WEBVTT 00:00:00.000 --> 00:00:01.943 align:middle line:90% [SQUEAKING] 00:00:01.943 --> 00:00:03.867 align:middle line:90% [RUSTLING] 00:00:03.867 --> 00:00:05.310 align:middle line:90% [CLICKING] 00:00:05.310 --> 00:00:08.915 align:middle line:90% 00:00:08.915 --> 00:00:11.040 align:middle line:84% IAN BALL: Today we're going to talk about something 00:00:11.040 --> 00:00:12.740 align:middle line:90% called revenue equivalence. 00:00:12.740 --> 00:00:25.080 align:middle line:90% 00:00:25.080 --> 00:00:27.873 align:middle line:84% So so far, we've studied a few different auction formats, 00:00:27.873 --> 00:00:30.040 align:middle line:84% and we've solved for the equilibria of these auction 00:00:30.040 --> 00:00:30.840 align:middle line:90% formats. 00:00:30.840 --> 00:00:34.480 align:middle line:84% And we found this surprising or peculiar finding 00:00:34.480 --> 00:00:38.320 align:middle line:84% that the amount of revenue that these auctions generate 00:00:38.320 --> 00:00:41.060 align:middle line:84% doesn't seem to depend on which auction format we use. 00:00:41.060 --> 00:00:43.460 align:middle line:84% So I want to first review a few of those examples. 00:00:43.460 --> 00:00:44.918 align:middle line:84% And then, today, we're going to try 00:00:44.918 --> 00:00:46.800 align:middle line:84% to understand, in general, why this happens 00:00:46.800 --> 00:00:51.120 align:middle line:84% and what are the limitations and scope of this result. 00:00:51.120 --> 00:00:55.520 align:middle line:84% So remember, we first looked at the really simple case 00:00:55.520 --> 00:00:59.880 align:middle line:84% where we had only two bidders, and each bidder's valuation 00:00:59.880 --> 00:01:03.760 align:middle line:84% was uniformly and independently drawn from the 0, 1 interval. 00:01:03.760 --> 00:01:06.080 align:middle line:84% So in the simple case, we had v1 and v2. 00:01:06.080 --> 00:01:07.900 align:middle line:90% This is bidder 1's valuation. 00:01:07.900 --> 00:01:09.680 align:middle line:90% This is bidder 2's valuation. 00:01:09.680 --> 00:01:14.060 align:middle line:84% They were in 0, 1, and they were uniform and independent. 00:01:14.060 --> 00:01:19.600 align:middle line:90% 00:01:19.600 --> 00:01:22.240 align:middle line:84% And in this really simple case, we 00:01:22.240 --> 00:01:24.005 align:middle line:84% looked at the first-price auction, 00:01:24.005 --> 00:01:25.880 align:middle line:84% and we said that, in the first-price auction, 00:01:25.880 --> 00:01:26.940 align:middle line:90% there was an equilibrium. 00:01:26.940 --> 00:01:30.680 align:middle line:84% Maybe I'll write b first-price auction 00:01:30.680 --> 00:01:35.560 align:middle line:84% where bidder i's bid, if their valuation was vi, 00:01:35.560 --> 00:01:37.580 align:middle line:84% was half of their true valuation. 00:01:37.580 --> 00:01:39.100 align:middle line:84% We know, in the first-price auction, 00:01:39.100 --> 00:01:43.380 align:middle line:84% people bid in equilibrium less than their true valuation. 00:01:43.380 --> 00:01:45.700 align:middle line:84% This is sometimes called bid shading. 00:01:45.700 --> 00:01:48.200 align:middle line:84% You shade your bid below your valuation. 00:01:48.200 --> 00:01:52.240 align:middle line:84% And we saw that this was exactly vi over 2. 00:01:52.240 --> 00:01:54.880 align:middle line:84% And then we also found an equilibrium of the Second-Price 00:01:54.880 --> 00:01:57.170 align:middle line:90% Auction, so I'll write SPA. 00:01:57.170 --> 00:01:59.650 align:middle line:84% And in the second-price auction, we 00:01:59.650 --> 00:02:04.510 align:middle line:84% observed that the amount you pay for the item, if you win, 00:02:04.510 --> 00:02:05.710 align:middle line:90% doesn't depend on your bid. 00:02:05.710 --> 00:02:07.512 align:middle line:84% It only depends on other people's bids. 00:02:07.512 --> 00:02:08.970 align:middle line:84% And we gave this argument as to why 00:02:08.970 --> 00:02:11.830 align:middle line:84% you should always bid truthfully in a second-price auction. 00:02:11.830 --> 00:02:17.970 align:middle line:84% So here, we had bi of vi equals vi. 00:02:17.970 --> 00:02:21.370 align:middle line:84% But what we noticed is, well, let's try 00:02:21.370 --> 00:02:23.850 align:middle line:84% to compute in each case how much you 00:02:23.850 --> 00:02:27.490 align:middle line:84% are expected to pay in each of these auctions. 00:02:27.490 --> 00:02:31.010 align:middle line:84% Well, in a first-price auction, let's say your true valuation is 00:02:31.010 --> 00:02:35.010 align:middle line:84% vi and this is how much you bid, well, 00:02:35.010 --> 00:02:42.250 align:middle line:84% your expected payment is going to be 00:02:42.250 --> 00:02:45.610 align:middle line:84% your bid times the probability that you actually 00:02:45.610 --> 00:02:48.770 align:middle line:84% win the good because you only pay your bid when you win. 00:02:48.770 --> 00:02:51.770 align:middle line:84% So your expected payment is vi over 2. 00:02:51.770 --> 00:02:52.630 align:middle line:90% That's your bid. 00:02:52.630 --> 00:02:55.610 align:middle line:84% And in a first-price auction, you pay your bid if you win. 00:02:55.610 --> 00:02:59.750 align:middle line:84% So it's the amount you pay times the probability that you win. 00:02:59.750 --> 00:03:03.290 align:middle line:84% What is the probability that you win in this auction 00:03:03.290 --> 00:03:05.670 align:middle line:90% if your valuation is vi? 00:03:05.670 --> 00:03:08.730 align:middle line:90% 00:03:08.730 --> 00:03:12.050 align:middle line:84% So maybe I'll say this is going to be 00:03:12.050 --> 00:03:13.670 align:middle line:90% the probability of winning. 00:03:13.670 --> 00:03:17.050 align:middle line:90% 00:03:17.050 --> 00:03:20.150 align:middle line:84% Well, this was a symmetric equilibrium. 00:03:20.150 --> 00:03:22.530 align:middle line:84% So the bidder who wins is the bidder 00:03:22.530 --> 00:03:24.130 align:middle line:90% who has the highest valuation. 00:03:24.130 --> 00:03:27.590 align:middle line:90% So I'm going to win-- yeah. 00:03:27.590 --> 00:03:28.423 align:middle line:90% AUDIENCE: v1, right? 00:03:28.423 --> 00:03:30.382 align:middle line:84% IAN BALL: Exactly, and I guess here we have vi, 00:03:30.382 --> 00:03:31.890 align:middle line:84% so it's going to be vi, but exactly. 00:03:31.890 --> 00:03:33.450 align:middle line:90% Yeah, it could be 1. 00:03:33.450 --> 00:03:34.310 align:middle line:90% Exactly right. 00:03:34.310 --> 00:03:35.790 align:middle line:90% Why is it vi? 00:03:35.790 --> 00:03:38.490 align:middle line:84% Well, I'm going to win if my value is higher 00:03:38.490 --> 00:03:40.890 align:middle line:84% than everyone else's because we're all 00:03:40.890 --> 00:03:42.910 align:middle line:84% using the same increasing bidding function. 00:03:42.910 --> 00:03:45.410 align:middle line:84% What's the probability that my value is higher than everyone 00:03:45.410 --> 00:03:46.110 align:middle line:90% else's? 00:03:46.110 --> 00:03:48.330 align:middle line:84% Well, there's only two of us, so it's the probability 00:03:48.330 --> 00:03:51.410 align:middle line:84% that the other bidder has a value below mine. 00:03:51.410 --> 00:03:54.770 align:middle line:84% But for a uniform distribution, if my value is vi, 00:03:54.770 --> 00:03:57.450 align:middle line:84% the probability that the other bidder's value is below vi 00:03:57.450 --> 00:03:58.330 align:middle line:90% is just vi. 00:03:58.330 --> 00:04:03.290 align:middle line:90% This is just, for uniform, 0, 1. 00:04:03.290 --> 00:04:04.410 align:middle line:90% Here's my bid. 00:04:04.410 --> 00:04:06.210 align:middle line:90% Here's my value vi. 00:04:06.210 --> 00:04:08.850 align:middle line:84% I'm going to win if the other bidders value lies 00:04:08.850 --> 00:04:13.170 align:middle line:84% in this interval, and the length of this interval is exactly vi. 00:04:13.170 --> 00:04:15.590 align:middle line:84% So we multiply it by the probability of winning, 00:04:15.590 --> 00:04:17.190 align:middle line:90% and we get vi squared over 2. 00:04:17.190 --> 00:04:20.730 align:middle line:84% This is the amount that I expect to pay if my valuation is vi. 00:04:20.730 --> 00:04:24.810 align:middle line:84% And then we can do a similar calculation here. 00:04:24.810 --> 00:04:27.890 align:middle line:84% What is my expected payment in this case? 00:04:27.890 --> 00:04:31.850 align:middle line:84% Well, we're still going to multiply by vi, 00:04:31.850 --> 00:04:33.690 align:middle line:90% the probability of winning. 00:04:33.690 --> 00:04:36.890 align:middle line:84% But now we can't just multiply my probability of winning times 00:04:36.890 --> 00:04:39.290 align:middle line:84% the amount I pay if I win because the amount that I 00:04:39.290 --> 00:04:42.170 align:middle line:84% pay if I win depends on what other people bid. 00:04:42.170 --> 00:04:44.210 align:middle line:84% So what we have to multiply it by, 00:04:44.210 --> 00:04:52.370 align:middle line:84% is not what I pay if I win, but my expected payment 00:04:52.370 --> 00:04:53.550 align:middle line:90% conditional on winning. 00:04:53.550 --> 00:04:59.900 align:middle line:90% 00:04:59.900 --> 00:05:01.650 align:middle line:84% And this is a general way of breaking down 00:05:01.650 --> 00:05:03.070 align:middle line:90% how much I expect to pay. 00:05:03.070 --> 00:05:06.450 align:middle line:84% I only pay if I win, so the expected amount that I pay 00:05:06.450 --> 00:05:09.810 align:middle line:84% is my probability of winning times 00:05:09.810 --> 00:05:11.890 align:middle line:90% my expected payment if I win. 00:05:11.890 --> 00:05:14.570 align:middle line:84% My expected payment conditional on winning. 00:05:14.570 --> 00:05:17.210 align:middle line:84% Now, if I win, how much do I pay? 00:05:17.210 --> 00:05:21.530 align:middle line:84% Will I pay the second highest bid? 00:05:21.530 --> 00:05:25.650 align:middle line:84% If I win, the second highest bid is the bit of the other person, 00:05:25.650 --> 00:05:28.530 align:middle line:84% and I know the other person is bidding truthfully. 00:05:28.530 --> 00:05:33.850 align:middle line:84% So the amount that I'm going to pay is my opponent's valuation. 00:05:33.850 --> 00:05:38.510 align:middle line:84% And it turns out what this is going to be is, well, let's see. 00:05:38.510 --> 00:05:44.930 align:middle line:84% It's the expectation of maybe vj conditional on vj 00:05:44.930 --> 00:05:48.050 align:middle line:90% being less than or equal to vi. 00:05:48.050 --> 00:05:50.060 align:middle line:90% So I'm bidder i. 00:05:50.060 --> 00:05:51.940 align:middle line:90% I know that my valuation is vi. 00:05:51.940 --> 00:05:53.160 align:middle line:90% When do I win? 00:05:53.160 --> 00:05:57.180 align:middle line:84% I win exactly if my opponent's value vj is less than mine. 00:05:57.180 --> 00:05:59.900 align:middle line:84% And if I do win, I pay exactly their valuation 00:05:59.900 --> 00:06:02.180 align:middle line:84% because I pay their bid and their bid 00:06:02.180 --> 00:06:04.380 align:middle line:90% equals their valuation. 00:06:04.380 --> 00:06:07.780 align:middle line:84% So we're going to get this formula here. 00:06:07.780 --> 00:06:10.540 align:middle line:84% Maybe I'll erase this arrow just to give myself some more space. 00:06:10.540 --> 00:06:13.700 align:middle line:84% So this is my probability of winning. 00:06:13.700 --> 00:06:16.780 align:middle line:84% This is my expected payment conditional on winning. 00:06:16.780 --> 00:06:20.360 align:middle line:84% And can we compute this conditional expectation here? 00:06:20.360 --> 00:06:20.860 align:middle line:90% Yes. 00:06:20.860 --> 00:06:22.235 align:middle line:84% AUDIENCE: Yes, we can compute it. 00:06:22.235 --> 00:06:24.580 align:middle line:84% And actually, we can also think about how 00:06:24.580 --> 00:06:26.900 align:middle line:84% vj is uniformly distributed from 0 00:06:26.900 --> 00:06:29.580 align:middle line:84% to vi so that the expectation would just be half of vi. 00:06:29.580 --> 00:06:31.080 align:middle line:84% IAN BALL: Exactly, let's be careful. 00:06:31.080 --> 00:06:33.720 align:middle line:84% So vj is uniformly distributed from 0 to 1. 00:06:33.720 --> 00:06:36.660 align:middle line:84% I mean, you're right, but you're saying conditional on vj 00:06:36.660 --> 00:06:38.900 align:middle line:84% being less than or equal to vi, it's uniformly 00:06:38.900 --> 00:06:40.520 align:middle line:90% distributed on 0 to vi. 00:06:40.520 --> 00:06:43.463 align:middle line:84% And therefore, here's the example, 00:06:43.463 --> 00:06:45.380 align:middle line:84% if I know that the other player's valuation is 00:06:45.380 --> 00:06:48.020 align:middle line:84% below mine, well, I know it's somewhere in here. 00:06:48.020 --> 00:06:51.320 align:middle line:84% And if it's uniform, Well, it turns out, conditionally, 00:06:51.320 --> 00:06:52.653 align:middle line:90% it's still uniform over here. 00:06:52.653 --> 00:06:53.320 align:middle line:90% What's its mean? 00:06:53.320 --> 00:06:55.903 align:middle line:84% Well, it's just going to be the middle of this interval, which 00:06:55.903 --> 00:06:57.440 align:middle line:90% is exactly vi over 2. 00:06:57.440 --> 00:07:01.300 align:middle line:90% So exactly here, vi over 2. 00:07:01.300 --> 00:07:03.800 align:middle line:84% So if I compute this out, what do I get? 00:07:03.800 --> 00:07:07.160 align:middle line:84% I also get vi over 2 times vi, and what do I get? 00:07:07.160 --> 00:07:10.420 align:middle line:90% vi squared over 2. 00:07:10.420 --> 00:07:15.020 align:middle line:84% So here, this is just a recap of this finding we had before that, 00:07:15.020 --> 00:07:17.280 align:middle line:84% in both the first-price and the second-price auction, 00:07:17.280 --> 00:07:19.260 align:middle line:84% at least in the equilibrium that we looked at, 00:07:19.260 --> 00:07:21.540 align:middle line:84% the expected payment made by bidder i, 00:07:21.540 --> 00:07:25.340 align:middle line:84% when her valuation is vi, is exactly the same 00:07:25.340 --> 00:07:26.580 align:middle line:90% in the two auctions. 00:07:26.580 --> 00:07:29.500 align:middle line:84% And that seemed kind of puzzling. 00:07:29.500 --> 00:07:31.600 align:middle line:84% And you wonder if that's a coincidence. 00:07:31.600 --> 00:07:37.460 align:middle line:84% Well, more generally, let's look at the general case. 00:07:37.460 --> 00:07:45.880 align:middle line:84% So in the general case, the general case, 00:07:45.880 --> 00:07:47.903 align:middle line:84% we had n bidders instead of just 2, 00:07:47.903 --> 00:07:50.320 align:middle line:84% and we didn't assume that they were uniformly distributed. 00:07:50.320 --> 00:07:52.040 align:middle line:84% We allowed more general distribution. 00:07:52.040 --> 00:07:55.820 align:middle line:84% So here, we had the valuations were v1 up to vn. 00:07:55.820 --> 00:07:59.420 align:middle line:84% And I think what we said is they lived in some set v 00:07:59.420 --> 00:08:08.660 align:middle line:84% lower bar to v upper bar where everything is nonnegative. 00:08:08.660 --> 00:08:10.740 align:middle line:84% So everyone's value lies somewhere 00:08:10.740 --> 00:08:13.980 align:middle line:84% between this lower bound v lower bar and this upper bound v 00:08:13.980 --> 00:08:14.940 align:middle line:90% upper bar. 00:08:14.940 --> 00:08:17.240 align:middle line:84% Up here, those bounds were just 0 and 1, 00:08:17.240 --> 00:08:19.140 align:middle line:84% but here we're going to be more general. 00:08:19.140 --> 00:08:22.260 align:middle line:84% And we're going to assume that these valuations are 00:08:22.260 --> 00:08:23.200 align:middle line:90% independent. 00:08:23.200 --> 00:08:26.140 align:middle line:90% 00:08:26.140 --> 00:08:30.500 align:middle line:84% That is, the distribution of v2 or v3 00:08:30.500 --> 00:08:33.700 align:middle line:84% is statistically independent of the distribution of v1. 00:08:33.700 --> 00:08:36.580 align:middle line:84% But now they're drawn from a different distribution. 00:08:36.580 --> 00:08:38.940 align:middle line:84% Instead of the uniform, we said they 00:08:38.940 --> 00:08:46.910 align:middle line:84% have density f over this interval, the lower bar 00:08:46.910 --> 00:08:49.150 align:middle line:90% to the upper bar. 00:08:49.150 --> 00:08:51.790 align:middle line:84% And then we introduced one more piece of notation. 00:08:51.790 --> 00:08:56.110 align:middle line:84% It was nice to have notation for the distribution of the maximum 00:08:56.110 --> 00:08:58.270 align:middle line:90% of everyone else's valuation. 00:08:58.270 --> 00:09:01.910 align:middle line:84% So we looked at the max over, let's say, 00:09:01.910 --> 00:09:07.830 align:middle line:90% j not equal to i of vj. 00:09:07.830 --> 00:09:09.930 align:middle line:90% So I'm holding out one bidder. 00:09:09.930 --> 00:09:11.630 align:middle line:84% I'm taking the perspective of bidder i, 00:09:11.630 --> 00:09:14.445 align:middle line:84% and I'm thinking of all the bidders other than bidder i. 00:09:14.445 --> 00:09:16.070 align:middle line:84% I'm looking at all of their valuations, 00:09:16.070 --> 00:09:17.470 align:middle line:90% and I'm taking the maximum. 00:09:17.470 --> 00:09:19.410 align:middle line:90% This is another random variable. 00:09:19.410 --> 00:09:21.050 align:middle line:84% As bidder i, I don't know what this is. 00:09:21.050 --> 00:09:22.910 align:middle line:90% I can form beliefs about this. 00:09:22.910 --> 00:09:25.490 align:middle line:84% And what we said is we just had notation. 00:09:25.490 --> 00:09:28.590 align:middle line:84% We said this has cumulative distribution function 00:09:28.590 --> 00:09:32.725 align:middle line:90% big G and density little g. 00:09:32.725 --> 00:09:34.350 align:middle line:84% And we had formulas for what these are, 00:09:34.350 --> 00:09:36.690 align:middle line:84% but the exact formulas don't matter. 00:09:36.690 --> 00:09:39.070 align:middle line:84% It's just that we can compute the distribution of this. 00:09:39.070 --> 00:09:40.250 align:middle line:90% And we gave a name to it. 00:09:40.250 --> 00:09:42.110 align:middle line:84% The cumulative distribution is big G, 00:09:42.110 --> 00:09:45.270 align:middle line:90% and the density is little g. 00:09:45.270 --> 00:09:49.150 align:middle line:84% Notice here that everything here is symmetric. 00:09:49.150 --> 00:09:51.790 align:middle line:84% So the value of i here doesn't matter. 