WEBVTT 00:00:00.000 --> 00:00:04.865 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:04.865 --> 00:00:09.490 align:middle line:90% 00:00:09.490 --> 00:00:11.240 align:middle line:84% IAN BALL: So today we're going to continue 00:00:11.240 --> 00:00:14.118 align:middle line:90% with Bayes-Nash equilibrium. 00:00:14.118 --> 00:00:16.160 align:middle line:84% Last class was a bit of a theoretical class where 00:00:16.160 --> 00:00:19.715 align:middle line:84% we defined the concept, and we went over the foundations. 00:00:19.715 --> 00:00:21.840 align:middle line:84% Today, we're actually going to solve some examples. 00:00:21.840 --> 00:00:24.507 align:middle line:84% So we're going to solve an example of Cournot competition, 00:00:24.507 --> 00:00:26.840 align:middle line:84% and then we're going to get started with auctions, which 00:00:26.840 --> 00:00:28.465 align:middle line:84% is one of my favorite topics, and we'll 00:00:28.465 --> 00:00:30.840 align:middle line:90% cover that for a few lectures. 00:00:30.840 --> 00:00:36.640 align:middle line:84% So let's start with an example of Cournot competition 00:00:36.640 --> 00:00:41.700 align:middle line:90% with unknown demand. 00:00:41.700 --> 00:00:46.680 align:middle line:90% 00:00:46.680 --> 00:00:49.000 align:middle line:84% So you recall that the model of Cournot competition 00:00:49.000 --> 00:00:53.520 align:middle line:84% we've introduced so far, it was assumed that both firms exactly 00:00:53.520 --> 00:00:55.340 align:middle line:90% knew the demand curve. 00:00:55.340 --> 00:00:57.920 align:middle line:84% They exactly knew what the price would 00:00:57.920 --> 00:01:01.205 align:middle line:84% be as a function of both of the quantities that was set. 00:01:01.205 --> 00:01:03.080 align:middle line:84% But that's kind of an unrealistic assumption. 00:01:03.080 --> 00:01:09.460 align:middle line:84% So let's now suppose that the demand or, more precisely, 00:01:09.460 --> 00:01:15.400 align:middle line:84% the inverse demand, P of Q. Before we just 00:01:15.400 --> 00:01:17.340 align:middle line:90% said it was 1 minus q. 00:01:17.340 --> 00:01:22.200 align:middle line:84% Let me write what we had before, and then I'll change it. 00:01:22.200 --> 00:01:24.240 align:middle line:84% Before, we wrote something like this, 00:01:24.240 --> 00:01:28.800 align:middle line:84% so that if the quantity Q was produced on the market, 00:01:28.800 --> 00:01:31.680 align:middle line:84% then the price was either 1 minus Q, 00:01:31.680 --> 00:01:34.100 align:middle line:84% unless Q was really, really high, in which case, 00:01:34.100 --> 00:01:35.698 align:middle line:90% the price was 0. 00:01:35.698 --> 00:01:37.240 align:middle line:84% But what we're going to look at today 00:01:37.240 --> 00:01:41.960 align:middle line:84% is what if this intercept that used to be a 1 is now a theta, 00:01:41.960 --> 00:01:46.320 align:middle line:84% and theta is unknown, or theta is going 00:01:46.320 --> 00:01:47.620 align:middle line:90% to be private information. 00:01:47.620 --> 00:01:52.802 align:middle line:90% 00:01:52.802 --> 00:01:54.260 align:middle line:84% So we're going to look at, I guess, 00:01:54.260 --> 00:01:56.760 align:middle line:84% the simplest possible case, where one of the firms 00:01:56.760 --> 00:01:59.640 align:middle line:84% knows the value of theta, but the other firm does not. 00:01:59.640 --> 00:02:01.720 align:middle line:90% So let's assume that-- 00:02:01.720 --> 00:02:05.140 align:middle line:84% let's say theta can take two possible values. 00:02:05.140 --> 00:02:08.180 align:middle line:84% And let's assume they each occur with probability 1/2, 00:02:08.180 --> 00:02:09.500 align:middle line:90% so kind of the easiest case. 00:02:09.500 --> 00:02:16.520 align:middle line:84% So theta is an element of, well, either theta or theta high, 00:02:16.520 --> 00:02:18.180 align:middle line:90% and each with probability 1/2. 00:02:18.180 --> 00:02:27.100 align:middle line:90% 00:02:27.100 --> 00:02:27.600 align:middle line:90% Great. 00:02:27.600 --> 00:02:30.720 align:middle line:84% So of course, we're going to assume here 00:02:30.720 --> 00:02:32.340 align:middle line:90% that the labels make sense. 00:02:32.340 --> 00:02:39.000 align:middle line:84% So theta L is smaller than theta H. So if the state is theta H, 00:02:39.000 --> 00:02:41.962 align:middle line:90% this is a high-demand state. 00:02:41.962 --> 00:02:43.920 align:middle line:84% And it means that for a given level of quantity 00:02:43.920 --> 00:02:45.712 align:middle line:84% that we put on the market, the market price 00:02:45.712 --> 00:02:46.700 align:middle line:90% is going to be higher. 00:02:46.700 --> 00:02:49.160 align:middle line:84% Because it turns out that more people like the good than we 00:02:49.160 --> 00:02:50.800 align:middle line:90% necessarily expected. 00:02:50.800 --> 00:02:53.560 align:middle line:84% If the demand state is theta L, then that's 00:02:53.560 --> 00:02:54.840 align:middle line:90% the lower number here. 00:02:54.840 --> 00:02:56.923 align:middle line:84% And for a given quantity Q that's produced, 00:02:56.923 --> 00:02:58.840 align:middle line:84% the price is going to be lower because there's 00:02:58.840 --> 00:03:00.580 align:middle line:84% going to be less demand for the product. 00:03:00.580 --> 00:03:02.288 align:middle line:84% So this is certainly relevant when a firm 00:03:02.288 --> 00:03:03.665 align:middle line:90% is introducing a new product. 00:03:03.665 --> 00:03:05.540 align:middle line:84% They don't know how popular it's going to be. 00:03:05.540 --> 00:03:07.290 align:middle line:84% They may have some uncertainty about that, 00:03:07.290 --> 00:03:09.960 align:middle line:84% and that's what this is capturing here. 00:03:09.960 --> 00:03:20.240 align:middle line:84% And we're going to assume that firm one observes theta, 00:03:20.240 --> 00:03:21.460 align:middle line:90% but firm two does not. 00:03:21.460 --> 00:03:26.280 align:middle line:90% 00:03:26.280 --> 00:03:28.640 align:middle line:84% So one story could be that firm one is kind 00:03:28.640 --> 00:03:30.140 align:middle line:90% of an experienced incumbent. 00:03:30.140 --> 00:03:32.120 align:middle line:84% They know how fashions change, and they're 00:03:32.120 --> 00:03:33.920 align:middle line:84% able to predict what demand is going to be. 00:03:33.920 --> 00:03:35.320 align:middle line:84% Firm two might be this new entrant 00:03:35.320 --> 00:03:37.540 align:middle line:84% who doesn't have as much data, doesn't have as much experience, 00:03:37.540 --> 00:03:38.957 align:middle line:84% and they don't know what theta is. 00:03:38.957 --> 00:03:42.520 align:middle line:84% Now, of course, what is assumed is that this distribution 00:03:42.520 --> 00:03:43.960 align:middle line:90% is common knowledge. 00:03:43.960 --> 00:03:49.640 align:middle line:84% So firm two knows, or believes, that the state is equal to theta 00:03:49.640 --> 00:03:52.540 align:middle line:84% L with probability 1/2 and theta H with probability 1/2. 00:03:52.540 --> 00:03:55.600 align:middle line:84% They know that, but they don't know exactly which value it is. 00:03:55.600 --> 00:03:57.840 align:middle line:84% And moreover, everything I've written on the board 00:03:57.840 --> 00:03:58.880 align:middle line:90% is common knowledge. 00:03:58.880 --> 00:04:00.320 align:middle line:90% This is this subtle thing. 00:04:00.320 --> 00:04:05.120 align:middle line:84% So firm two knows that firm one knows the state. 00:04:05.120 --> 00:04:10.000 align:middle line:84% And we're going to now write this down for you. 00:04:10.000 --> 00:04:13.480 align:middle line:90% So now let's try to solve for-- 00:04:13.480 --> 00:04:15.720 align:middle line:84% and I guess the last component I need to say, 00:04:15.720 --> 00:04:18.000 align:middle line:84% maybe I'll just say it here, is let's 00:04:18.000 --> 00:04:21.380 align:middle line:84% assume costs are 0 just to make it the algebra a bit easier. 00:04:21.380 --> 00:04:30.360 align:middle line:90% So marginal costs equals 0. 00:04:30.360 --> 00:04:33.740 align:middle line:84% So as usual with our Cournot specification, 00:04:33.740 --> 00:04:37.420 align:middle line:84% your profit is your quantity times your profit per unit, 00:04:37.420 --> 00:04:39.060 align:middle line:90% which is price minus cost. 00:04:39.060 --> 00:04:40.420 align:middle line:90% In this case, cost is 0. 00:04:40.420 --> 00:04:44.680 align:middle line:84% So your payoff is just your quantity times the market price. 00:04:44.680 --> 00:04:50.880 align:middle line:84% So now I think the first step, so now our goal is solving for B 00:04:50.880 --> 00:04:55.940 align:middle line:84% and E. A Bayes-Nash equilibrium-- 00:04:55.940 --> 00:04:57.420 align:middle line:84% I think sometimes the hardest part, 00:04:57.420 --> 00:04:59.680 align:middle line:84% often the hardest part of this exercise 00:04:59.680 --> 00:05:01.250 align:middle line:84% is figuring out what a strategy is. 00:05:01.250 --> 00:05:03.500 align:middle line:84% So the first step is to figure out what a strategy is. 00:05:03.500 --> 00:05:04.480 align:middle line:84% And once we've done that, I think 00:05:04.480 --> 00:05:05.605 align:middle line:90% we're in pretty good shape. 00:05:05.605 --> 00:05:07.320 align:middle line:84% So remember, a strategy is going to need 00:05:07.320 --> 00:05:09.800 align:middle line:84% to specify an action a player is going 00:05:09.800 --> 00:05:13.260 align:middle line:84% to take as a function of whatever their mental state is. 00:05:13.260 --> 00:05:14.980 align:middle line:84% If they have two distinct mental states, 00:05:14.980 --> 00:05:17.080 align:middle line:84% they can take different actions in those states. 00:05:17.080 --> 00:05:20.780 align:middle line:90% So any ideas here? 00:05:20.780 --> 00:05:26.397 align:middle line:84% Let's look at firm one and firm two. 00:05:26.397 --> 00:05:28.980 align:middle line:84% What are the strategies going to be for each of these players? 00:05:28.980 --> 00:05:30.880 align:middle line:84% So not what the equilibrium is, but just what kind of object 00:05:30.880 --> 00:05:31.660 align:middle line:90% is it going to be? 00:05:31.660 --> 00:05:32.320 align:middle line:90% Yeah? 00:05:32.320 --> 00:05:35.880 align:middle line:84% STUDENT: The strategy will be some non-negative yield number. 00:05:35.880 --> 00:05:37.400 align:middle line:90% IAN BALL: So a quantity. 00:05:37.400 --> 00:05:39.020 align:middle line:84% And that will be a strategy for whom? 00:05:39.020 --> 00:05:41.348 align:middle line:84% STUDENT: [INAUDIBLE] the strategy. 00:05:41.348 --> 00:05:42.140 align:middle line:90% IAN BALL: So close. 00:05:42.140 --> 00:05:43.240 align:middle line:90% So not quite. 00:05:43.240 --> 00:05:45.782 align:middle line:84% So I think that's right for firm two. 00:05:45.782 --> 00:05:47.240 align:middle line:84% So firm two is going to be choosing 00:05:47.240 --> 00:05:48.718 align:middle line:90% a non-negative real number. 00:05:48.718 --> 00:05:49.260 align:middle line:90% You're right. 00:05:49.260 --> 00:05:50.180 align:middle line:90% We have to be careful here. 00:05:50.180 --> 00:05:50.847 align:middle line:90% So you're right. 00:05:50.847 --> 00:05:52.423 align:middle line:90% Firm one will wake up. 00:05:52.423 --> 00:05:54.340 align:middle line:84% They'll see that demand is either high or low, 00:05:54.340 --> 00:05:56.363 align:middle line:84% and they will actually choose either-- they'll 00:05:56.363 --> 00:05:57.780 align:middle line:84% choose a non-negative real number. 00:05:57.780 --> 00:06:00.020 align:middle line:84% But their strategy has to be a complete contingent plan. 00:06:00.020 --> 00:06:01.060 align:middle line:84% Yeah, you want to follow up on that? 00:06:01.060 --> 00:06:01.260 align:middle line:90% Yeah? 00:06:01.260 --> 00:06:02.320 align:middle line:84% STUDENT: There should be [INAUDIBLE]. 00:06:02.320 --> 00:06:03.100 align:middle line:90% IAN BALL: Exactly. 00:06:03.100 --> 00:06:05.392 align:middle line:84% STUDENT: There should be two values in the [INAUDIBLE], 00:06:05.392 --> 00:06:08.060 align:middle line:84% one based on theta low and one-- so one for theta low, one for 00:06:08.060 --> 00:06:08.560 align:middle line:90% [INAUDIBLE]. 00:06:08.560 --> 00:06:08.820 align:middle line:90% IAN BALL: Great. 00:06:08.820 --> 00:06:09.860 align:middle line:84% So I'm glad you answered that way. 00:06:09.860 --> 00:06:11.680 align:middle line:84% Your first answer was a really common thing 00:06:11.680 --> 00:06:12.580 align:middle line:90% that we see in exams. 00:06:12.580 --> 00:06:14.760 align:middle line:84% And also, it's good that this came up. 00:06:14.760 --> 00:06:17.480 align:middle line:84% It's tricky because firm one will actually, in reality, 00:06:17.480 --> 00:06:19.560 align:middle line:84% choose a single number, but their strategy 00:06:19.560 --> 00:06:21.520 align:middle line:84% has to specify two numbers because it must 00:06:21.520 --> 00:06:23.080 align:middle line:90% be a complete contingent plan. 00:06:23.080 --> 00:06:24.900 align:middle line:84% And if it didn't specify two numbers, 00:06:24.900 --> 00:06:26.600 align:middle line:84% there's no way firm two could figure out 00:06:26.600 --> 00:06:28.640 align:middle line:84% what to do because firm two doesn't 00:06:28.640 --> 00:06:30.140 align:middle line:90% know the state of the world. 00:06:30.140 --> 00:06:32.080 align:middle line:84% So if they don't know, or don't form beliefs 00:06:32.080 --> 00:06:34.060 align:middle line:84% about how firm one is behaving in each state of the world, 00:06:34.060 --> 00:06:35.352 align:middle line:90% they're going to be in trouble. 00:06:35.352 --> 00:06:37.580 align:middle line:84% So for firm two, it is just a number. 00:06:37.580 --> 00:06:39.560 align:middle line:90% Maybe we'll call it Q2. 00:06:39.560 --> 00:06:42.800 align:middle line:84% Maybe I'll just say it's a non-negative number here. 00:06:42.800 --> 00:06:47.200 align:middle line:84% But firm one can, in principle, produce a different amount 00:06:47.200 --> 00:06:51.010 align:middle line:84% in situations or in states when demand is high and in states 00:06:51.010 --> 00:06:52.250 align:middle line:90% when demand is low. 00:06:52.250 --> 00:06:59.250 align:middle line:84% So they're going to choose two numbers, Q1 of theta L 00:06:59.250 --> 00:07:03.130 align:middle line:84% and Q1 of theta H. And these are both 00:07:03.130 --> 00:07:05.030 align:middle line:84% going to be non-negative quantity numbers. 00:07:05.030 --> 00:07:08.530 align:middle line:84% If we want to think of this in terms of our formalism 00:07:08.530 --> 00:07:11.550 align:middle line:84% from last week, firm two has a single type. 00:07:11.550 --> 00:07:13.070 align:middle line:90% They have a single mental state. 00:07:13.070 --> 00:07:15.130 align:middle line:90% Sometimes we call that T bar. 00:07:15.130 --> 00:07:18.430 align:middle line:84% And when I just write Q2, that's really Q2 of T bar. 00:07:18.430 --> 00:07:22.450 align:middle line:84% It's the quantity they produce for their single mental state. 00:07:22.450 --> 00:07:25.590 align:middle line:84% Firm one actually has two mental states. 00:07:25.590 --> 00:07:27.320 align:middle line:84% And the way we wrote it last time is we 00:07:27.320 --> 00:07:29.070 align:middle line:84% would have written it something like this. 00:07:29.070 --> 00:07:32.090 align:middle line:84% We would have said, firm one has type TL. 00:07:32.090 --> 00:07:35.370 align:middle line:84% And when their type is TL, they know the state is theta L. 00:07:35.370 --> 00:07:38.390 align:middle line:84% And when their type is TH, they know the state is theta H. 00:07:38.390 --> 00:07:39.990 align:middle line:84% And if we wanted to be really formal, 00:07:39.990 --> 00:07:42.685 align:middle line:84% we would write Q1 of TL and Q1 of TH. 00:07:42.685 --> 00:07:45.310 align:middle line:84% But then you just have to keep track of the T's and the thetas, 00:07:45.310 --> 00:07:48.307 align:middle line:84% and we're going to take a shortcut and write it this way. 00:07:48.307 --> 00:07:50.890 align:middle line:84% But the understanding is that the reason we write theta L here 00:07:50.890 --> 00:07:54.010 align:middle line:84% this is the type of firm one who knows for certain 00:07:54.010 --> 00:07:56.410 align:middle line:84% that the state is theta L. I want 00:07:56.410 --> 00:07:58.410 align:middle line:84% to be clear about that interpretation. 00:07:58.410 --> 00:08:02.398 align:middle line:90% 00:08:02.398 --> 00:08:06.210 align:middle line:84% So let me put these in parentheses. 00:08:06.210 --> 00:08:09.130 align:middle line:84% So now we want to solve for B and E. 00:08:09.130 --> 00:08:11.330 align:middle line:84% But I think first it's kind of good to always try 00:08:11.330 --> 00:08:12.913 align:middle line:84% to guess and form intuition about what 00:08:12.913 --> 00:08:14.090 align:middle line:90% our answer is going to be. 00:08:14.090 --> 00:08:17.890 align:middle line:84% So in equilibrium, how do you think Q1 of theta L, 00:08:17.890 --> 00:08:20.990 align:middle line:84% Q1 of theta H, and Q2 are going to be ordered? 00:08:20.990 --> 00:08:24.230 align:middle line:84% Any guesses about how those three numbers might be ordered? 00:08:24.230 --> 00:08:29.130 align:middle line:84% See if we can form some intuition in advance. 00:08:29.130 --> 00:08:30.730 align:middle line:90% So imagine you're a firm. 00:08:30.730 --> 00:08:34.610 align:middle line:84% If you think this product is very, very popular, 00:08:34.610 --> 00:08:36.850 align:middle line:84% and the market price is going to be very, very high, 00:08:36.850 --> 00:08:39.017 align:middle line:84% are you going to produce more of the product or less 00:08:39.017 --> 00:08:39.716 align:middle line:90% of the product? 00:08:39.716 --> 00:08:40.549 align:middle line:90% More of the product. 00:08:40.549 --> 00:08:42.809 align:middle line:90% So let's start up here. 00:08:42.809 --> 00:08:45.510 align:middle line:84% Between Q1 theta L and Q1 theta H, 00:08:45.510 --> 00:08:47.590 align:middle line:84% which one do you think is going to be higher? 