WEBVTT 00:00:00.000 --> 00:00:01.961 align:middle line:90% [SQUEAKING] 00:00:01.961 --> 00:00:03.909 align:middle line:90% [RUSTLING] 00:00:03.909 --> 00:00:09.050 align:middle line:90% [CLICKING] 00:00:09.050 --> 00:00:10.800 align:middle line:84% IAN BALL: So today we're going to continue 00:00:10.800 --> 00:00:12.380 align:middle line:90% with our study of auctions. 00:00:12.380 --> 00:00:14.440 align:middle line:84% And we're going to study a bit of a richer 00:00:14.440 --> 00:00:16.600 align:middle line:84% model than the example that we looked at so far. 00:00:16.600 --> 00:00:21.560 align:middle line:84% But let's just recall what we've done so far with auctions. 00:00:21.560 --> 00:00:24.640 align:middle line:84% So far, we looked at this very simple example 00:00:24.640 --> 00:00:27.480 align:middle line:84% where we just had two bidders, and each bidder's valuation 00:00:27.480 --> 00:00:30.520 align:middle line:84% for the item was uniformly distributed 00:00:30.520 --> 00:00:32.180 align:middle line:90% on the 0, 1 interval. 00:00:32.180 --> 00:00:33.720 align:middle line:90% So let's look at this example. 00:00:33.720 --> 00:00:43.040 align:middle line:84% We had our example where we had two bidders, 00:00:43.040 --> 00:00:57.680 align:middle line:84% and their valuations v1 and v2 were independent and uniform 00:00:57.680 --> 00:01:00.940 align:middle line:90% on the interval 0, 1. 00:01:00.940 --> 00:01:03.620 align:middle line:84% And we looked at two different auction 00:01:03.620 --> 00:01:06.300 align:middle line:90% formats in this simple example. 00:01:06.300 --> 00:01:09.760 align:middle line:84% The first auction format would be a first-price auction. 00:01:09.760 --> 00:01:16.260 align:middle line:90% 00:01:16.260 --> 00:01:17.960 align:middle line:84% Remember, in a first-price auction, 00:01:17.960 --> 00:01:19.900 align:middle line:90% each bidder submits a bid. 00:01:19.900 --> 00:01:23.500 align:middle line:84% The person who submits a higher bid wins the item. 00:01:23.500 --> 00:01:26.638 align:middle line:84% And the amount that they pay is the amount of their bid. 00:01:26.638 --> 00:01:29.180 align:middle line:84% And what we showed is that in a first-price auction, at least 00:01:29.180 --> 00:01:31.940 align:middle line:84% in this example, we had an equilibrium 00:01:31.940 --> 00:01:36.340 align:middle line:84% where each bidder bid exactly half of her valuation. 00:01:36.340 --> 00:01:41.380 align:middle line:84% So we had an equilibrium where maybe b1 star of v1 00:01:41.380 --> 00:01:50.020 align:middle line:84% equals v1 over m and b2 star of v2 equals v2 over 2. 00:01:50.020 --> 00:01:52.780 align:middle line:84% So this was a symmetric equilibrium. 00:01:52.780 --> 00:01:56.940 align:middle line:84% Each bidder bids exactly the same way, given their valuation 00:01:56.940 --> 00:01:57.520 align:middle line:90% evaluation. 00:01:57.520 --> 00:02:00.680 align:middle line:84% And of course, we noticed that each bidder is always 00:02:00.680 --> 00:02:03.620 align:middle line:84% going to bid strictly less than their true valuation. 00:02:03.620 --> 00:02:05.180 align:middle line:84% If they bid their true valuation, 00:02:05.180 --> 00:02:06.680 align:middle line:84% well, they can never come out ahead, 00:02:06.680 --> 00:02:08.303 align:middle line:84% because even if they won the auction, 00:02:08.303 --> 00:02:10.720 align:middle line:84% they'd end up paying exactly what they value the good for. 00:02:10.720 --> 00:02:13.480 align:middle line:84% So this way, there's a chance that when they win-- 00:02:13.480 --> 00:02:15.000 align:middle line:84% or when they do win, they're always 00:02:15.000 --> 00:02:19.520 align:middle line:84% paying strictly less than their valuation for the good. 00:02:19.520 --> 00:02:23.660 align:middle line:84% Another auction you could run is a second-price auction. 00:02:23.660 --> 00:02:27.360 align:middle line:90% 00:02:27.360 --> 00:02:30.520 align:middle line:84% And in a second-price auction, the bidders submit their bids, 00:02:30.520 --> 00:02:32.360 align:middle line:84% just like in a first-price auction, 00:02:32.360 --> 00:02:34.560 align:middle line:84% and the rule for allocating the good 00:02:34.560 --> 00:02:36.540 align:middle line:84% is the same as in a first-price auction. 00:02:36.540 --> 00:02:38.960 align:middle line:84% The person who bids the most gets the item. 00:02:38.960 --> 00:02:41.000 align:middle line:84% The difference from a first-price auction 00:02:41.000 --> 00:02:42.320 align:middle line:90% is the payment rule. 00:02:42.320 --> 00:02:45.560 align:middle line:84% So in a second-price auction, the winning bidder 00:02:45.560 --> 00:02:47.280 align:middle line:84% doesn't pay their own bid, but they 00:02:47.280 --> 00:02:49.600 align:middle line:84% pay the second-highest bid-- that is the highest 00:02:49.600 --> 00:02:52.240 align:middle line:90% bid among everyone else. 00:02:52.240 --> 00:02:55.020 align:middle line:84% And we argued informally that in the second-price auction, 00:02:55.020 --> 00:02:57.550 align:middle line:84% it's optimal to bid your true valuation. 00:02:57.550 --> 00:03:03.610 align:middle line:84% So here, we have an equilibrium where b1 star of b1 equals v1, 00:03:03.610 --> 00:03:07.650 align:middle line:90% and b2 star of v2 equals v2. 00:03:07.650 --> 00:03:11.850 align:middle line:84% And I want to start by arguing why this is equilibrium, why you 00:03:11.850 --> 00:03:14.450 align:middle line:84% always want to bid exactly your true valuation 00:03:14.450 --> 00:03:15.670 align:middle line:90% in a second-price auction. 00:03:15.670 --> 00:03:18.690 align:middle line:90% 00:03:18.690 --> 00:03:22.050 align:middle line:84% So to see this, let's consider a Second-Price Auction, 00:03:22.050 --> 00:03:26.110 align:middle line:84% SPA, and let's take the perspective of bidder 1. 00:03:26.110 --> 00:03:28.770 align:middle line:90% 00:03:28.770 --> 00:03:31.690 align:middle line:84% So suppose I'm bidder 1, and suppose my valuation 00:03:31.690 --> 00:03:32.990 align:middle line:90% for the good is v1. 00:03:32.990 --> 00:03:38.650 align:middle line:90% 00:03:38.650 --> 00:03:42.290 align:middle line:84% And we want to look at different ways I could bid. 00:03:42.290 --> 00:03:44.130 align:middle line:84% So we want to argue that it's always better 00:03:44.130 --> 00:03:48.130 align:middle line:84% to bid my true valuation v1 than to make a different bid. 00:03:48.130 --> 00:03:49.670 align:middle line:84% Well, how could I bid differently? 00:03:49.670 --> 00:03:51.930 align:middle line:84% Well, there's two rough ways I could bid differently. 00:03:51.930 --> 00:03:54.750 align:middle line:84% I could either bid more than my valuation 00:03:54.750 --> 00:03:56.170 align:middle line:90% or less than my valuation. 00:03:56.170 --> 00:03:58.870 align:middle line:84% So we're going to analyze those two cases separately. 00:03:58.870 --> 00:04:05.190 align:middle line:84% So let's think about our bid space here. 00:04:05.190 --> 00:04:07.110 align:middle line:90% So this is 0. 00:04:07.110 --> 00:04:12.830 align:middle line:84% And let's fix our valuation here, v1. 00:04:12.830 --> 00:04:14.390 align:middle line:84% And we want to first see, well what 00:04:14.390 --> 00:04:17.149 align:middle line:84% if instead of bidding my true valuation v1, 00:04:17.149 --> 00:04:24.790 align:middle line:84% I chose an alternative bid, b1 prime that was, say, here. 00:04:24.790 --> 00:04:30.390 align:middle line:84% So I'm contemplating a deviation where instead of bidding v1, 00:04:30.390 --> 00:04:31.950 align:middle line:90% I bid b1 prime. 00:04:31.950 --> 00:04:34.790 align:middle line:84% So I bid strictly higher than my true valuation. 00:04:34.790 --> 00:04:36.910 align:middle line:84% And we want to argue that this can never help me, 00:04:36.910 --> 00:04:40.110 align:middle line:84% that I'm never better off by bidding b1 00:04:40.110 --> 00:04:42.710 align:middle line:90% prime than I am bidding v1. 00:04:42.710 --> 00:04:45.350 align:middle line:84% And to see this, well, what happens in this auction 00:04:45.350 --> 00:04:48.190 align:middle line:84% is going to depend on what the other bidder bids. 00:04:48.190 --> 00:04:52.570 align:middle line:84% So I want to think of this axis here as representing what 00:04:52.570 --> 00:04:54.390 align:middle line:90% the other bidder is bidding. 00:04:54.390 --> 00:04:57.050 align:middle line:84% And then I want to compare the performance of this bid 00:04:57.050 --> 00:05:01.130 align:middle line:84% to this bid for each possible way that the other player bids. 00:05:01.130 --> 00:05:03.250 align:middle line:90% So let's first start down here. 00:05:03.250 --> 00:05:06.770 align:middle line:84% Suppose the other player b2, the other player 00:05:06.770 --> 00:05:09.210 align:middle line:84% makes a bid b2 that is, down here. 00:05:09.210 --> 00:05:11.450 align:middle line:90% It's strictly smaller than v1. 00:05:11.450 --> 00:05:14.570 align:middle line:90% OK, so what happens if I bid v1? 00:05:14.570 --> 00:05:18.970 align:middle line:84% And what happens if I bid b1 prime? 00:05:18.970 --> 00:05:19.510 align:middle line:90% Let's start. 00:05:19.510 --> 00:05:22.710 align:middle line:84% What if I bid v1 and the other player bids b2? 00:05:22.710 --> 00:05:23.592 align:middle line:90% What happens? 00:05:23.592 --> 00:05:25.550 align:middle line:84% What's the outcome of the second-price auction? 00:05:25.550 --> 00:05:26.690 align:middle line:90% Yeah, in the front? 00:05:26.690 --> 00:05:28.250 align:middle line:90% AUDIENCE: You pay b2, right? 00:05:28.250 --> 00:05:29.730 align:middle line:84% IAN BALL: You pay b2, and you win. 00:05:29.730 --> 00:05:30.930 align:middle line:90% Good. 00:05:30.930 --> 00:05:32.370 align:middle line:90% And you win the good. 00:05:32.370 --> 00:05:35.650 align:middle line:84% What if you had instead bid b1 prime? 00:05:35.650 --> 00:05:36.290 align:middle line:90% Yeah. 00:05:36.290 --> 00:05:36.870 align:middle line:90% AUDIENCE: The same thing. 00:05:36.870 --> 00:05:37.828 align:middle line:90% IAN BALL: Same outcome. 00:05:37.828 --> 00:05:40.150 align:middle line:84% So if the opponent's bid is down here, 00:05:40.150 --> 00:05:43.590 align:middle line:84% it doesn't make a difference whether I bid v1 or b1 prime. 00:05:43.590 --> 00:05:47.890 align:middle line:84% Either way, I win the item, and I pay the bid b2. 00:05:47.890 --> 00:05:54.870 align:middle line:84% So here, we'll say no difference between v1 and b1 prime. 00:05:54.870 --> 00:05:56.330 align:middle line:90% What if b2 is up here? 00:05:56.330 --> 00:06:00.190 align:middle line:90% 00:06:00.190 --> 00:06:04.230 align:middle line:84% Now what happens if I bid v1, and if I bid b1 prime? 00:06:04.230 --> 00:06:04.870 align:middle line:90% Yeah? 00:06:04.870 --> 00:06:05.790 align:middle line:90% AUDIENCE: You lose either way. 00:06:05.790 --> 00:06:07.040 align:middle line:90% IAN BALL: You lose either way. 00:06:07.040 --> 00:06:09.270 align:middle line:84% So here, it also doesn't make a difference. 00:06:09.270 --> 00:06:13.292 align:middle line:84% They're outbidding me whether I bid b1 prime or v1, so I lose. 00:06:13.292 --> 00:06:15.250 align:middle line:84% I don't get the item, and I don't pay anything, 00:06:15.250 --> 00:06:16.333 align:middle line:90% so it makes no difference. 00:06:16.333 --> 00:06:20.550 align:middle line:90% 00:06:20.550 --> 00:06:22.610 align:middle line:90% What if they bid in here? 00:06:22.610 --> 00:06:27.950 align:middle line:90% 00:06:27.950 --> 00:06:30.870 align:middle line:84% And we could worry about them bidding exactly V1 or exactly b1 00:06:30.870 --> 00:06:33.010 align:middle line:84% prime, but that just adds some complications. 00:06:33.010 --> 00:06:36.110 align:middle line:84% So let's say it's strictly between v1 and b1 prime. 00:06:36.110 --> 00:06:39.150 align:middle line:90% Now what happens if I bid v1? 00:06:39.150 --> 00:06:44.670 align:middle line:84% I lose, because I've been outbid by my opponent, so I get 0. 00:06:44.670 --> 00:06:48.610 align:middle line:84% If I instead bid b1 prime, well, this sounds pretty good, 00:06:48.610 --> 00:06:49.670 align:middle line:90% because now I win. 00:06:49.670 --> 00:06:51.170 align:middle line:90% I win the auction. 00:06:51.170 --> 00:06:53.970 align:middle line:84% But what happens if I win by bidding b1 prime 00:06:53.970 --> 00:06:56.730 align:middle line:90% when my opponent bids b2 here? 00:06:56.730 --> 00:06:57.450 align:middle line:90% Yeah? 00:06:57.450 --> 00:07:00.062 align:middle line:84% AUDIENCE: You're paying more than your actual valuation. 00:07:00.062 --> 00:07:01.770 align:middle line:84% IAN BALL: So I'm actually worse off here, 00:07:01.770 --> 00:07:05.770 align:middle line:84% because I could bid my true valuation v1, 00:07:05.770 --> 00:07:08.970 align:middle line:84% pay nothing, and win nothing, and my payoff is 0. 00:07:08.970 --> 00:07:12.850 align:middle line:84% Or if I bid b1 prime, I outbid my opponent. 00:07:12.850 --> 00:07:16.130 align:middle line:84% I win the item, but now I have to pay the second-highest bid, 00:07:16.130 --> 00:07:17.690 align:middle line:90% which in this case is b2. 00:07:17.690 --> 00:07:20.550 align:middle line:84% So I'm paying more for the item than it's actually worth to me. 00:07:20.550 --> 00:07:24.070 align:middle line:84% So I'm strictly worse off by bidding b1 prime than v1. 00:07:24.070 --> 00:07:28.130 align:middle line:84% So in this case, I'll say v1 is strictly preferred-- 00:07:28.130 --> 00:07:30.290 align:middle line:90% this is kind of a squiggly-- 00:07:30.290 --> 00:07:32.330 align:middle line:90% to b1 prime. 00:07:32.330 --> 00:07:34.390 align:middle line:84% So there's three separate cases we've analyzed. 00:07:34.390 --> 00:07:36.090 align:middle line:84% In two of the cases, it doesn't matter 00:07:36.090 --> 00:07:38.250 align:middle line:90% whether I bid v1 or b1 prime. 00:07:38.250 --> 00:07:40.530 align:middle line:84% And in one of the cases I strictly prefer 00:07:40.530 --> 00:07:43.890 align:middle line:84% bidding my true valuation of v1 to bidding b1 prime. 00:07:43.890 --> 00:07:46.620 align:middle line:84% So the argument is almost done, but we only 00:07:46.620 --> 00:07:49.348 align:middle line:84% considered bids strictly higher than my valuation. 00:07:49.348 --> 00:07:51.140 align:middle line:84% We now have to go through the same argument 00:07:51.140 --> 00:07:53.260 align:middle line:84% and make sure that I never want to bid strictly 00:07:53.260 --> 00:07:54.280 align:middle line:90% less than my valuation. 00:07:54.280 --> 00:07:58.220 align:middle line:84% So let's go through the same thing over here. 00:07:58.220 --> 00:07:59.580 align:middle line:90% So same figure. 00:07:59.580 --> 00:08:04.860 align:middle line:84% Here's v1, but now we're going to contemplate a downward bid. 00:08:04.860 --> 00:08:07.100 align:middle line:84% And you might think this is the more common bid when 00:08:07.100 --> 00:08:09.180 align:middle line:84% we played the-- or the more common mistake. 00:08:09.180 --> 00:08:12.460 align:middle line:84% When we played this first-price or the second-price auction, 00:08:12.460 --> 00:08:14.720 align:middle line:84% some people-- actually, most people bid pretty well, 00:08:14.720 --> 00:08:17.960 align:middle line:84% but some people bid strictly less than their valuation. 00:08:17.960 --> 00:08:20.320 align:middle line:84% So let's see whether this is a good idea. 00:08:20.320 --> 00:08:24.272 align:middle line:84% And let's again analyze our three cases. 00:08:24.272 --> 00:08:25.480 align:middle line:90% And again, I say three cases. 00:08:25.480 --> 00:08:27.563 align:middle line:84% We have to be a little careful if it's right here, 00:08:27.563 --> 00:08:29.400 align:middle line:84% but it's not going to change anything. 00:08:29.400 --> 00:08:33.179 align:middle line:90% 00:08:33.179 --> 00:08:37.580 align:middle line:84% So case 1, what if my opponent's bid is down here? 00:08:37.580 --> 00:08:39.500 align:middle line:84% It's strictly lower than b1 prime 00:08:39.500 --> 00:08:41.299 align:middle line:90% and it's strictly lower than v1? 00:08:41.299 --> 00:08:42.700 align:middle line:90% Well, same thing. 00:08:42.700 --> 00:08:45.600 align:middle line:84% I lose either way-- sorry, I win either way. 00:08:45.600 --> 00:08:48.880 align:middle line:84% And I pay b2 either way, so there's no difference. 00:08:48.880 --> 00:08:54.080 align:middle line:90% 00:08:54.080 --> 00:08:56.520 align:middle line:90% Up here, well, I'm being outbid. 00:08:56.520 --> 00:09:00.320 align:middle line:84% b2 is strictly higher than v1, and it's strictly higher than b1 00:09:00.320 --> 00:09:01.000 align:middle line:90% prime. 00:09:01.000 --> 00:09:04.120 align:middle line:84% So whether I bid my true valuation or b1 prime, 00:09:04.120 --> 00:09:05.480 align:middle line:90% I lose the auction. 00:09:05.480 --> 00:09:07.720 align:middle line:90% I pay nothing, I get nothing. 00:09:07.720 --> 00:09:09.580 align:middle line:84% And therefore, it makes no difference. 00:09:09.580 --> 00:09:12.160 align:middle line:90% 00:09:12.160 --> 00:09:16.440 align:middle line:84% But again, the critical region is this middle region, 00:09:16.440 --> 00:09:20.560 align:middle line:84% where my opponent's bid b2 is strictly between b1 00:09:20.560 --> 00:09:22.080 align:middle line:90% prime and v1. 00:09:22.080 --> 00:09:25.460 align:middle line:84% So what happens here if I bid truthfully and bid v1? 00:09:25.460 --> 00:09:30.600 align:middle line:90% 00:09:30.600 --> 00:09:33.265 align:middle line:90% AUDIENCE: You win it and pay b2. 00:09:33.265 --> 00:09:33.932 align:middle line:90% IAN BALL: Right. 00:09:33.932 --> 00:09:36.720 align:middle line:84% And what if I had instead bid b1 prime? 