WEBVTT 00:00:00.000 --> 00:00:02.444 align:middle line:90% [SQUEAKING] 00:00:02.444 --> 00:00:04.388 align:middle line:90% [RUSTLING] 00:00:04.388 --> 00:00:06.332 align:middle line:90% [CLICKING] 00:00:06.332 --> 00:00:10.827 align:middle line:90% 00:00:10.827 --> 00:00:13.160 align:middle line:84% IAN BALL: So today, I want to say just a little bit more 00:00:13.160 --> 00:00:15.840 align:middle line:84% about revenue equivalence, or the revenue formula 00:00:15.840 --> 00:00:16.700 align:middle line:90% that we wrote down. 00:00:16.700 --> 00:00:19.920 align:middle line:84% And I think this will be useful for the problem set and the quiz 00:00:19.920 --> 00:00:21.400 align:middle line:90% on Friday. 00:00:21.400 --> 00:00:23.880 align:middle line:84% And then I'm going to talk about an application of auctions 00:00:23.880 --> 00:00:27.140 align:middle line:84% to display advertising auctions, like the kinds of auctions-- 00:00:27.140 --> 00:00:31.009 align:middle line:84% the keyword auctions that Google does commonly today. 00:00:31.009 --> 00:00:32.759 align:middle line:84% Just to remind you of this revenue formula 00:00:32.759 --> 00:00:36.520 align:middle line:84% that we computed, we were looking at a single item 00:00:36.520 --> 00:00:37.520 align:middle line:90% auction. 00:00:37.520 --> 00:00:41.080 align:middle line:84% And we had this formula-- maybe I'll write it down here-- 00:00:41.080 --> 00:00:44.520 align:middle line:84% where we said Ti hat of Vi, which 00:00:44.520 --> 00:00:47.360 align:middle line:84% was the equilibrium expected payment 00:00:47.360 --> 00:00:54.980 align:middle line:84% by a bidder with valuation Vi, was equal to this. 00:00:54.980 --> 00:01:01.602 align:middle line:90% 00:01:01.602 --> 00:01:04.360 align:middle line:84% So we said the expected payment of a bidder with valuation 00:01:04.360 --> 00:01:08.040 align:middle line:84% Vi in this equilibrium is equal to the expected payment 00:01:08.040 --> 00:01:12.320 align:middle line:84% of bidder i when she has the lowest possible valuation 00:01:12.320 --> 00:01:16.400 align:middle line:84% plus this integral of maybe this mysterious term of x 00:01:16.400 --> 00:01:19.920 align:middle line:84% times the derivative of this function, q hat-- 00:01:19.920 --> 00:01:21.600 align:middle line:84% where, remember, this is the derivative 00:01:21.600 --> 00:01:25.620 align:middle line:90% of the allocation probability. 00:01:25.620 --> 00:01:33.000 align:middle line:90% 00:01:33.000 --> 00:01:36.160 align:middle line:84% So I just want to highlight a few assumptions that 00:01:36.160 --> 00:01:37.575 align:middle line:84% were required to get this result. 00:01:37.575 --> 00:01:39.200 align:middle line:84% And the problem set is going to explore 00:01:39.200 --> 00:01:41.242 align:middle line:84% some ways this assumption, this result can break. 00:01:41.242 --> 00:01:41.760 align:middle line:90% Yes. 00:01:41.760 --> 00:01:43.080 align:middle line:84% AUDIENCE: Is there a reason why the lower 00:01:43.080 --> 00:01:44.160 align:middle line:90% bound is on the integral? 00:01:44.160 --> 00:01:44.910 align:middle line:90% IAN BALL: Ah, yes. 00:01:44.910 --> 00:01:46.840 align:middle line:84% That should be V lower bar, sorry. 00:01:46.840 --> 00:01:49.000 align:middle line:90% Yeah, thanks. 00:01:49.000 --> 00:01:52.440 align:middle line:84% So a few assumptions we made-- one was the assumption 00:01:52.440 --> 00:01:59.910 align:middle line:90% of independent private values. 00:01:59.910 --> 00:02:03.730 align:middle line:90% 00:02:03.730 --> 00:02:05.530 align:middle line:84% Statistical independence, it says 00:02:05.530 --> 00:02:07.410 align:middle line:84% one bidder's valuation is statistically 00:02:07.410 --> 00:02:10.490 align:middle line:84% independent of another bidder's valuation. 00:02:10.490 --> 00:02:13.710 align:middle line:84% Private values-- private is a confusing term. 00:02:13.710 --> 00:02:20.010 align:middle line:84% I think maybe known values is maybe a more descriptive term. 00:02:20.010 --> 00:02:22.090 align:middle line:84% It means that each bidder knows exactly how 00:02:22.090 --> 00:02:23.450 align:middle line:90% much they value the good. 00:02:23.450 --> 00:02:25.490 align:middle line:84% Information that another bidder has 00:02:25.490 --> 00:02:28.043 align:middle line:84% is not relevant to my valuation of the item. 00:02:28.043 --> 00:02:29.710 align:middle line:84% And I still think that's a bit abstract, 00:02:29.710 --> 00:02:30.730 align:middle line:84% so I'm going to go over an example 00:02:30.730 --> 00:02:32.450 align:middle line:84% where this fails in a second, and I 00:02:32.450 --> 00:02:36.050 align:middle line:84% think that should clarify the role of this assumption. 00:02:36.050 --> 00:02:37.690 align:middle line:90% The other assumption we made-- 00:02:37.690 --> 00:02:41.350 align:middle line:84% notice we took an integral here over some range of values. 00:02:41.350 --> 00:02:45.050 align:middle line:84% So another important assumption is that the values-- 00:02:45.050 --> 00:02:46.870 align:middle line:84% maybe I'll say live in an interval. 00:02:46.870 --> 00:02:51.490 align:middle line:90% 00:02:51.490 --> 00:02:53.450 align:middle line:84% So if instead a player's valuation 00:02:53.450 --> 00:02:56.190 align:middle line:84% could only be, say, $1 or $2-- it's not an interval, 00:02:56.190 --> 00:02:57.470 align:middle line:90% it's just 2 points. 00:02:57.470 --> 00:03:00.350 align:middle line:84% Then we wouldn't be able to take an integral between $1 and $2, 00:03:00.350 --> 00:03:02.190 align:middle line:84% and this formula would no longer hold. 00:03:02.190 --> 00:03:05.570 align:middle line:84% And the problem set will show you an example of this. 00:03:05.570 --> 00:03:08.290 align:middle line:84% One other maybe more technical assumption you might notice 00:03:08.290 --> 00:03:12.170 align:middle line:84% is that, well, for this to make sense, this function, q hat, 00:03:12.170 --> 00:03:13.230 align:middle line:90% has to be well behaved. 00:03:13.230 --> 00:03:16.610 align:middle line:84% We have to be able to take the derivative of Q hat. 00:03:16.610 --> 00:03:18.530 align:middle line:84% That's a more technical thing, so I'll just 00:03:18.530 --> 00:03:25.070 align:middle line:84% say technical Qi hat is differentiable. 00:03:25.070 --> 00:03:29.440 align:middle line:90% 00:03:29.440 --> 00:03:31.190 align:middle line:84% I call this technical because it turns out 00:03:31.190 --> 00:03:32.890 align:middle line:84% if you use a more involved proof, 00:03:32.890 --> 00:03:35.210 align:middle line:84% you can get a substantially similar result 00:03:35.210 --> 00:03:36.272 align:middle line:90% without this assumption. 00:03:36.272 --> 00:03:38.730 align:middle line:84% So that's not something we're going to cover in the course, 00:03:38.730 --> 00:03:43.090 align:middle line:84% but I'll just highlight that our proof used differentiability 00:03:43.090 --> 00:03:45.730 align:middle line:84% of q hat, but that's not really necessary for the result. 00:03:45.730 --> 00:03:48.170 align:middle line:84% But these assumptions really are substantive and important 00:03:48.170 --> 00:03:51.730 align:middle line:90% for this revenue formula. 00:03:51.730 --> 00:03:55.620 align:middle line:84% So to understand what private values mean, 00:03:55.620 --> 00:03:59.940 align:middle line:84% let's look at a classic example without private values, which 00:03:59.940 --> 00:04:02.300 align:middle line:90% is called a wallet auction. 00:04:02.300 --> 00:04:04.640 align:middle line:84% I don't think anyone actually does auctions like this, 00:04:04.640 --> 00:04:06.130 align:middle line:90% but it's a good illustration. 00:04:06.130 --> 00:04:07.880 align:middle line:84% So I think there's a few stories for this. 00:04:07.880 --> 00:04:11.140 align:middle line:84% The basic story is that I guess this happened at a bar. 00:04:11.140 --> 00:04:12.600 align:middle line:90% You have two people at a bar. 00:04:12.600 --> 00:04:14.975 align:middle line:84% They each get out their wallet, and they put their wallet 00:04:14.975 --> 00:04:18.060 align:middle line:84% in a bowl, say, or in a pile on the table. 00:04:18.060 --> 00:04:24.100 align:middle line:84% So you have player one's wallet and player two's wallet. 00:04:24.100 --> 00:04:29.500 align:middle line:84% And then the players bid for the union of the two wallets. 00:04:29.500 --> 00:04:32.420 align:middle line:84% So we have both our wallets together in a bowl. 00:04:32.420 --> 00:04:34.100 align:middle line:84% We're now each going to submit a bid. 00:04:34.100 --> 00:04:38.540 align:middle line:84% Whoever bids more gets the bowl, but has to pay that amount. 00:04:38.540 --> 00:04:40.520 align:middle line:84% Now, what's special about this auction? 00:04:40.520 --> 00:04:43.420 align:middle line:84% The story maybe is far-fetched, but the important thing 00:04:43.420 --> 00:04:46.460 align:middle line:84% is I know as player one what's in my wallet, 00:04:46.460 --> 00:04:48.803 align:middle line:84% but I don't know what's in the other player's wallet. 00:04:48.803 --> 00:04:50.720 align:middle line:84% The other player knows what's in their wallet, 00:04:50.720 --> 00:04:52.387 align:middle line:84% but they don't know what's in my wallet. 00:04:52.387 --> 00:04:55.460 align:middle line:84% So maybe we'll call the amount in my wallet theta 1 00:04:55.460 --> 00:04:58.220 align:middle line:84% and the amount in the other person's wallet theta 2. 00:04:58.220 --> 00:05:02.980 align:middle line:84% And just to make it simple, let's say theta 1 and theta 2-- 00:05:02.980 --> 00:05:05.800 align:middle line:84% just to clarify exactly which assumption we're relaxing, 00:05:05.800 --> 00:05:11.580 align:middle line:84% let's say theta 1 and theta 2 are independent and uniform. 00:05:11.580 --> 00:05:17.100 align:middle line:84% Independent and uniform over 0, 1. 00:05:17.100 --> 00:05:18.280 align:middle line:90% That's just a normalization. 00:05:18.280 --> 00:05:21.500 align:middle line:84% One could be $1 billion or something. 00:05:21.500 --> 00:05:31.100 align:middle line:84% But the key point is that their 1 knows theta 1 but not theta 2 00:05:31.100 --> 00:05:33.600 align:middle line:84% because they don't know what's in the other person's wallet. 00:05:33.600 --> 00:05:36.860 align:middle line:84% So if I'm bidder one, I say, I know my wallet, say, has 0.3. 00:05:36.860 --> 00:05:39.700 align:middle line:84% But all for all I know, the other person's wallet 00:05:39.700 --> 00:05:41.525 align:middle line:84% is uniformly distributed between 0 and 1. 00:05:41.525 --> 00:05:42.900 align:middle line:84% That's all I know because I can't 00:05:42.900 --> 00:05:44.220 align:middle line:90% see what's in their wallet. 00:05:44.220 --> 00:05:55.150 align:middle line:84% And symmetrically, bidder two knows theta 2, but not theta 1. 00:05:55.150 --> 00:05:57.650 align:middle line:84% Now, why is this not a setting of private values? 00:05:57.650 --> 00:05:59.150 align:middle line:90% Well, I do know something. 00:05:59.150 --> 00:06:00.370 align:middle line:90% I know theta 1. 00:06:00.370 --> 00:06:01.750 align:middle line:90% I know it's in my wallet. 00:06:01.750 --> 00:06:05.790 align:middle line:84% But I don't know the joint value of our two wallets. 00:06:05.790 --> 00:06:09.110 align:middle line:84% And more specifically, something the other player knows 00:06:09.110 --> 00:06:13.830 align:middle line:84% is relevant for my valuation of the bowl, of the item. 00:06:13.830 --> 00:06:15.730 align:middle line:84% So what makes it not private values? 00:06:15.730 --> 00:06:18.850 align:middle line:84% So maybe I'll say here, it's a failure of private values. 00:06:18.850 --> 00:06:24.930 align:middle line:90% 00:06:24.930 --> 00:06:25.430 align:middle line:90% Why? 00:06:25.430 --> 00:06:28.150 align:middle line:84% Because my willingness to pay, my valuation 00:06:28.150 --> 00:06:30.690 align:middle line:84% depends on information that another player has-- 00:06:30.690 --> 00:06:36.710 align:middle line:84% or maybe more precisely, another player and other players 00:06:36.710 --> 00:06:41.462 align:middle line:84% information influences or affects my valuation 00:06:41.462 --> 00:06:43.670 align:middle line:84% for the item, where here, the item is the combination 00:06:43.670 --> 00:06:44.730 align:middle line:90% of the two wallets. 00:06:44.730 --> 00:06:48.270 align:middle line:90% 00:06:48.270 --> 00:06:49.970 align:middle line:84% Maybe I'll say my willingness to pay. 00:06:49.970 --> 00:06:55.488 align:middle line:90% 00:06:55.488 --> 00:06:57.030 align:middle line:84% If the other player's wallet contains 00:06:57.030 --> 00:06:59.750 align:middle line:84% $1, which only the other player knows, 00:06:59.750 --> 00:07:02.350 align:middle line:84% I'd be willing to pay more for this bowl that 00:07:02.350 --> 00:07:03.770 align:middle line:90% contains both of our wallets. 00:07:03.770 --> 00:07:06.390 align:middle line:90% 00:07:06.390 --> 00:07:08.370 align:middle line:90% Now, you might say, well, OK. 00:07:08.370 --> 00:07:11.290 align:middle line:84% I don't know exactly what's in the other wallet, 00:07:11.290 --> 00:07:13.010 align:middle line:90% but I can compute expectations. 00:07:13.010 --> 00:07:15.890 align:middle line:84% So the naive approach might be, well, 00:07:15.890 --> 00:07:18.695 align:middle line:84% let's try to treat this like a normal private values auction. 00:07:18.695 --> 00:07:20.070 align:middle line:84% So here I'll write naive approach 00:07:20.070 --> 00:07:21.445 align:middle line:84% to say this is going to be wrong. 00:07:21.445 --> 00:07:24.310 align:middle line:90% But let's see what happens. 00:07:24.310 --> 00:07:28.190 align:middle line:84% To the naive approach, if I'm player one, 00:07:28.190 --> 00:07:33.390 align:middle line:84% what would I say is my willingness to pay for the bowl? 00:07:33.390 --> 00:07:36.387 align:middle line:84% If I just looked at it, I know theta 1. 00:07:36.387 --> 00:07:38.970 align:middle line:84% What's a reasonable way to say, what is my willingness to pay? 00:07:38.970 --> 00:07:42.800 align:middle line:84% What is my expected value from the union of the two wallets 00:07:42.800 --> 00:07:44.050 align:middle line:90% or the sum of the two wallets? 00:07:44.050 --> 00:07:48.230 align:middle line:90% 00:07:48.230 --> 00:07:49.200 align:middle line:90% Yeah. 00:07:49.200 --> 00:07:51.240 align:middle line:84% AUDIENCE: Since the other player's value 00:07:51.240 --> 00:07:53.420 align:middle line:84% is uniformly distributed between 0 and 1, 00:07:53.420 --> 00:07:56.520 align:middle line:84% the expected value of the other player's value is 0.5. 00:07:56.520 --> 00:07:59.660 align:middle line:84% So the naive approach would be to add 0.5 to [INAUDIBLE]. 00:07:59.660 --> 00:08:00.660 align:middle line:90% IAN BALL: Exactly right. 00:08:00.660 --> 00:08:03.040 align:middle line:84% So you might say, OK, this is just a normal auction where 00:08:03.040 --> 00:08:07.240 align:middle line:84% player one's value is theta 1 plus 0.5 or plus 1/2. 00:08:07.240 --> 00:08:13.960 align:middle line:84% And player two's value is theta 2 plus 1/2. 00:08:13.960 --> 00:08:16.920 align:middle line:84% So I might say, look, we have some item, 00:08:16.920 --> 00:08:18.740 align:middle line:84% and I have some private information. 00:08:18.740 --> 00:08:19.700 align:middle line:90% I know my theta 1. 00:08:19.700 --> 00:08:21.660 align:middle line:84% I use that to compute my valuation. 00:08:21.660 --> 00:08:23.280 align:middle line:84% And let's just go ahead like this. 00:08:23.280 --> 00:08:25.680 align:middle line:84% And what I want to show is that this auction 00:08:25.680 --> 00:08:27.800 align:middle line:84% is going to behave very differently 00:08:27.800 --> 00:08:30.640 align:middle line:84% from a private auction, where this is my value 00:08:30.640 --> 00:08:32.360 align:middle line:84% and this is the other player's value. 00:08:32.360 --> 00:08:35.159 align:middle line:90% So to see this, let's look at-- 00:08:35.159 --> 00:08:37.260 align:middle line:84% so fully solving this auction is quite involved, 00:08:37.260 --> 00:08:39.059 align:middle line:84% and I'll leave that to the notes. 00:08:39.059 --> 00:08:41.020 align:middle line:84% And we won't really cover that in this course. 00:08:41.020 --> 00:08:42.880 align:middle line:84% We will cover it in 1416 in the spring, 00:08:42.880 --> 00:08:44.088 align:middle line:90% if you're interested in that. 00:08:44.088 --> 00:08:47.760 align:middle line:84% But let's just imagine we did a second price auction. 00:08:47.760 --> 00:08:51.840 align:middle line:84% And let's imagine that we looked for an equilibrium 00:08:51.840 --> 00:08:55.340 align:middle line:84% where each player bids truthfully. 00:08:55.340 --> 00:09:03.840 align:middle line:84% So let's look at a strategy profile where player one bids-- 00:09:03.840 --> 00:09:09.760 align:middle line:84% we could say b1 of theta 1 plus theta 1 plus 1/2 would be 2 00:09:09.760 --> 00:09:14.920 align:middle line:84% of theta 2 equals theta 2, et cetera. 00:09:14.920 --> 00:09:19.480 align:middle line:84% Now, if this were a private values auction, then 00:09:19.480 --> 00:09:22.600 align:middle line:84% this strategy profile would be in equilibrium. 00:09:22.600 --> 00:09:25.000 align:middle line:84% We already argued that in a classical private value 00:09:25.000 --> 00:09:27.920 align:middle line:84% single item auction, it was an equilibrium for each player 00:09:27.920 --> 00:09:31.360 align:middle line:84% to bid truthfully, to bid their valuation for the item. 00:09:31.360 --> 00:09:33.600 align:middle line:90% And in fact, we argued more. 00:09:33.600 --> 00:09:36.880 align:middle line:84% We said however the other player bids, it's optimal for me 00:09:36.880 --> 00:09:40.800 align:middle line:84% to bid my valuation in a second price auction. 