WEBVTT 00:00:00.000 --> 00:00:01.955 align:middle line:90% [SQUEAKING] 00:00:01.955 --> 00:00:03.895 align:middle line:90% [RUSTLING] 00:00:03.895 --> 00:00:05.835 align:middle line:90% [CLICKING] 00:00:05.835 --> 00:00:09.785 align:middle line:90% 00:00:09.785 --> 00:00:11.160 align:middle line:84% IAN BALL: So today I want to talk 00:00:11.160 --> 00:00:13.240 align:middle line:84% about-- we're going to move to the final phase 00:00:13.240 --> 00:00:16.820 align:middle line:84% of the course, which is going to be on dynamic Bayesian games. 00:00:16.820 --> 00:00:19.980 align:middle line:84% But I want to start with maybe one example of a VCG mechanism, 00:00:19.980 --> 00:00:21.800 align:middle line:84% because I think, yesterday, that was maybe 00:00:21.800 --> 00:00:23.467 align:middle line:84% a bit abstract and a bit hard to follow. 00:00:23.467 --> 00:00:26.020 align:middle line:84% So let's start with one final example. 00:00:26.020 --> 00:00:28.580 align:middle line:84% And it's going to be a VCG mechanism, 00:00:28.580 --> 00:00:32.580 align:middle line:84% or maybe we'll call it a VCG auction for two items. 00:00:32.580 --> 00:00:40.520 align:middle line:90% 00:00:40.520 --> 00:00:42.738 align:middle line:84% So a lot of the classical work that we did so far 00:00:42.738 --> 00:00:45.280 align:middle line:84% in the course on auctions, we assumed there was just one item 00:00:45.280 --> 00:00:46.392 align:middle line:90% being auctioned off. 00:00:46.392 --> 00:00:48.100 align:middle line:84% Then we looked at the ad auction setting, 00:00:48.100 --> 00:00:50.160 align:middle line:84% which had this special structure where there was 00:00:50.160 --> 00:00:51.457 align:middle line:90% this ranking between the items. 00:00:51.457 --> 00:00:53.540 align:middle line:84% Here, we're just going to have two abstract items. 00:00:53.540 --> 00:01:00.470 align:middle line:84% So I'm going to call these maybe item A and item B. 00:01:00.470 --> 00:01:03.970 align:middle line:84% So what the auction needs to do is it needs to solicit bids. 00:01:03.970 --> 00:01:05.630 align:middle line:84% Maybe I'll write what it's going to do. 00:01:05.630 --> 00:01:12.450 align:middle line:84% It's going to collect bids from the bidders. 00:01:12.450 --> 00:01:17.030 align:middle line:90% 00:01:17.030 --> 00:01:19.210 align:middle line:84% And then, based upon these bids, as usual, 00:01:19.210 --> 00:01:20.710 align:middle line:90% we have to decide two things. 00:01:20.710 --> 00:01:22.870 align:middle line:84% We have to decide how we allocate the goods 00:01:22.870 --> 00:01:24.750 align:middle line:90% and how much everyone pays. 00:01:24.750 --> 00:01:30.250 align:middle line:84% So then it specifies an allocation, which, in this case, 00:01:30.250 --> 00:01:33.270 align:middle line:84% it says who gets item A if someone gets it 00:01:33.270 --> 00:01:36.423 align:middle line:84% and who gets item B if anyone gets it. 00:01:36.423 --> 00:01:38.090 align:middle line:84% And then, in addition to the allocation, 00:01:38.090 --> 00:01:42.050 align:middle line:84% we have to specify a payment by each bidder. 00:01:42.050 --> 00:01:46.950 align:middle line:90% 00:01:46.950 --> 00:01:50.030 align:middle line:84% Now, I put bids in quotes because the kind 00:01:50.030 --> 00:01:53.750 align:middle line:84% of surprising or maybe somewhat confusing 00:01:53.750 --> 00:01:57.870 align:middle line:84% aspect of a VCG auction is that a bid is essentially 00:01:57.870 --> 00:02:01.590 align:middle line:84% a report or claim about how much you value the items. 00:02:01.590 --> 00:02:03.850 align:middle line:84% So you think of my bid as the auctioneer saying, 00:02:03.850 --> 00:02:05.450 align:middle line:84% tell me how much you value everything. 00:02:05.450 --> 00:02:07.808 align:middle line:84% And then the bidders say, I value it this much. 00:02:07.808 --> 00:02:09.350 align:middle line:84% And that may seem strange, but if you 00:02:09.350 --> 00:02:11.009 align:middle line:84% think about a second-price auction, 00:02:11.009 --> 00:02:12.230 align:middle line:84% that's exactly what happens because, 00:02:12.230 --> 00:02:13.830 align:middle line:84% in a second-price auction, people are 00:02:13.830 --> 00:02:15.265 align:middle line:90% bidding their true valuation. 00:02:15.265 --> 00:02:17.890 align:middle line:84% So it's as if the auctioneer is saying, tell me your valuation. 00:02:17.890 --> 00:02:21.590 align:middle line:84% And then the bidders are submitting their valuation. 00:02:21.590 --> 00:02:25.590 align:middle line:84% So let's put a little more structure on this. 00:02:25.590 --> 00:02:28.130 align:middle line:84% So let's imagine that we have two bidders. 00:02:28.130 --> 00:02:31.910 align:middle line:90% 00:02:31.910 --> 00:02:34.390 align:middle line:84% And it's a little complicated to specify the whole auction 00:02:34.390 --> 00:02:36.110 align:middle line:84% because we have to specify the allocation 00:02:36.110 --> 00:02:38.588 align:middle line:84% and payment for any vector of bids that's submitted. 00:02:38.588 --> 00:02:40.130 align:middle line:84% So let's just go through one example. 00:02:40.130 --> 00:02:42.950 align:middle line:84% Let's look at some vector of bids that could be submitted. 00:02:42.950 --> 00:02:46.508 align:middle line:84% So, well, we first have to say, well, what is a valuation? 00:02:46.508 --> 00:02:48.050 align:middle line:84% What do I need to tell the auctioneer 00:02:48.050 --> 00:02:50.290 align:middle line:84% if the auctioneer is asking me how much I value things? 00:02:50.290 --> 00:02:52.040 align:middle line:84% Well, I actually have to say three things. 00:02:52.040 --> 00:02:54.390 align:middle line:84% I have to say, how much do I value item A if I just 00:02:54.390 --> 00:02:55.820 align:middle line:90% get item A? 00:02:55.820 --> 00:02:58.580 align:middle line:84% How much do I value item B If I just get item B? 00:02:58.580 --> 00:03:01.300 align:middle line:84% And then how much do I value getting both items together? 00:03:01.300 --> 00:03:03.260 align:middle line:84% And depending on these items, it may not 00:03:03.260 --> 00:03:06.700 align:middle line:84% be the value of the bundle of the two items 00:03:06.700 --> 00:03:09.300 align:middle line:84% is exactly the same as some of your valuations for the two 00:03:09.300 --> 00:03:10.860 align:middle line:90% items. 00:03:10.860 --> 00:03:13.340 align:middle line:84% If it's shoes, a left shoe and a right shoe 00:03:13.340 --> 00:03:15.900 align:middle line:84% together are much more valuable than the two shoes 00:03:15.900 --> 00:03:19.540 align:middle line:84% separately whereas other items that are substitutes, the bundle 00:03:19.540 --> 00:03:23.717 align:middle line:84% may be less valuable than the sum of its parts, 00:03:23.717 --> 00:03:25.800 align:middle line:84% if there's some duplication between the two parts. 00:03:25.800 --> 00:03:29.580 align:middle line:84% Maybe 2 apples is not twice as valuable as 1 apple. 00:03:29.580 --> 00:03:36.140 align:middle line:84% So we're going to think of it as a table where 00:03:36.140 --> 00:03:39.120 align:middle line:84% my valuation is going to say, if I get item A alone, 00:03:39.120 --> 00:03:40.000 align:middle line:90% what's my value? 00:03:40.000 --> 00:03:42.080 align:middle line:84% If I get item B alone, what's my value? 00:03:42.080 --> 00:03:44.180 align:middle line:84% And if I get the bundle, what's my value? 00:03:44.180 --> 00:03:47.160 align:middle line:84% So bidder 1 is going to make some report. 00:03:47.160 --> 00:03:49.460 align:middle line:84% And bidder 2 is going to make some report. 00:03:49.460 --> 00:03:51.540 align:middle line:84% I've made the numbers here so it works out. 00:03:51.540 --> 00:03:55.170 align:middle line:84% So let's suppose that the report beta 1 makes is they say, 00:03:55.170 --> 00:03:59.330 align:middle line:84% if I get only item A, I value that at 5. 00:03:59.330 --> 00:04:03.890 align:middle line:84% If I get only item B, I valued at 3, and if I get both items, 00:04:03.890 --> 00:04:05.490 align:middle line:90% I valued at 8. 00:04:05.490 --> 00:04:09.050 align:middle line:84% So for bidder 1, it turns out, their value of the bundle 00:04:09.050 --> 00:04:11.550 align:middle line:84% is exactly the sum of the value of the two parts, 00:04:11.550 --> 00:04:13.690 align:middle line:84% but it doesn't have to be that way. 00:04:13.690 --> 00:04:16.450 align:middle line:90% Now let's look at bidder 2. 00:04:16.450 --> 00:04:20.649 align:middle line:84% Let's suppose bidder 2 says, I value item A alone at 2. 00:04:20.649 --> 00:04:24.610 align:middle line:84% I value B alone at 6, but I only value the bundle at 7. 00:04:24.610 --> 00:04:26.970 align:middle line:84% So notice for bidder 2, the bundle 00:04:26.970 --> 00:04:30.150 align:middle line:84% is not worth as much to them as the sum of its parts. 00:04:30.150 --> 00:04:32.210 align:middle line:84% If they just got A and if they just 00:04:32.210 --> 00:04:34.710 align:middle line:84% got B, that's different from if they get the bundle together 00:04:34.710 --> 00:04:38.867 align:middle line:84% because, for them, these two items, maybe if item B is 00:04:38.867 --> 00:04:41.450 align:middle line:84% a better version of item A, then getting both of them together 00:04:41.450 --> 00:04:44.250 align:middle line:84% is basically just as good as getting item B. 00:04:44.250 --> 00:04:48.410 align:middle line:84% So let's suppose these are the reported valuations for the two 00:04:48.410 --> 00:04:49.170 align:middle line:90% bidders. 00:04:49.170 --> 00:04:52.620 align:middle line:84% And now we have to specify who gets and what the payments are. 00:04:52.620 --> 00:04:54.000 align:middle line:90% So let's go through this. 00:04:54.000 --> 00:04:55.620 align:middle line:84% Maybe we'll do allocation over here. 00:04:55.620 --> 00:05:00.080 align:middle line:90% 00:05:00.080 --> 00:05:02.640 align:middle line:84% So remember, the rule of the VCG mechanism 00:05:02.640 --> 00:05:05.040 align:middle line:84% is that we always allocate things efficiently. 00:05:05.040 --> 00:05:07.520 align:middle line:84% So that is, we want to maximize the sum 00:05:07.520 --> 00:05:09.480 align:middle line:90% of the players' utilities. 