WEBVTT 00:00:00.000 --> 00:00:01.985 align:middle line:90% [SQUEAKING] 00:00:01.985 --> 00:00:03.965 align:middle line:90% [RUSTLING] 00:00:03.965 --> 00:00:06.440 align:middle line:90% [CLICKING] 00:00:06.440 --> 00:00:10.400 align:middle line:90% 00:00:10.400 --> 00:00:15.760 align:middle line:84% IAN BALL: Today, we're going to start with finitely repeated 00:00:15.760 --> 00:00:16.420 align:middle line:90% games. 00:00:16.420 --> 00:00:23.360 align:middle line:90% 00:00:23.360 --> 00:00:25.740 align:middle line:84% And I want to just start with a really simple example, 00:00:25.740 --> 00:00:27.460 align:middle line:84% and we'll actually play a game together. 00:00:27.460 --> 00:00:30.920 align:middle line:84% And then we'll develop the general theory of this. 00:00:30.920 --> 00:00:35.480 align:middle line:84% So let's recall the standard prisoner's dilemma game where 00:00:35.480 --> 00:00:36.998 align:middle line:90% each player had two choices. 00:00:36.998 --> 00:00:39.040 align:middle line:84% They could either cooperate with the other player 00:00:39.040 --> 00:00:40.520 align:middle line:90% or they could defect. 00:00:40.520 --> 00:00:42.520 align:middle line:84% And the payoffs that we wrote down 00:00:42.520 --> 00:00:49.680 align:middle line:84% are going to be 2, 2; 3 negative 1; negative 1, 3. 00:00:49.680 --> 00:00:52.480 align:middle line:84% And what we saw in this game is that for both players, 00:00:52.480 --> 00:00:57.080 align:middle line:84% defecting is a strictly dominant strategy. 00:00:57.080 --> 00:01:00.360 align:middle line:84% So if the opponent cooperates, I'm 00:01:00.360 --> 00:01:03.120 align:middle line:84% better off if I defect because 3 is better than 2. 00:01:03.120 --> 00:01:05.200 align:middle line:84% And if the opponent defects, I'm still better off 00:01:05.200 --> 00:01:09.040 align:middle line:84% if I defect because 0 is greater than negative 1. 00:01:09.040 --> 00:01:11.960 align:middle line:84% But so far, we've assumed that this game is played only once. 00:01:11.960 --> 00:01:13.520 align:middle line:84% And in reality, a lot of interactions 00:01:13.520 --> 00:01:14.780 align:middle line:90% take place over time. 00:01:14.780 --> 00:01:17.220 align:middle line:84% You play a game with someone, you see what happens, 00:01:17.220 --> 00:01:18.500 align:middle line:90% then you play the game again. 00:01:18.500 --> 00:01:19.583 align:middle line:90% Then you see what happens. 00:01:19.583 --> 00:01:21.800 align:middle line:90% Then you play the game again. 00:01:21.800 --> 00:01:24.440 align:middle line:84% And today, we're going to study what happens 00:01:24.440 --> 00:01:26.440 align:middle line:84% and how our conclusions change when 00:01:26.440 --> 00:01:28.720 align:middle line:84% we're in this kind of repeated situation. 00:01:28.720 --> 00:01:31.020 align:middle line:84% So let's start with a really simple example. 00:01:31.020 --> 00:01:35.440 align:middle line:84% Let's play the prisoner's dilemma three times. 00:01:35.440 --> 00:01:37.260 align:middle line:90% So how does this work? 00:01:37.260 --> 00:01:39.640 align:middle line:84% You're going to simultaneously play the game. 00:01:39.640 --> 00:01:42.680 align:middle line:84% Then both players observe the actions that were taken. 00:01:42.680 --> 00:01:44.320 align:middle line:90% Then the game is played again. 00:01:44.320 --> 00:01:46.460 align:middle line:84% Both players observe the actions that were taken. 00:01:46.460 --> 00:01:48.800 align:middle line:84% And then finally, the game is played a third time. 00:01:48.800 --> 00:01:53.400 align:middle line:84% And we'll assume that your payoffs in this grand game 00:01:53.400 --> 00:01:55.480 align:middle line:84% are just the sum of your payoffs, 00:01:55.480 --> 00:01:58.060 align:middle line:84% or maybe the average payoffs across the games. 00:01:58.060 --> 00:02:05.000 align:middle line:90% 00:02:05.000 --> 00:02:08.419 align:middle line:84% So if you get 2 in each repetition of the game, 00:02:08.419 --> 00:02:10.028 align:middle line:90% your average payoff is 2. 00:02:10.028 --> 00:02:10.820 align:middle line:90% I think it's good-- 00:02:10.820 --> 00:02:11.945 align:middle line:90% It's fun to just play this. 00:02:11.945 --> 00:02:13.400 align:middle line:84% So we could do it on MobLab, but I 00:02:13.400 --> 00:02:15.400 align:middle line:84% feel that if you look someone in the eye 00:02:15.400 --> 00:02:17.858 align:middle line:84% when you're playing the game, it changes the results a bit. 00:02:17.858 --> 00:02:20.683 align:middle line:84% So I want you to pair off, and you need a way to play this. 00:02:20.683 --> 00:02:22.600 align:middle line:84% I think the easiest way is to think of this as 00:02:22.600 --> 00:02:23.725 align:middle line:90% like rock, paper, scissors. 00:02:23.725 --> 00:02:25.480 align:middle line:90% So just agree, maybe rock-- 00:02:25.480 --> 00:02:28.440 align:middle line:84% if you need help-- let's say playing rock is defecting 00:02:28.440 --> 00:02:31.620 align:middle line:84% and maybe playing paper is cooperating. 00:02:31.620 --> 00:02:34.360 align:middle line:84% So then it's just a variant of rock, paper, scissors. 00:02:34.360 --> 00:02:34.860 align:middle line:90% Play it. 00:02:34.860 --> 00:02:35.700 align:middle line:90% Observe the outcome. 00:02:35.700 --> 00:02:36.980 align:middle line:84% You play it again, observe the outcome. 00:02:36.980 --> 00:02:37.500 align:middle line:90% You play it again. 00:02:37.500 --> 00:02:38.600 align:middle line:90% So three times in total. 00:02:38.600 --> 00:02:39.140 align:middle line:90% Pair up. 00:02:39.140 --> 00:02:40.682 align:middle line:84% Maybe one of you will be by yourself. 00:02:40.682 --> 00:02:42.520 align:middle line:90% You can observe other people. 00:02:42.520 --> 00:02:45.500 align:middle line:84% And let's do it and see what happens. 00:02:45.500 --> 00:02:51.000 align:middle line:90% 00:02:51.000 --> 00:02:51.500 align:middle line:90% All right. 00:02:51.500 --> 00:02:52.250 align:middle line:90% So let's see how-- 00:02:52.250 --> 00:02:53.560 align:middle line:90% I'm curious how people did. 00:02:53.560 --> 00:02:58.580 align:middle line:84% So I hope most groups got a chance to play. 00:02:58.580 --> 00:03:00.693 align:middle line:90% So tell me, how did people play? 00:03:00.693 --> 00:03:02.360 align:middle line:84% Does anyone want to share what you found 00:03:02.360 --> 00:03:04.480 align:middle line:90% when you played this game? 00:03:04.480 --> 00:03:05.180 align:middle line:90% Yes. 00:03:05.180 --> 00:03:07.360 align:middle line:84% AUDIENCE: We both cooperated on first try, 00:03:07.360 --> 00:03:09.800 align:middle line:84% so I think it has to do with cooperating. 00:03:09.800 --> 00:03:13.240 align:middle line:84% IAN BALL: So you cooperated in all three rounds, or-- 00:03:13.240 --> 00:03:14.540 align:middle line:90% OK, interesting. 00:03:14.540 --> 00:03:16.840 align:middle line:84% That's pretty good payoffs for you guys. 00:03:16.840 --> 00:03:19.360 align:middle line:84% So you each got an average payoff of 2. 00:03:19.360 --> 00:03:20.760 align:middle line:90% That's pretty good 00:03:20.760 --> 00:03:23.180 align:middle line:84% I would argue maybe someone was a little irrational there, 00:03:23.180 --> 00:03:23.763 align:middle line:90% but we'll see. 00:03:23.763 --> 00:03:25.415 align:middle line:90% Anyone else? 00:03:25.415 --> 00:03:26.540 align:middle line:90% Did other people cooperate? 00:03:26.540 --> 00:03:26.800 align:middle line:90% Yeah? 00:03:26.800 --> 00:03:28.467 align:middle line:84% AUDIENCE: We also cooperated every time, 00:03:28.467 --> 00:03:31.200 align:middle line:84% but then after we discussed that because we 00:03:31.200 --> 00:03:33.960 align:middle line:84% know each other in real life, like technically the game is 00:03:33.960 --> 00:03:35.540 align:middle line:90% infinitely repeated. 00:03:35.540 --> 00:03:37.453 align:middle line:84% So maybe that's why we cooperated. 00:03:37.453 --> 00:03:38.120 align:middle line:90% IAN BALL: Great. 00:03:38.120 --> 00:03:39.060 align:middle line:90% That's a great point. 00:03:39.060 --> 00:03:41.160 align:middle line:84% So we'll see actually that the theory of infinitely repeated 00:03:41.160 --> 00:03:43.280 align:middle line:84% games is much different than the theory of finitely repeated 00:03:43.280 --> 00:03:43.780 align:middle line:90% games. 00:03:43.780 --> 00:03:46.400 align:middle line:84% And Paul Milgrom loves to say the game is always larger 00:03:46.400 --> 00:03:47.100 align:middle line:90% than you think. 00:03:47.100 --> 00:03:48.140 align:middle line:84% So when you write down these games, 00:03:48.140 --> 00:03:50.040 align:middle line:84% often we're missing additional interactions 00:03:50.040 --> 00:03:52.153 align:middle line:90% that we might not be modeling. 00:03:52.153 --> 00:03:53.320 align:middle line:90% OK, so a lot of cooperation. 00:03:53.320 --> 00:03:56.480 align:middle line:84% Did anyone just defect every time? 00:03:56.480 --> 00:03:57.668 align:middle line:90% Yeah? 00:03:57.668 --> 00:03:59.460 align:middle line:84% AUDIENCE: Yeah, I defected all three times. 00:03:59.460 --> 00:04:00.240 align:middle line:90% IAN BALL: OK. 00:04:00.240 --> 00:04:00.780 align:middle line:90% Great. 00:04:00.780 --> 00:04:02.880 align:middle line:90% A paragon of rationality. 00:04:02.880 --> 00:04:04.443 align:middle line:90% And what was your reasoning? 00:04:04.443 --> 00:04:05.360 align:middle line:90% Did you have a reason? 00:04:05.360 --> 00:04:08.160 align:middle line:84% AUDIENCE: Well, I thought that Live would also defect all three 00:04:08.160 --> 00:04:10.920 align:middle line:84% times because I feel like that's what you're supposed to do. 00:04:10.920 --> 00:04:11.760 align:middle line:90% IAN BALL: OK. 00:04:11.760 --> 00:04:12.260 align:middle line:90% Great. 00:04:12.260 --> 00:04:14.427 align:middle line:84% So it seems like we have a few different approaches. 00:04:14.427 --> 00:04:16.279 align:middle line:84% Some people said, well, we understand 00:04:16.279 --> 00:04:19.120 align:middle line:84% that in the staged game, defecting is optimal. 00:04:19.120 --> 00:04:22.160 align:middle line:84% So let's just expect that the other player is going to defect 00:04:22.160 --> 00:04:24.060 align:middle line:84% and we're both going to defect every period. 00:04:24.060 --> 00:04:27.620 align:middle line:84% Other people seem to be able to sustain some cooperation. 00:04:27.620 --> 00:04:30.520 align:middle line:84% And maybe what you had in mind is if we cooperate today, 00:04:30.520 --> 00:04:33.000 align:middle line:84% then tomorrow, my opponent will reward me 00:04:33.000 --> 00:04:34.500 align:middle line:90% by cooperating in the future. 00:04:34.500 --> 00:04:37.280 align:middle line:84% We'll see that in the finite repeated game, 00:04:37.280 --> 00:04:40.320 align:middle line:84% the unique subgame perfect Nash equilibrium is actually 00:04:40.320 --> 00:04:42.578 align:middle line:84% always defecting, so as one of these groups played. 00:04:42.578 --> 00:04:44.120 align:middle line:84% And the way these other groups played 00:04:44.120 --> 00:04:46.520 align:middle line:84% is not actually consistent with subgame perfect Nash 00:04:46.520 --> 00:04:47.180 align:middle line:90% equilibrium. 00:04:47.180 --> 00:04:49.120 align:middle line:90% But let's understand why. 00:04:49.120 --> 00:04:51.960 align:middle line:84% So I think the first key observation here is that when 00:04:51.960 --> 00:04:54.430 align:middle line:90% we play a game repeatedly-- 00:04:54.430 --> 00:05:05.815 align:middle line:84% so repetition-- I guess I should say, together with observability 00:05:05.815 --> 00:05:09.170 align:middle line:84% creates the opportunity for rewards and punishments. 00:05:09.170 --> 00:05:22.438 align:middle line:90% 00:05:22.438 --> 00:05:24.230 align:middle line:84% I think what some of you might have thought 00:05:24.230 --> 00:05:27.270 align:middle line:84% is maybe if I cooperate today, I'll be rewarded, 00:05:27.270 --> 00:05:29.990 align:middle line:84% and my opponent will cooperate with me in the future 00:05:29.990 --> 00:05:31.650 align:middle line:84% and I'll get some benefit from that. 00:05:31.650 --> 00:05:33.990 align:middle line:84% Or conversely, if I defect today, 00:05:33.990 --> 00:05:37.590 align:middle line:84% my opponent might punish me by defecting in the future. 00:05:37.590 --> 00:05:40.030 align:middle line:84% Notice that we need both repetition and observability 00:05:40.030 --> 00:05:40.830 align:middle line:90% for this to happen. 00:05:40.830 --> 00:05:42.497 align:middle line:84% Of course, we need repetition because we 00:05:42.497 --> 00:05:45.190 align:middle line:84% need some future for people to reward or punish. 00:05:45.190 --> 00:05:47.350 align:middle line:84% And we need observability because the only way 00:05:47.350 --> 00:05:48.990 align:middle line:84% to reward and punish people is if you 00:05:48.990 --> 00:05:52.693 align:middle line:84% make your future play contingent on what was done in the past. 00:05:52.693 --> 00:05:54.610 align:middle line:84% If you couldn't observe the way people played, 00:05:54.610 --> 00:05:58.990 align:middle line:84% then the repetition wouldn't really be so important. 00:05:58.990 --> 00:05:59.670 align:middle line:90% Great. 00:05:59.670 --> 00:06:04.190 align:middle line:84% So now let's see if we can formally model this game 00:06:04.190 --> 00:06:07.430 align:middle line:84% and solve for a subgame perfect Nash equilibrium. 00:06:07.430 --> 00:06:10.570 align:middle line:84% The three-copy version actually gets a bit messy, 00:06:10.570 --> 00:06:12.910 align:middle line:84% so let me just go over the two-copy version, 00:06:12.910 --> 00:06:14.570 align:middle line:84% and then we'll generalize from there. 00:06:14.570 --> 00:06:16.730 align:middle line:84% So let's imagine that the game is played twice. 00:06:16.730 --> 00:06:20.030 align:middle line:84% So maybe I'll say this is prisoner's dilemma times 2. 00:06:20.030 --> 00:06:23.008 align:middle line:84% And even with this very simple game and just two repetitions, 00:06:23.008 --> 00:06:25.050 align:middle line:84% the game tree is going to get pretty complicated, 00:06:25.050 --> 00:06:26.592 align:middle line:84% but let's see if we can write it out. 00:06:26.592 --> 00:06:29.670 align:middle line:84% So we'll say, maybe the first player 00:06:29.670 --> 00:06:34.150 align:middle line:84% is choosing between cooperate and defect. 00:06:34.150 --> 00:06:37.510 align:middle line:84% And then the second player, without observing 00:06:37.510 --> 00:06:39.190 align:middle line:84% what the first player did, is also 00:06:39.190 --> 00:06:43.410 align:middle line:84% going to choose between cooperate and defect. 00:06:43.410 --> 00:06:45.990 align:middle line:90% 00:06:45.990 --> 00:06:48.030 align:middle line:84% Now, remember, we've chosen to model 00:06:48.030 --> 00:06:51.322 align:middle line:84% it this way as the first player going first, 00:06:51.322 --> 00:06:52.530 align:middle line:90% but it doesn't really matter. 00:06:52.530 --> 00:06:54.190 align:middle line:84% We could also have the second player going first, 00:06:54.190 --> 00:06:55.730 align:middle line:84% and then the first player moving, 00:06:55.730 --> 00:06:58.210 align:middle line:84% not having observed what the second player did right. 00:06:58.210 --> 00:07:01.090 align:middle line:84% The key thing is that this is really a simultaneous move game, 00:07:01.090 --> 00:07:03.190 align:middle line:84% because neither player's action can depend on what 00:07:03.190 --> 00:07:05.050 align:middle line:90% the other player is doing. 00:07:05.050 --> 00:07:06.770 align:middle line:84% So this was kind of the first period. 00:07:06.770 --> 00:07:08.750 align:middle line:90% Often we call this period 0. 00:07:08.750 --> 00:07:10.990 align:middle line:84% We often start our labeling at 0. 00:07:10.990 --> 00:07:12.650 align:middle line:84% And now we have the second period. 00:07:12.650 --> 00:07:21.290 align:middle line:84% So now player 1 chooses between cooperate and defect. 00:07:21.290 --> 00:07:27.510 align:middle line:90% 00:07:27.510 --> 00:07:30.150 align:middle line:84% And notice these are in separate information 00:07:30.150 --> 00:07:31.630 align:middle line:90% sets to indicate the fact that. 00:07:31.630 --> 00:07:33.710 align:middle line:84% At this point, the players observe 00:07:33.710 --> 00:07:35.090 align:middle line:90% what happened in period 0. 00:07:35.090 --> 00:07:37.050 align:middle line:84% You know whether both players cooperated, 00:07:37.050 --> 00:07:39.550 align:middle line:84% both players defected, one cooperated, the other defected, 00:07:39.550 --> 00:07:41.470 align:middle line:90% or vice versa. 00:07:41.470 --> 00:07:45.310 align:middle line:84% And then these are going to be in the same information set 00:07:45.310 --> 00:07:49.590 align:middle line:84% because this is really a simultaneous move game. 00:07:49.590 --> 00:07:50.850 align:middle line:90% And this is player 2. 00:07:50.850 --> 00:07:56.270 align:middle line:90% 00:07:56.270 --> 00:07:58.330 align:middle line:84% And player 2 chooses cooperate or defect. 00:07:58.330 --> 00:08:03.910 align:middle line:90% 00:08:03.910 --> 00:08:07.110 align:middle line:90% You see this gets a bit messy. 00:08:07.110 --> 00:08:09.018 align:middle line:90% And I think you see the pattern. 