00:08:47.590 --> 00:08:49.090 align:middle line:84% We think this is going to be higher. 00:08:49.090 --> 00:08:49.990 align:middle line:90% And it will be. 00:08:49.990 --> 00:08:53.230 align:middle line:84% Now maybe a trickier question, how do these two compare to Q2? 00:08:53.230 --> 00:08:55.115 align:middle line:90% That's maybe a bit harder. 00:08:55.115 --> 00:08:56.490 align:middle line:84% STUDENT: Somewhere in the middle. 00:08:56.490 --> 00:08:57.230 align:middle line:84% IAN BALL: Somewhere in the middle. 00:08:57.230 --> 00:08:57.830 align:middle line:90% Yeah, exactly. 00:08:57.830 --> 00:09:00.130 align:middle line:84% So it turns out, and this is maybe not obvious 00:09:00.130 --> 00:09:02.230 align:middle line:84% and depends a bit more in the details the problem, 00:09:02.230 --> 00:09:04.330 align:middle line:84% but if I'm firm two, I know there's 00:09:04.330 --> 00:09:07.450 align:middle line:84% a half chance the state is theta L and a half chance the state is 00:09:07.450 --> 00:09:10.570 align:middle line:84% theta H. So it's intuitive that the amount I'm going to produce 00:09:10.570 --> 00:09:12.110 align:middle line:90% is somewhere in between. 00:09:12.110 --> 00:09:15.188 align:middle line:84% Now, it's a bit trickier because when the state is theta L, 00:09:15.188 --> 00:09:17.230 align:middle line:84% it's also going to change what firm one produces. 00:09:17.230 --> 00:09:20.390 align:middle line:84% So that conclusion is a little sensitive to the specification. 00:09:20.390 --> 00:09:22.330 align:middle line:84% But in this case, that's going to be true. 00:09:22.330 --> 00:09:24.490 align:middle line:90% So let's try to write it down. 00:09:24.490 --> 00:09:26.210 align:middle line:84% So we've gone through step one which 00:09:26.210 --> 00:09:27.975 align:middle line:84% is identifying the kinds of objects 00:09:27.975 --> 00:09:30.350 align:middle line:84% that we're choosing-- how many numbers we have to choose, 00:09:30.350 --> 00:09:31.892 align:middle line:84% how many functions we have to choose. 00:09:31.892 --> 00:09:34.950 align:middle line:84% The next step is to write down the equilibrium conditions. 00:09:34.950 --> 00:09:38.323 align:middle line:84% And the crucial thing about BNE, Bayes-Nash Equilibrium, 00:09:38.323 --> 00:09:39.990 align:middle line:84% is that we're going to have a condition, 00:09:39.990 --> 00:09:43.410 align:middle line:84% an optimality condition for every type of every player. 00:09:43.410 --> 00:09:53.490 align:middle line:84% So remember, BNE, Bayes-Nash Equilibrium, 00:09:53.490 --> 00:10:00.990 align:middle line:84% requires optimality for every type of every player. 00:10:00.990 --> 00:10:09.297 align:middle line:90% 00:10:09.297 --> 00:10:11.630 align:middle line:84% And that means we're going to have a different equation, 00:10:11.630 --> 00:10:13.650 align:middle line:84% a different optimality condition for every type 00:10:13.650 --> 00:10:16.528 align:middle line:84% of every player that says that type is behaving optimally. 00:10:16.528 --> 00:10:19.070 align:middle line:84% So in this problem, how many conditions are we going to have? 00:10:19.070 --> 00:10:21.610 align:middle line:90% 00:10:21.610 --> 00:10:22.670 align:middle line:90% Three, right? 00:10:22.670 --> 00:10:24.570 align:middle line:84% Because there's one type of player two. 00:10:24.570 --> 00:10:26.050 align:middle line:84% So we're going to have one condition for player two 00:10:26.050 --> 00:10:27.930 align:middle line:84% that says player two is behaving optimally. 00:10:27.930 --> 00:10:30.513 align:middle line:84% And then we're going to have two different conditions for firm 00:10:30.513 --> 00:10:32.650 align:middle line:84% one saying if firm one is the high type, 00:10:32.650 --> 00:10:34.290 align:middle line:84% and they observe that demand is high, 00:10:34.290 --> 00:10:35.645 align:middle line:90% then they're behaving optimally. 00:10:35.645 --> 00:10:37.270 align:middle line:84% And if they observe that demand is low, 00:10:37.270 --> 00:10:38.810 align:middle line:84% then they're also behaving optimally. 00:10:38.810 --> 00:10:40.268 align:middle line:84% Now, to write out these conditions, 00:10:40.268 --> 00:10:42.188 align:middle line:84% it's nice to have a little math notation. 00:10:42.188 --> 00:10:43.730 align:middle line:84% People may have seen this, but I want 00:10:43.730 --> 00:10:45.897 align:middle line:84% to introduce it to make sure we're on the same page. 00:10:45.897 --> 00:10:48.710 align:middle line:84% So we often write things like max-- 00:10:48.710 --> 00:10:52.130 align:middle line:84% let's just go through a simple example to define this notation. 00:10:52.130 --> 00:10:55.410 align:middle line:84% We might want to maximize some function f over x. 00:10:55.410 --> 00:10:56.790 align:middle line:84% And let's put some range in here. 00:10:56.790 --> 00:11:02.810 align:middle line:84% Let's say x is between negative 1 and 1. 00:11:02.810 --> 00:11:05.970 align:middle line:84% So what this expression denotes is the maximum value 00:11:05.970 --> 00:11:08.130 align:middle line:84% of this function over this range. 00:11:08.130 --> 00:11:10.690 align:middle line:84% There's a different expression, a different notation that we 00:11:10.690 --> 00:11:14.530 align:middle line:90% want, which is the argmax-- 00:11:14.530 --> 00:11:17.570 align:middle line:84% let's use this-- of f of x, which 00:11:17.570 --> 00:11:20.290 align:middle line:84% is the set of x values, the set of inputs 00:11:20.290 --> 00:11:21.525 align:middle line:90% that achieve this maximum. 00:11:21.525 --> 00:11:24.150 align:middle line:84% And the easiest way to see it is just to go through an example. 00:11:24.150 --> 00:11:27.550 align:middle line:84% So let's say f of x is x squared. 00:11:27.550 --> 00:11:31.610 align:middle line:90% 00:11:31.610 --> 00:11:34.250 align:middle line:84% I think just doing a numerical example is the easiest thing. 00:11:34.250 --> 00:11:37.070 align:middle line:84% So maybe I'll just erase the f of x to avoid confusion. 00:11:37.070 --> 00:11:41.450 align:middle line:84% Let's just-- I don't have too many equal signs. 00:11:41.450 --> 00:11:45.450 align:middle line:84% So what is the maximum value of x squared over the domain where 00:11:45.450 --> 00:11:47.710 align:middle line:90% x ranges from negative 1 to 1? 00:11:47.710 --> 00:11:48.910 align:middle line:90% This is just 1, right? 00:11:48.910 --> 00:11:51.470 align:middle line:84% So this is not testing the math, but just the notation. 00:11:51.470 --> 00:11:52.970 align:middle line:84% This is the number 1, because that's 00:11:52.970 --> 00:11:55.130 align:middle line:84% the highest value that this-- actually, 00:11:55.130 --> 00:11:56.610 align:middle line:84% let me use two to avoid confusion. 00:11:56.610 --> 00:11:59.290 align:middle line:90% Sorry. 00:11:59.290 --> 00:12:01.690 align:middle line:84% Because otherwise, that would not be a good choice. 00:12:01.690 --> 00:12:03.410 align:middle line:90% So the answer is 2. 00:12:03.410 --> 00:12:06.410 align:middle line:84% What about here, the argmax of 2x squared? 00:12:06.410 --> 00:12:09.930 align:middle line:84% Where is this maximum value of 2 achieved, at what x points? 00:12:09.930 --> 00:12:11.390 align:middle line:90% STUDENT: 1 and minus 1. 00:12:11.390 --> 00:12:12.390 align:middle line:90% IAN BALL: 1 and minus 1. 00:12:12.390 --> 00:12:14.370 align:middle line:84% So when you write argmax, this is actually 00:12:14.370 --> 00:12:19.850 align:middle line:84% a set consisting of the two numbers 1 and minus 1. 00:12:19.850 --> 00:12:21.210 align:middle line:90% So it's good to distinguish. 00:12:21.210 --> 00:12:25.470 align:middle line:84% This max says we take in a function, we take in a domain, 00:12:25.470 --> 00:12:27.810 align:middle line:84% and it gives us the highest value that function 00:12:27.810 --> 00:12:29.690 align:middle line:90% takes over the domain. 00:12:29.690 --> 00:12:32.290 align:middle line:84% When we write argmax, now what we're returning 00:12:32.290 --> 00:12:34.503 align:middle line:84% is not the value, but an input to the function, 00:12:34.503 --> 00:12:35.670 align:middle line:90% an argument of the function. 00:12:35.670 --> 00:12:39.170 align:middle line:84% And we're saying at which arguments in the domain 00:12:39.170 --> 00:12:41.170 align:middle line:84% does the function achieve this maximum? 00:12:41.170 --> 00:12:43.850 align:middle line:84% So this is helpful notation because often we'll 00:12:43.850 --> 00:12:48.830 align:middle line:84% write things like x star is an element of the argmax. 00:12:48.830 --> 00:12:51.330 align:middle line:84% And what this means is if you were someone who were choosing 00:12:51.330 --> 00:12:53.990 align:middle line:84% x, and you wanted to maximize this function, 00:12:53.990 --> 00:12:56.270 align:middle line:84% you could choose either negative 1 or 1, 00:12:56.270 --> 00:12:58.070 align:middle line:84% and those would both maximize the function. 00:12:58.070 --> 00:13:00.737 align:middle line:84% And the way we would write down that condition is to say x star, 00:13:00.737 --> 00:13:03.310 align:middle line:84% the thing that you choose, is an element of the argmax. 00:13:03.310 --> 00:13:04.970 align:middle line:84% It's an element of the set of things 00:13:04.970 --> 00:13:06.190 align:middle line:90% that maximize the function. 00:13:06.190 --> 00:13:08.090 align:middle line:84% So we'll use that notation when we write down our equilibrium 00:13:08.090 --> 00:13:09.010 align:middle line:90% conditions. 00:13:09.010 --> 00:13:11.170 align:middle line:90% Any questions on this? 00:13:11.170 --> 00:13:13.050 align:middle line:84% Basically, the advantage of this is 00:13:13.050 --> 00:13:14.890 align:middle line:84% we only have to write the function once. 00:13:14.890 --> 00:13:16.890 align:middle line:84% If we had to say f of x is greater than or equal to f 00:13:16.890 --> 00:13:19.270 align:middle line:84% of x prime for all x prime, you just have to write things twice. 00:13:19.270 --> 00:13:20.770 align:middle line:84% And when the formulas get long, this 00:13:20.770 --> 00:13:24.530 align:middle line:90% is a nice way of writing it. 00:13:24.530 --> 00:13:28.490 align:middle line:84% So now after that digression, let's now 00:13:28.490 --> 00:13:31.782 align:middle line:84% write out our optimality conditions for this game. 00:13:31.782 --> 00:13:33.990 align:middle line:84% And as we said, there's going to be three conditions. 00:13:33.990 --> 00:13:36.365 align:middle line:84% So we're going to have the condition for maybe, I'll say, 00:13:36.365 --> 00:13:37.650 align:middle line:90% F1-- this is firm one-- 00:13:37.650 --> 00:13:39.810 align:middle line:90% when they know the state is low. 00:13:39.810 --> 00:13:44.170 align:middle line:90% There's F1 theta H-- 00:13:44.170 --> 00:13:46.930 align:middle line:84% firm one when they know the state is high. 00:13:46.930 --> 00:13:49.670 align:middle line:84% And then there's F2, firm two, and they don't know anything. 00:13:49.670 --> 00:13:50.920 align:middle line:90% There's only one type of them. 00:13:50.920 --> 00:13:52.970 align:middle line:84% So we just need a single optimality condition 00:13:52.970 --> 00:13:53.710 align:middle line:90% for the firm. 00:13:53.710 --> 00:13:58.862 align:middle line:90% 00:13:58.862 --> 00:14:03.870 align:middle line:84% And we're looking for an equilibrium, 00:14:03.870 --> 00:14:06.610 align:middle line:84% so we're looking for values of this that form an equilibrium. 00:14:06.610 --> 00:14:08.110 align:middle line:84% So we'll probably put stars in here. 00:14:08.110 --> 00:14:11.330 align:middle line:84% So what we're going to get is this system of equations 00:14:11.330 --> 00:14:15.410 align:middle line:84% is going to depend on, maybe I'll call it Q1 star of theta L, 00:14:15.410 --> 00:14:17.910 align:middle line:90% Q1 star of theta H, and Q2 star. 00:14:17.910 --> 00:14:20.610 align:middle line:84% Those are the three things we're trying to solve for. 00:14:20.610 --> 00:14:26.210 align:middle line:84% So let's start with theta L. So what is firm one choosing? 00:14:26.210 --> 00:14:28.030 align:middle line:84% They're choosing a non-negative quantity. 00:14:28.030 --> 00:14:33.530 align:middle line:84% So what they're choosing is Q1 star of theta L. 00:14:33.530 --> 00:14:35.790 align:middle line:84% And we want this to be in the argmax of something 00:14:35.790 --> 00:14:38.930 align:middle line:84% because we want this to be an optimal choice for this firm 00:14:38.930 --> 00:14:40.970 align:middle line:84% when they know that demand is low. 00:14:40.970 --> 00:14:49.850 align:middle line:84% So we're going to say this is in the argmax over Q1 prime. 00:14:49.850 --> 00:14:53.730 align:middle line:84% So what I'm saying here is I'm-- what I want to fill in here is, 00:14:53.730 --> 00:14:56.010 align:middle line:90% what is the payoff to firm one? 00:14:56.010 --> 00:14:58.490 align:middle line:84% If they know the state is theta L, 00:14:58.490 --> 00:15:00.530 align:middle line:90% they choose quantity Q1 prime. 00:15:00.530 --> 00:15:03.130 align:middle line:84% And they believe that the other firm is following 00:15:03.130 --> 00:15:05.170 align:middle line:90% their equilibrium strategy. 00:15:05.170 --> 00:15:08.810 align:middle line:84% So here, we're going to have the expected revenue 00:15:08.810 --> 00:15:10.530 align:middle line:90% or profit of firm one. 00:15:10.530 --> 00:15:12.290 align:middle line:84% And to say that Q1 star of theta L 00:15:12.290 --> 00:15:14.930 align:middle line:84% is in the argmax is to say this level of production 00:15:14.930 --> 00:15:17.890 align:middle line:84% maximizes the revenue of firm one. 00:15:17.890 --> 00:15:19.130 align:middle line:90% So what is it going to be? 00:15:19.130 --> 00:15:23.770 align:middle line:84% Well, if I produce Q1 prime, then my quantity is Q1 prime. 00:15:23.770 --> 00:15:27.190 align:middle line:84% But then I have to multiply that by my revenue-- 00:15:27.190 --> 00:15:31.015 align:middle line:90% 00:15:31.015 --> 00:15:33.530 align:middle line:84% let's see-- let me just write it like this-- 00:15:33.530 --> 00:15:36.890 align:middle line:90% by my revenue per unit. 00:15:36.890 --> 00:15:41.690 align:middle line:84% So that means I need to plug in the market price. 00:15:41.690 --> 00:15:43.930 align:middle line:84% And we know that the market price 00:15:43.930 --> 00:15:47.250 align:middle line:90% is going to be theta minus Q. 00:15:47.250 --> 00:15:50.650 align:middle line:84% So if I'm firm one, I know the state is theta L. 00:15:50.650 --> 00:15:53.370 align:middle line:84% So I'm going to put in theta L here. 00:15:53.370 --> 00:15:56.290 align:middle line:84% And then I need to subtract the amount 00:15:56.290 --> 00:16:00.770 align:middle line:90% that I produce minus Q1 prime. 00:16:00.770 --> 00:16:03.930 align:middle line:90% What else do I have to subtract? 00:16:03.930 --> 00:16:05.228 align:middle line:90% Yeah? 00:16:05.228 --> 00:16:05.770 align:middle line:90% Say it again? 00:16:05.770 --> 00:16:06.370 align:middle line:90% STUDENT: Q2 star. 00:16:06.370 --> 00:16:07.120 align:middle line:90% IAN BALL: Q2 star. 00:16:07.120 --> 00:16:10.910 align:middle line:90% Exactly And this is it. 00:16:10.910 --> 00:16:15.050 align:middle line:84% So let's understand because I know the state is theta L, 00:16:15.050 --> 00:16:17.210 align:middle line:84% I use theta L when I'm computing what the market 00:16:17.210 --> 00:16:18.490 align:middle line:90% price is going to be. 00:16:18.490 --> 00:16:22.170 align:middle line:84% If I'm firm one, and I choose quantity Q1 prime, 00:16:22.170 --> 00:16:24.730 align:middle line:84% and I believe that my opponent is following the equilibrium 00:16:24.730 --> 00:16:27.290 align:middle line:84% strategy of producing Q2 star, then this 00:16:27.290 --> 00:16:28.703 align:middle line:90% is going to be the market price. 00:16:28.703 --> 00:16:30.870 align:middle line:84% And I'm going to make that for every unit I produce, 00:16:30.870 --> 00:16:33.190 align:middle line:84% which is Q1 prime, and that's going to give me revenue. 00:16:33.190 --> 00:16:45.860 align:middle line:84% Now we're going to get a similar formula for theta H. 00:16:45.860 --> 00:16:47.610 align:middle line:84% And I want to make it clear here-- sorry-- 00:16:47.610 --> 00:16:48.830 align:middle line:90% Q1 prime is a number here. 00:16:48.830 --> 00:16:49.970 align:middle line:84% So this might look like a function 00:16:49.970 --> 00:16:51.030 align:middle line:90% being evaluated somewhere. 00:16:51.030 --> 00:16:51.988 align:middle line:90% That's not what I mean. 00:16:51.988 --> 00:16:53.727 align:middle line:84% It's just a number times another number. 00:16:53.727 --> 00:16:55.310 align:middle line:84% And the primes don't mean derivatives. 00:16:55.310 --> 00:16:57.650 align:middle line:90% They're just notation. 00:16:57.650 --> 00:17:00.050 align:middle line:90% So here, it's exactly the same. 00:17:00.050 --> 00:17:03.470 align:middle line:84% But now the market demand is different. 00:17:03.470 --> 00:17:06.170 align:middle line:90% 00:17:06.170 --> 00:17:08.849 align:middle line:84% So the key difference here is even if we both produce 00:17:08.849 --> 00:17:11.450 align:middle line:84% the same amount, the market price that we face 00:17:11.450 --> 00:17:13.210 align:middle line:84% is going to be different in the high state 00:17:13.210 --> 00:17:15.329 align:middle line:84% than it would have been in the low state. 00:17:15.329 --> 00:17:18.990 align:middle line:84% And now from two, this is a bit of a harder one-- 00:17:18.990 --> 00:17:24.849 align:middle line:90% 00:17:24.849 --> 00:17:27.170 align:middle line:84% so they're going to say the quantity I'm actually 00:17:27.170 --> 00:17:31.010 align:middle line:84% producing is optimal among all the quantities I could produce. 