00:09:51.790 --> 00:09:54.150 align:middle line:84% If I'm bidder 1 and I'm thinking about, 00:09:54.150 --> 00:09:56.710 align:middle line:84% what's the highest value of everyone other than me? 00:09:56.710 --> 00:09:58.345 align:middle line:84% Or if I'm bidder 2 and I'm saying, 00:09:58.345 --> 00:10:00.470 align:middle line:84% what's the highest value of everyone other than me? 00:10:00.470 --> 00:10:03.090 align:middle line:84% That distribution is going to be the same. 00:10:03.090 --> 00:10:05.470 align:middle line:84% Maybe just make sure we're on the same page. 00:10:05.470 --> 00:10:13.190 align:middle line:84% If n equals 2, what is the density g going to be? 00:10:13.190 --> 00:10:16.530 align:middle line:84% If n equals 2, so there's two bidders. 00:10:16.530 --> 00:10:17.030 align:middle line:90% Yeah. 00:10:17.030 --> 00:10:17.550 align:middle line:90% AUDIENCE: f. 00:10:17.550 --> 00:10:19.950 align:middle line:84% IAN BALL: Just f, because if there's two bidders, 00:10:19.950 --> 00:10:23.970 align:middle line:84% well, what is the maximum of every other bidder except me? 00:10:23.970 --> 00:10:25.550 align:middle line:84% Well, it's just the other person. 00:10:25.550 --> 00:10:27.470 align:middle line:90% So this is just vj. 00:10:27.470 --> 00:10:31.230 align:middle line:84% And we already assumed that the distribution of each bidder's 00:10:31.230 --> 00:10:33.630 align:middle line:90% valuation follows density f. 00:10:33.630 --> 00:10:36.350 align:middle line:84% So in this case, g is just going to reduce to f because there's 00:10:36.350 --> 00:10:39.050 align:middle line:90% only one other bidder. 00:10:39.050 --> 00:10:40.950 align:middle line:84% But when n is bigger than 2, then this 00:10:40.950 --> 00:10:42.670 align:middle line:84% is going to be a more complicated form. 00:10:42.670 --> 00:10:44.430 align:middle line:84% But the simple thing is the highest bidder 00:10:44.430 --> 00:10:46.530 align:middle line:84% among my opponents, if I only have one opponent, 00:10:46.530 --> 00:10:49.150 align:middle line:84% is just the value of my opponent. 00:10:49.150 --> 00:10:50.810 align:middle line:90% OK, great. 00:10:50.810 --> 00:10:52.750 align:middle line:84% So here was the general case, and we also 00:10:52.750 --> 00:10:54.390 align:middle line:84% looked at the first-price auction 00:10:54.390 --> 00:10:56.330 align:middle line:84% and the second-price auction in this case. 00:10:56.330 --> 00:10:57.830 align:middle line:90% And we computed equilibria. 00:10:57.830 --> 00:11:00.150 align:middle line:84% And maybe I'll start here with a second-price auction 00:11:00.150 --> 00:11:01.310 align:middle line:90% because that was-- 00:11:01.310 --> 00:11:03.830 align:middle line:84% maybe that's a bit easier to write down. 00:11:03.830 --> 00:11:07.030 align:middle line:84% So bi vi, and the second price-auction-- people 00:11:07.030 --> 00:11:11.050 align:middle line:84% remember what our equilibrium was here? 00:11:11.050 --> 00:11:17.110 align:middle line:90% 00:11:17.110 --> 00:11:18.050 align:middle line:90% I think people know. 00:11:18.050 --> 00:11:21.230 align:middle line:90% They just don't want to say. 00:11:21.230 --> 00:11:22.850 align:middle line:84% If we have a second-price auction, 00:11:22.850 --> 00:11:27.230 align:middle line:84% how much should you bid if your valuation is vi? 00:11:27.230 --> 00:11:27.870 align:middle line:90% Yeah. 00:11:27.870 --> 00:11:28.150 align:middle line:90% AUDIENCE: Vi 00:11:28.150 --> 00:11:29.690 align:middle line:84% IAN BALL: Just Vi, you should just bid your valuation. 00:11:29.690 --> 00:11:31.470 align:middle line:84% So it turned out, we gave this argument 00:11:31.470 --> 00:11:33.262 align:middle line:84% that, in a second-price auction, you always 00:11:33.262 --> 00:11:34.290 align:middle line:90% want to bid truthfully. 00:11:34.290 --> 00:11:36.630 align:middle line:84% You always want to bid your valuation, exactly. 00:11:36.630 --> 00:11:39.270 align:middle line:84% And that didn't depend on the details 00:11:39.270 --> 00:11:41.080 align:middle line:84% of whether there's two bidders or n bidders 00:11:41.080 --> 00:11:42.330 align:middle line:90% or what the distributions are. 00:11:42.330 --> 00:11:43.730 align:middle line:90% That's a very robust result. 00:11:43.730 --> 00:11:47.070 align:middle line:90% So we've got vi. 00:11:47.070 --> 00:11:50.510 align:middle line:84% And we can, again, try to compute, 00:11:50.510 --> 00:11:54.952 align:middle line:90% what is your expected payment? 00:11:54.952 --> 00:11:56.410 align:middle line:84% OK, this is a little trickier here, 00:11:56.410 --> 00:11:57.660 align:middle line:90% but let's see if we can do it. 00:11:57.660 --> 00:12:03.110 align:middle line:90% 00:12:03.110 --> 00:12:05.750 align:middle line:84% Well, first, let's just write it abstractly. 00:12:05.750 --> 00:12:07.950 align:middle line:90% What is my expected payment? 00:12:07.950 --> 00:12:10.450 align:middle line:84% Well, we first have to say, well, 00:12:10.450 --> 00:12:12.450 align:middle line:84% there's some chance I don't pay anything at all. 00:12:12.450 --> 00:12:14.470 align:middle line:84% If I lose the auction, I pay nothing. 00:12:14.470 --> 00:12:16.230 align:middle line:84% So my expected payment, it's always 00:12:16.230 --> 00:12:18.830 align:middle line:84% helpful to break it down into the probability 00:12:18.830 --> 00:12:22.990 align:middle line:84% that I win the auction times the expected amount that I pay, 00:12:22.990 --> 00:12:24.090 align:middle line:90% given that I win. 00:12:24.090 --> 00:12:26.550 align:middle line:84% So we'll break it down just like we did before. 00:12:26.550 --> 00:12:30.150 align:middle line:84% So let me first write down the probability that I win. 00:12:30.150 --> 00:12:33.670 align:middle line:84% So this is, the probability that I win 00:12:33.670 --> 00:12:37.160 align:middle line:84% is the probability-- well, in this equilibrium, everyone's 00:12:37.160 --> 00:12:38.560 align:middle line:90% bidding their valuation. 00:12:38.560 --> 00:12:40.920 align:middle line:84% So I'm going to win if my bid is highest, 00:12:40.920 --> 00:12:43.880 align:middle line:84% which is exactly the case if my value is higher than everyone 00:12:43.880 --> 00:12:44.480 align:middle line:90% else's. 00:12:44.480 --> 00:12:48.040 align:middle line:84% So I'm going to win if the probability of the max of j 00:12:48.040 --> 00:12:52.743 align:middle line:84% not equal to i of vj is less than or equal to vi. 00:12:52.743 --> 00:12:54.660 align:middle line:84% And as usual, we don't have to be too careful. 00:12:54.660 --> 00:12:55.702 align:middle line:90% Is it less than or equal? 00:12:55.702 --> 00:12:56.782 align:middle line:90% Is it strictly less than. 00:12:56.782 --> 00:12:59.240 align:middle line:84% The probability is going to be the same, so we don't really 00:12:59.240 --> 00:13:00.120 align:middle line:90% have to-- 00:13:00.120 --> 00:13:02.740 align:middle line:84% it doesn't really matter what we write here. 00:13:02.740 --> 00:13:06.320 align:middle line:84% So this is the probability that I win. 00:13:06.320 --> 00:13:10.420 align:middle line:84% Now what is the expected payment conditional on winning? 00:13:10.420 --> 00:13:15.240 align:middle line:84% So it's going to be an analog of this formula here. 00:13:15.240 --> 00:13:17.300 align:middle line:84% The expected payment conditional on winning, 00:13:17.300 --> 00:13:21.220 align:middle line:84% well it's going to be an expectation-- 00:13:21.220 --> 00:13:24.618 align:middle line:90% 00:13:24.618 --> 00:13:25.660 align:middle line:90% a little more space here. 00:13:25.660 --> 00:13:29.320 align:middle line:90% Let's see how small I can write. 00:13:29.320 --> 00:13:31.580 align:middle line:84% Well, it's expectation conditional on me winning. 00:13:31.580 --> 00:13:37.680 align:middle line:84% So I'm exactly going to win if the max over j not equal to i vj 00:13:37.680 --> 00:13:40.058 align:middle line:90% is less than or equal to vi. 00:13:40.058 --> 00:13:41.600 align:middle line:84% This is exactly the event that I win. 00:13:41.600 --> 00:13:45.180 align:middle line:84% I'm going to win if everyone else's valuation is below mine. 00:13:45.180 --> 00:13:48.000 align:middle line:84% Another way of saying that is the maximum over everyone else's 00:13:48.000 --> 00:13:51.360 align:middle line:84% valuation is weakly smaller than mine. 00:13:51.360 --> 00:13:54.280 align:middle line:84% And when that happens, how much am 00:13:54.280 --> 00:13:57.280 align:middle line:84% I going to pay in the second-price auction? 00:13:57.280 --> 00:13:57.880 align:middle line:90% Yeah. 00:13:57.880 --> 00:13:59.680 align:middle line:90% AUDIENCE: vj. 00:13:59.680 --> 00:14:01.120 align:middle line:90% Like, the second highest. 00:14:01.120 --> 00:14:03.147 align:middle line:84% IAN BALL: Right, and we have some notation 00:14:03.147 --> 00:14:03.980 align:middle line:90% for that right here. 00:14:03.980 --> 00:14:06.140 align:middle line:84% It's exactly-- so you're exactly right. 00:14:06.140 --> 00:14:08.120 align:middle line:84% I'm going to pay the second highest valuation. 00:14:08.120 --> 00:14:09.920 align:middle line:84% But what that is is it's precisely 00:14:09.920 --> 00:14:13.100 align:middle line:84% the highest among everyone other than me because I'm the highest. 00:14:13.100 --> 00:14:14.100 align:middle line:90% So you're exactly right. 00:14:14.100 --> 00:14:19.000 align:middle line:84% It is-- let's write it as this, max vj not equal to i of vj. 00:14:19.000 --> 00:14:22.040 align:middle line:84% So it's crucial here, yes, that this is the second highest 00:14:22.040 --> 00:14:22.900 align:middle line:90% value. 00:14:22.900 --> 00:14:25.727 align:middle line:84% But in the event that I have the highest value, well, 00:14:25.727 --> 00:14:27.060 align:middle line:90% what's the second highest value? 00:14:27.060 --> 00:14:29.020 align:middle line:84% It's just the highest among everyone else. 00:14:29.020 --> 00:14:31.700 align:middle line:84% So we exactly get this formula here. 00:14:31.700 --> 00:14:36.160 align:middle line:90% 00:14:36.160 --> 00:14:42.400 align:middle line:84% And this gets maybe a little messy, 00:14:42.400 --> 00:14:46.240 align:middle line:84% but we're actually able to compute it. 00:14:46.240 --> 00:14:47.140 align:middle line:90% Let's see. 00:14:47.140 --> 00:14:54.440 align:middle line:84% So it's going to be an integral from v lower bar to vi. 00:14:54.440 --> 00:14:56.780 align:middle line:84% So technically, we get into some conditional densities. 00:14:56.780 --> 00:14:59.460 align:middle line:84% It gets a little messy, but let me just write out what we get. 00:14:59.460 --> 00:15:01.720 align:middle line:90% We're going to get t-- 00:15:01.720 --> 00:15:09.900 align:middle line:84% maybe I'll write x, g of x for g of vi dx. 00:15:09.900 --> 00:15:16.720 align:middle line:90% 00:15:16.720 --> 00:15:21.120 align:middle line:84% So I just wrote this probability as g of vi 00:15:21.120 --> 00:15:25.360 align:middle line:84% because big G is the cumulative distribution function associated 00:15:25.360 --> 00:15:26.660 align:middle line:90% with this random variable. 00:15:26.660 --> 00:15:29.320 align:middle line:84% So the probability that this random variable is below vi 00:15:29.320 --> 00:15:30.970 align:middle line:90% is, by definition, g of vi. 00:15:30.970 --> 00:15:33.410 align:middle line:84% So here, I'm just using a definition. 00:15:33.410 --> 00:15:35.490 align:middle line:84% For this part, it's a little trickier. 00:15:35.490 --> 00:15:37.750 align:middle line:84% I'm using what's called a conditional density. 00:15:37.750 --> 00:15:40.090 align:middle line:84% I'm saying, what is the conditional density 00:15:40.090 --> 00:15:42.730 align:middle line:84% of the highest value of my opponents, 00:15:42.730 --> 00:15:46.210 align:middle line:84% given that their valuation is below vi? 00:15:46.210 --> 00:15:49.170 align:middle line:84% And this here is my conditional density. 00:15:49.170 --> 00:15:52.050 align:middle line:84% I'm integrating this between v lower bar and vi 00:15:52.050 --> 00:15:53.810 align:middle line:84% because I'm focusing on the event 00:15:53.810 --> 00:15:57.530 align:middle line:84% that the highest value of my opponents is lower than mine. 00:15:57.530 --> 00:16:00.530 align:middle line:84% And maybe we don't need to get too caught up in the algebra 00:16:00.530 --> 00:16:03.970 align:middle line:84% here, but we can just see that this G of vi and this G of vi 00:16:03.970 --> 00:16:06.390 align:middle line:84% cancel, and we're going to get a really simple formula. 00:16:06.390 --> 00:16:16.930 align:middle line:90% 00:16:16.930 --> 00:16:23.930 align:middle line:84% So this is my general formula for how much 00:16:23.930 --> 00:16:26.810 align:middle line:84% I expect to pay in the second price auction 00:16:26.810 --> 00:16:29.930 align:middle line:90% if my valuation is vi. 00:16:29.930 --> 00:16:31.110 align:middle line:90% Let's do one more. 00:16:31.110 --> 00:16:32.710 align:middle line:90% Let's finish things off here. 00:16:32.710 --> 00:16:36.750 align:middle line:90% 00:16:36.750 --> 00:16:39.250 align:middle line:84% And as I said, if this conditional density is 00:16:39.250 --> 00:16:41.150 align:middle line:84% new to you, it won't come up again. 00:16:41.150 --> 00:16:44.350 align:middle line:84% So the final answer is maybe what's important here. 00:16:44.350 --> 00:16:51.490 align:middle line:84% Let's just do one more, the bi first-price auction of vi. 00:16:51.490 --> 00:16:52.630 align:middle line:90% We actually computed this. 00:16:52.630 --> 00:16:55.295 align:middle line:90% Anyone remember what this was? 00:16:55.295 --> 00:16:56.670 align:middle line:84% We computed this in class before. 00:16:56.670 --> 00:16:58.030 align:middle line:90% Maybe I'll just write it down. 00:16:58.030 --> 00:17:01.730 align:middle line:84% It turns out we had a formula that actually looked exactly 00:17:01.730 --> 00:17:02.230 align:middle line:90% like this. 00:17:02.230 --> 00:17:04.327 align:middle line:90% This was actually our formula. 00:17:04.327 --> 00:17:06.410 align:middle line:84% So it turns out we derived this in class, I think, 00:17:06.410 --> 00:17:07.089 align:middle line:90% last Thursday. 00:17:07.089 --> 00:17:09.210 align:middle line:84% We used a differential equation, and we 00:17:09.210 --> 00:17:13.150 align:middle line:90% found that it's exactly this. 00:17:13.150 --> 00:17:18.170 align:middle line:90% 00:17:18.170 --> 00:17:22.290 align:middle line:84% So this is just a formula that we had. 00:17:22.290 --> 00:17:24.890 align:middle line:84% But now if this is how much I'm bidding 00:17:24.890 --> 00:17:28.089 align:middle line:84% in a first-price auction, what is the expected payment that I'm 00:17:28.089 --> 00:17:30.170 align:middle line:90% going to make? 00:17:30.170 --> 00:17:34.410 align:middle line:84% Well, in first-price auction, what I pay is my bid? 00:17:34.410 --> 00:17:36.930 align:middle line:84% So my expected payment is just my bid, 00:17:36.930 --> 00:17:39.170 align:middle line:84% how much I pay, times the probability 00:17:39.170 --> 00:17:41.170 align:middle line:84% that I actually have to pay my bid. 00:17:41.170 --> 00:17:42.710 align:middle line:90% But when do I pay my bid? 00:17:42.710 --> 00:17:45.010 align:middle line:90% I pay my bid when I win. 00:17:45.010 --> 00:17:53.650 align:middle line:84% So my expected payment, well, it's 00:17:53.650 --> 00:18:02.050 align:middle line:84% just bi FPA of vi, my bid times the probability that I win. 00:18:02.050 --> 00:18:03.590 align:middle line:84% What is the probability that I win? 00:18:03.590 --> 00:18:06.703 align:middle line:90% 00:18:06.703 --> 00:18:08.370 align:middle line:84% Well, it's the probability that my value 00:18:08.370 --> 00:18:09.650 align:middle line:90% is higher than everyone else. 00:18:09.650 --> 00:18:12.230 align:middle line:84% It's exactly this thing we already computed down here, 00:18:12.230 --> 00:18:13.790 align:middle line:84% and we have a simple formula for it. 00:18:13.790 --> 00:18:14.830 align:middle line:90% It's just g of vi. 00:18:14.830 --> 00:18:18.890 align:middle line:90% 00:18:18.890 --> 00:18:22.550 align:middle line:84% And this is probability of winning. 00:18:22.550 --> 00:18:25.130 align:middle line:90% 00:18:25.130 --> 00:18:29.770 align:middle line:84% OK, well, we already have a formula for bi FPA of vi, 00:18:29.770 --> 00:18:32.250 align:middle line:90% so let's plug in this here. 00:18:32.250 --> 00:18:34.890 align:middle line:84% And we see that the G of vi cancels, 00:18:34.890 --> 00:18:37.350 align:middle line:90% and we get the same integral. 00:18:37.350 --> 00:18:44.770 align:middle line:90% 00:18:44.770 --> 00:18:46.050 align:middle line:90% And we get this. 00:18:46.050 --> 00:18:51.550 align:middle line:84% So this seems like it probably is not a coincidence, 00:18:51.550 --> 00:18:54.990 align:middle line:84% that we've gotten exactly the same answer in both cases. 00:18:54.990 --> 00:18:55.770 align:middle line:90% Yes. 00:18:55.770 --> 00:19:00.090 align:middle line:84% AUDIENCE: So why is bi of vi this integral when we had 00:19:00.090 --> 00:19:03.730 align:middle line:90% already said it was just vi? 00:19:03.730 --> 00:19:06.770 align:middle line:84% IAN BALL: The key is the superscript here, SPA. 00:19:06.770 --> 00:19:10.690 align:middle line:84% So in a second-price auction, the equilibrium bidding strategy 00:19:10.690 --> 00:19:13.250 align:middle line:84% is for me to bid exactly my valuation. 00:19:13.250 --> 00:19:16.470 align:middle line:84% So everything we did here was for the second-price auction. 00:19:16.470 --> 00:19:18.690 align:middle line:84% Now we're changing cases here-- and I should've 00:19:18.690 --> 00:19:19.830 align:middle line:90% been more clear about this. 00:19:19.830 --> 00:19:21.780 align:middle line:84% Now we're moving to the first-price auction. 00:19:21.780 --> 00:19:24.620 align:middle line:84% In the first-price auction, the equilibrium is different. 00:19:24.620 --> 00:19:26.220 align:middle line:84% In the first price auction, I don't 00:19:26.220 --> 00:19:27.600 align:middle line:90% want to bid my true valuation. 00:19:27.600 --> 00:19:30.120 align:middle line:84% If I bid my true valuation in a first-price auction, 00:19:30.120 --> 00:19:34.260 align:middle line:84% then I can never come out ahead because either I don't win, 00:19:34.260 --> 00:19:36.620 align:middle line:84% or I win, but then I have to pay my bid. 00:19:36.620 --> 00:19:38.520 align:middle line:84% So deriving this was a lot harder. 00:19:38.520 --> 00:19:40.860 align:middle line:84% We spent basically all of class deriving a differential 00:19:40.860 --> 00:19:41.420 align:middle line:90% equation. 00:19:41.420 --> 00:19:42.920 align:middle line:84% And in the end, we got this formula. 00:19:42.920 --> 00:19:45.597 align:middle line:84% So if you don't-- this is hard to see. 00:19:45.597 --> 00:19:48.180 align:middle line:84% This took us an hour to get to, but it should be in your notes 00:19:48.180 --> 00:19:49.620 align:middle line:90% from last class. 