00:05:09.480 --> 00:05:14.120 align:middle line:84% So we need to think, who should item A go to? 00:05:14.120 --> 00:05:15.623 align:middle line:90% And who should I be go to? 00:05:15.623 --> 00:05:17.040 align:middle line:84% Any thoughts here on what would be 00:05:17.040 --> 00:05:20.633 align:middle line:84% the efficient way of allocating things, given these reports? 00:05:20.633 --> 00:05:22.800 align:middle line:84% Remember, when we say we allocated efficiently, what 00:05:22.800 --> 00:05:25.080 align:middle line:84% we mean is, we collect the reports, 00:05:25.080 --> 00:05:27.960 align:middle line:84% and we allocate in a way that would be efficient if these were 00:05:27.960 --> 00:05:29.920 align:middle line:90% the players true valuations. 00:05:29.920 --> 00:05:31.920 align:middle line:84% So taking this as the truth, let's think 00:05:31.920 --> 00:05:34.600 align:middle line:84% about the efficient way of allocating it? 00:05:34.600 --> 00:05:37.900 align:middle line:90% So how would you do this? 00:05:37.900 --> 00:05:38.800 align:middle line:90% Yeah. 00:05:38.800 --> 00:05:41.160 align:middle line:84% AUDIENCE: You could [INAUDIBLE] person 1. 00:05:41.160 --> 00:05:43.767 align:middle line:90% 00:05:43.767 --> 00:05:46.100 align:middle line:84% IAN BALL: Right, that seems like a good way of doing it. 00:05:46.100 --> 00:05:46.840 align:middle line:90% So let's see. 00:05:46.840 --> 00:05:51.440 align:middle line:84% If we give item A to person 1, they get a valuation of 5. 00:05:51.440 --> 00:05:54.040 align:middle line:84% If we give item B to person 2, they 00:05:54.040 --> 00:05:56.980 align:middle line:84% get evaluation of 6, which means the sum is 11. 00:05:56.980 --> 00:05:57.980 align:middle line:90% That sounds pretty good. 00:05:57.980 --> 00:06:00.920 align:middle line:84% If we did the opposite, we would only get a sum of 5. 00:06:00.920 --> 00:06:02.860 align:middle line:84% And if we gave both items to the same player, 00:06:02.860 --> 00:06:04.140 align:middle line:90% we'd either get 8 or 7. 00:06:04.140 --> 00:06:05.540 align:middle line:90% But that's still less than 11. 00:06:05.540 --> 00:06:07.180 align:middle line:84% So this is going to be the efficient allocation. 00:06:07.180 --> 00:06:08.805 align:middle line:84% This is going to be the allocation that 00:06:08.805 --> 00:06:12.660 align:middle line:84% maximizes the sum of the reported utilities. 00:06:12.660 --> 00:06:15.400 align:middle line:84% But now we have to do the payments. 00:06:15.400 --> 00:06:18.460 align:middle line:84% And I think the payments part is a bit trickier. 00:06:18.460 --> 00:06:22.530 align:middle line:84% So we have to specify two payments, t1 and t2. 00:06:22.530 --> 00:06:24.280 align:middle line:84% And maybe if I'm more precise, I would say 00:06:24.280 --> 00:06:28.400 align:middle line:90% this is t1 of v1 hat, v2 hat. 00:06:28.400 --> 00:06:31.920 align:middle line:84% And this is t2 of v1 hat, v2 hat. 00:06:31.920 --> 00:06:35.040 align:middle line:84% So what the mechanism is saying is if these are the submitted 00:06:35.040 --> 00:06:37.900 align:middle line:84% valuation reports that the two players submit, 00:06:37.900 --> 00:06:40.163 align:middle line:84% this is how much I'm going to ask player 1 to pay. 00:06:40.163 --> 00:06:42.580 align:middle line:84% And if these are the valuation reports that are submitted, 00:06:42.580 --> 00:06:44.935 align:middle line:84% this is how much I'm going to ask player 2 to pay. 00:06:44.935 --> 00:06:46.560 align:middle line:84% And the full mechanism is going to have 00:06:46.560 --> 00:06:49.470 align:middle line:84% to specify a function that says what these payments are 00:06:49.470 --> 00:06:51.270 align:middle line:84% for any submitted reports, but writing out 00:06:51.270 --> 00:06:52.650 align:middle line:84% that full function's a bit complicated. 00:06:52.650 --> 00:06:53.950 align:middle line:90% So we'll just do this example. 00:06:53.950 --> 00:06:57.510 align:middle line:84% Now, what we said in class last time 00:06:57.510 --> 00:07:01.870 align:middle line:84% is what bidder 1 is supposed to pay is the externality that they 00:07:01.870 --> 00:07:03.495 align:middle line:90% impose on the other bidders. 00:07:03.495 --> 00:07:04.870 align:middle line:84% So we have to compute two things. 00:07:04.870 --> 00:07:08.210 align:middle line:84% We have to say if we just cared about the other bidders 00:07:08.210 --> 00:07:09.990 align:middle line:84% and we allocated the items to maximize 00:07:09.990 --> 00:07:12.990 align:middle line:84% the utility of the other bidders, how well would they do? 00:07:12.990 --> 00:07:14.710 align:middle line:84% And then we have to subtract from that 00:07:14.710 --> 00:07:17.650 align:middle line:84% how well the bidders actually do under this allocation. 00:07:17.650 --> 00:07:20.110 align:middle line:84% So let's start with the first part. 00:07:20.110 --> 00:07:22.930 align:middle line:84% And this is how well the other bidders could do. 00:07:22.930 --> 00:07:24.910 align:middle line:90% So let's say I'm bidder 1. 00:07:24.910 --> 00:07:27.150 align:middle line:84% These are the submitted valuations. 00:07:27.150 --> 00:07:30.870 align:middle line:84% If we gave bidder 2, if we just allocated things in the best way 00:07:30.870 --> 00:07:35.310 align:middle line:84% for bidder 2, what would bidder 2's valuation be? 00:07:35.310 --> 00:07:36.750 align:middle line:84% Well, what we would do is we would 00:07:36.750 --> 00:07:39.270 align:middle line:90% give both items to bidder 2. 00:07:39.270 --> 00:07:41.350 align:middle line:84% And if we gave both items to bidder 2, 00:07:41.350 --> 00:07:43.530 align:middle line:84% then bidder 2's valuation would be 7. 00:07:43.530 --> 00:07:45.910 align:middle line:84% So we're going to first start with a 7 00:07:45.910 --> 00:07:49.020 align:middle line:84% here because this is how well the other bidders could 00:07:49.020 --> 00:07:51.657 align:middle line:84% do optimally if bidder 1 weren't here, or in other words, 00:07:51.657 --> 00:07:53.740 align:middle line:84% if we didn't care about bidder 1 and we only tried 00:07:53.740 --> 00:07:56.100 align:middle line:90% to maximize bidder 2's payoff. 00:07:56.100 --> 00:08:00.980 align:middle line:84% Now we want to subtract from that how well bidder 2 actually 00:08:00.980 --> 00:08:02.020 align:middle line:90% does. 00:08:02.020 --> 00:08:04.840 align:middle line:84% How well does bidder 2 actually do under the given allocation? 00:08:04.840 --> 00:08:07.020 align:middle line:90% What's their actual valuation? 00:08:07.020 --> 00:08:08.100 align:middle line:90% 6, right? 00:08:08.100 --> 00:08:11.500 align:middle line:90% So we subtract 6, and we get 1. 00:08:11.500 --> 00:08:15.220 align:middle line:84% So this is the payment that bidder 1 is asked to make. 00:08:15.220 --> 00:08:16.360 align:middle line:90% Why is it 1? 00:08:16.360 --> 00:08:20.140 align:middle line:84% Well, because what is the effect that they're 00:08:20.140 --> 00:08:21.520 align:middle line:90% having on the other bidders? 00:08:21.520 --> 00:08:24.020 align:middle line:84% The other bidder, bidder 2, could've gotten the bundle A, 00:08:24.020 --> 00:08:26.900 align:middle line:84% B, but because of the presence of bidder 1, 00:08:26.900 --> 00:08:29.820 align:middle line:84% bidder 2 is only getting the item B by itself. 00:08:29.820 --> 00:08:33.700 align:middle line:84% And the difference in utility for bidder 2, between the bundle 00:08:33.700 --> 00:08:36.880 align:middle line:84% and getting only B by itself is exactly 1. 00:08:36.880 --> 00:08:38.179 align:middle line:90% And that's why we get this. 00:08:38.179 --> 00:08:40.659 align:middle line:90% Now, let's do it for bidder 2. 00:08:40.659 --> 00:08:42.700 align:middle line:84% So if I'm bidder 2, we first need 00:08:42.700 --> 00:08:44.460 align:middle line:84% to compute how well bidder 1 would 00:08:44.460 --> 00:08:46.410 align:middle line:90% do if bidder 2 weren't here. 00:08:46.410 --> 00:08:49.810 align:middle line:84% If bidder 2 weren't here, bidder 1 would get everything, 00:08:49.810 --> 00:08:51.550 align:middle line:90% and their utility would be 8. 00:08:51.550 --> 00:08:54.650 align:middle line:90% 00:08:54.650 --> 00:08:57.090 align:middle line:84% How well does bidder 1 actually do? 00:08:57.090 --> 00:09:00.970 align:middle line:84% Well, bidder 1 actually gets item A, which 00:09:00.970 --> 00:09:02.890 align:middle line:90% gives them a utility of 5. 00:09:02.890 --> 00:09:05.250 align:middle line:90% So we get 8 minus 5. 00:09:05.250 --> 00:09:08.090 align:middle line:90% And we see the payment is 3. 00:09:08.090 --> 00:09:10.250 align:middle line:84% Why does bidder 2 have to pay more? 00:09:10.250 --> 00:09:13.850 align:middle line:84% Because bidder 2 has a bigger externality on bidder 1. 00:09:13.850 --> 00:09:17.930 align:middle line:84% Bidder 1 would've been able to get everything together A and B, 00:09:17.930 --> 00:09:19.890 align:middle line:84% but because of the presence of bidder 2, 00:09:19.890 --> 00:09:22.370 align:middle line:84% they only get item A by itself, and the difference 00:09:22.370 --> 00:09:26.268 align:middle line:84% in utility between 8 and 5 is 3, which is a larger number. 00:09:26.268 --> 00:09:27.810 align:middle line:84% And this is going to-- this completes 00:09:27.810 --> 00:09:31.790 align:middle line:84% the specification of VCG for this particular profile 00:09:31.790 --> 00:09:32.690 align:middle line:90% [INAUDIBLE]. 00:09:32.690 --> 00:09:34.658 align:middle line:84% And the result that I didn't prove 00:09:34.658 --> 00:09:36.450 align:middle line:84% but that I claimed yesterday is that if you 00:09:36.450 --> 00:09:38.930 align:middle line:84% use this kind of rule, in a VCG auction, 00:09:38.930 --> 00:09:41.330 align:middle line:84% it's going to be optimal for each bidder 00:09:41.330 --> 00:09:42.550 align:middle line:90% to report truthfully. 00:09:42.550 --> 00:09:46.050 align:middle line:84% So I should maybe make a note on this. 00:09:46.050 --> 00:09:56.170 align:middle line:84% There's an optimal strategy for each bidder. 