00:08:09.018 --> 00:08:10.810 align:middle line:84% And then I could write out all the payoffs, 00:08:10.810 --> 00:08:13.170 align:middle line:84% but that's going to be a bit time consuming, so I 00:08:13.170 --> 00:08:15.350 align:middle line:90% won't write out the payoffs. 00:08:15.350 --> 00:08:22.238 align:middle line:84% So we want to understand what are the SPNE of this game? 00:08:22.238 --> 00:08:24.030 align:middle line:84% And what's the first question we always ask 00:08:24.030 --> 00:08:26.735 align:middle line:90% when we try to solve for SPNE? 00:08:26.735 --> 00:08:28.610 align:middle line:84% Well, first we have to identify the subgames. 00:08:28.610 --> 00:08:30.670 align:middle line:90% That's always step one. 00:08:30.670 --> 00:08:31.885 align:middle line:90% So how many subgames? 00:08:31.885 --> 00:08:33.010 align:middle line:90% Maybe I'll write down here. 00:08:33.010 --> 00:08:34.090 align:middle line:90% This is period 1. 00:08:34.090 --> 00:08:37.870 align:middle line:90% 00:08:37.870 --> 00:08:42.510 align:middle line:84% So which nodes of this tree start a new subgame? 00:08:42.510 --> 00:08:43.929 align:middle line:90% Well, there's one easy answer. 00:08:43.929 --> 00:08:44.690 align:middle line:90% It's this node. 00:08:44.690 --> 00:08:46.482 align:middle line:84% I don't know if I have colored chalk today. 00:08:46.482 --> 00:08:47.510 align:middle line:90% I'll just use arrows. 00:08:47.510 --> 00:08:50.490 align:middle line:84% So this node starts one subgame that's just the entire game. 00:08:50.490 --> 00:08:51.550 align:middle line:90% That's easy. 00:08:51.550 --> 00:08:55.110 align:middle line:84% And what other nodes start subgames? 00:08:55.110 --> 00:08:57.610 align:middle line:84% Well, we know that if a node is going to start a subgame, 00:08:57.610 --> 00:08:58.910 align:middle line:90% it has to be by itself. 00:08:58.910 --> 00:09:01.317 align:middle line:84% If a node is with another node in the information set, 00:09:01.317 --> 00:09:03.150 align:middle line:84% that node can't start a subgame because that 00:09:03.150 --> 00:09:05.150 align:middle line:84% goes against the definition of subgame. 00:09:05.150 --> 00:09:07.910 align:middle line:84% And it has to be a decision node, so it can't 00:09:07.910 --> 00:09:09.330 align:middle line:90% be any of the terminal nodes. 00:09:09.330 --> 00:09:10.872 align:middle line:84% It can't be any of the nodes that are 00:09:10.872 --> 00:09:12.690 align:middle line:90% in multinode information sets. 00:09:12.690 --> 00:09:17.470 align:middle line:84% So actually, we immediately see that it's just these four. 00:09:17.470 --> 00:09:20.930 align:middle line:84% So what we see is, first, let's look at the subgames. 00:09:20.930 --> 00:09:26.070 align:middle line:90% 00:09:26.070 --> 00:09:27.730 align:middle line:84% And it turns out there's five subgames. 00:09:27.730 --> 00:09:29.670 align:middle line:84% And here, we see the general pattern. 00:09:29.670 --> 00:09:32.510 align:middle line:84% There's one subgame starting in period 0. 00:09:32.510 --> 00:09:35.150 align:middle line:84% And then there's four subgame starting in period 1 00:09:35.150 --> 00:09:37.430 align:middle line:84% where each of those subgames corresponds 00:09:37.430 --> 00:09:40.350 align:middle line:84% to the history of play that the players have observed. 00:09:40.350 --> 00:09:43.930 align:middle line:84% And in period 1, there's four possible histories-- cooperate, 00:09:43.930 --> 00:09:47.290 align:middle line:84% cooperate; cooperate defect; defect, cooperate; and defect, 00:09:47.290 --> 00:09:47.790 align:middle line:90% defect. 00:09:47.790 --> 00:09:50.750 align:middle line:90% 00:09:50.750 --> 00:09:53.250 align:middle line:84% So now let's use our usual approach, 00:09:53.250 --> 00:09:54.770 align:middle line:90% and let's try to work backwards. 00:09:54.770 --> 00:10:01.550 align:middle line:90% 00:10:01.550 --> 00:10:05.510 align:middle line:84% So let's start with our smallest subgames, the subgames that 00:10:05.510 --> 00:10:07.470 align:middle line:90% come at the end. 00:10:07.470 --> 00:10:10.150 align:middle line:84% And in the subgame starting here, 00:10:10.150 --> 00:10:13.370 align:middle line:84% if we're looking for a subgame perfect Nash equilibrium, 00:10:13.370 --> 00:10:16.470 align:middle line:84% the restriction of the strategy profile to this subgame 00:10:16.470 --> 00:10:18.730 align:middle line:84% must be a Nash equilibrium of this subgame. 00:10:18.730 --> 00:10:21.710 align:middle line:84% So how must we play in this subgame 00:10:21.710 --> 00:10:23.650 align:middle line:84% to be consistent with Nash equilibrium? 00:10:23.650 --> 00:10:26.590 align:middle line:90% 00:10:26.590 --> 00:10:28.630 align:middle line:84% Defect, defect, because we know-- 00:10:28.630 --> 00:10:32.430 align:middle line:90% maybe I'll do it here-- 00:10:32.430 --> 00:10:39.070 align:middle line:84% that if we fill in this game, we see 00:10:39.070 --> 00:10:46.670 align:middle line:84% that DD is the unique Nash equilibrium of this stage game. 00:10:46.670 --> 00:10:50.503 align:middle line:90% 00:10:50.503 --> 00:10:52.170 align:middle line:84% Now, you might say, well, wait a second. 00:10:52.170 --> 00:10:54.552 align:middle line:84% In this subgame, we're not quite playing the stage game, 00:10:54.552 --> 00:10:56.510 align:middle line:84% because the payoffs I'm going to put at the end 00:10:56.510 --> 00:10:59.230 align:middle line:84% here don't just depend on what happens here. 00:10:59.230 --> 00:11:01.610 align:middle line:84% They also depend on what happened in the first game. 00:11:01.610 --> 00:11:03.510 align:middle line:84% But that's not going to change incentives 00:11:03.510 --> 00:11:06.850 align:middle line:84% because the payoffs we got from the first game are sunk. 00:11:06.850 --> 00:11:08.330 align:middle line:90% We've already got those payoffs. 00:11:08.330 --> 00:11:11.728 align:middle line:84% So the only effect that my play has today is on the game 00:11:11.728 --> 00:11:12.770 align:middle line:90% that we're playing today. 00:11:12.770 --> 00:11:14.710 align:middle line:84% The fact that we add in yesterday's payoffs 00:11:14.710 --> 00:11:18.190 align:middle line:84% doesn't change the return to my actions. 00:11:18.190 --> 00:11:22.428 align:middle line:84% So indeed, we must be defect, defect here. 00:11:22.428 --> 00:11:24.470 align:middle line:84% And the same reasoning says we have to be defect, 00:11:24.470 --> 00:11:30.670 align:middle line:84% defect here, and defect, defect here, and defect, defect here. 00:11:30.670 --> 00:11:34.430 align:middle line:84% So what we've seen is that in each of the stage-- 00:11:34.430 --> 00:11:37.870 align:middle line:84% maybe I'll say the stage period 1, or-- yeah-- 00:11:37.870 --> 00:11:45.870 align:middle line:84% in each of the period 1 subgames, 00:11:45.870 --> 00:11:50.350 align:middle line:84% we know that defect, defect must be what's played. 00:11:50.350 --> 00:11:52.850 align:middle line:84% And now let's go to the period 0 subgame. 00:11:52.850 --> 00:12:01.745 align:middle line:90% 00:12:01.745 --> 00:12:03.870 align:middle line:84% Well, now this is where things are more complicated 00:12:03.870 --> 00:12:05.870 align:middle line:84% because here, there's the possibility of rewards 00:12:05.870 --> 00:12:06.830 align:middle line:90% and punishments. 00:12:06.830 --> 00:12:10.030 align:middle line:84% But I argue that in this period 0 subgame, 00:12:10.030 --> 00:12:14.010 align:middle line:84% we still must both play defect, defect in period 0. 00:12:14.010 --> 00:12:14.810 align:middle line:90% Why is that? 00:12:14.810 --> 00:12:18.710 align:middle line:84% What would happen if we weren't playing defect, defect? 00:12:18.710 --> 00:12:22.090 align:middle line:90% What's the key observation here? 00:12:22.090 --> 00:12:25.070 align:middle line:90% 00:12:25.070 --> 00:12:28.030 align:middle line:84% Well, I guess the first thing I'd point out 00:12:28.030 --> 00:12:31.570 align:middle line:84% is we don't have any rewards or punishments. 00:12:31.570 --> 00:12:32.070 align:middle line:90% Why? 00:12:32.070 --> 00:12:33.528 align:middle line:84% Because we've already computed what 00:12:33.528 --> 00:12:35.603 align:middle line:90% happens in the period 1 subgame. 00:12:35.603 --> 00:12:37.270 align:middle line:84% And what we've seen is that both players 00:12:37.270 --> 00:12:39.530 align:middle line:84% are going to play defect in period 1 00:12:39.530 --> 00:12:42.070 align:middle line:84% regardless of what was done in period 0. 00:12:42.070 --> 00:12:45.870 align:middle line:84% So because the future play is independent of the past play, 00:12:45.870 --> 00:12:48.010 align:middle line:84% there's no scope for rewards and punishments. 00:12:48.010 --> 00:12:49.970 align:middle line:84% So this is kind of an important point. 00:12:49.970 --> 00:12:57.270 align:middle line:90% 00:12:57.270 --> 00:13:00.230 align:middle line:84% Another way of saying this is, how I behave today 00:13:00.230 --> 00:13:03.150 align:middle line:84% has no effect on how we're going to behave tomorrow. 00:13:03.150 --> 00:13:07.230 align:middle line:84% Because whatever we play today, the same behavior, D, D, 00:13:07.230 --> 00:13:09.590 align:middle line:90% is going to be played tomorrow. 00:13:09.590 --> 00:13:11.710 align:middle line:90% So how should I play today? 00:13:11.710 --> 00:13:14.670 align:middle line:84% If how I play today has no effect on what happens tomorrow, 00:13:14.670 --> 00:13:18.590 align:middle line:84% I should simply do today what maximizes my payoff today. 00:13:18.590 --> 00:13:21.470 align:middle line:84% And we know that the unique Nash equilibrium of the prisoner's 00:13:21.470 --> 00:13:23.510 align:middle line:84% dilemma is D, D, so in the period 00:13:23.510 --> 00:13:26.790 align:middle line:84% 0 subgame we both must play D, D. So 00:13:26.790 --> 00:13:30.550 align:middle line:84% that was kind of a bit of a verbal argument, but let's see, 00:13:30.550 --> 00:13:31.890 align:middle line:90% any questions on that? 00:13:31.890 --> 00:13:34.830 align:middle line:90% 00:13:34.830 --> 00:13:39.670 align:middle line:84% So just to make sure we understand what's going on, 00:13:39.670 --> 00:13:44.820 align:middle line:84% what is the strategy-- maybe I'll put it over here. 00:13:44.820 --> 00:13:48.460 align:middle line:84% What is the strategy that each player plays? 00:13:48.460 --> 00:13:50.180 align:middle line:84% So I might say the outcome of this game 00:13:50.180 --> 00:13:51.800 align:middle line:90% is we just always defect. 00:13:51.800 --> 00:13:54.100 align:middle line:84% But remember, there's a difference between the outcome 00:13:54.100 --> 00:13:55.460 align:middle line:90% and the strategy. 00:13:55.460 --> 00:13:57.420 align:middle line:84% So what is a strategy of a player in this game 00:13:57.420 --> 00:13:58.900 align:middle line:90% or in this Nash equilibrium? 00:13:58.900 --> 00:14:01.860 align:middle line:84% Well, a strategy is a complete contingent plan. 00:14:01.860 --> 00:14:04.380 align:middle line:84% And the number of information sets 00:14:04.380 --> 00:14:07.400 align:middle line:84% is actually exactly the same as the histories here. 00:14:07.400 --> 00:14:09.320 align:middle line:84% Because when am I called upon to play? 00:14:09.320 --> 00:14:11.460 align:middle line:84% Either I know it's period 0 or I'm 00:14:11.460 --> 00:14:13.260 align:middle line:84% called upon to play in period 1, and I 00:14:13.260 --> 00:14:14.940 align:middle line:90% know what happened in period 0. 00:14:14.940 --> 00:14:18.940 align:middle line:90% So a strategy needs to specify-- 00:14:18.940 --> 00:14:20.160 align:middle line:90% we'll write it like this. 00:14:20.160 --> 00:14:25.900 align:middle line:90% 00:14:25.900 --> 00:14:30.540 align:middle line:84% So we have one spot for each possible information set. 00:14:30.540 --> 00:14:33.980 align:middle line:84% Notice I'm being-- if we look really carefully at the game, 00:14:33.980 --> 00:14:35.980 align:middle line:84% then the information sets for player 1 00:14:35.980 --> 00:14:41.260 align:middle line:84% are this node, this node, this node, this node, and this node. 00:14:41.260 --> 00:14:43.940 align:middle line:84% The information sets for player 2 look a little different. 00:14:43.940 --> 00:14:47.262 align:middle line:84% It's this dotted line, this dotted line, this dotted line, 00:14:47.262 --> 00:14:48.720 align:middle line:84% this dotted line, this dotted line. 00:14:48.720 --> 00:14:51.740 align:middle line:84% But we can really think of them as being the same, 00:14:51.740 --> 00:14:55.820 align:middle line:84% because in both cases, the player either were in period 0 00:14:55.820 --> 00:14:57.858 align:middle line:84% or were in period 1, and all we know 00:14:57.858 --> 00:14:59.400 align:middle line:84% is what happened in the first period. 00:14:59.400 --> 00:15:01.620 align:middle line:84% So the fact that player 2 comes later 00:15:01.620 --> 00:15:03.980 align:middle line:84% and it's dotted versus not, it doesn't actually 00:15:03.980 --> 00:15:05.700 align:middle line:90% change things at all. 00:15:05.700 --> 00:15:08.700 align:middle line:84% So what is the Nash equilibrium, or the subgame perfect Nash 00:15:08.700 --> 00:15:10.020 align:middle line:90% equilibrium we found? 00:15:10.020 --> 00:15:13.380 align:middle line:90% It looks like this. 00:15:13.380 --> 00:15:15.300 align:middle line:84% This is a full description of the strategy. 00:15:15.300 --> 00:15:18.580 align:middle line:84% So it says initially, before we played it all, 00:15:18.580 --> 00:15:20.300 align:middle line:90% I'm going to play defect. 00:15:20.300 --> 00:15:22.900 align:middle line:90% Then in the second period-- 00:15:22.900 --> 00:15:24.500 align:middle line:90% the first period, the second-- 00:15:24.500 --> 00:15:28.020 align:middle line:84% in the next game, if I observe that we both played C, 00:15:28.020 --> 00:15:31.060 align:middle line:84% C last game, then I'm going to play defect. 00:15:31.060 --> 00:15:33.620 align:middle line:84% If I observe that we played C, D, I'm going to play defect. 00:15:33.620 --> 00:15:36.120 align:middle line:84% If we observe that we played D, C, I'm going to play defect, 00:15:36.120 --> 00:15:37.600 align:middle line:90% and if D, D, play defect. 00:15:37.600 --> 00:15:42.380 align:middle line:84% So this specifies a complete contingent plan in the game. 00:15:42.380 --> 00:15:45.460 align:middle line:84% Notice one kind of subtlety here-- 00:15:45.460 --> 00:15:46.680 align:middle line:90% and this will come up. 00:15:46.680 --> 00:15:49.660 align:middle line:84% We normally think that how you play in future games 00:15:49.660 --> 00:15:52.415 align:middle line:84% will be contingent on how your opponent played in the past. 00:15:52.415 --> 00:15:54.540 align:middle line:84% This is how we think about rewards and punishments. 00:15:54.540 --> 00:15:57.780 align:middle line:84% If you play C today, I might behave differently tomorrow. 00:15:57.780 --> 00:15:59.300 align:middle line:90% And that's possible. 00:15:59.300 --> 00:16:01.260 align:middle line:84% But in general, we can actually do 00:16:01.260 --> 00:16:03.420 align:middle line:84% even more, that what a strategy allows us to do 00:16:03.420 --> 00:16:05.360 align:middle line:84% is make our future play contingent not only 00:16:05.360 --> 00:16:07.860 align:middle line:84% on what the other player did, but actually what we ourselves 00:16:07.860 --> 00:16:10.480 align:middle line:84% did, which may seem a bit weird, but this will also come up. 00:16:10.480 --> 00:16:12.300 align:middle line:84% I might say, I'll play C tomorrow 00:16:12.300 --> 00:16:14.585 align:middle line:84% if I played C in the first period. 00:16:14.585 --> 00:16:15.960 align:middle line:84% But I'll play something different 00:16:15.960 --> 00:16:17.160 align:middle line:90% if I play D in the first period. 00:16:17.160 --> 00:16:17.785 align:middle line:90% That's allowed. 00:16:17.785 --> 00:16:20.620 align:middle line:84% We can contingent-- make future play contingent on our own play, 00:16:20.620 --> 00:16:25.240 align:middle line:84% and that will actually play an important role below. 00:16:25.240 --> 00:16:26.960 align:middle line:84% So now that we've done an example, 00:16:26.960 --> 00:16:30.060 align:middle line:84% let's introduce a general framework for finitely repeated 00:16:30.060 --> 00:16:30.800 align:middle line:90% games. 00:16:30.800 --> 00:16:32.877 align:middle line:84% You can see this was the simplest possible game. 00:16:32.877 --> 00:16:33.960 align:middle line:90% It was prisoner's dilemma. 00:16:33.960 --> 00:16:35.260 align:middle line:90% And we only played it twice. 00:16:35.260 --> 00:16:37.862 align:middle line:84% And already, the extensive form got really messy. 00:16:37.862 --> 00:16:40.320 align:middle line:84% So we're not really going to draw any more extensive forms, 00:16:40.320 --> 00:16:42.195 align:middle line:84% and we're going to introduce special notation 00:16:42.195 --> 00:16:44.220 align:middle line:84% for repeated games that's a bit more convenient. 00:16:44.220 --> 00:16:53.060 align:middle line:84% So maybe I'll call this the general framework 00:16:53.060 --> 00:16:54.560 align:middle line:90% for finitely repeated games. 00:16:54.560 --> 00:17:03.720 align:middle line:90% 00:17:03.720 --> 00:17:05.240 align:middle line:90% So what's the first ingredient? 00:17:05.240 --> 00:17:07.520 align:middle line:84% Well, the first ingredient is the game that we repeat. 