00:19:49.620 --> 00:19:51.700 align:middle line:84% So the answer to, why are these different? 00:19:51.700 --> 00:19:56.180 align:middle line:84% These are different equilibria of different auctions. 00:19:56.180 --> 00:20:00.260 align:middle line:84% But the surprising thing is we get 00:20:00.260 --> 00:20:05.300 align:middle line:84% exactly the same formula for the expected payment in both cases. 00:20:05.300 --> 00:20:07.100 align:middle line:84% And let's just see if that's consistent. 00:20:07.100 --> 00:20:11.420 align:middle line:84% Up here, I got vi squared over 2. 00:20:11.420 --> 00:20:13.220 align:middle line:84% Is this formula going to give me vi squared 00:20:13.220 --> 00:20:14.900 align:middle line:84% over 2 in the really simple example, 00:20:14.900 --> 00:20:17.540 align:middle line:90% in the simple uniform example? 00:20:17.540 --> 00:20:21.580 align:middle line:84% Can we see this little calculus exercise? 00:20:21.580 --> 00:20:24.260 align:middle line:84% Well, in the uniform example, what is g? 00:20:24.260 --> 00:20:27.500 align:middle line:84% In our example, where we had two bidders, and the values 00:20:27.500 --> 00:20:30.500 align:middle line:84% were uniformly distributed over 0 1? 00:20:30.500 --> 00:20:32.820 align:middle line:84% It's just 1 because the first observation 00:20:32.820 --> 00:20:36.660 align:middle line:84% that you made was that d equals f when we have two bidders. 00:20:36.660 --> 00:20:41.100 align:middle line:84% But if our valuations are uniformly distributed, 00:20:41.100 --> 00:20:42.180 align:middle line:90% what is f? 00:20:42.180 --> 00:20:43.660 align:middle line:90% It's just 1. 00:20:43.660 --> 00:20:46.900 align:middle line:84% So if we just plug in 1 here, we get the integral 00:20:46.900 --> 00:20:50.500 align:middle line:90% from v lower bar to vi. 00:20:50.500 --> 00:20:53.540 align:middle line:84% Well, so let me go through our example. 00:20:53.540 --> 00:20:59.540 align:middle line:84% So in our simple example, everything works out 00:20:59.540 --> 00:21:03.020 align:middle line:90% because g of x equals 1. 00:21:03.020 --> 00:21:06.200 align:middle line:90% And what is v lower bar? 00:21:06.200 --> 00:21:06.700 align:middle line:90% AUDIENCE: 0. 00:21:06.700 --> 00:21:07.880 align:middle line:90% IAN BALL: 0, right? 00:21:07.880 --> 00:21:10.100 align:middle line:84% And v upper bar is 1, but that doesn't show up. 00:21:10.100 --> 00:21:18.300 align:middle line:84% So now we just get the integral from 0 to vi of xdx. 00:21:18.300 --> 00:21:20.540 align:middle line:84% And this is something people should know. 00:21:20.540 --> 00:21:21.880 align:middle line:90% This is just the antiderivative. 00:21:21.880 --> 00:21:23.860 align:middle line:84% This is just x squared over 2, so we just 00:21:23.860 --> 00:21:27.660 align:middle line:90% get vi squared over 2. 00:21:27.660 --> 00:21:31.515 align:middle line:84% And this just confirms the two formulas that we got up here. 00:21:31.515 --> 00:21:32.640 align:middle line:90% So everything's consistent. 00:21:32.640 --> 00:21:33.460 align:middle line:90% That's good. 00:21:33.460 --> 00:21:37.040 align:middle line:84% And we have this kind of surprising finding. 00:21:37.040 --> 00:21:47.780 align:middle line:84% So note the equilibrium expected payment 00:21:47.780 --> 00:21:56.540 align:middle line:84% is the same in the first price auction and the second price 00:21:56.540 --> 00:21:58.870 align:middle line:90% auction. 00:21:58.870 --> 00:22:00.540 align:middle line:84% To be a little more precise, I'm talking 00:22:00.540 --> 00:22:02.560 align:middle line:84% about a particular equilibrium that we computed. 00:22:02.560 --> 00:22:04.282 align:middle line:90% But I'll be a little loose here. 00:22:04.282 --> 00:22:05.740 align:middle line:84% The equilibrium expected payment is 00:22:05.740 --> 00:22:10.020 align:middle line:84% the same in the first-price auction 00:22:10.020 --> 00:22:13.340 align:middle line:90% and the second-price auction. 00:22:13.340 --> 00:22:15.320 align:middle line:84% And notice how this happened came about, 00:22:15.320 --> 00:22:17.870 align:middle line:84% getting back to the question, the rules of these auctions 00:22:17.870 --> 00:22:19.270 align:middle line:90% are different. 00:22:19.270 --> 00:22:22.490 align:middle line:84% So if the players bid the same way in these two auctions, 00:22:22.490 --> 00:22:24.430 align:middle line:84% their payments would definitely be different. 00:22:24.430 --> 00:22:27.660 align:middle line:84% But the equilibrium bidding behavior is also different. 00:22:27.660 --> 00:22:29.410 align:middle line:84% So the rules of the auction are different, 00:22:29.410 --> 00:22:31.050 align:middle line:84% the bidding behavior is different, 00:22:31.050 --> 00:22:34.150 align:middle line:84% and somehow, these things offset in exactly the right way 00:22:34.150 --> 00:22:37.830 align:middle line:84% that we get exactly the same formula in both cases. 00:22:37.830 --> 00:22:39.395 align:middle line:84% And I think it feels, to me at least, 00:22:39.395 --> 00:22:40.770 align:middle line:84% that this can't be a coincidence. 00:22:40.770 --> 00:22:42.910 align:middle line:84% It seems like there's something else going on here. 00:22:42.910 --> 00:22:44.750 align:middle line:84% And what's going on is exactly what's 00:22:44.750 --> 00:22:45.958 align:middle line:90% called revenue equivalence. 00:22:45.958 --> 00:22:47.750 align:middle line:84% And today, we're going to try to understand 00:22:47.750 --> 00:22:51.410 align:middle line:84% more generally why and under what conditions this is true. 00:22:51.410 --> 00:22:54.290 align:middle line:84% So I think this should be kind of a puzzle. 00:22:54.290 --> 00:22:56.610 align:middle line:84% It's not at all clear why we get the same answer here. 00:22:56.610 --> 00:22:58.235 align:middle line:84% And today, we want to get to the bottom 00:22:58.235 --> 00:22:59.790 align:middle line:90% and understand this puzzle. 00:22:59.790 --> 00:23:02.490 align:middle line:84% But this was to be motivation for what we do. 00:23:02.490 --> 00:23:04.850 align:middle line:84% So if this is unclear, what we do won't be motivated. 00:23:04.850 --> 00:23:07.190 align:middle line:84% So let me pause for some questions here. 00:23:07.190 --> 00:23:08.490 align:middle line:90% Everything clear so far? 00:23:08.490 --> 00:23:12.030 align:middle line:90% 00:23:12.030 --> 00:23:13.150 align:middle line:90% OK. 00:23:13.150 --> 00:23:16.110 align:middle line:84% Well, so far, we said that the equilibrium 00:23:16.110 --> 00:23:20.258 align:middle line:84% expected payment was the same in these two auctions. 00:23:20.258 --> 00:23:22.050 align:middle line:84% What if we used a different auction format? 00:23:22.050 --> 00:23:23.350 align:middle line:84% Maybe we could use a third-price auction 00:23:23.350 --> 00:23:25.090 align:middle line:84% or a fourth-price auction, all-pay auction. 00:23:25.090 --> 00:23:27.270 align:middle line:84% There's many different auction formats we could use. 00:23:27.270 --> 00:23:29.550 align:middle line:84% And we'd like our theorem to speak to any auction 00:23:29.550 --> 00:23:30.953 align:middle line:90% format you could come up with. 00:23:30.953 --> 00:23:33.370 align:middle line:84% So to do that, we have to think, well, what is an auction? 00:23:33.370 --> 00:23:36.670 align:middle line:84% How can I define a general abstract auction? 00:23:36.670 --> 00:23:38.050 align:middle line:90% So let's be really general. 00:23:38.050 --> 00:23:40.290 align:middle line:84% What is-- maybe I'll call this a generalized auction. 00:23:40.290 --> 00:23:44.590 align:middle line:90% 00:23:44.590 --> 00:23:47.690 align:middle line:84% And I'm still going to be in this setting over here. 00:23:47.690 --> 00:23:51.418 align:middle line:84% So I'm still assuming that these assumptions 00:23:51.418 --> 00:23:53.710 align:middle line:84% about the environment, these assumptions about how much 00:23:53.710 --> 00:23:56.670 align:middle line:84% each person values the item, but what's going to be general 00:23:56.670 --> 00:23:58.890 align:middle line:84% is the auction format that we use. 00:23:58.890 --> 00:24:01.590 align:middle line:90% 00:24:01.590 --> 00:24:03.470 align:middle line:84% Well, at bottom, what is an auction? 00:24:03.470 --> 00:24:06.410 align:middle line:84% Well, first, I have to specify what bids are allowed. 00:24:06.410 --> 00:24:08.890 align:middle line:84% So far, we've allowed bids to be real numbers. 00:24:08.890 --> 00:24:10.910 align:middle line:84% But you can have really weird auctions 00:24:10.910 --> 00:24:13.770 align:middle line:84% where bids can only be multiples of 5. 00:24:13.770 --> 00:24:16.270 align:middle line:84% Or maybe you can only say yes or no. 00:24:16.270 --> 00:24:19.332 align:middle line:84% Or maybe you say I really like it, or I don't really like it. 00:24:19.332 --> 00:24:21.290 align:middle line:84% There's a lot of different ways we could do it. 00:24:21.290 --> 00:24:23.040 align:middle line:84% So a generalized auction format, first, we 00:24:23.040 --> 00:24:24.570 align:middle line:90% have to specify the bid sets. 00:24:24.570 --> 00:24:27.270 align:middle line:90% 00:24:27.270 --> 00:24:31.630 align:middle line:90% B1, B2, all the way up to Bn. 00:24:31.630 --> 00:24:35.770 align:middle line:84% So this is specifying what each bidder is allowed to bid. 00:24:35.770 --> 00:24:38.395 align:middle line:84% You could think of these as messages. 00:24:38.395 --> 00:24:40.770 align:middle line:84% Or there's a lot of different ways we can think about it, 00:24:40.770 --> 00:24:43.150 align:middle line:84% but this is just the set of choices 00:24:43.150 --> 00:24:45.110 align:middle line:84% that each bidder has in the auction. 00:24:45.110 --> 00:24:50.610 align:middle line:84% And then we need to specify an allocation rule. 00:24:50.610 --> 00:24:53.630 align:middle line:90% 00:24:53.630 --> 00:24:55.990 align:middle line:84% An allocation rule says, well, if we 00:24:55.990 --> 00:25:00.230 align:middle line:84% collect some bid by bidder 1 and some bid by bidder 2 00:25:00.230 --> 00:25:03.430 align:middle line:84% and all the way up to a bid by bidder n, what do we do? 00:25:03.430 --> 00:25:05.670 align:middle line:84% Well, the auction is to specify two things. 00:25:05.670 --> 00:25:08.310 align:middle line:84% The allocation rule is, who gets the good? 00:25:08.310 --> 00:25:11.200 align:middle line:84% And then the payment rule is, how much does everyone pay? 00:25:11.200 --> 00:25:13.520 align:middle line:84% So we'll call it the allocation rule and maybe 00:25:13.520 --> 00:25:15.180 align:middle line:90% the payment or transfer rule. 00:25:15.180 --> 00:25:23.080 align:middle line:90% 00:25:23.080 --> 00:25:24.800 align:middle line:84% Now, for instance, so far, we've only 00:25:24.800 --> 00:25:27.600 align:middle line:84% looked at auctions where, all the auctions we've looked at, 00:25:27.600 --> 00:25:30.400 align:middle line:90% you only pay if you win. 00:25:30.400 --> 00:25:32.640 align:middle line:84% But that's actually not a requirement of an auction. 00:25:32.640 --> 00:25:35.000 align:middle line:84% In fact, there's something called an all-pay auction 00:25:35.000 --> 00:25:36.440 align:middle line:84% where everyone submits their bid, 00:25:36.440 --> 00:25:38.520 align:middle line:84% and you have to pay your bid whether or not 00:25:38.520 --> 00:25:40.200 align:middle line:90% you win the item or not. 00:25:40.200 --> 00:25:42.300 align:middle line:84% That may sound like an unappealing auction, 00:25:42.300 --> 00:25:44.818 align:middle line:84% but of course, you're going to bid less in response to that. 00:25:44.818 --> 00:25:47.360 align:middle line:84% So we're actually going to allow pretty general payment rules 00:25:47.360 --> 00:25:48.400 align:middle line:90% and transfer rules. 00:25:48.400 --> 00:25:50.160 align:middle line:84% We've generally looked at allocation rules 00:25:50.160 --> 00:25:52.483 align:middle line:84% where the highest bidder gets the item. 00:25:52.483 --> 00:25:53.900 align:middle line:84% But we could be even more general. 00:25:53.900 --> 00:25:55.483 align:middle line:84% Maybe we like one of the bidders more, 00:25:55.483 --> 00:25:57.360 align:middle line:84% and we want to always give them the item. 00:25:57.360 --> 00:25:59.180 align:middle line:84% So in general, what is an allocation rule? 00:25:59.180 --> 00:26:02.160 align:middle line:84% It's going to be a function-- maybe I'll call it q-- 00:26:02.160 --> 00:26:11.000 align:middle line:84% from B1 plus Bn to, well, it's going 00:26:11.000 --> 00:26:13.735 align:middle line:84% to take in a profile of bids by all the players, 00:26:13.735 --> 00:26:16.360 align:middle line:84% and what it's going to spit out is it should spit out something 00:26:16.360 --> 00:26:18.480 align:middle line:84% like, I give the good to player 1, 00:26:18.480 --> 00:26:21.745 align:middle line:84% or I give the good to player 2, or I give the good to player n. 00:26:21.745 --> 00:26:23.120 align:middle line:84% The problem is sometimes you have 00:26:23.120 --> 00:26:25.240 align:middle line:84% to randomize because sometimes two bidders will 00:26:25.240 --> 00:26:27.040 align:middle line:84% bid the same amount, and then we might 00:26:27.040 --> 00:26:29.080 align:middle line:84% have to flip a coin to see who gets the item. 00:26:29.080 --> 00:26:31.280 align:middle line:84% So we have to allow these kinds of lotteries. 00:26:31.280 --> 00:26:34.080 align:middle line:84% So I'm going to write-- it's going to map to what I'll 00:26:34.080 --> 00:26:37.160 align:middle line:84% call delta, where delta is just going 00:26:37.160 --> 00:26:38.460 align:middle line:90% to be the set of lotteries. 00:26:38.460 --> 00:26:44.560 align:middle line:84% It's going to be the set of all vectors q1 up to qn, where 00:26:44.560 --> 00:26:47.840 align:middle line:84% the interpretation is, if these are the submitted bids, 00:26:47.840 --> 00:26:50.760 align:middle line:84% then q1 is going to tell us the probability 00:26:50.760 --> 00:26:52.960 align:middle line:90% that player 1 gets the item. 00:26:52.960 --> 00:26:55.280 align:middle line:90% So these are going to be-- 00:26:55.280 --> 00:26:59.500 align:middle line:84% so q1 to qn is greater than or equal to 0. 00:26:59.500 --> 00:27:06.560 align:middle line:90% 00:27:06.560 --> 00:27:10.280 align:middle line:84% Now, I want to emphasize this here. 00:27:10.280 --> 00:27:12.720 align:middle line:90% I don't require equality. 00:27:12.720 --> 00:27:14.180 align:middle line:84% So what's the interpretation here? 00:27:14.180 --> 00:27:16.388 align:middle line:84% What if I were to say, OK, if these are the bids that 00:27:16.388 --> 00:27:18.640 align:middle line:84% are submitted, I'm going to allocate the good 00:27:18.640 --> 00:27:22.582 align:middle line:84% with these probabilities that sum to strictly less than 1. 00:27:22.582 --> 00:27:23.540 align:middle line:90% Could this ever happen? 00:27:23.540 --> 00:27:24.740 align:middle line:90% What's the interpretation? 00:27:24.740 --> 00:27:25.240 align:middle line:90% Yeah. 00:27:25.240 --> 00:27:26.990 align:middle line:84% AUDIENCE: Well, there's also the situation 00:27:26.990 --> 00:27:28.520 align:middle line:90% in which no one gets it. 00:27:28.520 --> 00:27:31.840 align:middle line:84% IAN BALL: Right, exactly, so why do I have to allocate the good? 00:27:31.840 --> 00:27:33.980 align:middle line:84% I might say-- and we see this in auctions-- 00:27:33.980 --> 00:27:35.360 align:middle line:90% no one's bid was high enough. 00:27:35.360 --> 00:27:37.380 align:middle line:84% I'm not happy with the bids people submitted. 00:27:37.380 --> 00:27:39.320 align:middle line:84% I'm going to keep the good myself. 00:27:39.320 --> 00:27:43.340 align:middle line:84% And that would correspond to all of these being 0, actually. 00:27:43.340 --> 00:27:47.580 align:middle line:84% So we allow the auctioneer to keep the good. 00:27:47.580 --> 00:27:54.400 align:middle line:90% 00:27:54.400 --> 00:27:59.280 align:middle line:84% So more formally, 1 minus the sum of the qi's is exactly 00:27:59.280 --> 00:28:01.412 align:middle line:84% the probability that the auctioneer keeps the good. 00:28:01.412 --> 00:28:03.120 align:middle line:84% So one way of thinking about this is it's 00:28:03.120 --> 00:28:06.220 align:middle line:84% a probability distribution, not over bidders 1 through n, 00:28:06.220 --> 00:28:08.583 align:middle line:84% but over the auctioneer and bidders 1 through n. 00:28:08.583 --> 00:28:10.000 align:middle line:84% And it doesn't have to sum up to 1 00:28:10.000 --> 00:28:13.200 align:middle line:84% because the auctioneer could get the good with some probability. 00:28:13.200 --> 00:28:15.583 align:middle line:84% So this is just a definition of this set delta 00:28:15.583 --> 00:28:16.500 align:middle line:90% that I'm going to use. 00:28:16.500 --> 00:28:18.880 align:middle line:84% So delta is the set of, sometimes it's called, 00:28:18.880 --> 00:28:20.140 align:middle line:90% subprobabilities. 00:28:20.140 --> 00:28:22.500 align:middle line:84% It's like distributions over who gets the good. 00:28:22.500 --> 00:28:25.840 align:middle line:84% But they might sum to less than 1. 00:28:25.840 --> 00:28:28.680 align:middle line:84% Now what about a payment or transfer rule? 00:28:28.680 --> 00:28:41.080 align:middle line:84% That's going to be a function t to Rn where, 00:28:41.080 --> 00:28:44.600 align:middle line:84% as a function of the bids, I'm going to specify t1 00:28:44.600 --> 00:28:47.920 align:middle line:84% how much player 1 has to pay me; T2, how much player 2 has 00:28:47.920 --> 00:28:52.200 align:middle line:90% to pay me; and so on. 00:28:52.200 --> 00:28:55.920 align:middle line:84% So just to make sure we understand what's going on, 00:28:55.920 --> 00:28:58.380 align:middle line:84% let's look at a really simple example. 00:28:58.380 --> 00:29:02.730 align:middle line:90% 00:29:02.730 --> 00:29:05.530 align:middle line:84% Let's look at the first-price auction. 00:29:05.530 --> 00:29:08.490 align:middle line:84% So in the standard first-price auction, 00:29:08.490 --> 00:29:11.290 align:middle line:84% what were these bid sets, B1 through Bn? 00:29:11.290 --> 00:29:13.390 align:middle line:90% What did we allow people to bid? 00:29:13.390 --> 00:29:15.990 align:middle line:90% 00:29:15.990 --> 00:29:16.490 align:middle line:90% Yeah. 00:29:16.490 --> 00:29:18.110 align:middle line:84% AUDIENCE: Any real number that's positive. 00:29:18.110 --> 00:29:20.030 align:middle line:84% IAN BALL: Any real number that's positive or maybe nonnegative. 