00:09:56.170 --> 00:09:57.350 align:middle line:90% And what is that strategy? 00:09:57.350 --> 00:09:58.710 align:middle line:90% Well, it's reporting truthfully. 00:09:58.710 --> 00:10:04.450 align:middle line:90% 00:10:04.450 --> 00:10:09.010 align:middle line:84% And we might call that vi hat equals vi. 00:10:09.010 --> 00:10:10.642 align:middle line:84% So whatever my true valuation is, 00:10:10.642 --> 00:10:12.850 align:middle line:84% I look, and I say, this is how much I really value A, 00:10:12.850 --> 00:10:14.930 align:middle line:84% this is how much I really value B, 00:10:14.930 --> 00:10:16.890 align:middle line:84% and this is how much I really value the bundle. 00:10:16.890 --> 00:10:20.530 align:middle line:84% It's optimal for me to tell the designer exactly how I really 00:10:20.530 --> 00:10:21.910 align:middle line:90% feel and be honest about it. 00:10:21.910 --> 00:10:23.930 align:middle line:84% And I can't benefit by misreporting my valuation 00:10:23.930 --> 00:10:26.850 align:middle line:90% to the designer. 00:10:26.850 --> 00:10:29.530 align:middle line:84% All right, any questions on that? 00:10:29.530 --> 00:10:30.290 align:middle line:90% Yes. 00:10:30.290 --> 00:10:31.957 align:middle line:84% AUDIENCE: Will you ever have a situation 00:10:31.957 --> 00:10:33.850 align:middle line:84% where your externalities are going 00:10:33.850 --> 00:10:38.410 align:middle line:84% to be less than the valuation that you have for the good? 00:10:38.410 --> 00:10:41.030 align:middle line:84% That payment would be negative in essence? 00:10:41.030 --> 00:10:42.090 align:middle line:90% IAN BALL: Good question. 00:10:42.090 --> 00:10:43.840 align:middle line:84% So you're never going to have a case where 00:10:43.840 --> 00:10:45.560 align:middle line:90% the payments are negative. 00:10:45.560 --> 00:10:51.560 align:middle line:84% And this follows from the way that we define an externality. 00:10:51.560 --> 00:10:57.500 align:middle line:84% So, yeah, we have to be a little careful about this. 00:10:57.500 --> 00:11:00.065 align:middle line:84% So it is true that, more broadly, in economics, 00:11:00.065 --> 00:11:01.440 align:middle line:84% when we talk about externalities, 00:11:01.440 --> 00:11:03.232 align:middle line:84% some things do have positive externalities. 00:11:03.232 --> 00:11:05.388 align:middle line:84% There are things I can do that make activities 00:11:05.388 --> 00:11:07.180 align:middle line:84% I can do that make other people better off. 00:11:07.180 --> 00:11:08.722 align:middle line:84% So I guess I should be maybe a little 00:11:08.722 --> 00:11:10.740 align:middle line:84% careful about the terminology of externality. 00:11:10.740 --> 00:11:15.440 align:middle line:84% So here, what we mean in this context by externality is, 00:11:15.440 --> 00:11:19.320 align:middle line:84% well, we're comparing how well the other bidder would 00:11:19.320 --> 00:11:21.720 align:middle line:84% do if we just cared about maximizing that bidder's 00:11:21.720 --> 00:11:24.160 align:middle line:84% utility to how well that bidder actually 00:11:24.160 --> 00:11:25.880 align:middle line:90% does under this allocation. 00:11:25.880 --> 00:11:28.720 align:middle line:84% And there's no way that the current allocation can 00:11:28.720 --> 00:11:30.680 align:middle line:84% give more to the other bidder than if we just 00:11:30.680 --> 00:11:32.412 align:middle line:90% maximize that bidder's utility. 00:11:32.412 --> 00:11:34.120 align:middle line:84% So for that reason, this number is always 00:11:34.120 --> 00:11:35.180 align:middle line:90% going to be nonnegative. 00:11:35.180 --> 00:11:36.680 align:middle line:84% And you're right, if you're thinking 00:11:36.680 --> 00:11:39.180 align:middle line:84% of other externality context, this is a bit more special. 00:11:39.180 --> 00:11:41.470 align:middle line:84% So it does turn out here that the payments will always 00:11:41.470 --> 00:11:42.130 align:middle line:90% be nonnegative. 00:11:42.130 --> 00:11:44.350 align:middle line:84% They could be 0, but they would never be negative 00:11:44.350 --> 00:11:45.570 align:middle line:84% because, in order for them to be negative, 00:11:45.570 --> 00:11:47.870 align:middle line:84% it would have to be that the allocation we're choosing 00:11:47.870 --> 00:11:50.550 align:middle line:84% makes the other bidder strictly better off than the best 00:11:50.550 --> 00:11:51.770 align:middle line:90% allocation for that bidder. 00:11:51.770 --> 00:11:54.570 align:middle line:84% And that goes against the definition of best. 00:11:54.570 --> 00:11:55.070 align:middle line:90% Yeah. 00:11:55.070 --> 00:11:58.150 align:middle line:84% AUDIENCE: 7 is maximizing their utility 00:11:58.150 --> 00:12:01.550 align:middle line:84% when everyone besides myself is in the game? 00:12:01.550 --> 00:12:03.930 align:middle line:84% Or is it just their utility alone? 00:12:03.930 --> 00:12:04.930 align:middle line:90% IAN BALL: Good question. 00:12:04.930 --> 00:12:08.550 align:middle line:84% So the 7 here is-- what we're maximizing 00:12:08.550 --> 00:12:14.190 align:middle line:84% is the sum of everyone else's utility, 00:12:14.190 --> 00:12:16.130 align:middle line:84% and we're ignoring my own utility. 00:12:16.130 --> 00:12:18.693 align:middle line:84% So in this case, because there's only two bidders, 00:12:18.693 --> 00:12:20.110 align:middle line:84% the sum of everyone else's utility 00:12:20.110 --> 00:12:22.410 align:middle line:84% is just maximizing the other bidder's utility. 00:12:22.410 --> 00:12:24.213 align:middle line:84% But if we had three bidders, then we 00:12:24.213 --> 00:12:26.630 align:middle line:84% would think about how could we allocate the goods in a way 00:12:26.630 --> 00:12:30.550 align:middle line:84% to maximize the sum of, say, bidders 2 and 3's utility, 00:12:30.550 --> 00:12:33.110 align:middle line:84% ignoring the effect of the presence of bidder one. 00:12:33.110 --> 00:12:34.910 align:middle line:90% Yeah, great. 00:12:34.910 --> 00:12:36.210 align:middle line:90% Any other questions on this? 00:12:36.210 --> 00:12:39.660 align:middle line:90% 00:12:39.660 --> 00:12:44.540 align:middle line:84% So now let's change tacks a little bit. 00:12:44.540 --> 00:12:46.620 align:middle line:84% And we're now going to move to the final phase 00:12:46.620 --> 00:12:50.640 align:middle line:84% of the course, which is called dynamic Bayesian games. 00:12:50.640 --> 00:12:56.780 align:middle line:90% 00:12:56.780 --> 00:12:59.020 align:middle line:84% And I think it's good at this point to pause and see 00:12:59.020 --> 00:13:00.480 align:middle line:90% where we've come. 00:13:00.480 --> 00:13:03.260 align:middle line:84% So we can think of the course as organized into four parts. 00:13:03.260 --> 00:13:07.520 align:middle line:84% We have looked at static games and dynamic games. 00:13:07.520 --> 00:13:11.340 align:middle line:90% 00:13:11.340 --> 00:13:17.100 align:middle line:84% And we've also looked at games of complete information 00:13:17.100 --> 00:13:18.880 align:middle line:84% and games of incomplete information. 00:13:18.880 --> 00:13:21.500 align:middle line:90% 00:13:21.500 --> 00:13:24.000 align:middle line:84% And basically, so far, we've filled in three of these boxes, 00:13:24.000 --> 00:13:25.917 align:middle line:84% and now we're going to fill in the fourth box. 00:13:25.917 --> 00:13:27.758 align:middle line:84% And for each box, each class of games, 00:13:27.758 --> 00:13:30.300 align:middle line:84% we have a different solution concept, a different equilibrium 00:13:30.300 --> 00:13:30.960 align:middle line:90% notion. 00:13:30.960 --> 00:13:32.040 align:middle line:90% So let's fill this in. 00:13:32.040 --> 00:13:37.340 align:middle line:90% 00:13:37.340 --> 00:13:40.120 align:middle line:84% So in the first part of the course, we started here. 00:13:40.120 --> 00:13:43.420 align:middle line:84% This would just be your basic strategic-form game 00:13:43.420 --> 00:13:44.740 align:middle line:90% with complete information. 00:13:44.740 --> 00:13:47.420 align:middle line:84% And our solution concept there was just Nash equilibrium. 00:13:47.420 --> 00:13:55.580 align:middle line:90% 00:13:55.580 --> 00:13:57.380 align:middle line:90% Next, we said, well-- 00:13:57.380 --> 00:13:59.460 align:middle line:84% we looked at extensive-form games 00:13:59.460 --> 00:14:01.580 align:middle line:84% where players make moves over time. 00:14:01.580 --> 00:14:04.260 align:middle line:84% And we said, one problem of the Nash equilibrium 00:14:04.260 --> 00:14:07.580 align:middle line:84% is it allows players to make non-credible threats. 00:14:07.580 --> 00:14:10.140 align:middle line:84% And we said we wanted to refine Nash equilibrium 00:14:10.140 --> 00:14:13.580 align:middle line:84% and impose a stronger solution concept that rules out 00:14:13.580 --> 00:14:15.280 align:middle line:90% threats that are not credible. 00:14:15.280 --> 00:14:17.980 align:middle line:84% And that solution concept was called subgame-perfect Nash 00:14:17.980 --> 00:14:19.180 align:middle line:90% equilibrium. 00:14:19.180 --> 00:14:28.780 align:middle line:90% 00:14:28.780 --> 00:14:31.820 align:middle line:84% So that was a stronger notion than Nash 00:14:31.820 --> 00:14:33.980 align:middle line:84% that ruled out some Nash equilibria 00:14:33.980 --> 00:14:36.570 align:middle line:84% that we didn't find compelling because they involved threats 00:14:36.570 --> 00:14:38.490 align:middle line:84% that were not credible at nodes that 00:14:38.490 --> 00:14:41.330 align:middle line:90% were not reached in the game. 00:14:41.330 --> 00:14:43.645 align:middle line:84% Then we stepped back, and we said, wait a second, 00:14:43.645 --> 00:14:46.270 align:middle line:84% our complete information seems like a really strong assumption. 