00:17:31.010 --> 00:17:33.530 align:middle line:84% But now they're in a bit of a tricky situation 00:17:33.530 --> 00:17:36.650 align:middle line:84% because they don't know what the state is. 00:17:36.650 --> 00:17:39.770 align:middle line:84% So firm two needs to take expectations 00:17:39.770 --> 00:17:41.770 align:middle line:90% over the state of the world. 00:17:41.770 --> 00:17:48.170 align:middle line:84% So there's a half probability that the state is high. 00:17:48.170 --> 00:17:52.210 align:middle line:84% And if the state is high, and firm two chooses quantity Q2 00:17:52.210 --> 00:17:54.030 align:middle line:84% prime-- let's write that in here-- 00:17:54.030 --> 00:17:57.930 align:middle line:90% 00:17:57.930 --> 00:17:59.810 align:middle line:84% the question is, what is the market price 00:17:59.810 --> 00:18:02.130 align:middle line:84% going to be in the case that the state is 00:18:02.130 --> 00:18:04.410 align:middle line:90% high in this equilibrium? 00:18:04.410 --> 00:18:07.130 align:middle line:84% Well, now we have, I guess, two-- the market-- the state 00:18:07.130 --> 00:18:08.530 align:middle line:90% matters in two ways. 00:18:08.530 --> 00:18:12.850 align:middle line:84% Because it's going to affect the actual intercept in the market 00:18:12.850 --> 00:18:14.470 align:middle line:84% price or the inverse demand curve, 00:18:14.470 --> 00:18:17.010 align:middle line:84% but it's also going to affect what firm one chooses. 00:18:17.010 --> 00:18:20.330 align:middle line:84% Because firm one's choice is contingent on the state. 00:18:20.330 --> 00:18:25.020 align:middle line:84% So if the state is high, we're going to get theta H minus Q2 00:18:25.020 --> 00:18:36.300 align:middle line:90% prime minus Q1 star of theta H. 00:18:36.300 --> 00:18:39.722 align:middle line:84% So you'll see actually here a tricky thing for firm two 00:18:39.722 --> 00:18:41.180 align:middle line:84% is these effects are actually going 00:18:41.180 --> 00:18:42.388 align:middle line:90% to go in opposite directions. 00:18:42.388 --> 00:18:48.340 align:middle line:84% So if firm two-- if it turns out the state is high, 00:18:48.340 --> 00:18:51.180 align:middle line:84% that's good for firm one because demand is going to be higher. 00:18:51.180 --> 00:18:53.680 align:middle line:84% But it means the opponent is actually going to produce more, 00:18:53.680 --> 00:18:57.620 align:middle line:84% and that kind of creates a bit of a those countervailing 00:18:57.620 --> 00:18:58.700 align:middle line:90% directions. 00:18:58.700 --> 00:19:00.280 align:middle line:84% And then we'll do the other thing. 00:19:00.280 --> 00:19:01.880 align:middle line:84% What if what if the state is low? 00:19:01.880 --> 00:19:04.255 align:middle line:84% Well, we're going to have to change this and change this. 00:19:04.255 --> 00:19:05.120 align:middle line:90% So we have-- 00:19:05.120 --> 00:19:18.020 align:middle line:90% 00:19:18.020 --> 00:19:20.180 align:middle line:90% Notice a crucial point here. 00:19:20.180 --> 00:19:24.317 align:middle line:84% It's the same Q2 prime here as here. 00:19:24.317 --> 00:19:25.900 align:middle line:84% Of course, if I were firm two, I would 00:19:25.900 --> 00:19:27.580 align:middle line:84% love to choose a different quantity 00:19:27.580 --> 00:19:30.272 align:middle line:84% when the state is high than when the state is low. 00:19:30.272 --> 00:19:31.980 align:middle line:84% But I'm unable to do that because I'm not 00:19:31.980 --> 00:19:33.240 align:middle line:90% able to observe the state. 00:19:33.240 --> 00:19:35.320 align:middle line:84% So I choose a single quantity, Q2 prime, 00:19:35.320 --> 00:19:38.040 align:middle line:84% and I take expectations over the state. 00:19:38.040 --> 00:19:41.020 align:middle line:90% So here's our conditions. 00:19:41.020 --> 00:19:43.360 align:middle line:84% We won't fully solve it out because it gets a bit messy. 00:19:43.360 --> 00:19:45.380 align:middle line:84% But let's just reduce each of these 00:19:45.380 --> 00:19:46.600 align:middle line:90% to a first-order condition. 00:19:46.600 --> 00:19:48.920 align:middle line:84% So here, we've stated it as an argmax property. 00:19:48.920 --> 00:19:52.740 align:middle line:84% Let's just write the corresponding equalities. 00:19:52.740 --> 00:19:55.820 align:middle line:84% Well, in this top row, we can use our trick. 00:19:55.820 --> 00:20:00.446 align:middle line:84% So I'll just go through these one more time. 00:20:00.446 --> 00:20:02.060 align:middle line:84% In the top row, we're going to see 00:20:02.060 --> 00:20:10.216 align:middle line:84% that Q1 prime should be half of theta L minus Q2 star 00:20:10.216 --> 00:20:18.900 align:middle line:90% of theta L. 2. 00:20:18.900 --> 00:20:21.420 align:middle line:90% 00:20:21.420 --> 00:20:34.180 align:middle line:84% F1 theta H, we can immediately see 00:20:34.180 --> 00:20:36.820 align:middle line:84% that our initial intuitive guess was correct, 00:20:36.820 --> 00:20:41.780 align:middle line:84% in that Q1 star of theta H is higher than Q1 star of theta L. 00:20:41.780 --> 00:20:44.480 align:middle line:84% Because Q2 star is the same in these two expressions. 00:20:44.480 --> 00:20:47.440 align:middle line:84% The only difference is theta L is here, and theta H is here, 00:20:47.440 --> 00:20:49.860 align:middle line:84% and we know theta H is higher than theta L. 00:20:49.860 --> 00:20:52.720 align:middle line:84% Now, the harder one, let's say F2. 00:20:52.720 --> 00:20:56.780 align:middle line:90% 00:20:56.780 --> 00:20:59.060 align:middle line:90% What's the trick here? 00:20:59.060 --> 00:21:03.380 align:middle line:84% It turns out we can distribute a little bit. 00:21:03.380 --> 00:21:06.300 align:middle line:84% So I guess one nice observation to keep in mind 00:21:06.300 --> 00:21:09.448 align:middle line:84% is the half is important to get the right value, 00:21:09.448 --> 00:21:11.240 align:middle line:84% but it's not going to affect the maximizer. 00:21:11.240 --> 00:21:14.980 align:middle line:84% Just like up here, if I maximize 3x squared instead of 2x 00:21:14.980 --> 00:21:18.040 align:middle line:84% squared, the set of maximizers would have been the same, 00:21:18.040 --> 00:21:19.900 align:middle line:84% though the maximum value would have changed. 00:21:19.900 --> 00:21:22.320 align:middle line:84% So I don't actually have to really worry about the half. 00:21:22.320 --> 00:21:28.020 align:middle line:84% And then I can factor the Q2 prime, 00:21:28.020 --> 00:21:35.900 align:middle line:84% and I think we will get Q2 star equals-- 00:21:35.900 --> 00:21:37.280 align:middle line:90% it's going to be an average. 00:21:37.280 --> 00:21:47.940 align:middle line:84% So it's going to be maybe theta L plus theta H minus Q1 star 00:21:47.940 --> 00:21:59.680 align:middle line:90% theta L I think over four. 00:21:59.680 --> 00:22:03.820 align:middle line:90% 00:22:03.820 --> 00:22:05.120 align:middle line:90% Make sure I did that right. 00:22:05.120 --> 00:22:15.123 align:middle line:90% 00:22:15.123 --> 00:22:16.540 align:middle line:84% Yeah, I think that should be good. 00:22:16.540 --> 00:22:20.460 align:middle line:90% 00:22:20.460 --> 00:22:23.380 align:middle line:84% And you can immediately make one-- 00:22:23.380 --> 00:22:25.760 align:middle line:84% I'm not going to solve-- this is how you got that. 00:22:25.760 --> 00:22:26.760 align:middle line:90% I mean, this is algebra. 00:22:26.760 --> 00:22:28.552 align:middle line:84% I'm not going to bore you with the details. 00:22:28.552 --> 00:22:30.360 align:middle line:84% I guess one nice observation, though, 00:22:30.360 --> 00:22:32.080 align:middle line:84% to make sure we see what's going on, 00:22:32.080 --> 00:22:35.180 align:middle line:84% is what if we added half of this equation 00:22:35.180 --> 00:22:36.560 align:middle line:90% to half of this equation? 00:22:36.560 --> 00:22:42.100 align:middle line:90% 00:22:42.100 --> 00:22:46.060 align:middle line:84% If we did that, what we would actually see 00:22:46.060 --> 00:22:49.540 align:middle line:84% is it can be shown that, in fact, Q2 star is 00:22:49.540 --> 00:22:54.860 align:middle line:84% going to equal Q1 star of theta L plus Q1 star 00:22:54.860 --> 00:22:57.923 align:middle line:90% of theta H over 2. 00:22:57.923 --> 00:22:59.340 align:middle line:84% So it turns out exactly consistent 00:22:59.340 --> 00:23:01.860 align:middle line:90% with our original intuition. 00:23:01.860 --> 00:23:04.740 align:middle line:84% The equilibrium quantity that Q2 star is going to choose 00:23:04.740 --> 00:23:06.757 align:middle line:84% is exactly going to be half of the quantity 00:23:06.757 --> 00:23:07.840 align:middle line:90% that the two firms choose. 00:23:07.840 --> 00:23:09.298 align:middle line:84% And if we did it out, we'd actually 00:23:09.298 --> 00:23:15.860 align:middle line:84% find that it turns out the amount Q2 star will actually 00:23:15.860 --> 00:23:20.700 align:middle line:84% be-- well, if we did a complete information game where 00:23:20.700 --> 00:23:25.000 align:middle line:84% the intercept was the average between theta L and theta H, 00:23:25.000 --> 00:23:27.180 align:middle line:84% and we solve for the equilibrium of that game, 00:23:27.180 --> 00:23:29.580 align:middle line:84% we'd actually get the same number for Q2 star. 00:23:29.580 --> 00:23:30.880 align:middle line:90% But that's just a detail. 00:23:30.880 --> 00:23:33.880 align:middle line:84% Any questions on how we solve this? 00:23:33.880 --> 00:23:35.520 align:middle line:90% So this is a good thing to know. 00:23:35.520 --> 00:23:37.900 align:middle line:90% This will come up on some exams. 00:23:37.900 --> 00:23:42.100 align:middle line:84% So now we're going to change course a little bit 00:23:42.100 --> 00:23:45.080 align:middle line:84% and start talking about maybe a more substantive application, 00:23:45.080 --> 00:23:47.680 align:middle line:84% really, what I think motivated us to set up Bayesian games, 00:23:47.680 --> 00:23:49.940 align:middle line:84% and that's the theory of auctions. 00:23:49.940 --> 00:23:52.758 align:middle line:84% So now we're going to start with auctions, 00:23:52.758 --> 00:23:54.800 align:middle line:84% and we'll play a little auction game in a second. 00:23:54.800 --> 00:23:57.480 align:middle line:84% But I want to start with just some examples of auctions. 00:23:57.480 --> 00:24:01.200 align:middle line:84% Have any of you been to an auction, played an auction, 00:24:01.200 --> 00:24:02.920 align:middle line:84% participated or know of auctions? 00:24:02.920 --> 00:24:04.637 align:middle line:84% What are some auctions that you've 00:24:04.637 --> 00:24:05.720 align:middle line:90% heard about or seen about? 00:24:05.720 --> 00:24:06.120 align:middle line:90% Yeah? 00:24:06.120 --> 00:24:06.820 align:middle line:90% STUDENT: Treasury auctions. 00:24:06.820 --> 00:24:08.820 align:middle line:84% IAN BALL: Treasury auctions are a great example. 00:24:08.820 --> 00:24:11.060 align:middle line:84% Yeah, maybe one of the biggest ones. 00:24:11.060 --> 00:24:13.660 align:middle line:90% So what's a Treasury auction? 00:24:13.660 --> 00:24:16.500 align:middle line:84% STUDENT: When the government basically auctions off 00:24:16.500 --> 00:24:19.860 align:middle line:84% government bonds to different retailers, 00:24:19.860 --> 00:24:22.160 align:middle line:84% to different institutional investors. 00:24:22.160 --> 00:24:23.220 align:middle line:90% IAN BALL: Great. 00:24:23.220 --> 00:24:25.562 align:middle line:84% And this is the way that the price and the interest 00:24:25.562 --> 00:24:27.520 align:middle line:84% rates that you see for government debt are set. 00:24:27.520 --> 00:24:29.280 align:middle line:84% They're set endogenously by this auction. 00:24:29.280 --> 00:24:29.918 align:middle line:90% Yeah. 00:24:29.918 --> 00:24:30.960 align:middle line:90% So that's a good example. 00:24:30.960 --> 00:24:31.600 align:middle line:90% That's the government. 00:24:31.600 --> 00:24:33.880 align:middle line:84% When the government wants to borrow and raise money, 00:24:33.880 --> 00:24:34.640 align:middle line:90% they sell debt. 00:24:34.640 --> 00:24:36.720 align:middle line:84% And the price that they sell it at is through an auction. 00:24:36.720 --> 00:24:37.340 align:middle line:90% Yes? 00:24:37.340 --> 00:24:40.420 align:middle line:84% STUDENT: Fundraiser auctions, where you actually get items. 00:24:40.420 --> 00:24:41.430 align:middle line:90% IAN BALL: Yeah. 00:24:41.430 --> 00:24:42.680 align:middle line:90% Yeah, so fundraising auctions. 00:24:42.680 --> 00:24:44.360 align:middle line:84% Yeah, what kinds of items are they auctioning off? 00:24:44.360 --> 00:24:44.860 align:middle line:90% Yeah. 00:24:44.860 --> 00:24:46.540 align:middle line:90% STUDENT: Cars, vacations. 00:24:46.540 --> 00:24:49.100 align:middle line:90% IAN BALL: Yeah. 00:24:49.100 --> 00:24:49.600 align:middle line:90% It's good. 00:24:49.600 --> 00:24:51.480 align:middle line:84% Yeah, and I think there's kind of a game element to it, 00:24:51.480 --> 00:24:53.313 align:middle line:84% and people show off about how much they bid. 00:24:53.313 --> 00:24:55.700 align:middle line:84% And they feel good because they show off 00:24:55.700 --> 00:24:58.380 align:middle line:84% that they're rich and charitable at the same time. 00:24:58.380 --> 00:25:00.833 align:middle line:84% So that's, I think, why we see auctions used there 00:25:00.833 --> 00:25:03.000 align:middle line:84% rather than just prices and saying, make a donation. 00:25:03.000 --> 00:25:05.500 align:middle line:84% You to get the warm glow, and you take the car. 00:25:05.500 --> 00:25:09.540 align:middle line:84% Any other kind of common auctions? 00:25:09.540 --> 00:25:10.040 align:middle line:90% Oh, yeah? 00:25:10.040 --> 00:25:10.720 align:middle line:90% Sorry, I didn't see you. 00:25:10.720 --> 00:25:11.620 align:middle line:90% STUDENT: Google Ad space. 00:25:11.620 --> 00:25:12.560 align:middle line:90% IAN BALL: Oh, great. 00:25:12.560 --> 00:25:15.280 align:middle line:84% So I think when I've taught this course in the past, 00:25:15.280 --> 00:25:18.940 align:middle line:84% no one's given that example, and I always have to bring it up. 00:25:18.940 --> 00:25:20.120 align:middle line:90% Google Ads is the biggest-- 00:25:20.120 --> 00:25:22.460 align:middle line:84% Google is the biggest private auctioneer in the world. 00:25:22.460 --> 00:25:26.880 align:middle line:84% They make $250 billion a year on auctions, roughly, in revenue. 00:25:26.880 --> 00:25:33.197 align:middle line:90% 00:25:33.197 --> 00:25:35.280 align:middle line:84% So normally, yeah, every year I ask this question. 00:25:35.280 --> 00:25:37.340 align:middle line:84% Everyone always thinks it's like Christie's or Sotheby's or one 00:25:37.340 --> 00:25:39.260 align:middle line:84% of these people that's-- one of these firms that's auctioning 00:25:39.260 --> 00:25:39.960 align:middle line:90% off paintings. 00:25:39.960 --> 00:25:42.800 align:middle line:84% But indeed, Google Ads is a much, much bigger market. 00:25:42.800 --> 00:25:45.000 align:middle line:84% So I guess great to see that answer. 00:25:45.000 --> 00:25:46.740 align:middle line:90% So how does Google Ads work? 00:25:46.740 --> 00:25:49.180 align:middle line:84% Or if you don't, that's OK, but just some thoughts. 00:25:49.180 --> 00:25:50.640 align:middle line:84% STUDENT: Automated at this point. 00:25:50.640 --> 00:25:54.620 align:middle line:84% But there's like a second-price auction, where everyone 00:25:54.620 --> 00:25:57.540 align:middle line:84% gets on an ad space, and then the top bidder 00:25:57.540 --> 00:26:00.020 align:middle line:84% pays the second highest bidder's price. 00:26:00.020 --> 00:26:02.885 align:middle line:84% And then it's ranked by the prominence of the ad. 00:26:02.885 --> 00:26:03.760 align:middle line:90% IAN BALL: Yeah, yeah. 00:26:03.760 --> 00:26:04.940 align:middle line:84% So you're describing really a lot 00:26:04.940 --> 00:26:05.980 align:middle line:84% of details which we're exactly going 00:26:05.980 --> 00:26:06.960 align:middle line:90% to go into in a few classes. 00:26:06.960 --> 00:26:07.440 align:middle line:90% That's great. 00:26:07.440 --> 00:26:08.500 align:middle line:90% So have you-- do you have-- 00:26:08.500 --> 00:26:09.667 align:middle line:90% did you work at Google, or-- 00:26:09.667 --> 00:26:11.372 align:middle line:84% STUDENT: I'm taking 1419 right now. 00:26:11.372 --> 00:26:12.080 align:middle line:90% IAN BALL: Oh, OK. 00:26:12.080 --> 00:26:13.740 align:middle line:90% Great. 00:26:13.740 --> 00:26:14.240 align:middle line:90% Good. 00:26:14.240 --> 00:26:16.200 align:middle line:84% So a little bit of overlap with that course. 00:26:16.200 --> 00:26:19.320 align:middle line:84% So I think Google Ads is two main categories. 00:26:19.320 --> 00:26:21.580 align:middle line:84% One is when you search on Google you 00:26:21.580 --> 00:26:23.500 align:middle line:84% see those sponsored advertisements at the top. 00:26:23.500 --> 00:26:25.780 align:middle line:84% Those are keyword searches where firms are bidding for the right 00:26:25.780 --> 00:26:26.760 align:middle line:90% to be displayed. 00:26:26.760 --> 00:26:29.660 align:middle line:84% So if you type in vacation, a hotel, or a flight 00:26:29.660 --> 00:26:31.380 align:middle line:84% might want to show their ad up there. 00:26:31.380 --> 00:26:33.963 align:middle line:84% The other way they make money is through third-party websites. 00:26:33.963 --> 00:26:36.100 align:middle line:84% So if you go to the New York Times and you see ads, 00:26:36.100 --> 00:26:38.317 align:middle line:84% those ads are also managed through Google. 00:26:38.317 --> 00:26:40.900 align:middle line:84% And the New York Times, if they want to raise revenue and post 00:26:40.900 --> 00:26:43.000 align:middle line:84% ads on their site, they'll use Google as a third party. 00:26:43.000 --> 00:26:44.380 align:middle line:84% And then Google will hold an auction 00:26:44.380 --> 00:26:46.580 align:middle line:84% on their behalf, where firms might be bidding for the right 00:26:46.580 --> 00:26:48.780 align:middle line:84% to show an ad to someone of a certain demographic 00:26:48.780 --> 00:26:51.080 align:middle line:84% in a certain place who's doing certain things. 