00:09:40.800 --> 00:09:44.040 align:middle line:84% So let's see what goes wrong in this case. 00:09:44.040 --> 00:09:49.890 align:middle line:84% So let's suppose that to make things simple, 00:09:49.890 --> 00:09:55.930 align:middle line:90% let's say theta 1 is 0.4. 00:09:55.930 --> 00:09:59.290 align:middle line:90% And theta 2 equals 0.3. 00:09:59.290 --> 00:10:08.410 align:middle line:84% So if that happens, then player 1 is supposed to bid 0.9. 00:10:08.410 --> 00:10:11.530 align:middle line:84% And player 2 is supposed to bid 0.80-- 00:10:11.530 --> 00:10:14.770 align:middle line:90% 0.5 plus 0.3. 00:10:14.770 --> 00:10:17.730 align:middle line:84% So this would be player two's bid. 00:10:17.730 --> 00:10:22.370 align:middle line:84% And this would be player one's bid. 00:10:22.370 --> 00:10:29.410 align:middle line:84% Now, how would our reasoning for why bidding b1 is optimal go? 00:10:29.410 --> 00:10:31.010 align:middle line:84% Well, if this were private values, 00:10:31.010 --> 00:10:35.210 align:middle line:84% we would say, OK, I'm bidder one. 00:10:35.210 --> 00:10:37.890 align:middle line:90% My valuation is 0.9. 00:10:37.890 --> 00:10:40.430 align:middle line:84% If I bid truthfully, what happens? 00:10:40.430 --> 00:10:43.610 align:middle line:84% Well, I get the item and I pay 0.8. 00:10:43.610 --> 00:10:48.450 align:middle line:84% And 0.8 is less than 0.9, so I'm doing well. 00:10:48.450 --> 00:10:51.370 align:middle line:84% And if I deviated, well, if I deviate 00:10:51.370 --> 00:10:53.978 align:middle line:84% to bid any amount more than 0.8, it's 00:10:53.978 --> 00:10:55.270 align:middle line:90% not going to make a difference. 00:10:55.270 --> 00:10:56.970 align:middle line:90% I'm still going to win. 00:10:56.970 --> 00:10:58.690 align:middle line:90% And I'm still going to play 0.8. 00:10:58.690 --> 00:11:00.850 align:middle line:84% And if I deviate to any amount less than 0.8, well, 00:11:00.850 --> 00:11:03.100 align:middle line:84% I'm going to lose, and I'm going to get a payoff of 0. 00:11:03.100 --> 00:11:08.190 align:middle line:84% So the naive argument might be, well, I'm bidding optimally. 00:11:08.190 --> 00:11:08.870 align:middle line:90% Why? 00:11:08.870 --> 00:11:13.210 align:middle line:90% Because 0.1 is greater than 0. 00:11:13.210 --> 00:11:15.430 align:middle line:84% And maybe I should explain where this comes from-- 00:11:15.430 --> 00:11:23.217 align:middle line:90% 0.9 minus 0.8. 00:11:23.217 --> 00:11:24.550 align:middle line:90% So it looks like I'm doing well. 00:11:24.550 --> 00:11:26.410 align:middle line:90% I get a payoff of 0.1. 00:11:26.410 --> 00:11:27.833 align:middle line:90% I could do no better than 0.1. 00:11:27.833 --> 00:11:28.750 align:middle line:90% I'm bidding optimally. 00:11:28.750 --> 00:11:30.650 align:middle line:84% That would be the private values intuition. 00:11:30.650 --> 00:11:32.210 align:middle line:90% What's wrong with this? 00:11:32.210 --> 00:11:35.130 align:middle line:90% So everything I said was wrong. 00:11:35.130 --> 00:11:35.730 align:middle line:90% Yeah. 00:11:35.730 --> 00:11:37.730 align:middle line:84% AUDIENCE: Your actual value is negative 0.1 00:11:37.730 --> 00:11:38.790 align:middle line:90% because the value is 0.7. 00:11:38.790 --> 00:11:42.040 align:middle line:84% When you're bidding 0.8, you're paying 0.9. 00:11:42.040 --> 00:11:43.040 align:middle line:90% IAN BALL: Exactly right. 00:11:43.040 --> 00:11:44.540 align:middle line:84% And this is getting at the key point 00:11:44.540 --> 00:11:47.900 align:middle line:84% that it's not really the case that my value is 00:11:47.900 --> 00:11:49.640 align:middle line:90% known to be 0.9. 00:11:49.640 --> 00:11:51.777 align:middle line:84% That was just my expected value without getting 00:11:51.777 --> 00:11:53.860 align:middle line:84% any additional information about the other player. 00:11:53.860 --> 00:11:56.980 align:middle line:84% The problem is if the other player bids 0.8, 00:11:56.980 --> 00:12:00.060 align:middle line:84% and the reason they bid 0.8 is that the value of their wallet 00:12:00.060 --> 00:12:03.940 align:middle line:84% is 0.3, that changes the amount that I value the wallet. 00:12:03.940 --> 00:12:05.900 align:middle line:90% I no longer value it at 0.9. 00:12:05.900 --> 00:12:07.200 align:middle line:90% What do I value it at? 00:12:07.200 --> 00:12:09.100 align:middle line:84% Well, the true value of the wallet to me 00:12:09.100 --> 00:12:12.340 align:middle line:84% is the sum of theta 1 plus theta 2, which in this case 00:12:12.340 --> 00:12:16.700 align:middle line:90% is going to be 0.7. 00:12:16.700 --> 00:12:18.980 align:middle line:84% So if I bid truthfully here, I'm going 00:12:18.980 --> 00:12:24.100 align:middle line:84% to pay 0.8 for something that's only worth 0.7 to me. 00:12:24.100 --> 00:12:25.355 align:middle line:90% What changed? 00:12:25.355 --> 00:12:27.480 align:middle line:84% And why is it different from the classical auction? 00:12:27.480 --> 00:12:32.180 align:middle line:84% It's that, well, this 0.9 wasn't my true valuation of the item. 00:12:32.180 --> 00:12:36.740 align:middle line:84% It was my belief or my expected value of the item. 00:12:36.740 --> 00:12:39.420 align:middle line:84% And unfortunately, my assessment of the value of the item 00:12:39.420 --> 00:12:42.740 align:middle line:84% is influenced by the other player's private information. 00:12:42.740 --> 00:12:45.940 align:middle line:84% So the issue is, when I won, I'm winning precisely 00:12:45.940 --> 00:12:47.900 align:middle line:84% because the other player's wallet doesn't 00:12:47.900 --> 00:12:49.020 align:middle line:90% have much money in it. 00:12:49.020 --> 00:12:52.540 align:middle line:84% And this phenomenon, which this is a very simple example. 00:12:52.540 --> 00:12:54.540 align:middle line:84% Maybe you think, oh, this doesn't have relevance 00:12:54.540 --> 00:12:57.420 align:middle line:84% to the real world, but this is actually 00:12:57.420 --> 00:13:00.445 align:middle line:84% a really serious problem in a lot of real auctions, 00:13:00.445 --> 00:13:02.320 align:middle line:84% and it's something called the winner's curse. 00:13:02.320 --> 00:13:06.180 align:middle line:90% 00:13:06.180 --> 00:13:07.880 align:middle line:84% So it sounds good for player one here. 00:13:07.880 --> 00:13:09.320 align:middle line:90% Player one won the auction. 00:13:09.320 --> 00:13:12.340 align:middle line:84% The problem is we have to think, why did player one win? 00:13:12.340 --> 00:13:15.260 align:middle line:84% Well, when will player one win this auction? 00:13:15.260 --> 00:13:16.540 align:middle line:90% Yeah, go ahead. 00:13:16.540 --> 00:13:17.715 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:13:17.715 --> 00:13:19.340 align:middle line:84% IAN BALL: Yeah, they'll win the auction 00:13:19.340 --> 00:13:21.540 align:middle line:84% when the other person bids a low amount, which 00:13:21.540 --> 00:13:23.992 align:middle line:84% is precisely when the other person's valuation is low. 00:13:23.992 --> 00:13:26.200 align:middle line:84% And the other person's wallet doesn't have very much. 00:13:26.200 --> 00:13:28.460 align:middle line:84% So the winner's curse is that you 00:13:28.460 --> 00:13:36.040 align:middle line:84% tend to win when the item is less valuable. 00:13:36.040 --> 00:13:47.510 align:middle line:90% 00:13:47.510 --> 00:13:49.330 align:middle line:84% And if you don't take this into account, 00:13:49.330 --> 00:13:50.612 align:middle line:90% people will tend to overbid. 00:13:50.612 --> 00:13:52.070 align:middle line:84% And the person who wins the auction 00:13:52.070 --> 00:13:53.987 align:middle line:84% will tend to come away a loser because they'll 00:13:53.987 --> 00:13:56.270 align:middle line:84% overpay for an item that's not worth very much. 00:13:56.270 --> 00:13:59.470 align:middle line:84% Now, just a few examples where this has happened in practice. 00:13:59.470 --> 00:14:03.350 align:middle line:84% And firms in the '70s and '80s lost tons of money making 00:14:03.350 --> 00:14:04.550 align:middle line:90% mistakes like this. 00:14:04.550 --> 00:14:07.830 align:middle line:84% One example would be oil drilling. 00:14:07.830 --> 00:14:10.510 align:middle line:84% Could someone maybe walk me through how 00:14:10.510 --> 00:14:13.970 align:middle line:84% oil drilling has anything to do with this wallet auction? 00:14:13.970 --> 00:14:18.410 align:middle line:84% And why firms might be subject to the winner's curse? 00:14:18.410 --> 00:14:20.990 align:middle line:84% So here we're drilling for oil, and we're 00:14:20.990 --> 00:14:23.750 align:middle line:90% bidding for attractive land. 00:14:23.750 --> 00:14:28.930 align:middle line:84% So oil drilling-- and let's say an auction for land. 00:14:28.930 --> 00:14:31.710 align:middle line:84% So it's for land rights, I'll say. 00:14:31.710 --> 00:14:33.910 align:middle line:84% So there's some big plot of land, say, in Alaska, 00:14:33.910 --> 00:14:35.950 align:middle line:90% in the Arctic. 00:14:35.950 --> 00:14:39.990 align:middle line:84% We are bidding for the right to drill for oil in that land 00:14:39.990 --> 00:14:41.370 align:middle line:90% and with different oil firms. 00:14:41.370 --> 00:14:42.045 align:middle line:90% Yeah. 00:14:42.045 --> 00:14:43.670 align:middle line:84% AUDIENCE: Maybe a lot of firms ended up 00:14:43.670 --> 00:14:46.050 align:middle line:84% paying more than they would actually get out. 00:14:46.050 --> 00:14:47.800 align:middle line:84% IAN BALL: So that's exactly what happened. 00:14:47.800 --> 00:14:50.190 align:middle line:90% And why did that happen? 00:14:50.190 --> 00:14:52.930 align:middle line:84% AUDIENCE: I guess they thought it was worth more. 00:14:52.930 --> 00:14:54.950 align:middle line:84% They got in the auction, and were just trying 00:14:54.950 --> 00:14:57.808 align:middle line:84% to pay the most that person was selling it at. 00:14:57.808 --> 00:14:59.850 align:middle line:84% IAN BALL: So I think we're along the right track. 00:14:59.850 --> 00:15:02.790 align:middle line:84% Any other thoughts about maybe a little more specificity. 00:15:02.790 --> 00:15:03.490 align:middle line:90% Yeah, in blue. 00:15:03.490 --> 00:15:04.005 align:middle line:90% Yeah. 00:15:04.005 --> 00:15:05.630 align:middle line:84% AUDIENCE: The person who's going to win 00:15:05.630 --> 00:15:07.005 align:middle line:84% is going to be the one who thinks 00:15:07.005 --> 00:15:08.170 align:middle line:90% there's the most oil there. 00:15:08.170 --> 00:15:12.668 align:middle line:84% And assuming there's a range that's of people's beliefs, 00:15:12.668 --> 00:15:14.210 align:middle line:84% and it falls somewhere in the middle, 00:15:14.210 --> 00:15:15.770 align:middle line:84% they're going to have overvalued it. 00:15:15.770 --> 00:15:16.110 align:middle line:90% IAN BALL: Exactly. 00:15:16.110 --> 00:15:17.818 align:middle line:84% So the crucial thing here is, what are we 00:15:17.818 --> 00:15:19.890 align:middle line:84% doing when we're a firm bidding on this? 00:15:19.890 --> 00:15:22.390 align:middle line:84% We're going to make a noisy estimate of the amount of oil 00:15:22.390 --> 00:15:23.050 align:middle line:90% in the land. 00:15:23.050 --> 00:15:26.930 align:middle line:84% So the issue here is noisy or imprecise estimates. 00:15:26.930 --> 00:15:32.870 align:middle line:90% 00:15:32.870 --> 00:15:34.850 align:middle line:84% So I'm going to do a little testing right. 00:15:34.850 --> 00:15:38.360 align:middle line:84% I'm going to try to guess or estimate 00:15:38.360 --> 00:15:40.800 align:middle line:90% how much oil is in this land. 00:15:40.800 --> 00:15:43.400 align:middle line:84% Every other firm is going to do the same thing. 00:15:43.400 --> 00:15:45.960 align:middle line:84% If I win, what does that mean about my estimate relative 00:15:45.960 --> 00:15:47.872 align:middle line:90% to everyone else's? 00:15:47.872 --> 00:15:50.080 align:middle line:84% It means my estimate was higher than everyone else's. 00:15:50.080 --> 00:15:52.330 align:middle line:84% And if my estimate was higher than everyone else's, it 00:15:52.330 --> 00:15:54.600 align:middle line:84% probably means my estimate just was 00:15:54.600 --> 00:15:57.500 align:middle line:84% an overestimate of the true value of the land, 00:15:57.500 --> 00:16:00.360 align:middle line:84% and I'm going to end up paying too much. 00:16:00.360 --> 00:16:01.920 align:middle line:84% Another example where this happens 00:16:01.920 --> 00:16:12.120 align:middle line:84% is auctions for, say, government procurement. 00:16:12.120 --> 00:16:13.660 align:middle line:90% So this is the analog of this. 00:16:13.660 --> 00:16:14.500 align:middle line:90% What happens there? 00:16:14.500 --> 00:16:18.440 align:middle line:84% The government says, we want to build a subway. 00:16:18.440 --> 00:16:20.902 align:middle line:84% And now the firms are going to bid on the amount 00:16:20.902 --> 00:16:22.860 align:middle line:84% that they need to get paid to build the subway. 00:16:22.860 --> 00:16:24.940 align:middle line:84% So I might say, I'll build it for $100 million. 00:16:24.940 --> 00:16:26.860 align:middle line:84% Someone else has outbid it for $110 million, 00:16:26.860 --> 00:16:28.500 align:middle line:84% someone else has outbid it for $90 million. 00:16:28.500 --> 00:16:30.280 align:middle line:84% And the government chooses the lowest bidder. 00:16:30.280 --> 00:16:32.155 align:middle line:84% So these, you always have to be careful here. 00:16:32.155 --> 00:16:37.640 align:middle line:84% So the winner is the lowest bidder 00:16:37.640 --> 00:16:40.260 align:middle line:84% because the government wants it built as cheaply as possible. 00:16:40.260 --> 00:16:43.240 align:middle line:84% Now, I think something else-- there can be a similar winner's 00:16:43.240 --> 00:16:43.740 align:middle line:90% curse here. 00:16:43.740 --> 00:16:47.480 align:middle line:84% What's the story in a government procurement auction? 00:16:47.480 --> 00:16:49.420 align:middle line:84% Say we're building a subway in New York. 00:16:49.420 --> 00:16:50.320 align:middle line:90% Yeah. 00:16:50.320 --> 00:16:53.220 align:middle line:84% AUDIENCE: To build a subway, there's unknown costs. 00:16:53.220 --> 00:16:54.340 align:middle line:90% You're not fully certain. 00:16:54.340 --> 00:16:56.215 align:middle line:84% And so, again, you're making a noisy estimate 00:16:56.215 --> 00:16:59.140 align:middle line:84% of exactly how much it's going to cost to make the subway. 00:16:59.140 --> 00:17:00.640 align:middle line:84% And then whoever wins is going to be 00:17:00.640 --> 00:17:02.900 align:middle line:84% whoever had the lowest estimate of cost, 00:17:02.900 --> 00:17:03.900 align:middle line:90% which is probably wrong. 00:17:03.900 --> 00:17:04.900 align:middle line:90% IAN BALL: Exactly right. 00:17:04.900 --> 00:17:06.560 align:middle line:84% So it's exactly parallel to this. 00:17:06.560 --> 00:17:08.000 align:middle line:90% We're making noisy estimates. 00:17:08.000 --> 00:17:10.140 align:middle line:84% Here we're estimating the value of something. 00:17:10.140 --> 00:17:12.599 align:middle line:84% If I win, I probably overestimated the value. 00:17:12.599 --> 00:17:14.800 align:middle line:84% Here we're estimating the cost of doing something. 00:17:14.800 --> 00:17:17.380 align:middle line:84% If I won, I probably underestimated the cost. 00:17:17.380 --> 00:17:19.460 align:middle line:84% So I thought it would only cost me $100 million, 00:17:19.460 --> 00:17:23.140 align:middle line:84% then I build it, and it ends up costing a lot more. 00:17:23.140 --> 00:17:25.920 align:middle line:84% So it's the same basic idea that we have imprecise estimates that 00:17:25.920 --> 00:17:26.940 align:middle line:90% are all correlated. 00:17:26.940 --> 00:17:28.800 align:middle line:84% There's actually a really nice story. 00:17:28.800 --> 00:17:30.440 align:middle line:84% So Paul Milgrom, who did some famous work on this 00:17:30.440 --> 00:17:31.900 align:middle line:84% and ultimately won the Nobel Prize, 00:17:31.900 --> 00:17:34.810 align:middle line:84% said his dad was a painter and would always complain to him 00:17:34.810 --> 00:17:36.950 align:middle line:84% that when he would bid on painting contracts, 00:17:36.950 --> 00:17:38.428 align:middle line:84% he'd always find that when he won, 00:17:38.428 --> 00:17:40.470 align:middle line:84% it cost more to paint the house than he expected. 00:17:40.470 --> 00:17:42.450 align:middle line:84% It was always a bit of a harder job than he thought. 00:17:42.450 --> 00:17:43.950 align:middle line:84% And Paul Milgrom said, oh, one day I 00:17:43.950 --> 00:17:45.790 align:middle line:90% wanted to tell him the answer. 00:17:45.790 --> 00:17:47.150 align:middle line:84% It turned out his father died before he could tell him, 00:17:47.150 --> 00:17:48.150 align:middle line:90% so it's a sad story. 00:17:48.150 --> 00:17:52.690 align:middle line:84% But I think it's a nice illustration of this phenomenon. 00:17:52.690 --> 00:17:54.190 align:middle line:90% Any questions on that? 00:17:54.190 --> 00:17:57.890 align:middle line:84% And then we'll move to another application. 00:17:57.890 --> 00:18:04.330 align:middle line:84% So now let's move to big application 00:18:04.330 --> 00:18:07.170 align:middle line:84% of auctions, which is going to be advertising auctions. 00:18:07.170 --> 00:18:08.930 align:middle line:84% And by advertising, what I really mean 00:18:08.930 --> 00:18:11.830 align:middle line:90% is keyword search auctions. 00:18:11.830 --> 00:18:20.730 align:middle line:90% 00:18:20.730 --> 00:18:23.250 align:middle line:84% And you probably see this all the time. 00:18:23.250 --> 00:18:26.773 align:middle line:84% If you Google something, you get the organic results. 