00:14:46.270 --> 00:14:48.690 align:middle line:84% A lot of times when people come to the game and they start 00:14:48.690 --> 00:14:50.950 align:middle line:84% the game, they know things that other players don't. 00:14:50.950 --> 00:14:52.690 align:middle line:84% They know things about their preferences that other players 00:14:52.690 --> 00:14:53.650 align:middle line:90% in the game don't. 00:14:53.650 --> 00:14:57.110 align:middle line:84% So we introduced Bayesian games, that is, 00:14:57.110 --> 00:14:58.735 align:middle line:84% games that have incomplete information, 00:14:58.735 --> 00:15:00.193 align:middle line:84% and we introduce a solution concept 00:15:00.193 --> 00:15:02.570 align:middle line:84% for that called Bayes-Nash equilibrium or Bayesian Nash 00:15:02.570 --> 00:15:03.350 align:middle line:90% equilibrium. 00:15:03.350 --> 00:15:15.610 align:middle line:90% 00:15:15.610 --> 00:15:19.190 align:middle line:84% Now this dealt with the issue of incomplete information. 00:15:19.190 --> 00:15:22.210 align:middle line:84% But it turns out that Bayesian Nash equilibrium 00:15:22.210 --> 00:15:25.745 align:middle line:84% doesn't deal with the issue of non-credible threats. 00:15:25.745 --> 00:15:27.370 align:middle line:84% And if we want to look at dynamic games 00:15:27.370 --> 00:15:28.913 align:middle line:84% with incomplete information, we have 00:15:28.913 --> 00:15:30.330 align:middle line:84% to both take into account the fact 00:15:30.330 --> 00:15:32.110 align:middle line:84% that players have incomplete information. 00:15:32.110 --> 00:15:34.800 align:middle line:84% But we also may want to rule out equilibria that 00:15:34.800 --> 00:15:36.100 align:middle line:90% involve non-credible threats. 00:15:36.100 --> 00:15:37.475 align:middle line:84% So we're going to introduce today 00:15:37.475 --> 00:15:40.380 align:middle line:84% our final solution concept, which it's not very creative. 00:15:40.380 --> 00:15:42.325 align:middle line:84% We basically combine the perfect from here 00:15:42.325 --> 00:15:44.200 align:middle line:84% and the Bayes from here, and it's just called 00:15:44.200 --> 00:15:45.660 align:middle line:90% perfect Bayesian equilibrium. 00:15:45.660 --> 00:16:01.600 align:middle line:90% 00:16:01.600 --> 00:16:04.800 align:middle line:84% I guess maybe I'll give somewhat of a Warning about this. 00:16:04.800 --> 00:16:07.640 align:middle line:84% These first three concepts are very-- 00:16:07.640 --> 00:16:09.420 align:middle line:90% are well-agreed upon. 00:16:09.420 --> 00:16:11.360 align:middle line:84% Everyone agrees what's the right notion. 00:16:11.360 --> 00:16:14.280 align:middle line:84% Once we get to dynamic games of incomplete information, 00:16:14.280 --> 00:16:16.040 align:middle line:84% there's been a lot of recent work arguing 00:16:16.040 --> 00:16:17.780 align:middle line:84% about what's the right notion of equilibrium. 00:16:17.780 --> 00:16:19.440 align:middle line:84% So we're going to present a particular notion called 00:16:19.440 --> 00:16:20.893 align:middle line:90% perfect Bayesian equilibrium. 00:16:20.893 --> 00:16:22.560 align:middle line:84% But there's 10 other equilibrium notions 00:16:22.560 --> 00:16:24.855 align:middle line:84% that could fill this box that are more exotic 00:16:24.855 --> 00:16:26.480 align:middle line:84% and involve more constraints and people 00:16:26.480 --> 00:16:28.563 align:middle line:84% are arguing to this day about what's the right way 00:16:28.563 --> 00:16:29.740 align:middle line:90% to define equilibrium. 00:16:29.740 --> 00:16:31.115 align:middle line:84% So I'll just warn you that things 00:16:31.115 --> 00:16:35.060 align:middle line:84% are a little maybe trickier or less clear cut in this case, 00:16:35.060 --> 00:16:37.600 align:middle line:84% but we'll try to focus on well-trodden ground 00:16:37.600 --> 00:16:39.000 align:middle line:90% in this class. 00:16:39.000 --> 00:16:41.400 align:middle line:84% OK, so I want to first, today, before defining 00:16:41.400 --> 00:16:43.048 align:middle line:84% perfect Bayesian equilibrium, I want 00:16:43.048 --> 00:16:44.840 align:middle line:84% to give a few examples that try to motivate 00:16:44.840 --> 00:16:48.320 align:middle line:84% why we need a new definition, why our existing equilibrium 00:16:48.320 --> 00:16:49.650 align:middle line:90% notions are insufficient. 00:16:49.650 --> 00:16:51.400 align:middle line:84% And then we'll go over how we can actually 00:16:51.400 --> 00:16:52.983 align:middle line:84% solve for perfect Bayesian equilibrium 00:16:52.983 --> 00:16:55.320 align:middle line:84% and go over the formal definition. 00:16:55.320 --> 00:16:59.200 align:middle line:84% So let's start with maybe a reminder 00:16:59.200 --> 00:17:04.119 align:middle line:84% about non-credible threats and a weakness of Nash equilibrium 00:17:04.119 --> 00:17:06.560 align:middle line:84% with complete information, and then we'll build on that. 00:17:06.560 --> 00:17:09.720 align:middle line:84% So let's go back to maybe our classic example 00:17:09.720 --> 00:17:11.800 align:middle line:90% where we had two firms. 00:17:11.800 --> 00:17:15.200 align:middle line:84% We had firm 1, and we had firm 2. 00:17:15.200 --> 00:17:20.960 align:middle line:84% And we'll think of firm 1 as a potential entrant. 00:17:20.960 --> 00:17:23.099 align:middle line:84% And we'll think of firm 2 as an incumbent. 00:17:23.099 --> 00:17:27.760 align:middle line:90% 00:17:27.760 --> 00:17:29.430 align:middle line:84% So firm 2's already in the market. 00:17:29.430 --> 00:17:32.330 align:middle line:84% Firm 1 is deciding whether to enter the market or not. 00:17:32.330 --> 00:17:33.630 align:middle line:84% So they're called an entrant, but really, they're 00:17:33.630 --> 00:17:34.463 align:middle line:90% a potential entrant. 00:17:34.463 --> 00:17:35.830 align:middle line:90% They may or may not enter. 00:17:35.830 --> 00:17:39.130 align:middle line:84% And firm 1 is choosing whether to enter the market-- 00:17:39.130 --> 00:17:40.350 align:middle line:90% we'll call that E-- 00:17:40.350 --> 00:17:42.030 align:middle line:90% or not enter the market. 00:17:42.030 --> 00:17:45.390 align:middle line:84% We'll call that N. And this is an example we've done before. 00:17:45.390 --> 00:17:46.990 align:middle line:90% Firm 2 is the incumbent. 00:17:46.990 --> 00:17:49.750 align:middle line:84% If firm 1 enters, firm 2 is the incumbent, 00:17:49.750 --> 00:17:52.030 align:middle line:84% can choose whether to accommodate their entry 00:17:52.030 --> 00:17:53.910 align:middle line:84% or to fight them by starting some price war 00:17:53.910 --> 00:17:55.702 align:middle line:84% or trying to cut prices and doing something 00:17:55.702 --> 00:17:56.950 align:middle line:90% that hurts both players. 00:17:56.950 --> 00:18:02.270 align:middle line:84% So firm 2 is choosing A or F, a for accommodate the new entrant 00:18:02.270 --> 00:18:04.870 align:middle line:84% and f for fight against the new entrant. 00:18:04.870 --> 00:18:07.790 align:middle line:84% So let's write this as an extensive-form game. 00:18:07.790 --> 00:18:09.210 align:middle line:90% We have firm 1 is here. 00:18:09.210 --> 00:18:12.190 align:middle line:90% 00:18:12.190 --> 00:18:14.830 align:middle line:84% They're choosing whether to-- maybe I'll put N here, not 00:18:14.830 --> 00:18:16.790 align:middle line:90% enter or enter. 00:18:16.790 --> 00:18:19.990 align:middle line:84% And then if they do enter, we have firm 2. 00:18:19.990 --> 00:18:23.030 align:middle line:84% And they're choosing to either accommodate the entrant 00:18:23.030 --> 00:18:24.090 align:middle line:90% or fight. 00:18:24.090 --> 00:18:27.860 align:middle line:84% Let me make sure I get the numbers right. 00:18:27.860 --> 00:18:37.940 align:middle line:84% So what we did here, let's put 0, 5, 1, 1, and negative 10, 10 00:18:37.940 --> 00:18:39.180 align:middle line:90% to make this easy. 00:18:39.180 --> 00:18:41.220 align:middle line:90% So what's the idea? 00:18:41.220 --> 00:18:43.687 align:middle line:84% If firm 1 does not enter, well, that's 00:18:43.687 --> 00:18:45.020 align:middle line:90% great for firm 2, the incumbent. 00:18:45.020 --> 00:18:45.740 align:middle line:90% They're the monopolist. 00:18:45.740 --> 00:18:46.823 align:middle line:90% They're all by themselves. 00:18:46.823 --> 00:18:48.220 align:middle line:90% So they get a payoff of 5. 00:18:48.220 --> 00:18:51.380 align:middle line:84% Firm 1 isn't entering, so they get a payoff of 0. 00:18:51.380 --> 00:18:54.140 align:middle line:84% If firm 1 does enter, well, then the payoffs of the firms 00:18:54.140 --> 00:18:57.700 align:middle line:84% depend on whether to accommodate the entrant or fights. 00:18:57.700 --> 00:19:00.080 align:middle line:84% If they accommodate, then they both do pretty well. 00:19:00.080 --> 00:19:02.900 align:middle line:84% They both get 1, but of course, not as well as firm 2 00:19:02.900 --> 00:19:06.565 align:middle line:84% would've gotten if firm 2 were by himself as monopolist. 00:19:06.565 --> 00:19:07.940 align:middle line:84% And then if firm 2 fights, that's 00:19:07.940 --> 00:19:09.680 align:middle line:90% really bad for both sides. 00:19:09.680 --> 00:19:10.560 align:middle line:90% They a price war. 00:19:10.560 --> 00:19:12.080 align:middle line:90% Prices go, say, really low. 00:19:12.080 --> 00:19:14.100 align:middle line:90% And this is bad for everyone. 00:19:14.100 --> 00:19:16.560 align:middle line:84% We're thinking of fighting in an abstract sense. 00:19:16.560 --> 00:19:18.060 align:middle line:84% It could literally involve violence, 00:19:18.060 --> 00:19:19.340 align:middle line:84% or it could be cutting prices, or it 00:19:19.340 --> 00:19:20.320 align:middle line:90% could be flooding the market. 00:19:20.320 --> 00:19:21.900 align:middle line:84% There's a lot of things it could be. 00:19:21.900 --> 00:19:26.650 align:middle line:84% So let's now just write this as a bit of a review 00:19:26.650 --> 00:19:30.750 align:middle line:84% today of our standard strategic form game box. 00:19:30.750 --> 00:19:34.330 align:middle line:84% Player 1 has two strategies, N, E. Player 2 00:19:34.330 --> 00:19:41.210 align:middle line:84% has two strategies, A, F. N, E to keep the order consistent. 