00:26:51.080 --> 00:26:52.680 align:middle line:84% So these ads are very, very targeted. 00:26:52.680 --> 00:26:54.020 align:middle line:84% And that's been sort of-- that's where 00:26:54.020 --> 00:26:54.840 align:middle line:90% Google makes all their money. 00:26:54.840 --> 00:26:57.340 align:middle line:84% You might think, oh, they have a great algorithm for search. 00:26:57.340 --> 00:26:58.740 align:middle line:84% But that algorithm just enabled them 00:26:58.740 --> 00:27:01.073 align:middle line:84% to get attention, which they then monetized through ads. 00:27:01.073 --> 00:27:01.700 align:middle line:90% Yeah? 00:27:01.700 --> 00:27:03.825 align:middle line:84% STUDENT: So when you said about the New York Times, 00:27:03.825 --> 00:27:06.108 align:middle line:84% so the New York Times basically pays Google to take 00:27:06.108 --> 00:27:07.400 align:middle line:90% care of their ads through them? 00:27:07.400 --> 00:27:08.900 align:middle line:84% IAN BALL: Oh, Google would pay them. 00:27:08.900 --> 00:27:09.420 align:middle line:90% Yeah. 00:27:09.420 --> 00:27:11.680 align:middle line:84% So they want to raise revenue on their site. 00:27:11.680 --> 00:27:14.208 align:middle line:90% So what would happen is-- 00:27:14.208 --> 00:27:16.000 align:middle line:84% I mean, they could try to do it themselves, 00:27:16.000 --> 00:27:17.200 align:middle line:90% but that's kind of hard. 00:27:17.200 --> 00:27:19.040 align:middle line:84% So they would work together with Google. 00:27:19.040 --> 00:27:21.520 align:middle line:84% And then Google is going to conduct these auctions. 00:27:21.520 --> 00:27:22.720 align:middle line:84% The advertisers are going to pay Google, 00:27:22.720 --> 00:27:24.420 align:middle line:84% and then Google is going to give some cut to the New York Times 00:27:24.420 --> 00:27:26.180 align:middle line:90% and then keep some cut. 00:27:26.180 --> 00:27:27.780 align:middle line:90% So the advertisers pay. 00:27:27.780 --> 00:27:28.520 align:middle line:90% Why do they pay? 00:27:28.520 --> 00:27:29.900 align:middle line:84% Because they get to show their ad, and they hope 00:27:29.900 --> 00:27:31.380 align:middle line:90% they make money off that. 00:27:31.380 --> 00:27:33.947 align:middle line:84% Google is the intermediary that manages this, 00:27:33.947 --> 00:27:35.280 align:middle line:90% and they're going to take a cut. 00:27:35.280 --> 00:27:37.828 align:middle line:84% And then the New York Times is able to monetize some 00:27:37.828 --> 00:27:39.120 align:middle line:90% of the attention they generate. 00:27:39.120 --> 00:27:41.620 align:middle line:84% Because if the New York Times does well, and a lot of people 00:27:41.620 --> 00:27:43.800 align:middle line:84% go to their site and look at their ads, 00:27:43.800 --> 00:27:45.620 align:middle line:84% then they want to get some money for this. 00:27:45.620 --> 00:27:46.120 align:middle line:90% Yeah. 00:27:46.120 --> 00:27:48.320 align:middle line:84% STUDENT: They're like, not New York Times ads. 00:27:48.320 --> 00:27:48.680 align:middle line:90% IAN BALL: No, no. 00:27:48.680 --> 00:27:50.020 align:middle line:84% What I mean is if you go to the New York Times, 00:27:50.020 --> 00:27:52.270 align:middle line:84% you'll see ads on the side, which have nothing 00:27:52.270 --> 00:27:53.520 align:middle line:90% to do with the New York Times. 00:27:53.520 --> 00:27:54.960 align:middle line:84% I mean, the New York Times might sometimes-- 00:27:54.960 --> 00:27:57.127 align:middle line:84% the New York Times also shows some of their own ads, 00:27:57.127 --> 00:27:58.677 align:middle line:90% but that's just-- 00:27:58.677 --> 00:28:00.260 align:middle line:84% And then the New York Times would also 00:28:00.260 --> 00:28:01.520 align:middle line:90% be a player on other websites. 00:28:01.520 --> 00:28:03.160 align:middle line:84% So if you go to the Washington Post, 00:28:03.160 --> 00:28:05.165 align:middle line:84% you might see a New York Times ad next to it. 00:28:05.165 --> 00:28:06.540 align:middle line:84% That's because the New York Times 00:28:06.540 --> 00:28:09.260 align:middle line:84% has bid through Google on an ad to display on the Washington 00:28:09.260 --> 00:28:09.760 align:middle line:90% Post. 00:28:09.760 --> 00:28:12.010 align:middle line:84% So some firms are playing on both sides of the market. 00:28:12.010 --> 00:28:13.940 align:middle line:84% They're both buying ads through Google 00:28:13.940 --> 00:28:16.040 align:middle line:84% and also advertising themselves on their site. 00:28:16.040 --> 00:28:17.940 align:middle line:84% STUDENT: So the New York Times is getting-- 00:28:17.940 --> 00:28:19.540 align:middle line:84% some of the money that's getting paid 00:28:19.540 --> 00:28:22.180 align:middle line:84% to Google by the advertisers go to the New York Times? 00:28:22.180 --> 00:28:23.115 align:middle line:90% [INAUDIBLE] 00:28:23.115 --> 00:28:23.740 align:middle line:90% IAN BALL: Yeah. 00:28:23.740 --> 00:28:26.020 align:middle line:84% I mean, I don't know exactly how the accounting-- 00:28:26.020 --> 00:28:27.480 align:middle line:84% you could imagine it going different ways. 00:28:27.480 --> 00:28:29.360 align:middle line:84% But at the end, the money that's paid by the advertiser-- 00:28:29.360 --> 00:28:30.660 align:middle line:84% some is going to go to Google, and some is going 00:28:30.660 --> 00:28:31.560 align:middle line:90% to go to the New York Times. 00:28:31.560 --> 00:28:32.060 align:middle line:90% Yeah. 00:28:32.060 --> 00:28:34.340 align:middle line:90% Yeah, exactly. 00:28:34.340 --> 00:28:35.180 align:middle line:90% Great. 00:28:35.180 --> 00:28:37.280 align:middle line:84% And then, of course, if it's on Google's own site, 00:28:37.280 --> 00:28:39.197 align:middle line:84% then there's no third party, and Google's just 00:28:39.197 --> 00:28:43.660 align:middle line:84% taking the money directly if it's a sponsored ad on the site. 00:28:43.660 --> 00:28:46.100 align:middle line:84% I guess the classic one that we often talk about-- 00:28:46.100 --> 00:28:51.860 align:middle line:90% paintings, antiques. 00:28:51.860 --> 00:28:54.160 align:middle line:84% And I think a basic question we should ask-- 00:28:54.160 --> 00:28:55.580 align:middle line:90% I mean, why-- 00:28:55.580 --> 00:28:57.040 align:middle line:90% I've given some examples here. 00:28:57.040 --> 00:29:00.180 align:middle line:84% Most things we buy in the world are not auctioned off. 00:29:00.180 --> 00:29:02.700 align:middle line:84% When you buy clothes at a store, when 00:29:02.700 --> 00:29:04.860 align:middle line:84% you go to a coffee shop, when you watch 00:29:04.860 --> 00:29:08.053 align:middle line:84% a movie, when you buy a drink, these things-- generally, 00:29:08.053 --> 00:29:08.720 align:middle line:90% there's a price. 00:29:08.720 --> 00:29:09.345 align:middle line:90% You get dinner. 00:29:09.345 --> 00:29:11.580 align:middle line:84% There's a price that's posted, and you just pay it. 00:29:11.580 --> 00:29:13.925 align:middle line:84% Whereas, in these examples we see auctions used. 00:29:13.925 --> 00:29:15.300 align:middle line:84% Do have any thoughts about why we 00:29:15.300 --> 00:29:21.380 align:middle line:84% might see auctions in some contexts and prices in another 00:29:21.380 --> 00:29:21.880 align:middle line:90% maybe? 00:29:21.880 --> 00:29:22.740 align:middle line:90% Yeah? 00:29:22.740 --> 00:29:23.838 align:middle line:90% STUDENT: Price discovery. 00:29:23.838 --> 00:29:24.380 align:middle line:90% IAN BALL: OK. 00:29:24.380 --> 00:29:25.000 align:middle line:90% So good word. 00:29:25.000 --> 00:29:25.500 align:middle line:90% Yes. 00:29:25.500 --> 00:29:26.440 align:middle line:90% Say more about it. 00:29:26.440 --> 00:29:27.662 align:middle line:90% Yeah, it's a keyword. 00:29:27.662 --> 00:29:29.620 align:middle line:84% STUDENT: For example, like in a credit auction, 00:29:29.620 --> 00:29:31.900 align:middle line:84% like the government's unsure how to value 00:29:31.900 --> 00:29:34.300 align:middle line:90% like a certain type of bond. 00:29:34.300 --> 00:29:38.300 align:middle line:84% So an auction, there's kind of a bunch of literature 00:29:38.300 --> 00:29:41.180 align:middle line:84% that suggests that it's like the most efficient way to both raise 00:29:41.180 --> 00:29:43.780 align:middle line:84% revenue for the auctioneer, but then also to efficiently 00:29:43.780 --> 00:29:45.880 align:middle line:90% allocate to all the auctions. 00:29:45.880 --> 00:29:47.893 align:middle line:84% And it's like a price-revealing mechanism. 00:29:47.893 --> 00:29:48.560 align:middle line:90% IAN BALL: Great. 00:29:48.560 --> 00:29:49.060 align:middle line:90% Great. 00:29:49.060 --> 00:29:51.763 align:middle line:84% So let's say I have one of a kind antique. 00:29:51.763 --> 00:29:54.180 align:middle line:84% It might be pretty hard for me to know how much people are 00:29:54.180 --> 00:29:55.320 align:middle line:90% willing to pay for that. 00:29:55.320 --> 00:29:56.560 align:middle line:90% So I could just post a price. 00:29:56.560 --> 00:29:57.840 align:middle line:90% I say it's $10 million. 00:29:57.840 --> 00:29:58.600 align:middle line:90% But I'd worry. 00:29:58.600 --> 00:30:01.120 align:middle line:84% What if someone in the audience ended up valuing that a lot? 00:30:01.120 --> 00:30:03.412 align:middle line:84% For some reason, they really, really like this antique. 00:30:03.412 --> 00:30:05.550 align:middle line:84% Now I've wasted a lot of revenue. 00:30:05.550 --> 00:30:07.790 align:middle line:84% So we see auctions generally when 00:30:07.790 --> 00:30:11.007 align:middle line:84% firms are uncertain about how much people are willing to pay. 00:30:11.007 --> 00:30:12.590 align:middle line:84% With a coffee, that's a bit different. 00:30:12.590 --> 00:30:14.448 align:middle line:84% Dunkin' Donuts sells millions of coffees. 00:30:14.448 --> 00:30:15.990 align:middle line:84% They pretty much know what the market 00:30:15.990 --> 00:30:17.550 align:middle line:84% price-- they pretty much know what people 00:30:17.550 --> 00:30:18.890 align:middle line:90% are willing to pay for coffee. 00:30:18.890 --> 00:30:20.858 align:middle line:90% And also, it would be-- 00:30:20.858 --> 00:30:22.650 align:middle line:84% if you had live auctions at Dunkin' Donuts, 00:30:22.650 --> 00:30:23.570 align:middle line:84% I think it would be pretty costly. 00:30:23.570 --> 00:30:24.730 align:middle line:84% We're talking about pretty small amounts. 00:30:24.730 --> 00:30:26.830 align:middle line:84% Though, on the other hand, Google Ads are very small 00:30:26.830 --> 00:30:27.210 align:middle line:90% amounts. 00:30:27.210 --> 00:30:28.210 align:middle line:90% So maybe there is a way. 00:30:28.210 --> 00:30:31.670 align:middle line:84% And maybe eventually, we will see auctions for coffee 00:30:31.670 --> 00:30:33.670 align:middle line:90% as everything gets automated, 00:30:33.670 --> 00:30:36.150 align:middle line:84% I guess one final point I'll make 00:30:36.150 --> 00:30:38.510 align:middle line:84% that I think is kind of a new, exciting area. 00:30:38.510 --> 00:30:40.790 align:middle line:84% Google Ads used to be kind of the hot new thing. 00:30:40.790 --> 00:30:42.373 align:middle line:84% Google's losing a lot of money on this 00:30:42.373 --> 00:30:44.030 align:middle line:84% now because everyone's just using LLMs. 00:30:44.030 --> 00:30:45.572 align:middle line:84% And I think there's a lot of interest 00:30:45.572 --> 00:30:48.410 align:middle line:84% now in how LLMs are going to monetize this through auctions. 00:30:48.410 --> 00:30:51.950 align:middle line:84% So there's some talk now about Grok or OpenAI 00:30:51.950 --> 00:30:53.430 align:middle line:84% might let firms bid for the right 00:30:53.430 --> 00:30:55.210 align:middle line:84% to choose the next token that appears. 00:30:55.210 --> 00:30:58.150 align:middle line:84% So if you google-- if you ask an LLM, 00:30:58.150 --> 00:30:59.450 align:middle line:90% where should I go on vacation? 00:30:59.450 --> 00:31:01.950 align:middle line:84% And the LLM's response says oh, a great vacation destination 00:31:01.950 --> 00:31:04.070 align:middle line:84% is, you might have firms bidding for the right 00:31:04.070 --> 00:31:05.353 align:middle line:90% to select that next token. 00:31:05.353 --> 00:31:06.770 align:middle line:84% And I think we're not quite there. 00:31:06.770 --> 00:31:08.312 align:middle line:84% Well, we don't think we're there yet. 00:31:08.312 --> 00:31:10.770 align:middle line:84% It's a little unclear, but we may be going that direction. 00:31:10.770 --> 00:31:14.310 align:middle line:84% I think that's an interesting domain for future auction work. 00:31:14.310 --> 00:31:17.510 align:middle line:84% So we're going to start with some models 00:31:17.510 --> 00:31:18.843 align:middle line:90% of very simple auctions. 00:31:18.843 --> 00:31:20.510 align:middle line:84% Obviously, in these applications there's 00:31:20.510 --> 00:31:22.710 align:middle line:84% some complexity that we're not going to capture. 00:31:22.710 --> 00:31:28.990 align:middle line:84% And we're going to start with a simple sealed bid 00:31:28.990 --> 00:31:30.170 align:middle line:90% first-price auction. 00:31:30.170 --> 00:31:34.270 align:middle line:90% 00:31:34.270 --> 00:31:36.110 align:middle line:90% So we'll play this on MobLab. 00:31:36.110 --> 00:31:37.750 align:middle line:90% What does a sealed bid mean? 00:31:37.750 --> 00:31:40.815 align:middle line:84% It means-- in the old days, you had an envelope, 00:31:40.815 --> 00:31:42.190 align:middle line:84% and you gave it to the auctioneer 00:31:42.190 --> 00:31:44.810 align:middle line:84% that said that was the amount of your bid. 00:31:44.810 --> 00:31:55.630 align:middle line:90% So each bidder submits a bid. 00:31:55.630 --> 00:31:57.810 align:middle line:84% It's sealed because that bid is not observed. 00:31:57.810 --> 00:31:59.190 align:middle line:84% It's not like what you might see in the movies 00:31:59.190 --> 00:32:00.398 align:middle line:90% where people are calling out. 00:32:00.398 --> 00:32:01.940 align:middle line:84% That's called an open outcry auction. 00:32:01.940 --> 00:32:02.650 align:middle line:90% Oh, I'll bid 100. 00:32:02.650 --> 00:32:04.310 align:middle line:84% I'll bid 120, something like that. 00:32:04.310 --> 00:32:04.810 align:middle line:90% It's sealed. 00:32:04.810 --> 00:32:07.870 align:middle line:84% You're just submitting your bid in an envelope. 00:32:07.870 --> 00:32:10.390 align:middle line:84% First-price-- well, what's going to happen 00:32:10.390 --> 00:32:12.550 align:middle line:84% is the highest bidder is going to win, 00:32:12.550 --> 00:32:14.510 align:middle line:84% and they're going to pay their bid. 00:32:14.510 --> 00:32:28.390 align:middle line:84% So the highest bidder wins the good and pays their bid. 00:32:28.390 --> 00:32:30.870 align:middle line:84% The first-price nature is really referring to the fact 00:32:30.870 --> 00:32:32.410 align:middle line:84% that the winner pays their own bid. 00:32:32.410 --> 00:32:33.910 align:middle line:84% We'll also talk about auctions where 00:32:33.910 --> 00:32:35.190 align:middle line:84% the winner might pay the second highest 00:32:35.190 --> 00:32:36.450 align:middle line:90% bid, or something like that. 00:32:36.450 --> 00:32:37.950 align:middle line:84% But for now, we're going to just say 00:32:37.950 --> 00:32:39.670 align:middle line:84% the winner is going to pay their bid. 00:32:39.670 --> 00:32:41.370 align:middle line:84% What is the private information here? 00:32:41.370 --> 00:32:43.015 align:middle line:90% What are these things that-- 00:32:43.015 --> 00:32:44.390 align:middle line:84% the information that some players 00:32:44.390 --> 00:32:48.490 align:middle line:84% know that other players don't in the context of an auction? 00:32:48.490 --> 00:32:48.990 align:middle line:90% Yeah? 00:32:48.990 --> 00:32:50.250 align:middle line:84% STUDENT: How much they value their goods. 00:32:50.250 --> 00:32:51.250 align:middle line:90% IAN BALL: Exactly right. 00:32:51.250 --> 00:33:03.670 align:middle line:84% So here, each bidder privately knows what 00:33:03.670 --> 00:33:05.050 align:middle line:90% we might call their valuation. 00:33:05.050 --> 00:33:07.790 align:middle line:90% 00:33:07.790 --> 00:33:09.790 align:middle line:84% The valuation is going to be the most they would 00:33:09.790 --> 00:33:11.210 align:middle line:90% be willing to pay for the good. 00:33:11.210 --> 00:33:12.710 align:middle line:84% The way to think about the valuation 00:33:12.710 --> 00:33:14.850 align:middle line:84% is if they were to win the auction, when the good, 00:33:14.850 --> 00:33:16.830 align:middle line:84% and pay exactly their valuation, they'd 00:33:16.830 --> 00:33:19.293 align:middle line:84% be just as well off as if they didn't win the item at all. 00:33:19.293 --> 00:33:20.710 align:middle line:84% So it's the amount that they could 00:33:20.710 --> 00:33:23.710 align:middle line:84% pay that would make them exactly break even. 00:33:23.710 --> 00:33:26.430 align:middle line:84% So these games, it kind of helps to have MobLab 00:33:26.430 --> 00:33:28.970 align:middle line:84% because we have to generate randomly people's valuations. 00:33:28.970 --> 00:33:30.950 align:middle line:90% So let's log into MobLab. 00:33:30.950 --> 00:33:34.190 align:middle line:90% I'm going to try to set this up. 00:33:34.190 --> 00:33:37.683 align:middle line:84% May have been a while, so I'll give you a second. 00:33:37.683 --> 00:33:38.850 align:middle line:90% It actually did pretty well. 00:33:38.850 --> 00:33:41.050 align:middle line:90% We'll put that on here. 00:33:41.050 --> 00:33:44.470 align:middle line:90% So here, we see the groups. 00:33:44.470 --> 00:33:47.750 align:middle line:84% So this is giving us a few different pieces 00:33:47.750 --> 00:33:50.350 align:middle line:90% of information here. 00:33:50.350 --> 00:33:53.050 align:middle line:84% Let's understand what each of these columns mean. 00:33:53.050 --> 00:33:54.877 align:middle line:84% So you were split into six groups. 