00:09:36.720 --> 00:09:37.920 align:middle line:90% AUDIENCE: You'd lose. 00:09:37.920 --> 00:09:38.712 align:middle line:90% IAN BALL: I'd lose. 00:09:38.712 --> 00:09:41.000 align:middle line:84% So I'm better off bidding truthfully, 00:09:41.000 --> 00:09:43.800 align:middle line:84% because I win and pay less than my valuation, 00:09:43.800 --> 00:09:45.780 align:middle line:84% so I make a strictly positive utility. 00:09:45.780 --> 00:09:48.600 align:middle line:84% Whereas if I had deviated and bid b1 prime, 00:09:48.600 --> 00:09:51.720 align:middle line:84% I would have lost the auction, won nothing, paid nothing, 00:09:51.720 --> 00:09:53.240 align:middle line:84% and therefore got a utility of 0. 00:09:53.240 --> 00:09:59.820 align:middle line:84% So once again, we see that v1 is strictly better than b1 prime. 00:09:59.820 --> 00:10:02.140 align:middle line:84% So now we've completed our graphical argument 00:10:02.140 --> 00:10:06.860 align:middle line:84% to show that bidding truthfully is better than bidding anything 00:10:06.860 --> 00:10:07.580 align:middle line:90% else. 00:10:07.580 --> 00:10:10.020 align:middle line:84% And in fact, what we've shown is that bidding truthfully 00:10:10.020 --> 00:10:13.260 align:middle line:84% is actually a weakly dominant strategy. 00:10:13.260 --> 00:10:15.140 align:middle line:84% So normally, with auction equilibria 00:10:15.140 --> 00:10:18.220 align:middle line:84% in, say, first-price auctions, the best way for you to bid 00:10:18.220 --> 00:10:20.540 align:middle line:84% depends on the way your opponent bids. 00:10:20.540 --> 00:10:23.540 align:middle line:84% In this case, bidding truthfully is best no matter 00:10:23.540 --> 00:10:25.660 align:middle line:84% how your opponent bids, no matter what strategy 00:10:25.660 --> 00:10:26.600 align:middle line:90% your opponent uses. 00:10:26.600 --> 00:10:27.100 align:middle line:90% Yes. 00:10:27.100 --> 00:10:28.340 align:middle line:90% Question? 00:10:28.340 --> 00:10:30.345 align:middle line:84% AUDIENCE: I may have missed it, but-- 00:10:30.345 --> 00:10:31.220 align:middle line:90% IAN BALL: Yeah, fine. 00:10:31.220 --> 00:10:34.380 align:middle line:84% AUDIENCE: --why isn't bidding b1 prime 00:10:34.380 --> 00:10:38.420 align:middle line:84% when your opponent bids b2 all the way at the bottom better 00:10:38.420 --> 00:10:39.360 align:middle line:90% than v1? 00:10:39.360 --> 00:10:40.400 align:middle line:90% Like don't you pay-- 00:10:40.400 --> 00:10:40.780 align:middle line:90% IAN BALL: Let's see. 00:10:40.780 --> 00:10:42.020 align:middle line:90% So there's six cases here. 00:10:42.020 --> 00:10:45.000 align:middle line:90% So you're in this one, case 1? 00:10:45.000 --> 00:10:47.100 align:middle line:84% AUDIENCE: It's the graph on the right. 00:10:47.100 --> 00:10:48.808 align:middle line:90% And so you see-- 00:10:48.808 --> 00:10:49.600 align:middle line:90% IAN BALL: And here? 00:10:49.600 --> 00:10:49.960 align:middle line:90% AUDIENCE: Yeah. 00:10:49.960 --> 00:10:50.280 align:middle line:90% IAN BALL: OK. 00:10:50.280 --> 00:10:50.780 align:middle line:90% Great. 00:10:50.780 --> 00:10:51.580 align:middle line:90% Let's go over this. 00:10:51.580 --> 00:10:55.800 align:middle line:84% So let's suppose my opponent bids b2 here. 00:10:55.800 --> 00:10:57.253 align:middle line:90% What happens if I bid v1? 00:10:57.253 --> 00:10:58.920 align:middle line:84% AUDIENCE: Oh, because it's second price. 00:10:58.920 --> 00:10:59.980 align:middle line:84% IAN BALL: Because the second-price auction. 00:10:59.980 --> 00:11:02.147 align:middle line:84% It wouldn't be true if it was a first-price auction. 00:11:02.147 --> 00:11:02.920 align:middle line:90% Either way, I win. 00:11:02.920 --> 00:11:05.040 align:middle line:84% And the crucial point about a second-price auction 00:11:05.040 --> 00:11:08.518 align:middle line:84% is when I win, the amount I pay doesn't depend on what I bid. 00:11:08.518 --> 00:11:10.560 align:middle line:84% So the crucial property of a second-price auction 00:11:10.560 --> 00:11:13.720 align:middle line:84% is my bid can affect whether I win. 00:11:13.720 --> 00:11:17.020 align:middle line:84% But if I win, the amount I pay does not depend on my bid. 00:11:17.020 --> 00:11:19.360 align:middle line:84% It only depends on my opponent's bid. 00:11:19.360 --> 00:11:23.000 align:middle line:84% So in this case, if I bid here or here, I win either way, 00:11:23.000 --> 00:11:24.360 align:middle line:90% and I pay b2. 00:11:24.360 --> 00:11:25.240 align:middle line:90% Good. 00:11:25.240 --> 00:11:27.640 align:middle line:90% Great. 00:11:27.640 --> 00:11:30.523 align:middle line:84% So now we've analyzed this example. 00:11:30.523 --> 00:11:32.440 align:middle line:84% I think you can see that this kind of argument 00:11:32.440 --> 00:11:33.607 align:middle line:90% is actually going to extend. 00:11:33.607 --> 00:11:35.780 align:middle line:84% We didn't really use the uniform distribution here. 00:11:35.780 --> 00:11:37.520 align:middle line:84% This is a much more general property 00:11:37.520 --> 00:11:40.480 align:middle line:84% of second-price auctions that it's optimal to bid truthfully. 00:11:40.480 --> 00:11:45.900 align:middle line:84% But I want to make one kind of interesting observation here. 00:11:45.900 --> 00:11:47.900 align:middle line:84% We've looked at two different auction formats, 00:11:47.900 --> 00:11:50.025 align:middle line:84% and we've solved for an equilibrium in each auction 00:11:50.025 --> 00:11:50.620 align:middle line:90% format. 00:11:50.620 --> 00:11:53.460 align:middle line:84% So a natural question to ask is, which auction format 00:11:53.460 --> 00:11:54.880 align:middle line:90% is better for the auctioneer? 00:11:54.880 --> 00:11:58.220 align:middle line:90% 00:11:58.220 --> 00:12:01.600 align:middle line:84% Which auction does the auctioneer prefer? 00:12:01.600 --> 00:12:07.280 align:middle line:90% 00:12:07.280 --> 00:12:09.780 align:middle line:84% Well, to say "prefer," we have to talk about the auctioneers 00:12:09.780 --> 00:12:10.680 align:middle line:90% preferences. 00:12:10.680 --> 00:12:12.780 align:middle line:84% Let's assume that the auctioneer likes money 00:12:12.780 --> 00:12:14.060 align:middle line:90% and they're risk neutral. 00:12:14.060 --> 00:12:16.460 align:middle line:84% So the goal of the auctioneer is to maximize 00:12:16.460 --> 00:12:18.580 align:middle line:84% their expected revenue in the auction. 00:12:18.580 --> 00:12:22.620 align:middle line:84% So now let's compute their expected revenue 00:12:22.620 --> 00:12:24.440 align:middle line:90% in the first-price auction. 00:12:24.440 --> 00:12:34.500 align:middle line:90% 00:12:34.500 --> 00:12:37.570 align:middle line:84% Well, let's first just look at their revenue. 00:12:37.570 --> 00:12:38.690 align:middle line:90% Maybe we'll call it pi. 00:12:38.690 --> 00:12:40.270 align:middle line:84% Pi is a good notation for revenue. 00:12:40.270 --> 00:12:45.270 align:middle line:84% So what is their revenue in the first-price auction 00:12:45.270 --> 00:12:48.590 align:middle line:84% if bidder 1's value for the good is v1, 00:12:48.590 --> 00:12:51.030 align:middle line:84% and bidder two's value for the good is v2, 00:12:51.030 --> 00:12:54.190 align:middle line:84% and the bidders are following the first-price auction 00:12:54.190 --> 00:12:56.990 align:middle line:90% strategy. 00:12:56.990 --> 00:13:01.790 align:middle line:84% Well, that means bidder 1 is going to bid v1 over 2. 00:13:01.790 --> 00:13:04.545 align:middle line:84% Bidder 2 is going to bid v2 over 2. 00:13:04.545 --> 00:13:07.170 align:middle line:84% And how much revenue then am I going to make as the auctioneer? 00:13:07.170 --> 00:13:12.730 align:middle line:90% 00:13:12.730 --> 00:13:13.230 align:middle line:90% Yeah? 00:13:13.230 --> 00:13:14.670 align:middle line:84% AUDIENCE: Whichever of those is larger. 00:13:14.670 --> 00:13:15.130 align:middle line:90% IAN BALL: Right. 00:13:15.130 --> 00:13:16.838 align:middle line:84% So one way of writing that mathematically 00:13:16.838 --> 00:13:18.670 align:middle line:84% is just the maximum of these two. 00:13:18.670 --> 00:13:22.970 align:middle line:84% I'm going to make the maximum of v1 over 2 and v2 over 2. 00:13:22.970 --> 00:13:23.630 align:middle line:90% Why? 00:13:23.630 --> 00:13:26.990 align:middle line:84% Well, this is how much player 1 bids. 00:13:26.990 --> 00:13:28.910 align:middle line:90% This is how much player 2 bids. 00:13:28.910 --> 00:13:30.630 align:middle line:90% In a first-price auction. 00:13:30.630 --> 00:13:34.370 align:middle line:84% I collect the higher bid because the higher bidder wins the item 00:13:34.370 --> 00:13:35.530 align:middle line:90% and pays their bid. 00:13:35.530 --> 00:13:37.570 align:middle line:84% So if v1 over 2 is bigger, they're 00:13:37.570 --> 00:13:41.210 align:middle line:84% going to-- player 1 wins the auction and pays v1 over 2. 00:13:41.210 --> 00:13:43.210 align:middle line:90% If bidder 2's bid is higher-- 00:13:43.210 --> 00:13:46.630 align:middle line:84% that is if v2 over 2 is higher, then bidder 2 wins the auction. 00:13:46.630 --> 00:13:50.970 align:middle line:84% They pay their bid v2 over 2, and this is what I get. 00:13:50.970 --> 00:13:52.610 align:middle line:84% So this is telling me what my revenue 00:13:52.610 --> 00:13:56.530 align:middle line:84% is for a given realization of the valuations of the two 00:13:56.530 --> 00:13:57.230 align:middle line:90% bidders. 00:13:57.230 --> 00:13:59.490 align:middle line:84% But now I want to take expectations here. 00:13:59.490 --> 00:14:08.690 align:middle line:84% So my expected revenue is equal to the expectation of this. 00:14:08.690 --> 00:14:10.750 align:middle line:84% Maybe as a trick, I'll just pull out the half. 00:14:10.750 --> 00:14:14.116 align:middle line:84% I can always-- the max of v1 over 2 and v2 over 2 00:14:14.116 --> 00:14:17.070 align:middle line:84% is just half of the maximum of v1 and v2. 00:14:17.070 --> 00:14:19.730 align:middle line:84% So maybe I'll write this as one half 00:14:19.730 --> 00:14:23.790 align:middle line:84% of the expectation of the max of v1, v2. 00:14:23.790 --> 00:14:29.030 align:middle line:90% 00:14:29.030 --> 00:14:32.750 align:middle line:84% And this is an expression that comes up 00:14:32.750 --> 00:14:34.730 align:middle line:84% a lot for uniform random variables. 00:14:34.730 --> 00:14:37.810 align:middle line:84% It turns out the answer to this expectation is 2/3. 00:14:37.810 --> 00:14:40.310 align:middle line:84% But I want to give a little geometric argument for that. 00:14:40.310 --> 00:14:41.530 align:middle line:90% But let's just fill it in. 00:14:41.530 --> 00:14:43.990 align:middle line:84% If this is 2/3, and I multiply it by a half, 00:14:43.990 --> 00:14:48.470 align:middle line:84% then what I'm going to get is one third. 00:14:48.470 --> 00:14:51.630 align:middle line:84% Where did this 2/3 magically come from? 00:14:51.630 --> 00:14:52.370 align:middle line:90% Well let's think. 00:14:52.370 --> 00:14:54.230 align:middle line:84% Well, first it's the number makes sense. 00:14:54.230 --> 00:14:57.530 align:middle line:84% I mean, v1 and v2 are both uniformly distributed on 0, 1. 00:14:57.530 --> 00:15:00.090 align:middle line:84% So this number should be between 0 and 1. 00:15:00.090 --> 00:15:01.190 align:middle line:90% That makes sense. 00:15:01.190 --> 00:15:05.330 align:middle line:84% Now, the expectation of v1 by itself is a half. 00:15:05.330 --> 00:15:07.070 align:middle line:84% If you're uniformly distributed on 0, 1, 00:15:07.070 --> 00:15:08.470 align:middle line:90% the expectation is a half. 00:15:08.470 --> 00:15:10.590 align:middle line:84% So certainly, it makes sense that the expectation 00:15:10.590 --> 00:15:12.230 align:middle line:84% of the maximum of these two numbers 00:15:12.230 --> 00:15:14.810 align:middle line:84% should be higher than the expectation of just one of them. 00:15:14.810 --> 00:15:17.570 align:middle line:84% So we'd expect to get a number between one half and 1. 00:15:17.570 --> 00:15:18.970 align:middle line:90% So 2/3 makes sense. 00:15:18.970 --> 00:15:20.748 align:middle line:90% Why do we get exactly 2/3? 00:15:20.748 --> 00:15:22.790 align:middle line:84% There's actually a really nice geometric argument 00:15:22.790 --> 00:15:23.790 align:middle line:90% that I want to show you. 00:15:23.790 --> 00:15:28.170 align:middle line:84% So we could think of v1 and v2 as living on this line. 00:15:28.170 --> 00:15:31.490 align:middle line:84% Maybe here is v2, maybe here is v1. 00:15:31.490 --> 00:15:34.010 align:middle line:84% And these are uniformly distributed. 00:15:34.010 --> 00:15:38.610 align:middle line:84% The trick is actually to connect this line to make it a circle. 00:15:38.610 --> 00:15:42.210 align:middle line:84% So now let's view this as a circle, 00:15:42.210 --> 00:15:46.590 align:middle line:90% where this point is 0 and 1. 00:15:46.590 --> 00:15:49.770 align:middle line:84% So I've basically taken this, and I've just bent it around 00:15:49.770 --> 00:15:52.130 align:middle line:90% to look like this. 00:15:52.130 --> 00:16:00.290 align:middle line:90% So now I have v2 and v1 here. 00:16:00.290 --> 00:16:15.050 align:middle line:84% Well, now, the maximum of v1 and v2 is going to be this length. 00:16:15.050 --> 00:16:17.570 align:middle line:84% But now, if I think about the circle, 00:16:17.570 --> 00:16:19.630 align:middle line:84% because things are uniformly distributed, 00:16:19.630 --> 00:16:21.970 align:middle line:84% I basically just have three uniformly distributed points 00:16:21.970 --> 00:16:22.870 align:middle line:90% on the circle. 00:16:22.870 --> 00:16:24.953 align:middle line:84% I mean, this one is a fixed point. 00:16:24.953 --> 00:16:26.870 align:middle line:84% It's not uniform, but it's as if it's uniform. 00:16:26.870 --> 00:16:28.120 align:middle line:90% It wouldn't make a difference. 00:16:28.120 --> 00:16:30.390 align:middle line:84% So if I have three points on a circle, 00:16:30.390 --> 00:16:33.188 align:middle line:84% the expected distance between any consecutive ones 00:16:33.188 --> 00:16:34.230 align:middle line:90% is going to be one third. 00:16:34.230 --> 00:16:36.790 align:middle line:84% In expectation, this is a third, this is a third, 00:16:36.790 --> 00:16:37.950 align:middle line:90% and this is a third. 00:16:37.950 --> 00:16:41.102 align:middle line:84% So the expectation of two of the segments is going to be 2/3. 00:16:41.102 --> 00:16:42.810 align:middle line:84% It's maybe not quite a rigorous argument, 00:16:42.810 --> 00:16:45.670 align:middle line:84% but I think that should give you intuition for why this 00:16:45.670 --> 00:16:49.510 align:middle line:84% expectation is 2/3, because you just effectively have three 00:16:49.510 --> 00:16:51.070 align:middle line:90% points on the circle. 00:16:51.070 --> 00:16:53.790 align:middle line:90% 00:16:53.790 --> 00:16:56.230 align:middle line:84% So now we see that in the first-price auction, 00:16:56.230 --> 00:16:58.670 align:middle line:84% the designer's expected revenue is one third. 00:16:58.670 --> 00:17:01.950 align:middle line:84% Let's now look at the second-price auction. 00:17:01.950 --> 00:17:09.990 align:middle line:84% So in the second-price auction, if bidder 1's valuation is v1, 00:17:09.990 --> 00:17:15.150 align:middle line:84% bidder 2's valuation is v2, and both bidders 00:17:15.150 --> 00:17:18.390 align:middle line:84% follow their equilibrium strategy of bidding truthfully, 00:17:18.390 --> 00:17:20.510 align:middle line:84% then what is the auctioneer's revenue going 00:17:20.510 --> 00:17:22.230 align:middle line:90% to be in this case? 00:17:22.230 --> 00:17:22.730 align:middle line:90% Yeah? 00:17:22.730 --> 00:17:23.992 align:middle line:90% AUDIENCE: The minimum of v1. 00:17:23.992 --> 00:17:25.450 align:middle line:84% IAN BALL: The minimum of v1 and v2. 00:17:25.450 --> 00:17:29.662 align:middle line:90% 00:17:29.662 --> 00:17:32.920 align:middle line:84% Because in this case, bidder 1 bids v1. 00:17:32.920 --> 00:17:35.080 align:middle line:90% Bidder 2 bids v2. 00:17:35.080 --> 00:17:37.080 align:middle line:84% The higher bidder wins the item, but they 00:17:37.080 --> 00:17:38.220 align:middle line:90% don't pay their own bid. 00:17:38.220 --> 00:17:40.040 align:middle line:84% They pay the second-highest bid, which 00:17:40.040 --> 00:17:42.100 align:middle line:84% in this case, what's the second-highest bid? 00:17:42.100 --> 00:17:43.620 align:middle line:84% It's the minimum of these two bids. 00:17:43.620 --> 00:17:45.520 align:middle line:84% If there's only two bids, the second-highest 00:17:45.520 --> 00:17:47.000 align:middle line:90% is the minimum of the two. 00:17:47.000 --> 00:17:49.640 align:middle line:84% So here, we nicely see the trade-off. 00:17:49.640 --> 00:17:53.600 align:middle line:84% In a first-price auction, you get the higher of the two bids, 00:17:53.600 --> 00:17:57.505 align:middle line:84% but each of the bids is smaller, because the bidders anticipate 00:17:57.505 --> 00:17:59.880 align:middle line:84% that it's a first-price auction, so they shade their bids 00:17:59.880 --> 00:18:01.240 align:middle line:90% downward. 00:18:01.240 --> 00:18:04.460 align:middle line:84% In a second-price auction, you get the lower of the two bids. 00:18:04.460 --> 00:18:06.140 align:middle line:90% That hurts the auctioneer. 00:18:06.140 --> 00:18:08.640 align:middle line:84% But the benefit is people are bidding higher because they're 00:18:08.640 --> 00:18:10.100 align:middle line:90% bidding their true valuations. 00:18:10.100 --> 00:18:13.840 align:middle line:84% So the question is, which of these two forces wins? 00:18:13.840 --> 00:18:16.340 align:middle line:90% Well let's compute this. 00:18:16.340 --> 00:18:19.720 align:middle line:90% We get the expectation of pi-- 00:18:19.720 --> 00:18:20.840 align:middle line:90% big pi, I guess-- 00:18:20.840 --> 00:18:28.300 align:middle line:84% of v1, v2 equals, well, just the expectation of the min 00:18:28.300 --> 00:18:30.180 align:middle line:90% of v1, v2. 