00:17:07.520 --> 00:17:09.079 align:middle line:84% So this is called the stage game. 00:17:09.079 --> 00:17:13.260 align:middle line:90% 00:17:13.260 --> 00:17:16.040 align:middle line:84% So in our example, the stage game was the prisoner's dilemma. 00:17:16.040 --> 00:17:20.339 align:middle line:84% But it could be any simultaneous move or any strategic form game. 00:17:20.339 --> 00:17:23.099 align:middle line:84% And we're going to call the stage game-- we're 00:17:23.099 --> 00:17:25.000 align:middle line:90% going to specify it like this. 00:17:25.000 --> 00:17:34.860 align:middle line:90% 00:17:34.860 --> 00:17:40.240 align:middle line:84% And we might call this game G, which is a strategic form game. 00:17:40.240 --> 00:17:46.483 align:middle line:90% 00:17:46.483 --> 00:17:47.400 align:middle line:90% So what does this say? 00:17:47.400 --> 00:17:49.580 align:middle line:84% It says in the stage game, player 1 00:17:49.580 --> 00:17:52.020 align:middle line:84% can choose any action from the set A1. 00:17:52.020 --> 00:17:54.980 align:middle line:84% Player n can choose any action from the set An. 00:17:54.980 --> 00:17:57.020 align:middle line:84% And then we have a payoff function that says, 00:17:57.020 --> 00:17:59.540 align:middle line:84% what is player i's utility as a function 00:17:59.540 --> 00:18:02.780 align:middle line:84% of the profile of actions that's played in the stage game? 00:18:02.780 --> 00:18:05.180 align:middle line:84% And we specify that function for all players 00:18:05.180 --> 00:18:07.180 align:middle line:90% I equals 1 through n. 00:18:07.180 --> 00:18:09.700 align:middle line:84% Now, we might call the stage game G. 00:18:09.700 --> 00:18:11.758 align:middle line:84% And this is just a strategic form game. 00:18:11.758 --> 00:18:14.300 align:middle line:84% You may notice, though, that the notation is a bit different. 00:18:14.300 --> 00:18:16.300 align:middle line:84% In the past, earlier in the course, 00:18:16.300 --> 00:18:18.400 align:middle line:84% when we wrote a strategic form game down, 00:18:18.400 --> 00:18:20.483 align:middle line:84% what was different about the way we wrote it down? 00:18:20.483 --> 00:18:22.500 align:middle line:90% We used different notation here. 00:18:22.500 --> 00:18:24.380 align:middle line:90% Remember? 00:18:24.380 --> 00:18:25.380 align:middle line:90% These A's are different. 00:18:25.380 --> 00:18:29.220 align:middle line:84% Before we wrote Ses, we had S1 up to Sn. 00:18:29.220 --> 00:18:31.740 align:middle line:84% And we can think of these as basically the same-- playing 00:18:31.740 --> 00:18:33.380 align:middle line:84% the same role as the Ses, but we don't 00:18:33.380 --> 00:18:36.660 align:middle line:84% want to get confused between strategies in the stage game 00:18:36.660 --> 00:18:38.890 align:middle line:84% and strategies in the grand repeated game. 00:18:38.890 --> 00:18:40.640 align:middle line:84% So now I'm going to change notation a bit, 00:18:40.640 --> 00:18:42.723 align:middle line:84% and I'm going to call what we do in the stage game 00:18:42.723 --> 00:18:44.480 align:middle line:84% an action rather than a strategy. 00:18:44.480 --> 00:18:46.400 align:middle line:90% But that's just terminology. 00:18:46.400 --> 00:18:49.100 align:middle line:84% So here, before, we said C and D were 00:18:49.100 --> 00:18:50.440 align:middle line:90% strategies of the stage game. 00:18:50.440 --> 00:18:52.940 align:middle line:84% Now we're going to say that C and D are actions in the stage 00:18:52.940 --> 00:18:54.700 align:middle line:84% game, so that we can keep track of what 00:18:54.700 --> 00:18:57.300 align:middle line:90% a strategy is in the full game. 00:18:57.300 --> 00:19:00.720 align:middle line:84% Then we have to specify the timing. 00:19:00.720 --> 00:19:04.900 align:middle line:90% 00:19:04.900 --> 00:19:07.240 align:middle line:84% And remember this is finitely repeated, 00:19:07.240 --> 00:19:09.120 align:middle line:84% so we're going to say that time goes-- 00:19:09.120 --> 00:19:10.543 align:middle line:90% we're going to start time at 0. 00:19:10.543 --> 00:19:11.960 align:middle line:84% And that will be a bit convenient, 00:19:11.960 --> 00:19:13.380 align:middle line:90% we'll see a bit later. 00:19:13.380 --> 00:19:16.860 align:middle line:84% And we're going to have period 0, period 1, period 2, all 00:19:16.860 --> 00:19:23.140 align:middle line:84% the way up to period T, where T is finite. 00:19:23.140 --> 00:19:25.420 align:middle line:84% So T is sometimes called-- big T is sometimes 00:19:25.420 --> 00:19:27.220 align:middle line:90% called the horizon of the game. 00:19:27.220 --> 00:19:31.140 align:middle line:84% And notice we actually play the game big T plus 1 times, 00:19:31.140 --> 00:19:33.620 align:middle line:90% because we start at 0. 00:19:33.620 --> 00:19:35.020 align:middle line:84% And for this reason, we sometimes 00:19:35.020 --> 00:19:36.940 align:middle line:90% call the repeated game-- 00:19:36.940 --> 00:19:39.420 align:middle line:90% maybe I'll call it G of T. 00:19:39.420 --> 00:19:43.140 align:middle line:84% So the idea is G is our notation for the stage game. 00:19:43.140 --> 00:19:47.460 align:middle line:84% G of T denotes the repeated game where we play the stage game 00:19:47.460 --> 00:19:50.140 align:middle line:84% G in period 0 all the way up to T. 00:19:50.140 --> 00:19:56.500 align:middle line:84% So just so we're on the same page, G of 0 is equal to G, 00:19:56.500 --> 00:19:59.480 align:middle line:84% because if we play the repeated game, but a horizon is 0, 00:19:59.480 --> 00:20:01.140 align:middle line:84% we just play it once, and that just 00:20:01.140 --> 00:20:02.680 align:middle line:90% coincides with the stage game. 00:20:02.680 --> 00:20:05.380 align:middle line:84% But in general, this game is going to be more complicated. 00:20:05.380 --> 00:20:06.880 align:middle line:90% So what did we play here? 00:20:06.880 --> 00:20:07.900 align:middle line:90% We played g? 00:20:07.900 --> 00:20:09.740 align:middle line:90% Of 1. 00:20:09.740 --> 00:20:11.340 align:middle line:84% Our horizon was 1 because we played it 00:20:11.340 --> 00:20:15.140 align:middle line:84% in period 0 and period 1, and G was the prisoner's dilemma. 00:20:15.140 --> 00:20:21.780 align:middle line:84% So as an example, just to remind ourselves what we played so far 00:20:21.780 --> 00:20:24.420 align:middle line:90% could maybe be called PD of 1. 00:20:24.420 --> 00:20:28.740 align:middle line:84% The game is the prisoner's dilemma, and our horizon was 1. 00:20:28.740 --> 00:20:32.880 align:middle line:84% We next have to say something about information. 00:20:32.880 --> 00:20:38.405 align:middle line:90% 00:20:38.405 --> 00:20:39.780 align:middle line:84% And what we're going to assume is 00:20:39.780 --> 00:20:41.920 align:middle line:90% that past actions are observed. 00:20:41.920 --> 00:20:50.127 align:middle line:90% 00:20:50.127 --> 00:20:51.960 align:middle line:84% And this was already assumed in our example. 00:20:51.960 --> 00:20:54.640 align:middle line:84% What we're saying is whenever we play the game together, 00:20:54.640 --> 00:20:57.780 align:middle line:84% we've observed exactly how people played in the past. 00:20:57.780 --> 00:20:59.820 align:middle line:84% So for instance, here, when we played the game 00:20:59.820 --> 00:21:02.380 align:middle line:84% for the second time, we observe how people played it 00:21:02.380 --> 00:21:03.320 align:middle line:90% the first time. 00:21:03.320 --> 00:21:05.120 align:middle line:84% And we pointed out that was crucial. 00:21:05.120 --> 00:21:07.820 align:middle line:84% This observability was crucial to create the opportunity 00:21:07.820 --> 00:21:09.400 align:middle line:90% for rewards and punishments. 00:21:09.400 --> 00:21:15.540 align:middle line:90% 00:21:15.540 --> 00:21:19.320 align:middle line:84% And now we have to say what our payoffs are. 00:21:19.320 --> 00:21:23.660 align:middle line:90% 00:21:23.660 --> 00:21:25.460 align:middle line:90% Let's use average payoffs. 00:21:25.460 --> 00:21:27.600 align:middle line:90% So maybe I'll just write Ui-- 00:21:27.600 --> 00:21:32.250 align:middle line:90% 00:21:32.250 --> 00:21:33.790 align:middle line:84% maybe I'll be a little more precise. 00:21:33.790 --> 00:21:41.050 align:middle line:90% 00:21:41.050 --> 00:21:43.650 align:middle line:84% So what I'm saying is, let's say I'm player i, 00:21:43.650 --> 00:21:45.432 align:middle line:90% and this is how we play-- 00:21:45.432 --> 00:21:46.890 align:middle line:84% or this is the outcome of the game. 00:21:46.890 --> 00:21:49.050 align:middle line:84% In period 0, we play the action profile 00:21:49.050 --> 00:21:51.310 align:middle line:84% a0, and all the way into period T, 00:21:51.310 --> 00:21:53.050 align:middle line:90% we play the action profile aT. 00:21:53.050 --> 00:21:56.250 align:middle line:84% I want to write down what is the payoff of player i? 00:21:56.250 --> 00:21:59.050 align:middle line:84% And this is just going to be the sum-- 00:21:59.050 --> 00:22:00.130 align:middle line:90% but write it here. 00:22:00.130 --> 00:22:12.450 align:middle line:84% It's going to be 1 over T plus 1 of Ui of a0 00:22:12.450 --> 00:22:18.370 align:middle line:90% plus all the way up to Ui of aT 00:22:18.370 --> 00:22:21.230 align:middle line:84% So I look what happens in stage 0, or period 0, 00:22:21.230 --> 00:22:22.930 align:middle line:90% I compute my payoff there. 00:22:22.930 --> 00:22:26.970 align:middle line:84% I go all the way up to period T, I compute my payoff there. 00:22:26.970 --> 00:22:29.250 align:middle line:84% Altogether I have T plus 1 terms, 00:22:29.250 --> 00:22:31.650 align:middle line:84% and it's convenient to just divide it by T plus 1. 00:22:31.650 --> 00:22:33.550 align:middle line:84% That doesn't have any effect on incentives. 00:22:33.550 --> 00:22:35.525 align:middle line:84% We can always scale utilities however we want. 00:22:35.525 --> 00:22:37.650 align:middle line:84% But it means that we're kind of computing utilities 00:22:37.650 --> 00:22:39.770 align:middle line:84% in the multiperiod game, in the repeated 00:22:39.770 --> 00:22:42.750 align:middle line:84% game, in the same utility units as the stage game. 00:22:42.750 --> 00:22:44.510 align:middle line:84% And it turns out that that's convenient, 00:22:44.510 --> 00:22:46.970 align:middle line:84% so that our payoff will be something like 2 or 3 00:22:46.970 --> 00:22:50.650 align:middle line:84% rather than 7 or 8, which isn't in the same terms as the stage 00:22:50.650 --> 00:22:51.150 align:middle line:90% game. 00:22:51.150 --> 00:22:55.610 align:middle line:90% 00:22:55.610 --> 00:22:57.390 align:middle line:84% So now let's talk about strategies. 00:22:57.390 --> 00:23:02.650 align:middle line:90% 00:23:02.650 --> 00:23:07.290 align:middle line:84% And maybe before we define strategies, let me just 00:23:07.290 --> 00:23:14.890 align:middle line:84% make one note that a finitely repeated game, 00:23:14.890 --> 00:23:21.870 align:middle line:84% this is a special case of a multistage game. 00:23:21.870 --> 00:23:30.682 align:middle line:90% 00:23:30.682 --> 00:23:32.890 align:middle line:84% Remember, in a multistage game, we separated the game 00:23:32.890 --> 00:23:36.770 align:middle line:84% into stages, and in each stage, some subset of players 00:23:36.770 --> 00:23:38.970 align:middle line:84% moved simultaneously having observed what 00:23:38.970 --> 00:23:41.010 align:middle line:90% happened in previous stages. 00:23:41.010 --> 00:23:45.330 align:middle line:84% This is a special case because the subset of players who move 00:23:45.330 --> 00:23:48.370 align:middle line:84% is always all the players, and the game they play 00:23:48.370 --> 00:23:49.970 align:middle line:90% is always the stage game. 00:23:49.970 --> 00:23:52.410 align:middle line:84% So more generally, in a multistage game, 00:23:52.410 --> 00:23:54.380 align:middle line:84% the stage game could vary over time. 00:23:54.380 --> 00:23:56.130 align:middle line:84% We might play a different game in period 0 00:23:56.130 --> 00:23:57.410 align:middle line:90% as we do in period 1. 00:23:57.410 --> 00:24:01.170 align:middle line:84% This is a special case in that we always play the same stage 00:24:01.170 --> 00:24:03.090 align:middle line:90% game in every period. 00:24:03.090 --> 00:24:05.250 align:middle line:84% So because this is a multistage game, 00:24:05.250 --> 00:24:06.750 align:middle line:84% what does that tell us we can apply? 00:24:06.750 --> 00:24:10.650 align:middle line:84% Let's just keep in mind, why might this be useful for us? 00:24:10.650 --> 00:24:13.430 align:middle line:84% What's a general result we know about multistage games? 00:24:13.430 --> 00:24:20.890 align:middle line:90% 00:24:20.890 --> 00:24:21.833 align:middle line:90% Remember? 00:24:21.833 --> 00:24:23.250 align:middle line:84% If we want to check that something 00:24:23.250 --> 00:24:25.150 align:middle line:84% is a subgame perfect Nash equilibrium 00:24:25.150 --> 00:24:27.450 align:middle line:84% in a multistage game, it's a bit easier 00:24:27.450 --> 00:24:29.190 align:middle line:90% to check because of what result? 00:24:29.190 --> 00:24:32.290 align:middle line:90% 00:24:32.290 --> 00:24:35.130 align:middle line:84% IAN BALL: The one-shot deviation principle. 00:24:35.130 --> 00:24:44.476 align:middle line:90% 00:24:44.476 --> 00:24:46.290 align:middle line:84% So the one-shot deviation principle 00:24:46.290 --> 00:24:49.450 align:middle line:84% gave us an easy way to verify that a given strategy 00:24:49.450 --> 00:24:53.450 align:middle line:84% profile was, in fact, a subgame perfect Nash equilibrium. 00:24:53.450 --> 00:24:57.970 align:middle line:84% The one-shot deviation principle applies to any multistage game. 00:24:57.970 --> 00:25:00.850 align:middle line:84% And finally, repeated games are a special case 00:25:00.850 --> 00:25:03.130 align:middle line:84% of multistage games, and therefore, 00:25:03.130 --> 00:25:06.650 align:middle line:84% the one-shot deviation principle applies to these games as well. 00:25:06.650 --> 00:25:08.650 align:middle line:90% And we'll use that fact below. 00:25:08.650 --> 00:25:14.570 align:middle line:84% So now let's define what a strategy is, 00:25:14.570 --> 00:25:15.870 align:middle line:90% let's say for player i. 00:25:15.870 --> 00:25:20.810 align:middle line:90% 00:25:20.810 --> 00:25:25.450 align:middle line:84% And as usual, a strategy is a complete contingent plan. 00:25:25.450 --> 00:25:27.890 align:middle line:84% So we have to say how player i will 00:25:27.890 --> 00:25:30.650 align:middle line:84% behave at every possible contingency 00:25:30.650 --> 00:25:33.570 align:middle line:90% player i finds herself at. 00:25:33.570 --> 00:25:35.510 align:middle line:84% And again, let's separate this into periods. 00:25:35.510 --> 00:25:43.730 align:middle line:84% We have period 0, period 1, to a generic period t, 00:25:43.730 --> 00:25:51.370 align:middle line:84% and then eventually we get to period T. 00:25:51.370 --> 00:25:54.370 align:middle line:84% Well, at period 0, nothing has happened. 00:25:54.370 --> 00:25:55.910 align:middle line:84% So we just have the trivial history. 00:25:55.910 --> 00:26:03.330 align:middle line:84% So at period 0, my strategy must say Si of the empty set in Ai. 00:26:03.330 --> 00:26:06.930 align:middle line:84% So my strategy Si is going to say, what action do I 00:26:06.930 --> 00:26:08.538 align:middle line:90% take in period 0? 00:26:08.538 --> 00:26:10.830 align:middle line:84% Well, at the null history we haven't observed anything. 00:26:10.830 --> 00:26:12.050 align:middle line:90% Nothing has happened so far. 00:26:12.050 --> 00:26:14.250 align:middle line:84% So we're going to use the empty set to denote 00:26:14.250 --> 00:26:17.290 align:middle line:90% the history in period 0. 00:26:17.290 --> 00:26:22.930 align:middle line:84% Then in period 1, we need to specify Si of h1. 00:26:22.930 --> 00:26:25.970 align:middle line:84% This is going to be an action in Ai. 00:26:25.970 --> 00:26:35.690 align:middle line:84% And we have to specify this for every history h1. 00:26:35.690 --> 00:26:37.470 align:middle line:90% This is a period 1 history-- 00:26:37.470 --> 00:26:41.130 align:middle line:90% 00:26:41.130 --> 00:26:42.130 align:middle line:90% h1. 00:26:42.130 --> 00:26:43.890 align:middle line:84% Well, where do these things live? 00:26:43.890 --> 00:26:46.410 align:middle line:84% Well, a period 1 history just specifies 00:26:46.410 --> 00:26:48.210 align:middle line:90% what happened in period 0. 00:26:48.210 --> 00:26:50.630 align:middle line:84% And crucially, not just what I did in period 0, 00:26:50.630 --> 00:26:52.130 align:middle line:90% but what my opponents did. 00:26:52.130 --> 00:26:55.330 align:middle line:84% So this is going to be an element of A. It's 00:26:55.330 --> 00:26:56.750 align:middle line:90% going to be an action profile. 00:26:56.750 --> 00:27:02.850 align:middle line:84% Remember, A-- let me write is A1 up to An. 00:27:02.850 --> 00:27:05.410 align:middle line:84% So what this says is in period 1, as player i, 00:27:05.410 --> 00:27:07.930 align:middle line:84% I'm going to choose some action to play. 00:27:07.930 --> 00:27:10.610 align:middle line:84% And the action I choose can depend upon the way 00:27:10.610 --> 00:27:14.450 align:middle line:84% that we played in period 0, namely the action profile 00:27:14.450 --> 00:27:16.470 align:middle line:90% that we all took in period 0. 