00:29:20.030 --> 00:29:21.890 align:middle line:84% Yeah, so I think it doesn't really-- 00:29:21.890 --> 00:29:24.390 align:middle line:84% the details don't matter too much if we allow negative bids. 00:29:24.390 --> 00:29:26.848 align:middle line:84% But you're right, so far, we haven't allowed negative bids. 00:29:26.848 --> 00:29:33.190 align:middle line:84% So in that case, we had B1 is equal to Bn is equal to, 00:29:33.190 --> 00:29:35.232 align:middle line:84% maybe I'll write, R plus, or maybe I'll 00:29:35.232 --> 00:29:36.190 align:middle line:90% be a bit more explicit. 00:29:36.190 --> 00:29:40.450 align:middle line:90% I'll write 0 to infinity. 00:29:40.450 --> 00:29:43.190 align:middle line:84% So what we said is, what is the bid set for each player? 00:29:43.190 --> 00:29:45.753 align:middle line:84% The bid set is just the set of all nonnegative real numbers. 00:29:45.753 --> 00:29:47.170 align:middle line:84% That's what we allowed you to bid. 00:29:47.170 --> 00:29:49.450 align:middle line:84% Maybe, in reality, maybe you can only 00:29:49.450 --> 00:29:51.335 align:middle line:84% bid multiples of a set or something, 00:29:51.335 --> 00:29:52.710 align:middle line:84% and we could add more complexity, 00:29:52.710 --> 00:29:55.570 align:middle line:84% but we'll just allow this for now. 00:29:55.570 --> 00:29:59.450 align:middle line:84% Now, specifying the allocation rule and transfer rule, 00:29:59.450 --> 00:30:01.210 align:middle line:84% it's just a bit messy to write it all out 00:30:01.210 --> 00:30:03.190 align:middle line:90% because their whole function. 00:30:03.190 --> 00:30:05.370 align:middle line:84% So let's just go through an example, one example. 00:30:05.370 --> 00:30:11.310 align:middle line:90% What if, say, n equals 2? 00:30:11.310 --> 00:30:13.530 align:middle line:90% So we have two bidders. 00:30:13.530 --> 00:30:20.450 align:middle line:84% Let's see if we can compute q of 3, 4 and t 00:30:20.450 --> 00:30:23.130 align:middle line:84% of 3, 4, so really just to make sure we understand 00:30:23.130 --> 00:30:24.610 align:middle line:90% the notation that's going on. 00:30:24.610 --> 00:30:31.250 align:middle line:84% So if n equals 2, the bid set for player 1 00:30:31.250 --> 00:30:33.130 align:middle line:84% is just a nonnegative real number. 00:30:33.130 --> 00:30:34.850 align:middle line:84% The bid set for player 2 is just the set 00:30:34.850 --> 00:30:36.690 align:middle line:90% of nonnegative real numbers. 00:30:36.690 --> 00:30:39.750 align:middle line:84% So the domain of the allocation rule and the transfer rule, 00:30:39.750 --> 00:30:43.290 align:middle line:84% they have to say if bidder 1 bids this number 00:30:43.290 --> 00:30:46.570 align:middle line:84% and bidder 2 bids this number, who do we allocate the good to? 00:30:46.570 --> 00:30:48.810 align:middle line:84% And how much do we make them pay? 00:30:48.810 --> 00:30:52.570 align:middle line:84% So when I write q of 3, 4, the interpretation of this 00:30:52.570 --> 00:30:56.810 align:middle line:90% is 3 is the bid by bidder 1. 00:30:56.810 --> 00:30:59.210 align:middle line:90% 4 is the bid by bidder 2. 00:30:59.210 --> 00:31:03.210 align:middle line:84% And we need to say what happens if bidder 1 bids 3 and bidder 2 00:31:03.210 --> 00:31:04.050 align:middle line:90% bids 4? 00:31:04.050 --> 00:31:06.175 align:middle line:84% And this is really just to check our notation here. 00:31:06.175 --> 00:31:08.490 align:middle line:90% So what is q of 3, 4? 00:31:08.490 --> 00:31:09.050 align:middle line:90% Yeah. 00:31:09.050 --> 00:31:10.585 align:middle line:84% AUDIENCE: Would it be a 0 comma 1. 00:31:10.585 --> 00:31:11.710 align:middle line:90% IAN BALL: 0 comma 1, right? 00:31:11.710 --> 00:31:15.530 align:middle line:84% Because here, because bidder 2's bid is higher than bidder 00:31:15.530 --> 00:31:18.770 align:middle line:84% 1's bid, we need to allocate the good to bidder 2. 00:31:18.770 --> 00:31:20.350 align:middle line:90% And we have notation for that. 00:31:20.350 --> 00:31:22.530 align:middle line:84% Our notation just tells us the probability 00:31:22.530 --> 00:31:24.070 align:middle line:90% that each bidder gets the good. 00:31:24.070 --> 00:31:25.910 align:middle line:84% Now probabilities seem a bit weird here. 00:31:25.910 --> 00:31:26.750 align:middle line:90% They're just 0, 1. 00:31:26.750 --> 00:31:28.730 align:middle line:84% We always give the good to bidder 2. 00:31:28.730 --> 00:31:30.590 align:middle line:90% If bidder 2's bid is higher. 00:31:30.590 --> 00:31:32.810 align:middle line:84% So this is just going to be the vector 0 comma 00:31:32.810 --> 00:31:37.810 align:middle line:84% 1 that says we give the good to bidder 1 with probability 0, 00:31:37.810 --> 00:31:42.950 align:middle line:84% and we give the good to bidder 2 with probability 1. 00:31:42.950 --> 00:31:48.330 align:middle line:84% What about the vector t, t of 3, 4. 00:31:48.330 --> 00:31:49.550 align:middle line:90% What is that going to be? 00:31:49.550 --> 00:31:50.050 align:middle line:90% Yeah. 00:31:50.050 --> 00:31:50.770 align:middle line:90% AUDIENCE: 0, 4. 00:31:50.770 --> 00:31:51.470 align:middle line:90% IAN BALL: 0, 4. 00:31:51.470 --> 00:31:55.235 align:middle line:90% 00:31:55.235 --> 00:31:58.380 align:middle line:84% This says, well, this first component 00:31:58.380 --> 00:32:00.140 align:middle line:90% we might call t1 of 3, 4. 00:32:00.140 --> 00:32:01.967 align:middle line:84% This is how much bidder 1 has to pay. 00:32:01.967 --> 00:32:04.300 align:middle line:84% Well, bidder 1 got outbid, they're not getting the item, 00:32:04.300 --> 00:32:05.940 align:middle line:90% and therefore they don't pay. 00:32:05.940 --> 00:32:07.500 align:middle line:90% So they pay 0. 00:32:07.500 --> 00:32:10.390 align:middle line:90% Bidder 2 was the highest bidder. 00:32:10.390 --> 00:32:12.140 align:middle line:84% They're winning the item, and in the rules 00:32:12.140 --> 00:32:15.260 align:middle line:84% of the first-price auction, they pay their bid. 00:32:15.260 --> 00:32:17.580 align:middle line:90% So we get 0, 4. 00:32:17.580 --> 00:32:21.260 align:middle line:84% Now, what if I were to change this to a second-price auction? 00:32:21.260 --> 00:32:23.300 align:middle line:84% If I change this to a second-price auction, 00:32:23.300 --> 00:32:24.520 align:middle line:90% what would change here? 00:32:24.520 --> 00:32:25.255 align:middle line:90% Would q change? 00:32:25.255 --> 00:32:25.880 align:middle line:90% Would t change? 00:32:25.880 --> 00:32:29.060 align:middle line:90% What would be different? 00:32:29.060 --> 00:32:29.700 align:middle line:90% Yeah. 00:32:29.700 --> 00:32:31.242 align:middle line:84% AUDIENCE: q would not change because, 00:32:31.242 --> 00:32:33.480 align:middle line:90% still, the person bidding 4-- 00:32:33.480 --> 00:32:34.580 align:middle line:90% well, yeah, they'll bid 4. 00:32:34.580 --> 00:32:35.580 align:middle line:90% In this case, would win. 00:32:35.580 --> 00:32:38.363 align:middle line:84% T would change, however, because they would only have to pay 3. 00:32:38.363 --> 00:32:39.780 align:middle line:84% IAN BALL: Exactly, so notice, when 00:32:39.780 --> 00:32:41.888 align:middle line:84% we compare the first- and second-price auction, 00:32:41.888 --> 00:32:43.680 align:middle line:84% they actually use the same allocation rule. 00:32:43.680 --> 00:32:45.280 align:middle line:84% And a lot of auctions use this allocation rule. 00:32:45.280 --> 00:32:47.488 align:middle line:84% The allocation rule is pretty simple-- we allocate it 00:32:47.488 --> 00:32:48.700 align:middle line:90% to the highest bidder. 00:32:48.700 --> 00:32:50.600 align:middle line:84% They both use that allocation rule. 00:32:50.600 --> 00:32:52.540 align:middle line:84% The difference between the first-price auction 00:32:52.540 --> 00:32:54.860 align:middle line:84% and the second-price auction is the payment rule, 00:32:54.860 --> 00:32:56.180 align:middle line:90% how much people have to pay. 00:32:56.180 --> 00:32:59.980 align:middle line:84% And indeed, down here, what we would get is 0, 3. 00:32:59.980 --> 00:33:01.740 align:middle line:84% If it were a second-price auction, 00:33:01.740 --> 00:33:03.640 align:middle line:90% the first bidder got outbid. 00:33:03.640 --> 00:33:06.740 align:middle line:84% They still lose the item and pay nothing. 00:33:06.740 --> 00:33:08.800 align:middle line:84% The second bidder, in a second-price auction, 00:33:08.800 --> 00:33:12.500 align:middle line:84% though, pays the-- well, bidder 2 is the highest bidder. 00:33:12.500 --> 00:33:14.160 align:middle line:90% They pay the second highest bid. 00:33:14.160 --> 00:33:16.520 align:middle line:84% Well, what's the second highest bid between 3 and 4? 00:33:16.520 --> 00:33:17.500 align:middle line:90% It would just be 3. 00:33:17.500 --> 00:33:19.380 align:middle line:84% So we'd have to change this 4 to a 3 00:33:19.380 --> 00:33:22.230 align:middle line:84% if we wanted to do the first-price auction. 00:33:22.230 --> 00:33:23.480 align:middle line:90% What about an all-pay auction? 00:33:23.480 --> 00:33:25.522 align:middle line:84% What would this be if it were an all-pay auction? 00:33:25.522 --> 00:33:28.430 align:middle line:90% 00:33:28.430 --> 00:33:31.580 align:middle line:84% Well, then it would just be 3, 4 because bidder 1 would 00:33:31.580 --> 00:33:34.420 align:middle line:84% have to pay 3 even though they didn't when the good and bidder 00:33:34.420 --> 00:33:35.400 align:middle line:90% 2 would pay 4. 00:33:35.400 --> 00:33:36.460 align:middle line:90% They'd pay their own bid. 00:33:36.460 --> 00:33:39.220 align:middle line:84% OK, so I think we understand the notation here. 00:33:39.220 --> 00:33:41.960 align:middle line:84% And now that we have this generalized auction format, 00:33:41.960 --> 00:33:46.860 align:middle line:84% we want to try to understand these expected payments. 00:33:46.860 --> 00:33:50.380 align:middle line:84% Now, to talk about how much people expect to pay, 00:33:50.380 --> 00:33:52.400 align:middle line:84% we have to first have an equilibrium in mind. 00:33:52.400 --> 00:33:56.018 align:middle line:84% This is a key conceptual point that, when I say-- 00:33:56.018 --> 00:33:57.560 align:middle line:84% notice when I said expected payments, 00:33:57.560 --> 00:33:59.260 align:middle line:84% I said equilibrium-expected payments 00:33:59.260 --> 00:34:01.482 align:middle line:84% because, just from the auction format alone, 00:34:01.482 --> 00:34:02.940 align:middle line:84% we have no idea how much people are 00:34:02.940 --> 00:34:05.860 align:middle line:84% going to pay until we specify how people actually bid. 00:34:05.860 --> 00:34:09.219 align:middle line:84% So first, we need to choose or focus 00:34:09.219 --> 00:34:11.560 align:middle line:84% our attention on an equilibrium of the auction. 00:34:11.560 --> 00:34:13.199 align:middle line:84% And then, given that equilibrium, 00:34:13.199 --> 00:34:16.139 align:middle line:84% we can compute how much people are expected to pay. 00:34:16.139 --> 00:34:22.820 align:middle line:84% So we're going to consider an equilibrium, 00:34:22.820 --> 00:34:28.340 align:middle line:84% a Bayes-Nash equilibrium of some generalized auction. 00:34:28.340 --> 00:34:31.219 align:middle line:84% And let's just make sure we understand what kind of object 00:34:31.219 --> 00:34:32.380 align:middle line:90% this is. 00:34:32.380 --> 00:34:34.580 align:middle line:84% So a Bayes-Nash equilibrium is going 00:34:34.580 --> 00:34:39.620 align:middle line:84% to specify functions-- let's see. 00:34:39.620 --> 00:34:41.100 align:middle line:84% Usually, we use star, but I think 00:34:41.100 --> 00:34:42.260 align:middle line:84% I'm actually going to use hats today 00:34:42.260 --> 00:34:43.960 align:middle line:84% because we're going to have to take some derivatives, 00:34:43.960 --> 00:34:45.739 align:middle line:84% and with the stars, it's a bit messy. 00:34:45.739 --> 00:34:48.420 align:middle line:84% So bidder 1's equilibrium strategy 00:34:48.420 --> 00:34:52.300 align:middle line:84% is going to be a mapping from v lower 00:34:52.300 --> 00:34:58.680 align:middle line:84% bar to v upper bar to B1, all the way down to Bn hat. 00:34:58.680 --> 00:35:05.740 align:middle line:90% 00:35:05.740 --> 00:35:08.260 align:middle line:84% So remember, Bayes-Nash equilibrium is a strategy 00:35:08.260 --> 00:35:10.740 align:middle line:84% profile, and a strategy profile just 00:35:10.740 --> 00:35:13.300 align:middle line:84% specifies a strategy or a bidding function for each 00:35:13.300 --> 00:35:14.700 align:middle line:90% of the players. 00:35:14.700 --> 00:35:17.140 align:middle line:84% Notice here that, in general, these bid sets 00:35:17.140 --> 00:35:20.120 align:middle line:84% could be different, and our strategies have to reflect that. 00:35:20.120 --> 00:35:22.340 align:middle line:90% So what does bidder 1 do? 00:35:22.340 --> 00:35:24.420 align:middle line:84% They know what their true value is. 00:35:24.420 --> 00:35:27.060 align:middle line:84% It's somewhere between v lower bar and v upper bar. 00:35:27.060 --> 00:35:29.180 align:middle line:84% And to each true value, their bidding function 00:35:29.180 --> 00:35:31.220 align:middle line:90% specifies how much they bid. 00:35:31.220 --> 00:35:32.420 align:middle line:90% Well, what is a bid? 00:35:32.420 --> 00:35:34.425 align:middle line:84% A bid is just something in the set B1. 00:35:34.425 --> 00:35:35.800 align:middle line:84% Remember this is really abstract. 00:35:35.800 --> 00:35:38.940 align:middle line:84% B1 could be, I like it, or I don't like it. 00:35:38.940 --> 00:35:41.460 align:middle line:84% And then my strategy would say, for which values, 00:35:41.460 --> 00:35:43.080 align:middle line:84% do I tell the auctioneer, I like it? 00:35:43.080 --> 00:35:46.100 align:middle line:84% And for which values, do I tell the auctioneer, I don't like it? 00:35:46.100 --> 00:35:49.510 align:middle line:84% And then, all the way down to bidder n, whose bidding function 00:35:49.510 --> 00:35:51.510 align:middle line:84% takes each value they could possibly have 00:35:51.510 --> 00:35:53.667 align:middle line:84% and assigns it to some bid that's feasible, 00:35:53.667 --> 00:35:55.750 align:middle line:84% some bid that's in the bid set that the auctioneer 00:35:55.750 --> 00:35:57.830 align:middle line:90% has presented. 00:35:57.830 --> 00:36:02.750 align:middle line:84% OK, so we're considering a Bayes-Nash equilibrium, 00:36:02.750 --> 00:36:05.850 align:middle line:84% and what we're interested in is the payments. 00:36:05.850 --> 00:36:09.130 align:middle line:84% We're interested in the expected payment that each bidder makes. 00:36:09.130 --> 00:36:11.630 align:middle line:84% And we might also be interested in how likely each bidder is 00:36:11.630 --> 00:36:13.030 align:middle line:90% to win the good. 00:36:13.030 --> 00:36:18.350 align:middle line:84% So given this, we want to define what 00:36:18.350 --> 00:36:26.190 align:middle line:84% we might call the equilibrium-expected allocation 00:36:26.190 --> 00:36:27.010 align:middle line:90% and payments. 00:36:27.010 --> 00:36:34.390 align:middle line:90% 00:36:34.390 --> 00:36:39.290 align:middle line:84% So let's first define the allocation. 00:36:39.290 --> 00:36:44.790 align:middle line:90% Maybe I'll call it Qi hat. 00:36:44.790 --> 00:36:45.825 align:middle line:90% Let's focus on bidder 1. 00:36:45.825 --> 00:36:47.950 align:middle line:84% I think it just makes the notation a little easier. 00:36:47.950 --> 00:36:50.690 align:middle line:90% Q1 hat of v1. 00:36:50.690 --> 00:36:54.510 align:middle line:90% 00:36:54.510 --> 00:36:56.150 align:middle line:84% So the hat reminds us that this depends 00:36:56.150 --> 00:36:58.030 align:middle line:90% on the equilibrium v1 hat. 00:36:58.030 --> 00:36:59.950 align:middle line:84% And what I want to compute here is-- 00:36:59.950 --> 00:37:03.510 align:middle line:84% let's say we're playing this generalized auction, 00:37:03.510 --> 00:37:06.870 align:middle line:84% and we're following this Bayes-Nash equilibrium. 00:37:06.870 --> 00:37:08.910 align:middle line:90% Suppose I'm bidder 1. 00:37:08.910 --> 00:37:12.030 align:middle line:84% Suppose my true valuation for the good is v1. 00:37:12.030 --> 00:37:17.550 align:middle line:84% What is my probability or my expected probability 00:37:17.550 --> 00:37:20.590 align:middle line:90% of winning the item? 00:37:20.590 --> 00:37:23.670 align:middle line:84% Well, whether I win the item depends on what 00:37:23.670 --> 00:37:25.130 align:middle line:90% other people's valuations are. 00:37:25.130 --> 00:37:27.310 align:middle line:84% So I need to take an expectation. 00:37:27.310 --> 00:37:33.150 align:middle line:84% So I'm going to take the expectation of-- 00:37:33.150 --> 00:37:36.670 align:middle line:90% 00:37:36.670 --> 00:37:38.090 align:middle line:90% well, let's write this. 00:37:38.090 --> 00:37:48.245 align:middle line:90% 00:37:48.245 --> 00:37:50.370 align:middle line:84% Let's make sure we understand what's going on here. 00:37:50.370 --> 00:37:54.110 align:middle line:90% 00:37:54.110 --> 00:38:01.350 align:middle line:84% And I'll say this expectation is over v2 to vn. 00:38:01.350 --> 00:38:06.070 align:middle line:84% Why do I not take expectations over v1? 00:38:06.070 --> 00:38:07.130 align:middle line:90% Well, I know v1. 00:38:07.130 --> 00:38:07.790 align:middle line:90% I'm bidder 1. 00:38:07.790 --> 00:38:08.730 align:middle line:90% I know what V1 is. 00:38:08.730 --> 00:38:09.590 align:middle line:84% So this is what's, I think, really 00:38:09.590 --> 00:38:10.990 align:middle line:90% tricky about this notation. 00:38:10.990 --> 00:38:13.630 align:middle line:84% When you see v1 in here, that's just a number. 00:38:13.630 --> 00:38:15.030 align:middle line:90% That's something I know. 00:38:15.030 --> 00:38:17.110 align:middle line:84% When you see vn, that's something I don't know, 00:38:17.110 --> 00:38:18.630 align:middle line:84% and that's something that's random. 00:38:18.630 --> 00:38:19.885 align:middle line:90% So think of v1. 00:38:19.885 --> 00:38:21.010 align:middle line:90% It's just something like 2. 00:38:21.010 --> 00:38:21.770 align:middle line:90% It's a number. 00:38:21.770 --> 00:38:24.030 align:middle line:84% We're not taking an expectation over it, 00:38:24.030 --> 00:38:26.970 align:middle line:84% but everyone else's bid-- everyone else's value, 00:38:26.970 --> 00:38:28.470 align:middle line:84% we are taking an expectation over it 00:38:28.470 --> 00:38:30.390 align:middle line:84% because we don't know what it is. 00:38:30.390 --> 00:38:36.870 align:middle line:84% So what this is saying is if my value is v1 as bidder 1, well, 00:38:36.870 --> 00:38:40.470 align:middle line:84% then the amount I'm going to bid in equilibrium is B1 hat of v1. 