00:18:30.180 --> 00:18:32.960 align:middle line:84% And if we use our little graph over there, 00:18:32.960 --> 00:18:36.390 align:middle line:84% can we guess what this expectation is? 00:18:36.390 --> 00:18:36.890 align:middle line:90% Yeah? 00:18:36.890 --> 00:18:37.723 align:middle line:90% AUDIENCE: One third. 00:18:37.723 --> 00:18:39.460 align:middle line:84% IAN BALL: One third, because now I 00:18:39.460 --> 00:18:42.240 align:middle line:84% have these three points on the circle. 00:18:42.240 --> 00:18:43.740 align:middle line:84% The minimum of the two is just going 00:18:43.740 --> 00:18:48.480 align:middle line:84% to be the length of the segment between two consecutive points. 00:18:48.480 --> 00:18:51.900 align:middle line:84% So it's going to be half as big as the length between-- 00:18:51.900 --> 00:18:53.300 align:middle line:90% over two segments. 00:18:53.300 --> 00:18:55.080 align:middle line:84% So I'm exactly going to get one third. 00:18:55.080 --> 00:18:59.260 align:middle line:90% 00:18:59.260 --> 00:19:01.500 align:middle line:84% So what you see here is actually these two effects 00:19:01.500 --> 00:19:03.580 align:middle line:90% I described exactly offset. 00:19:03.580 --> 00:19:06.720 align:middle line:84% In the first-price auction, you get paid the highest bid, 00:19:06.720 --> 00:19:08.580 align:middle line:90% but people choose to bid less. 00:19:08.580 --> 00:19:11.380 align:middle line:84% In the second-price auction, you get the smaller bid, 00:19:11.380 --> 00:19:13.340 align:middle line:90% but people choose to bid more. 00:19:13.340 --> 00:19:15.420 align:middle line:84% And those two effects exactly offset. 00:19:15.420 --> 00:19:17.020 align:middle line:84% And the designer's expected revenue 00:19:17.020 --> 00:19:19.500 align:middle line:84% is exactly one third in each case. 00:19:19.500 --> 00:19:21.100 align:middle line:84% Now, this was a very simple example 00:19:21.100 --> 00:19:22.615 align:middle line:90% with uniform distributions. 00:19:22.615 --> 00:19:24.740 align:middle line:84% So you might think, oh, this is just a coincidence. 00:19:24.740 --> 00:19:25.640 align:middle line:90% This is a fluke. 00:19:25.640 --> 00:19:28.120 align:middle line:84% It turns out this is a very deep and general phenomenon, 00:19:28.120 --> 00:19:31.160 align:middle line:84% that many different auction formats will 00:19:31.160 --> 00:19:34.260 align:middle line:84% generate the same expected revenue for the auctioneer. 00:19:34.260 --> 00:19:36.480 align:middle line:84% And that's something that we'll talk about a bit 00:19:36.480 --> 00:19:38.120 align:middle line:90% later in the course. 00:19:38.120 --> 00:19:42.540 align:middle line:90% Any questions on this? 00:19:42.540 --> 00:19:43.040 align:middle line:90% Yes? 00:19:43.040 --> 00:19:46.400 align:middle line:84% AUDIENCE: The trick for finding the maximum and minimum, 00:19:46.400 --> 00:19:47.980 align:middle line:84% is that applicable if you had more? 00:19:47.980 --> 00:19:48.320 align:middle line:90% IAN BALL: Exactly. 00:19:48.320 --> 00:19:48.820 align:middle line:90% Yeah. 00:19:48.820 --> 00:19:50.280 align:middle line:90% So maybe let's write this down. 00:19:50.280 --> 00:19:56.140 align:middle line:84% So if I did-- if everything was uniform 0, 1 and I had n-- 00:19:56.140 --> 00:20:05.040 align:middle line:84% I had v1 through vn, then the expectation of the max of v1 00:20:05.040 --> 00:20:08.520 align:middle line:84% through vn-- and this is kind of a good fact to know-- 00:20:08.520 --> 00:20:11.500 align:middle line:84% guess, can you see what-- if we complete the pattern, 00:20:11.500 --> 00:20:13.380 align:middle line:90% we got 2/3 when n was 2. 00:20:13.380 --> 00:20:16.180 align:middle line:84% What do you think this is going to be now? 00:20:16.180 --> 00:20:17.500 align:middle line:90% AUDIENCE: n over n plus 1. 00:20:17.500 --> 00:20:18.500 align:middle line:90% IAN BALL: Exactly right. 00:20:18.500 --> 00:20:22.460 align:middle line:90% 00:20:22.460 --> 00:20:24.980 align:middle line:84% And in fact, you can actually go further. 00:20:24.980 --> 00:20:27.920 align:middle line:84% If I wanted the expectation of the second highest, 00:20:27.920 --> 00:20:30.520 align:middle line:84% it would be n minus 1 over n plus 1. 00:20:30.520 --> 00:20:33.140 align:middle line:84% The third highest, n minus 2 over n plus 1 all 00:20:33.140 --> 00:20:41.660 align:middle line:84% the way down to the expectation of the min 00:20:41.660 --> 00:20:45.420 align:middle line:90% equals 1 over n plus 1. 00:20:45.420 --> 00:20:47.313 align:middle line:84% So there's a very nice structure. 00:20:47.313 --> 00:20:49.980 align:middle line:84% I could look at the lowest, the second lowest, the third lowest, 00:20:49.980 --> 00:20:51.240 align:middle line:90% all the way up to the highest. 00:20:51.240 --> 00:20:54.100 align:middle line:84% And those are exactly going to be equally spaced. 00:20:54.100 --> 00:20:57.660 align:middle line:84% And they're separated exactly by 1 over n plus 1. 00:20:57.660 --> 00:21:00.190 align:middle line:84% And you can use calculus to verify this, 00:21:00.190 --> 00:21:02.440 align:middle line:84% but I think this is a nice intuition for what goes on. 00:21:02.440 --> 00:21:05.160 align:middle line:84% And you could try to formalize that intuition. 00:21:05.160 --> 00:21:07.780 align:middle line:90% 00:21:07.780 --> 00:21:09.480 align:middle line:90% Good. 00:21:09.480 --> 00:21:10.440 align:middle line:90% Any other questions? 00:21:10.440 --> 00:21:12.960 align:middle line:90% 00:21:12.960 --> 00:21:17.620 align:middle line:84% So now, as I said, this was a very simple example. 00:21:17.620 --> 00:21:19.120 align:middle line:84% You might worry that our conclusions 00:21:19.120 --> 00:21:21.640 align:middle line:90% are really sensitive to that. 00:21:21.640 --> 00:21:25.200 align:middle line:84% What we want to do now is study a more general auction setup, 00:21:25.200 --> 00:21:27.830 align:middle line:84% where instead of two bidders, we have n bidders, 00:21:27.830 --> 00:21:30.080 align:middle line:84% and instead of assuming their valuations are uniformly 00:21:30.080 --> 00:21:32.700 align:middle line:84% distributed, we allow for more general valuations. 00:21:32.700 --> 00:21:35.440 align:middle line:84% So we're going to look at a more general model. 00:21:35.440 --> 00:21:37.320 align:middle line:84% So we're going to look at, in general, 00:21:37.320 --> 00:21:54.880 align:middle line:84% what's maybe called the symmetric independent private 00:21:54.880 --> 00:21:55.580 align:middle line:90% values model. 00:21:55.580 --> 00:22:00.560 align:middle line:84% Let me explain what these words mean as we go through. 00:22:00.560 --> 00:22:02.460 align:middle line:84% So now we're going to have n bidders. 00:22:02.460 --> 00:22:08.500 align:middle line:84% So we have bidders i equals 1 up to n. 00:22:08.500 --> 00:22:13.040 align:middle line:90% 00:22:13.040 --> 00:22:16.700 align:middle line:84% And the bidders are symmetric in the sense 00:22:16.700 --> 00:22:20.900 align:middle line:84% that the distribution from which each bidder's valuation is drawn 00:22:20.900 --> 00:22:22.060 align:middle line:90% is the same. 00:22:22.060 --> 00:22:25.440 align:middle line:84% So of course, when we actually run the auction, 00:22:25.440 --> 00:22:27.640 align:middle line:84% someone might value the good more than someone else. 00:22:27.640 --> 00:22:29.375 align:middle line:84% But before we start the auction, we 00:22:29.375 --> 00:22:31.500 align:middle line:84% don't believe that the distribution of one player's 00:22:31.500 --> 00:22:34.060 align:middle line:84% valuation is higher than the distribution of another player's 00:22:34.060 --> 00:22:34.633 align:middle line:90% valuation. 00:22:34.633 --> 00:22:36.800 align:middle line:84% In reality, this is probably not a great assumption, 00:22:36.800 --> 00:22:38.500 align:middle line:84% but this simplifies things, and we can talk 00:22:38.500 --> 00:22:39.600 align:middle line:90% later about how it extends. 00:22:39.600 --> 00:22:40.558 align:middle line:90% So what does that mean? 00:22:40.558 --> 00:22:47.220 align:middle line:84% It means that for each bidder i the valuation vi-- 00:22:47.220 --> 00:22:49.720 align:middle line:84% well, before we said the valuation was in 0, 1. 00:22:49.720 --> 00:22:51.160 align:middle line:90% Let's be a little more general. 00:22:51.160 --> 00:22:54.720 align:middle line:84% Let's say this lives in the interval between v lower bar 00:22:54.720 --> 00:22:55.620 align:middle line:90% and v upper bar. 00:22:55.620 --> 00:22:58.660 align:middle line:84% So this is the lowest possible valuation a bidder could have. 00:22:58.660 --> 00:23:00.540 align:middle line:84% And this is the highest possible valuation 00:23:00.540 --> 00:23:02.020 align:middle line:90% that a bidder could have. 00:23:02.020 --> 00:23:03.820 align:middle line:84% Notice I don't have subscripts here 00:23:03.820 --> 00:23:07.120 align:middle line:84% because the range of valuations is the same for all the players. 00:23:07.120 --> 00:23:10.420 align:middle line:84% It doesn't depend on the identity of the bidder. 00:23:10.420 --> 00:23:22.510 align:middle line:84% And we'll say this is distributed according to a CDF-- 00:23:22.510 --> 00:23:24.790 align:middle line:84% that's a Cumulative Distribution Function-- 00:23:24.790 --> 00:23:29.670 align:middle line:84% capital F and a PDF, a Probability Density Function, 00:23:29.670 --> 00:23:30.170 align:middle line:90% little f. 00:23:30.170 --> 00:23:33.350 align:middle line:84% And I'll explain what these mean. 00:23:33.350 --> 00:23:36.170 align:middle line:84% So this is just a way to describe a distribution. 00:23:36.170 --> 00:23:39.390 align:middle line:84% If you've taken a probability class, which is, I think, 00:23:39.390 --> 00:23:40.970 align:middle line:84% technically a prerequisite for this, 00:23:40.970 --> 00:23:42.370 align:middle line:90% you would have gone over this. 00:23:42.370 --> 00:23:43.770 align:middle line:90% But what are these things? 00:23:43.770 --> 00:23:51.030 align:middle line:84% Well, F of v just tells us the probability 00:23:51.030 --> 00:23:54.190 align:middle line:84% that the valuation is below v. So 00:23:54.190 --> 00:23:59.390 align:middle line:84% F of v is defined to be the probability that, say, vi 00:23:59.390 --> 00:24:03.790 align:middle line:90% is less than or equal to v. 00:24:03.790 --> 00:24:06.550 align:middle line:90% So F is an increasing function. 00:24:06.550 --> 00:24:10.150 align:middle line:84% If I plug In v lower bar, I'm going to get 0, 00:24:10.150 --> 00:24:14.450 align:middle line:84% because the probability that vi is below v lower bar is 0. 00:24:14.450 --> 00:24:18.410 align:middle line:84% If I plug in v upper bar, then I'm 00:24:18.410 --> 00:24:21.890 align:middle line:84% going to get 1, because the probability that vi 00:24:21.890 --> 00:24:24.250 align:middle line:84% is less than or equal to v upper bar is 1. 00:24:24.250 --> 00:24:26.490 align:middle line:84% And then it's going to fill in between. 00:24:26.490 --> 00:24:29.210 align:middle line:84% When I say it has a PDF f, What that means is I 00:24:29.210 --> 00:24:33.410 align:middle line:84% can express this probability as an integral from v lower bar 00:24:33.410 --> 00:24:37.410 align:middle line:90% to v of f of x dx. 00:24:37.410 --> 00:24:40.410 align:middle line:90% 00:24:40.410 --> 00:24:45.370 align:middle line:84% So that means we can visualize the distribution. 00:24:45.370 --> 00:24:48.050 align:middle line:84% Here's v lower bar, here's v upper bar 00:24:48.050 --> 00:24:52.570 align:middle line:90% by some density like this. 00:24:52.570 --> 00:24:54.970 align:middle line:84% And what this says, the peaks say 00:24:54.970 --> 00:24:58.885 align:middle line:84% these are places where the valuation is very likely to be. 00:24:58.885 --> 00:25:01.010 align:middle line:84% And then the valleys are places where the valuation 00:25:01.010 --> 00:25:02.530 align:middle line:90% is less likely to be. 00:25:02.530 --> 00:25:10.910 align:middle line:84% And what we're saying is if we fix v here, 00:25:10.910 --> 00:25:14.670 align:middle line:84% the area under this curve is exactly the probability 00:25:14.670 --> 00:25:17.630 align:middle line:84% that the valuation lies between v lower bar and v. 00:25:17.630 --> 00:25:21.138 align:middle line:84% So what that means, one implication of that, 00:25:21.138 --> 00:25:23.430 align:middle line:84% is that it must be that the area under the entire curve 00:25:23.430 --> 00:25:26.390 align:middle line:84% is 1, because the probability that v lives somewhere 00:25:26.390 --> 00:25:26.910 align:middle line:90% is just 1. 00:25:26.910 --> 00:25:28.410 align:middle line:84% That's just a fact of probabilities. 00:25:28.410 --> 00:25:30.030 align:middle line:84% So this is probably a little quick 00:25:30.030 --> 00:25:31.388 align:middle line:90% if you haven't seen this before. 00:25:31.388 --> 00:25:33.930 align:middle line:84% That is a prereq for the course, and I'm happy to chat after, 00:25:33.930 --> 00:25:35.910 align:middle line:84% or maybe you can self-study if you 00:25:35.910 --> 00:25:38.150 align:middle line:84% haven't seen probability distributions 00:25:38.150 --> 00:25:41.390 align:middle line:90% or densities before. 00:25:41.390 --> 00:25:44.950 align:middle line:84% So symmetry refers to the fact that we're 00:25:44.950 --> 00:25:47.790 align:middle line:84% using the same distribution F and little 00:25:47.790 --> 00:25:49.290 align:middle line:90% f for all the bidders. 00:25:49.290 --> 00:25:52.550 align:middle line:84% It's not varying across the bidders. 00:25:52.550 --> 00:25:57.990 align:middle line:84% Independence means that v1 through vn 00:25:57.990 --> 00:25:59.770 align:middle line:90% are statistically independent. 00:25:59.770 --> 00:26:09.065 align:middle line:90% 00:26:09.065 --> 00:26:11.190 align:middle line:84% This is the same assumption we made in the example. 00:26:11.190 --> 00:26:13.690 align:middle line:84% It's that if I know my valuation v1, 00:26:13.690 --> 00:26:16.650 align:middle line:84% that doesn't allow me to update my beliefs about other players' 00:26:16.650 --> 00:26:17.430 align:middle line:90% valuations. 00:26:17.430 --> 00:26:19.850 align:middle line:84% So what it rules out is things like positive or negative 00:26:19.850 --> 00:26:22.810 align:middle line:84% correlation, where when one bidder's valuation is high, 00:26:22.810 --> 00:26:25.010 align:middle line:84% everyone else's valuation tends to be high. 00:26:25.010 --> 00:26:27.010 align:middle line:84% Or when one bidder's valuation is low, 00:26:27.010 --> 00:26:29.690 align:middle line:84% everyone else's valuation tends to be low. 00:26:29.690 --> 00:26:32.270 align:middle line:84% This is also maybe a strong assumption, 00:26:32.270 --> 00:26:34.050 align:middle line:84% and we can, again, maybe talk later 00:26:34.050 --> 00:26:35.310 align:middle line:90% about how it could be relaxed. 00:26:35.310 --> 00:26:38.670 align:middle line:84% But this is what we're going to assume for today. 00:26:38.670 --> 00:26:42.600 align:middle line:90% 00:26:42.600 --> 00:26:47.750 align:middle line:84% And I should maybe say one thing about these endpoints. 00:26:47.750 --> 00:26:54.010 align:middle line:84% I'm going to assume that 0 is less than or equal to v bar, 00:26:54.010 --> 00:26:55.470 align:middle line:90% strictly less than v upper bar. 00:26:55.470 --> 00:26:58.082 align:middle line:84% So the agents always assign a nonnegative value to the good. 00:26:58.082 --> 00:26:59.790 align:middle line:84% I think this is a pretty safe assumption. 00:26:59.790 --> 00:27:01.850 align:middle line:84% If they really hated the good, they could just throw it away. 00:27:01.850 --> 00:27:04.343 align:middle line:84% So we're always assuming that they value it at least 0. 00:27:04.343 --> 00:27:06.010 align:middle line:84% They could value it at 0, but they never 00:27:06.010 --> 00:27:07.530 align:middle line:90% value it at a negative number. 00:27:07.530 --> 00:27:13.870 align:middle line:90% 00:27:13.870 --> 00:27:15.990 align:middle line:90% So we've talked about symmetry. 00:27:15.990 --> 00:27:18.710 align:middle line:90% We've talked about independence. 00:27:18.710 --> 00:27:20.670 align:middle line:84% What private values refers to-- and maybe 00:27:20.670 --> 00:27:21.730 align:middle line:90% I'll give us the acronym. 00:27:21.730 --> 00:27:25.150 align:middle line:90% This is SIPV. 00:27:25.150 --> 00:27:26.870 align:middle line:84% Private values refers to the fact 00:27:26.870 --> 00:27:31.550 align:middle line:84% that each bidder knows exactly how much he values the good. 00:27:31.550 --> 00:27:35.270 align:middle line:84% And bidder i's valuation doesn't depend 00:27:35.270 --> 00:27:37.430 align:middle line:90% on other players' valuations. 00:27:37.430 --> 00:27:41.430 align:middle line:84% So you might imagine, if we were in an auction for a painting, 00:27:41.430 --> 00:27:44.110 align:middle line:84% and there was a chance that I could resell the painting, 00:27:44.110 --> 00:27:46.230 align:middle line:84% I might care about how much other people value 00:27:46.230 --> 00:27:48.630 align:middle line:84% the painting, because if they value the painting more, 00:27:48.630 --> 00:27:50.630 align:middle line:84% I might think, wow, I could resell it to them 00:27:50.630 --> 00:27:52.750 align:middle line:84% after the auction, and therefore, my valuation 00:27:52.750 --> 00:27:54.090 align:middle line:90% for the painting goes up. 00:27:54.090 --> 00:27:56.890 align:middle line:84% So we're ruling that out by assuming private values. 