00:19:41.210 --> 00:19:42.250 align:middle line:90% And let's see. 00:19:42.250 --> 00:19:46.190 align:middle line:84% If player 1 doesn't enter, then they both get 0, 5 either way. 00:19:46.190 --> 00:19:49.930 align:middle line:90% 00:19:49.930 --> 00:19:52.290 align:middle line:84% If player 1 does enter, then the payoffs 00:19:52.290 --> 00:19:53.550 align:middle line:90% depend on what player 2 does. 00:19:53.550 --> 00:19:58.590 align:middle line:84% And we get 1, 1 here and negative 10, negative 10 here. 00:19:58.590 --> 00:20:01.655 align:middle line:90% 00:20:01.655 --> 00:20:03.530 align:middle line:84% Well, now let's just do our standard approach 00:20:03.530 --> 00:20:05.582 align:middle line:84% to try to find the Nash equilibria of this. 00:20:05.582 --> 00:20:06.790 align:middle line:90% This is a good little review. 00:20:06.790 --> 00:20:08.498 align:middle line:84% The final exam is going to be cumulative. 00:20:08.498 --> 00:20:11.988 align:middle line:84% So you do have to remember how to solve the basic games. 00:20:11.988 --> 00:20:12.530 align:middle line:90% So let's see. 00:20:12.530 --> 00:20:13.510 align:middle line:90% We go down here. 00:20:13.510 --> 00:20:16.250 align:middle line:84% If firm 2 is going to accommodate, 00:20:16.250 --> 00:20:18.730 align:middle line:84% then it's better for firm 1 to enter. 00:20:18.730 --> 00:20:21.970 align:middle line:84% If firm 2 is going to fight, it's better for firm 1 00:20:21.970 --> 00:20:23.210 align:middle line:90% to not enter. 00:20:23.210 --> 00:20:25.370 align:middle line:84% And then we can do things the other way. 00:20:25.370 --> 00:20:27.110 align:middle line:84% If firm 1 doesn't enter-- actually, 00:20:27.110 --> 00:20:28.890 align:middle line:84% it doesn't matter what firm 2 does. 00:20:28.890 --> 00:20:32.330 align:middle line:84% If firm 1 enters, then firm 2 would rather accommodate. 00:20:32.330 --> 00:20:34.650 align:middle line:90% So we get two equilibria. 00:20:34.650 --> 00:20:46.350 align:middle line:84% We get EA and NF, which we can see on our graph here. 00:20:46.350 --> 00:20:51.210 align:middle line:90% 00:20:51.210 --> 00:20:54.370 align:middle line:84% Now, both of these are Nash equilibria. 00:20:54.370 --> 00:20:57.970 align:middle line:84% But we precisely introduced subgame-perfect Nash equilibria 00:20:57.970 --> 00:21:00.050 align:middle line:84% as a more demanding solution concept 00:21:00.050 --> 00:21:02.730 align:middle line:84% to rule out one of these equilibria. 00:21:02.730 --> 00:21:06.450 align:middle line:84% So which of these equilibria is not a subgame-perfect Nash 00:21:06.450 --> 00:21:07.070 align:middle line:90% equilibrium? 00:21:07.070 --> 00:21:08.170 align:middle line:90% Which one did we-- 00:21:08.170 --> 00:21:10.470 align:middle line:84% does subgame perfection rule out? 00:21:10.470 --> 00:21:15.990 align:middle line:90% 00:21:15.990 --> 00:21:16.490 align:middle line:90% Yeah. 00:21:16.490 --> 00:21:19.010 align:middle line:84% AUDIENCE: [INAUDIBLE] because you can't really-- 00:21:19.010 --> 00:21:23.960 align:middle line:90% IAN BALL: Yeah, sorry, NF or EA. 00:21:23.960 --> 00:21:25.140 align:middle line:90% It's one of these two. 00:21:25.140 --> 00:21:26.375 align:middle line:90% It's EA and NF. 00:21:26.375 --> 00:21:27.000 align:middle line:90% AUDIENCE: Yeah. 00:21:27.000 --> 00:21:30.280 align:middle line:90% 00:21:30.280 --> 00:21:31.380 align:middle line:90% IAN BALL: Maybe, yeah. 00:21:31.380 --> 00:21:32.440 align:middle line:90% AUDIENCE: NF. 00:21:32.440 --> 00:21:33.720 align:middle line:90% IAN BALL: NF, I agree, OK. 00:21:33.720 --> 00:21:35.400 align:middle line:90% And why? 00:21:35.400 --> 00:21:38.560 align:middle line:84% AUDIENCE: Because if the entrant enters, 00:21:38.560 --> 00:21:44.000 align:middle line:84% it doesn't make sense for the incumbent to fight. 00:21:44.000 --> 00:21:45.000 align:middle line:90% IAN BALL: Exactly right. 00:21:45.000 --> 00:21:48.160 align:middle line:84% So both of these are Nash equilibria. 00:21:48.160 --> 00:21:51.020 align:middle line:84% Only EA is a subgame-perfect Nash equilibrium. 00:21:51.020 --> 00:21:53.380 align:middle line:84% As you said, NF is not subgame-perfect. 00:21:53.380 --> 00:22:00.520 align:middle line:90% 00:22:00.520 --> 00:22:03.160 align:middle line:84% And it's not subgame-perfect because it involves a threat 00:22:03.160 --> 00:22:06.200 align:middle line:90% by firm 2 that is not credible. 00:22:06.200 --> 00:22:11.120 align:middle line:84% Firm 2 says, if you enter, I'm going to fight. 00:22:11.120 --> 00:22:13.522 align:middle line:84% But if firm 1 actually enters, well, firm 2 is here, 00:22:13.522 --> 00:22:14.980 align:middle line:84% and they look, well, wait a second. 00:22:14.980 --> 00:22:17.040 align:middle line:90% If I fight, I get negative 10. 00:22:17.040 --> 00:22:18.740 align:middle line:90% If I accommodate, I get 1. 00:22:18.740 --> 00:22:20.870 align:middle line:84% I'm much better off accommodating. 00:22:20.870 --> 00:22:24.830 align:middle line:84% So it's not optimal for firm 2 to actually carry out 00:22:24.830 --> 00:22:26.990 align:middle line:84% this punishment or this threat that they 00:22:26.990 --> 00:22:29.070 align:middle line:90% threatened to carry out. 00:22:29.070 --> 00:22:33.190 align:middle line:84% Formally, well, we have to think of the definition 00:22:33.190 --> 00:22:36.050 align:middle line:84% of subgame-perfect Nash equilibrium. 00:22:36.050 --> 00:22:38.390 align:middle line:84% In the definition, we have to look at every subgame. 00:22:38.390 --> 00:22:40.470 align:middle line:84% We have the full game here, but we also 00:22:40.470 --> 00:22:42.070 align:middle line:90% have subgame that starts here. 00:22:42.070 --> 00:22:47.350 align:middle line:84% So remember, this is one subgame of the full game. 00:22:47.350 --> 00:22:51.070 align:middle line:84% And the problem with NF is that fighting is not 00:22:51.070 --> 00:22:52.750 align:middle line:90% an equilibrium of this subgame. 00:22:52.750 --> 00:22:54.750 align:middle line:84% Now, it may be weird to think of this as subgame 00:22:54.750 --> 00:22:56.375 align:middle line:84% because there's only one player moving. 00:22:56.375 --> 00:22:59.110 align:middle line:84% But remember, we can have a game with a single player, 00:22:59.110 --> 00:23:01.190 align:middle line:84% and an equilibrium of a single-player game 00:23:01.190 --> 00:23:04.030 align:middle line:84% just means that player's behaving optimally. 00:23:04.030 --> 00:23:08.690 align:middle line:84% So here F is not a Nash equilibrium of this subgame 00:23:08.690 --> 00:23:10.990 align:middle line:84% because F is strictly worse for player 2 00:23:10.990 --> 00:23:13.470 align:middle line:84% than A. So this doesn't satisfy our definition 00:23:13.470 --> 00:23:18.260 align:middle line:84% of subgame-perfect Nash equilibrium. 00:23:18.260 --> 00:23:21.840 align:middle line:84% But let's maybe change the game slightly. 00:23:21.840 --> 00:23:25.530 align:middle line:90% 00:23:25.530 --> 00:23:27.780 align:middle line:84% So what if we change the game and say, well, actually, 00:23:27.780 --> 00:23:32.920 align:middle line:84% firm 1 has two different ways of entering. 00:23:32.920 --> 00:23:35.400 align:middle line:84% They can enter the market and produce one kind of good, 00:23:35.400 --> 00:23:37.260 align:middle line:84% or they can enter the market and produce a slightly different 00:23:37.260 --> 00:23:38.080 align:middle line:90% kind of good. 00:23:38.080 --> 00:23:39.747 align:middle line:84% So we're now going to say that firm 1 is 00:23:39.747 --> 00:23:45.445 align:middle line:84% choosing between not entering or I'll call it E1 or E2. 00:23:45.445 --> 00:23:46.820 align:middle line:84% These are just two different ways 00:23:46.820 --> 00:23:48.780 align:middle line:84% they can enter and maybe use a slightly different marketing 00:23:48.780 --> 00:23:50.960 align:middle line:84% strategy or slightly different pricing strategy, 00:23:50.960 --> 00:23:54.620 align:middle line:84% but they're effectively entering either way. 00:23:54.620 --> 00:23:58.620 align:middle line:84% And now let's write out how this looks. 00:23:58.620 --> 00:24:01.780 align:middle line:90% So we have firm 1 here. 00:24:01.780 --> 00:24:09.700 align:middle line:84% Firm 1 is now choosing enter, not enter-- sorry, not enter, 00:24:09.700 --> 00:24:12.740 align:middle line:90% E1 or E2. 00:24:12.740 --> 00:24:16.060 align:middle line:84% Again, if they don't enter, the payoff is 0, 5. 00:24:16.060 --> 00:24:18.860 align:middle line:84% If they enter in this way, firm 2 00:24:18.860 --> 00:24:21.060 align:middle line:90% chooses to accommodate a fight. 00:24:21.060 --> 00:24:22.180 align:middle line:90% They enter in this way. 00:24:22.180 --> 00:24:24.300 align:middle line:84% Firm 2 chooses to commentator or fight. 00:24:24.300 --> 00:24:27.220 align:middle line:90% This is still firm 2, firm 2. 00:24:27.220 --> 00:24:30.580 align:middle line:84% And maybe the exact pricing strategy 00:24:30.580 --> 00:24:32.200 align:middle line:84% they use makes a bit of a difference. 00:24:32.200 --> 00:24:33.783 align:middle line:84% But just to make things simpler, let's 00:24:33.783 --> 00:24:36.583 align:middle line:84% say it doesn't really affect their payoffs whether they sell 00:24:36.583 --> 00:24:38.000 align:middle line:84% them in one way or a different way 00:24:38.000 --> 00:24:39.460 align:middle line:84% or whether they use this advertising firm 00:24:39.460 --> 00:24:40.837 align:middle line:90% or this other advertising firm. 00:24:40.837 --> 00:24:42.420 align:middle line:84% So let's say the payoffs down here are 00:24:42.420 --> 00:24:46.660 align:middle line:84% going to be 1, 1, negative 10, negative 10, 1, 1, 00:24:46.660 --> 00:24:49.100 align:middle line:90% negative 10, negative 10. 00:24:49.100 --> 00:24:51.000 align:middle line:84% So these are different ways of entering. 00:24:51.000 --> 00:24:54.155 align:middle line:84% But the effect on the payoffs is actually 0. 00:24:54.155 --> 00:24:56.280 align:middle line:84% It doesn't actually change the way the payoffs are. 00:24:56.280 --> 00:24:58.280 align:middle line:84% And we could change these slightly if we wanted, 00:24:58.280 --> 00:25:00.500 align:middle line:90% but it doesn't matter. 00:25:00.500 --> 00:25:05.180 align:middle line:84% And notice, the way I'm writing this, 00:25:05.180 --> 00:25:08.420 align:middle line:84% just let me slightly modify this. 00:25:08.420 --> 00:25:10.140 align:middle line:84% We have to make a choice, does firm 2 00:25:10.140 --> 00:25:14.080 align:middle line:84% observe the advertising strategy that firm 1 is using? 