00:27:16.470 --> 00:27:19.350 align:middle line:90% 00:27:19.350 --> 00:27:21.870 align:middle line:84% If we extend this and we get to period t, what happens? 00:27:21.870 --> 00:27:27.850 align:middle line:90% We're choosing Si of ht in Ai. 00:27:27.850 --> 00:27:29.770 align:middle line:84% And again, for every history, but now 00:27:29.770 --> 00:27:31.510 align:middle line:90% we have every history, ht. 00:27:31.510 --> 00:27:35.790 align:middle line:90% 00:27:35.790 --> 00:27:36.290 align:middle line:90% In what? 00:27:36.290 --> 00:27:39.050 align:middle line:90% Where does ht live? 00:27:39.050 --> 00:27:43.650 align:middle line:84% Well in period t, I observe how everyone behaved in period 0, 00:27:43.650 --> 00:27:47.210 align:middle line:84% in period 1, all the way up to period t minus 1. 00:27:47.210 --> 00:27:49.630 align:middle line:84% And altogether that's actually t periods. 00:27:49.630 --> 00:27:52.930 align:middle line:90% So this is going to be in At. 00:27:52.930 --> 00:27:56.170 align:middle line:84% It might be a little confusing why it's At because it only 00:27:56.170 --> 00:27:57.670 align:middle line:90% goes up to T minus 1. 00:27:57.670 --> 00:27:59.270 align:middle line:90% But remember, we started at 0. 00:27:59.270 --> 00:28:03.250 align:middle line:84% So actions profiles in 0, 1, 2, all the way up to T minus 1. 00:28:03.250 --> 00:28:06.730 align:middle line:90% There's t of those altogether. 00:28:06.730 --> 00:28:12.090 align:middle line:90% And then we could go down to T. 00:28:12.090 --> 00:28:13.970 align:middle line:84% So maybe if we wanted to say, really, 00:28:13.970 --> 00:28:18.730 align:middle line:84% to be really formal here, what is it? 00:28:18.730 --> 00:28:27.770 align:middle line:84% A strategy Si is a function that specifies what player i does 00:28:27.770 --> 00:28:30.690 align:middle line:90% at every possible history. 00:28:30.690 --> 00:28:34.810 align:middle line:84% So maybe we could write this as a function from the union from t 00:28:34.810 --> 00:28:41.910 align:middle line:90% equals 0 to T of At to Ai. 00:28:41.910 --> 00:28:49.970 align:middle line:90% 00:28:49.970 --> 00:28:51.950 align:middle line:84% So this is, I think, the way to interpret it. 00:28:51.950 --> 00:28:54.150 align:middle line:84% But if you want to think formally mathematically, 00:28:54.150 --> 00:28:56.430 align:middle line:84% well, I have to specify what I do at every history. 00:28:56.430 --> 00:28:58.950 align:middle line:84% So I'm going to take the union of all the sets of histories. 00:28:58.950 --> 00:29:02.162 align:middle line:84% And this will specify the period 0 history, all the period 1 00:29:02.162 --> 00:29:04.370 align:middle line:84% histories, all the period 2 histories, all the way up 00:29:04.370 --> 00:29:06.090 align:middle line:90% to all the period T histories. 00:29:06.090 --> 00:29:08.670 align:middle line:84% And this is going to be the formal way I write it down. 00:29:08.670 --> 00:29:10.750 align:middle line:90% But that's the math way. 00:29:10.750 --> 00:29:13.370 align:middle line:84% This is, I think, the intuitive way to do it. 00:29:13.370 --> 00:29:15.110 align:middle line:84% And you'll notice, this is maybe the way 00:29:15.110 --> 00:29:16.610 align:middle line:84% that we'll write it down in examples 00:29:16.610 --> 00:29:17.790 align:middle line:90% that will be a bit clearer. 00:29:17.790 --> 00:29:24.830 align:middle line:90% 00:29:24.830 --> 00:29:28.890 align:middle line:84% Any questions on the setup on how we've 00:29:28.890 --> 00:29:32.970 align:middle line:90% defined a finite repeated game? 00:29:32.970 --> 00:29:33.607 align:middle line:90% Yeah? 00:29:33.607 --> 00:29:34.690 align:middle line:90% AUDIENCE: Just a question. 00:29:34.690 --> 00:29:37.030 align:middle line:84% So for that game, we're only going for two two rounds? 00:29:37.030 --> 00:29:38.332 align:middle line:90% IAN BALL: Yeah. 00:29:38.332 --> 00:29:40.290 align:middle line:84% AUDIENCE: It just so happened that both players 00:29:40.290 --> 00:29:41.990 align:middle line:90% had identical strategies. 00:29:41.990 --> 00:29:44.350 align:middle line:84% But normally, we need to specify two of those. 00:29:44.350 --> 00:29:45.350 align:middle line:90% IAN BALL: Exactly right. 00:29:45.350 --> 00:29:49.810 align:middle line:84% So I should have said-- yeah, this is a strategy Si. 00:29:49.810 --> 00:29:51.290 align:middle line:90% Exactly right. 00:29:51.290 --> 00:29:51.790 align:middle line:90% Good point. 00:29:51.790 --> 00:29:53.690 align:middle line:84% So the equilibrium that we solved for 00:29:53.690 --> 00:29:55.390 align:middle line:90% has to be a strategy profile. 00:29:55.390 --> 00:29:57.210 align:middle line:84% So maybe I would say it's actually 00:29:57.210 --> 00:30:00.610 align:middle line:90% S1 equals S2 equals this. 00:30:00.610 --> 00:30:01.170 align:middle line:90% Yeah. 00:30:01.170 --> 00:30:01.670 align:middle line:90% Great. 00:30:01.670 --> 00:30:04.170 align:middle line:90% Thanks for clarifying. 00:30:04.170 --> 00:30:05.510 align:middle line:90% That's a symmetric equilibrium. 00:30:05.510 --> 00:30:07.590 align:middle line:84% We could also have potentially asymmetric equilibria. 00:30:07.590 --> 00:30:07.890 align:middle line:90% Yeah. 00:30:07.890 --> 00:30:08.150 align:middle line:90% Great. 00:30:08.150 --> 00:30:09.150 align:middle line:90% Any other questions? 00:30:09.150 --> 00:30:14.210 align:middle line:90% 00:30:14.210 --> 00:30:19.000 align:middle line:84% So now we want to understand what's kind of special 00:30:19.000 --> 00:30:20.320 align:middle line:90% and what happened here. 00:30:20.320 --> 00:30:21.800 align:middle line:84% We wrote down this model thinking 00:30:21.800 --> 00:30:24.950 align:middle line:84% that there would be maybe a potential for cooperation 00:30:24.950 --> 00:30:27.200 align:middle line:84% using rewards and punishments, where people might play 00:30:27.200 --> 00:30:28.800 align:middle line:84% differently early on in the game, 00:30:28.800 --> 00:30:31.440 align:middle line:84% anticipating what happens later on in the game. 00:30:31.440 --> 00:30:34.047 align:middle line:84% But then we looked at the prisoner's dilemma, 00:30:34.047 --> 00:30:35.380 align:middle line:90% and we found that didn't happen. 00:30:35.380 --> 00:30:37.240 align:middle line:84% We always just played the Nash equilibrium of the prisoner's 00:30:37.240 --> 00:30:38.440 align:middle line:90% dilemma every period. 00:30:38.440 --> 00:30:41.440 align:middle line:84% So let's try to understand how general that phenomenon is. 00:30:41.440 --> 00:30:45.160 align:middle line:84% And that will be what our first result tells us. 00:30:45.160 --> 00:30:48.400 align:middle line:84% So one special property of the prisoner's dilemma 00:30:48.400 --> 00:30:51.420 align:middle line:84% is that the stage game only has one Nash equilibrium. 00:30:51.420 --> 00:30:53.240 align:middle line:84% It had a unique Nash equilibrium. 00:30:53.240 --> 00:30:55.040 align:middle line:84% And we'll see that when the stage game only 00:30:55.040 --> 00:30:58.000 align:middle line:84% has a unique Nash equilibrium, that puts a lot of structure 00:30:58.000 --> 00:31:00.700 align:middle line:84% on the subgame perfect Nash equilibria of the repeated game. 00:31:00.700 --> 00:31:04.520 align:middle line:90% So let's state this theorem. 00:31:04.520 --> 00:31:09.122 align:middle line:84% So let G be a strategic form game. 00:31:09.122 --> 00:31:10.580 align:middle line:84% This is going to be our stage game. 00:31:10.580 --> 00:31:17.640 align:middle line:90% 00:31:17.640 --> 00:31:20.080 align:middle line:84% So this is an arbitrary strategic form game. 00:31:20.080 --> 00:31:26.240 align:middle line:84% May be finite, but I don't think we need finiteness. 00:31:26.240 --> 00:31:54.200 align:middle line:84% If this game G has a unique Nash equilibrium, then-- 00:31:54.200 --> 00:31:54.700 align:middle line:90% OK. 00:31:54.700 --> 00:31:57.698 align:middle line:84% So we've started with an assumption about our stage game. 00:31:57.698 --> 00:31:59.240 align:middle line:84% And we've assumed that our stage game 00:31:59.240 --> 00:32:01.460 align:middle line:90% G has unique Nash equilibrium. 00:32:01.460 --> 00:32:04.040 align:middle line:84% We want to draw a conclusion about the repeated 00:32:04.040 --> 00:32:05.425 align:middle line:90% version of this game. 00:32:05.425 --> 00:32:06.800 align:middle line:84% And we actually want a conclusion 00:32:06.800 --> 00:32:08.480 align:middle line:84% to hold no matter how many finitely 00:32:08.480 --> 00:32:09.500 align:middle line:90% many times we repeat it. 00:32:09.500 --> 00:32:16.840 align:middle line:84% So I'm going to say then for every horizon 00:32:16.840 --> 00:32:24.602 align:middle line:90% T, the game G of T-- 00:32:24.602 --> 00:32:26.060 align:middle line:84% so I'm going to finish my sentence, 00:32:26.060 --> 00:32:28.102 align:middle line:84% but let's just make sure we see what we're doing. 00:32:28.102 --> 00:32:30.960 align:middle line:84% We've started with a single game G. 00:32:30.960 --> 00:32:32.680 align:middle line:84% We've made an assumption about it, 00:32:32.680 --> 00:32:34.680 align:middle line:84% that it has unique Nash equilibrium. 00:32:34.680 --> 00:32:37.480 align:middle line:84% And now we're going to draw a conclusion about the T 00:32:37.480 --> 00:32:42.040 align:middle line:84% repeated version of this game for every horizon T, 00:32:42.040 --> 00:32:51.480 align:middle line:84% namely, the game G of T has a unique subgame perfect Nash 00:32:51.480 --> 00:32:52.260 align:middle line:90% equilibrium. 00:32:52.260 --> 00:32:55.120 align:middle line:90% 00:32:55.120 --> 00:32:56.380 align:middle line:90% We can actually say more. 00:32:56.380 --> 00:33:00.560 align:middle line:84% Any guesses on what this unique subgame perfect Nash 00:33:00.560 --> 00:33:01.660 align:middle line:90% equilibrium might be? 00:33:01.660 --> 00:33:06.520 align:middle line:90% 00:33:06.520 --> 00:33:07.200 align:middle line:90% Yeah. 00:33:07.200 --> 00:33:08.840 align:middle line:84% AUDIENCE: It's G is Nash equilibrium. 00:33:08.840 --> 00:33:09.320 align:middle line:90% IAN BALL: Exactly. 00:33:09.320 --> 00:33:10.778 align:middle line:84% And we have to be a little precise, 00:33:10.778 --> 00:33:12.985 align:middle line:84% so the set of strategies is a bit different, 00:33:12.985 --> 00:33:13.860 align:middle line:90% though, in this game. 00:33:13.860 --> 00:33:14.820 align:middle line:90% So let's be a little-- 00:33:14.820 --> 00:33:17.942 align:middle line:84% so can you say that a bit more precisely? 00:33:17.942 --> 00:33:19.400 align:middle line:84% AUDIENCE: It's the Nash equilibrium 00:33:19.400 --> 00:33:22.215 align:middle line:84% where the action is G is the same as G is Nash equilibrium. 00:33:22.215 --> 00:33:22.840 align:middle line:90% IAN BALL: Yeah. 00:33:22.840 --> 00:33:26.800 align:middle line:84% So I would say it's the strategy profile where 00:33:26.800 --> 00:33:31.840 align:middle line:84% we play the Nash equilibrium of G at every history. 00:33:31.840 --> 00:33:33.840 align:middle line:84% But there's a key point there that we can't just 00:33:33.840 --> 00:33:35.340 align:middle line:84% say it's the Nash equilibrium of G, 00:33:35.340 --> 00:33:37.460 align:middle line:84% because the set of strategies in G are different. 00:33:37.460 --> 00:33:42.260 align:middle line:90% So namely, maybe I'll say-- 00:33:42.260 --> 00:33:52.760 align:middle line:90% 00:33:52.760 --> 00:33:53.820 align:middle line:90% maybe I'll give notation. 00:33:53.820 --> 00:33:54.862 align:middle line:90% Maybe that would be nice. 00:33:54.862 --> 00:33:59.940 align:middle line:84% So unique Nash equilibrium, just add some notation, a star. 00:33:59.940 --> 00:34:03.240 align:middle line:90% 00:34:03.240 --> 00:34:06.900 align:middle line:84% So I'm just giving a name to the unique Nash equilibrium. 00:34:06.900 --> 00:34:08.800 align:middle line:90% This is an action profile. 00:34:08.800 --> 00:34:14.400 align:middle line:84% Then this has unique Nash equilibrium, namely S of h 00:34:14.400 --> 00:34:19.739 align:middle line:84% equals a star for every history h. 00:34:19.739 --> 00:34:23.320 align:middle line:90% 00:34:23.320 --> 00:34:25.920 align:middle line:84% So now we're formalizing what you said. 00:34:25.920 --> 00:34:31.600 align:middle line:84% A strategy in the repeated game has to say how we play. 00:34:31.600 --> 00:34:34.520 align:middle line:84% A strategy profile needs to say how every player plays 00:34:34.520 --> 00:34:36.440 align:middle line:90% at every history. 00:34:36.440 --> 00:34:41.040 align:middle line:84% And we're going to specify that whatever the history is, we all 00:34:41.040 --> 00:34:45.463 align:middle line:84% follow the Nash equilibrium profile. 00:34:45.463 --> 00:34:47.380 align:middle line:84% Maybe if I wanted to be a little more precise, 00:34:47.380 --> 00:34:50.719 align:middle line:84% maybe I'll say Si equals ai star for every history 00:34:50.719 --> 00:34:51.980 align:middle line:90% h and every player i. 00:34:51.980 --> 00:34:58.040 align:middle line:90% 00:34:58.040 --> 00:35:00.200 align:middle line:84% So if I'm player i, what do I do? 00:35:00.200 --> 00:35:02.260 align:middle line:84% I have to specify what I do at every history. 00:35:02.260 --> 00:35:03.220 align:middle line:90% What am I going to do? 00:35:03.220 --> 00:35:05.120 align:middle line:84% Just what I'm supposed to do in the Nash 00:35:05.120 --> 00:35:06.840 align:middle line:90% equilibrium of the stage game. 00:35:06.840 --> 00:35:11.240 align:middle line:84% And in particular, this strategy is history independent. 00:35:11.240 --> 00:35:15.355 align:middle line:84% How I play does not depend on how I played in the past 00:35:15.355 --> 00:35:16.980 align:middle line:84% or how other people played in the past. 00:35:16.980 --> 00:35:19.605 align:middle line:84% And this means we don't really have any rewards or punishments. 00:35:19.605 --> 00:35:25.400 align:middle line:90% 00:35:25.400 --> 00:35:27.760 align:middle line:84% This result is true if the unique Nash equilibrium 00:35:27.760 --> 00:35:30.063 align:middle line:84% is mixed, but for the proof, let's think 00:35:30.063 --> 00:35:31.980 align:middle line:84% of this as being pure to make it a bit easier, 00:35:31.980 --> 00:35:34.260 align:middle line:84% but the result will be true either way. 00:35:34.260 --> 00:35:35.540 align:middle line:90% So let's try to prove this. 00:35:35.540 --> 00:35:43.720 align:middle line:90% 00:35:43.720 --> 00:35:51.648 align:middle line:84% And let's assume that a star is pure, 00:35:51.648 --> 00:35:53.940 align:middle line:84% just to make the proof easier, the result goes through, 00:35:53.940 --> 00:35:55.640 align:middle line:84% but we haven't really talked about mixed strategies 00:35:55.640 --> 00:35:57.390 align:middle line:84% and we don't really want to get into that. 00:35:57.390 --> 00:36:01.273 align:middle line:84% So let's just say that this unique Nash equilibrium is pure. 00:36:01.273 --> 00:36:01.940 align:middle line:90% Let me be clear. 00:36:01.940 --> 00:36:04.320 align:middle line:84% I'm not saying that it's not enough to just say 00:36:04.320 --> 00:36:06.420 align:middle line:84% it's unique among the pure Nash equilibrium. 00:36:06.420 --> 00:36:11.920 align:middle line:84% I'm saying there's only one equilibrium, 00:36:11.920 --> 00:36:13.700 align:middle line:84% and that equilibrium happens to be pure. 00:36:13.700 --> 00:36:16.705 align:middle line:90% 00:36:16.705 --> 00:36:19.080 align:middle line:84% Well, we basically want to replicate the argument that we 00:36:19.080 --> 00:36:22.680 align:middle line:90% used in the prisoner's dilemma. 00:36:22.680 --> 00:36:31.440 align:middle line:90% So let's suppose that S star-- 00:36:31.440 --> 00:36:40.920 align:middle line:84% or S equals S1 through Sn is a subgame perfect Nash 00:36:40.920 --> 00:36:41.660 align:middle line:90% equilibrium. 00:36:41.660 --> 00:36:49.560 align:middle line:90% 00:36:49.560 --> 00:36:52.580 align:middle line:84% So we're going to consider some strategy profile. 00:36:52.580 --> 00:36:54.980 align:middle line:84% And let's assume it's a subgame perfect Nash equilibrium, 00:36:54.980 --> 00:36:57.272 align:middle line:84% and we want to say that if it is a subgame perfect Nash 00:36:57.272 --> 00:37:00.020 align:middle line:84% equilibrium, it has to take this very special form. 00:37:00.020 --> 00:37:02.680 align:middle line:90% 00:37:02.680 --> 00:37:05.100 align:middle line:84% Well, what we want to do is we want to work backwards. 00:37:05.100 --> 00:37:08.632 align:middle line:90% So let's look at our periods-- 00:37:08.632 --> 00:37:20.200 align:middle line:84% 0, 1, 2-- and I'll give a verbal proof. 00:37:20.200 --> 00:37:25.620 align:middle line:84% What do we know must be true within period T. 00:37:25.620 --> 00:37:30.160 align:middle line:84% How must we all play in the final period, period T? 00:37:30.160 --> 00:37:31.080 align:middle line:90% Yeah? 00:37:31.080 --> 00:37:32.640 align:middle line:84% AUDIENCE: You just played [INAUDIBLE] 00:37:32.640 --> 00:37:33.307 align:middle line:90% IAN BALL: Right. 00:37:33.307 --> 00:37:36.560 align:middle line:84% And why do we know that that has to be the case? 00:37:36.560 --> 00:37:38.960 align:middle line:84% AUDIENCE: You talked about how all the-- because T is 00:37:38.960 --> 00:37:41.160 align:middle line:84% independent of the first t games, 00:37:41.160 --> 00:37:45.280 align:middle line:84% so then you just have the one stage 00:37:45.280 --> 00:37:46.840 align:middle line:90% played by the Nash equilibrium. 