00:33:54.877 --> 00:33:56.210 align:middle line:90% You weren't a multiple of three. 00:33:56.210 --> 00:33:58.650 align:middle line:84% So some of you, we added robots to the groups to bid. 00:33:58.650 --> 00:33:59.530 align:middle line:90% I'm not exactly sure. 00:33:59.530 --> 00:34:01.780 align:middle line:84% They probably follow the equilibrium bidding strategy. 00:34:01.780 --> 00:34:03.670 align:middle line:84% I'm not exactly sure how the robots play. 00:34:03.670 --> 00:34:06.030 align:middle line:84% The first column is optimal surplus. 00:34:06.030 --> 00:34:07.810 align:middle line:84% What that's referring to is, well, 00:34:07.810 --> 00:34:09.389 align:middle line:84% each of the three people in the group 00:34:09.389 --> 00:34:11.710 align:middle line:84% have some value for the good, and that's 00:34:11.710 --> 00:34:14.270 align:middle line:84% referring to the maximum of those three values. 00:34:14.270 --> 00:34:16.270 align:middle line:84% Because the ideal thing, or the efficient thing, 00:34:16.270 --> 00:34:18.478 align:middle line:84% would be to give it to the person who values it most, 00:34:18.478 --> 00:34:20.489 align:middle line:84% and it's saying how much value would be created. 00:34:20.489 --> 00:34:23.770 align:middle line:84% So here you see these numbers are generally higher than 50 00:34:23.770 --> 00:34:26.989 align:middle line:84% because the maximum of three uniformly distributed numbers 00:34:26.989 --> 00:34:29.190 align:middle line:90% tends to be pretty high. 00:34:29.190 --> 00:34:31.010 align:middle line:90% And we see the optimal surplus. 00:34:31.010 --> 00:34:34.150 align:middle line:84% Then we see the surplus that was actually achieved. 00:34:34.150 --> 00:34:36.190 align:middle line:84% So we see in three of the groups, 00:34:36.190 --> 00:34:39.790 align:middle line:84% in groups three through five, we got 100%. 00:34:39.790 --> 00:34:42.190 align:middle line:84% What does that mean if we got 100% precisely? 00:34:42.190 --> 00:34:46.190 align:middle line:90% What does it mean exactly? 00:34:46.190 --> 00:34:47.010 align:middle line:90% Yeah, on the front? 00:34:47.010 --> 00:34:47.510 align:middle line:90% Yeah. 00:34:47.510 --> 00:34:49.685 align:middle line:84% STUDENT: Surplus [INAUDIBLE] surplus [INAUDIBLE]. 00:34:49.685 --> 00:34:50.310 align:middle line:90% IAN BALL: Yeah. 00:34:50.310 --> 00:34:51.909 align:middle line:84% So in words, what's another way of saying 00:34:51.909 --> 00:34:53.117 align:middle line:90% what happened in the auction? 00:34:53.117 --> 00:34:54.030 align:middle line:90% Who won the auction? 00:34:54.030 --> 00:34:55.429 align:middle line:84% STUDENT: The person who valued it the most. 00:34:55.429 --> 00:34:56.110 align:middle line:90% IAN BALL: The person-- 00:34:56.110 --> 00:34:57.230 align:middle line:84% exactly what it means is the person 00:34:57.230 --> 00:34:59.910 align:middle line:84% who valued the item the most is the one who won the auction 00:34:59.910 --> 00:35:01.070 align:middle line:90% and took the item home. 00:35:01.070 --> 00:35:03.750 align:middle line:84% So we see that the allocation was quite efficient. 00:35:03.750 --> 00:35:07.053 align:middle line:84% In one case, we see efficiency was 93%. 00:35:07.053 --> 00:35:09.470 align:middle line:84% That means the item did not go to the person who valued it 00:35:09.470 --> 00:35:12.230 align:middle line:84% most, but it happened to be that someone else valued it almost as 00:35:12.230 --> 00:35:13.855 align:middle line:84% much as the person who valued it most-- 00:35:13.855 --> 00:35:14.830 align:middle line:90% 93%. 00:35:14.830 --> 00:35:17.430 align:middle line:84% And so not too much of an efficiency loss. 00:35:17.430 --> 00:35:19.570 align:middle line:84% In group five, we did a little bit worse-- 00:35:19.570 --> 00:35:20.190 align:middle line:90% 82%. 00:35:20.190 --> 00:35:22.950 align:middle line:84% And then the last group, I think we just didn't finish there. 00:35:22.950 --> 00:35:27.225 align:middle line:84% And then we can see what the revenue was for the auctioneer. 00:35:27.225 --> 00:35:29.350 align:middle line:84% So that's another way of saying, in this case, what 00:35:29.350 --> 00:35:30.430 align:middle line:90% is the revenue? 00:35:30.430 --> 00:35:33.830 align:middle line:84% Well, the winning bidder pays their bid. 00:35:33.830 --> 00:35:38.350 align:middle line:84% So the revenue is exactly the winning bid. 00:35:38.350 --> 00:35:44.390 align:middle line:84% And you can see a few differences here. 00:35:44.390 --> 00:35:46.630 align:middle line:90% So let's look at group three. 00:35:46.630 --> 00:35:50.890 align:middle line:84% In group three, the revenue is 94, and the surplus was 94, 00:35:50.890 --> 00:35:52.450 align:middle line:90% and the optimal surplus was 94. 00:35:52.450 --> 00:35:54.550 align:middle line:90% So what happened in group three? 00:35:54.550 --> 00:35:55.510 align:middle line:90% Yeah? 00:35:55.510 --> 00:35:58.570 align:middle line:84% STUDENT: The guy with the highest value bid the value. 00:35:58.570 --> 00:35:59.510 align:middle line:90% IAN BALL: Right. 00:35:59.510 --> 00:36:00.670 align:middle line:90% And won. 00:36:00.670 --> 00:36:03.210 align:middle line:84% And that highest value is 94, right? 00:36:03.210 --> 00:36:03.710 align:middle line:90% Exactly. 00:36:03.710 --> 00:36:05.288 align:middle line:84% So how did how did that person come-- 00:36:05.288 --> 00:36:05.830 align:middle line:90% I don't know. 00:36:05.830 --> 00:36:07.705 align:middle line:84% Who was the-- you don't need to out yourself, 00:36:07.705 --> 00:36:09.570 align:middle line:90% but it was someone here. 00:36:09.570 --> 00:36:12.877 align:middle line:84% And how do you feel about winning at a price of 94? 00:36:12.877 --> 00:36:14.210 align:middle line:90% I don't know who this person is. 00:36:14.210 --> 00:36:17.217 align:middle line:84% Do you feel like you came away ahead or behind or you won? 00:36:17.217 --> 00:36:18.050 align:middle line:90% How are you feeling? 00:36:18.050 --> 00:36:18.590 align:middle line:90% Yeah? 00:36:18.590 --> 00:36:20.298 align:middle line:84% STUDENT: Since there was only one period, 00:36:20.298 --> 00:36:23.295 align:middle line:84% and I wanted to win the auction, I just bid all I had. 00:36:23.295 --> 00:36:25.670 align:middle line:84% IAN BALL: All you had, meaning your value, or literally-- 00:36:25.670 --> 00:36:27.188 align:middle line:90% STUDENT: Like the 94. 00:36:27.188 --> 00:36:27.730 align:middle line:90% IAN BALL: 94. 00:36:27.730 --> 00:36:28.883 align:middle line:90% So you bid your valuation. 00:36:28.883 --> 00:36:30.050 align:middle line:90% Yeah, I guess-- all you had. 00:36:30.050 --> 00:36:31.675 align:middle line:84% I don't know how much have in the game. 00:36:31.675 --> 00:36:34.890 align:middle line:84% So this gets at some issues of preferences. 00:36:34.890 --> 00:36:36.790 align:middle line:84% If your goal was to win the auction, 00:36:36.790 --> 00:36:38.530 align:middle line:84% maybe that was a pretty good strategy. 00:36:38.530 --> 00:36:39.630 align:middle line:90% You did win. 00:36:39.630 --> 00:36:41.510 align:middle line:84% But remember, the way that we defined 00:36:41.510 --> 00:36:44.070 align:middle line:84% your valuation is that if you win the item 00:36:44.070 --> 00:36:45.990 align:middle line:84% and pay your valuation, it's just 00:36:45.990 --> 00:36:47.430 align:middle line:90% as good as not winning at all. 00:36:47.430 --> 00:36:48.732 align:middle line:90% So in a sense, you won. 00:36:48.732 --> 00:36:50.190 align:middle line:84% But in another sense, you didn't do 00:36:50.190 --> 00:36:53.070 align:middle line:84% so well because you got something that was worth $94. 00:36:53.070 --> 00:36:56.530 align:middle line:84% In the extreme case, think of it as you paid $94 to get $94. 00:36:56.530 --> 00:37:00.210 align:middle line:84% So it was kind of a wash for you, but you didn't lose. 00:37:00.210 --> 00:37:01.390 align:middle line:90% So that's good. 00:37:01.390 --> 00:37:04.330 align:middle line:84% I think-- let's see how other people did. 00:37:04.330 --> 00:37:06.230 align:middle line:90% Who did best? 00:37:06.230 --> 00:37:07.990 align:middle line:84% We had the same issue in group 2. 00:37:07.990 --> 00:37:12.750 align:middle line:84% The winner bid exactly 59, and I guess didn't do so well. 00:37:12.750 --> 00:37:20.417 align:middle line:84% It was only-- in group five they bid 72 when the value was 73. 00:37:20.417 --> 00:37:22.250 align:middle line:84% Does that person want to share your thought? 00:37:22.250 --> 00:37:23.150 align:middle line:90% OK, yeah? 00:37:23.150 --> 00:37:26.350 align:middle line:84% STUDENT: Well, I thought that that's 00:37:26.350 --> 00:37:29.190 align:middle line:84% the max I can bid to at least guarantee that I'll have 00:37:29.190 --> 00:37:31.470 align:middle line:90% a positive payoff [INAUDIBLE]. 00:37:31.470 --> 00:37:32.190 align:middle line:90% IAN BALL: Yeah. 00:37:32.190 --> 00:37:34.430 align:middle line:84% STUDENT: I mean, there's a chance I still 00:37:34.430 --> 00:37:38.550 align:middle line:84% would have won if I did say 70, 71, or something, 00:37:38.550 --> 00:37:40.610 align:middle line:84% and I would have a higher payoff. 00:37:40.610 --> 00:37:45.150 align:middle line:90% But I think that there's no-- 00:37:45.150 --> 00:37:48.790 align:middle line:84% the probability that I win does go lower every time I bid lower. 00:37:48.790 --> 00:37:51.470 align:middle line:84% And I still, like even with my bid, even if I do bid, 00:37:51.470 --> 00:37:53.248 align:middle line:84% my payoff would be positive [INAUDIBLE]. 00:37:53.248 --> 00:37:53.790 align:middle line:90% IAN BALL: OK. 00:37:53.790 --> 00:37:56.175 align:middle line:84% And indeed, you did come away with a positive payoff. 00:37:56.175 --> 00:37:57.550 align:middle line:84% But I would argue maybe you could 00:37:57.550 --> 00:37:59.217 align:middle line:84% have gotten an even more positive payoff 00:37:59.217 --> 00:38:00.490 align:middle line:90% if you bid a little bit lower. 00:38:00.490 --> 00:38:00.990 align:middle line:90% Let's see. 00:38:00.990 --> 00:38:04.850 align:middle line:84% What about the person-- someone bid 40 and won and valued at 44. 00:38:04.850 --> 00:38:07.590 align:middle line:84% So I think they were the overall winner. 00:38:07.590 --> 00:38:09.290 align:middle line:84% They had the most surplus of anyone. 00:38:09.290 --> 00:38:09.590 align:middle line:90% Yeah? 00:38:09.590 --> 00:38:11.330 align:middle line:84% STUDENT: [INAUDIBLE] my valuation is pretty low. 00:38:11.330 --> 00:38:12.510 align:middle line:90% So I thought I would lose. 00:38:12.510 --> 00:38:13.532 align:middle line:90% IAN BALL: OK. 00:38:13.532 --> 00:38:15.740 align:middle line:84% STUDENT: That seemed like a reasonable amount that [? 00:38:15.740 --> 00:38:17.222 align:middle line:90% I've got ?] to lose. 00:38:17.222 --> 00:38:18.930 align:middle line:84% IAN BALL: So some people would argue, OK, 00:38:18.930 --> 00:38:19.690 align:middle line:90% I think I'm going to lose. 00:38:19.690 --> 00:38:20.450 align:middle line:90% My value is low. 00:38:20.450 --> 00:38:21.670 align:middle line:84% Why don't I bid more than my value 00:38:21.670 --> 00:38:22.962 align:middle line:90% to have some chance of winning? 00:38:22.962 --> 00:38:24.310 align:middle line:90% A lot of people reason that way. 00:38:24.310 --> 00:38:24.970 align:middle line:90% You didn't do that. 00:38:24.970 --> 00:38:26.210 align:middle line:84% I think that would have been a bad idea. 00:38:26.210 --> 00:38:26.912 align:middle line:90% Why did you not? 00:38:26.912 --> 00:38:28.370 align:middle line:84% You thought you were going to lose. 00:38:28.370 --> 00:38:30.070 align:middle line:84% Why didn't you bid a lot more to try to win? 00:38:30.070 --> 00:38:31.310 align:middle line:84% STUDENT: Because then my-- if I did win, 00:38:31.310 --> 00:38:32.830 align:middle line:90% my valuation would be negative. 00:38:32.830 --> 00:38:35.190 align:middle line:84% So I thought, in this case, I would rather 00:38:35.190 --> 00:38:36.920 align:middle line:90% get a valuation of 4 or 0. 00:38:36.920 --> 00:38:37.670 align:middle line:90% IAN BALL: Exactly. 00:38:37.670 --> 00:38:38.170 align:middle line:90% Right. 00:38:38.170 --> 00:38:39.990 align:middle line:84% So even if your valuation is really low, 00:38:39.990 --> 00:38:41.730 align:middle line:84% it's true that you're unlikely to win. 00:38:41.730 --> 00:38:43.372 align:middle line:84% But that might tempt you to overbid. 00:38:43.372 --> 00:38:44.330 align:middle line:90% That's not a good idea. 00:38:44.330 --> 00:38:45.913 align:middle line:84% Because if you do win, you're actually 00:38:45.913 --> 00:38:49.670 align:middle line:84% a loser because your valuation is smaller than the amount 00:38:49.670 --> 00:38:50.290 align:middle line:90% that you bid. 00:38:50.290 --> 00:38:53.210 align:middle line:84% So you bid $4 less than your true valuation. 00:38:53.210 --> 00:38:55.843 align:middle line:84% In the end, you got-- well, I don't 00:38:55.843 --> 00:38:57.510 align:middle line:84% know who else was in your group, but you 00:38:57.510 --> 00:38:59.650 align:middle line:84% got-- there was someone who had a value of 47. 00:38:59.650 --> 00:39:03.870 align:middle line:84% So actually you got lucky, and you 00:39:03.870 --> 00:39:06.550 align:middle line:84% beat someone who actually had a valuation higher than you, 00:39:06.550 --> 00:39:08.130 align:middle line:84% and you came away with a surplus. 00:39:08.130 --> 00:39:10.070 align:middle line:84% So my overall, I would say, people 00:39:10.070 --> 00:39:13.150 align:middle line:84% are maybe not ready to bid for Google 00:39:13.150 --> 00:39:14.410 align:middle line:90% or bid for the New York Times. 00:39:14.410 --> 00:39:16.535 align:middle line:84% I think there was some poor bidding-- 00:39:16.535 --> 00:39:17.910 align:middle line:84% I don't know-- OK bidding here, I 00:39:17.910 --> 00:39:19.660 align:middle line:84% think people generally bid a bit too high. 00:39:19.660 --> 00:39:23.110 align:middle line:84% So if you see they've actually computed the Nash equilibrium 00:39:23.110 --> 00:39:25.302 align:middle line:84% line, which is the dotted green line. 00:39:25.302 --> 00:39:27.010 align:middle line:84% And you can see everyone's actually bid-- 00:39:27.010 --> 00:39:28.710 align:middle line:84% I think every single person bid higher 00:39:28.710 --> 00:39:31.630 align:middle line:90% than would be optimal to bid. 00:39:31.630 --> 00:39:33.550 align:middle line:84% And I think this is a very common finding when 00:39:33.550 --> 00:39:34.592 align:middle line:90% you do things in the lab. 00:39:34.592 --> 00:39:36.008 align:middle line:84% And I think it gets to, do we have 00:39:36.008 --> 00:39:37.330 align:middle line:90% the right model of preferences? 00:39:37.330 --> 00:39:39.590 align:middle line:84% I think there's a certain excitement 00:39:39.590 --> 00:39:40.787 align:middle line:90% people get for winning. 00:39:40.787 --> 00:39:42.870 align:middle line:84% And they want to win, even if it means they end up 00:39:42.870 --> 00:39:44.250 align:middle line:90% paying a little bit too much. 00:39:44.250 --> 00:39:46.490 align:middle line:84% So people tend to overbid in first-price auctions. 00:39:46.490 --> 00:39:47.948 align:middle line:84% And that might actually be a reason 00:39:47.948 --> 00:39:50.250 align:middle line:84% why we see a lot of first-price auctions in practice, 00:39:50.250 --> 00:39:51.270 align:middle line:84% even though the theory would tell us 00:39:51.270 --> 00:39:53.853 align:middle line:84% that second-price auctions have some more desirable properties 00:39:53.853 --> 00:39:56.380 align:middle line:90% that we'll talk about. 00:39:56.380 --> 00:39:58.630 align:middle line:84% Maybe let's do it one more time, but now let's make it 00:39:58.630 --> 00:40:06.270 align:middle line:84% a second-price auction and see what happens. 00:40:06.270 --> 00:40:07.570 align:middle line:90% How do I change it? 00:40:07.570 --> 00:40:10.510 align:middle line:90% 00:40:10.510 --> 00:40:11.570 align:middle line:90% I'm not going to have it. 00:40:11.570 --> 00:40:15.270 align:middle line:90% 00:40:15.270 --> 00:40:16.650 align:middle line:84% Now, I just have to add it again. 00:40:16.650 --> 00:40:17.250 align:middle line:90% That's silly. 00:40:17.250 --> 00:40:26.070 align:middle line:90% 00:40:26.070 --> 00:40:29.570 align:middle line:84% So now let's make it a second-price auction. 00:40:29.570 --> 00:40:34.910 align:middle line:90% 00:40:34.910 --> 00:40:37.597 align:middle line:84% So what does a second-price auction mean? 00:40:37.597 --> 00:40:39.930 align:middle line:84% Well, the highest bidder is still going to win the item, 00:40:39.930 --> 00:40:41.888 align:middle line:84% but now they're going to pay the second highest 00:40:41.888 --> 00:40:44.310 align:middle line:90% bid rather than their own bid. 00:40:44.310 --> 00:40:50.190 align:middle line:84% So this should change how you play. 00:40:50.190 --> 00:40:52.450 align:middle line:90% So curious how things went. 00:40:52.450 --> 00:40:54.350 align:middle line:84% So here, let's look at group one. 00:40:54.350 --> 00:40:56.210 align:middle line:84% Here, we actually were in groups of six. 00:40:56.210 --> 00:40:58.350 align:middle line:84% So the optimal surpluses tended to be very high 00:40:58.350 --> 00:41:00.190 align:middle line:84% because we're looking at the maximum of six 00:41:00.190 --> 00:41:02.310 align:middle line:90% different random variables. 00:41:02.310 --> 00:41:04.950 align:middle line:84% It was efficiently allocated every single time. 00:41:04.950 --> 00:41:09.410 align:middle line:84% And it looks like the revenue was-- 00:41:09.410 --> 00:41:12.163 align:middle line:90% 00:41:12.163 --> 00:41:14.830 align:middle line:84% I guess it's a bit hard to know from the data we're seeing here, 00:41:14.830 --> 00:41:16.570 align:middle line:90% but it looks pretty good. 00:41:16.570 --> 00:41:20.470 align:middle line:84% It actually looks like people are bidding basically 00:41:20.470 --> 00:41:21.193 align:middle line:90% their valuations. 