00:27:56.890 --> 00:27:59.510 align:middle line:84% We're assuming that I know my valuation for the good, 00:27:59.510 --> 00:28:03.012 align:middle line:84% and how much I'm willing to pay depends only on my signal 00:28:03.012 --> 00:28:04.720 align:middle line:84% about, say, how much I like the painting, 00:28:04.720 --> 00:28:10.280 align:middle line:84% and not on how much anyone else likes the painting. 00:28:10.280 --> 00:28:13.160 align:middle line:84% So here we are, and we're going to try to analyze 00:28:13.160 --> 00:28:15.840 align:middle line:90% the first-price auction. 00:28:15.840 --> 00:28:18.640 align:middle line:84% We're going to try to solve for an equilibrium 00:28:18.640 --> 00:28:20.340 align:middle line:90% of the first-price auction. 00:28:20.340 --> 00:28:30.360 align:middle line:84% So goal, solve for BNE, a Bayes' Nash equilibrium, 00:28:30.360 --> 00:28:34.280 align:middle line:84% of the first-price auction in the SIPV setting. 00:28:34.280 --> 00:28:36.840 align:middle line:90% 00:28:36.840 --> 00:28:38.680 align:middle line:90% I'll use a lot of acronyms-- 00:28:38.680 --> 00:28:48.778 align:middle line:84% of FPA, the First-Price Auction in the SIPV setting. 00:28:48.778 --> 00:28:50.820 align:middle line:84% Notice the setting and the auction are different. 00:28:50.820 --> 00:28:52.680 align:middle line:84% The setting is about who's involved 00:28:52.680 --> 00:28:54.600 align:middle line:84% and how much they value the good. 00:28:54.600 --> 00:28:56.825 align:middle line:84% The auction format, these are the rules 00:28:56.825 --> 00:28:58.200 align:middle line:84% that the designer gets to choose, 00:28:58.200 --> 00:29:00.140 align:middle line:84% or the auctioneer gets to choose. 00:29:00.140 --> 00:29:02.820 align:middle line:84% The auctioneer can't choose how much people value the good. 00:29:02.820 --> 00:29:04.200 align:middle line:84% This is just a fact of the world. 00:29:04.200 --> 00:29:05.540 align:middle line:90% This is a fact of the setting. 00:29:05.540 --> 00:29:07.580 align:middle line:84% But the auction format is something that 00:29:07.580 --> 00:29:10.380 align:middle line:90% is chosen by the auctioneer. 00:29:10.380 --> 00:29:13.020 align:middle line:84% Well, first, as always, we have to figure out, well, 00:29:13.020 --> 00:29:15.980 align:middle line:84% what is a strategy for each player? 00:29:15.980 --> 00:29:18.600 align:middle line:84% So what is a strategy for player i? 00:29:18.600 --> 00:29:22.580 align:middle line:90% 00:29:22.580 --> 00:29:24.940 align:middle line:84% Well, a strategy needs to say, how much does 00:29:24.940 --> 00:29:30.380 align:middle line:84% player i bid as a function of their valuation for the good. 00:29:30.380 --> 00:29:37.000 align:middle line:84% So a strategy for player i is going to be a function bi from-- 00:29:37.000 --> 00:29:42.060 align:middle line:90% 00:29:42.060 --> 00:29:45.940 align:middle line:90% maybe I'll write this. 00:29:45.940 --> 00:29:48.100 align:middle line:90% So what does this mean? 00:29:48.100 --> 00:29:51.540 align:middle line:84% It means that for each valuation of the item that I 00:29:51.540 --> 00:29:55.080 align:middle line:84% could actually have, bi is my bidding function. 00:29:55.080 --> 00:29:57.640 align:middle line:84% Sometimes that's the terminology we'll use. 00:29:57.640 --> 00:30:02.560 align:middle line:90% 00:30:02.560 --> 00:30:06.080 align:middle line:84% My bidding function says, for each valuation, 00:30:06.080 --> 00:30:09.560 align:middle line:84% say, vi that I could have, my bidding function 00:30:09.560 --> 00:30:13.760 align:middle line:84% spits out bi of vi, which is the bid that I will 00:30:13.760 --> 00:30:16.680 align:middle line:90% make if my valuation is vi. 00:30:16.680 --> 00:30:18.620 align:middle line:84% And just using the rules of the auction, 00:30:18.620 --> 00:30:21.300 align:middle line:84% we're assuming that bids can be any nonnegative number. 00:30:21.300 --> 00:30:23.318 align:middle line:90% I can't bid a negative amount. 00:30:23.318 --> 00:30:24.860 align:middle line:84% I can't ask the auctioneer to pay me. 00:30:24.860 --> 00:30:26.360 align:middle line:90% That would be a bit strange. 00:30:26.360 --> 00:30:28.980 align:middle line:84% But I can bid any nonnegative number that I want. 00:30:28.980 --> 00:30:32.680 align:middle line:90% 00:30:32.680 --> 00:30:37.240 align:middle line:84% And an equilibrium is going to specify a strategy profile. 00:30:37.240 --> 00:30:39.440 align:middle line:84% So it's going to specify a bidding function 00:30:39.440 --> 00:30:42.000 align:middle line:90% for each of the n bidders. 00:30:42.000 --> 00:30:46.320 align:middle line:84% So as before, this is kind of a difficult problem 00:30:46.320 --> 00:30:48.920 align:middle line:84% because we're trying to solve for functions, not just 00:30:48.920 --> 00:30:49.778 align:middle line:90% single numbers. 00:30:49.778 --> 00:30:50.820 align:middle line:90% And these are really big. 00:30:50.820 --> 00:30:51.700 align:middle line:90% There's a lot of-- 00:30:51.700 --> 00:30:53.700 align:middle line:84% each player is choosing infinitely many numbers. 00:30:53.700 --> 00:30:55.117 align:middle line:84% How they bid if their valuation is 00:30:55.117 --> 00:30:58.380 align:middle line:84% 1, how they'd bid if the valuation is 2, 2.3, 2.5, 00:30:58.380 --> 00:31:01.220 align:middle line:84% there's too many things to solve for. 00:31:01.220 --> 00:31:04.760 align:middle line:84% Before, recall that we got around this by guessing. 00:31:04.760 --> 00:31:09.220 align:middle line:84% We guessed that there would be an equilibrium where 00:31:09.220 --> 00:31:12.140 align:middle line:84% each bidding function was just a linear function. 00:31:12.140 --> 00:31:13.860 align:middle line:84% And that simplified things a lot. 00:31:13.860 --> 00:31:16.980 align:middle line:84% It reduced our problem to just solving for a single parameter. 00:31:16.980 --> 00:31:20.220 align:middle line:84% But that only worked in the special uniform example. 00:31:20.220 --> 00:31:22.180 align:middle line:84% So beyond the uniform distribution, 00:31:22.180 --> 00:31:23.860 align:middle line:84% there's no reason to think that people 00:31:23.860 --> 00:31:25.700 align:middle line:84% are going to use a linear bidding function 00:31:25.700 --> 00:31:28.468 align:middle line:84% and bid exactly one third of their valuation, 00:31:28.468 --> 00:31:29.760 align:middle line:90% or one half of their valuation. 00:31:29.760 --> 00:31:31.523 align:middle line:84% So we have to be a bit more general. 00:31:31.523 --> 00:31:32.940 align:middle line:84% So what we're going to do is we're 00:31:32.940 --> 00:31:34.420 align:middle line:90% going to also make a guess. 00:31:34.420 --> 00:31:39.940 align:middle line:84% So we're going to guess, or look for, 00:31:39.940 --> 00:32:00.400 align:middle line:84% a symmetric increasing differentiable equilibrium. 00:32:00.400 --> 00:32:02.480 align:middle line:84% Let's go through what all these words mean. 00:32:02.480 --> 00:32:05.720 align:middle line:84% Well, we're going to look for an equilibrium 00:32:05.720 --> 00:32:09.240 align:middle line:84% where every bidder uses the bidding function 00:32:09.240 --> 00:32:12.140 align:middle line:90% beta from v bar. 00:32:12.140 --> 00:32:26.880 align:middle line:90% 00:32:26.880 --> 00:32:30.800 align:middle line:84% so we're going to say, look, because the bidders are 00:32:30.800 --> 00:32:33.720 align:middle line:84% x antisymmetric, their valuations 00:32:33.720 --> 00:32:35.680 align:middle line:84% are drawn from the same distribution, 00:32:35.680 --> 00:32:37.280 align:middle line:84% we're going to look for an equilibrium 00:32:37.280 --> 00:32:39.240 align:middle line:84% where all the bidders use the same bidding 00:32:39.240 --> 00:32:41.000 align:middle line:90% function, the same strategy. 00:32:41.000 --> 00:32:44.400 align:middle line:84% So bidder 1, bidder 2, all the way up to bidder n, 00:32:44.400 --> 00:32:45.362 align:middle line:90% use this function beta. 00:32:45.362 --> 00:32:47.320 align:middle line:84% So if you wanted to be formal what we're saying 00:32:47.320 --> 00:32:52.980 align:middle line:84% is b1 equals b2 equals bn equals beta. 00:32:52.980 --> 00:32:56.610 align:middle line:90% 00:32:56.610 --> 00:32:59.650 align:middle line:84% So everyone uses the same bidding function. 00:32:59.650 --> 00:33:02.370 align:middle line:84% And we're assuming that this bidding function is increasing. 00:33:02.370 --> 00:33:03.530 align:middle line:90% That makes sense. 00:33:03.530 --> 00:33:05.730 align:middle line:84% If you value the good more, it seems intuitive 00:33:05.730 --> 00:33:09.450 align:middle line:84% that you should bid more for the item because you care more-- 00:33:09.450 --> 00:33:12.430 align:middle line:84% you assign more utility to winning the items. 00:33:12.430 --> 00:33:15.850 align:middle line:84% You want to increase your probability of winning. 00:33:15.850 --> 00:33:18.170 align:middle line:84% And this is just kind of a technical mathematical 00:33:18.170 --> 00:33:19.050 align:middle line:90% assumption. 00:33:19.050 --> 00:33:20.910 align:middle line:84% We want to be able to apply calculus. 00:33:20.910 --> 00:33:23.470 align:middle line:84% So we're going to assume that beta is differentiable. 00:33:23.470 --> 00:33:27.073 align:middle line:84% That is, we can always take the derivative of beta, beta prime, 00:33:27.073 --> 00:33:28.490 align:middle line:84% and that that derivative is always 00:33:28.490 --> 00:33:30.210 align:middle line:90% going to be strictly positive. 00:33:30.210 --> 00:33:32.910 align:middle line:84% That's basically increasing, a little stronger than increasing. 00:33:32.910 --> 00:33:35.530 align:middle line:90% 00:33:35.530 --> 00:33:38.130 align:middle line:84% And now we've kind of reduced our problem. 00:33:38.130 --> 00:33:41.510 align:middle line:84% Instead of looking for n arbitrary bidding functions, 00:33:41.510 --> 00:33:43.760 align:middle line:84% we're now looking for a single bidding function beta. 00:33:43.760 --> 00:33:46.010 align:middle line:84% And we want to see if we can find an equilibrium where 00:33:46.010 --> 00:33:50.050 align:middle line:84% every single bidder uses this bidding function beta. 00:33:50.050 --> 00:33:51.613 align:middle line:84% Any questions here on our approach? 00:33:51.613 --> 00:33:53.530 align:middle line:84% Because we're going to get into a little math, 00:33:53.530 --> 00:33:57.610 align:middle line:84% so I want to make sure the approach is clear. 00:33:57.610 --> 00:34:00.390 align:middle line:90% 00:34:00.390 --> 00:34:03.450 align:middle line:90% So what do we need to do? 00:34:03.450 --> 00:34:05.550 align:middle line:84% How are we going to solve for beta? 00:34:05.550 --> 00:34:08.070 align:middle line:84% Well, if this is going to be an equilibrium, 00:34:08.070 --> 00:34:11.230 align:middle line:84% it better be that if everyone else is falling the bidding 00:34:11.230 --> 00:34:13.949 align:middle line:84% strategy beta, the bidding function beta, 00:34:13.949 --> 00:34:17.570 align:middle line:84% it's optimal for me to also choose the bidding strategy 00:34:17.570 --> 00:34:18.630 align:middle line:90% beta. 00:34:18.630 --> 00:34:21.550 align:middle line:84% So what we want to say is let's consider-- 00:34:21.550 --> 00:34:22.635 align:middle line:90% let's focus on bidder 1. 00:34:22.635 --> 00:34:24.010 align:middle line:84% It just makes it a little easier. 00:34:24.010 --> 00:34:28.389 align:middle line:90% 00:34:28.389 --> 00:34:31.110 align:middle line:84% So put yourself in the shoes of bidder 1. 00:34:31.110 --> 00:34:35.670 align:middle line:84% Suppose that-- I'm going to write-- define a function here 00:34:35.670 --> 00:34:43.790 align:middle line:90% U1 of b1 given v1 beta. 00:34:43.790 --> 00:34:45.710 align:middle line:84% Let me describe in words what this means. 00:34:45.710 --> 00:34:47.630 align:middle line:90% I want to say I'm bidder 1. 00:34:47.630 --> 00:34:48.949 align:middle line:90% What do I know? 00:34:48.949 --> 00:34:51.909 align:middle line:84% Well, I know my true valuation for the good is v1. 00:34:51.909 --> 00:34:53.190 align:middle line:90% So this is my valuation. 00:34:53.190 --> 00:34:57.290 align:middle line:90% 00:34:57.290 --> 00:35:00.370 align:middle line:84% And let me suppose that all of my opponents 00:35:00.370 --> 00:35:02.810 align:middle line:84% are following the bidding strategy beta. 00:35:02.810 --> 00:35:10.130 align:middle line:84% So what I'm supposing here is that b2 equals 00:35:10.130 --> 00:35:14.430 align:middle line:90% b3 equals bn equals beta. 00:35:14.430 --> 00:35:17.930 align:middle line:84% So I assume that player 2 is using the bidding function beta. 00:35:17.930 --> 00:35:19.970 align:middle line:84% Player 3 is using the bidding function beta, 00:35:19.970 --> 00:35:22.512 align:middle line:84% all the way up to player n is also using the bidding function 00:35:22.512 --> 00:35:23.370 align:middle line:90% beta. 00:35:23.370 --> 00:35:25.370 align:middle line:84% And I want to say, OK, here I am. 00:35:25.370 --> 00:35:29.010 align:middle line:84% I believe that my opponent is using these bidding functions. 00:35:29.010 --> 00:35:32.050 align:middle line:84% I know my true valuation v1 for the good. 00:35:32.050 --> 00:35:35.030 align:middle line:84% What I want to compute is what is my utility 00:35:35.030 --> 00:35:36.770 align:middle line:90% if I were to bid b1. 00:35:36.770 --> 00:35:38.230 align:middle line:90% So b1 is just a number. 00:35:38.230 --> 00:35:42.410 align:middle line:90% 00:35:42.410 --> 00:35:46.210 align:middle line:84% This is not a function, it's just a number. 00:35:46.210 --> 00:35:47.887 align:middle line:84% And once I compute this function, 00:35:47.887 --> 00:35:49.470 align:middle line:84% then I can start thinking about-- then 00:35:49.470 --> 00:35:53.270 align:middle line:84% I can maximize this function and find the optimal bid for me. 00:35:53.270 --> 00:35:55.310 align:middle line:84% But I need to put in this information 00:35:55.310 --> 00:35:57.790 align:middle line:84% because the best way for me to bid in this auction 00:35:57.790 --> 00:35:59.930 align:middle line:84% is going to depend on how much I value the good, 00:35:59.930 --> 00:36:01.630 align:middle line:84% and it's also going to depend on what I 00:36:01.630 --> 00:36:02.970 align:middle line:90% believe my opponents are using. 00:36:02.970 --> 00:36:06.630 align:middle line:84% So that's why I need all this notation here. 00:36:06.630 --> 00:36:08.990 align:middle line:90% And this is the key quantity. 00:36:08.990 --> 00:36:15.630 align:middle line:90% So let's try to-- 00:36:15.630 --> 00:36:16.290 align:middle line:90% any guesses? 00:36:16.290 --> 00:36:22.202 align:middle line:84% Can we write out what this expression is going to be? 00:36:22.202 --> 00:36:23.410 align:middle line:90% Well, let's think it through. 00:36:23.410 --> 00:36:28.250 align:middle line:84% If I lose the auction, my payoff is 0, because if I lose, 00:36:28.250 --> 00:36:30.950 align:middle line:84% I don't get the good and I don't pay anything. 00:36:30.950 --> 00:36:32.450 align:middle line:84% So we just have to figure out, well, 00:36:32.450 --> 00:36:33.930 align:middle line:90% what is my utility going to be? 00:36:33.930 --> 00:36:35.270 align:middle line:90% Let's suppose I win. 00:36:35.270 --> 00:36:38.070 align:middle line:84% If I win, and my valuation is v1, 00:36:38.070 --> 00:36:43.630 align:middle line:84% and my bid is V1, then what will my utility be if I win? 00:36:43.630 --> 00:36:44.130 align:middle line:90% Yeah? 00:36:44.130 --> 00:36:45.328 align:middle line:90% AUDIENCE: V1 minus b1. 00:36:45.328 --> 00:36:46.370 align:middle line:90% IAN BALL: Exactly, right. 00:36:46.370 --> 00:36:49.890 align:middle line:90% 00:36:49.890 --> 00:36:52.250 align:middle line:90% So this is my utility if I win. 00:36:52.250 --> 00:36:54.770 align:middle line:84% But I only get that utility if I do win. 00:36:54.770 --> 00:36:56.970 align:middle line:84% So now I have to multiply that by the probability 00:36:56.970 --> 00:36:58.130 align:middle line:90% that I actually win. 00:36:58.130 --> 00:37:00.690 align:middle line:84% What is the probability that I win? 00:37:00.690 --> 00:37:03.057 align:middle line:84% Well, it's the probability that-- well, 00:37:03.057 --> 00:37:04.390 align:middle line:90% what does it mean for me to win? 00:37:04.390 --> 00:37:08.690 align:middle line:84% I win if my bid is higher than everyone else's bid. 00:37:08.690 --> 00:37:10.930 align:middle line:84% Now, there's always this question of ties. 00:37:10.930 --> 00:37:15.290 align:middle line:84% Because this bidding function is strictly increasing, 00:37:15.290 --> 00:37:18.158 align:middle line:84% any single bid by my opponents is going to have probability 0. 00:37:18.158 --> 00:37:19.950 align:middle line:84% So I don't really have to worry about ties. 00:37:19.950 --> 00:37:22.130 align:middle line:84% So let me just put in the probability 00:37:22.130 --> 00:37:25.770 align:middle line:84% maybe that my bid is strictly higher than all my opponents. 00:37:25.770 --> 00:37:30.650 align:middle line:84% So that's the probability that, b1 is greater than-- well, it 00:37:30.650 --> 00:37:33.610 align:middle line:84% has to be greater than how much b2 bids, b3 bids, 00:37:33.610 --> 00:37:35.107 align:middle line:84% b4-- well that's just the maximum. 00:37:35.107 --> 00:37:37.190 align:middle line:84% Let me just take the maximum of all of their bids. 00:37:37.190 --> 00:37:43.490 align:middle line:84% So I'm going to say the maximum over maybe j-- 00:37:43.490 --> 00:37:46.340 align:middle line:90% let me write it out explicitly-- 00:37:46.340 --> 00:37:54.020 align:middle line:84% of beta v2 all the way up to beta vn. 00:37:54.020 --> 00:37:57.043 align:middle line:90% 00:37:57.043 --> 00:37:58.960 align:middle line:84% I think, again, there's always this confusion. 