00:37:46.840 --> 00:37:47.840 align:middle line:90% IAN BALL: Exactly right. 00:37:47.840 --> 00:37:50.275 align:middle line:84% So once you've gotten to here, it's 00:37:50.275 --> 00:37:52.400 align:middle line:84% always easier to work backwards because at the end, 00:37:52.400 --> 00:37:53.920 align:middle line:90% there's less to worry about. 00:37:53.920 --> 00:37:55.445 align:middle line:84% How I play in the last game can't 00:37:55.445 --> 00:37:57.320 align:middle line:84% have any effect in the future because there's 00:37:57.320 --> 00:37:58.300 align:middle line:90% no future ahead of me. 00:37:58.300 --> 00:37:59.500 align:middle line:90% This is the last game. 00:37:59.500 --> 00:38:02.400 align:middle line:84% So my only choice today is, how do I 00:38:02.400 --> 00:38:07.320 align:middle line:84% want to influence my flow payoffs in the game today? 00:38:07.320 --> 00:38:11.640 align:middle line:90% And if this is supposed to be-- 00:38:11.640 --> 00:38:14.720 align:middle line:84% if this is a Nash equilibrium of every subgame, well, what 00:38:14.720 --> 00:38:22.040 align:middle line:84% we know is that we must have S of ht 00:38:22.040 --> 00:38:30.960 align:middle line:84% equals ai star for every ht in At. 00:38:30.960 --> 00:38:33.840 align:middle line:84% So I'm saying whatever history we're at, 00:38:33.840 --> 00:38:37.880 align:middle line:84% in this last period, period T, this history starts a subgame. 00:38:37.880 --> 00:38:40.560 align:middle line:84% We have to be playing a Nash equilibrium within this subgame. 00:38:40.560 --> 00:38:42.733 align:middle line:84% But this subgame is just the same as the stage game, 00:38:42.733 --> 00:38:44.900 align:middle line:84% because there's only one-- we're in the last period, 00:38:44.900 --> 00:38:47.480 align:middle line:84% all we're doing is choosing how to play the stage game once. 00:38:47.480 --> 00:38:49.757 align:middle line:84% And therefore, we must be playing a Nash equilibrium. 00:38:49.757 --> 00:38:51.340 align:middle line:84% But there's only one Nash equilibrium, 00:38:51.340 --> 00:38:52.960 align:middle line:90% so we must be playing that. 00:38:52.960 --> 00:38:55.820 align:middle line:84% And now we want to move backwards. 00:38:55.820 --> 00:38:58.742 align:middle line:84% So maybe now I'll go to period T minus 1. 00:38:58.742 --> 00:39:00.700 align:middle line:84% So formally, it's kind of a proof by induction, 00:39:00.700 --> 00:39:03.850 align:middle line:84% but I'm just going to try to do it verbally. 00:39:03.850 --> 00:39:06.470 align:middle line:90% 00:39:06.470 --> 00:39:09.430 align:middle line:84% So now what happens in period T minus 1? 00:39:09.430 --> 00:39:12.750 align:middle line:84% Well, in principle, how I behave in period T minus 1 00:39:12.750 --> 00:39:14.470 align:middle line:84% could be more complicated because I 00:39:14.470 --> 00:39:17.950 align:middle line:84% have to anticipate that the way I play in period T minus 1 00:39:17.950 --> 00:39:20.990 align:middle line:84% might change the way that we play in period T. The future is 00:39:20.990 --> 00:39:22.510 align:middle line:90% more complicated now. 00:39:22.510 --> 00:39:25.230 align:middle line:84% But because I know that this is what's happening in period T, 00:39:25.230 --> 00:39:26.990 align:middle line:84% I don't have to worry about that. 00:39:26.990 --> 00:39:30.590 align:middle line:84% What I know is that if this is how we're playing in period T, 00:39:30.590 --> 00:39:32.350 align:middle line:84% then how we're playing in period T 00:39:32.350 --> 00:39:35.910 align:middle line:84% is independent of how I play in period T minus 1. 00:39:35.910 --> 00:39:41.410 align:middle line:84% So now I just need to maximize my payoff in the stage game. 00:39:41.410 --> 00:39:43.030 align:middle line:84% And the only way that can be the case 00:39:43.030 --> 00:39:45.405 align:middle line:84% is if we're playing a Nash equilibrium of the stage game. 00:39:45.405 --> 00:39:51.510 align:middle line:84% So if we work backwards we see that Si of hT minus 1 equals 00:39:51.510 --> 00:40:00.750 align:middle line:84% ai star for every history ht minus 1 in AT minus 1. 00:40:00.750 --> 00:40:05.050 align:middle line:84% And then we can just keep going back until we get to period 0, 00:40:05.050 --> 00:40:09.910 align:middle line:84% when we'll see that Si of the empty set must equal ai star. 00:40:09.910 --> 00:40:20.110 align:middle line:90% 00:40:20.110 --> 00:40:21.972 align:middle line:84% But notice the ordering really was crucial. 00:40:21.972 --> 00:40:23.430 align:middle line:84% It was crucial that we knew what we 00:40:23.430 --> 00:40:25.638 align:middle line:84% were going to do in the future in order to figure out 00:40:25.638 --> 00:40:27.350 align:middle line:90% our incentives to play today. 00:40:27.350 --> 00:40:27.910 align:middle line:90% Yes? 00:40:27.910 --> 00:40:30.410 align:middle line:84% AUDIENCE: The idea that in time T minus 1, 00:40:30.410 --> 00:40:32.410 align:middle line:84% because what happens at time T is already fixed. 00:40:32.410 --> 00:40:37.830 align:middle line:84% You just play [INAUDIBLE] in and of itself, 00:40:37.830 --> 00:40:39.245 align:middle line:84% and then you just look backwards. 00:40:39.245 --> 00:40:39.870 align:middle line:90% IAN BALL: Yeah. 00:40:39.870 --> 00:40:41.328 align:middle line:84% So it's exactly backward induction, 00:40:41.328 --> 00:40:43.910 align:middle line:84% but I want to be careful when we say fixed. 00:40:43.910 --> 00:40:46.190 align:middle line:84% It's not just that it's fixed-- what's crucial is that 00:40:46.190 --> 00:40:49.030 align:middle line:84% it's-- yeah-- it's independent of the history that 00:40:49.030 --> 00:40:52.150 align:middle line:84% it's-- the crucial thing is that what we do in period t does not 00:40:52.150 --> 00:40:54.310 align:middle line:90% depend on hT. 00:40:54.310 --> 00:40:57.430 align:middle line:84% If it did depend on hT, then things 00:40:57.430 --> 00:40:59.230 align:middle line:84% would be more complicated, because what 00:40:59.230 --> 00:41:03.010 align:middle line:84% I do today could affect how we end up playing tomorrow. 00:41:03.010 --> 00:41:05.790 align:middle line:84% Because what I do today, I guess the key issue-- 00:41:05.790 --> 00:41:09.210 align:middle line:90% and maybe we'll start over here. 00:41:09.210 --> 00:41:13.350 align:middle line:90% 00:41:13.350 --> 00:41:14.930 align:middle line:90% So maybe let's go kind of the-- 00:41:14.930 --> 00:41:18.910 align:middle line:90% 00:41:18.910 --> 00:41:25.630 align:middle line:84% what's the key issue in repeated games? 00:41:25.630 --> 00:41:33.890 align:middle line:84% It's that my action ait has two effects. 00:41:33.890 --> 00:41:37.230 align:middle line:90% 00:41:37.230 --> 00:41:43.290 align:middle line:84% It affects my flow payoff, namely Ui. 00:41:43.290 --> 00:41:47.790 align:middle line:90% 00:41:47.790 --> 00:41:50.910 align:middle line:84% So if this is how my opponents are playing in period T, 00:41:50.910 --> 00:41:55.790 align:middle line:84% the action that I choose directly affects my flow payoff. 00:41:55.790 --> 00:41:58.050 align:middle line:90% But it also affects future play. 00:41:58.050 --> 00:42:01.710 align:middle line:90% 00:42:01.710 --> 00:42:03.270 align:middle line:90% Why? 00:42:03.270 --> 00:42:20.790 align:middle line:84% Well, ait changes the history hs for all S greater than 00:42:20.790 --> 00:42:27.910 align:middle line:84% or equal to t because any subsequent history specifies 00:42:27.910 --> 00:42:29.670 align:middle line:90% exactly what happened so far. 00:42:29.670 --> 00:42:32.310 align:middle line:84% So if I behave differently in period 9, 00:42:32.310 --> 00:42:35.630 align:middle line:84% then the history that the players observe in period 10 00:42:35.630 --> 00:42:39.790 align:middle line:84% is going to depend on what I did in period 9. 00:42:39.790 --> 00:42:44.490 align:middle line:84% And in particular, that might therefore change actions. 00:42:44.490 --> 00:42:51.830 align:middle line:90% 00:42:51.830 --> 00:42:53.750 align:middle line:84% And this is the fundamental trade-off 00:42:53.750 --> 00:42:55.530 align:middle line:84% that we always face in repeated games. 00:42:55.530 --> 00:42:59.630 align:middle line:84% I have to compare how my action today affects my flow 00:42:59.630 --> 00:43:03.390 align:middle line:84% payoffs today, and also how it affects what happens 00:43:03.390 --> 00:43:06.910 align:middle line:90% in future play of the game. 00:43:06.910 --> 00:43:09.310 align:middle line:84% If we go over here, let's look at what happened in period 00:43:09.310 --> 00:43:11.830 align:middle line:90% T minus 1. 00:43:11.830 --> 00:43:13.510 align:middle line:84% The first effect was still true-- 00:43:13.510 --> 00:43:16.070 align:middle line:90% or this effect is still true. 00:43:16.070 --> 00:43:18.370 align:middle line:84% It's still true that at period T minus 1, 00:43:18.370 --> 00:43:22.310 align:middle line:84% my action changes the history tomorrow. 00:43:22.310 --> 00:43:24.950 align:middle line:90% But this is what broke down. 00:43:24.950 --> 00:43:28.110 align:middle line:84% If the way that we play tomorrow is independent of the history, 00:43:28.110 --> 00:43:30.310 align:middle line:84% then the fact that my action changes the history 00:43:30.310 --> 00:43:32.730 align:middle line:84% doesn't mean that the actions are going to be different, 00:43:32.730 --> 00:43:35.830 align:middle line:84% because the actions tomorrow are independent of the history-- 00:43:35.830 --> 00:43:40.090 align:middle line:90% namely this property. 00:43:40.090 --> 00:43:42.750 align:middle line:90% 00:43:42.750 --> 00:43:47.610 align:middle line:84% So sure, if I change my action at T minus 1, I might change hT. 00:43:47.610 --> 00:43:50.150 align:middle line:84% I'm going to change the history that everyone sees tomorrow. 00:43:50.150 --> 00:43:52.195 align:middle line:84% But if the way we play is a star, 00:43:52.195 --> 00:43:54.070 align:middle line:84% no matter what that history is, then the fact 00:43:54.070 --> 00:43:56.830 align:middle line:84% that I change that history doesn't matter at all. 00:43:56.830 --> 00:43:59.710 align:middle line:84% And therefore, I can ignore the effect of my action 00:43:59.710 --> 00:44:03.030 align:middle line:84% on future play, and only focus on the effect of my action 00:44:03.030 --> 00:44:04.068 align:middle line:90% on my flow payoff. 00:44:04.068 --> 00:44:06.610 align:middle line:84% But that means we're effectively just playing the stage game, 00:44:06.610 --> 00:44:09.030 align:middle line:84% and we already know that has a Nash equilibrium. 00:44:09.030 --> 00:44:10.510 align:middle line:84% Again, we could do all the algebra, 00:44:10.510 --> 00:44:12.260 align:middle line:84% and you can look in the notes for the more 00:44:12.260 --> 00:44:14.150 align:middle line:84% formal algebraic argument, but I just 00:44:14.150 --> 00:44:16.150 align:middle line:84% find it easier just to think through it verbally 00:44:16.150 --> 00:44:16.858 align:middle line:90% than the algebra. 00:44:16.858 --> 00:44:17.450 align:middle line:90% Yes, question. 00:44:17.450 --> 00:44:18.908 align:middle line:84% AUDIENCE: Kind of the main point is 00:44:18.908 --> 00:44:24.390 align:middle line:84% that if future actions were not dependent on the history, then 00:44:24.390 --> 00:44:27.000 align:middle line:84% it might not impact the Nash equilibrium. 00:44:27.000 --> 00:44:27.750 align:middle line:90% IAN BALL: Exactly. 00:44:27.750 --> 00:44:29.890 align:middle line:84% And we'll do an example of that exactly next. 00:44:29.890 --> 00:44:30.390 align:middle line:90% Yeah. 00:44:30.390 --> 00:44:33.980 align:middle line:90% 00:44:33.980 --> 00:44:35.730 align:middle line:84% I guess technically what I've shown here-- 00:44:35.730 --> 00:44:37.020 align:middle line:90% I should be careful. 00:44:37.020 --> 00:44:39.270 align:middle line:84% I've shown that if something is a subgame perfect Nash 00:44:39.270 --> 00:44:42.515 align:middle line:84% equilibrium, we must play the stage game Nash every period. 00:44:42.515 --> 00:44:44.390 align:middle line:84% I technically also have to check that playing 00:44:44.390 --> 00:44:46.270 align:middle line:84% the subgame-- playing the Nash every period 00:44:46.270 --> 00:44:48.270 align:middle line:84% is a subgame perfect Nash equilibrium, 00:44:48.270 --> 00:44:50.970 align:middle line:84% but that's also not too hard to show. 00:44:50.970 --> 00:44:53.150 align:middle line:90% So I'll just say-- 00:44:53.150 --> 00:44:55.130 align:middle line:84% and this is maybe a good exercise. 00:44:55.130 --> 00:44:58.670 align:middle line:90% 00:44:58.670 --> 00:45:04.960 align:middle line:84% Use the one-shot deviation principle 00:45:04.960 --> 00:45:10.510 align:middle line:84% to show that this is a subgame perfect Nash equilibrium. 00:45:10.510 --> 00:45:13.070 align:middle line:84% I showed that the only possible subgame perfect Nash 00:45:13.070 --> 00:45:14.250 align:middle line:90% equilibrium is this. 00:45:14.250 --> 00:45:16.150 align:middle line:84% But of course, it could be that there's no subgame perfect Nash 00:45:16.150 --> 00:45:16.770 align:middle line:90% equilibrium. 00:45:16.770 --> 00:45:19.228 align:middle line:84% So you should actually check that this is a subgame perfect 00:45:19.228 --> 00:45:20.090 align:middle line:90% Nash equilibrium. 00:45:20.090 --> 00:45:21.770 align:middle line:84% And it's exactly the same reasoning. 00:45:21.770 --> 00:45:23.145 align:middle line:84% I just have to check that there's 00:45:23.145 --> 00:45:24.590 align:middle line:84% no profitable one-shot deviation, 00:45:24.590 --> 00:45:26.548 align:middle line:84% and we go through basically the same reasoning. 00:45:26.548 --> 00:45:28.330 align:middle line:84% But maybe I'll leave that as an exercise. 00:45:28.330 --> 00:45:33.060 align:middle line:90% 00:45:33.060 --> 00:45:35.310 align:middle line:84% So I don't want to give the impression that repetition 00:45:35.310 --> 00:45:36.870 align:middle line:90% can't have any effect. 00:45:36.870 --> 00:45:38.730 align:middle line:84% And it turns out that this result, 00:45:38.730 --> 00:45:41.190 align:middle line:84% it is very, very special that we have a unique Nash 00:45:41.190 --> 00:45:43.030 align:middle line:90% equilibrium in the stage game. 00:45:43.030 --> 00:45:44.730 align:middle line:84% And if we have multiple Nash equilibria, 00:45:44.730 --> 00:45:46.190 align:middle line:90% this result is no longer true. 00:45:46.190 --> 00:45:49.610 align:middle line:90% So let's go through that. 00:45:49.610 --> 00:46:14.110 align:middle line:90% 00:46:14.110 --> 00:46:15.950 align:middle line:84% So let's go through maybe the oldest 00:46:15.950 --> 00:46:19.870 align:middle line:84% game that's been formalized, is we're 00:46:19.870 --> 00:46:23.070 align:middle line:90% going to study the stag hunt. 00:46:23.070 --> 00:46:27.070 align:middle line:84% So I think Rousseau talked about this in maybe the 1600s. 00:46:27.070 --> 00:46:29.710 align:middle line:84% And the story is you have two hunters. 00:46:29.710 --> 00:46:32.890 align:middle line:84% They can either hunt hare, which is easy, you do it by yourself, 00:46:32.890 --> 00:46:36.410 align:middle line:84% or they can hunt a big stag, and they have to do that together. 00:46:36.410 --> 00:46:38.230 align:middle line:84% So can each of them chooses, do we 00:46:38.230 --> 00:46:42.110 align:middle line:84% try to cooperate and hunt stag together, 00:46:42.110 --> 00:46:45.390 align:middle line:84% or do we just hunt for hare on our own? 00:46:45.390 --> 00:46:48.490 align:middle line:84% And if we both hunt for stag, then we both succeed. 00:46:48.490 --> 00:46:51.750 align:middle line:84% We get the stag, and therefore, we each get a payoff of 2. 00:46:51.750 --> 00:46:53.830 align:middle line:90% Payoff is going to be 2, 2. 00:46:53.830 --> 00:46:56.250 align:middle line:84% Now, hare is not as good as stag. 00:46:56.250 --> 00:46:57.370 align:middle line:90% It's just a rabbit. 00:46:57.370 --> 00:47:01.110 align:middle line:84% If we both hunt for rabbits on our own, we each get 1. 00:47:01.110 --> 00:47:02.110 align:middle line:90% We each get a rabbit. 00:47:02.110 --> 00:47:03.730 align:middle line:84% But a rabbit is worse than a stag. 00:47:03.730 --> 00:47:06.230 align:middle line:90% So we each just get 1, 1. 00:47:06.230 --> 00:47:08.870 align:middle line:84% But then the tricky thing is what happens here. 00:47:08.870 --> 00:47:11.990 align:middle line:84% What if I hunt for hare as player 1, 00:47:11.990 --> 00:47:13.870 align:middle line:90% and player 2 hunts for stag? 00:47:13.870 --> 00:47:17.470 align:middle line:84% Well, I'll still get 1 because I can hunt for hare by myself. 00:47:17.470 --> 00:47:19.670 align:middle line:84% But now I've kind of screwed over the other person 00:47:19.670 --> 00:47:22.477 align:middle line:84% because they're hunting for stag by themselves now. 00:47:22.477 --> 00:47:24.810 align:middle line:84% They're not going to be able to successfully get a stag, 00:47:24.810 --> 00:47:26.435 align:middle line:84% and they're going to get a payoff of 0. 