00:18:26.773 --> 00:18:28.690 align:middle line:84% So let's just see what it looks like on a page 00:18:28.690 --> 00:18:30.130 align:middle line:90% if I go to Google. 00:18:30.130 --> 00:18:32.550 align:middle line:84% Well, I'm going to get some organic search results. 00:18:32.550 --> 00:18:35.850 align:middle line:90% 00:18:35.850 --> 00:18:36.990 align:middle line:90% What do I mean by organic? 00:18:36.990 --> 00:18:39.170 align:middle line:84% Organic just means Google's algorithm is saying, 00:18:39.170 --> 00:18:40.670 align:middle line:84% these are the websites that are most 00:18:40.670 --> 00:18:42.530 align:middle line:90% relevant for what I typed in. 00:18:42.530 --> 00:18:44.970 align:middle line:84% But then you're also going to have sponsored results. 00:18:44.970 --> 00:18:48.450 align:middle line:90% You might have something here-- 00:18:48.450 --> 00:18:50.610 align:middle line:90% I'll say S for sponsored one. 00:18:50.610 --> 00:18:52.670 align:middle line:90% Maybe something here, S2. 00:18:52.670 --> 00:18:57.690 align:middle line:84% Maybe you'll have a few over here, S3, S4. 00:18:57.690 --> 00:19:02.930 align:middle line:84% And these slots-- if you look, they'll 00:19:02.930 --> 00:19:06.650 align:middle line:84% be relevant to what you searched. 00:19:06.650 --> 00:19:08.110 align:middle line:90% And you may not even notice-- 00:19:08.110 --> 00:19:10.100 align:middle line:84% I think you probably know, but a lot of people 00:19:10.100 --> 00:19:11.850 align:middle line:84% just think of all the results as the same. 00:19:11.850 --> 00:19:15.410 align:middle line:84% They don't quite recognize what's sponsored and what's not. 00:19:15.410 --> 00:19:16.910 align:middle line:84% What does it mean if it's sponsored? 00:19:16.910 --> 00:19:19.210 align:middle line:84% Do people know what's going on there? 00:19:19.210 --> 00:19:21.930 align:middle line:84% Once they've seen organic and sponsored? 00:19:21.930 --> 00:19:23.330 align:middle line:90% Yeah. 00:19:23.330 --> 00:19:26.368 align:middle line:84% AUDIENCE: I don't know exactly, but [INAUDIBLE]. 00:19:26.368 --> 00:19:27.910 align:middle line:84% IAN BALL: They paid to have it there. 00:19:27.910 --> 00:19:28.970 align:middle line:90% So if you notice-- 00:19:28.970 --> 00:19:34.210 align:middle line:84% let's say you Google Uber, just as one example. 00:19:34.210 --> 00:19:38.730 align:middle line:84% What might you see in the first slot if you Google Uber? 00:19:38.730 --> 00:19:41.810 align:middle line:90% Yeah, you often see Lyft. 00:19:41.810 --> 00:19:47.130 align:middle line:84% And what's happening is Uber will probably 00:19:47.130 --> 00:19:49.895 align:middle line:84% show up as the first organic result. 00:19:49.895 --> 00:19:51.770 align:middle line:84% But the first sponsored result might actually 00:19:51.770 --> 00:19:54.370 align:middle line:84% be Lyft, because Lyft recognizes that someone 00:19:54.370 --> 00:19:58.530 align:middle line:84% who's searching for Uber is looking to rideshare. 00:19:58.530 --> 00:20:02.490 align:middle line:84% And maybe Lyft can undermine them and get them to buy that. 00:20:02.490 --> 00:20:04.490 align:middle line:84% Now, Uber might actually buy a slot here as well 00:20:04.490 --> 00:20:05.310 align:middle line:90% to try to do that. 00:20:05.310 --> 00:20:07.352 align:middle line:84% And now we might see-- so Uber might actually 00:20:07.352 --> 00:20:08.810 align:middle line:84% appear both in your organic results 00:20:08.810 --> 00:20:10.018 align:middle line:90% and in the sponsored results. 00:20:10.018 --> 00:20:13.170 align:middle line:84% And there's a lot of strategy that goes on into this. 00:20:13.170 --> 00:20:16.590 align:middle line:84% And this is actually how Google makes a lot of their money-- 00:20:16.590 --> 00:20:18.310 align:middle line:84% that by having the best search engine, 00:20:18.310 --> 00:20:19.810 align:middle line:84% they have a lot of people searching. 00:20:19.810 --> 00:20:22.630 align:middle line:84% And firms are willing to pay a lot for these slots. 00:20:22.630 --> 00:20:25.370 align:middle line:84% Now, a big insight, a big advance for Google 00:20:25.370 --> 00:20:29.150 align:middle line:84% was the fact that these slots are responsive to the keyword. 00:20:29.150 --> 00:20:37.700 align:middle line:84% So I'll say the sponsored slots depend 00:20:37.700 --> 00:20:41.740 align:middle line:84% on the keyword, the keyword meaning what you Googled. 00:20:41.740 --> 00:20:44.100 align:middle line:84% Before this, Google would just do banner advertisements, 00:20:44.100 --> 00:20:46.392 align:middle line:84% where they would just have something on the screen that 00:20:46.392 --> 00:20:47.317 align:middle line:90% was shown to everyone. 00:20:47.317 --> 00:20:48.900 align:middle line:84% But that's not very valuable because I 00:20:48.900 --> 00:20:51.067 align:middle line:84% might be showing an ad to someone who doesn't really 00:20:51.067 --> 00:20:52.420 align:middle line:90% care about it. 00:20:52.420 --> 00:20:55.412 align:middle line:84% These sponsored searches allow Google to hyper target 00:20:55.412 --> 00:20:57.620 align:middle line:84% their ads, and therefore, generate a lot more revenue 00:20:57.620 --> 00:20:59.020 align:middle line:90% from their ads. 00:20:59.020 --> 00:21:01.513 align:middle line:84% Now, how does Google decide who gets to be in this slot? 00:21:01.513 --> 00:21:02.680 align:middle line:90% Who gets to be in this slot? 00:21:02.680 --> 00:21:03.847 align:middle line:90% Who gets to be in this slot? 00:21:03.847 --> 00:21:05.800 align:middle line:84% They run an auction among the advertisers. 00:21:05.800 --> 00:21:08.820 align:middle line:84% And this is actually, maybe in numerical terms, 00:21:08.820 --> 00:21:11.160 align:middle line:84% the most important auctions we see in the world. 00:21:11.160 --> 00:21:13.660 align:middle line:84% Hundreds of billions are raised every year doing this. 00:21:13.660 --> 00:21:16.420 align:middle line:84% And today we're going to analyze a very simple model 00:21:16.420 --> 00:21:20.620 align:middle line:84% of this sponsored search auction. 00:21:20.620 --> 00:21:23.020 align:middle line:84% So just to be clear here, the entire auction 00:21:23.020 --> 00:21:25.420 align:middle line:84% is designed for a particular keyword. 00:21:25.420 --> 00:21:28.140 align:middle line:84% If people Google Lyft, then there's 00:21:28.140 --> 00:21:30.473 align:middle line:84% going to be an auction for showing slots 00:21:30.473 --> 00:21:31.640 align:middle line:90% to someone who googled Lyft. 00:21:31.640 --> 00:21:33.480 align:middle line:84% If someone googled Uber, there would be a different thing. 00:21:33.480 --> 00:21:35.560 align:middle line:84% If they googled hotel, there'd be something else. 00:21:35.560 --> 00:21:37.435 align:middle line:84% So everything we're going to be talking about 00:21:37.435 --> 00:21:38.860 align:middle line:90% is for a fixed keyword. 00:21:38.860 --> 00:21:41.860 align:middle line:84% Now, why can't we just off-the-shelf apply 00:21:41.860 --> 00:21:43.300 align:middle line:90% what we've done so far? 00:21:43.300 --> 00:21:48.120 align:middle line:84% The challenge this is not a single item auction. 00:21:48.120 --> 00:21:52.380 align:middle line:90% 00:21:52.380 --> 00:21:53.280 align:middle line:90% We have many items. 00:21:53.280 --> 00:21:54.840 align:middle line:84% We have the first slot, we have the second slot, 00:21:54.840 --> 00:21:55.940 align:middle line:90% we have the third slot. 00:21:55.940 --> 00:21:58.107 align:middle line:84% And the other thing is these items are not the same. 00:21:58.107 --> 00:21:59.580 align:middle line:84% So it's not a single item auction. 00:21:59.580 --> 00:22:04.860 align:middle line:84% We have multiple heterogeneous items. 00:22:04.860 --> 00:22:06.600 align:middle line:84% Or maybe I'll say differentiated items. 00:22:06.600 --> 00:22:12.933 align:middle line:90% 00:22:12.933 --> 00:22:15.100 align:middle line:84% Getting shown first is a lot better than being shown 00:22:15.100 --> 00:22:17.100 align:middle line:84% second, which is better than being shown third. 00:22:17.100 --> 00:22:18.120 align:middle line:90% So how do we model this? 00:22:18.120 --> 00:22:20.720 align:middle line:84% Well, we're going to write down a pretty simple model. 00:22:20.720 --> 00:22:24.140 align:middle line:90% And here's our general model. 00:22:24.140 --> 00:22:27.050 align:middle line:84% We're going to say that if your ad shows up in the first slot. 00:22:27.050 --> 00:22:29.430 align:middle line:90% It's more likely to be clicked. 00:22:29.430 --> 00:22:31.570 align:middle line:84% So in our general model, we have n bidders. 00:22:31.570 --> 00:22:35.310 align:middle line:90% 00:22:35.310 --> 00:22:38.830 align:middle line:90% I'll call them i equals 1 to n. 00:22:38.830 --> 00:22:42.470 align:middle line:90% 00:22:42.470 --> 00:22:44.667 align:middle line:84% And then we're going to have m advertising slots. 00:22:44.667 --> 00:22:46.250 align:middle line:84% So in my example up here, we had four. 00:22:46.250 --> 00:22:49.330 align:middle line:84% In general, we'll have m advertising slots. 00:22:49.330 --> 00:22:53.510 align:middle line:90% 00:22:53.510 --> 00:22:57.750 align:middle line:84% We'll call these j equals 1 to m. 00:22:57.750 --> 00:23:00.110 align:middle line:84% And what distinguishes these slots is what we'll 00:23:00.110 --> 00:23:02.590 align:middle line:90% call the click-through rate. 00:23:02.590 --> 00:23:14.310 align:middle line:84% So we'll say the Click-Through Rate, sometimes called CTR, 00:23:14.310 --> 00:23:20.070 align:middle line:90% are alpha 1, alpha 2, alpha m. 00:23:20.070 --> 00:23:22.550 align:middle line:90% So what does this mean? 00:23:22.550 --> 00:23:25.710 align:middle line:84% Alpha 1 is the click-through rate of the first position. 00:23:25.710 --> 00:23:28.070 align:middle line:84% If someone shows their ad in the first position, 00:23:28.070 --> 00:23:31.430 align:middle line:84% this is the probability that it's going to get clicked. 00:23:31.430 --> 00:23:33.610 align:middle line:84% If someone shows their ad in the second position, 00:23:33.610 --> 00:23:35.318 align:middle line:84% then the probability that it gets clicked 00:23:35.318 --> 00:23:38.190 align:middle line:84% is alpha 2 all the way down to alpha m if it's shown in the nth 00:23:38.190 --> 00:23:39.030 align:middle line:90% position. 00:23:39.030 --> 00:23:41.310 align:middle line:84% Now, there's a big simplifying assumption here 00:23:41.310 --> 00:23:43.190 align:middle line:84% that the probability that it's clicked 00:23:43.190 --> 00:23:45.250 align:middle line:84% doesn't depend on which ad is shown there. 00:23:45.250 --> 00:23:48.330 align:middle line:84% If I showed a really bad ad for a really undesirable product, 00:23:48.330 --> 00:23:51.150 align:middle line:84% even if it's first, probably no one's going to click on it. 00:23:51.150 --> 00:23:52.825 align:middle line:84% But if we're looking at products that 00:23:52.825 --> 00:23:54.450 align:middle line:84% are pretty similar, like Uber and Lyft, 00:23:54.450 --> 00:23:56.650 align:middle line:84% maybe this is a reasonable assumption. 00:23:56.650 --> 00:23:59.710 align:middle line:90% 00:23:59.710 --> 00:24:01.340 align:middle line:84% We're next going to assume that each 00:24:01.340 --> 00:24:03.090 align:middle line:84% of the bidders-- we have to specify, well, 00:24:03.090 --> 00:24:04.470 align:middle line:90% what do the bidders know? 00:24:04.470 --> 00:24:06.303 align:middle line:84% Well, we're going to assume the bidders have 00:24:06.303 --> 00:24:08.070 align:middle line:90% valuations per click. 00:24:08.070 --> 00:24:13.870 align:middle line:84% So the bidders-- so let's say bidder i has 00:24:13.870 --> 00:24:19.170 align:middle line:90% valuation Vi per click. 00:24:19.170 --> 00:24:25.040 align:middle line:90% 00:24:25.040 --> 00:24:27.300 align:middle line:84% The idea would be if someone clicks on my ad, well, 00:24:27.300 --> 00:24:28.925 align:middle line:84% maybe they'll buy it, maybe they won't. 00:24:28.925 --> 00:24:32.280 align:middle line:84% But the expected value to me of someone clicking on my ad is Vi. 00:24:32.280 --> 00:24:33.760 align:middle line:84% Another simplifying assumption here 00:24:33.760 --> 00:24:37.000 align:middle line:84% is that we're saying if someone clicks on it when it's 00:24:37.000 --> 00:24:39.200 align:middle line:84% in the first position versus someone clicks on it 00:24:39.200 --> 00:24:40.920 align:middle line:84% when it's in the second position, that's 00:24:40.920 --> 00:24:42.163 align:middle line:90% equally valuable to me. 00:24:42.163 --> 00:24:43.580 align:middle line:84% That may not be a good assumption. 00:24:43.580 --> 00:24:45.205 align:middle line:84% Maybe if someone clicks on it even when 00:24:45.205 --> 00:24:46.760 align:middle line:84% it's in a bad position, maybe it's 00:24:46.760 --> 00:24:48.860 align:middle line:84% more valuable because those people really like it, 00:24:48.860 --> 00:24:52.060 align:middle line:84% but we're not going to focus on that. 00:24:52.060 --> 00:24:53.920 align:middle line:90% We're going to keep it simple. 00:24:53.920 --> 00:24:58.200 align:middle line:90% So now I'm Google. 00:24:58.200 --> 00:24:59.920 align:middle line:90% I have these m slots. 00:24:59.920 --> 00:25:01.580 align:middle line:90% I have these i bidders. 00:25:01.580 --> 00:25:03.640 align:middle line:84% I'm going to ask them to submit bids. 00:25:03.640 --> 00:25:06.013 align:middle line:90% And I need to then choose-- 00:25:06.013 --> 00:25:07.180 align:middle line:90% so what do I need to choose? 00:25:07.180 --> 00:25:08.500 align:middle line:90% Let's go back over here. 00:25:08.500 --> 00:25:27.320 align:middle line:90% 00:25:27.320 --> 00:25:31.380 align:middle line:84% So Google has-- maybe I'll call this auction design choices. 00:25:31.380 --> 00:25:39.400 align:middle line:90% 00:25:39.400 --> 00:25:41.800 align:middle line:84% So one first choice is, well, I'm 00:25:41.800 --> 00:25:43.400 align:middle line:90% going to ask people to pay me. 00:25:43.400 --> 00:25:45.840 align:middle line:84% But am I going to ask them to pay me every time someone 00:25:45.840 --> 00:25:46.595 align:middle line:90% clicks on the ad? 00:25:46.595 --> 00:25:48.720 align:middle line:84% Or am I going to ask them to pay me just every time 00:25:48.720 --> 00:25:49.800 align:middle line:90% the ad is shown? 00:25:49.800 --> 00:25:52.220 align:middle line:84% And these are two common models for these auctions. 00:25:52.220 --> 00:25:53.740 align:middle line:90% One is called pay-per-click. 00:25:53.740 --> 00:25:58.080 align:middle line:90% 00:25:58.080 --> 00:26:00.140 align:middle line:84% And one is called pay-per-impression. 00:26:00.140 --> 00:26:05.280 align:middle line:90% 00:26:05.280 --> 00:26:07.340 align:middle line:84% An impression means the ad is shown to someone, 00:26:07.340 --> 00:26:10.755 align:middle line:84% but not necessarily clicked on-- may or may not be clicked on. 00:26:10.755 --> 00:26:12.880 align:middle line:84% In our model, this actually won't make a difference 00:26:12.880 --> 00:26:15.840 align:middle line:84% because we're assuming that the clickthrough rates are perfectly 00:26:15.840 --> 00:26:16.865 align:middle line:90% known. 00:26:16.865 --> 00:26:18.240 align:middle line:84% In a more realistic model, people 00:26:18.240 --> 00:26:21.150 align:middle line:84% might be uncertain about these clickthrough rates, 00:26:21.150 --> 00:26:22.970 align:middle line:84% and then it does make a big difference 00:26:22.970 --> 00:26:25.190 align:middle line:84% whether it's pay-per-click or pay-per-impression. 00:26:25.190 --> 00:26:26.610 align:middle line:84% I might want pay-per-click if I'm 00:26:26.610 --> 00:26:28.870 align:middle line:84% really uncertain about how popular my ad is, 00:26:28.870 --> 00:26:31.490 align:middle line:84% because then if my ad turns out to be a failure and no one 00:26:31.490 --> 00:26:33.050 align:middle line:84% clicks on it, well, then at least I 00:26:33.050 --> 00:26:34.230 align:middle line:90% don't have to pay very much. 00:26:34.230 --> 00:26:35.470 align:middle line:84% Whereas if it's pay-for-impression, 00:26:35.470 --> 00:26:37.550 align:middle line:84% well, then I'm on the hook to keep showing my ad. 00:26:37.550 --> 00:26:39.592 align:middle line:84% No one's clicking on it, I'm still paying for it. 00:26:39.592 --> 00:26:42.530 align:middle line:84% So you can see why it's pretty common to use pay-per-click. 00:26:42.530 --> 00:26:44.548 align:middle line:84% You can think of this as ensuring advertisers 00:26:44.548 --> 00:26:47.090 align:middle line:84% against having a really bad ad that no one wants to click on. 00:26:47.090 --> 00:26:48.670 align:middle line:84% But in this model, that won't make a difference. 00:26:48.670 --> 00:26:49.270 align:middle line:90% Yes, Amy. 00:26:49.270 --> 00:26:52.170 align:middle line:84% AUDIENCE: But then does it close out on pay-per-click 00:26:52.170 --> 00:26:55.245 align:middle line:84% if all the ads are pretty unpopular? 00:26:55.245 --> 00:26:55.870 align:middle line:90% IAN BALL: Yeah. 00:26:55.870 --> 00:26:59.410 align:middle line:84% I guess the firms would still have an incentive 00:26:59.410 --> 00:27:02.190 align:middle line:90% to have good ads. 00:27:02.190 --> 00:27:04.650 align:middle line:84% Yeah, I guess you might worry a little bit, maybe about 00:27:04.650 --> 00:27:05.270 align:middle line:90% moral hazard. 