00:37:58.960 --> 00:38:01.880 align:middle line:84% You see a probability, and then you see b1 and people think, 00:38:01.880 --> 00:38:02.720 align:middle line:90% b1 is random. 00:38:02.720 --> 00:38:05.020 align:middle line:84% No, I know what b1 is. b1 is just a number. 00:38:05.020 --> 00:38:08.452 align:middle line:84% What I don't know is how much my opponents are going to bid, 00:38:08.452 --> 00:38:10.660 align:middle line:84% because even if I believe they're using this strategy 00:38:10.660 --> 00:38:13.460 align:middle line:84% beta, I don't know what their actual valuations are. 00:38:13.460 --> 00:38:16.180 align:middle line:84% And their actual valuations affect how much they actually 00:38:16.180 --> 00:38:16.900 align:middle line:90% bid. 00:38:16.900 --> 00:38:22.980 align:middle line:84% So if I bid b1, I'm going to win if player 2's bid is small, 00:38:22.980 --> 00:38:25.520 align:middle line:84% player 3's bid is small, player n's bid is small, 00:38:25.520 --> 00:38:27.020 align:middle line:84% all of their bids are small enough. 00:38:27.020 --> 00:38:28.860 align:middle line:84% That is, the maximum of all their bids 00:38:28.860 --> 00:38:31.060 align:middle line:90% is smaller than my bid. 00:38:31.060 --> 00:38:32.460 align:middle line:90% Yeah? 00:38:32.460 --> 00:38:35.833 align:middle line:84% AUDIENCE: Because beta is strictly increasing? 00:38:35.833 --> 00:38:38.000 align:middle line:84% IAN BALL: Here, I'm actually about to use that fact. 00:38:38.000 --> 00:38:41.580 align:middle line:84% I don't actually need to use it here. 00:38:41.580 --> 00:38:44.680 align:middle line:84% So I guess this statement doesn't use the fact 00:38:44.680 --> 00:38:46.400 align:middle line:90% that beta is increasing. 00:38:46.400 --> 00:38:50.160 align:middle line:84% Where that will come in is, when will beta of v2 be small? 00:38:50.160 --> 00:38:52.480 align:middle line:90% It will be small if v2 is small. 00:38:52.480 --> 00:38:54.100 align:middle line:84% That's where increasingness comes in. 00:38:54.100 --> 00:38:55.960 align:middle line:84% If beta were not increasing, then maybe 00:38:55.960 --> 00:38:58.640 align:middle line:84% I'd win when my opponent's value was really high because that's 00:38:58.640 --> 00:38:59.560 align:middle line:90% when they bid low. 00:38:59.560 --> 00:39:01.040 align:middle line:84% So there will be a step where we use the fact 00:39:01.040 --> 00:39:02.180 align:middle line:90% that beta is increasing. 00:39:02.180 --> 00:39:03.560 align:middle line:84% But for now, we actually haven't used that fact. 00:39:03.560 --> 00:39:06.192 align:middle line:84% Because I win if my bid is higher than everyone else's bid, 00:39:06.192 --> 00:39:08.400 align:middle line:84% it doesn't matter whether their bidding functions are 00:39:08.400 --> 00:39:12.140 align:middle line:84% increasing or decreasing or crazy, but that will come up. 00:39:12.140 --> 00:39:13.440 align:middle line:90% Yeah? 00:39:13.440 --> 00:39:14.480 align:middle line:90% OK. 00:39:14.480 --> 00:39:18.500 align:middle line:84% So now we will exactly go the next step. 00:39:18.500 --> 00:39:20.960 align:middle line:84% Let's analyze this probability here. 00:39:20.960 --> 00:39:28.200 align:middle line:84% Well, since beta is increasing, which of these players 00:39:28.200 --> 00:39:29.972 align:middle line:90% is going to bid the most? 00:39:29.972 --> 00:39:31.680 align:middle line:84% Well, beta is increasing, and they're all 00:39:31.680 --> 00:39:33.237 align:middle line:90% using the same bidding function. 00:39:33.237 --> 00:39:35.320 align:middle line:84% So which of my opponents is going to bid the most? 00:39:35.320 --> 00:39:35.840 align:middle line:90% Yeah? 00:39:35.840 --> 00:39:37.500 align:middle line:84% AUDIENCE: The opponent that has the highest value. 00:39:37.500 --> 00:39:38.740 align:middle line:84% IAN BALL: The opponent who has the highest value. 00:39:38.740 --> 00:39:40.940 align:middle line:84% So actually, I don't need to go all the way down. 00:39:40.940 --> 00:39:42.220 align:middle line:90% Let me just go here. 00:39:42.220 --> 00:39:44.380 align:middle line:84% I can simplify this expression as just 00:39:44.380 --> 00:39:51.200 align:middle line:84% writing this is beta of the maximum of v2 up to vn. 00:39:51.200 --> 00:39:55.260 align:middle line:90% 00:39:55.260 --> 00:39:58.260 align:middle line:84% And here, I am using the fact that beta is increasing, 00:39:58.260 --> 00:40:01.340 align:middle line:84% because I'm saying the player who bids the most is exactly 00:40:01.340 --> 00:40:04.340 align:middle line:84% going to be the player who values the good the most. 00:40:04.340 --> 00:40:08.900 align:middle line:84% So in words, this is the maximum of the bids of my opponents. 00:40:08.900 --> 00:40:11.260 align:middle line:84% In words, this is the bid of my opponent 00:40:11.260 --> 00:40:12.832 align:middle line:90% who has the highest value. 00:40:12.832 --> 00:40:14.540 align:middle line:84% And because beta is increasing, these two 00:40:14.540 --> 00:40:17.980 align:middle line:90% are going to be the same. 00:40:17.980 --> 00:40:19.300 align:middle line:90% Great. 00:40:19.300 --> 00:40:21.640 align:middle line:84% Now we can simplify it a little bit more. 00:40:21.640 --> 00:40:24.580 align:middle line:90% 00:40:24.580 --> 00:40:27.560 align:middle line:84% This is the random thing, and we want to focus on that. 00:40:27.560 --> 00:40:29.760 align:middle line:84% So let's apply beta inverse to this. 00:40:29.760 --> 00:40:34.900 align:middle line:84% So what we're going to get is v1 minus b1 times the probability 00:40:34.900 --> 00:40:40.440 align:middle line:84% that beta inverse of b1 is greater than. 00:40:40.440 --> 00:40:48.462 align:middle line:90% 00:40:48.462 --> 00:40:49.920 align:middle line:84% So one way of thinking beta inverse 00:40:49.920 --> 00:40:53.560 align:middle line:84% is saying, OK, let's suppose I bid b1. 00:40:53.560 --> 00:41:00.720 align:middle line:90% What is the valuation that-- 00:41:00.720 --> 00:41:06.360 align:middle line:84% which valuation using bidding function beta would bid b1? 00:41:06.360 --> 00:41:09.280 align:middle line:84% And we're saying that valuation should be higher than everyone 00:41:09.280 --> 00:41:11.760 align:middle line:90% else's valuations. 00:41:11.760 --> 00:41:15.080 align:middle line:84% But mathematically, we're just taking an inverse 00:41:15.080 --> 00:41:17.160 align:middle line:84% So now we've simplified things a bit. 00:41:17.160 --> 00:41:19.400 align:middle line:84% I think one key observation to point out 00:41:19.400 --> 00:41:22.220 align:middle line:84% is the math has gotten a little more complicated, 00:41:22.220 --> 00:41:24.400 align:middle line:84% but the basic trade-off hasn't changed. 00:41:24.400 --> 00:41:27.600 align:middle line:84% When I bid more, when I increase b1, 00:41:27.600 --> 00:41:30.980 align:middle line:84% the first term gets smaller, because when I win, 00:41:30.980 --> 00:41:34.240 align:middle line:84% I win less because I have to pay more. 00:41:34.240 --> 00:41:36.700 align:middle line:84% What happens to the second term if beta 1 goes up? 00:41:36.700 --> 00:41:40.340 align:middle line:90% 00:41:40.340 --> 00:41:42.420 align:middle line:84% It's actually easier to see maybe here. 00:41:42.420 --> 00:41:44.780 align:middle line:84% If beta 1 goes up, this increases. 00:41:44.780 --> 00:41:47.800 align:middle line:84% Because if I bid more, it's more likely that I win. 00:41:47.800 --> 00:41:50.360 align:middle line:84% It's more likely that my bid is higher than everyone else's. 00:41:50.360 --> 00:41:52.740 align:middle line:84% So again, we face the basic trade-off, if I bid more, 00:41:52.740 --> 00:41:54.552 align:middle line:84% I increase the chance that I win, 00:41:54.552 --> 00:41:55.760 align:middle line:90% but I win less when I do win. 00:41:55.760 --> 00:41:58.700 align:middle line:84% And we have to balance those two effects. 00:41:58.700 --> 00:42:00.540 align:middle line:84% So we're almost there, but now we 00:42:00.540 --> 00:42:02.460 align:middle line:90% need to analyze this expression. 00:42:02.460 --> 00:42:04.020 align:middle line:84% And what we need to do is we need 00:42:04.020 --> 00:42:06.420 align:middle line:84% to understand the distribution of the maximum 00:42:06.420 --> 00:42:07.980 align:middle line:90% of these random variables. 00:42:07.980 --> 00:42:11.040 align:middle line:84% So this is where we're going to use a little bit of math. 00:42:11.040 --> 00:42:13.780 align:middle line:84% We basically need to compute, what is the distribution 00:42:13.780 --> 00:42:15.940 align:middle line:90% function of this maximum? 00:42:15.940 --> 00:42:22.660 align:middle line:84% So we'll define let's say G of v is equal to the probability 00:42:22.660 --> 00:42:28.740 align:middle line:84% that the maximum of v1 up to vn is less than 00:42:28.740 --> 00:42:31.860 align:middle line:84% or equal to v. This is a definition. 00:42:31.860 --> 00:42:34.605 align:middle line:84% And we want to try to understand this function G. 00:42:34.605 --> 00:42:36.230 align:middle line:84% Well, what does it mean for the maximum 00:42:36.230 --> 00:42:39.590 align:middle line:84% of these random variables to be less than or equal to v? 00:42:39.590 --> 00:42:43.122 align:middle line:84% Well, that's the same as saying v1 is less than v-- 00:42:43.122 --> 00:42:44.330 align:middle line:90% ah, I shouldn't have v1 here. 00:42:44.330 --> 00:42:45.080 align:middle line:90% This should be v2. 00:42:45.080 --> 00:42:47.590 align:middle line:90% I apologize. 00:42:47.590 --> 00:42:50.670 align:middle line:84% It's the same as saying v2 is less than v, v3 is less than v, 00:42:50.670 --> 00:42:52.487 align:middle line:84% all the way up to vn is less than v. 00:42:52.487 --> 00:42:55.070 align:middle line:84% For the maximum of things to be smaller than something, that's 00:42:55.070 --> 00:42:56.987 align:middle line:84% going to be true exactly if every single thing 00:42:56.987 --> 00:43:00.950 align:middle line:84% in the maximum is smaller than that thing. 00:43:00.950 --> 00:43:06.470 align:middle line:84% And the trick here is that we can use independence. 00:43:06.470 --> 00:43:09.670 align:middle line:84% So it turns out this is the equal to the probability 00:43:09.670 --> 00:43:13.510 align:middle line:84% that v2 is less than or equal to v times all the way up 00:43:13.510 --> 00:43:18.790 align:middle line:84% to the probability that vn is less than or equal to v, 00:43:18.790 --> 00:43:21.350 align:middle line:90% because this is the-- 00:43:21.350 --> 00:43:26.070 align:middle line:84% this occurs if and only if each of these events occurs. 00:43:26.070 --> 00:43:28.070 align:middle line:84% And because they're independent, the probability 00:43:28.070 --> 00:43:29.310 align:middle line:84% of the joint occurrence of things 00:43:29.310 --> 00:43:30.970 align:middle line:84% is just the product of the probabilities. 00:43:30.970 --> 00:43:32.428 align:middle line:84% This is like saying the probability 00:43:32.428 --> 00:43:34.670 align:middle line:84% that I flip two heads is just one half times a half. 00:43:34.670 --> 00:43:35.490 align:middle line:90% It's one quarter. 00:43:35.490 --> 00:43:38.530 align:middle line:84% That's the fact that we're using if they're independent. 00:43:38.530 --> 00:43:39.922 align:middle line:84% And again, I feel like whenever I 00:43:39.922 --> 00:43:41.630 align:middle line:84% go over this, if you've seen this before, 00:43:41.630 --> 00:43:42.547 align:middle line:90% this will be too slow. 00:43:42.547 --> 00:43:45.250 align:middle line:84% If you haven't seen it, it'll be too fast. 00:43:45.250 --> 00:43:48.850 align:middle line:84% I think there's not a good solution to that. 00:43:48.850 --> 00:43:51.850 align:middle line:84% Can I simplify this expression on the right side? 00:43:51.850 --> 00:43:53.830 align:middle line:84% Let's just be clear what my notation means. 00:43:53.830 --> 00:43:55.690 align:middle line:84% I have n minus 1 numbers here, and I'm 00:43:55.690 --> 00:43:56.950 align:middle line:90% taking the product of them. 00:43:56.950 --> 00:43:58.970 align:middle line:90% This is one number. 00:43:58.970 --> 00:44:01.930 align:middle line:84% Then I have a number for v3, a number for v4, all the way up 00:44:01.930 --> 00:44:02.470 align:middle line:90% to vn. 00:44:02.470 --> 00:44:04.630 align:middle line:84% I have n minus 1 numbers and I'm multiplying them together. 00:44:04.630 --> 00:44:05.130 align:middle line:90% Yes? 00:44:05.130 --> 00:44:09.708 align:middle line:84% AUDIENCE: So isn't that just the CDF at b to the n minus 1? 00:44:09.708 --> 00:44:10.750 align:middle line:90% IAN BALL: Exactly, right? 00:44:10.750 --> 00:44:12.650 align:middle line:84% Because what is the probability that v2 00:44:12.650 --> 00:44:14.170 align:middle line:90% is less than or equal to v? 00:44:14.170 --> 00:44:18.810 align:middle line:84% Well, that's just F of v. And the probability 00:44:18.810 --> 00:44:23.130 align:middle line:84% that v3 is less than or equal to v, that's just F of v. 00:44:23.130 --> 00:44:25.770 align:middle line:84% So I just get F of v multiplied by itself 00:44:25.770 --> 00:44:27.970 align:middle line:84% n minus 1 times because I'm going 00:44:27.970 --> 00:44:30.010 align:middle line:90% from 2 all the way up to n. 00:44:30.010 --> 00:44:36.130 align:middle line:84% So what I'm going to get is F of v to the n minus 1. 00:44:36.130 --> 00:44:40.550 align:middle line:90% 00:44:40.550 --> 00:44:44.150 align:middle line:84% Let's just make sure this makes sense. 00:44:44.150 --> 00:44:48.550 align:middle line:84% Is G of v going to be bigger or smaller than F of v? 00:44:48.550 --> 00:44:52.070 align:middle line:84% And let's see if this makes sense. 00:44:52.070 --> 00:44:55.070 align:middle line:84% So F of v is a number between 0 and 1. 00:44:55.070 --> 00:44:56.950 align:middle line:84% So when we multiply a number between-- when 00:44:56.950 --> 00:45:00.230 align:middle line:84% we raise a number between 0 and 1 to the power n minus 1, 00:45:00.230 --> 00:45:01.790 align:middle line:84% that number is going to get smaller. 00:45:01.790 --> 00:45:03.950 align:middle line:84% And that makes sense because the probability 00:45:03.950 --> 00:45:06.390 align:middle line:84% for all of these things to be less than v 00:45:06.390 --> 00:45:09.510 align:middle line:84% is certainly-- that's less likely than the probability 00:45:09.510 --> 00:45:12.110 align:middle line:84% that a single one of them is less than or equal to v. 00:45:12.110 --> 00:45:14.850 align:middle line:84% And we can also see another intuitive property of this. 00:45:14.850 --> 00:45:16.890 align:middle line:84% What happens when n gets really, really big? 00:45:16.890 --> 00:45:19.630 align:middle line:90% 00:45:19.630 --> 00:45:22.190 align:middle line:84% What happens to this right-hand side? 00:45:22.190 --> 00:45:22.750 align:middle line:90% Yeah? 00:45:22.750 --> 00:45:26.470 align:middle line:84% AUDIENCE: Lower probability that you're on [INAUDIBLE] 00:45:26.470 --> 00:45:28.990 align:middle line:84% IAN BALL: Yeah, so it gets really, really small, 00:45:28.990 --> 00:45:32.050 align:middle line:84% because the probability that everyone else is less than v 00:45:32.050 --> 00:45:33.030 align:middle line:90% goes down. 00:45:33.030 --> 00:45:36.130 align:middle line:84% So I just want to show this kind of makes intuitive sense. 00:45:36.130 --> 00:45:41.450 align:middle line:84% And then one final piece of notation that will be helpful 00:45:41.450 --> 00:45:44.730 align:middle line:90% is let me-- 00:45:44.730 --> 00:45:45.870 align:middle line:90% I'm out of space here. 00:45:45.870 --> 00:45:47.270 align:middle line:90% Let me go to a new board. 00:45:47.270 --> 00:45:58.890 align:middle line:90% 00:45:58.890 --> 00:46:01.630 align:middle line:84% Actually, realize it'll be good to be close to this board. 00:46:01.630 --> 00:46:03.190 align:middle line:84% So let's go to this board instead. 00:46:03.190 --> 00:46:25.710 align:middle line:90% 00:46:25.710 --> 00:46:29.070 align:middle line:84% So I just wanted to find the density, 00:46:29.070 --> 00:46:33.550 align:middle line:84% the probability density function, associated with G. 00:46:33.550 --> 00:46:40.390 align:middle line:84% So let's define one other definition, little g of v 00:46:40.390 --> 00:46:44.590 align:middle line:84% is equal to the derivative of big G. 00:46:44.590 --> 00:46:47.150 align:middle line:84% Well, if big G equals this, now I just 00:46:47.150 --> 00:46:50.710 align:middle line:84% have to apply the chain rule, and I'll 00:46:50.710 --> 00:47:01.170 align:middle line:84% get n minus 1 times F of v to the n minus 2 times f of v. 00:47:01.170 --> 00:47:04.772 align:middle line:84% So I'm just using the power rule to bring the n minus 1 down, 00:47:04.772 --> 00:47:06.230 align:middle line:84% and then the chain rule to multiply 00:47:06.230 --> 00:47:09.830 align:middle line:84% by little f using the fact that little f is 00:47:09.830 --> 00:47:14.090 align:middle line:84% the derivative of big F. If you're a little rusty on this, 00:47:14.090 --> 00:47:17.170 align:middle line:90% that's not so important. 00:47:17.170 --> 00:47:19.470 align:middle line:84% So with all these simplifications, 00:47:19.470 --> 00:47:23.170 align:middle line:84% I think we now have a pretty clean formula. 00:47:23.170 --> 00:47:35.200 align:middle line:84% We have U1 of b1 given v1 beta is equal to v1 minus b1 times-- 00:47:35.200 --> 00:47:46.520 align:middle line:84% well, we use our function G, And we get G of beta inverse of b1. 00:47:46.520 --> 00:47:50.440 align:middle line:84% So all I've done is used our notation G to apply to here. 00:47:50.440 --> 00:47:53.240 align:middle line:84% There's a little wrinkle because this was a strict inequality 00:47:53.240 --> 00:47:56.140 align:middle line:84% and big G is defined by a weak inequality. 