00:47:26.435 --> 00:47:29.550 align:middle line:90% 00:47:29.550 --> 00:47:32.990 align:middle line:84% So the idea is, do we pursue the small prey 00:47:32.990 --> 00:47:35.390 align:middle line:84% that we can achieve by ourselves or the big prey that 00:47:35.390 --> 00:47:36.990 align:middle line:90% requires cooperation? 00:47:36.990 --> 00:47:40.990 align:middle line:84% If you seek out the big prey by yourself, you get nothing 00:47:40.990 --> 00:47:44.830 align:middle line:90% and you fail. 00:47:44.830 --> 00:47:47.310 align:middle line:84% So this is going to have multiple equilibria. 00:47:47.310 --> 00:47:48.650 align:middle line:90% Let's just go through it. 00:47:48.650 --> 00:47:51.283 align:middle line:84% If my opponent is hunting for stag, 00:47:51.283 --> 00:47:52.700 align:middle line:84% then I also want to hunt for stag, 00:47:52.700 --> 00:47:54.240 align:middle line:84% because if I know they're hunting for stag, 00:47:54.240 --> 00:47:56.740 align:middle line:84% then I can join them and we'll get stag, and that's great. 00:47:56.740 --> 00:47:58.660 align:middle line:84% But if my opponent is hunting for hare, 00:47:58.660 --> 00:48:00.618 align:middle line:84% I also want to hunt for hare, because if I hunt 00:48:00.618 --> 00:48:03.180 align:middle line:90% stag by myself, I get nothing. 00:48:03.180 --> 00:48:06.420 align:middle line:84% And now we can do the symmetric thing. 00:48:06.420 --> 00:48:08.760 align:middle line:84% And Rousseau didn't really write it out quite this way, 00:48:08.760 --> 00:48:12.140 align:middle line:84% but he was trying to make the point that the nature of society 00:48:12.140 --> 00:48:13.120 align:middle line:90% can have a big impact. 00:48:13.120 --> 00:48:14.820 align:middle line:84% We can be a society of stag hunters, 00:48:14.820 --> 00:48:16.800 align:middle line:84% do really well because we all trust each other, 00:48:16.800 --> 00:48:19.300 align:middle line:84% or we can be a society of hare hunters who don't do as well. 00:48:19.300 --> 00:48:22.145 align:middle line:84% And these are both consistent and stable courses of action. 00:48:22.145 --> 00:48:24.520 align:middle line:84% Of course, he didn't have the notion of Nash equilibrium, 00:48:24.520 --> 00:48:28.642 align:middle line:84% but he talks about this in his book in the 1600s. 00:48:28.642 --> 00:48:30.100 align:middle line:84% But for our purposes, the key point 00:48:30.100 --> 00:48:36.860 align:middle line:84% is that this has two Nash equilibrium, namely S, S 00:48:36.860 --> 00:48:50.420 align:middle line:84% and H, H. So now let's consider the repeated version of this. 00:48:50.420 --> 00:48:59.327 align:middle line:84% Maybe let's look at stag hunt with T equals 1. 00:48:59.327 --> 00:49:01.160 align:middle line:84% So we're just going to play stag hunt twice. 00:49:01.160 --> 00:49:03.517 align:middle line:90% 00:49:03.517 --> 00:49:06.100 align:middle line:84% And there's actually going to be a lot of subgame perfect Nash 00:49:06.100 --> 00:49:07.540 align:middle line:90% equilibria. 00:49:07.540 --> 00:49:08.840 align:middle line:90% Let's look for a few. 00:49:08.840 --> 00:49:13.540 align:middle line:90% 00:49:13.540 --> 00:49:19.980 align:middle line:84% Let's first look for subgame perfect Nash equilibria 00:49:19.980 --> 00:49:22.680 align:middle line:84% where I might say there's no rewards and punishments. 00:49:22.680 --> 00:49:30.140 align:middle line:90% 00:49:30.140 --> 00:49:31.540 align:middle line:90% What do I mean by that? 00:49:31.540 --> 00:49:35.980 align:middle line:84% I mean let's look for SPNE where the way we play in T 00:49:35.980 --> 00:49:40.660 align:middle line:84% equals 1 doesn't depend on how we played in T equals 0. 00:49:40.660 --> 00:49:42.900 align:middle line:90% Our play is history independent. 00:49:42.900 --> 00:49:44.960 align:middle line:84% Again, if there's no contingency, 00:49:44.960 --> 00:49:47.355 align:middle line:84% if the way we play tomorrow doesn't depend 00:49:47.355 --> 00:49:49.480 align:middle line:84% on how we played today, then the way we play today, 00:49:49.480 --> 00:49:50.980 align:middle line:84% doesn't have any effect on tomorrow, 00:49:50.980 --> 00:49:53.620 align:middle line:84% and we could just focus on the stage games. 00:49:53.620 --> 00:49:58.100 align:middle line:84% So any guesses for some kind of simple SPNE 00:49:58.100 --> 00:50:00.040 align:middle line:90% that have this property? 00:50:00.040 --> 00:50:06.420 align:middle line:84% Maybe I'll write period 0 and period one. 00:50:06.420 --> 00:50:09.060 align:middle line:84% There's actually a number of choices here. 00:50:09.060 --> 00:50:11.860 align:middle line:84% Any guesses for some really simple SPNE 00:50:11.860 --> 00:50:13.400 align:middle line:90% where there's no contingency? 00:50:13.400 --> 00:50:14.800 align:middle line:90% We just kind of ignore the past. 00:50:14.800 --> 00:50:23.100 align:middle line:90% 00:50:23.100 --> 00:50:26.580 align:middle line:84% Well, one thing we could do is we could just always hunt stag. 00:50:26.580 --> 00:50:30.242 align:middle line:84% We know that's a Nash equilibrium of the stage game. 00:50:30.242 --> 00:50:31.700 align:middle line:84% So maybe I'll be a little informal, 00:50:31.700 --> 00:50:36.560 align:middle line:84% and I'll just write we could do S, S; S, S. So formally, 00:50:36.560 --> 00:50:39.940 align:middle line:84% what I mean is in period 0, we each hunt stag. 00:50:39.940 --> 00:50:43.140 align:middle line:84% And in period 1, we each hunt stag regardless of the history 00:50:43.140 --> 00:50:44.007 align:middle line:90% in period 0. 00:50:44.007 --> 00:50:46.340 align:middle line:84% So remember, if I really formally wrote out the strategy 00:50:46.340 --> 00:50:49.640 align:middle line:84% here, I'm actually saying S, S at four different contingencies. 00:50:49.640 --> 00:50:54.110 align:middle line:90% 00:50:54.110 --> 00:50:55.500 align:middle line:90% Great. 00:50:55.500 --> 00:50:57.500 align:middle line:90% So that's one thing we could do. 00:50:57.500 --> 00:50:59.880 align:middle line:84% Is this a subgame perfect Nash equilibrium? 00:50:59.880 --> 00:51:02.700 align:middle line:84% Well, yes, because in period 1, we're each 00:51:02.700 --> 00:51:03.960 align:middle line:90% playing a Nash equilibrium. 00:51:03.960 --> 00:51:05.100 align:middle line:90% That's great. 00:51:05.100 --> 00:51:07.700 align:middle line:84% In period 0, how I play today has no effect 00:51:07.700 --> 00:51:09.820 align:middle line:90% on what happens in period 1. 00:51:09.820 --> 00:51:12.140 align:middle line:84% So all we need to check is that I'm playing a Nash 00:51:12.140 --> 00:51:12.960 align:middle line:90% equilibrium today. 00:51:12.960 --> 00:51:14.140 align:middle line:90% And indeed, that works out. 00:51:14.140 --> 00:51:15.420 align:middle line:90% So here's one. 00:51:15.420 --> 00:51:17.598 align:middle line:84% Any other examples of SPNE that don't 00:51:17.598 --> 00:51:19.140 align:middle line:84% have any rewards or punishments, that 00:51:19.140 --> 00:51:20.760 align:middle line:90% don't have any contingencies? 00:51:20.760 --> 00:51:21.260 align:middle line:90% Yeah? 00:51:21.260 --> 00:51:22.500 align:middle line:84% AUDIENCE: We could always hunt hare. 00:51:22.500 --> 00:51:24.292 align:middle line:84% IAN BALL: We could always send hare, right? 00:51:24.292 --> 00:51:27.660 align:middle line:90% 00:51:27.660 --> 00:51:29.860 align:middle line:84% So again, in each of these subgames, 00:51:29.860 --> 00:51:31.337 align:middle line:84% it's obviously a Nash equilibrium. 00:51:31.337 --> 00:51:33.420 align:middle line:84% We just have to check that it's a Nash equilibrium 00:51:33.420 --> 00:51:34.680 align:middle line:90% of this main subgame. 00:51:34.680 --> 00:51:36.180 align:middle line:84% But because the way we play tomorrow 00:51:36.180 --> 00:51:37.555 align:middle line:84% doesn't depend on the way we play 00:51:37.555 --> 00:51:39.860 align:middle line:84% today, it's Nash equilibrium and we're good. 00:51:39.860 --> 00:51:41.640 align:middle line:84% I'd argue there's actually even some more. 00:51:41.640 --> 00:51:44.380 align:middle line:90% Any other things we could do? 00:51:44.380 --> 00:51:45.060 align:middle line:90% Yeah? 00:51:45.060 --> 00:51:47.720 align:middle line:84% AUDIENCE: Could you hunt stag in the first period, both of you, 00:51:47.720 --> 00:51:49.840 align:middle line:84% and then hunt hares in the next one? 00:51:49.840 --> 00:51:51.900 align:middle line:90% IAN BALL: Exactly. 00:51:51.900 --> 00:51:53.500 align:middle line:90% Great. 00:51:53.500 --> 00:51:54.940 align:middle line:84% So you might think, wait, oh, this 00:51:54.940 --> 00:51:57.600 align:middle line:84% has punishments because we're playing differently tomorrow. 00:51:57.600 --> 00:52:00.500 align:middle line:84% But no, punishments are about making what we do tomorrow 00:52:00.500 --> 00:52:02.140 align:middle line:90% contingent on what we do today. 00:52:02.140 --> 00:52:05.000 align:middle line:84% So what I'm saying, or what our friend here is saying, 00:52:05.000 --> 00:52:07.820 align:middle line:90% is we both hunt stag today. 00:52:07.820 --> 00:52:09.740 align:middle line:84% Tomorrow we hunt hare, but we hunt 00:52:09.740 --> 00:52:12.140 align:middle line:90% hare whatever happened today. 00:52:12.140 --> 00:52:14.480 align:middle line:84% If we both hunted hare today, we hunt hare tomorrow. 00:52:14.480 --> 00:52:16.647 align:middle line:84% If we both hunted stag today, we hunt hare tomorrow. 00:52:16.647 --> 00:52:19.400 align:middle line:84% If it was S, H or H, S we still do hare tomorrow. 00:52:19.400 --> 00:52:22.500 align:middle line:84% So again, there's no contingency and we can go through it. 00:52:22.500 --> 00:52:25.940 align:middle line:84% Clearly a Nash equilibrium of this subgame, and then 00:52:25.940 --> 00:52:28.980 align:middle line:84% clearly a Nash equilibrium here, because all I care 00:52:28.980 --> 00:52:30.540 align:middle line:84% about is my flow payoffs, and S, S 00:52:30.540 --> 00:52:32.660 align:middle line:84% is a Nash equilibrium of the stage game. 00:52:32.660 --> 00:52:36.580 align:middle line:84% And then as you pointed out, we have one final one, 00:52:36.580 --> 00:52:42.260 align:middle line:84% where we do H, H and S, S. And maybe 00:52:42.260 --> 00:52:45.260 align:middle line:84% if I want to be more precise, what I mean-- 00:52:45.260 --> 00:52:49.620 align:middle line:90% I mean, at every h1. 00:52:49.620 --> 00:52:52.220 align:middle line:84% When I write this down, I'm really specifying four things. 00:52:52.220 --> 00:52:54.660 align:middle line:84% I'm saying, at every history h1-- 00:52:54.660 --> 00:52:55.820 align:middle line:90% there's four of them-- 00:52:55.820 --> 00:53:00.140 align:middle line:90% S, S; H, H; H, S; S, H-- 00:53:00.140 --> 00:53:02.460 align:middle line:90% this is what happens. 00:53:02.460 --> 00:53:06.380 align:middle line:84% So here, we have four subgame perfect Nash equilibria 00:53:06.380 --> 00:53:08.560 align:middle line:84% that involve no rewards or punishments, 00:53:08.560 --> 00:53:10.700 align:middle line:84% and they're pretty easy to analyze. 00:53:10.700 --> 00:53:15.340 align:middle line:84% But now we want to see if we can do a little more. 00:53:15.340 --> 00:53:19.060 align:middle line:84% Any questions on this before we move on? 00:53:19.060 --> 00:53:22.280 align:middle line:84% So now maybe I'll go to this board. 00:53:22.280 --> 00:53:41.020 align:middle line:90% 00:53:41.020 --> 00:53:44.620 align:middle line:90% So what's the question? 00:53:44.620 --> 00:53:49.260 align:middle line:84% Can we find a subgame perfect Nash equilibrium 00:53:49.260 --> 00:53:53.240 align:middle line:84% in which we don't play a stage game Nash in period 0? 00:53:53.240 --> 00:54:18.620 align:middle line:90% 00:54:18.620 --> 00:54:24.060 align:middle line:84% So in each of these subgame perfect Nash equilibrium that we 00:54:24.060 --> 00:54:26.940 align:middle line:84% looked at, we played a Nash equilibrium of the stage game 00:54:26.940 --> 00:54:27.920 align:middle line:90% in period 0. 00:54:27.920 --> 00:54:30.540 align:middle line:84% I'm asking, can we find a different subgame perfect Nash 00:54:30.540 --> 00:54:36.260 align:middle line:84% equilibrium in which we do not play a Nash 00:54:36.260 --> 00:54:38.220 align:middle line:90% equilibrium of the stage game? 00:54:38.220 --> 00:54:41.060 align:middle line:84% So let's just try to think through this. 00:54:41.060 --> 00:54:43.180 align:middle line:84% Well, we might say, wait, that's impossible. 00:54:43.180 --> 00:54:44.520 align:middle line:90% We can apply our theorem. 00:54:44.520 --> 00:54:46.460 align:middle line:84% But why can't we apply our theorem here? 00:54:46.460 --> 00:54:48.740 align:middle line:84% Our theorem told us there's only one subgame 00:54:48.740 --> 00:54:49.955 align:middle line:90% perfect Nash equilibrium. 00:54:49.955 --> 00:54:52.080 align:middle line:84% We already found multiple, so something's going on. 00:54:52.080 --> 00:54:54.780 align:middle line:84% So why doesn't the theorem apply to this setting? 00:54:54.780 --> 00:54:55.900 align:middle line:90% Yeah. 00:54:55.900 --> 00:54:57.580 align:middle line:84% AUDIENCE: Because it doesn't have a unique Nash equilibrium. 00:54:57.580 --> 00:54:58.580 align:middle line:90% IAN BALL: Exactly right. 00:54:58.580 --> 00:55:01.700 align:middle line:84% Our theorem was only applied to games where the-- or stage games 00:55:01.700 --> 00:55:03.340 align:middle line:90% with unique Nash equilibria. 00:55:03.340 --> 00:55:06.352 align:middle line:84% This stag hunt game has multiple Nash equilibria, 00:55:06.352 --> 00:55:07.560 align:middle line:90% so our theorem doesn't apply. 00:55:07.560 --> 00:55:09.320 align:middle line:84% Of course, just because the theorem doesn't apply, 00:55:09.320 --> 00:55:10.960 align:middle line:84% it doesn't mean that the conclusions aren't true, 00:55:10.960 --> 00:55:12.400 align:middle line:90% so we have to be careful. 00:55:12.400 --> 00:55:14.260 align:middle line:84% And that's what we're going to do here. 00:55:14.260 --> 00:55:17.460 align:middle line:84% So we're looking for a subgame perfect Nash equilibrium 00:55:17.460 --> 00:55:20.180 align:middle line:84% in which we're not playing a Nash equilibrium of the stage 00:55:20.180 --> 00:55:21.522 align:middle line:90% game in period 0. 00:55:21.522 --> 00:55:22.480 align:middle line:90% So what does that mean? 00:55:22.480 --> 00:55:24.320 align:middle line:84% Let's just write out what that means. 00:55:24.320 --> 00:55:27.740 align:middle line:84% If we're not playing a Nash equilibrium, 00:55:27.740 --> 00:55:31.780 align:middle line:90% then that means some player-- 00:55:31.780 --> 00:55:33.660 align:middle line:90% what can they do in period 0? 00:55:33.660 --> 00:55:37.840 align:middle line:84% That means some player can deviate in period 0-- 00:55:37.840 --> 00:55:42.900 align:middle line:90% 00:55:42.900 --> 00:55:45.540 align:middle line:84% unilaterally deviate in period 0 and strictly increase 00:55:45.540 --> 00:55:46.520 align:middle line:90% their flow path. 00:55:46.520 --> 00:55:58.540 align:middle line:90% 00:55:58.540 --> 00:56:00.440 align:middle line:84% So I realize when I say flow payoff, 00:56:00.440 --> 00:56:02.920 align:middle line:84% I just mean the payoff you get in that period. 00:56:02.920 --> 00:56:05.460 align:middle line:84% So often we talk about stock and flow. 00:56:05.460 --> 00:56:09.120 align:middle line:84% What I mean here is each of these things are flow payoffs, 00:56:09.120 --> 00:56:12.900 align:middle line:84% and this is maybe your total stock payoff. 00:56:12.900 --> 00:56:14.260 align:middle line:90% Well, now we have an issue. 00:56:14.260 --> 00:56:17.020 align:middle line:84% We're looking for a subgame perfect Nash equilibrium. 00:56:17.020 --> 00:56:19.220 align:middle line:84% If it's a subgame perfect Nash equilibrium, 00:56:19.220 --> 00:56:21.240 align:middle line:84% in particular, it's a Nash equilibrium, 00:56:21.240 --> 00:56:23.460 align:middle line:84% so no one should be able to profitably deviate. 00:56:23.460 --> 00:56:28.700 align:middle line:84% But I've already said that some player can deviate in period 0 00:56:28.700 --> 00:56:31.060 align:middle line:84% and strictly increase their flow payoff. 00:56:31.060 --> 00:56:33.540 align:middle line:84% So how can we resurrect this as a subgame perfect Nash 00:56:33.540 --> 00:56:34.120 align:middle line:90% equilibrium? 00:56:34.120 --> 00:56:36.980 align:middle line:84% What has to happen if this player deviates in period 0 00:56:36.980 --> 00:56:40.410 align:middle line:84% and strictly increases their flow payoff? 00:56:40.410 --> 00:56:43.190 align:middle line:84% This can't be profitable, but it increases the flow payoff. 00:56:43.190 --> 00:56:46.410 align:middle line:90% So what must happen? 00:56:46.410 --> 00:56:47.810 align:middle line:90% Yeah? 00:56:47.810 --> 00:56:49.650 align:middle line:90% AUDIENCE: Maybe in a future one? 00:56:49.650 --> 00:56:52.650 align:middle line:84% Would it be a flow payoff is this one, right? 00:56:52.650 --> 00:56:54.850 align:middle line:84% IAN BALL: Yeah, so I'm behaving differently today. 00:56:54.850 --> 00:56:56.770 align:middle line:90% I'm increasing my payoff today. 00:56:56.770 --> 00:56:58.810 align:middle line:84% But we don't want this to be profitable. 00:56:58.810 --> 00:57:00.473 align:middle line:84% So what must happen in the future? 00:57:00.473 --> 00:57:01.890 align:middle line:84% I think you're on the right track. 