00:27:05.270 --> 00:27:09.290 align:middle line:90% 00:27:09.290 --> 00:27:10.890 align:middle line:84% If the pay-per-click was too high, 00:27:10.890 --> 00:27:12.682 align:middle line:84% then maybe firms wouldn't have an incentive 00:27:12.682 --> 00:27:13.650 align:middle line:90% to improve their ads. 00:27:13.650 --> 00:27:15.470 align:middle line:84% And I guess there is a bit of an incentive trade-off there, 00:27:15.470 --> 00:27:16.970 align:middle line:84% but I think the firms still have a-- 00:27:16.970 --> 00:27:20.250 align:middle line:84% Google can still tell the firms, no one's clicking on your ad, 00:27:20.250 --> 00:27:22.570 align:middle line:84% and the firms still want their ads to be clicked on. 00:27:22.570 --> 00:27:24.112 align:middle line:84% But I agree, their incentives may not 00:27:24.112 --> 00:27:26.090 align:middle line:84% be quite as strong in that case, yeah. 00:27:26.090 --> 00:27:29.050 align:middle line:84% So there's a trade-off there between maybe insurance 00:27:29.050 --> 00:27:31.983 align:middle line:90% and incentives, but yeah. 00:27:31.983 --> 00:27:33.650 align:middle line:84% And over time, I think the firms learned 00:27:33.650 --> 00:27:36.970 align:middle line:84% a lot about how often the likelihood of their ad being 00:27:36.970 --> 00:27:38.010 align:middle line:90% clicked on. 00:27:38.010 --> 00:27:39.808 align:middle line:84% So today, we're going to use pay-per-click. 00:27:39.808 --> 00:27:41.350 align:middle line:84% Again, in our model, it won't matter. 00:27:41.350 --> 00:27:43.100 align:middle line:84% But this will just simplify things, right? 00:27:43.100 --> 00:27:45.290 align:middle line:84% So what does the auction need to say? 00:27:45.290 --> 00:27:47.650 align:middle line:84% Well, the auction is going to collect the bids. 00:27:47.650 --> 00:27:49.670 align:middle line:84% And the bids are going to be payments per click. 00:27:49.670 --> 00:27:51.730 align:middle line:84% So my bid is going to say, think of it roughly 00:27:51.730 --> 00:27:54.970 align:middle line:84% as how much am I willing to pay each time my ad is clicked on? 00:27:54.970 --> 00:27:59.990 align:middle line:84% So we're going to collect these bids, b1 through bn. 00:27:59.990 --> 00:28:02.890 align:middle line:90% 00:28:02.890 --> 00:28:04.550 align:middle line:84% And then what do we need to decide? 00:28:04.550 --> 00:28:06.110 align:middle line:84% Well, we need to have an allocation. 00:28:06.110 --> 00:28:10.330 align:middle line:84% So we have slot 1, slot 2, up to slot m. 00:28:10.330 --> 00:28:15.050 align:middle line:84% For each slot, we have to say who gets it? 00:28:15.050 --> 00:28:17.320 align:middle line:84% So which ad is shown in this shot? 00:28:17.320 --> 00:28:24.340 align:middle line:90% 00:28:24.340 --> 00:28:25.920 align:middle line:90% And then the payment per click. 00:28:25.920 --> 00:28:33.300 align:middle line:90% 00:28:33.300 --> 00:28:35.520 align:middle line:84% So this is formally what we're choosing. 00:28:35.520 --> 00:28:38.420 align:middle line:84% We're choosing a function that says, if these are the n 00:28:38.420 --> 00:28:41.460 align:middle line:84% bids that the advertisers submit, who are we 00:28:41.460 --> 00:28:42.900 align:middle line:90% going to put in the first slot? 00:28:42.900 --> 00:28:45.340 align:middle line:84% And how much are we going to ask them to pay per click? 00:28:45.340 --> 00:28:46.920 align:middle line:84% Who are we going to put in the second slot? 00:28:46.920 --> 00:28:48.780 align:middle line:84% And how much are we going to ask them to pay per click? 00:28:48.780 --> 00:28:50.280 align:middle line:90% All the way down to the m slot. 00:28:50.280 --> 00:28:52.700 align:middle line:84% So that's the choice we have to make. 00:28:52.700 --> 00:28:55.422 align:middle line:84% And of course, the design of the auction that we use 00:28:55.422 --> 00:28:57.880 align:middle line:84% is going to affect the bidding behavior of the advertisers. 00:28:57.880 --> 00:29:00.640 align:middle line:84% We could use this various auction format. 00:29:00.640 --> 00:29:03.980 align:middle line:84% So today, the first auction format we're going to look at 00:29:03.980 --> 00:29:08.060 align:middle line:84% is what's called a GSP, or generalized second price 00:29:08.060 --> 00:29:08.640 align:middle line:90% auction. 00:29:08.640 --> 00:29:18.940 align:middle line:90% 00:29:18.940 --> 00:29:20.857 align:middle line:84% And I think we can maybe guess how this works. 00:29:20.857 --> 00:29:22.607 align:middle line:84% You know how a second price auction works? 00:29:22.607 --> 00:29:23.880 align:middle line:90% Why do we call it generalized? 00:29:23.880 --> 00:29:25.463 align:middle line:84% Well, the classic second price auction 00:29:25.463 --> 00:29:28.300 align:middle line:84% applies when there's a single item that we're auctioning off. 00:29:28.300 --> 00:29:30.910 align:middle line:84% Here, we have multiple items, so have to generalize it. 00:29:30.910 --> 00:29:33.160 align:middle line:84% And I think this is the most intuitive generalization. 00:29:33.160 --> 00:29:36.460 align:middle line:84% Any ideas about if you had to think how-- let's say you're 00:29:36.460 --> 00:29:37.420 align:middle line:90% working at Google. 00:29:37.420 --> 00:29:39.360 align:middle line:84% Someone said, let's generalize the second price auction. 00:29:39.360 --> 00:29:40.400 align:middle line:90% What might you do here? 00:29:40.400 --> 00:29:44.685 align:middle line:90% 00:29:44.685 --> 00:29:46.560 align:middle line:84% Well, let's first think about the allocation. 00:29:46.560 --> 00:29:47.600 align:middle line:90% I think that's pretty easy. 00:29:47.600 --> 00:29:49.933 align:middle line:84% Who do you think you're going to show in the first slot? 00:29:49.933 --> 00:29:50.600 align:middle line:90% Who's that? 00:29:50.600 --> 00:29:53.572 align:middle line:90% 00:29:53.572 --> 00:29:54.780 align:middle line:90% AUDIENCE: The highest bidder. 00:29:54.780 --> 00:29:55.720 align:middle line:84% IAN BALL: The highest bidder, right? 00:29:55.720 --> 00:29:57.600 align:middle line:84% So the allocation rule makes sense. 00:29:57.600 --> 00:29:59.460 align:middle line:84% We'll show the highest bidder here, 00:29:59.460 --> 00:30:01.460 align:middle line:84% the second highest bidder here, all the way down 00:30:01.460 --> 00:30:02.800 align:middle line:90% to the mth highest bidder here. 00:30:02.800 --> 00:30:07.120 align:middle line:84% So in general, maybe I'll put it here. 00:30:07.120 --> 00:30:08.260 align:middle line:90% We're in slot k. 00:30:08.260 --> 00:30:17.270 align:middle line:84% So slot k is going to go to the kth highest bidder, on the m. 00:30:17.270 --> 00:30:18.910 align:middle line:84% But now maybe the harder thing is, 00:30:18.910 --> 00:30:20.250 align:middle line:90% what's the payment per click? 00:30:20.250 --> 00:30:24.190 align:middle line:84% What's a reasonable amount to charge the kth highest bidder 00:30:24.190 --> 00:30:27.290 align:middle line:90% per click? 00:30:27.290 --> 00:30:28.363 align:middle line:90% AUDIENCE: k plus first. 00:30:28.363 --> 00:30:29.030 align:middle line:90% IAN BALL: Right. 00:30:29.030 --> 00:30:30.697 align:middle line:84% That's a pretty natural way of doing it. 00:30:30.697 --> 00:30:32.450 align:middle line:84% We say, OK, you're the kth highest bidder, 00:30:32.450 --> 00:30:33.670 align:middle line:90% so you get the k slot. 00:30:33.670 --> 00:30:35.770 align:middle line:84% But we're not going to charge you your bid. 00:30:35.770 --> 00:30:38.230 align:middle line:84% We're going to charge you the next highest bid. 00:30:38.230 --> 00:30:42.230 align:middle line:84% In a second price auction, you can think of the second price 00:30:42.230 --> 00:30:44.410 align:middle line:84% as the next highest bid after the winner. 00:30:44.410 --> 00:30:46.590 align:middle line:84% And here, we're charging the next highest bid 00:30:46.590 --> 00:30:49.670 align:middle line:84% after the person who wins the k slot. 00:30:49.670 --> 00:30:54.830 align:middle line:84% So the payment-- maybe I'll call it pk-- 00:30:54.830 --> 00:30:58.410 align:middle line:90% is equal to k plus first. 00:30:58.410 --> 00:31:03.950 align:middle line:90% 00:31:03.950 --> 00:31:06.883 align:middle line:90% Highest bid. 00:31:06.883 --> 00:31:08.050 align:middle line:90% This seems pretty intuitive. 00:31:08.050 --> 00:31:12.990 align:middle line:84% And let's try to get a feel for how this works in the example. 00:31:12.990 --> 00:31:16.935 align:middle line:84% AUDIENCE: What would happen in an event the last person? 00:31:16.935 --> 00:31:17.810 align:middle line:90% IAN BALL: Good point. 00:31:17.810 --> 00:31:21.630 align:middle line:84% So there's a question of what's the relationship between m 00:31:21.630 --> 00:31:23.710 align:middle line:90% and n? 00:31:23.710 --> 00:31:26.210 align:middle line:84% Usually the last person would pay-- 00:31:26.210 --> 00:31:30.030 align:middle line:84% so m may be smaller than n, but if m 00:31:30.030 --> 00:31:32.730 align:middle line:84% is equal to n, the next highest bid, we would usually call 0. 00:31:32.730 --> 00:31:34.290 align:middle line:84% So we can imagine we have the n bids. 00:31:34.290 --> 00:31:36.655 align:middle line:84% And then imagine the n plus first bid is just 0. 00:31:36.655 --> 00:31:38.030 align:middle line:84% AUDIENCE: So first of all, they-- 00:31:38.030 --> 00:31:39.572 align:middle line:84% IAN BALL: They wouldn't pay anything. 00:31:39.572 --> 00:31:42.390 align:middle line:84% Now in reality, m is probably much smaller than n. 00:31:42.390 --> 00:31:44.830 align:middle line:84% So if m is smaller than n, then the mth highest 00:31:44.830 --> 00:31:47.270 align:middle line:84% bidder is going to pay the m plus first highest bid, 00:31:47.270 --> 00:31:48.958 align:middle line:84% and then m plus 2, m plus 3, m plus 4, 00:31:48.958 --> 00:31:50.250 align:middle line:90% they don't get anything at all. 00:31:50.250 --> 00:31:51.670 align:middle line:84% So what you're describing could only 00:31:51.670 --> 00:31:53.610 align:middle line:84% happen if there's more bidders than ad slots. 00:31:53.610 --> 00:31:56.608 align:middle line:84% And that makes sense if there's more ad slots than bidders. 00:31:56.608 --> 00:31:58.650 align:middle line:84% If there's as many or more ad slots than bidders, 00:31:58.650 --> 00:32:02.230 align:middle line:84% then we can't really charge much because we can just 00:32:02.230 --> 00:32:03.070 align:middle line:90% give them away. 00:32:03.070 --> 00:32:04.930 align:middle line:90% But good question. 00:32:04.930 --> 00:32:08.630 align:middle line:90% 00:32:08.630 --> 00:32:09.130 align:middle line:90% OK. 00:32:09.130 --> 00:32:12.080 align:middle line:84% So let's go through a simple example 00:32:12.080 --> 00:32:21.180 align:middle line:90% to make sure we understand. 00:32:21.180 --> 00:32:31.480 align:middle line:84% So let's do an example with m equals maybe 2 and n equals 3. 00:32:31.480 --> 00:32:34.120 align:middle line:84% So we have three bidders and two ad slots. 00:32:34.120 --> 00:32:35.960 align:middle line:90% And let's say the ad slots-- 00:32:35.960 --> 00:32:40.600 align:middle line:84% we have ad slot 1, which has clickthrough rate alpha 1. 00:32:40.600 --> 00:32:44.840 align:middle line:84% And we have ad slot 2, which has clickthrough rate alpha 2. 00:32:44.840 --> 00:32:47.780 align:middle line:84% And we're always assuming that they're in descending order, 00:32:47.780 --> 00:32:50.040 align:middle line:84% so alpha 1 is strictly greater than alpha 2. 00:32:50.040 --> 00:32:52.440 align:middle line:84% And let's say the bidders submit these bids. 00:32:52.440 --> 00:33:04.320 align:middle line:84% Maybe b1 is $7, b2 is $8, and bid 3 is $5. 00:33:04.320 --> 00:33:06.520 align:middle line:84% So now what's going to happen in this auction? 00:33:06.520 --> 00:33:09.600 align:middle line:84% Well, the first slot is going to go to the bidder who 00:33:09.600 --> 00:33:10.760 align:middle line:90% bids the most. 00:33:10.760 --> 00:33:12.520 align:middle line:90% In this case, that's bidder 2. 00:33:12.520 --> 00:33:14.720 align:middle line:84% So bidder 2's add is going to be-- they're 00:33:14.720 --> 00:33:16.260 align:middle line:90% going to show bidder 2's add. 00:33:16.260 --> 00:33:21.800 align:middle line:90% 00:33:21.800 --> 00:33:23.872 align:middle line:84% And in the second slot, they're going 00:33:23.872 --> 00:33:26.080 align:middle line:84% to show the second highest bidder, which in this case 00:33:26.080 --> 00:33:26.993 align:middle line:90% is bidder 1. 00:33:26.993 --> 00:33:28.660 align:middle line:84% So they're going to show bidder 1's add. 00:33:28.660 --> 00:33:34.520 align:middle line:90% 00:33:34.520 --> 00:33:37.400 align:middle line:84% And what is the payment per click going to be? 00:33:37.400 --> 00:33:40.060 align:middle line:84% So maybe I'll call this p1 and p2. 00:33:40.060 --> 00:33:44.200 align:middle line:84% So I'll try to use superscripts for positions and subscripts 00:33:44.200 --> 00:33:44.940 align:middle line:90% for bidders. 00:33:44.940 --> 00:33:47.460 align:middle line:84% So p1 is not necessarily bidder 1. 00:33:47.460 --> 00:33:51.960 align:middle line:84% It's the payment made per click for the first slot. 00:33:51.960 --> 00:33:55.640 align:middle line:84% So in this case, the highest bidder bid 8. 00:33:55.640 --> 00:34:00.600 align:middle line:84% The second highest bid was 7, so p1 is going to be 7. 00:34:00.600 --> 00:34:05.640 align:middle line:84% So bidder 2 is going to have to pay $7 per click. 00:34:05.640 --> 00:34:09.770 align:middle line:84% And now if we go to 2, well, the second highest bidder 00:34:09.770 --> 00:34:12.850 align:middle line:84% is going to pay the third highest bid, which in this case 00:34:12.850 --> 00:34:13.690 align:middle line:90% is 5. 00:34:13.690 --> 00:34:17.210 align:middle line:90% So p2 is going to be 5. 00:34:17.210 --> 00:34:22.290 align:middle line:90% This all sounds good. 00:34:22.290 --> 00:34:27.350 align:middle line:84% But now we might ask, well, is it optimal to bid truthfully? 00:34:27.350 --> 00:34:35.650 align:middle line:90% 00:34:35.650 --> 00:34:37.870 align:middle line:84% We know in a standard second price auction, it is. 00:34:37.870 --> 00:34:39.429 align:middle line:90% You should bid truthfully. 00:34:39.429 --> 00:34:43.530 align:middle line:84% But now things are a little more complicated. 00:34:43.530 --> 00:34:46.510 align:middle line:84% So let's take the perspective of bidder 2. 00:34:46.510 --> 00:34:50.850 align:middle line:90% 00:34:50.850 --> 00:34:53.850 align:middle line:84% Well, what is their expected profit going 00:34:53.850 --> 00:34:56.429 align:middle line:90% to be from bidding truthfully? 00:34:56.429 --> 00:34:57.910 align:middle line:84% That is bidding 8 in this auction. 00:34:57.910 --> 00:35:05.210 align:middle line:90% 00:35:05.210 --> 00:35:06.430 align:middle line:90% I guess I haven't been clear. 00:35:06.430 --> 00:35:08.230 align:middle line:90% Let's assume that b2 equals 8. 00:35:08.230 --> 00:35:12.970 align:middle line:84% So let's suppose the true value is 8, and they bid 8. 00:35:12.970 --> 00:35:16.490 align:middle line:84% What are they going to get in expectation? 00:35:16.490 --> 00:35:19.370 align:middle line:90% Well, let's go through it. 00:35:19.370 --> 00:35:21.490 align:middle line:90% They're being shown in slot 1. 00:35:21.490 --> 00:35:25.890 align:middle line:84% The probability their ad gets clicked is alpha 1. 00:35:25.890 --> 00:35:30.010 align:middle line:84% And each time it's clicked, their valuation per click is 8, 00:35:30.010 --> 00:35:32.290 align:middle line:90% but they end up having to pay 7. 00:35:32.290 --> 00:35:37.830 align:middle line:84% So we get alpha 1 times 8 minus 7, which is just alpha 1. 00:35:37.830 --> 00:35:40.970 align:middle line:90% 00:35:40.970 --> 00:35:48.930 align:middle line:84% But let's suppose that instead they bid a bit less. 00:35:48.930 --> 00:35:55.090 align:middle line:84% And let's suppose that instead of bidding 8, they bid 6. 00:35:55.090 --> 00:36:02.550 align:middle line:84% So this is a bid of 8, but let's suppose instead they bid 6. 00:36:02.550 --> 00:36:06.500 align:middle line:90% 00:36:06.500 --> 00:36:08.580 align:middle line:84% Well, now, what's going to happen? 00:36:08.580 --> 00:36:10.740 align:middle line:84% Now what's their expected payment? 00:36:10.740 --> 00:36:13.560 align:middle line:84% Well, if they bid 6, where is their ad going to be shown? 00:36:13.560 --> 00:36:16.860 align:middle line:90% 00:36:16.860 --> 00:36:18.155 align:middle line:90% Yeah. 00:36:18.155 --> 00:36:19.280 align:middle line:90% It will be the second slot. 00:36:19.280 --> 00:36:22.500 align:middle line:84% If they bid 6, bidder 1 is going to get the first slot. 00:36:22.500 --> 00:36:24.440 align:middle line:84% Bidder 2 is now going to get the second slot, 00:36:24.440 --> 00:36:26.340 align:middle line:90% so they're going to get alpha 2. 00:36:26.340 --> 00:36:30.280 align:middle line:84% Their value is still 8, but now how much do they pay? 00:36:30.280 --> 00:36:34.260 align:middle line:90% 00:36:34.260 --> 00:36:35.660 align:middle line:90% Yeah. 00:36:35.660 --> 00:36:39.040 align:middle line:84% They're going to pay 5, because now they're 00:36:39.040 --> 00:36:40.040 align:middle line:90% getting the second slot. 00:36:40.040 --> 00:36:41.940 align:middle line:84% They're going to pay the third highest bid. 00:36:41.940 --> 00:36:43.400 align:middle line:90% The highest bid is 7. 00:36:43.400 --> 00:36:45.100 align:middle line:90% The second highest bid is 6. 00:36:45.100 --> 00:36:46.660 align:middle line:90% The third highest bid is 5. 00:36:46.660 --> 00:36:52.620 align:middle line:90% 00:36:52.620 --> 00:36:55.860 align:middle line:90% So we get 3 alpha 2. 00:36:55.860 --> 00:36:58.720 align:middle line:90% Well, what's better? 00:36:58.720 --> 00:37:01.220 align:middle line:84% Well, it's going to depend on the relative values of alpha 1 00:37:01.220 --> 00:37:02.700 align:middle line:90% and alpha 2. 