00:25:14.080 --> 00:25:15.370 align:middle line:90% And we're going to assume no. 00:25:15.370 --> 00:25:18.470 align:middle line:84% So we're going to put this in the same information set. 00:25:18.470 --> 00:25:20.793 align:middle line:84% So as firm 2, all I see is that firm 1 has entered. 00:25:20.793 --> 00:25:22.210 align:middle line:84% I don't really know under the Hood 00:25:22.210 --> 00:25:24.530 align:middle line:84% exactly what, say, production process they're using 00:25:24.530 --> 00:25:26.550 align:middle line:84% or what advertising firm they are. 00:25:26.550 --> 00:25:30.785 align:middle line:90% 00:25:30.785 --> 00:25:36.410 align:middle line:84% And now let's think of the strategies in this game. 00:25:36.410 --> 00:25:38.810 align:middle line:84% It turns out the strategies in this game 00:25:38.810 --> 00:25:41.730 align:middle line:84% are exactly the same as the strategies over here. 00:25:41.730 --> 00:25:44.050 align:middle line:90% Firm 1 is still choosing-- 00:25:44.050 --> 00:25:47.250 align:middle line:84% sorry, for firm 1, it's slightly different. 00:25:47.250 --> 00:25:52.890 align:middle line:84% So firm 1 is choosing between N, E1, and E2. 00:25:52.890 --> 00:25:57.833 align:middle line:84% And firm 2 is still choosing accommodate or fight. 00:25:57.833 --> 00:25:59.250 align:middle line:84% It's the same for firm 2, but it's 00:25:59.250 --> 00:26:02.622 align:middle line:90% slightly different from firm 1. 00:26:02.622 --> 00:26:04.330 align:middle line:84% Let's try to compute the Nash-- let's try 00:26:04.330 --> 00:26:15.400 align:middle line:84% to find the Nash equilibria of this game. 00:26:15.400 --> 00:26:21.900 align:middle line:84% And the idea is what are some Nash equilibria of this game? 00:26:21.900 --> 00:26:22.900 align:middle line:90% It looks really similar. 00:26:22.900 --> 00:26:27.920 align:middle line:84% So I think it shouldn't be too hard to figure this out. 00:26:27.920 --> 00:26:31.160 align:middle line:84% Well, let's organize it by what firm 2's doing. 00:26:31.160 --> 00:26:33.200 align:middle line:84% Let's say firm 2 is accommodating. 00:26:33.200 --> 00:26:36.020 align:middle line:84% And notice firm 2 makes a single choice-- accommodate or fight. 00:26:36.020 --> 00:26:38.120 align:middle line:84% It's a single choice because these two nodes are 00:26:38.120 --> 00:26:39.540 align:middle line:90% in the same information set. 00:26:39.540 --> 00:26:41.960 align:middle line:84% So I can't choose to accommodate E1 and fight 00:26:41.960 --> 00:26:44.360 align:middle line:84% against E2 because I can't distinguish those. 00:26:44.360 --> 00:26:47.920 align:middle line:84% So if firm 2 accommodates, what does firm 1 want to do? 00:26:47.920 --> 00:26:48.520 align:middle line:90% Yeah. 00:26:48.520 --> 00:26:49.500 align:middle line:90% AUDIENCE: E1 or E2. 00:26:49.500 --> 00:26:51.300 align:middle line:84% IAN BALL: E1 and E2 is going to be good for them. 00:26:51.300 --> 00:26:53.592 align:middle line:84% They want to enter, and they're happy to do either one. 00:26:53.592 --> 00:26:58.240 align:middle line:84% So we're going to get E1A and E2A are both 00:26:58.240 --> 00:27:01.500 align:middle line:90% going to be Nash equilibrium. 00:27:01.500 --> 00:27:04.040 align:middle line:90% What if firm 2's going to fight? 00:27:04.040 --> 00:27:05.380 align:middle line:90% What does firm 1 want to do? 00:27:05.380 --> 00:27:09.156 align:middle line:90% 00:27:09.156 --> 00:27:09.656 align:middle line:90% Yeah. 00:27:09.656 --> 00:27:10.600 align:middle line:90% AUDIENCE: Not enter. 00:27:10.600 --> 00:27:11.433 align:middle line:90% IAN BALL: Not enter. 00:27:11.433 --> 00:27:15.160 align:middle line:84% So now it looks similar to before. 00:27:15.160 --> 00:27:17.320 align:middle line:84% But now we actually have three equilibria. 00:27:17.320 --> 00:27:20.120 align:middle line:90% 00:27:20.120 --> 00:27:23.260 align:middle line:84% Same idea, if firm 1 is going to accommodate, then-- 00:27:23.260 --> 00:27:23.760 align:middle line:90% sorry. 00:27:23.760 --> 00:27:25.640 align:middle line:84% If firm 2 is going to accommodate, 00:27:25.640 --> 00:27:27.260 align:middle line:90% then firm 1 wants to enter. 00:27:27.260 --> 00:27:28.880 align:middle line:84% They're happy to enter either way. 00:27:28.880 --> 00:27:33.960 align:middle line:84% If firm 2's going to fight, then firm 1 doesn't want to enter. 00:27:33.960 --> 00:27:39.060 align:middle line:84% Now, I think that NF still feels like a non-credible threat. 00:27:39.060 --> 00:27:43.680 align:middle line:90% 00:27:43.680 --> 00:27:46.160 align:middle line:84% For exactly the same reason as before. 00:27:46.160 --> 00:27:48.265 align:middle line:90% Firm 2 is fighting. 00:27:48.265 --> 00:27:49.640 align:middle line:84% They know that, by fighting, they 00:27:49.640 --> 00:27:52.340 align:middle line:84% get a payoff of negative 10, and if they didn't fight, 00:27:52.340 --> 00:27:53.840 align:middle line:90% they would get a payoff of 1. 00:27:53.840 --> 00:27:56.040 align:middle line:84% So to me, this feels like a non-credible threat. 00:27:56.040 --> 00:27:59.680 align:middle line:84% But the question is, is NF a subgame-perfect Nash equilibrium 00:27:59.680 --> 00:28:00.380 align:middle line:90% of this game? 00:28:00.380 --> 00:28:03.310 align:middle line:90% 00:28:03.310 --> 00:28:04.623 align:middle line:90% What do people think? 00:28:04.623 --> 00:28:06.040 align:middle line:84% Well, the first question we always 00:28:06.040 --> 00:28:08.417 align:middle line:84% have to ask ourselves about subgame-perfect Nash 00:28:08.417 --> 00:28:10.250 align:middle line:84% is, first, we have to identify the subgames. 00:28:10.250 --> 00:28:12.167 align:middle line:84% Remember, maybe we're going a little bit back. 00:28:12.167 --> 00:28:13.550 align:middle line:90% We need to review a little bit. 00:28:13.550 --> 00:28:16.150 align:middle line:84% So if we go all the way back, what 00:28:16.150 --> 00:28:20.350 align:middle line:84% are the subgames of this game here? 00:28:20.350 --> 00:28:22.270 align:middle line:84% AUDIENCE: Only if the entire game is a subgame 00:28:22.270 --> 00:28:25.670 align:middle line:84% because the decision point for player 2 00:28:25.670 --> 00:28:27.290 align:middle line:90% is not a singleton node. 00:28:27.290 --> 00:28:28.290 align:middle line:90% IAN BALL: Exactly right. 00:28:28.290 --> 00:28:32.070 align:middle line:84% So we can't start a subgame here because we would 00:28:32.070 --> 00:28:33.790 align:middle line:90% break this information set. 00:28:33.790 --> 00:28:36.310 align:middle line:84% We can't start here because we would break this information 00:28:36.310 --> 00:28:36.850 align:middle line:90% set. 00:28:36.850 --> 00:28:40.350 align:middle line:84% And we can't include both of them as part of our subgame 00:28:40.350 --> 00:28:42.430 align:middle line:84% because we had a rule that a subgame has 00:28:42.430 --> 00:28:44.510 align:middle line:90% to start with a single node. 00:28:44.510 --> 00:28:47.070 align:middle line:84% So the only subgame of the full game 00:28:47.070 --> 00:28:49.430 align:middle line:90% is just the full game itself. 00:28:49.430 --> 00:28:53.170 align:middle line:84% And if there's no proper subgames, 00:28:53.170 --> 00:28:55.310 align:middle line:84% no subgames other than the game itself, 00:28:55.310 --> 00:28:57.270 align:middle line:84% then subgame-perfect Nash equilibrium 00:28:57.270 --> 00:29:00.990 align:middle line:84% doesn't have any additional bite because the additional strength 00:29:00.990 --> 00:29:04.067 align:middle line:84% of subgame-perfect Nash is it says, in every subgame, 00:29:04.067 --> 00:29:05.650 align:middle line:84% we must be playing a Nash equilibrium. 00:29:05.650 --> 00:29:06.850 align:middle line:84% But if there's no other subgames, 00:29:06.850 --> 00:29:08.260 align:middle line:90% well, it doesn't have any bite. 00:29:08.260 --> 00:29:12.586 align:middle line:84% So Nash equilibria and all of these-- 00:29:12.586 --> 00:29:16.440 align:middle line:90% all three are subgame-perfect. 00:29:16.440 --> 00:29:25.980 align:middle line:90% 00:29:25.980 --> 00:29:28.000 align:middle line:84% So this seems like maybe a problem. 00:29:28.000 --> 00:29:31.900 align:middle line:84% It shows that our notion of subgame perfection 00:29:31.900 --> 00:29:35.740 align:middle line:84% doesn't have the bite to rule out non-credible threats 00:29:35.740 --> 00:29:37.860 align:middle line:84% in games like this where we have multiple nodes 00:29:37.860 --> 00:29:39.235 align:middle line:90% in the same information set. 00:29:39.235 --> 00:29:40.860 align:middle line:84% And we're going to introduce our notion 00:29:40.860 --> 00:29:43.140 align:middle line:84% of perfect Bayesian equilibrium to rule out 00:29:43.140 --> 00:29:44.800 align:middle line:84% these kinds of non-credible threats, 00:29:44.800 --> 00:29:47.380 align:middle line:90% even in situations like this. 00:29:47.380 --> 00:29:50.680 align:middle line:84% So let's maybe go through a more involved example. 00:29:50.680 --> 00:29:55.140 align:middle line:90% 00:29:55.140 --> 00:29:58.700 align:middle line:84% So let's think of, maybe we'll call this, an application game. 00:29:58.700 --> 00:30:04.540 align:middle line:84% People have probably applied to internships or jobs or schools 00:30:04.540 --> 00:30:05.360 align:middle line:90% like this. 00:30:05.360 --> 00:30:07.632 align:middle line:84% And sometimes, there's an optional essay 00:30:07.632 --> 00:30:09.590 align:middle line:84% or an optional interview or something optional. 00:30:09.590 --> 00:30:11.190 align:middle line:84% And often, people feel it's not really optional. 00:30:11.190 --> 00:30:12.510 align:middle line:84% If I don't do it, I'm not going to get the job. 00:30:12.510 --> 00:30:13.390 align:middle line:90% I'm not going to get in. 00:30:13.390 --> 00:30:15.265 align:middle line:84% Well, let's try to explore that in this game. 00:30:15.265 --> 00:30:18.690 align:middle line:84% So let's say, well, it's an application game. 00:30:18.690 --> 00:30:19.910 align:middle line:90% We have an applicant. 