00:38:40.470 --> 00:38:42.800 align:middle line:84% That's what my equilibrium bidding strategy tells me 00:38:42.800 --> 00:38:44.440 align:middle line:90% I should bid. 00:38:44.440 --> 00:38:46.360 align:middle line:84% Now let's take the example of bidder n. 00:38:46.360 --> 00:38:49.560 align:middle line:84% Well, I don't know what bidder n's value is, 00:38:49.560 --> 00:38:52.760 align:middle line:84% but if their value is vn, then how much do they bid? 00:38:52.760 --> 00:38:55.440 align:middle line:90% They bid vn hat of vn. 00:38:55.440 --> 00:38:58.200 align:middle line:84% And we'll do the same for bidder n minus 1 and bidder n minus 3, 00:38:58.200 --> 00:39:00.000 align:middle line:90% all the way down to bidder 2. 00:39:00.000 --> 00:39:05.520 align:middle line:84% So what's in here is the profile of bids 00:39:05.520 --> 00:39:07.880 align:middle line:84% that are submitted when we're following the equilibrium 00:39:07.880 --> 00:39:08.740 align:middle line:90% strategy. 00:39:08.740 --> 00:39:10.680 align:middle line:84% And bidder 1's value is v1, bidder 00:39:10.680 --> 00:39:15.000 align:middle line:84% 2's value is V2 all the way up to bidder n's value is vn. 00:39:15.000 --> 00:39:18.840 align:middle line:84% Now, bidder 1 knows her own value, v1, 00:39:18.840 --> 00:39:21.240 align:middle line:84% but she doesn't know these other values. 00:39:21.240 --> 00:39:23.200 align:middle line:84% So that's why we take an expectation. 00:39:23.200 --> 00:39:28.120 align:middle line:84% And then notice we have to feed in this profile of bids 00:39:28.120 --> 00:39:29.798 align:middle line:90% into our allocation function. 00:39:29.798 --> 00:39:31.340 align:middle line:84% And I realized I made a mistake here. 00:39:31.340 --> 00:39:33.000 align:middle line:90% We need a 1 here. 00:39:33.000 --> 00:39:39.680 align:middle line:84% Should be q1 because this is the profile of bids. 00:39:39.680 --> 00:39:43.720 align:middle line:84% q1 of this profile is going to specify the probability 00:39:43.720 --> 00:39:47.120 align:middle line:84% that player 1-- or bidder 1 gets the item when this is 00:39:47.120 --> 00:39:50.037 align:middle line:90% the submitted profile of bids. 00:39:50.037 --> 00:39:51.620 align:middle line:84% And then we compute expectations here. 00:39:51.620 --> 00:39:53.203 align:middle line:84% But I think this is a little abstract. 00:39:53.203 --> 00:39:55.240 align:middle line:90% So clear-- yeah. 00:39:55.240 --> 00:39:59.120 align:middle line:84% AUDIENCE: So player 1 can't know her own allocation function, 00:39:59.120 --> 00:40:00.620 align:middle line:90% right? 00:40:00.620 --> 00:40:03.693 align:middle line:84% Because it depends on other people's values. 00:40:03.693 --> 00:40:04.860 align:middle line:90% IAN BALL: So let's be clear. 00:40:04.860 --> 00:40:06.200 align:middle line:84% There's a key difference here between the function 00:40:06.200 --> 00:40:07.620 align:middle line:90% and the value of the function. 00:40:07.620 --> 00:40:09.640 align:middle line:84% So she does know the allocation rule. 00:40:09.640 --> 00:40:12.280 align:middle line:84% She knows if these are the bids that are submitted, 00:40:12.280 --> 00:40:14.023 align:middle line:84% then this is the amount I'm going to get. 00:40:14.023 --> 00:40:15.440 align:middle line:84% But you're right, she doesn't know 00:40:15.440 --> 00:40:17.773 align:middle line:84% the bids that are actually submitted because she doesn't 00:40:17.773 --> 00:40:18.920 align:middle line:90% know how other people bid. 00:40:18.920 --> 00:40:21.680 align:middle line:84% So this thing in here, she doesn't 00:40:21.680 --> 00:40:23.200 align:middle line:90% know what value this takes. 00:40:23.200 --> 00:40:26.600 align:middle line:84% The value this takes is going to depend on these bids here. 00:40:26.600 --> 00:40:28.740 align:middle line:84% But she's able to form beliefs over that. 00:40:28.740 --> 00:40:31.500 align:middle line:84% She's able to say things like, well, I don't know-- 00:40:31.500 --> 00:40:32.900 align:middle line:90% let's say there's two bidders-- 00:40:32.900 --> 00:40:34.880 align:middle line:84% I don't know what bidder 2's going to bid, 00:40:34.880 --> 00:40:38.260 align:middle line:84% but if bidder 2 bids 5, this is the probability I get the good. 00:40:38.260 --> 00:40:40.980 align:middle line:84% If bidder 2 bids 7, this is the probability that I get the good. 00:40:40.980 --> 00:40:43.600 align:middle line:84% And because she has a belief about the distribution 00:40:43.600 --> 00:40:46.160 align:middle line:84% from which bidder 2's valuations are drawn 00:40:46.160 --> 00:40:49.040 align:middle line:84% and she believes that bidder 2's following bidder 2's equilibrium 00:40:49.040 --> 00:40:51.740 align:middle line:84% strategy, she's able to compute an expectation here. 00:40:51.740 --> 00:40:54.120 align:middle line:90% So she doesn't know the actual-- 00:40:54.120 --> 00:40:58.880 align:middle line:84% I guess what's also tricky here, it's weird to talk about-- 00:40:58.880 --> 00:41:00.655 align:middle line:84% this is a probability on the inside. 00:41:00.655 --> 00:41:02.780 align:middle line:84% And then we're also taking an expected probability. 00:41:02.780 --> 00:41:04.560 align:middle line:84% I know that's a little weird, but think 00:41:04.560 --> 00:41:07.420 align:middle line:84% of this number as basically always being 0, 1 on the inside. 00:41:07.420 --> 00:41:09.640 align:middle line:84% So we're basically saying, what's the probability 00:41:09.640 --> 00:41:10.800 align:middle line:90% that I get the good? 00:41:10.800 --> 00:41:12.800 align:middle line:84% So we're taking an expectation of something 00:41:12.800 --> 00:41:15.240 align:middle line:84% that's sometimes 1-- it's 1 if I win-- 00:41:15.240 --> 00:41:17.300 align:middle line:84% and sometimes 0-- it's 0 if I lose. 00:41:17.300 --> 00:41:20.747 align:middle line:84% This tie-breaking stuff is not really going to come up. 00:41:20.747 --> 00:41:21.580 align:middle line:90% Any other questions? 00:41:21.580 --> 00:41:25.040 align:middle line:90% Great, great question, OK. 00:41:25.040 --> 00:41:28.080 align:middle line:84% Great, so this is the equilibrium probability 00:41:28.080 --> 00:41:32.193 align:middle line:84% that I get the good if I'm bidder 1 and my valuation is v1. 00:41:32.193 --> 00:41:33.860 align:middle line:84% Now let's do the same with the transfer. 00:41:33.860 --> 00:41:37.360 align:middle line:84% So we wanted to find T1 hat of v1. 00:41:37.360 --> 00:41:40.060 align:middle line:84% So this is exactly the object that we are computing over here. 00:41:40.060 --> 00:41:42.240 align:middle line:84% We are saying if I'm bidder v1-- if I'm 00:41:42.240 --> 00:41:44.458 align:middle line:84% bidder 1 and my valuation is v1, what 00:41:44.458 --> 00:41:46.000 align:middle line:84% is the expected amount that I'm going 00:41:46.000 --> 00:41:48.040 align:middle line:84% to pay in this particular auction, 00:41:48.040 --> 00:41:50.320 align:middle line:84% in this particular equilibrium of the auction. 00:41:50.320 --> 00:41:53.080 align:middle line:84% So again, we're going to take expectations. 00:41:53.080 --> 00:41:56.440 align:middle line:84% Again, our expectations are going to be over v2 through vn 00:41:56.440 --> 00:42:00.560 align:middle line:84% because those are the values that bidder 1 does not know. 00:42:00.560 --> 00:42:07.320 align:middle line:84% And now we're going to write T1 of b1 hat v1 all the way up 00:42:07.320 --> 00:42:12.320 align:middle line:90% to bn hat vn here. 00:42:12.320 --> 00:42:14.600 align:middle line:84% And maybe I'll make it clear that this 00:42:14.600 --> 00:42:22.040 align:middle line:84% is known where these are uncertain from the perspective 00:42:22.040 --> 00:42:25.320 align:middle line:90% of bidder 1. 00:42:25.320 --> 00:42:30.120 align:middle line:84% So we have bidder 1's bid all the way up to bidder n's bid. 00:42:30.120 --> 00:42:32.200 align:middle line:90% That forms the profile of bids. 00:42:32.200 --> 00:42:35.070 align:middle line:84% The auctioneer says, if I get this profile of bids, 00:42:35.070 --> 00:42:37.610 align:middle line:84% this is how much I'm going to ask player 1 to pay. 00:42:37.610 --> 00:42:40.570 align:middle line:84% And then we're taking expectations. 00:42:40.570 --> 00:42:44.450 align:middle line:84% So we've now kind of formalized our question. 00:42:44.450 --> 00:42:47.290 align:middle line:84% Revenue equivalence tells us that this function, 00:42:47.290 --> 00:42:50.650 align:middle line:84% T1 hat of v1, seems to take the same form 00:42:50.650 --> 00:42:52.630 align:middle line:84% in a lot of different auction formats 00:42:52.630 --> 00:42:54.450 align:middle line:84% and a lot of different equilibria. 00:42:54.450 --> 00:42:56.770 align:middle line:84% In both the equilibrium of the first-price auction 00:42:56.770 --> 00:42:59.130 align:middle line:84% and the equilibrium of the second-price auction, 00:42:59.130 --> 00:43:00.530 align:middle line:90% this function was the same. 00:43:00.530 --> 00:43:02.810 align:middle line:84% And we'd like to understand why that is. 00:43:02.810 --> 00:43:04.930 align:middle line:84% So that's at a high level what we want to. 00:43:04.930 --> 00:43:08.050 align:middle line:90% 00:43:08.050 --> 00:43:11.290 align:middle line:84% OK, well, we're going to have to use the fact that this 00:43:11.290 --> 00:43:14.770 align:middle line:90% is a Bayes-Nash equilibrium. 00:43:14.770 --> 00:43:19.250 align:middle line:84% So the fact that this is Bayes-Nash equilibrium tells me 00:43:19.250 --> 00:43:19.790 align:middle line:90% that-- 00:43:19.790 --> 00:43:26.610 align:middle line:90% 00:43:26.610 --> 00:43:30.930 align:middle line:84% let's take the perspective of bidder 1 00:43:30.930 --> 00:43:32.390 align:middle line:90% who has a valuation of v1? 00:43:32.390 --> 00:43:38.610 align:middle line:90% 00:43:38.610 --> 00:43:51.150 align:middle line:84% What we know is that bidding b1 hat of v1 is optimal. 00:43:51.150 --> 00:43:53.850 align:middle line:90% 00:43:53.850 --> 00:43:55.770 align:middle line:84% Given the way everyone else is bidding. 00:43:55.770 --> 00:44:00.610 align:middle line:84% I could bid whatever I want, but bidding b1 hat of v1 00:44:00.610 --> 00:44:02.270 align:middle line:90% is the best possible bid for me. 00:44:02.270 --> 00:44:04.390 align:middle line:84% It maximizes my expected utility. 00:44:04.390 --> 00:44:07.610 align:middle line:90% 00:44:07.610 --> 00:44:14.010 align:middle line:84% And a weird way you want to think about this 00:44:14.010 --> 00:44:19.850 align:middle line:84% is that tells me that bidding b1 hat of v1 00:44:19.850 --> 00:44:22.250 align:middle line:84% is better than any other bid I could make. 00:44:22.250 --> 00:44:24.917 align:middle line:84% But these other bids, as you said, they're numbers. 00:44:24.917 --> 00:44:25.750 align:middle line:90% They could be words. 00:44:25.750 --> 00:44:27.542 align:middle line:84% We don't really know what these things are. 00:44:27.542 --> 00:44:33.490 align:middle line:84% But in particular, what this tells us is that bidding-- 00:44:33.490 --> 00:44:35.810 align:middle line:84% and this is the key observation-- 00:44:35.810 --> 00:44:46.010 align:middle line:84% bidding b1 hat of v1 is weakly better 00:44:46.010 --> 00:44:52.270 align:middle line:84% than bidding b1 hat of v1 prime for all v1 prime. 00:44:52.270 --> 00:44:55.450 align:middle line:90% 00:44:55.450 --> 00:44:57.430 align:middle line:84% So in a sense, I haven't done anything here, 00:44:57.430 --> 00:44:59.990 align:middle line:84% but this actually turns out to be the crucial observation. 00:44:59.990 --> 00:45:03.970 align:middle line:84% So the point is, bidding b1 hat of v1 00:45:03.970 --> 00:45:05.770 align:middle line:90% is better than any other bid. 00:45:05.770 --> 00:45:08.850 align:middle line:84% So in particular, what's another bid I could make? 00:45:08.850 --> 00:45:10.650 align:middle line:84% I could choose the bid that I'm supposed 00:45:10.650 --> 00:45:13.210 align:middle line:84% to make in this equilibrium if my two valuation is 00:45:13.210 --> 00:45:16.430 align:middle line:84% v1 prime. v1 prime is just a different valuation. 00:45:16.430 --> 00:45:18.770 align:middle line:90% I could do that. 00:45:18.770 --> 00:45:21.650 align:middle line:84% But following my equilibrium bidding strategy 00:45:21.650 --> 00:45:26.810 align:middle line:84% must be better than this alternative bid b1 of v prime. 00:45:26.810 --> 00:45:28.710 align:middle line:84% And then the name of the game here 00:45:28.710 --> 00:45:30.460 align:middle line:84% is just to try to write out mathematically 00:45:30.460 --> 00:45:34.620 align:middle line:84% what this means in terms of these functions here. 00:45:34.620 --> 00:45:36.820 align:middle line:84% So it's very useful to contemplate 00:45:36.820 --> 00:45:39.580 align:middle line:84% this particular alternative bid because we 00:45:39.580 --> 00:45:41.240 align:middle line:90% can use this alternative bid. 00:45:41.240 --> 00:45:44.420 align:middle line:84% We can express my utility from this alternative bid in terms 00:45:44.420 --> 00:45:45.780 align:middle line:90% of these functions here. 00:45:45.780 --> 00:45:47.480 align:middle line:90% And that's really the key step. 00:45:47.480 --> 00:45:49.760 align:middle line:90% So let's come back over here. 00:45:49.760 --> 00:46:06.660 align:middle line:90% 00:46:06.660 --> 00:46:09.260 align:middle line:84% So I really want to emphasize everything we're going to do now 00:46:09.260 --> 00:46:11.225 align:middle line:84% is just following from this simple observation. 00:46:11.225 --> 00:46:13.600 align:middle line:84% So if there's any questions on this, I think this is key. 00:46:13.600 --> 00:46:14.660 align:middle line:90% Yes, thank you. 00:46:14.660 --> 00:46:17.940 align:middle line:84% AUDIENCE: So does that imply that for every item in the bid 00:46:17.940 --> 00:46:21.373 align:middle line:84% set, there's some valuation that maps to it? 00:46:21.373 --> 00:46:23.040 align:middle line:84% IAN BALL: That's a very subtle question. 00:46:23.040 --> 00:46:23.998 align:middle line:90% It does not imply that. 00:46:23.998 --> 00:46:28.202 align:middle line:90% 00:46:28.202 --> 00:46:29.660 align:middle line:84% So I think you're basically asking, 00:46:29.660 --> 00:46:31.400 align:middle line:90% are the bidding functions onto? 00:46:31.400 --> 00:46:33.740 align:middle line:90% Are they subjective functions? 00:46:33.740 --> 00:46:36.540 align:middle line:84% If they were onto, then these statements would be equivalent. 00:46:36.540 --> 00:46:38.553 align:middle line:84% In general, the bidding function's not onto. 00:46:38.553 --> 00:46:40.220 align:middle line:84% So it's still the case that this implies 00:46:40.220 --> 00:46:41.760 align:middle line:90% this but not the converse. 00:46:41.760 --> 00:46:44.580 align:middle line:84% So we're not using that assumption. 00:46:44.580 --> 00:46:46.420 align:middle line:84% If that assumption is violated, then we're 00:46:46.420 --> 00:46:47.700 align:middle line:84% actually throwing away information 00:46:47.700 --> 00:46:49.492 align:middle line:84% because we're saying, we know that this bit 00:46:49.492 --> 00:46:51.660 align:middle line:84% is better than every other bid, even bids 00:46:51.660 --> 00:46:53.760 align:middle line:84% that no one ever chooses in equilibrium. 00:46:53.760 --> 00:46:57.100 align:middle line:84% But in particular, we know this, so we're throwing away 00:46:57.100 --> 00:46:58.640 align:middle line:90% information potentially. 00:46:58.640 --> 00:47:00.428 align:middle line:84% We're not even considering-- it's true 00:47:00.428 --> 00:47:02.720 align:middle line:84% that this bid is better than a bid that's never chosen, 00:47:02.720 --> 00:47:04.600 align:middle line:84% but we don't even need to use that fact. 00:47:04.600 --> 00:47:06.460 align:middle line:90% So we're not requiring that. 00:47:06.460 --> 00:47:07.100 align:middle line:90% Yes. 00:47:07.100 --> 00:47:11.580 align:middle line:84% AUDIENCE: So why can b1 hat of v1 prime technically 00:47:11.580 --> 00:47:14.880 align:middle line:84% just represent all other possible bidding strategies? 00:47:14.880 --> 00:47:17.620 align:middle line:90% 00:47:17.620 --> 00:47:19.100 align:middle line:90% IAN BALL: You said-- 00:47:19.100 --> 00:47:20.000 align:middle line:90% could you maybe-- 00:47:20.000 --> 00:47:21.580 align:middle line:90% I'm not sure if I quite follow. 00:47:21.580 --> 00:47:23.740 align:middle line:84% AUDIENCE: Because there's infinite other bids. 00:47:23.740 --> 00:47:24.698 align:middle line:90% IAN BALL: You're right. 00:47:24.698 --> 00:47:28.660 align:middle line:84% So I think the key here is I'm just using this forward arrow. 00:47:28.660 --> 00:47:30.023 align:middle line:90% I'm not saying that this-- 00:47:30.023 --> 00:47:32.440 align:middle line:84% that if this is true-- and this relates to this question-- 00:47:32.440 --> 00:47:33.660 align:middle line:84% I'm not saying if this is true, then 00:47:33.660 --> 00:47:34.907 align:middle line:90% this must be in equilibrium. 00:47:34.907 --> 00:47:36.240 align:middle line:90% I'm only going in one direction. 00:47:36.240 --> 00:47:38.780 align:middle line:84% I'm saying, we know that this is an equilibrium, 00:47:38.780 --> 00:47:40.780 align:middle line:84% and an implication of the fact that it's 00:47:40.780 --> 00:47:43.080 align:middle line:84% in equilibrium, because it's equilibrium, 00:47:43.080 --> 00:47:45.300 align:middle line:84% I'm better off bidding b hat of v1 00:47:45.300 --> 00:47:47.940 align:middle line:90% than making any alternative bid. 00:47:47.940 --> 00:47:50.500 align:middle line:84% So in particular, one implication of that is I 00:47:50.500 --> 00:47:51.560 align:middle line:90% don't want to bid 5. 00:47:51.560 --> 00:47:52.600 align:middle line:90% I don't a bid 3. 00:47:52.600 --> 00:47:53.787 align:middle line:90% I don't want to bid this. 00:47:53.787 --> 00:47:56.120 align:middle line:84% Maybe we're not considering all the other possible bids. 00:47:56.120 --> 00:47:58.180 align:middle line:90% That's fine. 00:47:58.180 --> 00:48:00.040 align:middle line:84% Maybe we're throwing away information here, 00:48:00.040 --> 00:48:03.180 align:middle line:84% but it's certainly an implication of equilibrium 00:48:03.180 --> 00:48:04.158 align:middle line:90% that this must hold. 00:48:04.158 --> 00:48:06.200 align:middle line:84% AUDIENCE: When you say throwing away information, 00:48:06.200 --> 00:48:08.660 align:middle line:90% what are you referring to? 00:48:08.660 --> 00:48:11.780 align:middle line:84% IAN BALL: What I mean is the definition of equilibrium 00:48:11.780 --> 00:48:13.820 align:middle line:84% tells a lot of things, and I'm only 00:48:13.820 --> 00:48:16.700 align:middle line:84% using some logical implications of the definition 00:48:16.700 --> 00:48:18.300 align:middle line:90% of equilibrium. 