00:47:56.140 --> 00:47:58.913 align:middle line:84% But the probability that the maximum takes a single value 00:47:58.913 --> 00:48:00.580 align:middle line:84% is 0, so that doesn't make a difference, 00:48:00.580 --> 00:48:03.300 align:middle line:90% so I can ignore that. 00:48:03.300 --> 00:48:06.640 align:middle line:84% So now this is the expression we want to maximize, once again 00:48:06.640 --> 00:48:09.040 align:middle line:84% recognizing the trade-off that as I increase b1, 00:48:09.040 --> 00:48:11.320 align:middle line:84% this gets smaller and this gets bigger. 00:48:11.320 --> 00:48:14.420 align:middle line:84% So in words, this is the probability that I win. 00:48:14.420 --> 00:48:20.825 align:middle line:90% 00:48:20.825 --> 00:48:23.200 align:middle line:84% The way we're expressing it is the probability that I win 00:48:23.200 --> 00:48:26.340 align:middle line:84% is the probability that everyone else's valuation is 00:48:26.340 --> 00:48:28.700 align:middle line:90% below this magical threshold. 00:48:28.700 --> 00:48:30.278 align:middle line:84% What's magical about this threshold? 00:48:30.278 --> 00:48:31.820 align:middle line:84% This is the valuation that they would 00:48:31.820 --> 00:48:35.060 align:middle line:90% have to have to bid exactly b1. 00:48:35.060 --> 00:48:37.980 align:middle line:84% So I'm going to win if every single player's valuation is 00:48:37.980 --> 00:48:41.080 align:middle line:84% below the valuation that they would need in order to bid b1. 00:48:41.080 --> 00:48:44.780 align:middle line:90% 00:48:44.780 --> 00:48:47.620 align:middle line:84% So now what I want to do is I want 00:48:47.620 --> 00:48:49.460 align:middle line:84% to take the derivative of this expression 00:48:49.460 --> 00:48:51.600 align:middle line:84% so that I can choose the b1 optimally. 00:48:51.600 --> 00:48:53.000 align:middle line:84% I want to maximize this function. 00:48:53.000 --> 00:48:54.900 align:middle line:84% This function is telling me what my payoff 00:48:54.900 --> 00:48:57.620 align:middle line:84% is as a function of how much I bid. 00:48:57.620 --> 00:49:00.620 align:middle line:84% So let me just take derivatives and maximize this. 00:49:00.620 --> 00:49:03.740 align:middle line:84% So if I take derivatives and evaluate the first-order 00:49:03.740 --> 00:49:07.000 align:middle line:84% condition, I'm going to get minus-- 00:49:07.000 --> 00:49:31.600 align:middle line:90% 00:49:31.600 --> 00:49:33.280 align:middle line:90% Let's go through what I've done. 00:49:33.280 --> 00:49:35.280 align:middle line:84% I've taken the derivative of the first term 00:49:35.280 --> 00:49:40.560 align:middle line:84% is negative 1 and multiplied it by the second term to get this. 00:49:40.560 --> 00:49:42.960 align:middle line:84% Then I've done the first term times the derivative 00:49:42.960 --> 00:49:44.200 align:middle line:90% of the second term. 00:49:44.200 --> 00:49:45.680 align:middle line:90% I've used the chain rule. 00:49:45.680 --> 00:49:47.640 align:middle line:84% So when I take the derivative of this, 00:49:47.640 --> 00:49:51.720 align:middle line:84% first I take the derivative of big G to get little g. 00:49:51.720 --> 00:49:54.360 align:middle line:84% And then I need to take the derivative of beta inverse. 00:49:54.360 --> 00:49:57.280 align:middle line:84% But here I'm using what's called the implicit function theorem. 00:49:57.280 --> 00:50:00.357 align:middle line:84% The derivative of the inverse is the inverse of the derivative. 00:50:00.357 --> 00:50:01.940 align:middle line:84% It's the reciprocal of the derivative. 00:50:01.940 --> 00:50:05.160 align:middle line:90% So this is just a calculus fact. 00:50:05.160 --> 00:50:06.680 align:middle line:84% The slope of the inverse is just 1 00:50:06.680 --> 00:50:08.680 align:middle line:84% over the slope of the original function. 00:50:08.680 --> 00:50:11.140 align:middle line:90% So I'm going to get this here. 00:50:11.140 --> 00:50:19.846 align:middle line:90% 00:50:19.846 --> 00:50:26.460 align:middle line:84% So now I have a formula for what my optimal bid b1 should be. 00:50:26.460 --> 00:50:30.620 align:middle line:84% So my optimal bid b1 has to satisfy this formula. 00:50:30.620 --> 00:50:34.820 align:middle line:84% But I'm looking for an equilibrium. 00:50:34.820 --> 00:50:42.580 align:middle line:84% So it must be the case that this holds when b1 equals. 00:50:42.580 --> 00:50:44.460 align:middle line:90% beta of V1. 00:50:44.460 --> 00:50:46.980 align:middle line:90% This is kind of a subtle point. 00:50:46.980 --> 00:50:48.940 align:middle line:84% Let's understand what we're doing here. 00:50:48.940 --> 00:50:52.780 align:middle line:84% This is a formula that's satisfied by my optimal bid. 00:50:52.780 --> 00:50:55.460 align:middle line:84% But in order for this to be an equilibrium, 00:50:55.460 --> 00:51:00.323 align:middle line:84% my optimal bid must be beta of v1, 00:51:00.323 --> 00:51:02.740 align:middle line:84% because we want it to be the case that if everyone else is 00:51:02.740 --> 00:51:06.340 align:middle line:84% using this bidding function beta, and my valuation is v1, 00:51:06.340 --> 00:51:11.100 align:middle line:84% then the optimal bid for me is exactly beta of v1. 00:51:11.100 --> 00:51:13.800 align:middle line:84% So we know that this must hold at beta v1. 00:51:13.800 --> 00:51:16.320 align:middle line:84% So maybe a more formal way of saying this, 00:51:16.320 --> 00:51:17.540 align:middle line:90% so why is this true? 00:51:17.540 --> 00:51:19.600 align:middle line:90% Let me give the justification. 00:51:19.600 --> 00:51:24.440 align:middle line:84% It's because beta of v1 must be an element 00:51:24.440 --> 00:51:29.540 align:middle line:90% of the argmax over b1 prime. 00:51:29.540 --> 00:51:36.680 align:middle line:90% 00:51:36.680 --> 00:51:38.360 align:middle line:84% So of all the possible bids I could 00:51:38.360 --> 00:51:42.040 align:middle line:84% choose as player 1, following my equilibrium strategy 00:51:42.040 --> 00:51:44.160 align:middle line:90% must be the best possible bid. 00:51:44.160 --> 00:51:45.720 align:middle line:84% So when I take derivatives of this 00:51:45.720 --> 00:51:47.840 align:middle line:84% and set them to 0 to find the maximum, 00:51:47.840 --> 00:51:50.200 align:middle line:84% that first-order condition must be satisfied 00:51:50.200 --> 00:51:53.400 align:middle line:90% by my maximizer, beta of v1. 00:51:53.400 --> 00:51:55.332 align:middle line:84% I think this is maybe a little subtle. 00:51:55.332 --> 00:51:56.540 align:middle line:90% This is kind of the key step. 00:51:56.540 --> 00:52:00.200 align:middle line:90% So any questions here? 00:52:00.200 --> 00:52:00.960 align:middle line:90% Yes? 00:52:00.960 --> 00:52:02.710 align:middle line:84% AUDIENCE: So are you basically just saying 00:52:02.710 --> 00:52:06.020 align:middle line:84% that this is only necessarily true if b1 equals beta? 00:52:06.020 --> 00:52:07.020 align:middle line:90% IAN BALL: Exactly right. 00:52:07.020 --> 00:52:07.520 align:middle line:90% Yeah. 00:52:07.520 --> 00:52:11.320 align:middle line:84% So I'm solving for the optimal-- this is a condition that must 00:52:11.320 --> 00:52:15.160 align:middle line:90% be satisfied by my optimal bid. 00:52:15.160 --> 00:52:18.870 align:middle line:84% But then I know that beta of v1 must be my optimal bid. 00:52:18.870 --> 00:52:22.750 align:middle line:84% So therefore, beta of v1 must satisfy this equation. 00:52:22.750 --> 00:52:24.750 align:middle line:84% And that's going to simplify things a lot right. 00:52:24.750 --> 00:52:29.310 align:middle line:84% So let me plug in beta of v1 and see what I get here. 00:52:29.310 --> 00:52:33.610 align:middle line:90% 00:52:33.610 --> 00:52:36.350 align:middle line:84% And of course, the details of this are all in the notes, 00:52:36.350 --> 00:52:39.950 align:middle line:84% but I think if you don't see it in real time, 00:52:39.950 --> 00:52:41.770 align:middle line:84% it can get a little overwhelming. 00:52:41.770 --> 00:52:46.690 align:middle line:90% 00:52:46.690 --> 00:52:50.698 align:middle line:84% So if we plug in beta of v1, we're 00:52:50.698 --> 00:52:52.490 align:middle line:84% actually going to get-- things are actually 00:52:52.490 --> 00:52:54.650 align:middle line:90% going to simplify a lot. 00:52:54.650 --> 00:53:03.850 align:middle line:84% We're going to get 0 equals negative G of v1, 00:53:03.850 --> 00:53:08.450 align:middle line:84% because beta inverse of beta of v1 is just v1. 00:53:08.450 --> 00:53:25.110 align:middle line:84% And then we get plus v1 minus beta of v1 times g of v1 times-- 00:53:25.110 --> 00:53:30.650 align:middle line:90% divided by beta prime of v1. 00:53:30.650 --> 00:53:33.390 align:middle line:90% 00:53:33.390 --> 00:53:35.430 align:middle line:84% So this is a much simpler formula. 00:53:35.430 --> 00:53:40.990 align:middle line:84% And we can actually simplify it a little further. 00:53:40.990 --> 00:53:43.670 align:middle line:84% We know we're allowed to divide by this because beta prime is 00:53:43.670 --> 00:53:45.190 align:middle line:90% strictly positive. 00:53:45.190 --> 00:53:47.230 align:middle line:84% That's an assumption we made over here. 00:53:47.230 --> 00:53:51.230 align:middle line:84% So let's multiply by beta prime and move this to the other side. 00:53:51.230 --> 00:53:59.890 align:middle line:84% And what we get is beta prime of v1, G of v1 equals-- 00:53:59.890 --> 00:54:14.770 align:middle line:90% 00:54:14.770 --> 00:54:18.250 align:middle line:84% and this is a necessary condition 00:54:18.250 --> 00:54:21.370 align:middle line:84% for-- that our beta function must satisfy in order for us 00:54:21.370 --> 00:54:25.690 align:middle line:84% to have an equilibrium where everyone uses the beta function. 00:54:25.690 --> 00:54:30.890 align:middle line:84% Now, I think we can actually intuit this pretty nicely. 00:54:30.890 --> 00:54:33.850 align:middle line:84% So here's some nice intuition for this. 00:54:33.850 --> 00:54:38.610 align:middle line:84% What I want to say is suppose my value is v1, 00:54:38.610 --> 00:54:42.370 align:middle line:84% and I'm contemplating, instead of bidding beta of v1, 00:54:42.370 --> 00:54:45.390 align:middle line:84% I could bid beta of v1 plus epsilon. 00:54:45.390 --> 00:54:50.450 align:middle line:90% 00:54:50.450 --> 00:54:53.730 align:middle line:84% Well, let's think about epsilon being very small. 00:54:53.730 --> 00:54:56.890 align:middle line:84% So I'm supposed to bid beta of v1. 00:54:56.890 --> 00:54:58.570 align:middle line:84% We want to see what happens if I instead 00:54:58.570 --> 00:55:00.530 align:middle line:90% bid beta of v1 plus epsilon. 00:55:00.530 --> 00:55:06.410 align:middle line:84% And this is approximately going to be beta of v1 plus epsilon 00:55:06.410 --> 00:55:10.090 align:middle line:90% times b prime of v1. 00:55:10.090 --> 00:55:11.930 align:middle line:90% This is just a Taylor expansion. 00:55:11.930 --> 00:55:16.710 align:middle line:90% 00:55:16.710 --> 00:55:25.990 align:middle line:84% So if epsilon is really small and positive, what I claim 00:55:25.990 --> 00:55:44.070 align:middle line:84% is that this is measuring the gain from winning more. 00:55:44.070 --> 00:55:53.070 align:middle line:84% And this is measuring the additional payment 00:55:53.070 --> 00:55:55.050 align:middle line:90% to understand what's going on. 00:55:55.050 --> 00:56:00.710 align:middle line:90% 00:56:00.710 --> 00:56:07.790 align:middle line:84% If I deviate from beta of v1 to beta of v1 plus epsilon, 00:56:07.790 --> 00:56:11.658 align:middle line:90% then I'm going to win-- 00:56:11.658 --> 00:56:13.450 align:middle line:84% how is that going to change the probability 00:56:13.450 --> 00:56:15.370 align:middle line:90% that I win the auction? 00:56:15.370 --> 00:56:28.730 align:middle line:84% Well it's only going to make a difference if my opponents-- 00:56:28.730 --> 00:56:30.810 align:middle line:84% if the maximum value of my opponent 00:56:30.810 --> 00:56:35.370 align:middle line:84% is between v1 and v1 plus epsilon. 00:56:35.370 --> 00:56:37.250 align:middle line:84% If the maximum value of my opponents 00:56:37.250 --> 00:56:39.490 align:middle line:84% is between v1 and v1 plus epsilon, 00:56:39.490 --> 00:56:42.590 align:middle line:84% then when I bid as if my evaluation were v1, I lost. 00:56:42.590 --> 00:56:46.070 align:middle line:84% But when I increase my bid to bidding beta v1 plus epsilon, 00:56:46.070 --> 00:56:46.830 align:middle line:90% now I win. 00:56:46.830 --> 00:56:51.850 align:middle line:90% 00:56:51.850 --> 00:56:59.530 align:middle line:84% The probability of that is roughly g of v1 times epsilon, 00:56:59.530 --> 00:57:02.490 align:middle line:84% because it's the density of my opponents' bid 00:57:02.490 --> 00:57:04.890 align:middle line:84% times the width of this interval. 00:57:04.890 --> 00:57:08.580 align:middle line:84% So what this right-hand term is exactly 00:57:08.580 --> 00:57:11.340 align:middle line:84% expressing is it's saying if I slightly increase my bid, 00:57:11.340 --> 00:57:15.100 align:middle line:84% I get my utility if I win times a slightly higher probability 00:57:15.100 --> 00:57:18.600 align:middle line:84% of winning, which is exactly g of v1 times epsilon. 00:57:18.600 --> 00:57:21.220 align:middle line:84% That's the amount by which my winning probability goes up 00:57:21.220 --> 00:57:22.820 align:middle line:90% by bidding a bit more. 00:57:22.820 --> 00:57:27.220 align:middle line:84% On the other hand, if I bid a little bit more, 00:57:27.220 --> 00:57:32.220 align:middle line:84% I have to pay epsilon beta prime of v1 whenever I win. 00:57:32.220 --> 00:57:34.740 align:middle line:84% So I lose this much in additional payments 00:57:34.740 --> 00:57:35.397 align:middle line:90% whenever I win. 00:57:35.397 --> 00:57:36.980 align:middle line:84% And what's the probability that I win? 00:57:36.980 --> 00:57:38.400 align:middle line:90% Exactly big G of v1. 00:57:38.400 --> 00:57:39.493 align:middle line:90% So there's a very-- 00:57:39.493 --> 00:57:41.160 align:middle line:84% if you could really had great intuition, 00:57:41.160 --> 00:57:43.260 align:middle line:84% you could maybe jump right away to this formula 00:57:43.260 --> 00:57:46.120 align:middle line:84% by thinking through this, but we derived it with calculus, 00:57:46.120 --> 00:57:47.280 align:middle line:90% so it doesn't matter. 00:57:47.280 --> 00:57:49.860 align:middle line:90% 00:57:49.860 --> 00:57:52.580 align:middle line:84% So to be very precise, what I'm saying 00:57:52.580 --> 00:57:55.320 align:middle line:84% is if instead of bidding beta of v1, 00:57:55.320 --> 00:57:57.780 align:middle line:84% I bid beta of v1 plus epsilon, then 00:57:57.780 --> 00:58:00.580 align:middle line:84% my expected additional payment would be approximately epsilon 00:58:00.580 --> 00:58:04.220 align:middle line:84% times the left-hand side, and my expected gain from winning more 00:58:04.220 --> 00:58:07.240 align:middle line:84% would be approximately epsilon times the right-hand side. 00:58:07.240 --> 00:58:09.360 align:middle line:84% And if I'm already in an optimum, 00:58:09.360 --> 00:58:13.060 align:middle line:84% then the effect of this perturbation must be 0. 00:58:13.060 --> 00:58:14.620 align:middle line:84% This is like a first-order condition. 00:58:14.620 --> 00:58:16.078 align:middle line:84% And therefore, I get this equation. 00:58:16.078 --> 00:58:18.340 align:middle line:84% But that may have been a little subtle. 00:58:18.340 --> 00:58:20.080 align:middle line:84% So that's how I like to think about it. 00:58:20.080 --> 00:58:22.580 align:middle line:84% But if you don't see that the first time, I think that's OK. 00:58:22.580 --> 00:58:23.080 align:middle line:90% Yeah? 00:58:23.080 --> 00:58:26.760 align:middle line:84% AUDIENCE: So the idea isn't that in the formula, you've-- 00:58:26.760 --> 00:58:30.220 align:middle line:84% so you have to multiply both sides by epsilon. 00:58:30.220 --> 00:58:30.926 align:middle line:90% IAN BALL: Right. 00:58:30.926 --> 00:58:31.280 align:middle line:90% Yeah. 00:58:31.280 --> 00:58:32.800 align:middle line:84% AUDIENCE: So epsilon is an implicit. 00:58:32.800 --> 00:58:33.467 align:middle line:90% IAN BALL: Right. 00:58:33.467 --> 00:58:38.460 align:middle line:84% So the argument is if I multiply both sides by epsilon, 00:58:38.460 --> 00:58:42.663 align:middle line:84% then I guess the more formal argument would 00:58:42.663 --> 00:58:44.080 align:middle line:84% be I could actually choose epsilon 00:58:44.080 --> 00:58:47.218 align:middle line:84% to be a small positive number or a small negative number. 00:58:47.218 --> 00:58:49.760 align:middle line:84% So because I can choose it to be a small positive or negative 00:58:49.760 --> 00:58:51.280 align:middle line:84% number, the only way that I won't 00:58:51.280 --> 00:58:54.720 align:middle line:84% have a profitable deviation is if both sides of this are equal. 00:58:54.720 --> 00:58:56.520 align:middle line:84% AUDIENCE: And then why does it make sense 00:58:56.520 --> 00:58:59.960 align:middle line:84% that some of this stuff is like derivative, some of it's 00:58:59.960 --> 00:59:02.260 align:middle line:90% like CDF, et cetera. 00:59:02.260 --> 00:59:05.460 align:middle line:84% IAN BALL: Yeah, so let's understand the role of it. 00:59:05.460 --> 00:59:07.780 align:middle line:84% I'm saying, what happens if I bid slightly higher? 00:59:07.780 --> 00:59:09.420 align:middle line:90% Let's take epsilon positive. 00:59:09.420 --> 00:59:13.380 align:middle line:84% Well, the amount more by which I bid 00:59:13.380 --> 00:59:15.480 align:middle line:84% is epsilon times beta prime of v1. 