00:30:19.910 --> 00:30:23.370 align:middle line:90% 00:30:23.370 --> 00:30:26.170 align:middle line:84% And we could call it a firm or a college or a school. 00:30:26.170 --> 00:30:26.950 align:middle line:90% Let's say school. 00:30:26.950 --> 00:30:30.690 align:middle line:84% Let's use school just in this example, so we have a school. 00:30:30.690 --> 00:30:35.730 align:middle line:84% And the applicant, let's assume, has private information. 00:30:35.730 --> 00:30:38.458 align:middle line:84% In this example, it could be private information about how 00:30:38.458 --> 00:30:39.750 align:middle line:90% strong of a candidate they are. 00:30:39.750 --> 00:30:42.350 align:middle line:84% But maybe, here, just to make it a little nicer, 00:30:42.350 --> 00:30:45.130 align:middle line:84% we'll say the applicant privately knows how excited they 00:30:45.130 --> 00:30:46.770 align:middle line:90% are about going to this school. 00:30:46.770 --> 00:30:50.190 align:middle line:90% So the applicant has two types. 00:30:50.190 --> 00:30:51.450 align:middle line:90% We'll call them types. 00:30:51.450 --> 00:30:59.850 align:middle line:84% Maybe t excited or passionate, let's say P, tP or tN. 00:30:59.850 --> 00:31:01.802 align:middle line:84% So tP says, I'm really passionate about going 00:31:01.802 --> 00:31:04.010 align:middle line:84% to this school about the thing that this school does. 00:31:04.010 --> 00:31:05.490 align:middle line:90% I'm I want to go. 00:31:05.490 --> 00:31:08.270 align:middle line:84% And tN doesn't really-- they're not quite as excited. 00:31:08.270 --> 00:31:12.610 align:middle line:84% They still want to go, but they're not quite as excited. 00:31:12.610 --> 00:31:16.073 align:middle line:84% And then we have the school and the school-- 00:31:16.073 --> 00:31:17.490 align:middle line:84% let's just say in words, what they 00:31:17.490 --> 00:31:28.855 align:middle line:84% do is they want to accept tP but not tN. 00:31:28.855 --> 00:31:30.230 align:middle line:84% So there's two kinds of students, 00:31:30.230 --> 00:31:31.470 align:middle line:90% two kinds of applicants. 00:31:31.470 --> 00:31:33.370 align:middle line:84% I'll be more precise about what the utilities are, but just 00:31:33.370 --> 00:31:35.198 align:middle line:84% at a high level to understand, if someone's 00:31:35.198 --> 00:31:36.990 align:middle line:84% really passionate and excited about coming, 00:31:36.990 --> 00:31:38.630 align:middle line:90% they want to accept that person. 00:31:38.630 --> 00:31:40.922 align:middle line:84% If they're not excited, they don't want to accept them. 00:31:40.922 --> 00:31:42.210 align:middle line:90% So let's try to-- 00:31:42.210 --> 00:31:45.890 align:middle line:90% let's draw out a game tree here. 00:31:45.890 --> 00:31:47.970 align:middle line:84% So the applicant has private information 00:31:47.970 --> 00:31:50.170 align:middle line:84% at the beginning of the game about whether they're 00:31:50.170 --> 00:31:52.090 align:middle line:84% excited or not about going to the school. 00:31:52.090 --> 00:31:54.450 align:middle line:84% And the way that we model this is we start with a move 00:31:54.450 --> 00:31:55.330 align:middle line:90% by nature. 00:31:55.330 --> 00:31:57.450 align:middle line:84% We first assume that nature makes a move 00:31:57.450 --> 00:31:59.930 align:middle line:90% and draws this applicant's type. 00:31:59.930 --> 00:32:02.030 align:middle line:84% And we need to specify some probabilities. 00:32:02.030 --> 00:32:06.980 align:middle line:84% So let's say that they're passionate with probability 1/2, 00:32:06.980 --> 00:32:08.980 align:middle line:84% and they're not passionate with probability 1/2, 00:32:08.980 --> 00:32:12.570 align:middle line:84% just to make the math a little bit easier. 00:32:12.570 --> 00:32:15.320 align:middle line:84% So I need a new board just to-- this 00:32:15.320 --> 00:32:17.960 align:middle line:84% is going to get a little big, this game. 00:32:17.960 --> 00:32:18.920 align:middle line:90% Let's start here. 00:32:18.920 --> 00:32:23.000 align:middle line:90% So we start with nature. 00:32:23.000 --> 00:32:25.920 align:middle line:84% And nature is first going to draw whether this person is 00:32:25.920 --> 00:32:27.480 align:middle line:90% passionate or not. 00:32:27.480 --> 00:32:30.720 align:middle line:84% So we're going to have nature making this first move, 00:32:30.720 --> 00:32:33.540 align:middle line:84% probability 1/2, and probability 1/2. 00:32:33.540 --> 00:32:37.900 align:middle line:84% And this is going to be the passionate student. 00:32:37.900 --> 00:32:40.682 align:middle line:90% 00:32:40.682 --> 00:32:42.640 align:middle line:84% This is going to be the non-passionate student. 00:32:42.640 --> 00:32:45.160 align:middle line:90% 00:32:45.160 --> 00:32:49.480 align:middle line:84% And then the student or the applicant-- 00:32:49.480 --> 00:32:52.040 align:middle line:84% I'm calling them the applicant, so I'll say A-- 00:32:52.040 --> 00:32:53.420 align:middle line:90% is going to make a choice. 00:32:53.420 --> 00:32:57.220 align:middle line:84% They can either submit the optional essay or not. 00:32:57.220 --> 00:32:58.800 align:middle line:90% They're applying to this job-- 00:32:58.800 --> 00:32:59.960 align:middle line:90% or this school. 00:32:59.960 --> 00:33:01.390 align:middle line:90% There's some optional essay. 00:33:01.390 --> 00:33:12.290 align:middle line:84% So let's write here, let's say I can include the essay or not. 00:33:12.290 --> 00:33:13.910 align:middle line:90% Maybe I'll use these words. 00:33:13.910 --> 00:33:17.350 align:middle line:84% So they're applying either way, but they're just choosing 00:33:17.350 --> 00:33:18.610 align:middle line:90% whether to submit the essay. 00:33:18.610 --> 00:33:27.650 align:middle line:90% 00:33:27.650 --> 00:33:37.470 align:middle line:84% And now we have the school is making a choice 00:33:37.470 --> 00:33:41.030 align:middle line:84% whether to accept or reject the candidate. 00:33:41.030 --> 00:33:46.310 align:middle line:84% Let's say accept, reject, and let's save 00:33:46.310 --> 00:33:57.750 align:middle line:90% some space over here, A, R. OK. 00:33:57.750 --> 00:34:00.700 align:middle line:84% Now, we said the school would rather 00:34:00.700 --> 00:34:04.220 align:middle line:84% accept someone who's passionate than someone who's not. 00:34:04.220 --> 00:34:11.580 align:middle line:84% And let's say that a passionate candidate gives the school 00:34:11.580 --> 00:34:14.820 align:middle line:84% a utility of 3, but someone who's not excited 00:34:14.820 --> 00:34:17.540 align:middle line:84% gives the school a payoff of negative 2, if they accept them. 00:34:17.540 --> 00:34:20.580 align:middle line:84% And if they don't accept them, they just get a payoff of 0. 00:34:20.580 --> 00:34:22.699 align:middle line:84% So let's go and fill that in here. 00:34:22.699 --> 00:34:24.880 align:middle line:84% On this side of the tree, the student is excited. 00:34:24.880 --> 00:34:25.820 align:middle line:90% They're passionate. 00:34:25.820 --> 00:34:27.860 align:middle line:84% So if the school accepts them, the school 00:34:27.860 --> 00:34:31.699 align:middle line:90% is going to get a payoff of 3. 00:34:31.699 --> 00:34:33.715 align:middle line:84% And if they reject them, the school's 00:34:33.715 --> 00:34:34.840 align:middle line:90% going to get a payoff of 0. 00:34:34.840 --> 00:34:36.423 align:middle line:84% I'm writing a comma here because we're 00:34:36.423 --> 00:34:38.320 align:middle line:84% going to fill in the applicants payoffs next. 00:34:38.320 --> 00:34:38.820 align:middle line:90% Yeah. 00:34:38.820 --> 00:34:42.460 align:middle line:84% AUDIENCE: Is there a reason why there are no-- 00:34:42.460 --> 00:34:44.020 align:middle line:90% [INAUDIBLE]? 00:34:44.020 --> 00:34:46.915 align:middle line:84% IAN BALL: Yes, I going to fill it out, but maybe I'll-- 00:34:46.915 --> 00:34:49.040 align:middle line:84% let me fill out the payoffs, and then I'll do that. 00:34:49.040 --> 00:34:52.540 align:middle line:84% So I'm going to fill in some information sets next. 00:34:52.540 --> 00:34:56.300 align:middle line:84% And then if we go over here, if we accept the not very 00:34:56.300 --> 00:35:01.420 align:middle line:84% excited student, then we get negative 2, negative 2, 00:35:01.420 --> 00:35:03.340 align:middle line:90% and then 0, 0. 00:35:03.340 --> 00:35:07.620 align:middle line:84% But as you pointed out, the school 00:35:07.620 --> 00:35:09.940 align:middle line:84% only observes whether the applicant has 00:35:09.940 --> 00:35:11.163 align:middle line:90% submitted the optional essay. 00:35:11.163 --> 00:35:13.580 align:middle line:84% They don't directly observe whether the student is excited 00:35:13.580 --> 00:35:14.500 align:middle line:90% or not. 00:35:14.500 --> 00:35:19.780 align:middle line:84% So we're going to need to define our information sets like this. 00:35:19.780 --> 00:35:20.400 align:middle line:90% Here. 00:35:20.400 --> 00:35:24.060 align:middle line:90% 00:35:24.060 --> 00:35:25.740 align:middle line:90% And like this. 00:35:25.740 --> 00:35:27.660 align:middle line:84% So the school has two information sets. 00:35:27.660 --> 00:35:30.940 align:middle line:84% At this information set, I know that you've not 00:35:30.940 --> 00:35:36.500 align:middle line:84% submitted the essay, but I don't know your true enthusiasm. 00:35:36.500 --> 00:35:40.420 align:middle line:84% At this information set, I know you've submitted the essay, 00:35:40.420 --> 00:35:44.620 align:middle line:84% but again, I don't know your true enthusiasm. 00:35:44.620 --> 00:35:48.940 align:middle line:84% Now we need to specify the student's payoff. 00:35:48.940 --> 00:35:51.740 align:middle line:84% And let's start, when the student 00:35:51.740 --> 00:35:55.180 align:middle line:84% doesn't submit the essay, let's suppose 00:35:55.180 --> 00:36:00.570 align:middle line:84% that if they're rejected, they get a payoff of 0. 00:36:00.570 --> 00:36:03.810 align:middle line:84% And actually, let's assume that everywhere. 