00:48:18.300 --> 00:48:19.820 align:middle line:84% So it's like I have 10 equations, 00:48:19.820 --> 00:48:21.460 align:middle line:90% and I'm only using nine of them. 00:48:21.460 --> 00:48:24.240 align:middle line:84% And the 10th equation's true, but I'm just not focusing on it. 00:48:24.240 --> 00:48:26.780 align:middle line:90% I'm not using that fact. 00:48:26.780 --> 00:48:29.378 align:middle line:84% And maybe this all abstract stuff 00:48:29.378 --> 00:48:31.420 align:middle line:84% makes more sense once we go through the argument. 00:48:31.420 --> 00:48:35.645 align:middle line:84% AUDIENCE: So you're saying like bidding b1 hat v1 is optimal. 00:48:35.645 --> 00:48:38.020 align:middle line:84% And there's other strategies that we're not comparing to. 00:48:38.020 --> 00:48:39.900 align:middle line:90% We're just comparing two. 00:48:39.900 --> 00:48:42.400 align:middle line:84% The bidding of b1 hat but with a different market. 00:48:42.400 --> 00:48:45.380 align:middle line:84% IAN BALL: Exactly, so the fact that bidding b1 hat of v1 00:48:45.380 --> 00:48:47.240 align:middle line:90% is optimal tells me a lot. 00:48:47.240 --> 00:48:48.940 align:middle line:84% It tells me that bidding b1 hat of v1 00:48:48.940 --> 00:48:52.020 align:middle line:84% is better than any other bid I could come up with. 00:48:52.020 --> 00:48:55.780 align:middle line:84% But it's certainly better than these special bids, 00:48:55.780 --> 00:48:57.520 align:middle line:84% and I'm only going to use that fact here. 00:48:57.520 --> 00:48:59.280 align:middle line:84% Even though I know it's better than some other bits too, 00:48:59.280 --> 00:49:00.280 align:middle line:90% I'm not going to worry-- 00:49:00.280 --> 00:49:02.300 align:middle line:84% I don't even need to use that fact. 00:49:02.300 --> 00:49:06.360 align:middle line:84% OK, so now, as we said, not equivalent 00:49:06.360 --> 00:49:09.303 align:middle line:84% but an implication of equilibrium is this observation. 00:49:09.303 --> 00:49:11.220 align:middle line:84% And now we just want to write this observation 00:49:11.220 --> 00:49:14.260 align:middle line:84% down mathematically using our notation here. 00:49:14.260 --> 00:49:17.600 align:middle line:84% So what we want to say is, what is my expected utility 00:49:17.600 --> 00:49:18.940 align:middle line:90% if I bid this? 00:49:18.940 --> 00:49:20.800 align:middle line:84% What is my expected utility if I bid this? 00:49:20.800 --> 00:49:22.967 align:middle line:84% And then we want to have an inequality between them. 00:49:22.967 --> 00:49:25.070 align:middle line:90% So let's take the first. 00:49:25.070 --> 00:49:32.410 align:middle line:84% So what is my utility from bidding b1 hat of v1? 00:49:32.410 --> 00:49:35.310 align:middle line:90% 00:49:35.310 --> 00:49:41.670 align:middle line:84% Well, it's going to be v1 times the probability 00:49:41.670 --> 00:49:51.930 align:middle line:84% that I win minus my expected payment. 00:49:51.930 --> 00:49:56.430 align:middle line:90% 00:49:56.430 --> 00:49:57.770 align:middle line:90% This is what my utility is. 00:49:57.770 --> 00:49:59.870 align:middle line:84% When I enter an auction, I say, well, this 00:49:59.870 --> 00:50:01.550 align:middle line:90% is my probability of winning. 00:50:01.550 --> 00:50:04.110 align:middle line:84% Whenever I win, I get my valuation for the good, 00:50:04.110 --> 00:50:05.350 align:middle line:90% but then I have to pay. 00:50:05.350 --> 00:50:07.510 align:middle line:84% Well, we'll just subtract my expected payment 00:50:07.510 --> 00:50:10.310 align:middle line:84% because we're assuming here people are risk neutral. 00:50:10.310 --> 00:50:13.710 align:middle line:84% So what is my probability of winning if I bid b1 hat of v1 00:50:13.710 --> 00:50:15.010 align:middle line:90% using our notation over there? 00:50:15.010 --> 00:50:22.070 align:middle line:90% 00:50:22.070 --> 00:50:25.350 align:middle line:90% AUDIENCE: Just a few hat of 1. 00:50:25.350 --> 00:50:26.830 align:middle line:90% IAN BALL: Of v1, right? 00:50:26.830 --> 00:50:30.990 align:middle line:90% So this is q1 hat of v1. 00:50:30.990 --> 00:50:33.666 align:middle line:84% And then what is my expected payment? 00:50:33.666 --> 00:50:34.618 align:middle line:90% AUDIENCE: It would-- 00:50:34.618 --> 00:50:35.285 align:middle line:90% IAN BALL: Right. 00:50:35.285 --> 00:50:40.190 align:middle line:90% 00:50:40.190 --> 00:50:43.910 align:middle line:84% Now the tricky part, what if I instead bid-- 00:50:43.910 --> 00:50:45.710 align:middle line:84% I deviate, and instead of bidding 00:50:45.710 --> 00:50:47.450 align:middle line:84% what I'm supposed to bid in equilibrium, 00:50:47.450 --> 00:50:48.450 align:middle line:90% I make a different bid. 00:50:48.450 --> 00:50:55.470 align:middle line:84% Specifically, I bid v1 hat of v1 prime. 00:50:55.470 --> 00:50:58.810 align:middle line:84% Well, the crucial point is, what is my utility? 00:50:58.810 --> 00:51:01.410 align:middle line:84% Well, first, I still have a v1 here. 00:51:01.410 --> 00:51:02.770 align:middle line:90% Let's not get confused. 00:51:02.770 --> 00:51:03.888 align:middle line:90% My value is still v1. 00:51:03.888 --> 00:51:05.430 align:middle line:84% The fact that I'm bidding differently 00:51:05.430 --> 00:51:07.250 align:middle line:84% doesn't change my value to v1 prime. 00:51:07.250 --> 00:51:08.910 align:middle line:84% My value is still v1, so it's going 00:51:08.910 --> 00:51:12.470 align:middle line:84% to be my value here, times, again, my probability of winning 00:51:12.470 --> 00:51:16.190 align:middle line:90% minus my expected payment. 00:51:16.190 --> 00:51:19.690 align:middle line:84% What is the probability of winning now going to be? 00:51:19.690 --> 00:51:21.510 align:middle line:90% AUDIENCE: b1 hat v1 prime. 00:51:21.510 --> 00:51:23.110 align:middle line:84% IAN BALL: Right, so I just plug in-- 00:51:23.110 --> 00:51:26.250 align:middle line:84% if I look carefully at this formula, 00:51:26.250 --> 00:51:30.568 align:middle line:84% if I replace b1 hat of v1 with b1 hat of v1 prime, 00:51:30.568 --> 00:51:32.110 align:middle line:84% well, then that's just the definition 00:51:32.110 --> 00:51:35.310 align:middle line:84% of this function evaluated at v1 prime. 00:51:35.310 --> 00:51:45.910 align:middle line:84% So I'm going to get Q1 hat v1 prime plus T1 hat. 00:51:45.910 --> 00:51:46.810 align:middle line:90% And this makes sense. 00:51:46.810 --> 00:51:48.060 align:middle line:90% What's the definition of this? 00:51:48.060 --> 00:51:50.550 align:middle line:84% This tells me if I use bidding-- if I 00:51:50.550 --> 00:51:53.070 align:middle line:84% use the bid b1 hat of v1 prime, what 00:51:53.070 --> 00:51:56.270 align:middle line:90% is my probability of winning? 00:51:56.270 --> 00:51:57.750 align:middle line:84% And the crucial thing going on here 00:51:57.750 --> 00:51:59.630 align:middle line:84% is that my probability of winning only 00:51:59.630 --> 00:52:01.090 align:middle line:90% depends on how much I bid. 00:52:01.090 --> 00:52:02.930 align:middle line:84% It doesn't depend on my true valuation. 00:52:02.930 --> 00:52:04.290 align:middle line:90% It just depends on what I bid. 00:52:04.290 --> 00:52:07.230 align:middle line:84% So the v1 prime here is really reflecting my bid, not 00:52:07.230 --> 00:52:09.710 align:middle line:90% my true valuation. 00:52:09.710 --> 00:52:11.430 align:middle line:90% So now what do we know? 00:52:11.430 --> 00:52:14.010 align:middle line:84% Well, we know this is greater than or equal to this. 00:52:14.010 --> 00:52:17.060 align:middle line:84% And maybe I've interpreted things so much, 00:52:17.060 --> 00:52:18.720 align:middle line:84% it's nicer to start a clean board 00:52:18.720 --> 00:52:23.000 align:middle line:84% and just write down our inequality. 00:52:23.000 --> 00:52:23.980 align:middle line:90% So what do we know? 00:52:23.980 --> 00:52:34.040 align:middle line:90% 00:52:34.040 --> 00:52:41.280 align:middle line:84% So what we know is that for all v1, v1 prime-- 00:52:41.280 --> 00:52:44.640 align:middle line:84% remember, the interpretation here is v1 is my true value. 00:52:44.640 --> 00:52:49.200 align:middle line:84% v1 prime is the value that I pretend to have when I bid. 00:52:49.200 --> 00:53:00.560 align:middle line:84% Well, we must have v1 Q1 hat of v1 T1 hat of v1 greater than 00:53:00.560 --> 00:53:01.300 align:middle line:90% or equal to-- 00:53:01.300 --> 00:53:10.960 align:middle line:90% 00:53:10.960 --> 00:53:13.140 align:middle line:84% so I've just brought that equation down here. 00:53:13.140 --> 00:53:19.280 align:middle line:90% 00:53:19.280 --> 00:53:22.480 align:middle line:84% And maybe it's easier if we say for all v1, 00:53:22.480 --> 00:53:27.360 align:middle line:84% this holds for all v1 prime, just a slightly different 00:53:27.360 --> 00:53:28.560 align:middle line:90% perspective. 00:53:28.560 --> 00:53:34.120 align:middle line:90% 00:53:34.120 --> 00:53:38.320 align:middle line:90% Well, what does this tell me? 00:53:38.320 --> 00:53:41.480 align:middle line:84% Well, I have some function on the right side that 00:53:41.480 --> 00:53:45.080 align:middle line:84% depends on v1 prime, but it's always smaller 00:53:45.080 --> 00:53:47.960 align:middle line:90% than this left-hand side. 00:53:47.960 --> 00:53:50.320 align:middle line:84% So in particular, what that tells 00:53:50.320 --> 00:53:55.280 align:middle line:84% me is that the value of v1 prime that 00:53:55.280 --> 00:53:59.000 align:middle line:84% maximizes this right-hand side must be V1 itself. 00:53:59.000 --> 00:54:02.000 align:middle line:84% So that tells me that V1, let's use our arg max notation 00:54:02.000 --> 00:54:07.820 align:middle line:84% is an element of the arg max over v1 prime of v1. 00:54:07.820 --> 00:54:16.520 align:middle line:90% 00:54:16.520 --> 00:54:17.860 align:middle line:90% In words, what is this saying? 00:54:17.860 --> 00:54:19.960 align:middle line:84% It's saying if my true value is v1, 00:54:19.960 --> 00:54:21.880 align:middle line:84% I could bid as if my valuation is 00:54:21.880 --> 00:54:25.080 align:middle line:84% v1 prime for any other value of v1 prime. 00:54:25.080 --> 00:54:28.200 align:middle line:84% But because we know that I'm playing an equilibrium, 00:54:28.200 --> 00:54:31.440 align:middle line:84% the best thing for me to do is to bid as if I am myself, 00:54:31.440 --> 00:54:34.680 align:middle line:84% to bid as if my value v1 prime is exactly v1, 00:54:34.680 --> 00:54:36.700 align:middle line:84% and that's what this is telling us here. 00:54:36.700 --> 00:54:40.280 align:middle line:90% 00:54:40.280 --> 00:54:43.360 align:middle line:84% And now let's just compute the first-order condition. 00:54:43.360 --> 00:54:47.503 align:middle line:84% So if v1 is a maximizer of this, then we 00:54:47.503 --> 00:54:49.920 align:middle line:84% should be able to take the derivative of this with respect 00:54:49.920 --> 00:54:53.700 align:middle line:84% to v1 prime, and then plug in v1 prime equals v1, 00:54:53.700 --> 00:54:54.620 align:middle line:90% and we should get 0. 00:54:54.620 --> 00:54:56.053 align:middle line:90% So let's do that. 00:54:56.053 --> 00:54:58.220 align:middle line:84% So it's going to be a little tricky with the primes. 00:54:58.220 --> 00:55:00.012 align:middle line:84% You have to interpret the primes carefully. 00:55:00.012 --> 00:55:05.800 align:middle line:84% So this is v1 Q1 hat prime v1 prime minus T1 hat 00:55:05.800 --> 00:55:07.680 align:middle line:90% prime v1 prime. 00:55:07.680 --> 00:55:11.260 align:middle line:84% Evaluated at v1 prime equals v1, this should be 0. 00:55:11.260 --> 00:55:15.240 align:middle line:90% 00:55:15.240 --> 00:55:19.440 align:middle line:84% Sorry, the primes, this prime on the inside is just a symbol. 00:55:19.440 --> 00:55:21.400 align:middle line:84% And this prime on the outside is a derivative. 00:55:21.400 --> 00:55:24.400 align:middle line:84% Sorry, that can be a bit confusing here. 00:55:24.400 --> 00:55:26.420 align:middle line:84% So you could call v1 prime z if you wanted. 00:55:26.420 --> 00:55:29.640 align:middle line:90% It wouldn't make a difference. 00:55:29.640 --> 00:55:31.600 align:middle line:90% And what do we get? 00:55:31.600 --> 00:55:35.920 align:middle line:90% We get v1 Q1 prime v1. 00:55:35.920 --> 00:55:47.300 align:middle line:90% 00:55:47.300 --> 00:55:48.860 align:middle line:90% Minus, sorry. 00:55:48.860 --> 00:55:52.015 align:middle line:90% 00:55:52.015 --> 00:55:54.640 align:middle line:84% So this equals 0, or I could say it implies that they're equal. 00:55:54.640 --> 00:56:00.600 align:middle line:90% 00:56:00.600 --> 00:56:02.640 align:middle line:90% Well, so a lot of work. 00:56:02.640 --> 00:56:04.800 align:middle line:84% But now we've narrowed things down a lot. 00:56:04.800 --> 00:56:07.770 align:middle line:84% I have a formula for the derivative of this expected 00:56:07.770 --> 00:56:11.570 align:middle line:84% payment that I make in terms of my probability of getting 00:56:11.570 --> 00:56:13.330 align:middle line:90% the good. 00:56:13.330 --> 00:56:17.650 align:middle line:84% So now the trick is we're just going to integrate this. 00:56:17.650 --> 00:56:21.490 align:middle line:84% So if this equals 0, that tells me 00:56:21.490 --> 00:56:24.730 align:middle line:84% that this derivative equals this. 00:56:24.730 --> 00:56:27.035 align:middle line:84% So now we can just integrate this from v lower bar 00:56:27.035 --> 00:56:27.910 align:middle line:90% and see what happens. 00:56:27.910 --> 00:56:30.250 align:middle line:84% Actually, maybe I'll just do it right here. 00:56:30.250 --> 00:56:32.530 align:middle line:90% So what does this tell me? 00:56:32.530 --> 00:56:41.010 align:middle line:84% It tells me that T1 hat of V1 is equal to T1 00:56:41.010 --> 00:56:46.690 align:middle line:84% hat of v lower bar plus an integral from v lower bar 00:56:46.690 --> 00:56:50.590 align:middle line:90% to v1 of x-- 00:56:50.590 --> 00:56:56.510 align:middle line:90% I'll change to x. 00:56:56.510 --> 00:57:01.610 align:middle line:90% 00:57:01.610 --> 00:57:03.390 align:middle line:84% Again, when we start integrating, 00:57:03.390 --> 00:57:05.150 align:middle line:90% you have to be careful. 00:57:05.150 --> 00:57:08.323 align:middle line:84% I've used a new dummy variable here, x. 00:57:08.323 --> 00:57:09.490 align:middle line:90% And there's a prime missing. 00:57:09.490 --> 00:57:23.010 align:middle line:90% 00:57:23.010 --> 00:57:27.590 align:middle line:84% OK, so this is going to be the key revenue formula. 00:57:27.590 --> 00:57:32.170 align:middle line:90% 00:57:32.170 --> 00:57:36.222 align:middle line:90% Now, why is this so powerful? 00:57:36.222 --> 00:57:37.430 align:middle line:90% I think we want to step back. 00:57:37.430 --> 00:57:40.013 align:middle line:84% And at first, you might say, OK, we just did a sea of algebra. 00:57:40.013 --> 00:57:42.210 align:middle line:90% What's going on? 00:57:42.210 --> 00:57:44.450 align:middle line:90% Well, what is this giving me? 00:57:44.450 --> 00:57:49.850 align:middle line:84% It's giving me a formula for, if I'm player 1 and my valuation 00:57:49.850 --> 00:57:53.330 align:middle line:84% is v1, and we're considering an equilibrium 00:57:53.330 --> 00:57:54.930 align:middle line:90% of this generalized auction. 00:57:54.930 --> 00:57:58.330 align:middle line:84% It's giving me a formula for the expected payment 00:57:58.330 --> 00:57:59.930 align:middle line:90% that I make in this auction. 00:57:59.930 --> 00:58:04.100 align:middle line:84% But let's understand, in general, you 00:58:04.100 --> 00:58:05.850 align:middle line:84% might think that my expected payment would 00:58:05.850 --> 00:58:07.290 align:middle line:90% depend on a lot of things. 00:58:07.290 --> 00:58:09.207 align:middle line:84% You might think it would depend on the details 00:58:09.207 --> 00:58:11.490 align:middle line:84% of the auction and all these complicated things. 00:58:11.490 --> 00:58:14.410 align:middle line:84% But if we look here, what is it depending on? 00:58:14.410 --> 00:58:17.450 align:middle line:90% It's depending on two things. 00:58:17.450 --> 00:58:19.670 align:middle line:84% Can someone interpret these two terms? 00:58:19.670 --> 00:58:23.410 align:middle line:84% These are the only things that appear on the right-hand side. 00:58:23.410 --> 00:58:26.650 align:middle line:90% So what is this first term? 00:58:26.650 --> 00:58:30.610 align:middle line:84% AUDIENCE: The base cost of beginning the auction. 00:58:30.610 --> 00:58:33.810 align:middle line:84% IAN BALL: Yeah, so it's just the expected payment that the lowest 00:58:33.810 --> 00:58:37.190 align:middle line:90% possible valuation person pays. 00:58:37.190 --> 00:58:39.690 align:middle line:84% That's the one thing, and I think you could interpret it 00:58:39.690 --> 00:58:41.410 align:middle line:90% maybe that way. 00:58:41.410 --> 00:58:44.270 align:middle line:90% And then what is Q1 prime-- 00:58:44.270 --> 00:58:46.990 align:middle line:84% I mean, the prime, let's just focus on the function Q1 hat. 00:58:46.990 --> 00:58:49.130 align:middle line:90% What is this function Q1 hat? 00:58:49.130 --> 00:58:52.015 align:middle line:84% Can we interpret-- what does it mean? 00:58:52.015 --> 00:58:52.890 align:middle line:90% Yeah, maybe, I think. 00:58:52.890 --> 00:58:53.170 align:middle line:90% Yeah. 00:58:53.170 --> 00:58:54.753 align:middle line:84% AUDIENCE: It's the marginal likelihood 00:58:54.753 --> 00:58:58.490 align:middle line:84% of winning for each infinitesimal change 00:58:58.490 --> 00:59:01.327 align:middle line:90% in your valuation. 00:59:01.327 --> 00:59:03.660 align:middle line:84% IAN BALL: I think the integral is making you think about 00:59:03.660 --> 00:59:05.917 align:middle line:84% infinitesimal stuff, but really, it's-- oh, 00:59:05.917 --> 00:59:07.500 align:middle line:84% you're taking the derivative, exactly. 00:59:07.500 --> 00:59:08.220 align:middle line:90% Sorry, yes. 00:59:08.220 --> 00:59:11.082 align:middle line:84% So sorry, you said, yes, that's the derivative of it, exactly. 00:59:11.082 --> 00:59:12.540 align:middle line:84% And if we take the derivative, it's 00:59:12.540 --> 00:59:15.370 align:middle line:84% just my probability of winning, exactly. 00:59:15.370 --> 00:59:17.120 align:middle line:84% And this is the infinitesimal, sorry, yes. 00:59:17.120 --> 00:59:19.420 align:middle line:90% Exactly, I forgot, yeah. 00:59:19.420 --> 00:59:23.700 align:middle line:84% OK, so what it tells us is that a lot of the details 00:59:23.700 --> 00:59:25.680 align:middle line:90% of the auction don't matter. 