00:41:21.193 --> 00:41:22.610 align:middle line:84% This is basically a straight line. 00:41:22.610 --> 00:41:24.527 align:middle line:84% So people were actually very, very consistent. 00:41:24.527 --> 00:41:27.090 align:middle line:84% Do people want to share how they thought about this? 00:41:27.090 --> 00:41:30.550 align:middle line:90% 00:41:30.550 --> 00:41:32.150 align:middle line:84% I'm actually a bit surprised by this. 00:41:32.150 --> 00:41:33.650 align:middle line:90% This is not how it usually goes. 00:41:33.650 --> 00:41:34.630 align:middle line:90% Yeah, in the back? 00:41:34.630 --> 00:41:36.047 align:middle line:84% STUDENT: I think in my notes there 00:41:36.047 --> 00:41:38.050 align:middle line:84% was like an analysis of second-price auctions. 00:41:38.050 --> 00:41:39.590 align:middle line:84% And it says like it's most optimal 00:41:39.590 --> 00:41:40.698 align:middle line:90% to always bid [INAUDIBLE]. 00:41:40.698 --> 00:41:41.240 align:middle line:90% IAN BALL: OK. 00:41:41.240 --> 00:41:41.980 align:middle line:90% That's a good answer. 00:41:41.980 --> 00:41:43.140 align:middle line:90% Yeah, that's exactly right. 00:41:43.140 --> 00:41:44.260 align:middle line:90% So maybe people read the notes. 00:41:44.260 --> 00:41:46.302 align:middle line:84% I'm surprised so many people have read the notes. 00:41:46.302 --> 00:41:49.100 align:middle line:84% I thought-- so I think I've done this in past years. 00:41:49.100 --> 00:41:50.680 align:middle line:84% And normally when I tell people they 00:41:50.680 --> 00:41:53.180 align:middle line:84% bid too high in the first-price auction, we then go to this, 00:41:53.180 --> 00:41:54.788 align:middle line:90% and they bid really low. 00:41:54.788 --> 00:41:56.580 align:middle line:84% But indeed, if it's a second-price auction, 00:41:56.580 --> 00:41:58.840 align:middle line:84% it's actually optimal to bid your true valuation 00:41:58.840 --> 00:42:01.063 align:middle line:84% and not to bid less than your true valuation. 00:42:01.063 --> 00:42:01.980 align:middle line:90% What's different here? 00:42:01.980 --> 00:42:03.820 align:middle line:84% I said before, if you bid your valuation, 00:42:03.820 --> 00:42:05.220 align:middle line:84% then you can never win any money. 00:42:05.220 --> 00:42:07.420 align:middle line:84% So what's different in the second-price auction? 00:42:07.420 --> 00:42:07.920 align:middle line:90% Yeah? 00:42:07.920 --> 00:42:10.040 align:middle line:84% STUDENT: [INAUDIBLE] my valuation. 00:42:10.040 --> 00:42:15.160 align:middle line:84% I think someone said earlier, [INAUDIBLE] highest. 00:42:15.160 --> 00:42:16.980 align:middle line:84% I've got a pretty high chance to win, 00:42:16.980 --> 00:42:19.000 align:middle line:84% but then I'm actually getting a price that's 00:42:19.000 --> 00:42:20.840 align:middle line:84% below that because you're [? being ?] the second highest 00:42:20.840 --> 00:42:21.340 align:middle line:90% bidder. 00:42:21.340 --> 00:42:24.020 align:middle line:84% So you know you're getting a positive value back as well. 00:42:24.020 --> 00:42:24.580 align:middle line:90% IAN BALL: Exactly right. 00:42:24.580 --> 00:42:27.000 align:middle line:84% So in the first-price auction if I bid my valuation, then 00:42:27.000 --> 00:42:28.940 align:middle line:84% if I win the auction I'm actually going to pay my bid. 00:42:28.940 --> 00:42:30.080 align:middle line:84% So I'm going to pay my valuation. 00:42:30.080 --> 00:42:31.700 align:middle line:84% And therefore, I don't come out ahead. 00:42:31.700 --> 00:42:34.060 align:middle line:84% But in the second-price auction, if I bid my valuation, 00:42:34.060 --> 00:42:35.560 align:middle line:84% well, I'm not actually going to have 00:42:35.560 --> 00:42:36.768 align:middle line:90% to pay my valuation if I win. 00:42:36.768 --> 00:42:38.268 align:middle line:84% I don't know what I'll pay, but I'll 00:42:38.268 --> 00:42:40.720 align:middle line:84% pay the second highest bid, which will always be strictly 00:42:40.720 --> 00:42:41.860 align:middle line:90% lower than my valuation. 00:42:41.860 --> 00:42:43.480 align:middle line:84% And we'll go over the argument, I 00:42:43.480 --> 00:42:46.680 align:middle line:84% think, next class about exactly why bidding your own value 00:42:46.680 --> 00:42:49.420 align:middle line:84% is exactly optimal in a second-price auction. 00:42:49.420 --> 00:42:52.520 align:middle line:84% And this is why one nice thing about second-price auctions 00:42:52.520 --> 00:42:54.440 align:middle line:84% is you don't actually have to think too much 00:42:54.440 --> 00:42:56.440 align:middle line:84% about how much other people value the good. 00:42:56.440 --> 00:42:58.040 align:middle line:84% In the first-price auction, you really 00:42:58.040 --> 00:43:00.290 align:middle line:84% have to think hard about how much the other people are 00:43:00.290 --> 00:43:02.620 align:middle line:84% going to bid, and try to respond strategically to that. 00:43:02.620 --> 00:43:04.020 align:middle line:90% So let's close this down. 00:43:04.020 --> 00:43:07.920 align:middle line:84% And let's actually analyze the simplest possible example 00:43:07.920 --> 00:43:10.172 align:middle line:84% of a first-price sealed bid auction 00:43:10.172 --> 00:43:12.380 align:middle line:84% and try to solve for the equilibrium of this auction. 00:43:12.380 --> 00:43:23.920 align:middle line:90% 00:43:23.920 --> 00:43:27.380 align:middle line:84% So solving for auction equilibrium can get a bit messy. 00:43:27.380 --> 00:43:30.040 align:middle line:84% So let's start with the simplest possible example, which 00:43:30.040 --> 00:43:40.540 align:middle line:84% is a first-price sealed bid auction with two bidders. 00:43:40.540 --> 00:43:45.160 align:middle line:90% 00:43:45.160 --> 00:43:47.000 align:middle line:90% i equals 1 and 2. 00:43:47.000 --> 00:43:50.320 align:middle line:84% And let's assume that each bidder's valuation for the good 00:43:50.320 --> 00:43:53.520 align:middle line:84% is uniformly drawn from between 0 and 1. 00:43:53.520 --> 00:43:57.180 align:middle line:84% So before in the example, in the MobLab example, 00:43:57.180 --> 00:43:58.460 align:middle line:90% I think it 0 to 100. 00:43:58.460 --> 00:44:00.040 align:middle line:84% We're just going to change units now. 00:44:00.040 --> 00:44:09.260 align:middle line:84% So bidder i's valuation, we'll call that Vi-- 00:44:09.260 --> 00:44:11.720 align:middle line:90% So bidder i is valuation Vi. 00:44:11.720 --> 00:44:18.880 align:middle line:84% This is going to be in 01, and it's uniformly distributed 00:44:18.880 --> 00:44:19.940 align:middle line:90% and independent. 00:44:19.940 --> 00:44:23.920 align:middle line:90% 00:44:23.920 --> 00:44:30.080 align:middle line:84% Uniform just means that the probability that my value is 00:44:30.080 --> 00:44:32.560 align:middle line:84% in any subinterval of 0, 1 is just 00:44:32.560 --> 00:44:34.280 align:middle line:90% the length of that interval. 00:44:34.280 --> 00:44:38.560 align:middle line:84% So the probability that my value is between 0 and 0.2 00:44:38.560 --> 00:44:39.540 align:middle line:90% is just 0.2. 00:44:39.540 --> 00:44:43.400 align:middle line:84% So it's pretty easy to calculate these probabilities 00:44:43.400 --> 00:44:44.900 align:middle line:90% with uniform distributions. 00:44:44.900 --> 00:44:46.600 align:middle line:84% Another way of saying that if you've 00:44:46.600 --> 00:44:48.920 align:middle line:84% taken some probability is the PDF, 00:44:48.920 --> 00:44:53.200 align:middle line:84% the probability density function is just 1. 00:44:53.200 --> 00:44:55.400 align:middle line:84% Independent-- what I mean by that 00:44:55.400 --> 00:44:59.400 align:middle line:84% is that bitter one's valuation is independent statistically 00:44:59.400 --> 00:45:01.160 align:middle line:90% of bidder two's valuation. 00:45:01.160 --> 00:45:06.120 align:middle line:84% So what that means is if I know my own valuation, V1, I still 00:45:06.120 --> 00:45:08.080 align:middle line:84% believe that my opponent's valuation is 00:45:08.080 --> 00:45:09.420 align:middle line:90% uniformly distributed. 00:45:09.420 --> 00:45:12.385 align:middle line:84% My valuation provides no information about the valuation 00:45:12.385 --> 00:45:13.260 align:middle line:90% of the other bidders. 00:45:13.260 --> 00:45:15.440 align:middle line:84% At the extreme, other case would be we 00:45:15.440 --> 00:45:17.100 align:middle line:84% all valued exactly the same amount. 00:45:17.100 --> 00:45:21.240 align:middle line:84% But that's not what we're studying here 00:45:21.240 --> 00:45:26.620 align:middle line:84% And each bidder can submit a bid. 00:45:26.620 --> 00:45:29.440 align:middle line:90% 00:45:29.440 --> 00:45:31.040 align:middle line:90% We'll call it bid. 00:45:31.040 --> 00:45:35.400 align:middle line:84% Let's say bi, which is just a non-negative number 00:45:35.400 --> 00:45:36.460 align:middle line:90% simultaneously. 00:45:36.460 --> 00:45:37.620 align:middle line:90% So here are the bidders. 00:45:37.620 --> 00:45:39.700 align:middle line:84% They each privately draw their valuations. 00:45:39.700 --> 00:45:41.760 align:middle line:84% Each bidder knows her own valuation, but not 00:45:41.760 --> 00:45:46.760 align:middle line:84% the other player's valuation, and then submits the bid, bi, 00:45:46.760 --> 00:45:48.040 align:middle line:90% to the auctioneer. 00:45:48.040 --> 00:45:49.560 align:middle line:90% And then what happens? 00:45:49.560 --> 00:45:56.180 align:middle line:84% Well, the higher bid, the higher bidder, 00:45:56.180 --> 00:46:03.920 align:middle line:84% that is, the bidder who bids more, wins good, wins the item, 00:46:03.920 --> 00:46:04.900 align:middle line:90% and pays bid. 00:46:04.900 --> 00:46:09.280 align:middle line:90% 00:46:09.280 --> 00:46:12.440 align:middle line:84% So I think the first step to analyzing this auction 00:46:12.440 --> 00:46:15.240 align:middle line:84% is to try to understand what each player's payoff is 00:46:15.240 --> 00:46:15.980 align:middle line:90% in the auction. 00:46:15.980 --> 00:46:19.240 align:middle line:84% What is their utility as a function of their valuation 00:46:19.240 --> 00:46:20.800 align:middle line:90% and how much they bid? 00:46:20.800 --> 00:46:21.900 align:middle line:90% So let's go over here. 00:46:21.900 --> 00:46:42.400 align:middle line:90% 00:46:42.400 --> 00:46:46.720 align:middle line:84% So formally, this action induces a Bayesian game, where 00:46:46.720 --> 00:46:49.960 align:middle line:84% each bidder is-- the action they can choose is how much to bid, 00:46:49.960 --> 00:46:52.720 align:middle line:84% and their payoffs are going to be what we write down here. 00:46:52.720 --> 00:46:54.680 align:middle line:90% So let's look at player one. 00:46:54.680 --> 00:46:56.400 align:middle line:84% We could reason about player i, but I 00:46:56.400 --> 00:46:58.440 align:middle line:84% think that just makes it a little more abstract. 00:46:58.440 --> 00:47:03.060 align:middle line:84% So let's look at U1 player one's utility as a function of well, 00:47:03.060 --> 00:47:05.560 align:middle line:84% what are the things that can affect player one's utility? 00:47:05.560 --> 00:47:08.640 align:middle line:84% How much they bid, how much the other player bids, 00:47:08.640 --> 00:47:11.080 align:middle line:84% and also how much they value the good. 00:47:11.080 --> 00:47:16.560 align:middle line:84% So we're going to write U1 of b1, b2, v1. 00:47:16.560 --> 00:47:20.960 align:middle line:84% So these are the bids of both players. 00:47:20.960 --> 00:47:24.060 align:middle line:84% Why does the other player's bid affect my payoff? 00:47:24.060 --> 00:47:24.820 align:middle line:90% So I'm player one. 00:47:24.820 --> 00:47:27.220 align:middle line:84% Why do I care about the other player's bid? 00:47:27.220 --> 00:47:27.840 align:middle line:90% Yeah? 00:47:27.840 --> 00:47:29.590 align:middle line:84% STUDENT: Because if you you've been higher 00:47:29.590 --> 00:47:31.890 align:middle line:84% than the other player, then you have 00:47:31.890 --> 00:47:34.140 align:middle line:84% to actually consider your value in finding the payoff. 00:47:34.140 --> 00:47:35.807 align:middle line:84% Otherwise, that other player bids higher 00:47:35.807 --> 00:47:37.813 align:middle line:84% than your value [INAUDIBLE], or your utilities. 00:47:37.813 --> 00:47:38.480 align:middle line:90% IAN BALL: Right. 00:47:38.480 --> 00:47:41.560 align:middle line:84% So the other bidder's bid affects me 00:47:41.560 --> 00:47:43.480 align:middle line:84% because it affects whether I win. 00:47:43.480 --> 00:47:45.880 align:middle line:84% It doesn't affect how much I pay if I win. 00:47:45.880 --> 00:47:47.440 align:middle line:90% I always pay my own bid. 00:47:47.440 --> 00:47:48.780 align:middle line:90% But it affects whether I win. 00:47:48.780 --> 00:47:50.480 align:middle line:84% So that's relevant to my utility. 00:47:50.480 --> 00:47:55.840 align:middle line:84% Also, of course, my valuation, v1, is also relevant. 00:47:55.840 --> 00:47:58.020 align:middle line:90% This is bidder one's valuation. 00:47:58.020 --> 00:48:00.720 align:middle line:90% 00:48:00.720 --> 00:48:03.040 align:middle line:84% It turns out bidder two's valuation doesn't enter here 00:48:03.040 --> 00:48:05.200 align:middle line:84% because all I care about is what we bid 00:48:05.200 --> 00:48:06.740 align:middle line:90% and how much I value the good. 00:48:06.740 --> 00:48:09.120 align:middle line:84% Of course, in the game, the other player's valuation 00:48:09.120 --> 00:48:10.800 align:middle line:84% might affect how much they choose 00:48:10.800 --> 00:48:12.980 align:middle line:84% to bid under their equilibrium strategy. 00:48:12.980 --> 00:48:14.780 align:middle line:84% But in terms of just the payoffs, 00:48:14.780 --> 00:48:18.280 align:middle line:84% we just want to write payoffs as a function of actions and types, 00:48:18.280 --> 00:48:19.160 align:middle line:90% or state. 00:48:19.160 --> 00:48:21.903 align:middle line:90% So let's try to fill this in. 00:48:21.903 --> 00:48:23.320 align:middle line:84% So if we want to be really formal, 00:48:23.320 --> 00:48:26.080 align:middle line:84% then v1 would be playing the role of your type 00:48:26.080 --> 00:48:28.320 align:middle line:84% or of the state of nature, going back 00:48:28.320 --> 00:48:30.418 align:middle line:84% to our formalism about Bayesian games. 00:48:30.418 --> 00:48:32.460 align:middle line:84% So now I want to write down this payoff function. 00:48:32.460 --> 00:48:34.580 align:middle line:84% I think it's helpful to separate into three cases. 00:48:34.580 --> 00:48:43.217 align:middle line:90% 00:48:43.217 --> 00:48:44.800 align:middle line:84% And in a sense, I should say that what 00:48:44.800 --> 00:48:48.800 align:middle line:84% we're going to fill in here is basically 00:48:48.800 --> 00:48:52.220 align:middle line:84% what defines the first-price auction. 00:48:52.220 --> 00:48:54.200 align:middle line:84% The rules of the first-price auction 00:48:54.200 --> 00:48:56.540 align:middle line:84% are going to determine what the payoffs are here. 00:48:56.540 --> 00:48:58.300 align:middle line:84% If we had a different auction format, 00:48:58.300 --> 00:48:59.760 align:middle line:84% then our payoffs, as a function of the bids, 00:48:59.760 --> 00:49:01.677 align:middle line:84% would be different because those bids would be 00:49:01.677 --> 00:49:03.480 align:middle line:90% aggregated in a different way. 00:49:03.480 --> 00:49:05.085 align:middle line:90% So let's say b1 is less than b2. 00:49:05.085 --> 00:49:06.460 align:middle line:84% I think this is the easiest case. 00:49:06.460 --> 00:49:11.280 align:middle line:84% What is player one's utility going to be in this case. 00:49:11.280 --> 00:49:11.820 align:middle line:90% Zero, right? 00:49:11.820 --> 00:49:14.640 align:middle line:90% If their bid's lower they lose. 00:49:14.640 --> 00:49:15.940 align:middle line:90% If they lose, they win nothing. 00:49:15.940 --> 00:49:16.820 align:middle line:90% They pay nothing. 00:49:16.820 --> 00:49:17.960 align:middle line:90% The utility is zero. 00:49:17.960 --> 00:49:22.040 align:middle line:90% Let's now look at this case. 00:49:22.040 --> 00:49:24.880 align:middle line:84% What's their utility going to be here? 00:49:24.880 --> 00:49:25.560 align:middle line:90% Yeah? 00:49:25.560 --> 00:49:27.400 align:middle line:90% STUDENT: b1 minus b1. 00:49:27.400 --> 00:49:30.320 align:middle line:90% IAN BALL: Exactly. 00:49:30.320 --> 00:49:31.600 align:middle line:90% So what happens here-- 00:49:31.600 --> 00:49:32.720 align:middle line:90% I'm bidder one. 00:49:32.720 --> 00:49:34.800 align:middle line:90% I bid higher than my opponent. 00:49:34.800 --> 00:49:37.160 align:middle line:84% That means I'm the highest bidder, so I win. 00:49:37.160 --> 00:49:40.200 align:middle line:84% Because I win the good, I get my valuation v1, 00:49:40.200 --> 00:49:42.120 align:middle line:90% but I have to pay my bid. 00:49:42.120 --> 00:49:43.600 align:middle line:90% And my bid was b1. 00:49:43.600 --> 00:49:45.140 align:middle line:84% So v and b sound kind of similar, 00:49:45.140 --> 00:49:49.640 align:middle line:84% but my total payoff is v1 minus b1. 00:49:49.640 --> 00:49:53.100 align:middle line:90% What about this case here? 00:49:53.100 --> 00:49:55.478 align:middle line:84% Well, we have to make a choice about how we break ties. 00:49:55.478 --> 00:49:57.020 align:middle line:84% Normally, when things are continuous, 00:49:57.020 --> 00:49:58.273 align:middle line:90% this isn't so important. 