00:59:15.480 --> 00:59:16.860 align:middle line:90% That's how much more I bid. 00:59:16.860 --> 00:59:19.620 align:middle line:84% So you can think of that-- let's put the mini epsilon here. 00:59:19.620 --> 00:59:21.500 align:middle line:84% That's how much more I'm bidding. 00:59:21.500 --> 00:59:23.860 align:middle line:84% But what's the increase in how much I pay? 00:59:23.860 --> 00:59:26.820 align:middle line:84% Well, I don't actually pay that extra amount unless I win. 00:59:26.820 --> 00:59:28.340 align:middle line:84% What's the probability that I win? 00:59:28.340 --> 00:59:31.100 align:middle line:84% It's the probability that everyone else's value 00:59:31.100 --> 00:59:32.380 align:middle line:90% is below mine. 00:59:32.380 --> 00:59:34.080 align:middle line:90% That's exactly G of V1. 00:59:34.080 --> 00:59:36.700 align:middle line:84% So this is the probability I win times the extra payment I 00:59:36.700 --> 00:59:38.700 align:middle line:90% make if I do win. 00:59:38.700 --> 00:59:43.300 align:middle line:84% On the other hand, I'm looking at the extra probability 00:59:43.300 --> 00:59:47.760 align:middle line:90% that I win is g times epsilon. 00:59:47.760 --> 00:59:49.260 align:middle line:90% That's what I get down here. 00:59:49.260 --> 00:59:52.140 align:middle line:84% But then that marginal change in how much I win 00:59:52.140 --> 00:59:56.300 align:middle line:84% is multiplied by the utility I get when I do win. 00:59:56.300 --> 00:59:59.268 align:middle line:84% But I'll admit this is a little-- if you follow the math 00:59:59.268 --> 01:00:01.060 align:middle line:84% and you don't fully see all this intuition, 01:00:01.060 --> 01:00:02.500 align:middle line:84% I think that's OK the first time. 01:00:02.500 --> 01:00:04.460 align:middle line:90% Yeah. 01:00:04.460 --> 01:00:05.720 align:middle line:90% So now we want to solve this. 01:00:05.720 --> 01:00:06.720 align:middle line:90% We're close. 01:00:06.720 --> 01:00:08.420 align:middle line:90% We have this formula. 01:00:08.420 --> 01:00:12.120 align:middle line:90% Let me erase the epsilons now. 01:00:12.120 --> 01:00:13.800 align:middle line:84% And now we have a differential equation. 01:00:13.800 --> 01:00:15.175 align:middle line:84% This is not really a course where 01:00:15.175 --> 01:00:17.920 align:middle line:84% we're trying to test how well people can solve differential 01:00:17.920 --> 01:00:20.220 align:middle line:84% equations, but this one, there's a pretty nice trick. 01:00:20.220 --> 01:00:22.720 align:middle line:90% So let me just-- 01:00:22.720 --> 01:00:24.840 align:middle line:84% I want to get all the betas to one side 01:00:24.840 --> 01:00:27.100 align:middle line:84% because we now have a differential equation for beta. 01:00:27.100 --> 01:00:29.433 align:middle line:84% And now we want to find the function beta that satisfies 01:00:29.433 --> 01:00:30.760 align:middle line:90% this differential equation. 01:00:30.760 --> 01:00:33.160 align:middle line:84% So if we move all the betas to one side-- 01:00:33.160 --> 01:00:36.580 align:middle line:84% and let me also call my variable now v instead of v1, 01:00:36.580 --> 01:00:38.720 align:middle line:84% just to make things-- life a little easier. 01:00:38.720 --> 01:00:46.440 align:middle line:84% So then my differential equation becomes beta prime of v G of v 01:00:46.440 --> 01:00:55.360 align:middle line:90% plus beta of v g of v. 01:00:55.360 --> 01:00:56.940 align:middle line:84% So this is my differential equation. 01:00:56.940 --> 01:00:58.482 align:middle line:84% So formally, what we've done is we've 01:00:58.482 --> 01:01:00.580 align:middle line:84% said-- we've guessed that there exists 01:01:00.580 --> 01:01:05.300 align:middle line:84% an equilibrium in which everyone used the bidding function beta. 01:01:05.300 --> 01:01:07.980 align:middle line:84% And then we've used the properties of equilibrium 01:01:07.980 --> 01:01:09.780 align:middle line:84% to argue that if that's the case, 01:01:09.780 --> 01:01:12.280 align:middle line:84% then we know a lot about what this beta function must be. 01:01:12.280 --> 01:01:15.860 align:middle line:84% In fact, it must satisfy this special differential equation. 01:01:15.860 --> 01:01:18.600 align:middle line:84% And now we want to solve for this differential equation. 01:01:18.600 --> 01:01:20.410 align:middle line:84% Find the function that satisfies this. 01:01:20.410 --> 01:01:22.660 align:middle line:84% And if anyone's recently taken differential equations, 01:01:22.660 --> 01:01:25.557 align:middle line:84% they may see there's kind of something special 01:01:25.557 --> 01:01:26.640 align:middle line:90% about this left-hand side. 01:01:26.640 --> 01:01:28.640 align:middle line:84% Does anyone-- anything pop out at you. 01:01:28.640 --> 01:01:32.260 align:middle line:84% I don't know you just took a class on ODEs? 01:01:32.260 --> 01:01:32.897 align:middle line:90% Yeah? 01:01:32.897 --> 01:01:34.980 align:middle line:84% AUDIENCE: That looks like the product [INAUDIBLE]. 01:01:34.980 --> 01:01:35.740 align:middle line:90% IAN BALL: Exactly. 01:01:35.740 --> 01:01:42.883 align:middle line:84% So let's express the left-hand side as beta of v G of v-- 01:01:42.883 --> 01:01:45.300 align:middle line:84% this is kind of sloppy notation, but I mean the derivative 01:01:45.300 --> 01:01:46.133 align:middle line:90% of this whole thing. 01:01:46.133 --> 01:01:47.298 align:middle line:90% Maybe I'll write d/dv. 01:01:47.298 --> 01:01:47.840 align:middle line:90% I don't know. 01:01:47.840 --> 01:01:49.840 align:middle line:84% Everyone has different notation for derivatives. 01:01:49.840 --> 01:01:51.260 align:middle line:90% Let's write that. 01:01:51.260 --> 01:01:53.220 align:middle line:84% Well, now I have the derivative of something, 01:01:53.220 --> 01:01:54.500 align:middle line:90% and I have something else. 01:01:54.500 --> 01:01:57.560 align:middle line:84% So now I'm just going to integrate this from v 01:01:57.560 --> 01:02:03.260 align:middle line:84% lower bar to v. There's always an issue with what terms we use. 01:02:03.260 --> 01:02:07.513 align:middle line:84% But, OK, so let's integrate both sides from v lower bar to-- 01:02:07.513 --> 01:02:10.180 align:middle line:84% well, we don't want to call it v because we already have v here. 01:02:10.180 --> 01:02:11.880 align:middle line:90% So let's call it V hat 01:02:11.880 --> 01:02:14.800 align:middle line:84% So on the left-hand side, we get beta 01:02:14.800 --> 01:02:21.040 align:middle line:84% of v G of v evaluated from v equals v lower bar 01:02:21.040 --> 01:02:23.640 align:middle line:90% to let's say V hat. 01:02:23.640 --> 01:02:27.360 align:middle line:84% And on the right-hand side, we get the integral from v 01:02:27.360 --> 01:02:36.200 align:middle line:84% lower bar to V hat of v g of v dv. 01:02:36.200 --> 01:02:39.320 align:middle line:84% So if these two sides are equal, certainly the integrals 01:02:39.320 --> 01:02:40.720 align:middle line:90% of the two sides are equal. 01:02:40.720 --> 01:02:43.788 align:middle line:84% I've integrated the right side from v lower bar to v upper bar. 01:02:43.788 --> 01:02:46.080 align:middle line:84% And I've also integrated the left side from v lower bar 01:02:46.080 --> 01:02:47.020 align:middle line:90% to v upper bar. 01:02:47.020 --> 01:02:48.760 align:middle line:84% But this just takes a really simple form 01:02:48.760 --> 01:02:51.180 align:middle line:84% because I'm taking the integral of a derivative. 01:02:51.180 --> 01:02:54.360 align:middle line:84% The integral of a derivative is just the antiderivative 01:02:54.360 --> 01:02:56.690 align:middle line:90% at the endpoints. 01:02:56.690 --> 01:03:00.690 align:middle line:84% And this notation here means I plug in v equals V hat, 01:03:00.690 --> 01:03:03.210 align:middle line:84% and then I subtract the expression with v 01:03:03.210 --> 01:03:04.110 align:middle line:90% equals v lower bar. 01:03:04.110 --> 01:03:07.570 align:middle line:84% I think this is pretty standard notation. 01:03:07.570 --> 01:03:11.470 align:middle line:84% So maybe I'll put in arrows and I'll write integrate. 01:03:11.470 --> 01:03:15.170 align:middle line:90% 01:03:15.170 --> 01:03:17.050 align:middle line:90% So now we're getting closer. 01:03:17.050 --> 01:03:19.210 align:middle line:84% But I'd argue that this left-hand side simplifies 01:03:19.210 --> 01:03:20.050 align:middle line:90% even more. 01:03:20.050 --> 01:03:22.050 align:middle line:84% How can we simplify this left-hand side? 01:03:22.050 --> 01:03:23.590 align:middle line:90% What happens at v lower bar? 01:03:23.590 --> 01:03:26.450 align:middle line:90% 01:03:26.450 --> 01:03:26.970 align:middle line:90% Yeah? 01:03:26.970 --> 01:03:28.730 align:middle line:90% AUDIENCE: G of v lower bar is 0. 01:03:28.730 --> 01:03:30.230 align:middle line:84% IAN BALL: G of v lower bar is zero. 01:03:30.230 --> 01:03:31.563 align:middle line:90% So we can just ignore that term. 01:03:31.563 --> 01:03:34.250 align:middle line:84% That disappears, right, because we get 0 times something. 01:03:34.250 --> 01:03:36.330 align:middle line:90% So we only get the top term. 01:03:36.330 --> 01:03:42.730 align:middle line:84% So we get beta of V hat times G of V hat on the left 01:03:42.730 --> 01:03:45.170 align:middle line:84% equals the integral from v lower bar 01:03:45.170 --> 01:03:49.730 align:middle line:90% to v upper bar of v g of v dv. 01:03:49.730 --> 01:03:51.730 align:middle line:84% And now we've solved our differential equation 01:03:51.730 --> 01:03:53.890 align:middle line:90% because we have beta by itself. 01:03:53.890 --> 01:03:58.450 align:middle line:84% And now we can just divide by big G. So let's divide. 01:03:58.450 --> 01:04:15.310 align:middle line:90% 01:04:15.310 --> 01:04:22.050 align:middle line:84% So we get our solution is beta-- and this big V is annoying. 01:04:22.050 --> 01:04:26.070 align:middle line:84% So maybe I'll change notation a little bit and write beta of v-- 01:04:26.070 --> 01:04:28.470 align:middle line:84% let me just use v instead of V hat 01:04:28.470 --> 01:04:30.470 align:middle line:84% and then maybe x instead of v. So 01:04:30.470 --> 01:04:32.710 align:middle line:84% let me say this is the integral from v 01:04:32.710 --> 01:04:38.210 align:middle line:90% lower bar to v of x g of x. 01:04:38.210 --> 01:04:42.070 align:middle line:90% 01:04:42.070 --> 01:04:43.990 align:middle line:90% No. 01:04:43.990 --> 01:04:48.390 align:middle line:84% So let me check wherever I had a v hat I want a V. 01:04:48.390 --> 01:04:51.890 align:middle line:84% And wherever I had a little v I want an x. 01:04:51.890 --> 01:04:52.630 align:middle line:90% Looks good. 01:04:52.630 --> 01:04:56.090 align:middle line:90% 01:04:56.090 --> 01:04:57.150 align:middle line:90% So here's our solution. 01:04:57.150 --> 01:05:00.770 align:middle line:90% 01:05:00.770 --> 01:05:04.170 align:middle line:84% So what we've solved for is the bidding function 01:05:04.170 --> 01:05:06.510 align:middle line:84% that every player is going to use 01:05:06.510 --> 01:05:08.250 align:middle line:90% in this symmetric equilibrium. 01:05:08.250 --> 01:05:09.170 align:middle line:90% I should warn that-- 01:05:09.170 --> 01:05:11.850 align:middle line:84% I should caution that there's one kind of trick we've done. 01:05:11.850 --> 01:05:15.450 align:middle line:84% Technically, what we've shown is that if there's 01:05:15.450 --> 01:05:19.933 align:middle line:84% an equilibrium of this form, then beta must look like this. 01:05:19.933 --> 01:05:22.350 align:middle line:84% But really, we have to check that if beta looks like this, 01:05:22.350 --> 01:05:23.558 align:middle line:90% it's actually an equilibrium. 01:05:23.558 --> 01:05:25.290 align:middle line:84% We've shown a necessary condition, not 01:05:25.290 --> 01:05:26.470 align:middle line:90% a sufficient condition. 01:05:26.470 --> 01:05:28.350 align:middle line:84% But you can check that this actually is an equilibrium, 01:05:28.350 --> 01:05:28.892 align:middle line:90% so that's OK. 01:05:28.892 --> 01:05:31.290 align:middle line:84% It's sort of like when you solve a maximization problem 01:05:31.290 --> 01:05:32.932 align:middle line:84% and you set the derivative to 0, well, 01:05:32.932 --> 01:05:34.390 align:middle line:84% sometimes you don't have a maximum, 01:05:34.390 --> 01:05:35.490 align:middle line:90% you have a minimum or something. 01:05:35.490 --> 01:05:36.948 align:middle line:84% That could be going on here, but we 01:05:36.948 --> 01:05:38.810 align:middle line:84% can check that that's not what's going on. 01:05:38.810 --> 01:05:41.170 align:middle line:90% So let's try to interpret this. 01:05:41.170 --> 01:05:46.170 align:middle line:84% Well, we already have a formula for the equilibrium 01:05:46.170 --> 01:05:47.687 align:middle line:90% in the special uniform case. 01:05:47.687 --> 01:05:49.770 align:middle line:84% So let's see if this agrees with our answer there. 01:05:49.770 --> 01:05:52.550 align:middle line:84% If it doesn't agree there, then we're kind of in trouble. 01:05:52.550 --> 01:06:01.850 align:middle line:84% So in the uniform 0, 1 case, well, what is g of x, 01:06:01.850 --> 01:06:03.290 align:middle line:90% and what is G of v? 01:06:03.290 --> 01:06:05.990 align:middle line:90% 01:06:05.990 --> 01:06:07.830 align:middle line:84% Well, let's remember what g is. g 01:06:07.830 --> 01:06:14.750 align:middle line:84% is the distribution of the maximum value of my opponents. 01:06:14.750 --> 01:06:17.310 align:middle line:84% But if there's only two of us, then the maximum value 01:06:17.310 --> 01:06:19.450 align:middle line:84% of my opponents is just the value of my opponent. 01:06:19.450 --> 01:06:23.190 align:middle line:84% So in the uniform 0, 1, with two bidders, 01:06:23.190 --> 01:06:24.850 align:middle line:90% what is g of x going to be? 01:06:24.850 --> 01:06:30.510 align:middle line:90% 01:06:30.510 --> 01:06:33.790 align:middle line:84% It's going to be the density of the distribution 01:06:33.790 --> 01:06:35.390 align:middle line:90% of my opponent's value. 01:06:35.390 --> 01:06:37.670 align:middle line:84% And what is that density in the uniform case? 01:06:37.670 --> 01:06:38.250 align:middle line:90% Just 1. 01:06:38.250 --> 01:06:44.790 align:middle line:90% 01:06:44.790 --> 01:06:46.610 align:middle line:90% And what is G of v going to be? 01:06:46.610 --> 01:06:50.210 align:middle line:90% 01:06:50.210 --> 01:06:53.370 align:middle line:84% Well, it's the CDF of the uniform distribution. 01:06:53.370 --> 01:06:56.250 align:middle line:84% So that's just v. The probability 01:06:56.250 --> 01:06:59.970 align:middle line:84% that a uniform distribution is below v is just v assuming 01:06:59.970 --> 01:07:03.130 align:middle line:90% v is between 0 and 1. 01:07:03.130 --> 01:07:05.130 align:middle line:90% So let's plug this in. 01:07:05.130 --> 01:07:10.190 align:middle line:84% And I guess the final step is v lower bar is equal to 0. 01:07:10.190 --> 01:07:13.210 align:middle line:84% So these are all the special restrictions 01:07:13.210 --> 01:07:16.010 align:middle line:84% we get in this uniform example with two players-- 01:07:16.010 --> 01:07:19.450 align:middle line:90% I should say two bidders. 01:07:19.450 --> 01:07:20.298 align:middle line:90% Let's see. 01:07:20.298 --> 01:07:22.590 align:middle line:84% I guess we'll see in real time if we've made a mistake. 01:07:22.590 --> 01:07:24.290 align:middle line:84% I don't think we have, but we'll see. 01:07:24.290 --> 01:07:41.146 align:middle line:84% So we get the integral from 0 to v of x over v dx. 01:07:41.146 --> 01:07:47.020 align:middle line:90% x times 1 divided by v dx. 01:07:47.020 --> 01:07:51.400 align:middle line:84% Well, that's just 1 over v is a constant. 01:07:51.400 --> 01:07:57.660 align:middle line:84% We can take that out of the integral from 0 to v of xdx. 01:07:57.660 --> 01:07:59.400 align:middle line:90% What is the antiderivative of x? 01:07:59.400 --> 01:08:01.180 align:middle line:90% It's just x squared over 2. 01:08:01.180 --> 01:08:08.180 align:middle line:84% So we get v squared over 2 times 1 over v. 01:08:08.180 --> 01:08:09.760 align:middle line:84% And, phew, we didn't get it wrong. 01:08:09.760 --> 01:08:10.560 align:middle line:90% We got v over 2. 01:08:10.560 --> 01:08:11.660 align:middle line:90% So that's good. 01:08:11.660 --> 01:08:14.020 align:middle line:84% So this formula is consistent with our guess in the very 01:08:14.020 --> 01:08:14.840 align:middle line:90% special case. 01:08:14.840 --> 01:08:16.680 align:middle line:84% But this is a much more general formula. 01:08:16.680 --> 01:08:19.580 align:middle line:90% 01:08:19.580 --> 01:08:24.060 align:middle line:84% Now, how can we interpret this formula? 01:08:24.060 --> 01:08:27.260 align:middle line:84% There's actually quite a nice way of looking at this. 01:08:27.260 --> 01:08:30.378 align:middle line:84% Does this fraction remind you of anything? 01:08:30.378 --> 01:08:30.920 align:middle line:90% I don't know. 01:08:30.920 --> 01:08:39.740 align:middle line:84% Does anyone see a way of interpreting this integral 01:08:39.740 --> 01:08:41.859 align:middle line:84% in terms of conditional expectations? 01:08:41.859 --> 01:08:46.760 align:middle line:84% So G of v is the probability that all my opponent's 01:08:46.760 --> 01:08:50.240 align:middle line:84% valuations are below v. And g of x 01:08:50.240 --> 01:08:53.399 align:middle line:84% is the density of the distribution of the maximum 01:08:53.399 --> 01:08:55.018 align:middle line:90% of my opponent's valuations. 01:08:55.018 --> 01:08:56.560 align:middle line:84% So it turns out that this is actually 01:08:56.560 --> 01:08:58.100 align:middle line:90% a conditional expectation. 