00:59:25.680 --> 00:59:29.280 align:middle line:84% I mean, whether we're bidding what we name our bids-- 00:59:29.280 --> 00:59:31.920 align:middle line:84% notice the sets B1 through Bn don't even appear here. 00:59:31.920 --> 00:59:34.180 align:middle line:84% All that matters is the probability 00:59:34.180 --> 00:59:36.980 align:middle line:84% that I win the good in equilibrium and the amount 00:59:36.980 --> 00:59:39.860 align:middle line:84% I pay if I'm the lowest-value person. 00:59:39.860 --> 00:59:44.500 align:middle line:84% So we immediately see that only a few details of an auction 00:59:44.500 --> 00:59:48.380 align:middle line:84% equilibrium pin down or uniquely determine the expected payment 00:59:48.380 --> 00:59:49.660 align:middle line:90% that everyone makes. 00:59:49.660 --> 00:59:52.720 align:middle line:90% So maybe I'll come over here. 00:59:52.720 --> 01:00:12.220 align:middle line:90% 01:00:12.220 --> 01:00:14.220 align:middle line:90% So what's the implication? 01:00:14.220 --> 01:00:19.720 align:middle line:90% 01:00:19.720 --> 01:00:30.420 align:middle line:84% That the expected payment by, say, 01:00:30.420 --> 01:00:43.660 align:middle line:84% bidder 1 when the valuation is v1 depends only on-- 01:00:43.660 --> 01:00:51.620 align:middle line:84% well, T1 hat v lower bar, which is the expected payment 01:00:51.620 --> 01:01:01.740 align:middle line:90% of the lowest valuation-- 01:01:01.740 --> 01:01:03.340 align:middle line:84% more precisely, the expected payment 01:01:03.340 --> 01:01:06.500 align:middle line:84% that bidder 1 makes when her valuation is the lowest 01:01:06.500 --> 01:01:08.580 align:middle line:90% possible valuation. 01:01:08.580 --> 01:01:12.500 align:middle line:84% And we had the derivative here, but if it 01:01:12.500 --> 01:01:14.980 align:middle line:84% depends on the derivative of Q1 hat, well, 01:01:14.980 --> 01:01:16.400 align:middle line:90% it also depends on Q1 hat. 01:01:16.400 --> 01:01:21.420 align:middle line:84% So it's a bit cleaner to say just depends on Q1 hat-- 01:01:21.420 --> 01:01:29.800 align:middle line:84% this function Q1 hat, which is the win probability. 01:01:29.800 --> 01:01:33.860 align:middle line:90% 01:01:33.860 --> 01:01:35.160 align:middle line:90% And this is an entire function. 01:01:35.160 --> 01:01:38.040 align:middle line:84% So it's the win probability of every valuation. 01:01:38.040 --> 01:01:47.420 align:middle line:90% 01:01:47.420 --> 01:01:50.780 align:middle line:84% But the point is, in a lot of different equilibria 01:01:50.780 --> 01:01:53.340 align:middle line:84% of different auction formats, these two values 01:01:53.340 --> 01:01:55.850 align:middle line:84% take very-- these take very special values. 01:01:55.850 --> 01:01:59.110 align:middle line:84% So in many equilibria, in many auctions that we've 01:01:59.110 --> 01:02:01.870 align:middle line:90% looked at so far, what is-- 01:02:01.870 --> 01:02:03.170 align:middle line:90% so I'll just say often. 01:02:03.170 --> 01:02:06.270 align:middle line:90% 01:02:06.270 --> 01:02:14.350 align:middle line:84% Well, intuitively, what do you think T1 hat of v1 lower 01:02:14.350 --> 01:02:16.570 align:middle line:90% bar is often going to be? 01:02:16.570 --> 01:02:17.150 align:middle line:90% AUDIENCE: 0. 01:02:17.150 --> 01:02:19.550 align:middle line:84% IAN BALL: 0 because, in most auctions, 01:02:19.550 --> 01:02:22.470 align:middle line:84% if you have the lowest possible valuation for the item, 01:02:22.470 --> 01:02:25.755 align:middle line:84% in most equilibria, well, you're never going to win. 01:02:25.755 --> 01:02:27.130 align:middle line:84% And if you're never going to win, 01:02:27.130 --> 01:02:28.590 align:middle line:90% you don't want to pay anything. 01:02:28.590 --> 01:02:32.750 align:middle line:84% So often, auction equilibria have this property. 01:02:32.750 --> 01:02:36.150 align:middle line:84% And now maybe this is a more subtle one. 01:02:36.150 --> 01:02:38.910 align:middle line:90% What about Q1 hat? 01:02:38.910 --> 01:02:42.190 align:middle line:90% Maybe I'll write Q1 hat of v1? 01:02:42.190 --> 01:02:44.170 align:middle line:84% Well, in a lot of auction equilibria 01:02:44.170 --> 01:02:48.670 align:middle line:84% we've looked at, which bidder wins the item? 01:02:48.670 --> 01:02:49.690 align:middle line:90% It's the highest bidder. 01:02:49.690 --> 01:02:51.470 align:middle line:84% But another way of saying it is it's 01:02:51.470 --> 01:02:53.220 align:middle line:84% the bidder who bids the most, that's true. 01:02:53.220 --> 01:02:56.710 align:middle line:84% But it's also the bidder whose valuation is the highest. 01:02:56.710 --> 01:02:58.930 align:middle line:84% So often, when auctions work well, 01:02:58.930 --> 01:03:01.870 align:middle line:84% we might think that an outcome of the auction 01:03:01.870 --> 01:03:05.270 align:middle line:84% is that the bidder who values the item most gets the item. 01:03:05.270 --> 01:03:08.350 align:middle line:84% But if the bidder who values the item most gets the item, 01:03:08.350 --> 01:03:15.230 align:middle line:84% then what is Q hat 1 of v1 going to be? 01:03:15.230 --> 01:03:16.590 align:middle line:90% Well-- Yeah. 01:03:16.590 --> 01:03:19.150 align:middle line:90% AUDIENCE: Or as an expectation-- 01:03:19.150 --> 01:03:20.530 align:middle line:90% it will either be 0 or 1. 01:03:20.530 --> 01:03:22.190 align:middle line:84% So it's a kind of average of those probabilities. 01:03:22.190 --> 01:03:23.530 align:middle line:84% IAN BALL: Its an expectation between 0 and 1, 01:03:23.530 --> 01:03:25.613 align:middle line:84% so it's going to be a probability between 0 and 1. 01:03:25.613 --> 01:03:27.470 align:middle line:90% Yeah, that's true. 01:03:27.470 --> 01:03:28.650 align:middle line:90% Can we say more about it? 01:03:28.650 --> 01:03:30.790 align:middle line:90% Yeah. 01:03:30.790 --> 01:03:32.990 align:middle line:84% AUDIENCE: Maybe, but if there are n other players, 01:03:32.990 --> 01:03:35.395 align:middle line:84% then would be v1 to the power of n minus 1. 01:03:35.395 --> 01:03:36.770 align:middle line:84% IAN BALL: In the uniform example, 01:03:36.770 --> 01:03:38.010 align:middle line:90% that's exactly what it would be. 01:03:38.010 --> 01:03:39.510 align:middle line:84% But more generally, maybe let's just 01:03:39.510 --> 01:03:41.790 align:middle line:84% write it would be the probability that everyone else's 01:03:41.790 --> 01:03:43.517 align:middle line:84% value is below v1 because if it goes 01:03:43.517 --> 01:03:45.350 align:middle line:84% to the bidder with the highest value, what's 01:03:45.350 --> 01:03:46.990 align:middle line:84% the probability that bidder 1 gets it 01:03:46.990 --> 01:03:48.090 align:middle line:90% when the valuation is v1? 01:03:48.090 --> 01:03:49.715 align:middle line:84% It's the probability that everyone else 01:03:49.715 --> 01:03:50.650 align:middle line:90% is value is below v1. 01:03:50.650 --> 01:03:57.750 align:middle line:84% So it's the probability that the max of j not equal to 1 vj 01:03:57.750 --> 01:04:00.230 align:middle line:90% is less than or equal to v1. 01:04:00.230 --> 01:04:02.870 align:middle line:84% And we actually have notation for this. 01:04:02.870 --> 01:04:06.050 align:middle line:84% In the symmetric case, what is the notation? 01:04:06.050 --> 01:04:06.803 align:middle line:90% Yeah. 01:04:06.803 --> 01:04:07.470 align:middle line:90% AUDIENCE: Big G. 01:04:07.470 --> 01:04:08.530 align:middle line:90% IAN BALL: Big G, right? 01:04:08.530 --> 01:04:10.740 align:middle line:90% So this is also-- 01:04:10.740 --> 01:04:13.830 align:middle line:90% I'll write up here, big G of v1. 01:04:13.830 --> 01:04:17.590 align:middle line:90% 01:04:17.590 --> 01:04:20.070 align:middle line:90% So we often see that this holds. 01:04:20.070 --> 01:04:24.310 align:middle line:84% "Often" is in quotes, but what we see is that these tend 01:04:24.310 --> 01:04:24.810 align:middle line:90% to happen. 01:04:24.810 --> 01:04:27.910 align:middle line:84% And if that happens, let's plug into a formula over here. 01:04:27.910 --> 01:04:42.750 align:middle line:84% So under these conditions, what does T1 hat of v1 give us? 01:04:42.750 --> 01:04:48.050 align:middle line:84% Well, what is this first term under those conditions? 01:04:48.050 --> 01:04:48.550 align:middle line:90% AUDIENCE: 0. 01:04:48.550 --> 01:04:50.870 align:middle line:84% IAN BALL: 0, OK, so we can forget that. 01:04:50.870 --> 01:04:53.690 align:middle line:90% Q1 hat 1 of x is what? 01:04:53.690 --> 01:04:56.950 align:middle line:90% 01:04:56.950 --> 01:04:59.470 align:middle line:90% AUDIENCE: Small g of x. 01:04:59.470 --> 01:05:01.227 align:middle line:84% IAN BALL: Right, so Q1 hat is big G. 01:05:01.227 --> 01:05:02.810 align:middle line:84% And then, when we take the derivative, 01:05:02.810 --> 01:05:09.310 align:middle line:84% we get small G. So what we get is the integral from v lower bar 01:05:09.310 --> 01:05:15.070 align:middle line:90% to v1 of x g of x dx. 01:05:15.070 --> 01:05:17.550 align:middle line:90% Does this look familiar? 01:05:17.550 --> 01:05:19.750 align:middle line:84% This is exactly the formula we had. 01:05:19.750 --> 01:05:21.850 align:middle line:84% So now we understand where it comes from. 01:05:21.850 --> 01:05:24.830 align:middle line:84% So this is not a special property of first-price auctions 01:05:24.830 --> 01:05:26.090 align:middle line:90% or second-price auctions. 01:05:26.090 --> 01:05:28.790 align:middle line:84% Now we understand exactly when this is going to hold. 01:05:28.790 --> 01:05:31.870 align:middle line:84% This is going to hold in any equilibrium of any auction that 01:05:31.870 --> 01:05:34.230 align:middle line:90% satisfies these two properties. 01:05:34.230 --> 01:05:39.630 align:middle line:84% Whenever my value is the lowest possible value, I pay nothing. 01:05:39.630 --> 01:05:41.230 align:middle line:90% That's the condition here. 01:05:41.230 --> 01:05:43.790 align:middle line:84% And if the good is always allocated 01:05:43.790 --> 01:05:46.230 align:middle line:84% to the bidder with the highest valuation-- 01:05:46.230 --> 01:05:48.272 align:middle line:84% now, how that happens is going to depend 01:05:48.272 --> 01:05:50.480 align:middle line:84% on the details of the equilibrium, what the bids are. 01:05:50.480 --> 01:05:52.753 align:middle line:84% But as long as the auction has that property, 01:05:52.753 --> 01:05:54.920 align:middle line:84% then the expected payments are going to be the same. 01:05:54.920 --> 01:05:57.600 align:middle line:84% They're exactly going to take this form. 01:05:57.600 --> 01:05:59.540 align:middle line:84% So indeed, it wasn't a coincidence. 01:05:59.540 --> 01:06:01.960 align:middle line:84% It wasn't just something about the first- and second-price 01:06:01.960 --> 01:06:02.660 align:middle line:90% auction. 01:06:02.660 --> 01:06:03.980 align:middle line:90% This is a very general fact. 01:06:03.980 --> 01:06:07.080 align:middle line:84% So I think it's good to understand-- 01:06:07.080 --> 01:06:08.280 align:middle line:90% I think sometimes when-- 01:06:08.280 --> 01:06:09.760 align:middle line:84% I should say exactly how it's done 01:06:09.760 --> 01:06:11.480 align:middle line:84% in the notes-- revenue equivalents can be 01:06:11.480 --> 01:06:13.317 align:middle line:90% stated in two different ways. 01:06:13.317 --> 01:06:15.400 align:middle line:84% Sometimes people say only revenue equivalence only 01:06:15.400 --> 01:06:16.692 align:middle line:90% holds under certain conditions. 01:06:16.692 --> 01:06:17.640 align:middle line:90% Let's understand. 01:06:17.640 --> 01:06:19.280 align:middle line:90% This formula always holds-- 01:06:19.280 --> 01:06:20.580 align:middle line:90% or always. 01:06:20.580 --> 01:06:22.760 align:middle line:84% This formula holds quite generally, 01:06:22.760 --> 01:06:25.400 align:middle line:84% but it reduces to this special form 01:06:25.400 --> 01:06:26.980 align:middle line:90% under the additional conditions. 01:06:26.980 --> 01:06:29.160 align:middle line:84% So you can still apply this formula to an auction 01:06:29.160 --> 01:06:31.120 align:middle line:84% where the good isn't always allocated 01:06:31.120 --> 01:06:33.300 align:middle line:84% to the person with the highest valuation 01:06:33.300 --> 01:06:36.220 align:middle line:84% or where the lowest-valuation person pays a positive amount. 01:06:36.220 --> 01:06:39.240 align:middle line:84% This formula still holds, but it reduces and simplifies 01:06:39.240 --> 01:06:40.560 align:middle line:90% under those further conditions. 01:06:40.560 --> 01:06:41.740 align:middle line:90% Yes. 01:06:41.740 --> 01:06:43.240 align:middle line:84% AUDIENCE: Could you revisit the step 01:06:43.240 --> 01:06:45.400 align:middle line:84% of how we went through taking the derivative 01:06:45.400 --> 01:06:47.217 align:middle line:90% of our v1 equation? 01:06:47.217 --> 01:06:47.800 align:middle line:90% IAN BALL: Yes. 01:06:47.800 --> 01:06:49.717 align:middle line:84% AUDIENCE: --and getting to the revenue format? 01:06:49.717 --> 01:06:52.480 align:middle line:84% IAN BALL: Yes, so from here to there? 01:06:52.480 --> 01:06:56.040 align:middle line:84% OK, so you're good up to this point? 01:06:56.040 --> 01:06:58.020 align:middle line:90% OK, so we're at this point. 01:06:58.020 --> 01:07:00.860 align:middle line:84% And it's a little spread out, but this is 0 over here. 01:07:00.860 --> 01:07:10.680 align:middle line:84% So the 0 over here tells me that v1 times q1 hat prime of v1 01:07:10.680 --> 01:07:13.480 align:middle line:90% equals T1 hat prime of v1. 01:07:13.480 --> 01:07:16.440 align:middle line:84% And now, yeah, I think the way I changed variables is tricky. 01:07:16.440 --> 01:07:17.960 align:middle line:84% I think the best thing to do would 01:07:17.960 --> 01:07:21.960 align:middle line:84% be wherever you see a v1, let's just put an x. 01:07:21.960 --> 01:07:26.960 align:middle line:84% So this tells me x q1 prime of x equals T1 prime of x. 01:07:26.960 --> 01:07:28.960 align:middle line:84% Now I'm going to integrate both sides 01:07:28.960 --> 01:07:32.960 align:middle line:84% from x equals v lower bar to x equals v1. 01:07:32.960 --> 01:07:38.760 align:middle line:84% So when I integrate the derivative of T1 01:07:38.760 --> 01:07:44.480 align:middle line:84% from v lower bar to v1, I just get t hat of v1 minus t 01:07:44.480 --> 01:07:45.560 align:middle line:90% hat of v lower bar. 01:07:45.560 --> 01:07:48.240 align:middle line:84% So you could think of that coming to the left-hand side. 01:07:48.240 --> 01:07:52.568 align:middle line:84% And then, when I integrate this, well, I exactly get this. 01:07:52.568 --> 01:07:53.860 align:middle line:90% AUDIENCE: OK, that makes sense. 01:07:53.860 --> 01:07:54.120 align:middle line:90% IAN BALL: Clear? 01:07:54.120 --> 01:07:54.460 align:middle line:90% OK. 01:07:54.460 --> 01:07:55.210 align:middle line:90% Thanks for asking. 01:07:55.210 --> 01:07:56.880 align:middle line:90% Any other questions? 01:07:56.880 --> 01:07:58.600 align:middle line:90% So yeah, it was tricky. 01:07:58.600 --> 01:08:01.860 align:middle line:84% I used v1 here, but then v1 becomes my dummy variable x. 01:08:01.860 --> 01:08:03.140 align:middle line:90% And then I integrate up to v1. 01:08:03.140 --> 01:08:06.360 align:middle line:90% 01:08:06.360 --> 01:08:08.360 align:middle line:90% Any other questions? 01:08:08.360 --> 01:08:10.440 align:middle line:84% So you might say, OK, this is a nice theory, 01:08:10.440 --> 01:08:12.322 align:middle line:90% but how can we ever use this? 01:08:12.322 --> 01:08:14.280 align:middle line:84% Well, it turns out this can be a convenient way 01:08:14.280 --> 01:08:17.279 align:middle line:84% to solve for equilibria in auctions. 01:08:17.279 --> 01:08:19.840 align:middle line:90% So let's go over a mini-- 01:08:19.840 --> 01:08:21.700 align:middle line:84% an application of revenue equivalence. 01:08:21.700 --> 01:08:24.880 align:middle line:90% 01:08:24.880 --> 01:08:27.120 align:middle line:84% Maybe I'll call it RET, an application of the Revenue 01:08:27.120 --> 01:08:30.000 align:middle line:90% Equivalence Theorem. 01:08:30.000 --> 01:08:31.380 align:middle line:90% Let's look at all-pay auction. 01:08:31.380 --> 01:08:37.840 align:middle line:84% So let's do our simple case where we have just two bidders, 01:08:37.840 --> 01:08:40.920 align:middle line:90% v1 and v2. 01:08:40.920 --> 01:08:43.189 align:middle line:84% And they're in 0, 1, and they're uniform. 01:08:43.189 --> 01:08:50.330 align:middle line:90% 01:08:50.330 --> 01:08:52.675 align:middle line:84% And now, instead of considering the first-price auction 01:08:52.675 --> 01:08:55.050 align:middle line:84% or the second-price auction, let's consider what's called 01:08:55.050 --> 01:08:58.090 align:middle line:90% an All-Pay Auction, APA. 01:08:58.090 --> 01:09:01.010 align:middle line:84% So in an all-pay auction, the allocation rule 01:09:01.010 --> 01:09:03.270 align:middle line:84% is the same as the first- and second-price auction. 01:09:03.270 --> 01:09:06.689 align:middle line:84% We allocate the good to the highest bidder. 01:09:06.689 --> 01:09:10.510 align:middle line:84% So we allocate to highest bidder. 01:09:10.510 --> 01:09:16.210 align:middle line:90% 01:09:16.210 --> 01:09:19.350 align:middle line:84% But everyone has to pay their bids, whether they win or lose. 01:09:19.350 --> 01:09:28.457 align:middle line:90% 01:09:28.457 --> 01:09:30.290 align:middle line:84% So I think it should be quite intuitive here 01:09:30.290 --> 01:09:32.649 align:middle line:84% that you don't want to bid very much in this auction 01:09:32.649 --> 01:09:35.609 align:middle line:84% because you know you have to pay your bid whether or not you win. 01:09:35.609 --> 01:09:40.850 align:middle line:84% So intuitively, people will bid less. 01:09:40.850 --> 01:09:43.390 align:middle line:90% But we might guess-- 01:09:43.390 --> 01:09:46.410 align:middle line:90% 01:09:46.410 --> 01:09:53.323 align:middle line:84% so let's guess that there's a symmetric equilibrium 01:09:53.323 --> 01:09:53.990 align:middle line:90% of this auction. 01:09:53.990 --> 01:09:57.050 align:middle line:84% Or let's look for a symmetric equilibrium of this auction. 01:09:57.050 --> 01:09:58.590 align:middle line:90% So a symmetric equilibrium. 01:09:58.590 --> 01:10:04.930 align:middle line:90% 01:10:04.930 --> 01:10:06.770 align:middle line:84% Remember, a symmetric equilibrium 01:10:06.770 --> 01:10:09.510 align:middle line:84% means both players are using the same bidding function. 01:10:09.510 --> 01:10:11.490 align:middle line:84% So a symmetric equilibrium means we're 01:10:11.490 --> 01:10:20.250 align:middle line:84% looking for a function beta from 0, 1 to maybe 0 to infinity. 