00:57:01.890 --> 00:57:04.798 align:middle line:84% AUDIENCE: It must be profitable then. 00:57:04.798 --> 00:57:06.590 align:middle line:84% IAN BALL: I must be punished in the future. 00:57:06.590 --> 00:57:09.867 align:middle line:84% So if the player deviates today, they're increasing their flow 00:57:09.867 --> 00:57:11.450 align:middle line:84% payoff, but this is not supposed to be 00:57:11.450 --> 00:57:12.970 align:middle line:90% profitable in the entire game. 00:57:12.970 --> 00:57:15.610 align:middle line:84% It must be that the deviation they consider today 00:57:15.610 --> 00:57:17.330 align:middle line:84% that increased their flow payoff today 00:57:17.330 --> 00:57:18.947 align:middle line:84% eventually comes back to bite them, 00:57:18.947 --> 00:57:20.530 align:middle line:84% because they're eventually going to be 00:57:20.530 --> 00:57:21.697 align:middle line:90% punished in the next period. 00:57:21.697 --> 00:57:24.330 align:middle line:90% So what must be true-- 00:57:24.330 --> 00:57:34.550 align:middle line:84% so this deviation must be punished tomorrow. 00:57:34.550 --> 00:57:40.795 align:middle line:90% 00:57:40.795 --> 00:57:42.170 align:middle line:84% And this is exactly the intuition 00:57:42.170 --> 00:57:44.390 align:middle line:84% that I think we all have from just human affairs. 00:57:44.390 --> 00:57:46.130 align:middle line:84% If your friend does something that's 00:57:46.130 --> 00:57:49.010 align:middle line:84% selfish and hurts you or benefits them today, 00:57:49.010 --> 00:57:52.273 align:middle line:84% tomorrow you might punish them by behaving differently. 00:57:52.273 --> 00:57:53.690 align:middle line:84% And that's a way of deterring them 00:57:53.690 --> 00:57:55.288 align:middle line:90% from engaging in this action. 00:57:55.288 --> 00:57:57.830 align:middle line:84% Maybe this is a negative view of human affairs, I don't know. 00:57:57.830 --> 00:57:59.290 align:middle line:90% But this I think does happen. 00:57:59.290 --> 00:58:02.190 align:middle line:84% Certainly between firms colluding, this happens. 00:58:02.190 --> 00:58:04.750 align:middle line:90% 00:58:04.750 --> 00:58:08.410 align:middle line:90% So this is all kind of abstract. 00:58:08.410 --> 00:58:13.650 align:middle line:84% Now, what's tricky is we need to punish the deviation tomorrow. 00:58:13.650 --> 00:58:18.210 align:middle line:84% But we know that tomorrow is the last period. 00:58:18.210 --> 00:58:21.330 align:middle line:84% And we know that in the last period of a game, 00:58:21.330 --> 00:58:25.010 align:middle line:84% we have to be playing a Nash equilibrium of the stage game, 00:58:25.010 --> 00:58:27.530 align:middle line:84% because this is kind of a key property. 00:58:27.530 --> 00:58:29.230 align:middle line:84% In the last period, there's no future. 00:58:29.230 --> 00:58:31.170 align:middle line:84% So you must be playing a stage game Nash. 00:58:31.170 --> 00:58:34.050 align:middle line:84% So we need to punish you tomorrow, 00:58:34.050 --> 00:58:38.530 align:middle line:84% but we can only punish you by playing a Nash equilibrium. 00:58:38.530 --> 00:58:40.250 align:middle line:90% So this seems a little tricky. 00:58:40.250 --> 00:58:43.610 align:middle line:84% But the key is that we have multiple Nash equilibrium. 00:58:43.610 --> 00:58:46.050 align:middle line:90% So we can punish you tomorrow-- 00:58:46.050 --> 00:58:48.670 align:middle line:84% How? by playing a different Nash equilibrium. 00:58:48.670 --> 00:58:54.690 align:middle line:90% 00:58:54.690 --> 00:58:57.090 align:middle line:84% And this really gets to the core of the difference 00:58:57.090 --> 00:58:59.370 align:middle line:84% between our theorem and our game here. 00:58:59.370 --> 00:59:02.890 align:middle line:84% In the last period, we have to play a Nash equilibrium. 00:59:02.890 --> 00:59:04.703 align:middle line:84% If there's only one Nash equilibrium, 00:59:04.703 --> 00:59:06.870 align:middle line:84% then we know how we have to play in the last period, 00:59:06.870 --> 00:59:09.690 align:middle line:84% and there's no scope for punishments or rewards. 00:59:09.690 --> 00:59:13.170 align:middle line:84% If our game has multiple Nash equilibrium like the stag hunt, 00:59:13.170 --> 00:59:16.550 align:middle line:84% now we still have to play a Nash equilibrium in the last period, 00:59:16.550 --> 00:59:18.450 align:middle line:84% but which Nash equilibrium we play 00:59:18.450 --> 00:59:20.173 align:middle line:90% can depend on our past behavior. 00:59:20.173 --> 00:59:22.090 align:middle line:84% So there might be a good Nash equilibrium that 00:59:22.090 --> 00:59:24.650 align:middle line:84% can serve as a reward, and a bad Nash equilibrium 00:59:24.650 --> 00:59:26.250 align:middle line:90% that can serve as a punishment. 00:59:26.250 --> 00:59:29.810 align:middle line:90% So let's try to go through that. 00:59:29.810 --> 00:59:40.325 align:middle line:84% So the idea is we want a good Nash equilibrium as a reward. 00:59:40.325 --> 00:59:41.950 align:middle line:84% Things could be a bit more complicated, 00:59:41.950 --> 00:59:43.370 align:middle line:84% and Nash equilibrium could be good for some people 00:59:43.370 --> 00:59:44.162 align:middle line:90% and bad for others. 00:59:44.162 --> 00:59:48.370 align:middle line:84% But let's start with this really simple case. 00:59:48.370 --> 00:59:50.830 align:middle line:84% And a bad Nash equilibrium as a punishment. 00:59:50.830 --> 00:59:58.170 align:middle line:90% 00:59:58.170 --> 01:00:00.650 align:middle line:84% So in this stag hunt game, what is going to be the reward, 01:00:00.650 --> 01:00:05.090 align:middle line:84% and what is going to be the punishment? 01:00:05.090 --> 01:00:06.490 align:middle line:90% Well, we can look, right? 01:00:06.490 --> 01:00:10.430 align:middle line:84% S, S is the good equilibrium, and H, H is the bad equilibrium. 01:00:10.430 --> 01:00:14.130 align:middle line:84% So what we want is we want S, S to serve 01:00:14.130 --> 01:00:17.170 align:middle line:84% as a reward for good behavior, and we 01:00:17.170 --> 01:00:22.730 align:middle line:84% want H, H to serve as our punishment for bad behavior. 01:00:22.730 --> 01:00:26.090 align:middle line:84% So what we say is, if you don't do what you're 01:00:26.090 --> 01:00:27.890 align:middle line:84% supposed to today, then tomorrow, 01:00:27.890 --> 01:00:29.250 align:middle line:90% we're both going to hunt hare. 01:00:29.250 --> 01:00:30.350 align:middle line:90% That's bad for you. 01:00:30.350 --> 01:00:32.090 align:middle line:84% If you do do what you're supposed to do, 01:00:32.090 --> 01:00:33.715 align:middle line:84% then we're going to hunt stag tomorrow, 01:00:33.715 --> 01:00:35.250 align:middle line:90% and that's better for you. 01:00:35.250 --> 01:00:39.250 align:middle line:90% So let's see if this works out. 01:00:39.250 --> 01:00:42.010 align:middle line:90% So let's be a bit more specific. 01:00:42.010 --> 01:00:44.530 align:middle line:84% I want to find an SPNE in which in period 0 01:00:44.530 --> 01:00:47.050 align:middle line:84% we play something that's not a stage game Nash. 01:00:47.050 --> 01:00:51.050 align:middle line:84% Let's be more specific, and let's look for an SPNE 01:00:51.050 --> 01:00:53.430 align:middle line:90% where in period 0, we play-- 01:00:53.430 --> 01:00:56.010 align:middle line:90% 01:00:56.010 --> 01:00:57.582 align:middle line:84% well, we don't want to play a Nash. 01:00:57.582 --> 01:00:59.290 align:middle line:84% So that means we're playing one of these. 01:00:59.290 --> 01:01:00.730 align:middle line:84% And it's kind of symmetric. it doesn't really 01:01:00.730 --> 01:01:01.855 align:middle line:90% matter which one we choose. 01:01:01.855 --> 01:01:03.190 align:middle line:90% But let's choose this one. 01:01:03.190 --> 01:01:11.570 align:middle line:90% So let's play H, S. 01:01:11.570 --> 01:01:16.010 align:middle line:84% So let's try to fill in our table. 01:01:16.010 --> 01:01:18.810 align:middle line:84% I think this is kind of a good way to go about it. 01:01:18.810 --> 01:01:20.890 align:middle line:84% So let me-- remember, I think the hardest thing 01:01:20.890 --> 01:01:23.210 align:middle line:84% about sometimes finding equilibria is figuring out 01:01:23.210 --> 01:01:25.870 align:middle line:84% what kind of object they are, like how many positions, 01:01:25.870 --> 01:01:26.830 align:middle line:84% how many things you have to fill in. 01:01:26.830 --> 01:01:28.450 align:middle line:84% So let's make a blank table, and then we'll 01:01:28.450 --> 01:01:29.890 align:middle line:84% gradually fill it in and try to construct 01:01:29.890 --> 01:01:31.510 align:middle line:84% our subgame perfect Nash equilibrium. 01:01:31.510 --> 01:01:33.177 align:middle line:84% So what is our table going to look like? 01:01:33.177 --> 01:01:38.490 align:middle line:90% Well, we have our histories. 01:01:38.490 --> 01:01:40.890 align:middle line:90% We have the empty history. 01:01:40.890 --> 01:01:43.250 align:middle line:84% And then we have four histories after that. 01:01:43.250 --> 01:01:50.730 align:middle line:84% We have S, S; S, H; H, S; And H, H. So to be clear, 01:01:50.730 --> 01:01:58.170 align:middle line:84% these are period 0 histories, and these are period 01:01:58.170 --> 01:01:58.990 align:middle line:90% 1 histories. 01:01:58.990 --> 01:02:02.673 align:middle line:90% 01:02:02.673 --> 01:02:04.090 align:middle line:84% And then for each history, we have 01:02:04.090 --> 01:02:08.570 align:middle line:84% to specify how player 1 plays and how player 2 plays. 01:02:08.570 --> 01:02:12.270 align:middle line:84% So let's put player 1 here and player 2 here. 01:02:12.270 --> 01:02:14.710 align:middle line:90% And maybe let's put in a spot. 01:02:14.710 --> 01:02:19.290 align:middle line:90% 01:02:19.290 --> 01:02:22.530 align:middle line:84% So just to be clear, when I'm asking to solve for an SPNE, 01:02:22.530 --> 01:02:26.810 align:middle line:84% I'm asking to specify 10 things, five things for player 1, 01:02:26.810 --> 01:02:29.930 align:middle line:84% and five things for player 2, because player 1 has five 01:02:29.930 --> 01:02:32.570 align:middle line:84% information sets, and player 2 has five information sets, 01:02:32.570 --> 01:02:35.250 align:middle line:84% and we're looking for a strategy profile. 01:02:35.250 --> 01:02:37.530 align:middle line:84% So the goal is to fill this in such a way 01:02:37.530 --> 01:02:40.330 align:middle line:84% that we get a subgame perfect Nash equilibrium in which this 01:02:40.330 --> 01:02:41.890 align:middle line:90% is what happens in period 0. 01:02:41.890 --> 01:02:43.950 align:middle line:84% So we first have a very easy thing to fill in. 01:02:43.950 --> 01:02:45.710 align:middle line:84% We want this to happen in period 0. 01:02:45.710 --> 01:02:50.850 align:middle line:84% So we're going to fill in H, S. And now 01:02:50.850 --> 01:02:52.310 align:middle line:90% H, S is just written this way. 01:02:52.310 --> 01:02:54.230 align:middle line:84% So it's player 1 followed by player 2. 01:02:54.230 --> 01:03:02.190 align:middle line:90% 01:03:02.190 --> 01:03:05.130 align:middle line:84% So now we need to fill in what happens in period 1 01:03:05.130 --> 01:03:09.290 align:middle line:84% to try to make this a subgame perfect Nash equilibrium. 01:03:09.290 --> 01:03:14.530 align:middle line:84% So the first observation is that whatever we do in period 1, 01:03:14.530 --> 01:03:17.790 align:middle line:84% at every history, it must be a Nash equilibrium, because we 01:03:17.790 --> 01:03:20.958 align:middle line:84% know, in the last period, the subgame is 01:03:20.958 --> 01:03:23.250 align:middle line:84% just the same as the stage game because there's nothing 01:03:23.250 --> 01:03:24.690 align:middle line:90% else that comes in the future. 01:03:24.690 --> 01:03:27.010 align:middle line:84% So what we're choosing, basically, is we need to-- 01:03:27.010 --> 01:03:29.410 align:middle line:84% it's almost like kind of a word search. 01:03:29.410 --> 01:03:35.130 align:middle line:84% We have H, H and S, S. These are Nash equilibria. 01:03:35.130 --> 01:03:38.330 align:middle line:84% And we need to choose where to put H, H 01:03:38.330 --> 01:03:41.130 align:middle line:84% and where to put S, S. So we could be really mechanical 01:03:41.130 --> 01:03:43.610 align:middle line:90% about it, but 4-- 01:03:43.610 --> 01:03:46.452 align:middle line:84% I mean, then we have 16 different choices, 01:03:46.452 --> 01:03:47.910 align:middle line:84% so that's going to take us a while, 01:03:47.910 --> 01:03:49.243 align:middle line:90% so let's be a little thoughtful. 01:03:49.243 --> 01:03:52.430 align:middle line:84% We could put either S, S here or H, H here, 01:03:52.430 --> 01:03:54.730 align:middle line:84% either S, S here or H, H here, and so on. 01:03:54.730 --> 01:03:56.710 align:middle line:84% But that's kind of a silly way to do it. 01:03:56.710 --> 01:03:58.970 align:middle line:90% So let's try to think more. 01:03:58.970 --> 01:04:02.050 align:middle line:84% Any guesses about where we should put H, 01:04:02.050 --> 01:04:06.050 align:middle line:84% H and where we should put S, S. Maybe any thoughts 01:04:06.050 --> 01:04:08.110 align:middle line:90% about one position? 01:04:08.110 --> 01:04:12.984 align:middle line:90% 01:04:12.984 --> 01:04:14.450 align:middle line:90% Hmm? 01:04:14.450 --> 01:04:16.850 align:middle line:84% AUDIENCE: So if he have H, H, we could just-- 01:04:16.850 --> 01:04:20.410 align:middle line:84% I mean, if we have S, S, then both people cooperated 01:04:20.410 --> 01:04:21.970 align:middle line:90% and we wanted to do so. 01:04:21.970 --> 01:04:26.880 align:middle line:90% So like S, S goes under S, S. 01:04:26.880 --> 01:04:29.220 align:middle line:84% IAN BALL: S, S goes under S, S. I'm not so sure. 01:04:29.220 --> 01:04:30.780 align:middle line:90% So let's understand. 01:04:30.780 --> 01:04:34.640 align:middle line:84% We want a subgame perfect Nash equilibrium 01:04:34.640 --> 01:04:37.260 align:middle line:84% where we play H, S in the first period. 01:04:37.260 --> 01:04:39.040 align:middle line:84% So you might be thinking we want S, S, 01:04:39.040 --> 01:04:40.880 align:middle line:84% but we're trying to get people to play 01:04:40.880 --> 01:04:42.055 align:middle line:90% H, S in the first period. 01:04:42.055 --> 01:04:42.680 align:middle line:90% AUDIENCE: Yeah. 01:04:42.680 --> 01:04:46.800 align:middle line:84% IAN BALL: So you said put S, S under S, S? 01:04:46.800 --> 01:04:47.480 align:middle line:90% AUDIENCE: Yes. 01:04:47.480 --> 01:04:48.980 align:middle line:84% IAN BALL: So if we put S, S under S, 01:04:48.980 --> 01:04:53.600 align:middle line:84% S, that would make it appealing to play S, S today. 01:04:53.600 --> 01:04:57.720 align:middle line:90% So let's think-- so say again? 01:04:57.720 --> 01:05:00.280 align:middle line:84% AUDIENCE: Then H, H would be under S, S. 01:05:00.280 --> 01:05:01.060 align:middle line:90% IAN BALL: Right. 01:05:01.060 --> 01:05:01.560 align:middle line:90% Exactly. 01:05:01.560 --> 01:05:03.680 align:middle line:90% So let's actually start-- 01:05:03.680 --> 01:05:06.400 align:middle line:84% I think this is the kind of-- let's start here. 01:05:06.400 --> 01:05:09.480 align:middle line:84% We're trying to encourage people to play H, S today. 01:05:09.480 --> 01:05:12.400 align:middle line:84% So we want to reward them if they do play H, S today 01:05:12.400 --> 01:05:13.860 align:middle line:90% with a good equilibrium. 01:05:13.860 --> 01:05:16.903 align:middle line:84% So let's reward them by playing S, S tomorrow. 01:05:16.903 --> 01:05:19.320 align:middle line:84% And I want to be clear, when we're computing repeated game 01:05:19.320 --> 01:05:22.840 align:middle line:84% equilibria, it's very tempting and it's very convenient 01:05:22.840 --> 01:05:26.280 align:middle line:84% to think about choosing an action profile rather than just 01:05:26.280 --> 01:05:26.780 align:middle line:90% an action. 01:05:26.780 --> 01:05:28.880 align:middle line:84% So I'm saying we're going to play S, S tomorrow. 01:05:28.880 --> 01:05:31.192 align:middle line:84% But we have to really understand it's always the case 01:05:31.192 --> 01:05:33.400 align:middle line:84% that a single player is choosing how they're playing. 01:05:33.400 --> 01:05:35.220 align:middle line:84% We're never jointly choosing what to do. 01:05:35.220 --> 01:05:37.460 align:middle line:84% It's just, as the analyst, when I'm analyzing it, 01:05:37.460 --> 01:05:39.980 align:middle line:84% it's convenient to think about where we put S, S. 01:05:39.980 --> 01:05:42.820 align:middle line:84% But each player is making a decision for themselves. 01:05:42.820 --> 01:05:45.320 align:middle line:90% I just want to keep that clear. 01:05:45.320 --> 01:05:47.920 align:middle line:84% So what we've set up so far is we've made it so 01:05:47.920 --> 01:05:50.477 align:middle line:84% that if we play H, S, which is how we want people to play. 01:05:50.477 --> 01:05:52.060 align:middle line:84% We want this to be a Nash equilibrium. 01:05:52.060 --> 01:05:53.800 align:middle line:84% We're going to be rewarded tomorrow 01:05:53.800 --> 01:05:58.620 align:middle line:84% by playing S, S. We also need to think about punishments. 01:05:58.620 --> 01:06:01.560 align:middle line:84% Let's try to punish the players if anyone unilaterally 01:06:01.560 --> 01:06:03.960 align:middle line:90% deviates so. 01:06:03.960 --> 01:06:07.640 align:middle line:84% If player 1 unilaterally deviates, what happens? 