00:37:02.700 --> 00:37:07.380 align:middle line:84% And in particular, if alpha 2 is pretty close to alpha 1, 00:37:07.380 --> 00:37:10.060 align:middle line:84% they're going to be much better off shading their bit. 00:37:10.060 --> 00:37:20.080 align:middle line:84% So this is going to be preferred if alpha 2 is close to alpha 1. 00:37:20.080 --> 00:37:21.880 align:middle line:84% It's smaller than alpha 1, but it's close. 00:37:21.880 --> 00:37:30.900 align:middle line:90% 00:37:30.900 --> 00:37:32.900 align:middle line:84% And here, we see a fundamental difference 00:37:32.900 --> 00:37:35.740 align:middle line:84% between a standard second price auction 00:37:35.740 --> 00:37:37.780 align:middle line:84% and the generalized second price auction. 00:37:37.780 --> 00:37:39.630 align:middle line:84% In the standard second price auction, 00:37:39.630 --> 00:37:41.380 align:middle line:84% you want to bid truthfully, because if you 00:37:41.380 --> 00:37:44.140 align:middle line:84% don't get the item, you don't get anything. 00:37:44.140 --> 00:37:46.700 align:middle line:84% But in a multi-item auction, now it's less clear. 00:37:46.700 --> 00:37:49.600 align:middle line:84% If you shade your bid instead of getting nothing, 00:37:49.600 --> 00:37:50.745 align:middle line:90% you get the next slot. 00:37:50.745 --> 00:37:53.120 align:middle line:84% And if the next slot is almost as good as the first slot, 00:37:53.120 --> 00:37:54.200 align:middle line:84% you might as well shade your bid. 00:37:54.200 --> 00:37:55.617 align:middle line:84% You might as well reduce your bid. 00:37:55.617 --> 00:37:57.420 align:middle line:84% So we now have a much stronger temptation 00:37:57.420 --> 00:38:00.240 align:middle line:84% to reduce your bid when the penalty for losing is lower. 00:38:00.240 --> 00:38:02.730 align:middle line:84% Instead of getting nothing, I just go one slot down. 00:38:02.730 --> 00:38:04.855 align:middle line:84% And in particular, when these slots are very close, 00:38:04.855 --> 00:38:06.670 align:middle line:84% going one slot down is not very costly. 00:38:06.670 --> 00:38:11.150 align:middle line:84% So now we need to ask, well, we could use this bidding format, 00:38:11.150 --> 00:38:14.270 align:middle line:84% but then bidders have to spend a lot of money 00:38:14.270 --> 00:38:16.810 align:middle line:84% trying to figure out how much other people are going to bid. 00:38:16.810 --> 00:38:18.490 align:middle line:84% And their bid is going to depend on other people's bids, 00:38:18.490 --> 00:38:19.730 align:middle line:84% and it's going to be complicated. 00:38:19.730 --> 00:38:22.105 align:middle line:84% So maybe we want an auction where it's optimal for people 00:38:22.105 --> 00:38:23.250 align:middle line:90% to just bid truthfully. 00:38:23.250 --> 00:38:24.990 align:middle line:84% So we want a different generalization 00:38:24.990 --> 00:38:26.615 align:middle line:84% of the second price auction than what's 00:38:26.615 --> 00:38:28.365 align:middle line:84% called a generalized second price auction. 00:38:28.365 --> 00:38:28.870 align:middle line:90% Yes. 00:38:28.870 --> 00:38:31.630 align:middle line:84% AUDIENCE: This jsp, does this decrease revenue 00:38:31.630 --> 00:38:34.810 align:middle line:84% for the auctioneer because everyone's going to shade? 00:38:34.810 --> 00:38:38.395 align:middle line:90% 00:38:38.395 --> 00:38:39.770 align:middle line:84% IAN BALL: It's a subtle question. 00:38:39.770 --> 00:38:47.550 align:middle line:84% So there is an equilibrium of this auction that will give-- 00:38:47.550 --> 00:38:50.002 align:middle line:84% I guess you have to say, what is the comparison with? 00:38:50.002 --> 00:38:52.210 align:middle line:84% The auction format that we're going to describe next, 00:38:52.210 --> 00:38:54.150 align:middle line:84% the VCG auction, there will be an equilibrium 00:38:54.150 --> 00:38:56.570 align:middle line:84% of this auction that will give the same revenue in this. 00:38:56.570 --> 00:38:58.237 align:middle line:84% But that's in this very stylized example 00:38:58.237 --> 00:39:00.510 align:middle line:90% where there's no uncertainty. 00:39:00.510 --> 00:39:02.530 align:middle line:84% If you added some more realistic features, 00:39:02.530 --> 00:39:04.280 align:middle line:84% then you have to be a little more careful, 00:39:04.280 --> 00:39:05.590 align:middle line:90% and it's actually not so clear. 00:39:05.590 --> 00:39:07.048 align:middle line:84% AUDIENCE: So the auctioneer doesn't 00:39:07.048 --> 00:39:10.390 align:middle line:84% have a preference between the style of the second price 00:39:10.390 --> 00:39:10.890 align:middle line:90% auction? 00:39:10.890 --> 00:39:12.630 align:middle line:84% IAN BALL: So I think the question is-- 00:39:12.630 --> 00:39:14.450 align:middle line:84% some of the theorems say it doesn't matter. 00:39:14.450 --> 00:39:16.617 align:middle line:84% But those theorems make some very strong assumptions 00:39:16.617 --> 00:39:18.885 align:middle line:84% that people bid exactly according to the equilibrium. 00:39:18.885 --> 00:39:21.010 align:middle line:84% And we know in practice that doesn't really happen. 00:39:21.010 --> 00:39:24.750 align:middle line:84% So in fact, one concern there was, going back historically, 00:39:24.750 --> 00:39:28.110 align:middle line:84% is people were using basically a version of a first price 00:39:28.110 --> 00:39:29.110 align:middle line:90% auction of this. 00:39:29.110 --> 00:39:35.150 align:middle line:84% And there was a suggestion that we should move to a second price 00:39:35.150 --> 00:39:35.930 align:middle line:90% auction. 00:39:35.930 --> 00:39:37.550 align:middle line:84% But the concern was, well, if we jump right away 00:39:37.550 --> 00:39:40.230 align:middle line:84% to a second price auction and firms don't adjust their bidding 00:39:40.230 --> 00:39:42.410 align:middle line:84% behavior, now we're going to lose a lot of money. 00:39:42.410 --> 00:39:44.650 align:middle line:84% So the second price auction should give you as much revenue 00:39:44.650 --> 00:39:45.950 align:middle line:84% if we play the new equilibrium where 00:39:45.950 --> 00:39:47.130 align:middle line:90% people bid much differently. 00:39:47.130 --> 00:39:48.190 align:middle line:84% But if firms don't know that, and they 00:39:48.190 --> 00:39:49.670 align:middle line:84% don't react quickly enough, then there's 00:39:49.670 --> 00:39:51.890 align:middle line:84% really a strong disincentive to move to a second price auction. 00:39:51.890 --> 00:39:53.473 align:middle line:84% I think that's one issue that went on. 00:39:53.473 --> 00:39:55.870 align:middle line:84% So there's a gap between the theory and the applications 00:39:55.870 --> 00:39:57.470 align:middle line:90% here. 00:39:57.470 --> 00:40:02.750 align:middle line:84% I guess I should say maybe the fundamental issue 00:40:02.750 --> 00:40:04.510 align:middle line:84% that motivated a lot of this auction work 00:40:04.510 --> 00:40:07.030 align:middle line:84% was that firms are able to modify their bids quickly 00:40:07.030 --> 00:40:07.650 align:middle line:90% in real time. 00:40:07.650 --> 00:40:09.933 align:middle line:84% So the firm submits some bids, their ad is shown, 00:40:09.933 --> 00:40:11.850 align:middle line:84% and then they can change their bids over time. 00:40:11.850 --> 00:40:13.990 align:middle line:84% And what was happening is firms were constantly 00:40:13.990 --> 00:40:16.365 align:middle line:84% changing their bids, and these bids were going in cycles. 00:40:16.365 --> 00:40:18.510 align:middle line:84% So we're modeling this as a one-off interaction. 00:40:18.510 --> 00:40:21.323 align:middle line:84% If we use the auction format we're going to describe, 00:40:21.323 --> 00:40:22.490 align:middle line:90% I think that's a good model. 00:40:22.490 --> 00:40:24.833 align:middle line:84% But if people really care about how other people bid, 00:40:24.833 --> 00:40:26.750 align:middle line:84% then it can make things a lot more complicated 00:40:26.750 --> 00:40:28.590 align:middle line:84% when people can change their bids over time. 00:40:28.590 --> 00:40:29.950 align:middle line:90% Good question. 00:40:29.950 --> 00:40:34.270 align:middle line:84% So let's think about how else we can generalize the second price 00:40:34.270 --> 00:40:34.770 align:middle line:90% auction. 00:40:34.770 --> 00:40:37.910 align:middle line:84% And I think the key is to think about the second price 00:40:37.910 --> 00:40:39.250 align:middle line:90% auction a bit differently. 00:40:39.250 --> 00:40:50.710 align:middle line:90% 00:40:50.710 --> 00:40:53.590 align:middle line:84% So I think the most obvious way to generalize the second price 00:40:53.590 --> 00:40:55.410 align:middle line:84% auction is to say, well, what is a second price auction? 00:40:55.410 --> 00:40:57.780 align:middle line:84% You pay the second highest bid, so let's generalize it. 00:40:57.780 --> 00:40:58.900 align:middle line:90% You pay the next highest bid. 00:40:58.900 --> 00:40:59.780 align:middle line:90% But let's think about it. 00:40:59.780 --> 00:41:02.113 align:middle line:84% What's really going on in the second price auction a bit 00:41:02.113 --> 00:41:02.640 align:middle line:90% differently? 00:41:02.640 --> 00:41:07.320 align:middle line:90% 00:41:07.320 --> 00:41:09.720 align:middle line:84% Another way of thinking about a second price auction 00:41:09.720 --> 00:41:13.100 align:middle line:84% is that each bidder is asked to pay their externality. 00:41:13.100 --> 00:41:17.080 align:middle line:90% 00:41:17.080 --> 00:41:19.760 align:middle line:84% Let's understand what we mean by this. 00:41:19.760 --> 00:41:22.500 align:middle line:84% So let's say the valuations are ordered. 00:41:22.500 --> 00:41:25.260 align:middle line:90% It's v2 to vn. 00:41:25.260 --> 00:41:28.720 align:middle line:90% 00:41:28.720 --> 00:41:33.240 align:middle line:90% And let's say I'm this guy. 00:41:33.240 --> 00:41:34.960 align:middle line:90% And I win. 00:41:34.960 --> 00:41:36.360 align:middle line:90% So I'm the highest bidder. 00:41:36.360 --> 00:41:38.193 align:middle line:84% Let's suppose everyone's bidding truthfully, 00:41:38.193 --> 00:41:39.560 align:middle line:90% I guess, to make this argument. 00:41:39.560 --> 00:41:40.520 align:middle line:90% I win. 00:41:40.520 --> 00:41:43.380 align:middle line:84% I get my valuation of v1, and these guys all lose. 00:41:43.380 --> 00:41:47.502 align:middle line:90% 00:41:47.502 --> 00:41:51.160 align:middle line:84% Now, what if I weren't here and we just 00:41:51.160 --> 00:41:54.360 align:middle line:84% wanted to allocate the good in a way that was 00:41:54.360 --> 00:41:56.480 align:middle line:90% efficient for everyone else? 00:41:56.480 --> 00:41:59.200 align:middle line:84% What would we do if bidder 1 were not here? 00:41:59.200 --> 00:42:00.380 align:middle line:90% Who would we allocate it to? 00:42:00.380 --> 00:42:03.400 align:middle line:90% 00:42:03.400 --> 00:42:04.040 align:middle line:90% Bidder 2. 00:42:04.040 --> 00:42:05.060 align:middle line:90% And what would they get? 00:42:05.060 --> 00:42:06.800 align:middle line:90% Their evaluation would be v2. 00:42:06.800 --> 00:42:18.720 align:middle line:84% So we can say if bidder 1 were gone, 00:42:18.720 --> 00:42:22.720 align:middle line:84% then the total value that all the other bidders would get 00:42:22.720 --> 00:42:23.820 align:middle line:90% would be v2. 00:42:23.820 --> 00:42:29.100 align:middle line:84% If bidder 1 were gone, the item would go to bidder 2. 00:42:29.100 --> 00:42:32.320 align:middle line:90% 00:42:32.320 --> 00:42:37.400 align:middle line:90% And the value would be v2. 00:42:37.400 --> 00:42:42.040 align:middle line:84% So we can think of it as, well, the total value 00:42:42.040 --> 00:42:44.120 align:middle line:84% that everyone other than bidder 1 gets 00:42:44.120 --> 00:42:47.520 align:middle line:84% is 0 when bidder 1 is here and would have been 2 00:42:47.520 --> 00:42:49.520 align:middle line:90% if bidder 1 were not present. 00:42:49.520 --> 00:42:53.080 align:middle line:84% So the externality that bidder 1's presence or bitter 1's bid 00:42:53.080 --> 00:42:56.250 align:middle line:84% imposes on everyone else is exactly v2, 00:42:56.250 --> 00:42:58.705 align:middle line:84% because they take the item away from everyone else. 00:42:58.705 --> 00:43:00.330 align:middle line:84% And the most efficient allocation of it 00:43:00.330 --> 00:43:03.570 align:middle line:84% to everyone else would have been to bidder v2. 00:43:03.570 --> 00:43:05.650 align:middle line:84% So this is the aspect of the second price auction 00:43:05.650 --> 00:43:07.450 align:middle line:84% that we're next going to try to generalize. 00:43:07.450 --> 00:43:10.090 align:middle line:84% We're going to try to charge people the externality 00:43:10.090 --> 00:43:13.210 align:middle line:84% that their presence imposes on everyone else. 00:43:13.210 --> 00:43:15.130 align:middle line:84% In this case, that externality happens 00:43:15.130 --> 00:43:17.490 align:middle line:84% to be the second highest valuation, the second highest 00:43:17.490 --> 00:43:17.990 align:middle line:90% bid. 00:43:17.990 --> 00:43:19.590 align:middle line:84% But in general, it's not necessarily. 00:43:19.590 --> 00:43:23.650 align:middle line:84% And that's what we'll see with the next auction format. 00:43:23.650 --> 00:43:25.630 align:middle line:90% So let's look at this. 00:43:25.630 --> 00:43:46.050 align:middle line:90% 00:43:46.050 --> 00:43:57.250 align:middle line:84% So let's say that we bid b1, b2, b3. 00:43:57.250 --> 00:44:01.770 align:middle line:84% And let's suppose that these are ordered like this. 00:44:01.770 --> 00:44:04.050 align:middle line:90% And we have these slots-- 00:44:04.050 --> 00:44:08.250 align:middle line:90% 1, 2, 3. 00:44:08.250 --> 00:44:12.250 align:middle line:84% Now, if I'm the highest bidder, then I get slot 1. 00:44:12.250 --> 00:44:14.850 align:middle line:84% The second highest bidder gets slot 2 and the third highest 00:44:14.850 --> 00:44:16.130 align:middle line:90% bidder gets slot 3. 00:44:16.130 --> 00:44:17.612 align:middle line:84% So here, I'm just assuming bidder 1 00:44:17.612 --> 00:44:19.570 align:middle line:84% is the highest bidder, 2 is the second highest, 00:44:19.570 --> 00:44:22.250 align:middle line:90% and bidder 3 is next. 00:44:22.250 --> 00:44:27.570 align:middle line:84% Well, what would happen if bidder 1 were not present? 00:44:27.570 --> 00:44:29.630 align:middle line:84% If bidder 1 weren't here-- or in other words, 00:44:29.630 --> 00:44:32.700 align:middle line:84% if bidder 1 just bid 0 and didn't win anything, 00:44:32.700 --> 00:44:33.950 align:middle line:90% what would happen to bidder 2? 00:44:33.950 --> 00:44:36.250 align:middle line:90% Which slot would they get? 00:44:36.250 --> 00:44:38.250 align:middle line:84% They'd move up to the first slot. 00:44:38.250 --> 00:44:39.350 align:middle line:90% So they'd go up here. 00:44:39.350 --> 00:44:41.930 align:middle line:90% 00:44:41.930 --> 00:44:44.670 align:middle line:84% And bidder 3, similarly, would move up to the second slot. 00:44:44.670 --> 00:44:49.490 align:middle line:90% 00:44:49.490 --> 00:44:55.060 align:middle line:84% So let's now compute bidder 1's-- the highest 00:44:55.060 --> 00:44:56.660 align:middle line:90% bidder's externality. 00:44:56.660 --> 00:45:01.660 align:middle line:90% 00:45:01.660 --> 00:45:04.860 align:middle line:84% Well, bidder 2-- the externality is 00:45:04.860 --> 00:45:07.580 align:middle line:84% that they're pushed down from alpha 1 to alpha 2 00:45:07.580 --> 00:45:09.340 align:middle line:90% as their clickthrough rate. 00:45:09.340 --> 00:45:10.980 align:middle line:90% So we have b2. 00:45:10.980 --> 00:45:25.140 align:middle line:90% 00:45:25.140 --> 00:45:26.820 align:middle line:90% So what happens? 00:45:26.820 --> 00:45:29.900 align:middle line:84% Bidder 2 gets alpha 2 instead of alpha 1 00:45:29.900 --> 00:45:31.660 align:middle line:90% as their clickthrough rate. 00:45:31.660 --> 00:45:33.940 align:middle line:90% And their value per click-- 00:45:33.940 --> 00:45:36.300 align:middle line:90% let's treat it as their bid, b2. 00:45:36.300 --> 00:45:38.780 align:middle line:84% This is the externality imposed on bidder 2, 00:45:38.780 --> 00:45:41.060 align:middle line:84% and this is the externality imposed on bidder 3. 00:45:41.060 --> 00:45:43.380 align:middle line:84% Notice here, we have to be a little careful. 00:45:43.380 --> 00:45:45.780 align:middle line:84% To directly compute the externality, 00:45:45.780 --> 00:45:48.580 align:middle line:84% we'd have to know how much other people actually value the item. 00:45:48.580 --> 00:45:50.120 align:middle line:84% So when we compute the externality, 00:45:50.120 --> 00:45:52.540 align:middle line:84% we compute it as if people are bidding truthfully. 00:45:52.540 --> 00:45:55.972 align:middle line:84% So I don't actually know how much bidder 2 values the slot, 00:45:55.972 --> 00:45:58.180 align:middle line:84% but we're going to compute the externality as if they 00:45:58.180 --> 00:45:59.280 align:middle line:90% were bidding truthfully. 00:45:59.280 --> 00:46:01.080 align:middle line:84% So if bidder 2 were bidding truthfully, 00:46:01.080 --> 00:46:04.540 align:middle line:84% then the amount they value the slot per click is exactly b2, 00:46:04.540 --> 00:46:07.420 align:middle line:84% and their loss is the difference in clickthrough rate 00:46:07.420 --> 00:46:09.060 align:middle line:90% times the valuation per click. 00:46:09.060 --> 00:46:10.980 align:middle line:84% And similarly, b3-- their loss is 00:46:10.980 --> 00:46:15.900 align:middle line:84% the difference in clickthrough rate times their valuation here. 00:46:15.900 --> 00:46:20.500 align:middle line:84% So in general, the idea or the goal 00:46:20.500 --> 00:46:31.400 align:middle line:84% is to charge each bidder per externality. 