00:49:58.273 --> 00:50:00.940 align:middle line:84% Normally, what we imagine is if both people bid the same amount, 00:50:00.940 --> 00:50:03.200 align:middle line:90% we flip a coin to see who wins. 00:50:03.200 --> 00:50:05.400 align:middle line:90% So we flip a coin. 00:50:05.400 --> 00:50:07.700 align:middle line:84% If it's heads, player one wins and pays their bid. 00:50:07.700 --> 00:50:10.000 align:middle line:84% If it's tails, player two wins and pays their bid. 00:50:10.000 --> 00:50:13.380 align:middle line:84% So what is player one's expected payoff going to be in this case? 00:50:13.380 --> 00:50:15.640 align:middle line:90% STUDENT: v1 minus b1. 00:50:15.640 --> 00:50:16.640 align:middle line:90% IAN BALL: Exactly right. 00:50:16.640 --> 00:50:20.380 align:middle line:84% So there's a half chance they win the coin flip. 00:50:20.380 --> 00:50:22.440 align:middle line:84% If they lose the coin flip, they get 0. 00:50:22.440 --> 00:50:24.880 align:middle line:84% If they win the coin flip, they get this payoff. 00:50:24.880 --> 00:50:29.160 align:middle line:90% So we just get v1 minus b1. 00:50:29.160 --> 00:50:34.660 align:middle line:90% 00:50:34.660 --> 00:50:37.080 align:middle line:84% STUDENT: So like, how we model it isn't-- 00:50:37.080 --> 00:50:39.900 align:middle line:84% probably doesn't matter too much [INAUDIBLE]. 00:50:39.900 --> 00:50:40.900 align:middle line:90% IAN BALL: Exactly right. 00:50:40.900 --> 00:50:41.960 align:middle line:90% Exactly right. 00:50:41.960 --> 00:50:44.600 align:middle line:84% Let's just, as an exercise, just to make 00:50:44.600 --> 00:50:46.740 align:middle line:84% sure we understand things we're going to solve 00:50:46.740 --> 00:50:47.740 align:middle line:90% the first-price auction. 00:50:47.740 --> 00:50:50.260 align:middle line:84% But let's just imagine if it was a second-price auction, 00:50:50.260 --> 00:50:51.635 align:middle line:84% let's see how these payoffs would 00:50:51.635 --> 00:50:55.600 align:middle line:84% change, so just to make sure we're on the same page. 00:50:55.600 --> 00:50:59.160 align:middle line:84% If it were second-price, well, the number of bidders 00:50:59.160 --> 00:50:59.780 align:middle line:90% are the same. 00:50:59.780 --> 00:51:01.180 align:middle line:90% The valuations are the same. 00:51:01.180 --> 00:51:04.200 align:middle line:84% They're still bidding as before, except instead, the higher 00:51:04.200 --> 00:51:07.282 align:middle line:84% bidder wins the item and pays the second highest bid. 00:51:07.282 --> 00:51:09.240 align:middle line:84% Then we're going to have a different expression 00:51:09.240 --> 00:51:11.640 align:middle line:90% for this utility. 00:51:11.640 --> 00:51:12.760 align:middle line:90% What would it be? 00:51:12.760 --> 00:51:16.320 align:middle line:84% So let's just go through on the other side. 00:51:16.320 --> 00:51:18.020 align:middle line:84% I think this is still the easy case. 00:51:18.020 --> 00:51:20.280 align:middle line:84% What if this is second-price auction, then what's 00:51:20.280 --> 00:51:21.920 align:middle line:90% the payoff here for bidder one? 00:51:21.920 --> 00:51:24.010 align:middle line:84% STUDENT: We would have b2 instead of b1. 00:51:24.010 --> 00:51:24.760 align:middle line:90% IAN BALL: Exactly. 00:51:24.760 --> 00:51:25.260 align:middle line:90% Yeah. 00:51:25.260 --> 00:51:27.540 align:middle line:84% So otherwise, everything is exactly the same. 00:51:27.540 --> 00:51:28.040 align:middle line:90% Good. 00:51:28.040 --> 00:51:30.760 align:middle line:84% So it's still 0 here because if I bid less 00:51:30.760 --> 00:51:34.060 align:middle line:90% than the other player, I lose. 00:51:34.060 --> 00:51:35.100 align:middle line:90% Let's go down here. 00:51:35.100 --> 00:51:37.140 align:middle line:84% If I bid more than the other player, I win. 00:51:37.140 --> 00:51:39.600 align:middle line:84% But now I play the second highest bid, not 00:51:39.600 --> 00:51:40.880 align:middle line:90% the highest bid. 00:51:40.880 --> 00:51:43.520 align:middle line:84% If I'm bidder one and I bid more than the other player, then 00:51:43.520 --> 00:51:45.320 align:middle line:84% their bid is the second highest bid. 00:51:45.320 --> 00:51:49.280 align:middle line:90% So I'm going to get v1 minus b2. 00:51:49.280 --> 00:51:52.125 align:middle line:84% And indeed, in the middle we're going to flip a coin again. 00:51:52.125 --> 00:51:54.000 align:middle line:84% So there's a half chance I win, in which case 00:51:54.000 --> 00:51:57.520 align:middle line:84% I get v1 minus-- well, here it actually-- notice, 00:51:57.520 --> 00:51:59.380 align:middle line:90% in this case b1 equals b2. 00:51:59.380 --> 00:52:00.463 align:middle line:90% So I could write b1 or b2. 00:52:00.463 --> 00:52:01.880 align:middle line:84% It won't matter, but I'll write b2 00:52:01.880 --> 00:52:03.500 align:middle line:84% because I think it's more intuitive. 00:52:03.500 --> 00:52:05.560 align:middle line:90% So exactly right. 00:52:05.560 --> 00:52:09.200 align:middle line:84% So you'll see how this is going to fundamentally change 00:52:09.200 --> 00:52:11.420 align:middle line:90% the incentives of the bidder. 00:52:11.420 --> 00:52:12.920 align:middle line:84% A key difference, a special property 00:52:12.920 --> 00:52:14.640 align:middle line:84% of the second-price auction, notice, 00:52:14.640 --> 00:52:18.720 align:middle line:84% is that your bid b1 affects whether you win, but not 00:52:18.720 --> 00:52:21.360 align:middle line:90% your payoff if you do win. 00:52:21.360 --> 00:52:24.120 align:middle line:84% In a first-price auction My bid b1 00:52:24.120 --> 00:52:28.360 align:middle line:84% Affects whether I win, but also what my utility is if I do win. 00:52:28.360 --> 00:52:30.380 align:middle line:84% And these two counteracting forces. 00:52:30.380 --> 00:52:33.080 align:middle line:90% 00:52:33.080 --> 00:52:37.780 align:middle line:84% So now let's see if we can solve for an equilibrium. 00:52:37.780 --> 00:52:49.560 align:middle line:90% 00:52:49.560 --> 00:52:50.900 align:middle line:90% So let's go through our steps. 00:52:50.900 --> 00:52:53.360 align:middle line:84% The first step is to figure out what kind of objects 00:52:53.360 --> 00:52:54.173 align:middle line:90% the strategies are. 00:52:54.173 --> 00:52:56.340 align:middle line:84% We first have to figure out what the strategies are. 00:52:56.340 --> 00:53:04.200 align:middle line:84% So let's say solving for Bayes-Nash equilibrium. 00:53:04.200 --> 00:53:07.960 align:middle line:84% What is a strategy for-- let's say 00:53:07.960 --> 00:53:12.840 align:middle line:90% we have firm one and firm two. 00:53:12.840 --> 00:53:14.880 align:middle line:84% So as before, a strategy has to specify 00:53:14.880 --> 00:53:17.080 align:middle line:84% what action you're going to take as a function 00:53:17.080 --> 00:53:19.332 align:middle line:84% of your private information, as a function 00:53:19.332 --> 00:53:21.290 align:middle line:84% of what you know because you can only condition 00:53:21.290 --> 00:53:23.290 align:middle line:90% your action on things you know. 00:53:23.290 --> 00:53:28.350 align:middle line:84% So what can firm one condition their bid on? 00:53:28.350 --> 00:53:29.467 align:middle line:90% Their own valuation. 00:53:29.467 --> 00:53:31.550 align:middle line:84% And indeed, that's what a strategy is going to be. 00:53:31.550 --> 00:53:33.670 align:middle line:84% It's going to have to specify my action. 00:53:33.670 --> 00:53:37.088 align:middle line:84% That is, how much I bid as a function of my valuation. 00:53:37.088 --> 00:53:39.130 align:middle line:84% And this is where understanding function notation 00:53:39.130 --> 00:53:39.830 align:middle line:90% is really important. 00:53:39.830 --> 00:53:41.080 align:middle line:90% We have to careful about this. 00:53:41.080 --> 00:53:46.930 align:middle line:84% So it's going to be a function b1 from 0, 1 to-- 00:53:46.930 --> 00:53:48.690 align:middle line:84% maybe I'll write R plus to denote 00:53:48.690 --> 00:53:51.350 align:middle line:90% the non-negative real numbers. 00:53:51.350 --> 00:53:53.690 align:middle line:90% So what does this say? 00:53:53.690 --> 00:53:56.490 align:middle line:84% My strategy is a function, b1, that 00:53:56.490 --> 00:53:59.930 align:middle line:84% takes in any possible valuation I could have for the good. 00:53:59.930 --> 00:54:02.490 align:middle line:84% We said the valuations live in 01, 00:54:02.490 --> 00:54:06.930 align:middle line:84% and it assigns to each of those valuations an amount that I bid. 00:54:06.930 --> 00:54:11.570 align:middle line:84% So what's tricky here is that before, in our Cournot example, 00:54:11.570 --> 00:54:13.610 align:middle line:90% firm one only had two types. 00:54:13.610 --> 00:54:15.130 align:middle line:84% So we had to specify two numbers-- 00:54:15.130 --> 00:54:17.290 align:middle line:84% how much they produced in the high state, 00:54:17.290 --> 00:54:19.450 align:middle line:84% and how much they produced in the low state. 00:54:19.450 --> 00:54:20.970 align:middle line:90% Now we have a continuum. 00:54:20.970 --> 00:54:22.770 align:middle line:84% So one way of thinking about a function 00:54:22.770 --> 00:54:24.328 align:middle line:84% is it's like a continuous vector. 00:54:24.328 --> 00:54:26.370 align:middle line:84% We're having to specify infinitely many, in fact, 00:54:26.370 --> 00:54:27.735 align:middle line:90% continuously many numbers. 00:54:27.735 --> 00:54:29.610 align:middle line:84% And that's why solving for auction equilibria 00:54:29.610 --> 00:54:30.670 align:middle line:90% is going to be quite hard. 00:54:30.670 --> 00:54:32.970 align:middle line:84% Because we need to understand not just how each bit are 00:54:32.970 --> 00:54:34.730 align:middle line:84% actually bids, but how they would 00:54:34.730 --> 00:54:36.930 align:middle line:84% have bid if their valuation were different. 00:54:36.930 --> 00:54:39.390 align:middle line:84% And we're going to have the same thing for two. 00:54:39.390 --> 00:54:43.730 align:middle line:90% 00:54:43.730 --> 00:54:46.110 align:middle line:84% So we've got to solve for a lot of things here. 00:54:46.110 --> 00:54:50.290 align:middle line:90% 00:54:50.290 --> 00:54:53.330 align:middle line:84% We could write out the full equilibrium condition. 00:54:53.330 --> 00:54:56.900 align:middle line:84% But I think the trick with a lot of these auction equilibria, 00:54:56.900 --> 00:54:59.150 align:middle line:84% at least in this case, is we're going to make a guess. 00:54:59.150 --> 00:55:01.490 align:middle line:84% We're going to use guess and check. 00:55:01.490 --> 00:55:03.330 align:middle line:90% So what's our strategy? 00:55:03.330 --> 00:55:04.610 align:middle line:90% I shouldn't say strategy. 00:55:04.610 --> 00:55:06.990 align:middle line:90% That would cause confusion. 00:55:06.990 --> 00:55:08.330 align:middle line:90% We want the guess and check. 00:55:08.330 --> 00:55:09.390 align:middle line:90% What's our approach? 00:55:09.390 --> 00:55:13.090 align:middle line:90% 00:55:13.090 --> 00:55:15.770 align:middle line:84% So we're going to look for an equilibrium 00:55:15.770 --> 00:55:18.210 align:middle line:90% of a very specific form. 00:55:18.210 --> 00:55:24.470 align:middle line:84% So we're going to guess a symmetric equilibrium. 00:55:24.470 --> 00:55:36.158 align:middle line:90% 00:55:36.158 --> 00:55:38.450 align:middle line:84% I think it's pretty intuitive that the equilibrium will 00:55:38.450 --> 00:55:41.490 align:middle line:84% be symmetric here because the distributions of the valuations 00:55:41.490 --> 00:55:42.690 align:middle line:90% are symmetric. 00:55:42.690 --> 00:55:45.570 align:middle line:84% Now remember, symmetric doesn't mean the two players will always 00:55:45.570 --> 00:55:47.290 align:middle line:84% bid the same amount because the two 00:55:47.290 --> 00:55:49.430 align:middle line:84% players may have different valuations for the good. 00:55:49.430 --> 00:55:52.690 align:middle line:84% What symmetric means is the two players use the same strategy, 00:55:52.690 --> 00:55:55.990 align:middle line:84% which means if player two had a valuation of 0.7, 00:55:55.990 --> 00:55:58.530 align:middle line:84% she would bid the same amount as player one would bid 00:55:58.530 --> 00:56:00.530 align:middle line:90% if she had a valuation of 0.7. 00:56:00.530 --> 00:56:03.990 align:middle line:84% So in a symmetric equilibrium, b1 of v-- 00:56:03.990 --> 00:56:08.530 align:middle line:84% I'll just use v to denote a generic value-- 00:56:08.530 --> 00:56:15.850 align:middle line:84% equals b2 of v for all v. So this 00:56:15.850 --> 00:56:17.490 align:middle line:84% says if bidder one's valuation is 00:56:17.490 --> 00:56:20.410 align:middle line:84% v, the amount that she bids is the same that bidder two 00:56:20.410 --> 00:56:22.850 align:middle line:84% bids if her valuation is v. Maybe 00:56:22.850 --> 00:56:26.130 align:middle line:84% I'll put stars here to show that this is an equilibrium. 00:56:26.130 --> 00:56:29.330 align:middle line:84% But we want to guess a bit more about the structure 00:56:29.330 --> 00:56:31.480 align:middle line:84% of these bidding functions because now we 00:56:31.480 --> 00:56:33.230 align:middle line:84% have to solve for infinitely many numbers. 00:56:33.230 --> 00:56:34.310 align:middle line:90% That's a bit hard. 00:56:34.310 --> 00:56:36.435 align:middle line:84% So we want to guess that the bidding function takes 00:56:36.435 --> 00:56:37.690 align:middle line:90% kind of a nice form. 00:56:37.690 --> 00:56:39.770 align:middle line:84% And it turns out that the trick here 00:56:39.770 --> 00:56:43.130 align:middle line:84% is to guess what's called a linear bidding function, where 00:56:43.130 --> 00:56:46.050 align:middle line:84% everyone just bids their value times a fraction, 00:56:46.050 --> 00:56:47.690 align:middle line:90% times a constant. 00:56:47.690 --> 00:56:51.930 align:middle line:90% So let's call it alpha v. 00:56:51.930 --> 00:56:53.770 align:middle line:84% So we're guessing a symmetric equilibrium 00:56:53.770 --> 00:56:56.490 align:middle line:90% that takes this special form. 00:56:56.490 --> 00:56:59.350 align:middle line:84% And we're going to try to solve for alpha. 00:56:59.350 --> 00:57:09.370 align:middle line:90% 00:57:09.370 --> 00:57:13.610 align:middle line:84% Any guesses about the value that alpha might have? 00:57:13.610 --> 00:57:15.190 align:middle line:90% Is it going to be bigger than 1? 00:57:15.190 --> 00:57:15.690 align:middle line:90% Less than 1? 00:57:15.690 --> 00:57:17.610 align:middle line:90% Positive, negative? 00:57:17.610 --> 00:57:19.850 align:middle line:84% Any thoughts about this thing, alpha? 00:57:19.850 --> 00:57:20.350 align:middle line:90% Yeah? 00:57:20.350 --> 00:57:22.070 align:middle line:84% STUDENT: Something positive less than 1. 00:57:22.070 --> 00:57:24.028 align:middle line:84% IAN BALL: Positive less than 1 is a good guess. 00:57:24.028 --> 00:57:26.290 align:middle line:84% Because we know, and we saw from the example, 00:57:26.290 --> 00:57:28.350 align:middle line:84% that I want to what's called shade my bid. 00:57:28.350 --> 00:57:30.650 align:middle line:84% I want to bid less than my true valuation. 00:57:30.650 --> 00:57:33.570 align:middle line:84% So certainly, alpha should be less than 1. 00:57:33.570 --> 00:57:35.190 align:middle line:90% You can't bid negative amounts. 00:57:35.190 --> 00:57:36.610 align:middle line:84% And we'd like your bid to be-- we think it should 00:57:36.610 --> 00:57:37.990 align:middle line:90% be increasing in your value. 00:57:37.990 --> 00:57:39.948 align:middle line:84% So it makes sense for alpha to be non-negative, 00:57:39.948 --> 00:57:41.823 align:middle line:84% or it's really going to be strictly positive. 00:57:41.823 --> 00:57:43.290 align:middle line:84% So in general, the idea is you're 00:57:43.290 --> 00:57:45.882 align:middle line:84% going to get some value v. And instead of bidding 00:57:45.882 --> 00:57:48.590 align:middle line:84% your true value, you're going to bid some fraction of that value. 00:57:48.590 --> 00:57:52.170 align:middle line:84% And our goal is to solve for that alpha. 00:57:52.170 --> 00:57:54.170 align:middle line:90% So step one is our guess. 00:57:54.170 --> 00:57:56.650 align:middle line:84% And step two is we're going to check that this is actually 00:57:56.650 --> 00:57:57.510 align:middle line:90% an equilibrium. 00:57:57.510 --> 00:57:59.865 align:middle line:90% 00:57:59.865 --> 00:58:01.990 align:middle line:84% Now the checking is the hardest-- is a tricky part. 00:58:01.990 --> 00:58:06.970 align:middle line:84% And I want to make-- there's a really important point here. 00:58:06.970 --> 00:58:11.730 align:middle line:84% It's not enough to say suppose my opponent is using 00:58:11.730 --> 00:58:15.610 align:middle line:84% a linear bidding strategy, alpha v, then the best linear bidding 00:58:15.610 --> 00:58:17.690 align:middle line:84% strategy for me is also alpha v. That's 00:58:17.690 --> 00:58:21.050 align:middle line:84% not enough because I could deviate to a nonlinear strategy. 00:58:21.050 --> 00:58:24.130 align:middle line:84% So to understand, we're looking for an equilibrium 00:58:24.130 --> 00:58:26.550 align:middle line:84% where both firms use linear strategies. 00:58:26.550 --> 00:58:29.530 align:middle line:84% But the firms could deviate to nonlinear strategies as well. 00:58:29.530 --> 00:58:32.130 align:middle line:84% We're not restricting them to using linear strategies. 00:58:32.130 --> 00:58:36.650 align:middle line:84% So the way we check this is we want to say-- 00:58:36.650 --> 00:58:39.490 align:middle line:90% so check. 00:58:39.490 --> 00:58:46.610 align:middle line:84% if b2 of v equals alpha of v. So if bidder two is following 00:58:46.610 --> 00:58:50.370 align:middle line:84% this strategy, then the best bidding strategy for firm one, 00:58:50.370 --> 00:59:08.730 align:middle line:84% or for bidder one, is also alpha v. 00:59:08.730 --> 00:59:11.530 align:middle line:84% So we're going to suppose that my opponent is using 00:59:11.530 --> 00:59:14.410 align:middle line:84% this strategy and then compute, what's 00:59:14.410 --> 00:59:18.130 align:middle line:84% the best way for bidder one to bid and check if that happens 00:59:18.130 --> 00:59:20.570 align:middle line:90% to be the same strategy. 