34.570 --> 01:15:36.910 align:middle line:84% that's greater than or equal to delta over 1 plus delta. 01:15:36.910 --> 01:15:38.993 align:middle line:84% And over here, I'm going to get this amount that's 01:15:38.993 --> 01:15:42.290 align:middle line:84% less than delta over 1 plus delta. 01:15:42.290 --> 01:15:43.790 align:middle line:84% Now, the harder case is figuring out 01:15:43.790 --> 01:15:46.810 align:middle line:84% what happens if I reject because if I reject, 01:15:46.810 --> 01:15:48.870 align:middle line:84% my payoff depends on what happens tomorrow. 01:15:48.870 --> 01:15:51.850 align:middle line:84% But again, we're only looking at one-shot deviations. 01:15:51.850 --> 01:15:54.130 align:middle line:84% So if I deviate here by rejecting, 01:15:54.130 --> 01:15:56.510 align:middle line:84% we can consider what happens tomorrow when I follow 01:15:56.510 --> 01:15:58.230 align:middle line:90% the strategy I'm supposed to. 01:15:58.230 --> 01:16:04.430 align:middle line:84% So if I reject as the responder, tomorrow, I become the proposer. 01:16:04.430 --> 01:16:06.210 align:middle line:90% And now I follow my strategy. 01:16:06.210 --> 01:16:09.630 align:middle line:84% I propose to keep 1 over 1 plus delta for myself 01:16:09.630 --> 01:16:11.930 align:middle line:84% and give delta over 1 plus delta to the other player, 01:16:11.930 --> 01:16:13.972 align:middle line:84% and the other player has to accept because that's 01:16:13.972 --> 01:16:15.030 align:middle line:90% what their strategy says. 01:16:15.030 --> 01:16:23.230 align:middle line:84% So then, tomorrow, I'm going to get 1 over 1 plus delta 01:16:23.230 --> 01:16:25.350 align:middle line:90% as the proposer. 01:16:25.350 --> 01:16:28.390 align:middle line:84% But that's going to be-- the discounted 01:16:28.390 --> 01:16:35.870 align:middle line:84% value of that equals delta over 1 plus delta. 01:16:35.870 --> 01:16:38.530 align:middle line:90% 01:16:38.530 --> 01:16:41.450 align:middle line:84% So 1 over 1 plus delta tomorrow is only worth delta times 01:16:41.450 --> 01:16:42.690 align:middle line:90% that to me today. 01:16:42.690 --> 01:16:45.410 align:middle line:84% But if I accept, I'm doing weakly better than that. 01:16:45.410 --> 01:16:47.230 align:middle line:84% So this deviation's not profitable. 01:16:47.230 --> 01:16:47.870 align:middle line:90% So we're good. 01:16:47.870 --> 01:16:58.170 align:middle line:90% 01:16:58.170 --> 01:17:04.090 align:middle line:84% Let's look what happens if I reject the offer. 01:17:04.090 --> 01:17:05.975 align:middle line:84% Well, we know if I reject the offer, 01:17:05.975 --> 01:17:07.350 align:middle line:84% I'm going to become the proposer. 01:17:07.350 --> 01:17:08.975 align:middle line:84% Exactly the same argument goes through. 01:17:08.975 --> 01:17:10.590 align:middle line:84% In fact, this is not even a deviation. 01:17:10.590 --> 01:17:12.170 align:middle line:84% This is just saying, what happens 01:17:12.170 --> 01:17:16.030 align:middle line:84% if I follow my equilibrium strategy and reject? 01:17:16.030 --> 01:17:19.570 align:middle line:84% I'm exactly going to get a discounted value of delta 01:17:19.570 --> 01:17:22.530 align:middle line:90% over 1 plus delta. 01:17:22.530 --> 01:17:25.130 align:middle line:84% And if I deviate and accept this offer, 01:17:25.130 --> 01:17:27.470 align:middle line:84% I get strictly less than delta over 1 plus delta. 01:17:27.470 --> 01:17:29.550 align:middle line:90% So again, it's not profitable. 01:17:29.550 --> 01:17:30.688 align:middle line:90% So we're good. 01:17:30.688 --> 01:17:31.730 align:middle line:90% Let's just go through it. 01:17:31.730 --> 01:17:33.270 align:middle line:84% I think the algebra can sometimes 01:17:33.270 --> 01:17:34.562 align:middle line:90% make it seem harder than it is. 01:17:34.562 --> 01:17:35.750 align:middle line:90% So let's try to understand. 01:17:35.750 --> 01:17:38.350 align:middle line:84% Suppose I'm offered an amount less than delta 01:17:38.350 --> 01:17:39.870 align:middle line:90% over 1 plus delta. 01:17:39.870 --> 01:17:43.030 align:middle line:84% My equilibrium strategy says, I should reject that offer, 01:17:43.030 --> 01:17:47.030 align:middle line:84% propose the equilibrium split tomorrow, and therefore get 01:17:47.030 --> 01:17:49.470 align:middle line:84% 1 over 1 plus delta tomorrow, which has discounted 01:17:49.470 --> 01:17:51.390 align:middle line:90% value delta over 1 plus delta. 01:17:51.390 --> 01:17:53.182 align:middle line:84% If I deviate, well, here we don't even 01:17:53.182 --> 01:17:55.390 align:middle line:84% have to worry about whether it's a one-shot deviation 01:17:55.390 --> 01:17:58.150 align:middle line:84% or not because if I deviate here, the game ends immediately, 01:17:58.150 --> 01:18:00.450 align:middle line:84% and I get strictly less than delta over 1 plus delta. 01:18:00.450 --> 01:18:03.590 align:middle line:84% So my deviation is strictly unprofitable. 01:18:03.590 --> 01:18:06.750 align:middle line:84% And now we've confirmed that this 01:18:06.750 --> 01:18:10.830 align:middle line:84% constitutes a subgame-perfect Nash equilibrium. 01:18:10.830 --> 01:18:12.630 align:middle line:84% And in fact, you can show that this 01:18:12.630 --> 01:18:14.958 align:middle line:84% is the unique subgame-perfect Nash equilibrium. 01:18:14.958 --> 01:18:16.250 align:middle line:90% But that's a bit more involved. 01:18:16.250 --> 01:18:19.030 align:middle line:90% And I might show it next class. 01:18:19.030 --> 01:18:21.290 align:middle line:84% And I think I'm perfectly out of time. 01:18:21.290 --> 01:18:22.930 align:middle line:90% So let me stop there. 01:18:22.930 --> 01:18:24.480 align:middle line:90% Thanks. 01:18:24.480 --> 01:18:28.000 align:middle line:90%