00:46:31.400 --> 00:46:36.380 align:middle line:90% 00:46:36.380 --> 00:46:37.880 align:middle line:84% Motivated by this simple example, 00:46:37.880 --> 00:46:40.420 align:middle line:84% let's see if we can give a general formula for this auction 00:46:40.420 --> 00:46:41.040 align:middle line:90% format. 00:46:41.040 --> 00:46:55.630 align:middle line:90% 00:46:55.630 --> 00:46:59.490 align:middle line:84% And this is called a VCG auction, 00:46:59.490 --> 00:47:01.550 align:middle line:84% and this is named after three people. 00:47:01.550 --> 00:47:04.870 align:middle line:84% Vickrey won the Nobel Prize for this-- 00:47:04.870 --> 00:47:08.090 align:middle line:90% Clark and Groves. 00:47:08.090 --> 00:47:11.710 align:middle line:90% 00:47:11.710 --> 00:47:13.630 align:middle line:84% And we can think of this as a different 00:47:13.630 --> 00:47:18.470 align:middle line:84% or maybe the true generalization of the second price auction. 00:47:18.470 --> 00:47:22.150 align:middle line:84% So now we get some bids, b1 through bn. 00:47:22.150 --> 00:47:24.230 align:middle line:84% And we have to decide what to do with them. 00:47:24.230 --> 00:47:26.473 align:middle line:84% It's helpful to have some notation for what's 00:47:26.473 --> 00:47:28.890 align:middle line:84% the highest bid, what's the second highest bid, and so on. 00:47:28.890 --> 00:47:30.890 align:middle line:84% It's not necessarily the first bidder is the highest. 00:47:30.890 --> 00:47:31.848 align:middle line:90% So just some notation-- 00:47:31.848 --> 00:47:33.670 align:middle line:90% I'm going to write b1. 00:47:33.670 --> 00:47:37.190 align:middle line:84% In parentheses, this is going to say, which among these bids 00:47:37.190 --> 00:47:38.990 align:middle line:90% is the highest? 00:47:38.990 --> 00:47:41.670 align:middle line:84% And then I'll write b2 in parentheses to say, 00:47:41.670 --> 00:47:43.550 align:middle line:84% which among these bids is the second highest? 00:47:43.550 --> 00:47:46.470 align:middle line:84% All the way down to b in parentheses 00:47:46.470 --> 00:47:48.490 align:middle line:84% n for the nth highest among the bits. 00:47:48.490 --> 00:47:53.990 align:middle line:90% 00:47:53.990 --> 00:47:56.510 align:middle line:90% Now I'm going to have my slots-- 00:47:56.510 --> 00:48:00.310 align:middle line:90% say, slot 1 down to slot m. 00:48:00.310 --> 00:48:02.250 align:middle line:84% And instead of writing it many times, 00:48:02.250 --> 00:48:04.870 align:middle line:84% let me just describe what happens to the kth slot, where 00:48:04.870 --> 00:48:06.190 align:middle line:90% k is an arbitrary. 00:48:06.190 --> 00:48:07.610 align:middle line:90% Or I guess I was using j before. 00:48:07.610 --> 00:48:11.190 align:middle line:90% Maybe I'll call it the j slot. 00:48:11.190 --> 00:48:13.150 align:middle line:90% So who gets the j slot? 00:48:13.150 --> 00:48:15.990 align:middle line:84% Well, the j slot is going to go to the jth highest bidder, as 00:48:15.990 --> 00:48:16.790 align:middle line:90% before. 00:48:16.790 --> 00:48:19.150 align:middle line:84% So the allocation rule is going to be the same 00:48:19.150 --> 00:48:20.610 align:middle line:90% as the second price auction. 00:48:20.610 --> 00:48:25.470 align:middle line:84% I allocate the jth best good to the jth highest bidder. 00:48:25.470 --> 00:48:37.350 align:middle line:84% So it's allocated to jth high square. 00:48:37.350 --> 00:48:39.210 align:middle line:84% But the tricky thing is the payment. 00:48:39.210 --> 00:48:41.910 align:middle line:90% So maybe I'll call it pj. 00:48:41.910 --> 00:48:46.600 align:middle line:84% And remember, this is a payment per click. 00:48:46.600 --> 00:48:50.380 align:middle line:90% Well, it's easier, I think-- 00:48:50.380 --> 00:48:51.720 align:middle line:90% I guess a little more space-- 00:48:51.720 --> 00:48:54.920 align:middle line:90% to think about alpha j pj. 00:48:54.920 --> 00:48:58.240 align:middle line:84% Because remember, I want each bidder 00:48:58.240 --> 00:49:00.320 align:middle line:90% to be charged for externality. 00:49:00.320 --> 00:49:02.800 align:middle line:84% So what I want to set equal to the bidders' externality 00:49:02.800 --> 00:49:04.600 align:middle line:90% is the actual amount they pay. 00:49:04.600 --> 00:49:06.900 align:middle line:84% And the actual amount they pay is not pj. 00:49:06.900 --> 00:49:08.540 align:middle line:84% That's the amount they pay per click. 00:49:08.540 --> 00:49:11.005 align:middle line:84% So they pay pj per click times the probability 00:49:11.005 --> 00:49:12.380 align:middle line:84% that they're actually clicked on. 00:49:12.380 --> 00:49:14.200 align:middle line:84% That's the expected amount that they pay. 00:49:14.200 --> 00:49:16.300 align:middle line:84% So this is going to be the expected payment. 00:49:16.300 --> 00:49:21.080 align:middle line:90% 00:49:21.080 --> 00:49:23.900 align:middle line:84% So later we can divide by alpha j and get a formula for pj, 00:49:23.900 --> 00:49:25.520 align:middle line:84% but it's just when we're thinking of the externality, 00:49:25.520 --> 00:49:26.460 align:middle line:90% it's easier to think this way. 00:49:26.460 --> 00:49:27.080 align:middle line:90% Yes. 00:49:27.080 --> 00:49:29.640 align:middle line:90% AUDIENCE: [INAUDIBLE] right? 00:49:29.640 --> 00:49:30.995 align:middle line:90% IAN BALL: No. 00:49:30.995 --> 00:49:32.620 align:middle line:84% Well, I guess it depends what you mean. 00:49:32.620 --> 00:49:34.573 align:middle line:84% If alpha j is the probability per click 00:49:34.573 --> 00:49:36.240 align:middle line:84% of the probability of a click, then this 00:49:36.240 --> 00:49:38.367 align:middle line:84% is just the expected payment per click. 00:49:38.367 --> 00:49:39.200 align:middle line:90% AUDIENCE: Per click. 00:49:39.200 --> 00:49:42.015 align:middle line:84% IAN BALL: No, just pj is the payment per click, right? 00:49:42.015 --> 00:49:42.640 align:middle line:90% AUDIENCE: Yeah. 00:49:42.640 --> 00:49:44.320 align:middle line:84% IAN BALL: So if I multiply the payment 00:49:44.320 --> 00:49:47.360 align:middle line:84% per click times my probability of a click, 00:49:47.360 --> 00:49:49.262 align:middle line:84% that gives me my full expected payment. 00:49:49.262 --> 00:49:51.720 align:middle line:84% If I pay 100 per click, and there's a 1/2 chance of getting 00:49:51.720 --> 00:49:53.840 align:middle line:84% clicked, then my expected payment is 50 00:49:53.840 --> 00:49:56.200 align:middle line:84% because there's a half chance I get clicked and pay 100 00:49:56.200 --> 00:49:59.760 align:middle line:84% and a half chance I don't get clicked and pay nothing. 00:49:59.760 --> 00:50:01.618 align:middle line:84% If there's multiple clicks, then we 00:50:01.618 --> 00:50:03.160 align:middle line:84% could more generally think of alpha j 00:50:03.160 --> 00:50:04.768 align:middle line:84% as the expected number of clicks. 00:50:04.768 --> 00:50:07.060 align:middle line:84% And maybe the expected number of clicks is more than 1, 00:50:07.060 --> 00:50:09.140 align:middle line:84% but in reality, each time a keyword is searched, 00:50:09.140 --> 00:50:10.265 align:middle line:90% there's a different option. 00:50:10.265 --> 00:50:13.080 align:middle line:90% 00:50:13.080 --> 00:50:17.280 align:middle line:90% Well, that is how it used to be. 00:50:17.280 --> 00:50:19.175 align:middle line:84% Now people have a lot of automated bidding. 00:50:19.175 --> 00:50:21.300 align:middle line:84% Now people are using robots to bid on their behalf, 00:50:21.300 --> 00:50:22.717 align:middle line:84% so it's a little more complicated. 00:50:22.717 --> 00:50:24.420 align:middle line:90% But originally, yeah. 00:50:24.420 --> 00:50:27.860 align:middle line:84% Maybe they were batched a little bit, but roughly, yeah. 00:50:27.860 --> 00:50:29.943 align:middle line:84% You can change your bid very, very frequently. 00:50:29.943 --> 00:50:32.360 align:middle line:84% AUDIENCE: If two people are searching for the same keyword 00:50:32.360 --> 00:50:33.068 align:middle line:90% at the same time. 00:50:33.068 --> 00:50:33.860 align:middle line:90% IAN BALL: Yeah, OK. 00:50:33.860 --> 00:50:35.200 align:middle line:90% There's going to be some lag. 00:50:35.200 --> 00:50:37.160 align:middle line:90% I think formerly what happened-- 00:50:37.160 --> 00:50:39.640 align:middle line:84% I don't know, some of this may be proprietary. 00:50:39.640 --> 00:50:41.780 align:middle line:84% Firms are able to change their bid very quickly. 00:50:41.780 --> 00:50:43.738 align:middle line:84% I know firms are able to change their bid maybe 00:50:43.738 --> 00:50:44.790 align:middle line:90% at a second interval. 00:50:44.790 --> 00:50:47.915 align:middle line:84% So maybe all the keywords that were searched in a given second 00:50:47.915 --> 00:50:48.790 align:middle line:90% would use those bids. 00:50:48.790 --> 00:50:51.930 align:middle line:84% And then when your bid changed, all the auctions 00:50:51.930 --> 00:50:53.970 align:middle line:84% in the next second would use the new bid. 00:50:53.970 --> 00:50:56.210 align:middle line:90% And there was actually-- 00:50:56.210 --> 00:50:59.970 align:middle line:84% when they used to use things that were not strategy proof 00:50:59.970 --> 00:51:03.170 align:middle line:84% and people were gaming it, this is what would happen. 00:51:03.170 --> 00:51:05.530 align:middle line:84% People were investing in technology to change their bids 00:51:05.530 --> 00:51:08.230 align:middle line:84% and update their bids very, very quickly. 00:51:08.230 --> 00:51:10.450 align:middle line:84% But yeah, probably not literally per auction, 00:51:10.450 --> 00:51:13.572 align:middle line:84% but every auction in a second, which is a lot. 00:51:13.572 --> 00:51:15.530 align:middle line:84% AUDIENCE: So that's why we're just considering, 00:51:15.530 --> 00:51:17.350 align:middle line:90% is there one click or not? 00:51:17.350 --> 00:51:18.530 align:middle line:90% Instead of the-- 00:51:18.530 --> 00:51:20.002 align:middle line:90% IAN BALL: Yeah. 00:51:20.002 --> 00:51:21.710 align:middle line:84% Mathematically, it doesn't matter at all. 00:51:21.710 --> 00:51:24.570 align:middle line:84% If you want to think of alpha j, one way of thinking of this 00:51:24.570 --> 00:51:27.110 align:middle line:84% is this is the probability per click. 00:51:27.110 --> 00:51:29.910 align:middle line:84% And maybe our bid is relevant for a batch of 100 auctions. 00:51:29.910 --> 00:51:32.310 align:middle line:84% And then you could just multiply every alpha by 100, 00:51:32.310 --> 00:51:33.370 align:middle line:84% and then you get the expected number 00:51:33.370 --> 00:51:34.828 align:middle line:84% of clicks in this batch of auctions 00:51:34.828 --> 00:51:37.170 align:middle line:90% that's relevant for a bit. 00:51:37.170 --> 00:51:38.250 align:middle line:90% Great. 00:51:38.250 --> 00:51:40.630 align:middle line:84% OK, so here's our expected payment. 00:51:40.630 --> 00:51:43.990 align:middle line:84% And I was trying to be consistent about superscripts, 00:51:43.990 --> 00:51:45.163 align:middle line:90% but I wasn't-- oh, no. 00:51:45.163 --> 00:51:45.830 align:middle line:90% I was over here. 00:51:45.830 --> 00:51:46.330 align:middle line:90% OK, sorry. 00:51:46.330 --> 00:51:48.850 align:middle line:90% This should be alpha j, I guess. 00:51:48.850 --> 00:51:50.850 align:middle line:84% I don't know if I've been consistent. 00:51:50.850 --> 00:51:52.110 align:middle line:90% I wasn't consistent over here. 00:51:52.110 --> 00:51:53.150 align:middle line:90% I'll change this. 00:51:53.150 --> 00:52:02.170 align:middle line:90% 00:52:02.170 --> 00:52:05.230 align:middle line:84% So this is actually right. b3 is actually the third bidder, 00:52:05.230 --> 00:52:06.390 align:middle line:90% so that's a subscript. 00:52:06.390 --> 00:52:09.650 align:middle line:84% Alpha 2 is the second bid slot, so it's superscript. 00:52:09.650 --> 00:52:11.890 align:middle line:90% So I think my notation is good. 00:52:11.890 --> 00:52:14.410 align:middle line:84% So now we want to compute, what is 00:52:14.410 --> 00:52:19.570 align:middle line:84% the externality that bidder j is imposing on the other bidders? 00:52:19.570 --> 00:52:24.910 align:middle line:84% So the first question is, who is bidder j affecting? 00:52:24.910 --> 00:52:26.970 align:middle line:84% So over here, we thought bidder 1 was affecting 00:52:26.970 --> 00:52:30.330 align:middle line:90% bidder 2 and bidder 3. 00:52:30.330 --> 00:52:32.470 align:middle line:84% Which bidders is bidder j's bid affecting? 00:52:32.470 --> 00:52:36.157 align:middle line:84% If bidder j were gone, whose allocation would change? 00:52:36.157 --> 00:52:36.990 align:middle line:90% Everyone afterwards? 00:52:36.990 --> 00:52:37.690 align:middle line:90% OK, yeah. 00:52:37.690 --> 00:52:41.020 align:middle line:84% So what we need to compute is, what is the externality 00:52:41.020 --> 00:52:44.280 align:middle line:84% that bidder j is imposing on everyone after bidder j? 00:52:44.280 --> 00:52:46.820 align:middle line:84% Because these people above him are not going to be changed. 00:52:46.820 --> 00:52:49.740 align:middle line:84% If bidder j weren't there, those people are still going to win. 00:52:49.740 --> 00:52:51.820 align:middle line:90% So it's going to be a sum. 00:52:51.820 --> 00:52:55.260 align:middle line:84% And let's sum it over everyone after bidder j. 00:52:55.260 --> 00:52:57.760 align:middle line:84% So this is the jth highest bidder. 00:52:57.760 --> 00:53:04.523 align:middle line:84% We want to sum from the j plus first highest bidder to n. 00:53:04.523 --> 00:53:05.940 align:middle line:84% So within our sum, we want to say, 00:53:05.940 --> 00:53:10.845 align:middle line:84% what is the externality imposed on the kth highest bidder? 00:53:10.845 --> 00:53:12.220 align:middle line:84% So if I'm the kth highest bidder, 00:53:12.220 --> 00:53:19.380 align:middle line:84% where k is bigger than j, when j is here, I get the k slot. 00:53:19.380 --> 00:53:23.260 align:middle line:84% But if j were not here, what slot would I get? 00:53:23.260 --> 00:53:25.300 align:middle line:90% The k minus 1 slot. 00:53:25.300 --> 00:53:28.820 align:middle line:84% So the externality, the change in my clickthrough rate 00:53:28.820 --> 00:53:30.700 align:middle line:90% is alpha k minus. 00:53:30.700 --> 00:53:34.540 align:middle line:90% 00:53:34.540 --> 00:53:35.680 align:middle line:90% This is the externality. 00:53:35.680 --> 00:53:38.755 align:middle line:84% Without j, I would have become the k minus first highest 00:53:38.755 --> 00:53:41.380 align:middle line:84% bidder, and I would have gotten the k minus first highest slot. 00:53:41.380 --> 00:53:44.842 align:middle line:84% With j, I'm only the kth highest bidder, and I get the k slot. 00:53:44.842 --> 00:53:47.300 align:middle line:84% But I have to multiply that change in the clickthrough rate 00:53:47.300 --> 00:53:49.292 align:middle line:90% times my value per click. 00:53:49.292 --> 00:53:51.000 align:middle line:84% Again, we don't know that bidder's value, 00:53:51.000 --> 00:53:52.417 align:middle line:84% but the way we compute externality 00:53:52.417 --> 00:53:54.180 align:middle line:84% is as if people are bidding truthfully. 00:53:54.180 --> 00:53:55.640 align:middle line:90% And we have notation. 00:53:55.640 --> 00:54:00.660 align:middle line:84% What is our notation for the bid of the kth highest bidder? 00:54:00.660 --> 00:54:04.960 align:middle line:84% It's just b to the superscript k. 00:54:04.960 --> 00:54:09.740 align:middle line:90% 00:54:09.740 --> 00:54:10.720 align:middle line:90% So this is our formula. 00:54:10.720 --> 00:54:12.220 align:middle line:84% And this fully specifies the auction 00:54:12.220 --> 00:54:18.540 align:middle line:84% because we've specified which bidder each slot is assigned to. 00:54:18.540 --> 00:54:20.780 align:middle line:84% And we've specified the payment per click 00:54:20.780 --> 00:54:22.380 align:middle line:90% for each of these slots. 00:54:22.380 --> 00:54:24.460 align:middle line:84% But now let's try to understand a little bit what 00:54:24.460 --> 00:54:25.880 align:middle line:90% this formula looks like. 00:54:25.880 --> 00:54:27.780 align:middle line:90% So let's dig in a little more. 00:54:27.780 --> 00:54:30.600 align:middle line:90% Let's go over here. 00:54:30.600 --> 00:54:43.150 align:middle line:90% 00:54:43.150 --> 00:54:45.790 align:middle line:90% Let's try to interpret this. 00:54:45.790 --> 00:54:48.370 align:middle line:90% Let's divide both sides by. 00:54:48.370 --> 00:54:51.110 align:middle line:90% 00:54:51.110 --> 00:54:57.430 align:middle line:84% alpha J. So we're going to see that pj, the payment per click 00:54:57.430 --> 00:55:01.590 align:middle line:84% in the j slot, is going to equal the sum of k 00:55:01.590 --> 00:55:15.150 align:middle line:84% equals j plus 1 to n of bk times alpha k minus 1 00:55:15.150 --> 00:55:17.290 align:middle line:90% divided by alpha j. 00:55:17.290 --> 00:55:21.710 align:middle line:90% 00:55:21.710 --> 00:55:23.603 align:middle line:90% Good. 00:55:23.603 --> 00:55:29.030 align:middle line:84% And we can see that the amount that we pay in the j slot 00:55:29.030 --> 00:55:31.990 align:middle line:84% is a sum over all the lower bids. 00:55:31.990 --> 00:55:35.070 align:middle line:84% But each of the lower bids is multiplied by some coefficient 00:55:35.070 --> 00:55:35.690 align:middle line:90% here. 00:55:35.690 --> 00:55:37.550 align:middle line:84% So let's introduce notation for this. 00:55:37.550 --> 00:55:40.830 align:middle line:90% Let's call this wk. 00:55:40.830 --> 00:55:46.510 align:middle line:84% And let's see what we can say about wk. 00:55:46.510 --> 00:55:51.790 align:middle line:84% Well first, wk is greater than or equal to 0-- in fact, 00:55:51.790 --> 00:55:55.470 align:middle line:90% strictly greater than 0. 