01:06:07.640 --> 01:06:11.960 align:middle line:84% Player 1 is supposed to play H. If they deviate, they play S 01:06:11.960 --> 01:06:16.040 align:middle line:84% And therefore, the history is S, S. So because S, 01:06:16.040 --> 01:06:18.720 align:middle line:84% S is a history that could result from a unilateral deviation 01:06:18.720 --> 01:06:22.000 align:middle line:84% by player 1, in order to punish that, we want to put H, H here. 01:06:22.000 --> 01:06:27.200 align:middle line:90% 01:06:27.200 --> 01:06:31.503 align:middle line:84% Conversely, if player 2 deviates from S to H, 01:06:31.503 --> 01:06:32.920 align:middle line:84% then the history is going to be H, 01:06:32.920 --> 01:06:39.760 align:middle line:84% H. So we also want to punish H, H with H, H. 01:06:39.760 --> 01:06:44.600 align:middle line:84% What about this remaining history? 01:06:44.600 --> 01:06:48.160 align:middle line:84% Should we put H, H here or S, S here? 01:06:48.160 --> 01:06:51.480 align:middle line:90% Or It doesn't matter. 01:06:51.480 --> 01:06:54.340 align:middle line:84% It turns out it won't matter at all. 01:06:54.340 --> 01:06:54.840 align:middle line:90% Why? 01:06:54.840 --> 01:06:58.120 align:middle line:84% Well, remember, Nash equilibrium and subgame perfect Nash 01:06:58.120 --> 01:07:00.435 align:middle line:84% equilibrium is all about unilateral deviations. 01:07:00.435 --> 01:07:02.560 align:middle line:84% So if we're interested in the incentives for people 01:07:02.560 --> 01:07:05.040 align:middle line:84% to unilaterally deviate, we have to say 01:07:05.040 --> 01:07:08.760 align:middle line:84% what happens if player 1 unilaterally deviates to S, 01:07:08.760 --> 01:07:11.520 align:middle line:84% or player 2 unilaterally deviates to H? 01:07:11.520 --> 01:07:14.120 align:middle line:84% But there's no unilateral deviation from H, S 01:07:14.120 --> 01:07:16.540 align:middle line:90% that will result in S, H. 01:07:16.540 --> 01:07:19.680 align:middle line:84% That would mean that player 1 had deviated from H to S, 01:07:19.680 --> 01:07:23.200 align:middle line:84% and player 2 had deviated from S to H. But if they both deviate, 01:07:23.200 --> 01:07:25.020 align:middle line:84% well, that's not a unilateral deviation. 01:07:25.020 --> 01:07:28.200 align:middle line:84% So it turns out whether this is an equilibrium will not actually 01:07:28.200 --> 01:07:29.660 align:middle line:90% depend on what happens here. 01:07:29.660 --> 01:07:34.000 align:middle line:84% So we could either do H, H or S, S. Let's do H, H, but S, 01:07:34.000 --> 01:07:37.000 align:middle line:90% S would also work. 01:07:37.000 --> 01:07:38.440 align:middle line:90% It can't be S, H, though. 01:07:38.440 --> 01:07:40.200 align:middle line:84% It's either H, H or S, S. Or maybe 01:07:40.200 --> 01:07:43.980 align:middle line:84% I should have big parentheses to make that clear. 01:07:43.980 --> 01:07:54.300 align:middle line:90% 01:07:54.300 --> 01:07:59.360 align:middle line:84% So now let's try to check whether this is indeed 01:07:59.360 --> 01:08:03.060 align:middle line:84% a Nash equilibrium, a subgame perfect Nash equilibrium. 01:08:03.060 --> 01:08:17.880 align:middle line:90% 01:08:17.880 --> 01:08:20.380 align:middle line:90% So maybe I'll call this star. 01:08:20.380 --> 01:08:23.000 align:middle line:90% This is our strategy profile. 01:08:23.000 --> 01:08:27.120 align:middle line:84% We're not playing a stage game Nash in period 0, 01:08:27.120 --> 01:08:29.020 align:middle line:84% but we claim that this is still an SPNE. 01:08:29.020 --> 01:08:36.200 align:middle line:84% So I claim that star is a subgame perfect Nash 01:08:36.200 --> 01:08:36.979 align:middle line:90% equilibrium. 01:08:36.979 --> 01:08:40.720 align:middle line:90% 01:08:40.720 --> 01:08:42.399 align:middle line:90% Now, there's a lot to check. 01:08:42.399 --> 01:08:44.960 align:middle line:84% We have to check that this forms a Nash 01:08:44.960 --> 01:08:48.040 align:middle line:90% equilibrium in every subgame. 01:08:48.040 --> 01:08:50.300 align:middle line:84% So we have to check all five subgames. 01:08:50.300 --> 01:08:58.720 align:middle line:90% 01:08:58.720 --> 01:09:01.732 align:middle line:84% But the one-shot deviation principle is going to help us. 01:09:01.732 --> 01:09:03.899 align:middle line:84% Technically, we have to check that in every subgame, 01:09:03.899 --> 01:09:05.819 align:middle line:90% there's no profitable deviation. 01:09:05.819 --> 01:09:07.319 align:middle line:84% But the one-shot deviation principle 01:09:07.319 --> 01:09:08.944 align:middle line:84% says we just have to check that there's 01:09:08.944 --> 01:09:10.620 align:middle line:84% no profitable one-shot deviation. 01:09:10.620 --> 01:09:13.479 align:middle line:84% So check in all five subgames, there 01:09:13.479 --> 01:09:19.180 align:middle line:84% is no profitable one-shot deviation. 01:09:19.180 --> 01:09:40.080 align:middle line:90% 01:09:40.080 --> 01:09:42.520 align:middle line:84% So the period 1 subgames are pretty easy, 01:09:42.520 --> 01:09:44.899 align:middle line:84% but let's just do one of them to check as an example. 01:09:44.899 --> 01:09:47.760 align:middle line:90% 01:09:47.760 --> 01:09:50.800 align:middle line:84% So in period 1, we have four subgames. 01:09:50.800 --> 01:09:59.600 align:middle line:84% Let's look at the subgame following H, H just 01:09:59.600 --> 01:10:06.140 align:middle line:90% to make sure we understand. 01:10:06.140 --> 01:10:09.360 align:middle line:90% 01:10:09.360 --> 01:10:12.920 align:middle line:90% So let's consider player 1. 01:10:12.920 --> 01:10:17.280 align:middle line:84% Player 1 is supposed to play H. At history H, H, 01:10:17.280 --> 01:10:20.920 align:middle line:84% player 1 is supposed to play H. So let's compare two things. 01:10:20.920 --> 01:10:23.640 align:middle line:84% What happens if player 1 plays H? 01:10:23.640 --> 01:10:26.840 align:middle line:84% And what happens if player 1 plays S? 01:10:26.840 --> 01:10:28.600 align:middle line:84% So let's understand what I'm doing here. 01:10:28.600 --> 01:10:32.680 align:middle line:84% I'm contemplating a one-shot deviation 01:10:32.680 --> 01:10:35.760 align:middle line:84% by player 1 at history H, H. I'm saying 01:10:35.760 --> 01:10:38.880 align:middle line:84% suppose we're at history H, H, and player 1 unilaterally 01:10:38.880 --> 01:10:40.000 align:middle line:90% deviates. 01:10:40.000 --> 01:10:41.640 align:middle line:84% Instead of playing H, they play S, 01:10:41.640 --> 01:10:44.120 align:middle line:84% and we want to compare the payoffs. 01:10:44.120 --> 01:10:49.160 align:middle line:84% So what if player 1 plays H as he's supposed to do? 01:10:49.160 --> 01:10:52.160 align:middle line:84% What is player 1's average payoff in the entire game 01:10:52.160 --> 01:10:53.640 align:middle line:90% going to be? 01:10:53.640 --> 01:10:56.560 align:middle line:90% Well, we're at the H, H history. 01:10:56.560 --> 01:10:59.200 align:middle line:84% So the way we compute payoffs is we already 01:10:59.200 --> 01:11:01.740 align:middle line:84% assume that H, H was played in the first period. 01:11:01.740 --> 01:11:05.280 align:middle line:84% So that gives player 1 a payoff of 1. 01:11:05.280 --> 01:11:07.200 align:middle line:84% And then if I play H again, I also 01:11:07.200 --> 01:11:10.880 align:middle line:84% get 1 because my opponent is also playing H. 01:11:10.880 --> 01:11:15.040 align:middle line:84% So we're going to get a payoff of 1 plus 1 over 2, 01:11:15.040 --> 01:11:18.080 align:middle line:90% which equals 1. 01:11:18.080 --> 01:11:20.760 align:middle line:84% What if I unilaterally deviate to S? 01:11:20.760 --> 01:11:23.480 align:middle line:90% What is my payoff going to be? 01:11:23.480 --> 01:11:25.897 align:middle line:84% Well, if I unilaterally deviate to S-- 01:11:25.897 --> 01:11:27.480 align:middle line:84% and maybe let's write-- let me be very 01:11:27.480 --> 01:11:29.300 align:middle line:90% clear about what happens here. 01:11:29.300 --> 01:11:37.560 align:middle line:90% 01:11:37.560 --> 01:11:42.560 align:middle line:84% So if at history H, H, I do what I'm supposed to and I play H, 01:11:42.560 --> 01:11:44.563 align:middle line:84% and my opponent also plays H, then this 01:11:44.563 --> 01:11:45.980 align:middle line:84% is what our payoff is going to be. 01:11:45.980 --> 01:11:49.820 align:middle line:84% The path of play is H, H in period 0, H, H in period 1, 01:11:49.820 --> 01:11:52.800 align:middle line:84% and my payoff is an average of 1. 01:11:52.800 --> 01:11:57.640 align:middle line:84% If I unilaterally deviate to S, then after history H, H 01:11:57.640 --> 01:11:59.840 align:middle line:84% I unilaterally deviate to S, but my opponent 01:11:59.840 --> 01:12:04.400 align:middle line:84% stays at H, which means my payoff is now 1 plus 0 01:12:04.400 --> 01:12:07.160 align:middle line:90% over 2, which equals one half. 01:12:07.160 --> 01:12:11.000 align:middle line:84% And indeed, deviating is not profitable. 01:12:11.000 --> 01:12:14.560 align:middle line:84% So what I've checked is there's no one-shot profitable deviation 01:12:14.560 --> 01:12:18.440 align:middle line:84% for player 1 at H, H. I actually have to do that eight times 01:12:18.440 --> 01:12:20.498 align:middle line:84% because there's four histories and two players, 01:12:20.498 --> 01:12:22.040 align:middle line:84% so it's a bit of a mess, but it's not 01:12:22.040 --> 01:12:24.280 align:middle line:84% too hard to check that that's going to work out. 01:12:24.280 --> 01:12:25.640 align:middle line:90% So now let's go to period 0. 01:12:25.640 --> 01:12:27.140 align:middle line:84% This is where I think the action is. 01:12:27.140 --> 01:12:51.400 align:middle line:90% 01:12:51.400 --> 01:12:53.660 align:middle line:90% So let's look at period 0. 01:12:53.660 --> 01:12:58.820 align:middle line:90% 01:12:58.820 --> 01:13:00.580 align:middle line:90% So the history is just nothing. 01:13:00.580 --> 01:13:02.000 align:middle line:90% Nothing has happened. 01:13:02.000 --> 01:13:06.340 align:middle line:84% And let's look at player 1 and player 2. 01:13:06.340 --> 01:13:09.480 align:middle line:90% 01:13:09.480 --> 01:13:18.950 align:middle line:84% And in period 0, player 1 is supposed to play H, 01:13:18.950 --> 01:13:20.830 align:middle line:90% but they could also play S. 01:13:20.830 --> 01:13:25.682 align:middle line:84% And player 2 is supposed to play S, 01:13:25.682 --> 01:13:30.510 align:middle line:84% but they could also play H. So to check that neither player has 01:13:30.510 --> 01:13:34.270 align:middle line:84% a profitable one-shot deviation, I need to say, what if player 1, 01:13:34.270 --> 01:13:36.230 align:middle line:84% instead of playing H as they're supposed to, 01:13:36.230 --> 01:13:40.230 align:middle line:84% one-shot deviates to playing S at this history, 01:13:40.230 --> 01:13:43.390 align:middle line:84% and then follows their strategy from there on out. 01:13:43.390 --> 01:13:46.070 align:middle line:84% And ask the same about player 2, and check that these deviations 01:13:46.070 --> 01:13:48.350 align:middle line:90% are not going to be profitable. 01:13:48.350 --> 01:13:51.110 align:middle line:84% So I think the first thing, before we can compute payoffs, 01:13:51.110 --> 01:13:52.390 align:middle line:90% is figure out what happens. 01:13:52.390 --> 01:13:57.830 align:middle line:84% So if player 1 does what they're supposed to and plays H today, 01:13:57.830 --> 01:14:00.130 align:middle line:84% then what's going to happen is we're going to get H, 01:14:00.130 --> 01:14:07.550 align:middle line:84% S today followed by S, S tomorrow. 01:14:07.550 --> 01:14:09.510 align:middle line:84% This is what's supposed to happen. 01:14:09.510 --> 01:14:10.810 align:middle line:90% And the same is true down here. 01:14:10.810 --> 01:14:12.870 align:middle line:84% If player 2 does what they're supposed to, 01:14:12.870 --> 01:14:18.270 align:middle line:84% we get H, S followed by S, S. But what 01:14:18.270 --> 01:14:23.230 align:middle line:84% happens if player 1 unilaterally deviates to S, S? 01:14:23.230 --> 01:14:26.590 align:middle line:90% Then we get S, S today. 01:14:26.590 --> 01:14:28.050 align:middle line:90% But then what happens tomorrow? 01:14:28.050 --> 01:14:33.030 align:middle line:90% 01:14:33.030 --> 01:14:33.530 align:middle line:90% Yeah? 01:14:33.530 --> 01:14:34.155 align:middle line:90% AUDIENCE: H, H. 01:14:34.155 --> 01:14:35.530 align:middle line:90% IAN BALL: H, H. Exactly. 01:14:35.530 --> 01:14:38.430 align:middle line:90% 01:14:38.430 --> 01:14:39.970 align:middle line:84% And let's ask the same of player 2. 01:14:39.970 --> 01:14:42.910 align:middle line:84% If they deviate from S to H, then we're 01:14:42.910 --> 01:14:47.390 align:middle line:90% going to get H, H today. 01:14:47.390 --> 01:14:48.730 align:middle line:90% And then what happens tomorrow? 01:14:48.730 --> 01:14:52.810 align:middle line:90% 01:14:52.810 --> 01:14:53.310 align:middle line:90% Yeah? 01:14:53.310 --> 01:14:54.030 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:14:54.030 --> 01:14:54.947 align:middle line:90% IAN BALL: H, H, right? 01:14:54.947 --> 01:14:59.510 align:middle line:90% 01:14:59.510 --> 01:15:03.430 align:middle line:84% So now we can exactly see what we were discussing. 01:15:03.430 --> 01:15:05.310 align:middle line:84% If player 1 unilaterally deviates, 01:15:05.310 --> 01:15:09.190 align:middle line:84% they get a benefit today because S, S is better than H, S. 01:15:09.190 --> 01:15:10.710 align:middle line:84% But then they get punished tomorrow 01:15:10.710 --> 01:15:14.230 align:middle line:84% because they get H, H tomorrow rather than S, S. 01:15:14.230 --> 01:15:16.213 align:middle line:84% And the same thing goes down here. 01:15:16.213 --> 01:15:17.630 align:middle line:84% And instead of adding it all up, I 01:15:17.630 --> 01:15:20.350 align:middle line:84% think it's easier to see, what is 01:15:20.350 --> 01:15:24.910 align:middle line:84% the gain from the deviation in going from H, S to S, S? 01:15:24.910 --> 01:15:27.690 align:middle line:90% Well if they did-- 01:15:27.690 --> 01:15:30.190 align:middle line:84% oh, maybe let's just write it all out. 01:15:30.190 --> 01:15:38.350 align:middle line:84% Player 1 payoff from H, S is going to be 1, 2; 2, 1. 01:15:38.350 --> 01:15:44.490 align:middle line:84% And here, player 2's payoff is 1, ; 0, 2. 01:15:44.490 --> 01:15:49.030 align:middle line:90% 01:15:49.030 --> 01:15:53.550 align:middle line:84% So if we add it all up, we're going to get an average of 3/2, 01:15:53.550 --> 01:15:58.170 align:middle line:90% 3/2, 1, and 1. 01:15:58.170 --> 01:16:02.350 align:middle line:90% 01:16:02.350 --> 01:16:04.910 align:middle line:84% So I've just gone through and computed the flow payoff 01:16:04.910 --> 01:16:06.730 align:middle line:84% that player 1 gets in each situation. 01:16:06.730 --> 01:16:08.150 align:middle line:90% And then I've taken the average. 01:16:08.150 --> 01:16:09.733 align:middle line:84% And then I've computed the flow payoff 01:16:09.733 --> 01:16:11.750 align:middle line:84% player 2 gets in each situation, and then I've 01:16:11.750 --> 01:16:13.270 align:middle line:90% computed the average. 01:16:13.270 --> 01:16:15.688 align:middle line:84% And what we indeed see is that if player 1 unilaterally 01:16:15.688 --> 01:16:17.230 align:middle line:84% deviates, well, actually their payoff 01:16:17.230 --> 01:16:18.770 align:middle line:84% is going to be exactly unchanged. 01:16:18.770 --> 01:16:20.310 align:middle line:90% So it's not profitable. 01:16:20.310 --> 01:16:22.193 align:middle line:84% If player 2 unilaterally deviates, 01:16:22.193 --> 01:16:23.610 align:middle line:84% their payoff is exactly unchanged. 01:16:23.610 --> 01:16:24.730 align:middle line:90% So it's not profitable. 01:16:24.730 --> 01:16:27.490 align:middle line:84% And therefore, we see that these deviations are not profitable 01:16:27.490 --> 01:16:28.950 align:middle line:90% and we're done here. 01:16:28.950 --> 01:16:30.710 align:middle line:84% And we can exactly see the effect. 01:16:30.710 --> 01:16:32.770 align:middle line:90% If player 1 says, wait a second. 01:16:32.770 --> 01:16:35.190 align:middle line:84% You're asking me to hunt for hare 01:16:35.190 --> 01:16:37.010 align:middle line:84% when my opponent's hunting for stag. 01:16:37.010 --> 01:16:39.430 align:middle line:84% I'd much rather hunt for stag and get a higher payoff. 01:16:39.430 --> 01:16:42.170 align:middle line:84% Indeed, your flow payoff would jump up by 1. 01:16:42.170 --> 01:16:44.250 align:middle line:84% It would jump from 1 to 2 if you deviated. 01:16:44.250 --> 01:16:46.430 align:middle line:84% The problem is that tomorrow, you're 01:16:46.430 --> 01:16:50.390 align:middle line:84% going to be punished with H, H, which reduces your payoff by 1. 01:16:50.390 --> 01:16:52.470 align:middle line:84% So the punishment you experience tomorrow 01:16:52.470 --> 01:16:56.772 align:middle line:84% is just large enough to offset the gain you experienced today. 01:16:56.772 --> 01:16:58.730 align:middle line:84% And therefore, the deviation is not profitable. 01:16:58.730 --> 01:17:01.070 align:middle line:84% And the same happens for player 2. 01:17:01.070 --> 01:17:05.060 align:middle line:84% Let me stop there, and I'll see everyone on Thursday. 01:17:05.060 --> 01:17:08.000 align:middle line:90%