00:59:20.570 --> 00:59:22.132 align:middle line:84% If it doesn't work out, then that 00:59:22.132 --> 00:59:23.590 align:middle line:84% means our original guess was wrong, 00:59:23.590 --> 00:59:25.430 align:middle line:84% and there is an equilibrium of this form. 00:59:25.430 --> 00:59:29.130 align:middle line:84% If it does work out, then we'll be able to solve for alpha. 00:59:29.130 --> 00:59:32.730 align:middle line:84% So I guess that's a key step of guess and check. 00:59:32.730 --> 00:59:34.530 align:middle line:84% The output of guess and check will either 00:59:34.530 --> 00:59:36.630 align:middle line:90% be I made a good guess. 00:59:36.630 --> 00:59:37.650 align:middle line:90% Here's the equilibrium. 00:59:37.650 --> 00:59:39.270 align:middle line:84% Or it will be my guess was wrong. 00:59:39.270 --> 00:59:41.530 align:middle line:84% I guessed an equilibrium of a particular form 00:59:41.530 --> 00:59:44.610 align:middle line:84% that didn't exist, and then let me try again. 00:59:44.610 --> 00:59:46.860 align:middle line:84% Of course, in class, we're going to make good guesses. 00:59:46.860 --> 00:59:48.693 align:middle line:84% But on your homework, you might make a guess 00:59:48.693 --> 00:59:50.050 align:middle line:90% that it turns out to be wrong. 00:59:50.050 --> 00:59:52.550 align:middle line:90% OK, let's move over here. 00:59:52.550 --> 01:00:09.770 align:middle line:90% 01:00:09.770 --> 01:00:13.190 align:middle line:84% So following our guess and check strategy, 01:00:13.190 --> 01:00:21.010 align:middle line:84% we're going to suppose b2 of v equals alpha v for some alpha 01:00:21.010 --> 01:00:23.450 align:middle line:84% that we're going to try to solve for later, so 01:00:23.450 --> 01:00:26.570 align:middle line:90% for an arbitrary alpha. 01:00:26.570 --> 01:00:34.910 align:middle line:84% Let's say for some fixed alpha that's maybe between 0 and 1. 01:00:34.910 --> 01:00:38.330 align:middle line:90% 01:00:38.330 --> 01:00:47.350 align:middle line:84% And now we want to consider bidder one with valuation v1. 01:00:47.350 --> 01:00:53.113 align:middle line:90% 01:00:53.113 --> 01:00:54.530 align:middle line:84% And we want to say, put yourselves 01:00:54.530 --> 01:00:56.850 align:middle line:90% in the shoes of bidder 1. 01:00:56.850 --> 01:01:00.530 align:middle line:84% Suppose your valuation for the item is v1. 01:01:00.530 --> 01:01:04.410 align:middle line:84% Suppose you believe that the other bidder is following 01:01:04.410 --> 01:01:05.770 align:middle line:90% this bidding strategy. 01:01:05.770 --> 01:01:08.250 align:middle line:84% What is the optimal bid for you as bidder one? 01:01:08.250 --> 01:01:12.890 align:middle line:90% 01:01:12.890 --> 01:01:16.170 align:middle line:90% So what are they trying to do? 01:01:16.170 --> 01:01:25.790 align:middle line:84% Their goal is to maximize over b1 their expected payoff. 01:01:25.790 --> 01:01:28.450 align:middle line:90% 01:01:28.450 --> 01:01:31.850 align:middle line:84% Well, let's go over and look at our formula 01:01:31.850 --> 01:01:34.650 align:middle line:84% and try to compute our expected payoff. 01:01:34.650 --> 01:01:37.010 align:middle line:84% Now, the tricky thing is we can't just 01:01:37.010 --> 01:01:41.110 align:middle line:84% plug in a single number for firm two's bid, bidder two's bid. 01:01:41.110 --> 01:01:44.070 align:middle line:84% Because bidder two's bid depends on bidder two's valuation, 01:01:44.070 --> 01:01:45.730 align:middle line:90% and we don't know what that is. 01:01:45.730 --> 01:01:47.105 align:middle line:84% So what we're going to have to do 01:01:47.105 --> 01:01:53.157 align:middle line:84% is take expectations over what the value of bidder two is. 01:01:53.157 --> 01:01:54.490 align:middle line:90% So we're going to maximize what? 01:01:54.490 --> 01:02:12.302 align:middle line:84% We're going to take expectations of u1, b1, b2 of v2, v1. 01:02:12.302 --> 01:02:13.510 align:middle line:90% I think this is the key step. 01:02:13.510 --> 01:02:16.530 align:middle line:84% So let's understand what's going on here. 01:02:16.530 --> 01:02:19.810 align:middle line:84% What I'm taking expectations over is v2. 01:02:19.810 --> 01:02:23.570 align:middle line:90% This is the random thing. 01:02:23.570 --> 01:02:26.230 align:middle line:84% Because from my perspective as player one, 01:02:26.230 --> 01:02:28.810 align:middle line:84% I don't know what the valuation of bidder two is. 01:02:28.810 --> 01:02:31.370 align:middle line:84% So this is the thing I don't know about. 01:02:31.370 --> 01:02:35.190 align:middle line:84% v1 is not random. v1 is my valuation is bidder one, 01:02:35.190 --> 01:02:36.310 align:middle line:90% and I know what that is. 01:02:36.310 --> 01:02:37.270 align:middle line:90% So this is fixed. 01:02:37.270 --> 01:02:43.810 align:middle line:90% 01:02:43.810 --> 01:02:48.170 align:middle line:84% b2 of v2 is the amount that bidder two bids 01:02:48.170 --> 01:02:50.250 align:middle line:90% when their valuation is v2. 01:02:50.250 --> 01:02:52.770 align:middle line:84% And under our supposition, we know that this 01:02:52.770 --> 01:02:57.530 align:middle line:90% is going to be alpha v2. 01:02:57.530 --> 01:02:59.690 align:middle line:84% But even though we know what bidder two bids 01:02:59.690 --> 01:03:01.390 align:middle line:84% as a function of their valuation, 01:03:01.390 --> 01:03:02.930 align:middle line:84% we don't know what they actually bid because we don't 01:03:02.930 --> 01:03:04.150 align:middle line:90% know what the valuation is. 01:03:04.150 --> 01:03:07.170 align:middle line:84% So it's alpha v2, but we don't know what v2 is. 01:03:07.170 --> 01:03:10.370 align:middle line:90% And then b1 is a number. 01:03:10.370 --> 01:03:11.390 align:middle line:90% This is my bid. 01:03:11.390 --> 01:03:14.530 align:middle line:90% 01:03:14.530 --> 01:03:16.770 align:middle line:84% So just to be clear, what's inside 01:03:16.770 --> 01:03:20.450 align:middle line:84% the expectation is, what is my payoff going 01:03:20.450 --> 01:03:25.730 align:middle line:84% to be if it turns out that the valuation of bidder 2 is v2? 01:03:25.730 --> 01:03:27.030 align:middle line:90% And that's what we get in here. 01:03:27.030 --> 01:03:29.290 align:middle line:84% And then I'm taking expectations over that. 01:03:29.290 --> 01:03:34.370 align:middle line:84% So now let's try to simplify this. 01:03:34.370 --> 01:03:40.930 align:middle line:84% Well, in a lot of cases, this is just zero. 01:03:40.930 --> 01:03:43.250 align:middle line:84% This is only going to be non-zero 01:03:43.250 --> 01:03:48.850 align:middle line:84% when my bid is bigger than my opponent's bid. 01:03:48.850 --> 01:03:51.150 align:middle line:84% And when my bid is bigger than my opponent's bid, 01:03:51.150 --> 01:03:52.850 align:middle line:90% this is exactly what I get. 01:03:52.850 --> 01:04:13.487 align:middle line:84% So it turns out we can write this as this. 01:04:13.487 --> 01:04:15.070 align:middle line:84% Let's understand what's going on here. 01:04:15.070 --> 01:04:19.970 align:middle line:90% 01:04:19.970 --> 01:04:21.810 align:middle line:84% To compute my expected payoff, we just 01:04:21.810 --> 01:04:24.690 align:middle line:84% have to say, well, there's two possibilities. 01:04:24.690 --> 01:04:26.930 align:middle line:90% Either I win, or I lose. 01:04:26.930 --> 01:04:29.910 align:middle line:84% If I lose the auction, I pay nothing, and I get nothing. 01:04:29.910 --> 01:04:30.910 align:middle line:90% So my payoff is zero. 01:04:30.910 --> 01:04:32.660 align:middle line:84% So we don't even have to worry about that. 01:04:32.660 --> 01:04:35.725 align:middle line:84% You can think of it as adding plus 0 times 01:04:35.725 --> 01:04:37.850 align:middle line:84% the probability that my bid is lower than the other 01:04:37.850 --> 01:04:40.250 align:middle line:84% if you want, but that doesn't change things. 01:04:40.250 --> 01:04:42.790 align:middle line:84% v1 minus b1 is what I get if I win, 01:04:42.790 --> 01:04:46.010 align:middle line:84% but I have to multiply that by the probability that I do win. 01:04:46.010 --> 01:04:46.690 align:middle line:90% One second. 01:04:46.690 --> 01:04:48.570 align:middle line:84% What is the probability that I do win? 01:04:48.570 --> 01:04:51.570 align:middle line:84% Well, I'm going to win if my bid, b1, is 01:04:51.570 --> 01:04:54.462 align:middle line:84% higher than my opponent's bid, alpha v2. 01:04:54.462 --> 01:04:56.420 align:middle line:84% And we really have to understand crucially here 01:04:56.420 --> 01:04:59.620 align:middle line:84% in this probability, the random thing is v2. 01:04:59.620 --> 01:05:01.880 align:middle line:84% b1 is not random. b1 is a number. 01:05:01.880 --> 01:05:02.795 align:middle line:90% I know what b1 is. 01:05:02.795 --> 01:05:05.420 align:middle line:84% What's random is v2, and that's what we're taking probabilities 01:05:05.420 --> 01:05:05.840 align:middle line:90% over. 01:05:05.840 --> 01:05:06.380 align:middle line:90% Yes? 01:05:06.380 --> 01:05:07.500 align:middle line:84% STUDENT: Are we just not considering 01:05:07.500 --> 01:05:09.915 align:middle line:84% the type case because we have a continuous distribution 01:05:09.915 --> 01:05:11.040 align:middle line:90% to the probability or that? 01:05:11.040 --> 01:05:11.840 align:middle line:90% IAN BALL: Exactly right. 01:05:11.840 --> 01:05:13.420 align:middle line:84% So you could add a third-- really, 01:05:13.420 --> 01:05:15.003 align:middle line:84% we could think of it as three numbers. 01:05:15.003 --> 01:05:17.365 align:middle line:84% It's this times this probability-- 01:05:17.365 --> 01:05:19.240 align:middle line:84% maybe we'll write it out to make it clearer-- 01:05:19.240 --> 01:05:23.900 align:middle line:84% plus 0 times the probability that b1 is less than alpha v2, 01:05:23.900 --> 01:05:32.900 align:middle line:84% plus v1 minus b1 plus 1/2 the probability that b1 equals alpha 01:05:32.900 --> 01:05:33.400 align:middle line:90% v2. 01:05:33.400 --> 01:05:34.980 align:middle line:90% This is the full expression. 01:05:34.980 --> 01:05:39.380 align:middle line:84% But this disappears because we have a 0. 01:05:39.380 --> 01:05:43.060 align:middle line:84% And this disappears because the probability that b1 is exactly 01:05:43.060 --> 01:05:43.880 align:middle line:90% alpha v2 is 0. 01:05:43.880 --> 01:05:44.380 align:middle line:90% Why? 01:05:44.380 --> 01:05:45.780 align:middle line:84% It's always tricky to say the probability 01:05:45.780 --> 01:05:46.680 align:middle line:90% that b1 is something. 01:05:46.680 --> 01:05:48.000 align:middle line:90% Remember, b1 is just a number. 01:05:48.000 --> 01:05:49.667 align:middle line:84% So what I really mean is the probability 01:05:49.667 --> 01:05:52.600 align:middle line:84% that alpha v2 is equal to this fixed number b1. 01:05:52.600 --> 01:05:54.140 align:middle line:90% That's also 0. 01:05:54.140 --> 01:05:56.900 align:middle line:84% So that cancels, and I just get this expression. 01:05:56.900 --> 01:06:00.780 align:middle line:90% 01:06:00.780 --> 01:06:03.220 align:middle line:84% Now we see the basic trade-off that we always 01:06:03.220 --> 01:06:05.100 align:middle line:90% face in auction theory. 01:06:05.100 --> 01:06:09.500 align:middle line:84% If I increase b1, then when I win the item I pay more. 01:06:09.500 --> 01:06:14.340 align:middle line:84% And therefore, I win less when I win, but I win more often. 01:06:14.340 --> 01:06:18.020 align:middle line:84% If I reduce my bid, then winning is even better 01:06:18.020 --> 01:06:19.640 align:middle line:90% because I pay less when I win. 01:06:19.640 --> 01:06:22.480 align:middle line:84% But winning happens less often because I'm less likely to win. 01:06:22.480 --> 01:06:24.020 align:middle line:84% And the optimal bid is exactly going 01:06:24.020 --> 01:06:26.300 align:middle line:90% to trade off those two forces. 01:06:26.300 --> 01:06:29.620 align:middle line:84% Notice, I'm not putting weight on just whether I win or not. 01:06:29.620 --> 01:06:31.520 align:middle line:90% It's how well I do when I win. 01:06:31.520 --> 01:06:34.980 align:middle line:84% If my goal was just to win, or just to win and make 01:06:34.980 --> 01:06:37.660 align:middle line:84% positive profits, then I would just bid exactly or just 01:06:37.660 --> 01:06:38.520 align:middle line:90% below my valuation. 01:06:38.520 --> 01:06:40.600 align:middle line:84% But I care about how much I win when I do win. 01:06:40.600 --> 01:06:42.720 align:middle line:84% So let's go to a new board and finish this off. 01:06:42.720 --> 01:06:56.130 align:middle line:90% 01:06:56.130 --> 01:06:57.880 align:middle line:84% Well, let's write this in a different way. 01:06:57.880 --> 01:07:01.340 align:middle line:84% We have v1 minus b1 times the probability-- well, 01:07:01.340 --> 01:07:02.680 align:middle line:90% the random thing is v2. 01:07:02.680 --> 01:07:04.100 align:middle line:90% So let's focus on v2. 01:07:04.100 --> 01:07:09.080 align:middle line:84% This is v2 less than b1 over alpha. 01:07:09.080 --> 01:07:14.940 align:middle line:90% 01:07:14.940 --> 01:07:21.220 align:middle line:84% But what is this going to equal for a uniform distribution? 01:07:21.220 --> 01:07:21.780 align:middle line:90% Yeah? 01:07:21.780 --> 01:07:22.880 align:middle line:90% STUDENT: b1 over alpha. 01:07:22.880 --> 01:07:24.088 align:middle line:90% IAN BALL: Just b1 over alpha. 01:07:24.088 --> 01:07:28.420 align:middle line:84% We have to be a little careful that if b1 over alpha 01:07:28.420 --> 01:07:31.720 align:middle line:84% is-- if b1 is bigger than alpha, then it could be 1. 01:07:31.720 --> 01:07:34.720 align:middle line:84% But it turns out we usually aren't that careful about that. 01:07:34.720 --> 01:07:39.480 align:middle line:84% So let's just say if b1 less than or equal to alpha. 01:07:39.480 --> 01:07:43.900 align:middle line:84% It turns out you're never going to want to bid more than alpha. 01:07:43.900 --> 01:07:45.422 align:middle line:84% Because if you bid more than alpha, 01:07:45.422 --> 01:07:46.880 align:middle line:84% you're already going to win anyway, 01:07:46.880 --> 01:07:48.700 align:middle line:84% so you're just increasing the amount that you pay. 01:07:48.700 --> 01:07:49.580 align:middle line:90% Let's understand. 01:07:49.580 --> 01:07:52.260 align:middle line:84% If my opponent's strategy is alpha times v2, 01:07:52.260 --> 01:07:54.860 align:middle line:84% the highest they could possibly bid is alpha. 01:07:54.860 --> 01:07:56.943 align:middle line:84% So it would be crazy for me to bid more than alpha 01:07:56.943 --> 01:07:59.235 align:middle line:84% because I'm already guaranteed to win by bidding alpha. 01:07:59.235 --> 01:08:01.520 align:middle line:84% So it's a safe assumption that b1 is less than alpha. 01:08:01.520 --> 01:08:05.980 align:middle line:84% And in that case, then we indeed get b1 over alpha. 01:08:05.980 --> 01:08:15.220 align:middle line:84% So now if we simplify, we get v1 minus b1 times b1 over alpha. 01:08:15.220 --> 01:08:21.460 align:middle line:84% And if we maximize this, we're actually 01:08:21.460 --> 01:08:29.643 align:middle line:84% going to get b1 star equals v1 over 2. 01:08:29.643 --> 01:08:31.060 align:middle line:84% Because the alpha doesn't affect-- 01:08:31.060 --> 01:08:32.685 align:middle line:84% you could do the first-order condition. 01:08:32.685 --> 01:08:34.020 align:middle line:90% But it's our classic trick-- 01:08:34.020 --> 01:08:36.700 align:middle line:84% b1 times v1 minus b1 is maximized by b1 01:08:36.700 --> 01:08:40.140 align:middle line:90% equals v1 over 2. 01:08:40.140 --> 01:08:41.779 align:middle line:90% Wait a second. 01:08:41.779 --> 01:08:43.022 align:middle line:90% Was our guess wrong? 01:08:43.022 --> 01:08:44.939 align:middle line:84% We guessed that player two was using a bidding 01:08:44.939 --> 01:08:49.340 align:middle line:84% strategy of alpha v. And our best response for bidder two 01:08:49.340 --> 01:08:52.140 align:middle line:90% is to always bid v1 over 2. 01:08:52.140 --> 01:08:55.111 align:middle line:90% So was our guess right? 01:08:55.111 --> 01:08:57.319 align:middle line:84% How can we make sure that this agrees with our guess? 01:08:57.319 --> 01:09:00.180 align:middle line:84% How do we have a symmetric equilibrium? 01:09:00.180 --> 01:09:01.359 align:middle line:90% Alpha equals 1/2, right? 01:09:01.359 --> 01:09:04.979 align:middle line:90% 01:09:04.979 --> 01:09:07.442 align:middle line:84% So we conclude that alpha equals f. 01:09:07.442 --> 01:09:09.359 align:middle line:84% And let's just go through the reasoning right. 01:09:09.359 --> 01:09:12.460 align:middle line:84% If alpha equals 1/2, then both bidders are using the bidding 01:09:12.460 --> 01:09:15.340 align:middle line:84% strategy where they bid exactly half their value. 01:09:15.340 --> 01:09:17.960 align:middle line:84% Given that the other bidder is bidding half their value, 01:09:17.960 --> 01:09:20.460 align:middle line:84% well that's a special case of alpha times their value. 01:09:20.460 --> 01:09:22.021 align:middle line:84% So my best response is half my value. 01:09:22.021 --> 01:09:23.979 align:middle line:84% And that shows that I'm bidding optimally given 01:09:23.979 --> 01:09:25.271 align:middle line:90% the way my opponent is bidding. 01:09:25.271 --> 01:09:28.500 align:middle line:84% So the key property is that when we solve for the maximizer, 01:09:28.500 --> 01:09:30.640 align:middle line:84% we got something within the linear family. 01:09:30.640 --> 01:09:33.800 align:middle line:84% We got a new linear function with coefficient 1/2, 01:09:33.800 --> 01:09:35.779 align:middle line:84% and then we just matched our coefficients. 01:09:35.779 --> 01:09:37.620 align:middle line:90% And we said alpha equals 1/2. 01:09:37.620 --> 01:09:41.220 align:middle line:84% So we'll continue this on Thursday. 01:09:41.220 --> 01:09:42.770 align:middle line:90% Great. 01:09:42.770 --> 01:09:47.000 align:middle line:90%