00:55:55.470 --> 00:55:56.850 align:middle line:90% Why? 00:55:56.850 --> 00:56:00.250 align:middle line:84% Well, alpha k minus 1 is strictly higher than alpha k. 00:56:00.250 --> 00:56:01.330 align:middle line:90% That's our assumption. 00:56:01.330 --> 00:56:04.170 align:middle line:84% The higher slots-- higher slot meaning lower numbered slot. 00:56:04.170 --> 00:56:06.490 align:middle line:84% The higher slots have a higher clickthrough rate, 00:56:06.490 --> 00:56:08.470 align:middle line:84% so each of these numbers is strictly positive. 00:56:08.470 --> 00:56:11.150 align:middle line:84% Now let's try to see what the sum of these numbers are. 00:56:11.150 --> 00:56:19.830 align:middle line:84% So let's look at wj plus 1, all the way up to wn. 00:56:19.830 --> 00:56:26.830 align:middle line:90% 00:56:26.830 --> 00:56:32.430 align:middle line:84% Well, let's go through it. wj plus 1-- 00:56:32.430 --> 00:56:34.720 align:middle line:90% let's plug in j plus 1 for k. 00:56:34.720 --> 00:56:38.280 align:middle line:84% We're going to get alpha j minus alpha 00:56:38.280 --> 00:56:41.940 align:middle line:90% j plus 1 divided by alpha j. 00:56:41.940 --> 00:56:45.120 align:middle line:90% 00:56:45.120 --> 00:56:47.240 align:middle line:84% Plus-- well, the next term is going 00:56:47.240 --> 00:56:56.652 align:middle line:84% to be alpha j plus 1 minus alpha j plus 2 over alpha j. 00:56:56.652 --> 00:56:58.860 align:middle line:84% All the way-- well, what's the last term going to be? 00:56:58.860 --> 00:57:06.160 align:middle line:84% We have wn here, so we're going to have alpha n minus 1 00:57:06.160 --> 00:57:10.300 align:middle line:90% minus alpha n over alpha j. 00:57:10.300 --> 00:57:12.960 align:middle line:90% 00:57:12.960 --> 00:57:16.140 align:middle line:84% Well, alpha j is underneath everything, 00:57:16.140 --> 00:57:18.040 align:middle line:84% so we can make this into a single fraction. 00:57:18.040 --> 00:57:20.600 align:middle line:90% And we add alpha j plus 1-- 00:57:20.600 --> 00:57:23.480 align:middle line:84% sorry, we subtract alpha j plus 1, and then we add it back. 00:57:23.480 --> 00:57:25.647 align:middle line:84% We subtract alpha j plus 2, and then we add it back. 00:57:25.647 --> 00:57:27.938 align:middle line:84% So this is going to be what's called a telescoping sum. 00:57:27.938 --> 00:57:29.200 align:middle line:90% It's going to simplify a lot. 00:57:29.200 --> 00:57:34.180 align:middle line:84% And we're only going to get the first term and the last term. 00:57:34.180 --> 00:57:36.720 align:middle line:90% 00:57:36.720 --> 00:57:38.920 align:middle line:84% Every intermediate term is going to cancel. 00:57:38.920 --> 00:57:43.040 align:middle line:84% So what we're going to get is alpha 00:57:43.040 --> 00:57:48.140 align:middle line:90% j minus alpha n over alpha j. 00:57:48.140 --> 00:57:52.120 align:middle line:90% 00:57:52.120 --> 00:57:52.760 align:middle line:90% Yes. 00:57:52.760 --> 00:57:54.135 align:middle line:84% AUDIENCE: What's the significance 00:57:54.135 --> 00:57:56.400 align:middle line:84% of taking this sum if each of the terms 00:57:56.400 --> 00:57:58.600 align:middle line:84% is multiplied by a different value 00:57:58.600 --> 00:58:01.600 align:middle line:84% so it's not a convenience factor. 00:58:01.600 --> 00:58:03.100 align:middle line:84% IAN BALL: Just as an interpretation. 00:58:03.100 --> 00:58:05.340 align:middle line:84% so let's see in a second, and then I think it'll make sense. 00:58:05.340 --> 00:58:05.880 align:middle line:90% Yeah. 00:58:05.880 --> 00:58:09.080 align:middle line:84% So what can we say about this value? 00:58:09.080 --> 00:58:13.200 align:middle line:84% Well, in general, this value is less than or equal to 1. 00:58:13.200 --> 00:58:17.920 align:middle line:84% And in fact, it's going to equal 1 if alpha n equals 0. 00:58:17.920 --> 00:58:20.880 align:middle line:84% So if we have fewer slots than bidders, 00:58:20.880 --> 00:58:22.560 align:middle line:90% this is going to equal 1. 00:58:22.560 --> 00:58:24.920 align:middle line:84% You might worry if there's fewer slots than bidders. 00:58:24.920 --> 00:58:26.380 align:middle line:84% Are we ever going to divide by 0? 00:58:26.380 --> 00:58:28.280 align:middle line:84% It turns out that top's going to become 0 00:58:28.280 --> 00:58:29.680 align:middle line:84% before the bottom does. and we're not really 00:58:29.680 --> 00:58:31.263 align:middle line:84% going to have to worry about it, but I 00:58:31.263 --> 00:58:32.560 align:middle line:90% will go through the details. 00:58:32.560 --> 00:58:34.960 align:middle line:84% So it turns out this is going to be less than 00:58:34.960 --> 00:58:44.645 align:middle line:84% or equal to 1 with equality if m is strictly less than n-- 00:58:44.645 --> 00:58:46.520 align:middle line:84% if there's fewer slots than bidders, which is 00:58:46.520 --> 00:58:48.600 align:middle line:90% a pretty standard phenomenon. 00:58:48.600 --> 00:58:51.520 align:middle line:84% You have many advertisers wanting to advertise 00:58:51.520 --> 00:58:53.020 align:middle line:90% on just a few limited slots. 00:58:53.020 --> 00:58:56.040 align:middle line:84% So generally, n is going to be bigger than m. 00:58:56.040 --> 00:59:00.960 align:middle line:84% And I guess the trick I've used here is if m is bigger than n, 00:59:00.960 --> 00:59:04.360 align:middle line:84% well, let's just create null slots beyond the m. 00:59:04.360 --> 00:59:07.190 align:middle line:90% I guess I should have said this. 00:59:07.190 --> 00:59:09.440 align:middle line:84% We could have just stopped or we could say, yeah, well 00:59:09.440 --> 00:59:10.780 align:middle line:84% there's actually an n plus first slot, 00:59:10.780 --> 00:59:12.280 align:middle line:84% but it just has 0 clickthrough rate. 00:59:12.280 --> 00:59:13.300 align:middle line:90% It's just a fake slot. 00:59:13.300 --> 00:59:16.540 align:middle line:84% And if we do that, then all these alphas are well-defined. 00:59:16.540 --> 00:59:18.280 align:middle line:90% They're just all 0 after the m. 00:59:18.280 --> 00:59:20.160 align:middle line:84% And if we do that, then we see that this sum 00:59:20.160 --> 00:59:22.320 align:middle line:90% is going to be exactly 1. 00:59:22.320 --> 00:59:23.760 align:middle line:90% So why did we do this sum? 00:59:23.760 --> 00:59:25.760 align:middle line:90% The sum was for interpretation. 00:59:25.760 --> 00:59:28.570 align:middle line:84% What is the payment per click that I make 00:59:28.570 --> 00:59:30.890 align:middle line:90% if I'm the jth highest bidder? 00:59:30.890 --> 00:59:32.930 align:middle line:84% It's a weighted average of all the bids 00:59:32.930 --> 00:59:34.730 align:middle line:90% that are lower than mine. 00:59:34.730 --> 00:59:37.170 align:middle line:84% The weights are non-negative, and the sum 00:59:37.170 --> 00:59:39.930 align:middle line:84% of the weights-- well, in this general case here, 00:59:39.930 --> 00:59:41.050 align:middle line:90% it's exactly 1. 00:59:41.050 --> 00:59:42.670 align:middle line:84% So it's exactly a weighted average. 00:59:42.670 --> 00:59:45.690 align:middle line:90% 00:59:45.690 --> 00:59:50.810 align:middle line:84% And now we have a nice comparison because-- 00:59:50.810 --> 00:59:52.030 align:middle line:90% I'll go back over here. 00:59:52.030 --> 01:00:01.770 align:middle line:90% 01:00:01.770 --> 01:00:05.450 align:middle line:84% If we looked at the GSP, or the Generalized Second Price 01:00:05.450 --> 01:00:08.570 align:middle line:90% auction, pj was what? 01:00:08.570 --> 01:00:10.390 align:middle line:84% Just using our notation, what was pj? 01:00:10.390 --> 01:00:12.550 align:middle line:84% What was the payment per click for the jth slot? 01:00:12.550 --> 01:00:16.970 align:middle line:90% 01:00:16.970 --> 01:00:17.510 align:middle line:90% Right. 01:00:17.510 --> 01:00:26.290 align:middle line:84% It was the j plus first highest bid. 01:00:26.290 --> 01:00:32.410 align:middle line:90% Under the VCG mechanism, pj-- 01:00:32.410 --> 01:00:34.470 align:middle line:84% well, we have this form over here. 01:00:34.470 --> 01:00:37.010 align:middle line:90% But let's just check. 01:00:37.010 --> 01:00:42.210 align:middle line:84% K ranges from j plus 1 to n, so these range from bj plus 1 to n. 01:00:42.210 --> 01:00:57.150 align:middle line:84% So we get a weighted average of all the lower bids. 01:00:57.150 --> 01:01:00.650 align:middle line:90% 01:01:00.650 --> 01:01:04.730 align:middle line:84% And we can exactly see why the VCG-- or at least there's 01:01:04.730 --> 01:01:11.030 align:middle line:84% some intuition for why, under the GSP, we see bid shading. 01:01:11.030 --> 01:01:15.330 align:middle line:90% 01:01:15.330 --> 01:01:19.550 align:middle line:84% And under the VCG, bidding truthfully is optimal. 01:01:19.550 --> 01:01:29.260 align:middle line:90% 01:01:29.260 --> 01:01:31.840 align:middle line:84% In short, it turned out in the GSP-- 01:01:31.840 --> 01:01:35.060 align:middle line:84% while it seemed like the right generalization 01:01:35.060 --> 01:01:38.740 align:middle line:84% to have you pay the next highest bid, that was actually too high. 01:01:38.740 --> 01:01:41.340 align:middle line:84% And as a result, people wanted to shade their bids. 01:01:41.340 --> 01:01:44.300 align:middle line:84% And in a VCG, you're actually ask to pay 01:01:44.300 --> 01:01:46.780 align:middle line:90% less relative to your bids. 01:01:46.780 --> 01:01:48.823 align:middle line:84% Instead of paying the next highest bid, 01:01:48.823 --> 01:01:50.740 align:middle line:84% you pay a weighted average of the next highest 01:01:50.740 --> 01:01:53.460 align:middle line:84% bid, the next highest, the next highest, and so on. 01:01:53.460 --> 01:01:57.160 align:middle line:84% And these later bids are lower than this bid here. 01:01:57.160 --> 01:02:10.380 align:middle line:90% 01:02:10.380 --> 01:02:16.140 align:middle line:84% In particular, let's look at a special case I think 01:02:16.140 --> 01:02:17.320 align:middle line:90% is instructive. 01:02:17.320 --> 01:02:21.300 align:middle line:90% 01:02:21.300 --> 01:02:22.680 align:middle line:90% What if alpha 1-- 01:02:22.680 --> 01:02:28.533 align:middle line:90% 01:02:28.533 --> 01:02:30.700 align:middle line:84% so this technically isn't covered by our assumptions 01:02:30.700 --> 01:02:32.075 align:middle line:84% because we assumed it was strict. 01:02:32.075 --> 01:02:33.580 align:middle line:84% But we can think of this as a limit. 01:02:33.580 --> 01:02:36.900 align:middle line:84% So let's say there's n bidders, and the n slots 01:02:36.900 --> 01:02:39.420 align:middle line:90% are all basically the same. 01:02:39.420 --> 01:02:45.780 align:middle line:84% So we have n bidders and we have n identical slots. 01:02:45.780 --> 01:02:50.500 align:middle line:90% How would people bid in the GSP? 01:02:50.500 --> 01:02:53.280 align:middle line:84% Would you want to bid your evaluation for the slot? 01:02:53.280 --> 01:02:56.060 align:middle line:84% How would you bid if you were in the GSPN 01:02:56.060 --> 01:02:57.440 align:middle line:90% and you knew there were n slots? 01:02:57.440 --> 01:03:04.580 align:middle line:90% 01:03:04.580 --> 01:03:05.390 align:middle line:90% Yeah. 01:03:05.390 --> 01:03:07.140 align:middle line:84% AUDIENCE: You could end up bidding nothing 01:03:07.140 --> 01:03:09.580 align:middle line:84% and still you could just end up-- 01:03:09.580 --> 01:03:11.740 align:middle line:84% even if you're in the last slot, it 01:03:11.740 --> 01:03:14.300 align:middle line:84% doesn't really matter because you'll have the same value. 01:03:14.300 --> 01:03:14.967 align:middle line:90% IAN BALL: Right. 01:03:14.967 --> 01:03:16.760 align:middle line:84% So here, there's really no scarcity. 01:03:16.760 --> 01:03:17.760 align:middle line:90% There's n slots. 01:03:17.760 --> 01:03:19.120 align:middle line:90% They're all equally good. 01:03:19.120 --> 01:03:21.593 align:middle line:84% There's no competition between the bidders at all. 01:03:21.593 --> 01:03:23.510 align:middle line:84% It's not like we're fighting for that top slot 01:03:23.510 --> 01:03:25.470 align:middle line:84% because the next slot is just as good. 01:03:25.470 --> 01:03:30.950 align:middle line:84% So we see that the GSP is going to induce really intense bid 01:03:30.950 --> 01:03:31.450 align:middle line:90% shading. 01:03:31.450 --> 01:03:32.610 align:middle line:84% People are going to shade their bids a lot. 01:03:32.610 --> 01:03:34.870 align:middle line:84% In fact, people are generally going to bid 0. 01:03:34.870 --> 01:03:35.810 align:middle line:90% There's no shortage? 01:03:35.810 --> 01:03:36.650 align:middle line:90% I can just bid 0. 01:03:36.650 --> 01:03:41.022 align:middle line:84% I'll still get one of the n slots, and I'm happy with that. 01:03:41.022 --> 01:03:43.230 align:middle line:84% So you might think, well, how is the VCG going to get 01:03:43.230 --> 01:03:44.630 align:middle line:90% people to be truthful at all? 01:03:44.630 --> 01:03:46.630 align:middle line:84% How could we possibly get people to pay anything 01:03:46.630 --> 01:03:47.870 align:middle line:90% if there's no scarcity? 01:03:47.870 --> 01:03:53.910 align:middle line:84% Well, what is the payment rule going to be in the VCG? 01:03:53.910 --> 01:03:55.390 align:middle line:84% Well, what are these weights going 01:03:55.390 --> 01:03:58.030 align:middle line:84% to be if all the alphas are the same? 01:03:58.030 --> 01:03:59.510 align:middle line:90% AUDIENCE: 0. 01:03:59.510 --> 01:04:00.350 align:middle line:90% IAN BALL: 0. 01:04:00.350 --> 01:04:03.230 align:middle line:84% So in fact, in the VCG mechanism in this special case, 01:04:03.230 --> 01:04:05.470 align:middle line:84% people are always just going to pay 0. 01:04:05.470 --> 01:04:08.030 align:middle line:84% So you might think, oh, VCG-- how do we get something 01:04:08.030 --> 01:04:08.570 align:middle line:90% for nothing? 01:04:08.570 --> 01:04:11.430 align:middle line:84% Well, it's like if people had a really strong incentive to shade 01:04:11.430 --> 01:04:15.333 align:middle line:84% their bid in the GSP, then the VCG 01:04:15.333 --> 01:04:17.750 align:middle line:84% is going to shade on their behalf and just charge everyone 01:04:17.750 --> 01:04:19.208 align:middle line:84% 0 anyway, and then people are going 01:04:19.208 --> 01:04:20.430 align:middle line:90% to be willing to be truthful. 01:04:20.430 --> 01:04:22.790 align:middle line:90% So nothing comes for free. 01:04:22.790 --> 01:04:24.450 align:middle line:84% How do we get people to bid more? 01:04:24.450 --> 01:04:25.750 align:middle line:84% They're willing to bid more because it's 01:04:25.750 --> 01:04:26.770 align:middle line:90% a function of their bids. 01:04:26.770 --> 01:04:27.870 align:middle line:90% They're asked to pay less. 01:04:27.870 --> 01:04:30.790 align:middle line:84% And that's, I think, the key idea or key 01:04:30.790 --> 01:04:33.130 align:middle line:90% insight of the VCG mechanism. 01:04:33.130 --> 01:04:37.590 align:middle line:90% 01:04:37.590 --> 01:04:42.150 align:middle line:84% So maybe in the last 15 minutes, we'll try to-- 01:04:42.150 --> 01:04:46.710 align:middle line:84% I'd like to go over maybe the general formula for VCG 01:04:46.710 --> 01:04:47.690 align:middle line:90% beyond this setting. 01:04:47.690 --> 01:04:49.530 align:middle line:84% So it turns out the VCG mechanism, 01:04:49.530 --> 01:04:51.542 align:middle line:84% it was applied to this setting of ad auctions. 01:04:51.542 --> 01:04:54.250 align:middle line:84% But it makes sense, and it works in a much more abstract setting. 01:04:54.250 --> 01:04:56.450 align:middle line:84% And we're just going to go over how that works. 01:04:56.450 --> 01:05:10.190 align:middle line:90% 01:05:10.190 --> 01:05:12.850 align:middle line:90% So now I'll say abstract VCG. 01:05:12.850 --> 01:05:18.430 align:middle line:90% 01:05:18.430 --> 01:05:21.140 align:middle line:84% So the general setting of VCG is-- well, 01:05:21.140 --> 01:05:24.040 align:middle line:84% we're having to make some decision for all the bidders. 01:05:24.040 --> 01:05:26.560 align:middle line:84% Here, the decision was how we match the bidders 01:05:26.560 --> 01:05:28.080 align:middle line:90% to the ad slots. 01:05:28.080 --> 01:05:32.480 align:middle line:84% Maybe the decision is, do we build a bridge? 01:05:32.480 --> 01:05:34.720 align:middle line:84% There are many, many things it could be. 01:05:34.720 --> 01:05:36.700 align:middle line:84% How we allocate some item among the bidders, 01:05:36.700 --> 01:05:37.740 align:middle line:84% there could be many, many things. 01:05:37.740 --> 01:05:39.323 align:middle line:84% So we're going to be totally abstract. 01:05:39.323 --> 01:05:41.960 align:middle line:84% And the decision is just going to be 01:05:41.960 --> 01:05:45.480 align:middle line:84% some abstract x in some set X. But if you 01:05:45.480 --> 01:05:48.760 align:middle line:84% want to make it concrete, think of x as the allocation 01:05:48.760 --> 01:05:50.062 align:middle line:90% of bidders to slots. 01:05:50.062 --> 01:05:51.520 align:middle line:84% Maybe another classic example would 01:05:51.520 --> 01:05:56.940 align:middle line:84% be the allocation of planes to airport landing slots. 01:05:56.940 --> 01:05:59.440 align:middle line:84% This is a classic application of BCG. 01:05:59.440 --> 01:06:01.940 align:middle line:84% Or maybe routing internet protocols, 01:06:01.940 --> 01:06:03.920 align:middle line:84% routing packets through the network-- that 01:06:03.920 --> 01:06:05.260 align:middle line:90% might be what x represents. 01:06:05.260 --> 01:06:07.593 align:middle line:84% But the crucial thing is x is a collective decision that 01:06:07.593 --> 01:06:11.120 align:middle line:84% affects everyone. x does not represent bidder 1 gets slot 1. 