01:08:58.100 --> 01:08:59.920 align:middle line:84% And, again, this is a probability fact. 01:08:59.920 --> 01:09:01.279 align:middle line:90% So it turns out-- 01:09:01.279 --> 01:09:03.880 align:middle line:90% what is this? 01:09:03.880 --> 01:09:13.399 align:middle line:84% It's actually the expectation of the max of v2 01:09:13.399 --> 01:09:16.720 align:middle line:84% through vn conditional on the event 01:09:16.720 --> 01:09:38.359 align:middle line:84% that this maximum is less than or equal to v. 01:09:38.359 --> 01:09:42.000 align:middle line:84% So this gives us a nicely interpretable formula. 01:09:42.000 --> 01:09:43.660 align:middle line:90% What does this say? 01:09:43.660 --> 01:09:46.100 align:middle line:84% How much do I bid in the equilibrium 01:09:46.100 --> 01:09:49.340 align:middle line:84% of the first-price auction when my value is v? 01:09:49.340 --> 01:09:54.460 align:middle line:84% What I'm actually bidding is the expectation 01:09:54.460 --> 01:09:58.060 align:middle line:84% of the highest value of my opponents, 01:09:58.060 --> 01:10:00.340 align:middle line:84% conditional on the event that the highest 01:10:00.340 --> 01:10:04.740 align:middle line:84% value of my opponents is less than or equal to v, 01:10:04.740 --> 01:10:06.640 align:middle line:84% less than or equal to my true valuation. 01:10:06.640 --> 01:10:14.900 align:middle line:90% 01:10:14.900 --> 01:10:17.840 align:middle line:84% And there's a nice connection here. 01:10:17.840 --> 01:10:19.717 align:middle line:90% Maybe we'll try to do it. 01:10:19.717 --> 01:10:21.300 align:middle line:84% But let me pause, see I feel like I've 01:10:21.300 --> 01:10:22.342 align:middle line:90% lost people a little bit. 01:10:22.342 --> 01:10:24.085 align:middle line:90% So any questions? 01:10:24.085 --> 01:10:25.460 align:middle line:84% I know it's been a little mathier 01:10:25.460 --> 01:10:26.502 align:middle line:90% than we've normally done. 01:10:26.502 --> 01:10:27.480 align:middle line:90% So any questions here? 01:10:27.480 --> 01:10:28.360 align:middle line:90% Yeah, let's pause. 01:10:28.360 --> 01:10:29.620 align:middle line:90% Yeah? 01:10:29.620 --> 01:10:33.580 align:middle line:84% AUDIENCE: So that the condition is that your valuation is higher 01:10:33.580 --> 01:10:35.460 align:middle line:90% than the valuation is like-- 01:10:35.460 --> 01:10:38.080 align:middle line:84% higher than the valuation of the maximum opponent. 01:10:38.080 --> 01:10:40.678 align:middle line:90% What if that's not true? 01:10:40.678 --> 01:10:42.720 align:middle line:84% IAN BALL: So you're asking about this final step? 01:10:42.720 --> 01:10:45.960 align:middle line:84% So let's understand right here. v is just a number. 01:10:45.960 --> 01:10:48.640 align:middle line:84% So what I'm writing here is just a formula 01:10:48.640 --> 01:10:50.560 align:middle line:84% for just something that's defined 01:10:50.560 --> 01:10:53.440 align:middle line:84% as a function of this number v. So mathematically, all this 01:10:53.440 --> 01:10:56.320 align:middle line:84% is is a very special conditional expectation where 01:10:56.320 --> 01:10:58.560 align:middle line:90% this is a random variable. 01:10:58.560 --> 01:11:00.100 align:middle line:90% This is a random variable. 01:11:00.100 --> 01:11:02.880 align:middle line:84% And I'm just looking at a very special conditional expectation 01:11:02.880 --> 01:11:04.060 align:middle line:90% of a random variable. 01:11:04.060 --> 01:11:08.880 align:middle line:84% So it may be that their valuations are lower than mine. 01:11:08.880 --> 01:11:11.380 align:middle line:84% What we're just saying is it's kind of a thought experiment. 01:11:11.380 --> 01:11:13.005 align:middle line:84% It's saying how much am I going to bid? 01:11:13.005 --> 01:11:15.960 align:middle line:84% Well, one way I can compute my equilibrium bid 01:11:15.960 --> 01:11:20.960 align:middle line:84% is by computing this very special conditional expectation. 01:11:20.960 --> 01:11:23.560 align:middle line:84% But you're right that the true value of my opponents 01:11:23.560 --> 01:11:25.800 align:middle line:84% may be lower than v. This is just kind of a formula 01:11:25.800 --> 01:11:27.720 align:middle line:90% for what my bid will be. 01:11:27.720 --> 01:11:28.720 align:middle line:90% Yeah. 01:11:28.720 --> 01:11:30.600 align:middle line:84% AUDIENCE: But it could also be that the true value of your bid 01:11:30.600 --> 01:11:31.740 align:middle line:90% is higher than v, right? 01:11:31.740 --> 01:11:32.120 align:middle line:90% IAN BALL: It could be. 01:11:32.120 --> 01:11:32.680 align:middle line:90% Yeah. 01:11:32.680 --> 01:11:34.930 align:middle line:84% AUDIENCE: But you're just saying like you're basing it 01:11:34.930 --> 01:11:37.940 align:middle line:84% off of the idea that if everything were less than 01:11:37.940 --> 01:11:39.665 align:middle line:90% or equal v, like [INAUDIBLE]. 01:11:39.665 --> 01:11:42.040 align:middle line:84% IAN BALL: I can just define this conditional expectation. 01:11:42.040 --> 01:11:44.620 align:middle line:84% So for now, I mean, it's just a definition. 01:11:44.620 --> 01:11:48.648 align:middle line:84% It's just a function of v, and it's a number we can compute. 01:11:48.648 --> 01:11:50.940 align:middle line:84% But it gives us kind of a nice interpretation that I'll 01:11:50.940 --> 01:11:51.898 align:middle line:90% go through in a second. 01:11:51.898 --> 01:11:53.260 align:middle line:90% Yeah. 01:11:53.260 --> 01:11:53.760 align:middle line:90% Yes. 01:11:53.760 --> 01:11:54.360 align:middle line:90% In the back. 01:11:54.360 --> 01:11:58.460 align:middle line:84% AUDIENCE: Can you explain one more time why g of x over G of v 01:11:58.460 --> 01:11:59.195 align:middle line:90% is the-- 01:11:59.195 --> 01:11:59.820 align:middle line:90% IAN BALL: Yeah. 01:11:59.820 --> 01:12:00.580 align:middle line:90% OK. 01:12:00.580 --> 01:12:02.955 align:middle line:84% And this is something-- yeah, if you haven't seen before. 01:12:02.955 --> 01:12:05.972 align:middle line:90% So this is-- 01:12:05.972 --> 01:12:08.412 align:middle line:84% AUDIENCE: Wouldn't it also be the x 01:12:08.412 --> 01:12:09.380 align:middle line:90% IAN BALL: Say it again. 01:12:09.380 --> 01:12:10.755 align:middle line:84% AUDIENCE: Instead of just g of x, 01:12:10.755 --> 01:12:13.660 align:middle line:84% would it-- because x g of x is an expectation. 01:12:13.660 --> 01:12:16.435 align:middle line:84% IAN BALL: So this is an unconditional expectation. 01:12:16.435 --> 01:12:18.060 align:middle line:84% When I put the g over v here it becomes 01:12:18.060 --> 01:12:19.500 align:middle line:90% a conditional expectation. 01:12:19.500 --> 01:12:22.740 align:middle line:84% AUDIENCE: I wonder why x is [INAUDIBLE] 01:12:22.740 --> 01:12:24.020 align:middle line:90% IAN BALL: Oh, I guess-- 01:12:24.020 --> 01:12:26.000 align:middle line:84% so it's a question of what we circle. 01:12:26.000 --> 01:12:28.660 align:middle line:84% So I'm saying the conditional expectation 01:12:28.660 --> 01:12:32.580 align:middle line:84% can be expressed as x times the conditional density. 01:12:32.580 --> 01:12:34.820 align:middle line:84% So what I've circled here is the conditional density. 01:12:34.820 --> 01:12:37.510 align:middle line:84% So this is the conditional density 01:12:37.510 --> 01:12:40.613 align:middle line:84% given the event that everyone else's value is 01:12:40.613 --> 01:12:41.530 align:middle line:90% below a certain thing. 01:12:41.530 --> 01:12:43.710 align:middle line:84% So maybe I'll just say, what's the general property 01:12:43.710 --> 01:12:44.690 align:middle line:90% I'm using here? 01:12:44.690 --> 01:12:48.990 align:middle line:90% 01:12:48.990 --> 01:12:52.470 align:middle line:84% Let me go down here and write this kind of math fact. 01:12:52.470 --> 01:12:57.230 align:middle line:90% 01:12:57.230 --> 01:13:08.710 align:middle line:84% So if you have a random variable X distributed with a density g, 01:13:08.710 --> 01:13:12.690 align:middle line:84% then we can also compute the conditional density. 01:13:12.690 --> 01:13:15.190 align:middle line:84% So we can say what is the conditional density 01:13:15.190 --> 01:13:22.350 align:middle line:84% g of x given that X is less than or equal to v 01:13:22.350 --> 01:13:29.310 align:middle line:84% And that's defined to be g of x divided by the probability 01:13:29.310 --> 01:13:32.430 align:middle line:84% that X is less than or equal to v, which is exactly 01:13:32.430 --> 01:13:40.210 align:middle line:90% g of x times big G of v. 01:13:40.210 --> 01:13:44.410 align:middle line:84% One way to think about this is if I want to understand 01:13:44.410 --> 01:13:47.090 align:middle line:84% the conditional probability, or the conditional distribution 01:13:47.090 --> 01:13:51.210 align:middle line:84% of x, given that x is below v, well, it's still proportional 01:13:51.210 --> 01:13:53.330 align:middle line:84% to the density, but now it has to integrate 01:13:53.330 --> 01:13:56.030 align:middle line:90% to 1 over this condition of n. 01:13:56.030 --> 01:13:59.850 align:middle line:84% So conditional on my opponent's value being below v, 01:13:59.850 --> 01:14:02.290 align:middle line:84% the conditional density, if I integrate it 01:14:02.290 --> 01:14:04.530 align:middle line:84% from v lower bar to v, better give me 1, 01:14:04.530 --> 01:14:07.010 align:middle line:84% because that's how conditional distributions work. 01:14:07.010 --> 01:14:10.490 align:middle line:84% And if we do this integral from v lower bar to v of g of x, 01:14:10.490 --> 01:14:12.890 align:middle line:84% we exactly get big G of v. And then 01:14:12.890 --> 01:14:14.790 align:middle line:84% if we divide it by big G of v, we get 1. 01:14:14.790 --> 01:14:16.850 align:middle line:84% So this is just-- it's a normalization 01:14:16.850 --> 01:14:20.410 align:middle line:84% to ensure that the distribution, the conditional distribution, 01:14:20.410 --> 01:14:22.090 align:middle line:90% integrates to 1. 01:14:22.090 --> 01:14:24.992 align:middle line:84% But again, as I said, we're using a little more probability. 01:14:24.992 --> 01:14:26.450 align:middle line:84% This is maybe the one lecture where 01:14:26.450 --> 01:14:28.810 align:middle line:84% we're kind of leaning on some of these official prereqs 01:14:28.810 --> 01:14:30.250 align:middle line:90% that we haven't really enforced. 01:14:30.250 --> 01:14:34.450 align:middle line:84% And yeah, some of these things may take a little refresher. 01:14:34.450 --> 01:14:38.710 align:middle line:84% But this is the general fact that I'm using. 01:14:38.710 --> 01:14:41.990 align:middle line:90% Any other questions? 01:14:41.990 --> 01:14:45.750 align:middle line:90% So one final observation. 01:14:45.750 --> 01:14:47.770 align:middle line:84% If we try to look at this equation star-- 01:14:47.770 --> 01:14:54.190 align:middle line:90% 01:14:54.190 --> 01:14:55.210 align:middle line:90% let me go over here. 01:14:55.210 --> 01:15:14.150 align:middle line:90% 01:15:14.150 --> 01:15:20.270 align:middle line:84% So we said that if my valuation is v, this is how much I bid. 01:15:20.270 --> 01:15:24.470 align:middle line:84% So if I win the good, the amount that I'm going to pay 01:15:24.470 --> 01:15:28.510 align:middle line:84% is exactly beta of v, which is exactly this formula over here. 01:15:28.510 --> 01:15:31.850 align:middle line:84% But let's just go through a little thought experiment. 01:15:31.850 --> 01:15:33.590 align:middle line:90% Let's suppose that we instead-- 01:15:33.590 --> 01:15:37.250 align:middle line:90% 01:15:37.250 --> 01:15:38.890 align:middle line:84% we did a second-price auction instead 01:15:38.890 --> 01:15:40.110 align:middle line:90% of a first-price auction. 01:15:40.110 --> 01:15:48.770 align:middle line:90% 01:15:48.770 --> 01:15:52.350 align:middle line:84% Well, now everyone's going to bid truthfully. 01:15:52.350 --> 01:16:07.210 align:middle line:90% 01:16:07.210 --> 01:16:16.933 align:middle line:84% So let's say the profile of valuations is v1 through vn. 01:16:16.933 --> 01:16:18.850 align:middle line:84% If we're all bidding truthfully, and we're now 01:16:18.850 --> 01:16:22.330 align:middle line:84% in a second-price auction, when is bidder 1 going to win? 01:16:22.330 --> 01:16:26.210 align:middle line:84% What must be true about v1 through vn for bidder 1 01:16:26.210 --> 01:16:26.990 align:middle line:90% to win the item? 01:16:26.990 --> 01:16:27.490 align:middle line:90% Yeah? 01:16:27.490 --> 01:16:29.145 align:middle line:84% AUDIENCE: V1 is the max [INAUDIBLE] 01:16:29.145 --> 01:16:29.770 align:middle line:90% IAN BALL: Yeah. 01:16:29.770 --> 01:16:31.312 align:middle line:84% So maybe another way I'll say it is-- 01:16:31.312 --> 01:16:36.830 align:middle line:84% so bidder 1 wins if V1 is greater than or equal 01:16:36.830 --> 01:16:38.790 align:middle line:84% to-- or maybe I'll say greater than-- 01:16:38.790 --> 01:16:46.390 align:middle line:84% I'll say greater than or equal to the max of v2 up to vn. 01:16:46.390 --> 01:16:48.650 align:middle line:84% And if they win, what do they pay? 01:16:48.650 --> 01:16:54.190 align:middle line:90% 01:16:54.190 --> 01:16:57.210 align:middle line:84% How much does bidder 1 pay if they win the auction? 01:16:57.210 --> 01:17:02.590 align:middle line:90% 01:17:02.590 --> 01:17:06.270 align:middle line:84% Well, bidder 1 pays the second-highest bid, right? 01:17:06.270 --> 01:17:08.910 align:middle line:84% The second-highest bid is going to be the highest 01:17:08.910 --> 01:17:10.670 align:middle line:90% bid of his opponents. 01:17:10.670 --> 01:17:12.870 align:middle line:84% But his opponents are bidding truthfully, 01:17:12.870 --> 01:17:14.510 align:middle line:84% so the second-highest bid is exactly 01:17:14.510 --> 01:17:20.770 align:middle line:84% going to be the maximum of v2 up to vn. 01:17:20.770 --> 01:17:29.000 align:middle line:90% 01:17:29.000 --> 01:17:33.640 align:middle line:84% So in a second-price auction as opposed 01:17:33.640 --> 01:17:37.457 align:middle line:84% to a first-price auction, the amount that I pay is random. 01:17:37.457 --> 01:17:39.040 align:middle line:84% I don't know how much I'm going to pay 01:17:39.040 --> 01:17:41.960 align:middle line:84% because it's going to depend on the valuations of my opponents. 01:17:41.960 --> 01:17:47.040 align:middle line:84% But the amount that I'm going to pay is exactly this. 01:17:47.040 --> 01:17:50.200 align:middle line:84% And I'm going to pay that exactly if I win. 01:17:50.200 --> 01:17:53.320 align:middle line:84% So what is the expected amount that I'm 01:17:53.320 --> 01:17:57.040 align:middle line:90% going to pay given that I win? 01:17:57.040 --> 01:18:00.600 align:middle line:84% It's exactly going to be-- so suppose I win. 01:18:00.600 --> 01:18:03.080 align:middle line:84% What is my conditional expected payment? 01:18:03.080 --> 01:18:07.560 align:middle line:84% It's exactly the expectation of this, conditional on the event 01:18:07.560 --> 01:18:09.920 align:middle line:84% that this is less than or equal to v1. 01:18:09.920 --> 01:18:11.560 align:middle line:90% Wait a second. 01:18:11.560 --> 01:18:14.320 align:middle line:84% That's exactly the formula we got over here. 01:18:14.320 --> 01:18:16.460 align:middle line:84% So this is an expression of revenue equivalence. 01:18:16.460 --> 01:18:19.680 align:middle line:84% It says that in a first-price auction 01:18:19.680 --> 01:18:22.840 align:middle line:84% and a second-price auction, the expected amount 01:18:22.840 --> 01:18:25.820 align:middle line:84% that each type of bidder pays is actually going to be the same. 01:18:25.820 --> 01:18:28.980 align:middle line:84% Here, I bid B of v. It's just one number, 01:18:28.980 --> 01:18:31.100 align:middle line:90% and I always pay that if I win. 01:18:31.100 --> 01:18:34.082 align:middle line:84% Here, how much I pay depends on what other people bid. 01:18:34.082 --> 01:18:35.540 align:middle line:84% I don't know what I'm going to pay. 01:18:35.540 --> 01:18:37.748 align:middle line:84% If other people have valuations really close to mine, 01:18:37.748 --> 01:18:38.720 align:middle line:90% I'm going to pay a lot. 01:18:38.720 --> 01:18:40.160 align:middle line:84% If they all have valuations really low, 01:18:40.160 --> 01:18:41.340 align:middle line:90% I'm going to pay a little. 01:18:41.340 --> 01:18:43.250 align:middle line:84% But the expected amount that I'm going to pay 01:18:43.250 --> 01:18:45.500 align:middle line:84% is actually going to be exactly the same as the amount 01:18:45.500 --> 01:18:47.083 align:middle line:84% that I pay in the first-price auction. 01:18:47.083 --> 01:18:49.300 align:middle line:84% And this is the first indication of this revenue 01:18:49.300 --> 01:18:52.500 align:middle line:84% equivalence fact, that while the way that we bid 01:18:52.500 --> 01:18:55.500 align:middle line:84% is going to be different, the expected amount that each bidder 01:18:55.500 --> 01:18:56.920 align:middle line:90% pays is going to be the same. 01:18:56.920 --> 01:18:59.080 align:middle line:84% And for that reason, the auctioneer's revenue, 01:18:59.080 --> 01:19:01.038 align:middle line:84% expected revenue, is also going to be the same. 01:19:01.038 --> 01:19:02.780 align:middle line:84% So this is the first hint at something 01:19:02.780 --> 01:19:05.060 align:middle line:84% called revenue equivalence, which 01:19:05.060 --> 01:19:07.340 align:middle line:84% we saw in a special example before, 01:19:07.340 --> 01:19:09.720 align:middle line:84% and now we see is a much more general phenomenon. 01:19:09.720 --> 01:19:12.020 align:middle line:84% But we'll talk about that a bit more next class. 01:19:12.020 --> 01:19:13.920 align:middle line:84% I guess it's a holiday on Tuesday, 01:19:13.920 --> 01:19:15.900 align:middle line:84% so I guess I'll see everyone next Thursday. 01:19:15.900 --> 01:19:18.000 align:middle line:84% And I'm happy to stick around with any questions. 01:19:18.000 --> 01:19:19.550 align:middle line:90% Thanks. 01:19:19.550 --> 01:19:23.000 align:middle line:90%