01:10:20.250 --> 01:10:22.450 align:middle line:84% So we're looking for a symmetric equilibrium 01:10:22.450 --> 01:10:25.410 align:middle line:84% where both players follow the bidding strategy beta that 01:10:25.410 --> 01:10:27.290 align:middle line:84% says, if my valuation is this, I'm going 01:10:27.290 --> 01:10:29.690 align:middle line:90% to bid beta of my valuation. 01:10:29.690 --> 01:10:32.610 align:middle line:84% So this is what we're trying to solve for. 01:10:32.610 --> 01:10:33.870 align:middle line:90% I'll go over here. 01:10:33.870 --> 01:10:47.770 align:middle line:90% 01:10:47.770 --> 01:10:50.138 align:middle line:84% And we're going to guess a symmetric equilibrium, 01:10:50.138 --> 01:10:51.430 align:middle line:90% but with a few more properties. 01:10:51.430 --> 01:10:53.805 align:middle line:84% I think we're going to guess that there's an equilibrium, 01:10:53.805 --> 01:10:58.470 align:middle line:84% and also, beta of 0 equals 0, and beta is strictly increasing. 01:10:58.470 --> 01:11:03.370 align:middle line:90% 01:11:03.370 --> 01:11:06.450 align:middle line:84% These seem pretty reasonable guesses. 01:11:06.450 --> 01:11:09.150 align:middle line:84% So we think, because the environment's symmetric, 01:11:09.150 --> 01:11:11.410 align:middle line:84% maybe people use symmetric bidding functions. 01:11:11.410 --> 01:11:14.330 align:middle line:84% It seems to make sense that if your value is higher, 01:11:14.330 --> 01:11:15.450 align:middle line:90% you should bid more. 01:11:15.450 --> 01:11:16.730 align:middle line:90% That makes sense. 01:11:16.730 --> 01:11:20.070 align:middle line:84% And why do we expect beta of 0 to be 0? 01:11:20.070 --> 01:11:23.370 align:middle line:90% 01:11:23.370 --> 01:11:23.890 align:middle line:90% Yeah. 01:11:23.890 --> 01:11:26.090 align:middle line:84% AUDIENCE: You wouldn't want to bid anything positive 01:11:26.090 --> 01:11:28.022 align:middle line:84% if the item isn't worth anything. 01:11:28.022 --> 01:11:30.230 align:middle line:84% IAN BALL: Especially because it's an all-pay auction. 01:11:30.230 --> 01:11:31.750 align:middle line:84% I mean, that's always true, but in all-pay auction, 01:11:31.750 --> 01:11:33.810 align:middle line:84% if you bid something positive, what happens? 01:11:33.810 --> 01:11:34.630 align:middle line:90% You have to pay. 01:11:34.630 --> 01:11:35.830 align:middle line:84% So you're paying a positive amount. 01:11:35.830 --> 01:11:37.950 align:middle line:84% And even if you get the good, you don't value it. 01:11:37.950 --> 01:11:40.650 align:middle line:84% So these seem like pretty safe guesses. 01:11:40.650 --> 01:11:43.850 align:middle line:84% But let's say there is an equilibrium with this property. 01:11:43.850 --> 01:11:46.850 align:middle line:84% Will our revenue equivalent assumptions be satisfied? 01:11:46.850 --> 01:11:49.290 align:middle line:84% So let's check if there's an equilibrium of this property, 01:11:49.290 --> 01:12:01.750 align:middle line:84% what is the expected payment by someone whose valuation is 0? 01:12:01.750 --> 01:12:03.450 align:middle line:90% What's it going to be? 01:12:03.450 --> 01:12:04.790 align:middle line:90% Just 0. 01:12:04.790 --> 01:12:07.350 align:middle line:84% If there's an equilibrium, where, if my value is 0, 01:12:07.350 --> 01:12:11.090 align:middle line:84% I bid nothing, well, then I'm going to pay nothing. 01:12:11.090 --> 01:12:13.250 align:middle line:90% It's going to be 0. 01:12:13.250 --> 01:12:17.690 align:middle line:84% And then if we're both using the same strictly increasing bidding 01:12:17.690 --> 01:12:20.810 align:middle line:84% function, well, which bidder is the good going 01:12:20.810 --> 01:12:21.830 align:middle line:90% to be allocated to? 01:12:21.830 --> 01:12:27.350 align:middle line:90% 01:12:27.350 --> 01:12:27.850 align:middle line:90% Yeah. 01:12:27.850 --> 01:12:29.150 align:middle line:84% AUDIENCE: Whoever has a greater amount here. 01:12:29.150 --> 01:12:30.610 align:middle line:84% IAN BALL: It's going to go to whichever bidder has the highest 01:12:30.610 --> 01:12:32.902 align:middle line:84% value because it's going to go to whichever bidder bids 01:12:32.902 --> 01:12:35.380 align:middle line:84% the most, but we're both using the same increasing bidding 01:12:35.380 --> 01:12:36.160 align:middle line:90% strategy. 01:12:36.160 --> 01:12:37.120 align:middle line:90% So who's going to bid the most? 01:12:37.120 --> 01:12:38.860 align:middle line:84% It's going to be the person who has the higher value. 01:12:38.860 --> 01:12:40.400 align:middle line:90% So it's going to be allocated-- 01:12:40.400 --> 01:12:54.260 align:middle line:90% 01:12:54.260 --> 01:12:57.700 align:middle line:84% So the two conditions we needed to get the revenue formula 01:12:57.700 --> 01:13:00.020 align:middle line:90% to simplify are satisfied. 01:13:00.020 --> 01:13:02.300 align:middle line:84% And that means we immediately know 01:13:02.300 --> 01:13:04.740 align:middle line:84% what the expected payment of each bidder has to be. 01:13:04.740 --> 01:13:06.540 align:middle line:90% So what does that tell us? 01:13:06.540 --> 01:13:08.840 align:middle line:84% Well, what is-- let's just focus on bidder 1. 01:13:08.840 --> 01:13:10.540 align:middle line:84% It'll be the same for everyone else. 01:13:10.540 --> 01:13:14.637 align:middle line:90% What is T1 hat of v1 then? 01:13:14.637 --> 01:13:15.720 align:middle line:90% Well, let's go through it. 01:13:15.720 --> 01:13:18.140 align:middle line:90% It's the integral. 01:13:18.140 --> 01:13:19.680 align:middle line:90% Well, what is v lower bar? 01:13:19.680 --> 01:13:27.100 align:middle line:84% It's just 0, 0 to v1 of x g of x dx. 01:13:27.100 --> 01:13:28.280 align:middle line:90% What is g of x here? 01:13:28.280 --> 01:13:36.668 align:middle line:90% 01:13:36.668 --> 01:13:37.383 align:middle line:90% AUDIENCE: 1. 01:13:37.383 --> 01:13:38.300 align:middle line:90% IAN BALL: It's just 1. 01:13:38.300 --> 01:13:41.260 align:middle line:84% Remember g equals f if there's two bidders. 01:13:41.260 --> 01:13:43.860 align:middle line:84% And if they're uniform, g is just 1. 01:13:43.860 --> 01:13:45.900 align:middle line:90% So why? 01:13:45.900 --> 01:13:49.940 align:middle line:84% G is the density of the highest valuation among my opponents. 01:13:49.940 --> 01:13:51.040 align:middle line:90% There's only one of them. 01:13:51.040 --> 01:13:52.800 align:middle line:84% So it's just the density of the uniform distribution, 01:13:52.800 --> 01:13:53.580 align:middle line:90% which is 1. 01:13:53.580 --> 01:13:54.560 align:middle line:90% So what do we get? 01:13:54.560 --> 01:13:57.140 align:middle line:90% 01:13:57.140 --> 01:14:01.540 align:middle line:84% Same formula that we always got before-- 01:14:01.540 --> 01:14:03.180 align:middle line:90% v1 squared over 2. 01:14:03.180 --> 01:14:04.980 align:middle line:84% Notice, this is exactly what we got way 01:14:04.980 --> 01:14:06.240 align:middle line:90% at the beginning of class. 01:14:06.240 --> 01:14:08.940 align:middle line:84% So in general, this is always the formula 01:14:08.940 --> 01:14:11.220 align:middle line:84% for the expected payment by a bidder with valuation 01:14:11.220 --> 01:14:16.000 align:middle line:84% v1 in our simple two-bidder uniform distribution case 01:14:16.000 --> 01:14:17.900 align:middle line:84% as long as our assumptions are satisfied. 01:14:17.900 --> 01:14:19.360 align:middle line:90% But now I claim that we're done. 01:14:19.360 --> 01:14:22.400 align:middle line:84% What must the bidding function be? 01:14:22.400 --> 01:14:27.660 align:middle line:90% 01:14:27.660 --> 01:14:30.020 align:middle line:84% So let's step back and see what we've done. 01:14:30.020 --> 01:14:33.100 align:middle line:84% We've guessed that this is the bidding function beta. 01:14:33.100 --> 01:14:36.060 align:middle line:84% We've said that if this is the bidding function beta, then 01:14:36.060 --> 01:14:39.860 align:middle line:84% the expected payment by bidder 1 when the valuation is v1 01:14:39.860 --> 01:14:42.260 align:middle line:84% must exactly be v1 squared over 2. 01:14:42.260 --> 01:14:44.740 align:middle line:90% Yeah. 01:14:44.740 --> 01:14:46.243 align:middle line:90% AUDIENCE: Is it 1/2? 01:14:46.243 --> 01:14:47.160 align:middle line:90% IAN BALL: Even easier. 01:14:47.160 --> 01:14:50.540 align:middle line:90% It's an all-pay auction. 01:14:50.540 --> 01:14:52.640 align:middle line:84% So another way of writing the expected payment-- 01:14:52.640 --> 01:14:58.100 align:middle line:90% 01:14:58.100 --> 01:14:59.820 align:middle line:90% beta v1. 01:14:59.820 --> 01:15:02.220 align:middle line:84% If it's an all-pay auction, my expected payment 01:15:02.220 --> 01:15:04.780 align:middle line:84% is my bid because I always pay my bid. 01:15:04.780 --> 01:15:07.580 align:middle line:84% So of course, that's my expected payment. 01:15:07.580 --> 01:15:09.080 align:middle line:90% If I bid 2, I always pay 2. 01:15:09.080 --> 01:15:10.480 align:middle line:90% If I bid 3, I always pay 3. 01:15:10.480 --> 01:15:11.680 align:middle line:90% That's my expected payment. 01:15:11.680 --> 01:15:14.980 align:middle line:90% 01:15:14.980 --> 01:15:16.000 align:middle line:90% So what's the answer? 01:15:16.000 --> 01:15:18.855 align:middle line:90% 01:15:18.855 --> 01:15:19.980 align:middle line:90% AUDIENCE: j squared over 2. 01:15:19.980 --> 01:15:21.480 align:middle line:84% IAN BALL: That's it, and we're done. 01:15:21.480 --> 01:15:23.220 align:middle line:90% So we've now found-- 01:15:23.220 --> 01:15:25.240 align:middle line:84% I mean, technically we should verify this. 01:15:25.240 --> 01:15:28.000 align:middle line:84% We've only said, this is a necessary condition, 01:15:28.000 --> 01:15:28.890 align:middle line:90% but it is true. 01:15:28.890 --> 01:15:32.470 align:middle line:90% So you can check if it's right. 01:15:32.470 --> 01:15:37.790 align:middle line:90% Actually, it's v. We're done. 01:15:37.790 --> 01:15:41.270 align:middle line:84% So revenue equivalence can be a convenient way 01:15:41.270 --> 01:15:43.850 align:middle line:84% of actually solving for an equilibrium 01:15:43.850 --> 01:15:45.350 align:middle line:84% because revenue equivalence gives us 01:15:45.350 --> 01:15:47.550 align:middle line:84% a very simple formula for the expected payment 01:15:47.550 --> 01:15:49.470 align:middle line:90% that each player makes. 01:15:49.470 --> 01:15:51.150 align:middle line:84% This is from revenue equivalence. 01:15:51.150 --> 01:15:53.230 align:middle line:84% The auction format gives us a relationship 01:15:53.230 --> 01:15:55.630 align:middle line:84% between the expected payment and the bid. 01:15:55.630 --> 01:15:57.970 align:middle line:84% In the all-pay auction, it's a very simple relationship. 01:15:57.970 --> 01:16:00.510 align:middle line:84% The expected payment I make is exactly my bid. 01:16:00.510 --> 01:16:03.210 align:middle line:84% But now if we have two different formulas, 01:16:03.210 --> 01:16:05.950 align:middle line:84% two different expressions for T1 hat of v1, 01:16:05.950 --> 01:16:08.230 align:middle line:90% they must both be equal. 01:16:08.230 --> 01:16:09.170 align:middle line:90% And now we have this. 01:16:09.170 --> 01:16:12.830 align:middle line:90% 01:16:12.830 --> 01:16:14.430 align:middle line:84% Maybe just one more example to see 01:16:14.430 --> 01:16:21.710 align:middle line:84% how we could derive the first-price auction 01:16:21.710 --> 01:16:23.090 align:middle line:90% equilibrium using this. 01:16:23.090 --> 01:16:30.407 align:middle line:90% 01:16:30.407 --> 01:16:32.990 align:middle line:84% Basically, we're going to do the reverse of what we did before 01:16:32.990 --> 01:16:35.070 align:middle line:90% at the beginning of class. 01:16:35.070 --> 01:16:37.130 align:middle line:84% Let's consider a first-price auction, again, 01:16:37.130 --> 01:16:39.870 align:middle line:90% in our simple example-- 01:16:39.870 --> 01:16:46.530 align:middle line:84% in our simple two-bidder uniform example. 01:16:46.530 --> 01:16:52.190 align:middle line:90% 01:16:52.190 --> 01:16:56.990 align:middle line:84% Well, again, if we look for a symmetric equilibrium 01:16:56.990 --> 01:17:00.670 align:middle line:84% where if your value is 0, you bid nothing and beta is strictly 01:17:00.670 --> 01:17:02.510 align:middle line:84% increasing, then our two assumptions 01:17:02.510 --> 01:17:04.470 align:middle line:84% of revenue equivalence are going to hold. 01:17:04.470 --> 01:17:07.390 align:middle line:90% And therefore, we know-- 01:17:07.390 --> 01:17:12.790 align:middle line:84% so revenue equivalence tells us that T1 hat of v1 01:17:12.790 --> 01:17:14.860 align:middle line:90% equals v1 squared over 2. 01:17:14.860 --> 01:17:17.110 align:middle line:84% That's still going to be true as long as the good goes 01:17:17.110 --> 01:17:18.790 align:middle line:84% to the bidder who values it most, 01:17:18.790 --> 01:17:21.830 align:middle line:84% and the bidder with 0 value pays nothing. 01:17:21.830 --> 01:17:30.610 align:middle line:84% But now from the auction format, what is T1 hat of v1? 01:17:30.610 --> 01:17:33.670 align:middle line:90% 01:17:33.670 --> 01:17:34.170 align:middle line:90% Yeah. 01:17:34.170 --> 01:17:38.638 align:middle line:84% AUDIENCE: That's the probability of winning times the-- 01:17:38.638 --> 01:17:40.430 align:middle line:84% IAN BALL: Exactly, so this should be my bid 01:17:40.430 --> 01:17:45.950 align:middle line:84% is beta of v1 times the probability that I win, 01:17:45.950 --> 01:17:46.930 align:middle line:90% which is v1. 01:17:46.930 --> 01:17:50.070 align:middle line:90% 01:17:50.070 --> 01:17:51.550 align:middle line:90% So now we set these equal. 01:17:51.550 --> 01:17:52.730 align:middle line:90% And indeed, we get-- 01:17:52.730 --> 01:17:57.190 align:middle line:90% 01:17:57.190 --> 01:17:58.490 align:middle line:90% which we already derived. 01:17:58.490 --> 01:18:00.755 align:middle line:90% So we can go back and forth. 01:18:00.755 --> 01:18:02.130 align:middle line:84% We can solve for the equilibrium. 01:18:02.130 --> 01:18:04.130 align:middle line:84% We can see that revenue equivalence is satisfied. 01:18:04.130 --> 01:18:05.922 align:middle line:84% Or we can jump right to revenue equivalence 01:18:05.922 --> 01:18:08.210 align:middle line:84% and use that together with the rules of the auction 01:18:08.210 --> 01:18:10.430 align:middle line:84% to try to derive what the equilibrium bidding 01:18:10.430 --> 01:18:11.650 align:middle line:90% behavior will be. 01:18:11.650 --> 01:18:13.190 align:middle line:84% And this is useful when we confront 01:18:13.190 --> 01:18:15.270 align:middle line:84% new auctions we haven't looked at before because 01:18:15.270 --> 01:18:17.670 align:middle line:84% of how general revenue equivalence is. 01:18:17.670 --> 01:18:21.830 align:middle line:84% I want to end with maybe one caveat about revenue 01:18:21.830 --> 01:18:22.850 align:middle line:90% equivalence, though. 01:18:22.850 --> 01:18:28.070 align:middle line:90% 01:18:28.070 --> 01:18:31.830 align:middle line:84% And revenue equivalence is often misinterpreted. 01:18:31.830 --> 01:18:33.810 align:middle line:84% When people interpret revenue equivalence, 01:18:33.810 --> 01:18:35.790 align:middle line:84% they often say it's about the equivalence between auction 01:18:35.790 --> 01:18:36.410 align:middle line:90% formats. 01:18:36.410 --> 01:18:38.430 align:middle line:84% They say something like, the first price-auction 01:18:38.430 --> 01:18:40.970 align:middle line:84% generates the same revenue as the second-price auction. 01:18:40.970 --> 01:18:42.550 align:middle line:84% But if we want to be really precise, 01:18:42.550 --> 01:18:44.590 align:middle line:84% revenue equivalence is a comparison 01:18:44.590 --> 01:18:46.630 align:middle line:84% between an equilibrium of one auction 01:18:46.630 --> 01:18:48.450 align:middle line:84% and an equilibrium of another auction. 01:18:48.450 --> 01:18:51.670 align:middle line:90% So a caveat, it's not about-- 01:18:51.670 --> 01:18:58.670 align:middle line:84% so it's not about the auction per se. 01:18:58.670 --> 01:19:01.690 align:middle line:84% It's about a particular equilibrium of the auction. 01:19:01.690 --> 01:19:11.790 align:middle line:90% 01:19:11.790 --> 01:19:14.450 align:middle line:84% Maybe I'll say BNE of the auction. 01:19:14.450 --> 01:19:19.750 align:middle line:90% 01:19:19.750 --> 01:19:23.400 align:middle line:84% And in fact-- we'll give some examples-- in a given auction 01:19:23.400 --> 01:19:25.840 align:middle line:84% format, you could have multiple equilibria 01:19:25.840 --> 01:19:28.198 align:middle line:84% that generate different revenue for the auctioneer 01:19:28.198 --> 01:19:29.740 align:middle line:84% if those conditions aren't satisfied. 01:19:29.740 --> 01:19:33.480 align:middle line:84% So let's make sure we understand these conditions 01:19:33.480 --> 01:19:37.440 align:middle line:84% that the bidder with the lowest value pays nothing, 01:19:37.440 --> 01:19:40.280 align:middle line:84% and the good is allocated to the bidder with the highest value. 01:19:40.280 --> 01:19:43.040 align:middle line:84% These are not properties simply of the auction format. 01:19:43.040 --> 01:19:45.440 align:middle line:84% They're properties of the equilibrium 01:19:45.440 --> 01:19:47.160 align:middle line:84% together with the auction format. 01:19:47.160 --> 01:19:49.780 align:middle line:84% If everyone used a different bidding strategy, 01:19:49.780 --> 01:19:51.813 align:middle line:90% then this may no longer be true. 01:19:51.813 --> 01:19:53.480 align:middle line:84% So these assumptions are not assumptions 01:19:53.480 --> 01:19:54.980 align:middle line:90% about the rules of the auction. 01:19:54.980 --> 01:19:57.680 align:middle line:84% They're assumptions about a particular equilibrium 01:19:57.680 --> 01:19:58.740 align:middle line:90% of the auction. 01:19:58.740 --> 01:20:00.620 align:middle line:84% And whenever we apply revenue equivalence, 01:20:00.620 --> 01:20:02.912 align:middle line:84% we're not just comparing the revenue from two auctions. 01:20:02.912 --> 01:20:05.120 align:middle line:84% We're saying, let's look at equilibrium of one 01:20:05.120 --> 01:20:07.940 align:middle line:84% of this auction and equilibrium two of auction two, 01:20:07.940 --> 01:20:10.720 align:middle line:84% and let's compare the revenue there. 01:20:10.720 --> 01:20:12.100 align:middle line:84% So that's kind of a final caveat. 01:20:12.100 --> 01:20:14.548 align:middle line:84% And we'll see some implications of that next week. 01:20:14.548 --> 01:20:15.340 align:middle line:90% So I'll stop there. 01:20:15.340 --> 01:20:17.030 align:middle line:90% Thank you. 01:20:17.030 --> 01:20:20.000 align:middle line:90%