01:06:11.120 --> 01:06:13.680 align:middle line:84% It represents the complete mapping of bidders 01:06:13.680 --> 01:06:16.640 align:middle line:84% to slots or the complete allocation of airplanes 01:06:16.640 --> 01:06:18.140 align:middle line:90% to landing slots, and so on. 01:06:18.140 --> 01:06:21.600 align:middle line:90% 01:06:21.600 --> 01:06:24.520 align:middle line:84% We now need to describe the bidder's preferences 01:06:24.520 --> 01:06:25.740 align:middle line:90% or the player's preferences. 01:06:25.740 --> 01:06:35.080 align:middle line:84% So we have-- let's say player i has valuation vi. 01:06:35.080 --> 01:06:36.983 align:middle line:84% Now, before, vi was just a number. 01:06:36.983 --> 01:06:39.400 align:middle line:84% When we were in a setting where we were allocating a good, 01:06:39.400 --> 01:06:41.080 align:middle line:84% my valuation was just, how much am I 01:06:41.080 --> 01:06:42.288 align:middle line:90% willing to pay for that good? 01:06:42.288 --> 01:06:44.960 align:middle line:84% But now my valuation needs to specify 01:06:44.960 --> 01:06:48.700 align:middle line:84% how much I value every possible social decision that's made. 01:06:48.700 --> 01:06:50.860 align:middle line:84% Let me say social or collective decision. 01:06:50.860 --> 01:06:53.520 align:middle line:90% 01:06:53.520 --> 01:06:58.080 align:middle line:84% So now my valuation is a function, vi, from x to r. 01:06:58.080 --> 01:07:00.070 align:middle line:84% So I'm just going to call it vi, but remember, 01:07:00.070 --> 01:07:01.320 align:middle line:90% it's a much richer object now. 01:07:01.320 --> 01:07:02.778 align:middle line:84% It's not just one number that says, 01:07:02.778 --> 01:07:04.020 align:middle line:90% how much do I value an item? 01:07:04.020 --> 01:07:05.640 align:middle line:84% It's an entire mapping that says, 01:07:05.640 --> 01:07:10.240 align:middle line:84% if this is the allocation of bidders to auction slots, that's 01:07:10.240 --> 01:07:11.460 align:middle line:90% how much utility I get. 01:07:11.460 --> 01:07:12.560 align:middle line:84% If we have a different allocation 01:07:12.560 --> 01:07:14.840 align:middle line:84% of bidders to auction slots, here's how much utility I get, 01:07:14.840 --> 01:07:15.420 align:middle line:90% and so on. 01:07:15.420 --> 01:07:15.920 align:middle line:90% Yeah. 01:07:15.920 --> 01:07:19.450 align:middle line:84% AUDIENCE: What do you to consider how other bidders are 01:07:19.450 --> 01:07:21.330 align:middle line:90% put in different slots? 01:07:21.330 --> 01:07:22.950 align:middle line:84% If you're in a fixed auction slot, 01:07:22.950 --> 01:07:25.470 align:middle line:84% can it just be your auction slot to your utility? 01:07:25.470 --> 01:07:26.637 align:middle line:90% IAN BALL: It's a good point. 01:07:26.637 --> 01:07:30.090 align:middle line:84% So if we're willing to impose more structure on x, maybe 01:07:30.090 --> 01:07:32.390 align:middle line:84% x says, this is what you get, this is what you get, 01:07:32.390 --> 01:07:33.310 align:middle line:90% this is what you got. 01:07:33.310 --> 01:07:36.370 align:middle line:84% And my valuation only depends on my component of x. 01:07:36.370 --> 01:07:38.130 align:middle line:84% But at this level of generality, we're 01:07:38.130 --> 01:07:40.710 align:middle line:84% not even assuming that kind of structure. 01:07:40.710 --> 01:07:44.372 align:middle line:84% And it may be that we want to be-- 01:07:44.372 --> 01:07:45.830 align:middle line:84% the point is, this is more general, 01:07:45.830 --> 01:07:47.770 align:middle line:84% so it may be that if we're allocating planes 01:07:47.770 --> 01:07:50.370 align:middle line:84% to landing slots, I also really like if my competitor gets 01:07:50.370 --> 01:07:51.070 align:middle line:90% a bad slot. 01:07:51.070 --> 01:07:52.510 align:middle line:84% I don't just care about getting a good slot. 01:07:52.510 --> 01:07:54.250 align:middle line:84% So in principle, that could be the case. 01:07:54.250 --> 01:07:56.030 align:middle line:84% Maybe there are ways we can restrict it, 01:07:56.030 --> 01:07:58.150 align:middle line:84% but this is more general and it makes the notation cleaner, 01:07:58.150 --> 01:08:00.025 align:middle line:84% so we're not going to impose that assumption. 01:08:00.025 --> 01:08:02.450 align:middle line:84% But I think that's a very reasonable assumption. 01:08:02.450 --> 01:08:04.450 align:middle line:90% OK. 01:08:04.450 --> 01:08:07.130 align:middle line:90% So what are we going to do? 01:08:07.130 --> 01:08:10.350 align:middle line:84% Well, before, in a second price auction, 01:08:10.350 --> 01:08:12.590 align:middle line:84% we had this idea that people were bidding truthfully. 01:08:12.590 --> 01:08:14.110 align:middle line:84% We'd also like people to bid truthfully. 01:08:14.110 --> 01:08:15.750 align:middle line:84% But if we want people to bid truthfully, 01:08:15.750 --> 01:08:16.890 align:middle line:84% well, we have to just ask them what 01:08:16.890 --> 01:08:18.149 align:middle line:90% their valuation function is. 01:08:18.149 --> 01:08:19.630 align:middle line:84% And that's what the mechanism is going to do. 01:08:19.630 --> 01:08:21.172 align:middle line:84% We're going to say, everyone, tell me 01:08:21.172 --> 01:08:22.550 align:middle line:90% what your valuation function is. 01:08:22.550 --> 01:08:24.670 align:middle line:84% So instead of telling me just how much you value one item, 01:08:24.670 --> 01:08:26.587 align:middle line:84% we're going to say, tell me how much you value 01:08:26.587 --> 01:08:28.250 align:middle line:90% every possible social decision. 01:08:28.250 --> 01:08:39.330 align:middle line:84% So the mechanism, we'll say, elicits reports. 01:08:39.330 --> 01:08:42.930 align:middle line:84% And we'll often denote these v1 hat through vn hat. 01:08:42.930 --> 01:08:44.569 align:middle line:84% So what's the interpretation here? 01:08:44.569 --> 01:08:49.130 align:middle line:84% v1 is bidder 1's true valuation function. 01:08:49.130 --> 01:08:53.410 align:middle line:90% v1 hat is effectively a bid. 01:08:53.410 --> 01:08:55.290 align:middle line:84% It's the valuation function that bidder 1 01:08:55.290 --> 01:08:56.770 align:middle line:90% reports to the mechanism. 01:08:56.770 --> 01:08:58.250 align:middle line:90% And that may not be truthful. 01:08:58.250 --> 01:09:00.667 align:middle line:84% We want to make sure that people have the right incentives 01:09:00.667 --> 01:09:01.810 align:middle line:90% to report truthfully. 01:09:01.810 --> 01:09:03.550 align:middle line:84% But they're reporting an entire function, 01:09:03.550 --> 01:09:07.529 align:middle line:84% which may be a little bit confusing and abstract at first. 01:09:07.529 --> 01:09:11.149 align:middle line:90% So what does the mechanism do? 01:09:11.149 --> 01:09:26.300 align:middle line:90% 01:09:26.300 --> 01:09:27.120 align:middle line:90% We have 5 minutes. 01:09:27.120 --> 01:09:28.745 align:middle line:84% We're just going to write two formulas, 01:09:28.745 --> 01:09:30.819 align:middle line:90% but they're a little long. 01:09:30.819 --> 01:09:33.060 align:middle line:84% So just like an auction, a mechanism 01:09:33.060 --> 01:09:34.717 align:middle line:84% is going to need to specify two things. 01:09:34.717 --> 01:09:37.300 align:middle line:84% It's going to have to specify an allocation rule and a payment 01:09:37.300 --> 01:09:37.800 align:middle line:90% rule. 01:09:37.800 --> 01:09:42.340 align:middle line:90% 01:09:42.340 --> 01:09:50.500 align:middle line:84% We need an allocation rule and a payment rule. 01:09:50.500 --> 01:09:53.380 align:middle line:84% The allocation rule says, if these are the reported valuation 01:09:53.380 --> 01:09:55.740 align:middle line:84% functions that people send to me, what 01:09:55.740 --> 01:09:58.540 align:middle line:90% decision am I going to choose? 01:09:58.540 --> 01:10:00.980 align:middle line:84% And the payment rule says, if these are the valuation 01:10:00.980 --> 01:10:03.140 align:middle line:84% functions that are reported to me, how much am I 01:10:03.140 --> 01:10:07.260 align:middle line:90% going to ask each agent to pay? 01:10:07.260 --> 01:10:16.460 align:middle line:84% So the allocation rule, we need to define xv1 hat vn hat. 01:10:16.460 --> 01:10:20.020 align:middle line:84% These are the valuation functions that are reported. 01:10:20.020 --> 01:10:23.140 align:middle line:84% x of this is going to be the allocation or the decision-- 01:10:23.140 --> 01:10:27.340 align:middle line:90% allocation or decision rule. 01:10:27.340 --> 01:10:28.980 align:middle line:90% I'll use these interchangeably. 01:10:28.980 --> 01:10:32.860 align:middle line:84% Well, we want to allocate things efficiently. 01:10:32.860 --> 01:10:35.300 align:middle line:90% So what does that mean? 01:10:35.300 --> 01:10:37.620 align:middle line:84% This is going to be in the argmax-- 01:10:37.620 --> 01:10:40.620 align:middle line:84% I think the notation here is a little tricky-- of x 01:10:40.620 --> 01:10:45.240 align:middle line:90% prime in x of the sum. 01:10:45.240 --> 01:10:53.820 align:middle line:90% 01:10:53.820 --> 01:10:56.180 align:middle line:84% So let's just understand what this is. 01:10:56.180 --> 01:11:00.900 align:middle line:84% This says, if we choose decision x prime, what 01:11:00.900 --> 01:11:04.780 align:middle line:84% is the sum of the players' utilities when we compute them, 01:11:04.780 --> 01:11:06.280 align:middle line:84% with respect to what they report? 01:11:06.280 --> 01:11:08.940 align:middle line:90% 01:11:08.940 --> 01:11:11.310 align:middle line:84% Taking the argmax of this says we're 01:11:11.310 --> 01:11:13.350 align:middle line:84% going to choose the decision that 01:11:13.350 --> 01:11:16.285 align:middle line:84% maximizes the sum of the reported valuations. 01:11:16.285 --> 01:11:17.910 align:middle line:84% What we'd really like to do is maximize 01:11:17.910 --> 01:11:20.050 align:middle line:84% the sum of the true valuations, but we don't know those. 01:11:20.050 --> 01:11:21.250 align:middle line:90% We only know the reports. 01:11:21.250 --> 01:11:24.630 align:middle line:84% So all we can do is take in the reported valuation function 01:11:24.630 --> 01:11:27.390 align:middle line:84% and choose the decision that makes the sum of the valuations 01:11:27.390 --> 01:11:30.230 align:middle line:90% as large as possible. 01:11:30.230 --> 01:11:31.250 align:middle line:90% And that's what this is. 01:11:31.250 --> 01:11:32.790 align:middle line:84% And sometimes we call this x star 01:11:32.790 --> 01:11:34.430 align:middle line:84% to denote that this is efficient. 01:11:34.430 --> 01:11:37.710 align:middle line:90% 01:11:37.710 --> 01:11:43.970 align:middle line:84% I think this is a bit abstract, so any questions on this? 01:11:43.970 --> 01:11:44.470 align:middle line:90% No? 01:11:44.470 --> 01:11:45.350 align:middle line:90% Good. 01:11:45.350 --> 01:11:46.110 align:middle line:90% OK. 01:11:46.110 --> 01:11:47.753 align:middle line:84% There could be multiple maximizers. 01:11:47.753 --> 01:11:49.170 align:middle line:84% And we'll just choose one of them. 01:11:49.170 --> 01:11:49.710 align:middle line:90% It doesn't matter. 01:11:49.710 --> 01:11:51.770 align:middle line:84% If there's two things, that would be efficient. 01:11:51.770 --> 01:11:56.150 align:middle line:84% So as an example, if we were in the auction context, then 01:11:56.150 --> 01:12:00.170 align:middle line:84% this would be the allocation that gives the highest bidder 01:12:00.170 --> 01:12:02.870 align:middle line:84% the first slot, the second highest bidder the second slot, 01:12:02.870 --> 01:12:03.370 align:middle line:90% and so on. 01:12:03.370 --> 01:12:05.430 align:middle line:84% That's what x star would represent. 01:12:05.430 --> 01:12:08.870 align:middle line:84% And v1 hat would be the report of how much I 01:12:08.870 --> 01:12:13.270 align:middle line:90% value each slot, each click. 01:12:13.270 --> 01:12:16.510 align:middle line:84% Now, the payment rule is exactly what we said before. 01:12:16.510 --> 01:12:20.270 align:middle line:84% We want to charge each bidder their externality. 01:12:20.270 --> 01:12:27.430 align:middle line:84% So ti of v1 through vn hat is saying, 01:12:27.430 --> 01:12:29.150 align:middle line:84% if these are the valuation functions that 01:12:29.150 --> 01:12:30.990 align:middle line:84% are reported by the bidders, how much 01:12:30.990 --> 01:12:33.230 align:middle line:90% are we going to charge bidder i? 01:12:33.230 --> 01:12:42.230 align:middle line:84% We're going to charge bidder i informally the externality 01:12:42.230 --> 01:12:45.790 align:middle line:90% imposed on all others. 01:12:45.790 --> 01:12:50.230 align:middle line:84% So let's write that more formally. 01:12:50.230 --> 01:12:52.370 align:middle line:84% What is the externality imposed on all others? 01:12:52.370 --> 01:12:54.550 align:middle line:84% It's how well everyone else would 01:12:54.550 --> 01:12:59.630 align:middle line:84% do if bidder i weren't there minus how well they actually do. 01:12:59.630 --> 01:13:01.243 align:middle line:90% That's what the externality is. 01:13:01.243 --> 01:13:03.410 align:middle line:84% So how well would they do if bidder i weren't there? 01:13:03.410 --> 01:13:05.785 align:middle line:84% Well, let's just look at the best they could possibly do. 01:13:05.785 --> 01:13:10.272 align:middle line:84% So the best they could possibly do if bidder i weren't there 01:13:10.272 --> 01:13:11.380 align:middle line:90% would be this. 01:13:11.380 --> 01:13:16.680 align:middle line:90% 01:13:16.680 --> 01:13:18.540 align:middle line:84% I'm looking at everyone other than bidder i. 01:13:18.540 --> 01:13:22.160 align:middle line:84% I'm using their reported valuation functions. 01:13:22.160 --> 01:13:24.440 align:middle line:84% And I'm saying, if we chose x prime, 01:13:24.440 --> 01:13:27.480 align:middle line:84% this is how well everyone but player i would do. 01:13:27.480 --> 01:13:31.440 align:middle line:84% And let me choose the allocation that makes those people as well 01:13:31.440 --> 01:13:32.880 align:middle line:90% off as possible. 01:13:32.880 --> 01:13:35.360 align:middle line:84% So I want to maximize the welfare of everyone other 01:13:35.360 --> 01:13:36.960 align:middle line:90% than bidder i. 01:13:36.960 --> 01:13:39.180 align:middle line:84% That's what we could do if bitter i weren't here-- 01:13:39.180 --> 01:13:41.520 align:middle line:84% or in other words, if we didn't care about bitter i, 01:13:41.520 --> 01:13:44.320 align:middle line:84% this is how well everyone else could do. 01:13:44.320 --> 01:13:47.480 align:middle line:84% Now I want to subtract from that how well they actually do, 01:13:47.480 --> 01:13:50.800 align:middle line:84% how well they do given that bidder i is present. 01:13:50.800 --> 01:13:52.160 align:middle line:90% How well do they actually do? 01:13:52.160 --> 01:13:56.280 align:middle line:84% Well, it's the sum over j not equal to i-- 01:13:56.280 --> 01:13:57.560 align:middle line:90% vj hat. 01:13:57.560 --> 01:13:59.600 align:middle line:84% But now what I want to plug in here 01:13:59.600 --> 01:14:03.120 align:middle line:84% is the decision that's actually chosen. 01:14:03.120 --> 01:14:05.620 align:middle line:84% And the decision that's actually chosen is this one up here. 01:14:05.620 --> 01:14:16.240 align:middle line:90% 01:14:16.240 --> 01:14:18.160 align:middle line:90% And this is the formula. 01:14:18.160 --> 01:14:19.880 align:middle line:90% It's a little abstract. 01:14:19.880 --> 01:14:21.800 align:middle line:90% I'll put this in parentheses. 01:14:21.800 --> 01:14:24.800 align:middle line:84% And if you go to the notes, you can see a proof 01:14:24.800 --> 01:14:31.150 align:middle line:84% as to why using this mechanism, it's always optimal for bidders 01:14:31.150 --> 01:14:31.900 align:middle line:90% to bid truthfully. 01:14:31.900 --> 01:14:33.460 align:middle line:90% So let me just write that fact. 01:14:33.460 --> 01:14:35.140 align:middle line:84% This is a fact that you can prove. 01:14:35.140 --> 01:14:39.520 align:middle line:90% 01:14:39.520 --> 01:14:56.720 align:middle line:84% So for each bidder i and each valuation function vi, 01:14:56.720 --> 01:14:59.820 align:middle line:90% bidding vi hat equals vi. 01:14:59.820 --> 01:15:03.440 align:middle line:84% What I mean by that is reporting my true valuation 01:15:03.440 --> 01:15:11.310 align:middle line:84% to the mechanism is optimal, no matter how everyone else bids. 01:15:11.310 --> 01:15:18.642 align:middle line:90% 01:15:18.642 --> 01:15:20.350 align:middle line:84% And when I say no matter what others bid, 01:15:20.350 --> 01:15:27.370 align:middle line:84% well, that's basically vj hat for j not equal to i. 01:15:27.370 --> 01:15:30.130 align:middle line:84% So whatever everyone else other than me reports 01:15:30.130 --> 01:15:34.370 align:middle line:84% to the mechanism, if my true value is vi, 01:15:34.370 --> 01:15:36.585 align:middle line:84% bidding or reporting my true value vi 01:15:36.585 --> 01:15:37.710 align:middle line:90% is going to be best for me. 01:15:37.710 --> 01:15:41.010 align:middle line:84% It's going to be weakly better than any other bid. 01:15:41.010 --> 01:15:46.810 align:middle line:84% And this is an incredible property of the VCG mechanism. 01:15:46.810 --> 01:15:49.310 align:middle line:84% The proof is three lines, but it's a little dense, 01:15:49.310 --> 01:15:52.850 align:middle line:84% so I encourage you to check the notes to see the proof. 01:15:52.850 --> 01:15:56.588 align:middle line:84% And maybe I'll stop here a few minutes early. 01:15:56.588 --> 01:15:59.130 align:middle line:84% And I'm happy to stick around to go over any questions people 01:15:59.130 --> 01:16:01.200 align:middle line:90% have about VCG. 01:16:01.200 --> 01:16:04.000 align:middle line:90%