00:36:03.810 --> 00:36:06.650 align:middle line:84% So all I've said here is whenever the student is 00:36:06.650 --> 00:36:08.570 align:middle line:84% rejected, they get a payoff of 0 because they 00:36:08.570 --> 00:36:09.830 align:middle line:90% want to be accepted. 00:36:09.830 --> 00:36:12.970 align:middle line:90% 00:36:12.970 --> 00:36:17.610 align:middle line:84% Now, if they do get accepted, let's suppose that-- 00:36:17.610 --> 00:36:19.910 align:middle line:84% so let's talk about the value of acceptance. 00:36:19.910 --> 00:36:24.130 align:middle line:90% 00:36:24.130 --> 00:36:29.610 align:middle line:84% Well it's 2 to the passionate student. 00:36:29.610 --> 00:36:32.170 align:middle line:84% It's only 1 to the less passionate student. 00:36:32.170 --> 00:36:34.650 align:middle line:84% And let's assume that writing the essay is costly. 00:36:34.650 --> 00:36:37.830 align:middle line:84% So writing the essay also has a cost of c. 00:36:37.830 --> 00:36:41.530 align:middle line:90% 00:36:41.530 --> 00:36:43.990 align:middle line:84% And I'm just using c so I can vary this over the game. 00:36:43.990 --> 00:36:45.190 align:middle line:90% But c is a known parameter. 00:36:45.190 --> 00:36:47.510 align:middle line:84% We all know c is just like a number like 1 and 2, 00:36:47.510 --> 00:36:50.830 align:middle line:84% but I just want to have some flexibility over what this is. 00:36:50.830 --> 00:36:52.350 align:middle line:84% So let's see if we can fill this in. 00:36:52.350 --> 00:36:55.760 align:middle line:84% So down here, the student is passionate, 00:36:55.760 --> 00:36:59.160 align:middle line:84% and they don't submit the essay, and then they get accepted. 00:36:59.160 --> 00:37:00.560 align:middle line:90% Yeah. 00:37:00.560 --> 00:37:02.400 align:middle line:90% Question. 00:37:02.400 --> 00:37:04.065 align:middle line:84% AUDIENCE: I guess we'll modify it, 00:37:04.065 --> 00:37:06.440 align:middle line:84% but for the one where the student writes the essay and is 00:37:06.440 --> 00:37:08.940 align:middle line:84% rejected, would it be [INAUDIBLE]? 00:37:08.940 --> 00:37:11.320 align:middle line:84% IAN BALL: Good, good, yes, we should do that. 00:37:11.320 --> 00:37:12.135 align:middle line:90% Good point. 00:37:12.135 --> 00:37:13.260 align:middle line:90% They still had to write it. 00:37:13.260 --> 00:37:15.140 align:middle line:90% Yeah, great point. 00:37:15.140 --> 00:37:15.980 align:middle line:90% Let's do that. 00:37:15.980 --> 00:37:20.360 align:middle line:90% 00:37:20.360 --> 00:37:23.000 align:middle line:90% Good point. 00:37:23.000 --> 00:37:24.720 align:middle line:90% Thank you. 00:37:24.720 --> 00:37:26.380 align:middle line:90% OK, thanks. 00:37:26.380 --> 00:37:27.700 align:middle line:90% So let's have that in here. 00:37:27.700 --> 00:37:28.700 align:middle line:90% And now we go over here. 00:37:28.700 --> 00:37:29.242 align:middle line:90% So let's see. 00:37:29.242 --> 00:37:31.408 align:middle line:90% The student is passionate. 00:37:31.408 --> 00:37:33.700 align:middle line:84% They don't write the essay, and then they get accepted. 00:37:33.700 --> 00:37:37.760 align:middle line:84% So they get 2 for being passionate, and that's it. 00:37:37.760 --> 00:37:39.505 align:middle line:84% Over here, the student is passionate, 00:37:39.505 --> 00:37:40.880 align:middle line:84% but they have to write the essay, 00:37:40.880 --> 00:37:42.180 align:middle line:90% and then they're accepted. 00:37:42.180 --> 00:37:45.920 align:middle line:90% So we get 2 minus c. 00:37:45.920 --> 00:37:47.240 align:middle line:90% Then we go over here. 00:37:47.240 --> 00:37:48.818 align:middle line:90% The student is not passionate. 00:37:48.818 --> 00:37:50.860 align:middle line:84% They don't write the essay, and they're accepted. 00:37:50.860 --> 00:37:54.030 align:middle line:90% So they get 1. 00:37:54.030 --> 00:37:56.370 align:middle line:84% And then, over here, the student is not passionate. 00:37:56.370 --> 00:37:59.480 align:middle line:84% And they do write the essay so they get 1 minus. 00:37:59.480 --> 00:38:01.530 align:middle line:84% It's maybe getting a little messy, 00:38:01.530 --> 00:38:05.370 align:middle line:90% but I think we have everything. 00:38:05.370 --> 00:38:07.510 align:middle line:84% Maybe I'll put parentheses around these things just 00:38:07.510 --> 00:38:09.010 align:middle line:90% to make it a little clearer. 00:38:09.010 --> 00:38:16.190 align:middle line:90% 00:38:16.190 --> 00:38:22.070 align:middle line:90% OK, so here's our game. 00:38:22.070 --> 00:38:23.950 align:middle line:90% Let's go back over here. 00:38:23.950 --> 00:38:25.010 align:middle line:90% Start analyzing. 00:38:25.010 --> 00:38:44.127 align:middle line:90% 00:38:44.127 --> 00:38:46.710 align:middle line:84% So we're going to look at a few different versions of the game 00:38:46.710 --> 00:38:48.377 align:middle line:84% and a few different kinds of equilibria. 00:38:48.377 --> 00:38:55.030 align:middle line:84% But let's first assume that c is less than 1. 00:38:55.030 --> 00:38:57.453 align:middle line:84% so this means even if I'm not passionate, 00:38:57.453 --> 00:38:58.870 align:middle line:84% if I'm going to get accepted, it's 00:38:58.870 --> 00:39:01.350 align:middle line:90% still worth writing the essay. 00:39:01.350 --> 00:39:07.790 align:middle line:84% The benefit of being accepted outweighs the essay cost. 00:39:07.790 --> 00:39:11.550 align:middle line:84% And let's see if we can look for an equilibrium 00:39:11.550 --> 00:39:14.365 align:middle line:84% where every student applies and writes 00:39:14.365 --> 00:39:15.990 align:middle line:84% the essay because I think this is often 00:39:15.990 --> 00:39:16.970 align:middle line:90% what we see in practice. 00:39:16.970 --> 00:39:18.450 align:middle line:84% Even the students who aren't that excited, 00:39:18.450 --> 00:39:19.630 align:middle line:84% they write some essay about how excited 00:39:19.630 --> 00:39:20.838 align:middle line:90% they are to go to the school. 00:39:20.838 --> 00:39:22.970 align:middle line:84% So let's look for an equilibrium. 00:39:22.970 --> 00:39:33.710 align:middle line:90% 00:39:33.710 --> 00:39:43.090 align:middle line:84% In which both types of students submit the essay. 00:39:43.090 --> 00:39:48.656 align:middle line:90% 00:39:48.656 --> 00:39:53.820 align:middle line:84% So formally, let's look at the student strategy. 00:39:53.820 --> 00:39:56.510 align:middle line:84% Remember, a student strategy has to say what the student does 00:39:56.510 --> 00:39:58.260 align:middle line:84% as a function of their type, as a function 00:39:58.260 --> 00:40:00.100 align:middle line:90% of their private information. 00:40:00.100 --> 00:40:03.300 align:middle line:84% So we have the passionate student 00:40:03.300 --> 00:40:06.800 align:middle line:84% and the non-passionate student, and they both choose the essay. 00:40:06.800 --> 00:40:12.780 align:middle line:90% 00:40:12.780 --> 00:40:14.260 align:middle line:90% It's pretty easy. 00:40:14.260 --> 00:40:21.660 align:middle line:84% And now we have to go to the school and see what they do. 00:40:21.660 --> 00:40:26.500 align:middle line:84% So let's put ourselves in the position of the school. 00:40:26.500 --> 00:40:27.900 align:middle line:84% And let's look at the information 00:40:27.900 --> 00:40:30.420 align:middle line:84% set where they have received an essay. 00:40:30.420 --> 00:40:33.380 align:middle line:84% So remember, for the school, their information 00:40:33.380 --> 00:40:36.000 align:middle line:90% sets are essay and no essay. 00:40:36.000 --> 00:40:40.740 align:middle line:90% 00:40:40.740 --> 00:40:42.460 align:middle line:84% So the school needs to decide what 00:40:42.460 --> 00:40:45.413 align:middle line:84% to do if they see that the student has submitted an essay 00:40:45.413 --> 00:40:47.330 align:middle line:84% and what to do if they see the student has not 00:40:47.330 --> 00:40:48.190 align:middle line:90% submitted an essay. 00:40:48.190 --> 00:40:51.250 align:middle line:90% 00:40:51.250 --> 00:40:55.433 align:middle line:84% And if we think about our experience with how 00:40:55.433 --> 00:40:56.850 align:middle line:84% these things work, what is a guess 00:40:56.850 --> 00:40:58.488 align:middle line:84% about what the school strategy might 00:40:58.488 --> 00:41:00.530 align:middle line:84% be if we want to try to look for this equilibrium 00:41:00.530 --> 00:41:02.690 align:middle line:84% where all the students are writing the essay. 00:41:02.690 --> 00:41:04.750 align:middle line:84% What's your guess about what the school might do? 00:41:04.750 --> 00:41:07.682 align:middle line:90% 00:41:07.682 --> 00:41:09.390 align:middle line:84% If they see that you submitted the essay, 00:41:09.390 --> 00:41:11.650 align:middle line:90% what are they going to do? 00:41:11.650 --> 00:41:13.450 align:middle line:84% They're probably going to accept you. 00:41:13.450 --> 00:41:15.503 align:middle line:84% And if they see you didn't write the essay, 00:41:15.503 --> 00:41:16.670 align:middle line:90% they're going to reject you. 00:41:16.670 --> 00:41:19.468 align:middle line:90% Let's see how this works out. 00:41:19.468 --> 00:41:21.010 align:middle line:84% So first, we need to say, well, let's 00:41:21.010 --> 00:41:23.450 align:middle line:84% suppose they see that you've written the essay. 00:41:23.450 --> 00:41:26.267 align:middle line:84% What is the school believe about your passion for the position. 00:41:26.267 --> 00:41:28.350 align:middle line:84% Do they think you're passionate or not passionate? 00:41:28.350 --> 00:41:29.710 align:middle line:90% What probability do they assign? 00:41:29.710 --> 00:41:33.250 align:middle line:90% 00:41:33.250 --> 00:41:36.670 align:middle line:84% 1/2, 1/2, so technically, we're using Bayes' rule, 00:41:36.670 --> 00:41:39.977 align:middle line:84% but here it's pretty simple because the prior is 1/2, 1/2. 00:41:39.977 --> 00:41:41.310 align:middle line:90% 1/2 the students are passionate. 00:41:41.310 --> 00:41:42.590 align:middle line:84% 1/2 the students are not passionate. 00:41:42.590 --> 00:41:44.530 align:middle line:84% If all the students submit the essay, well, 00:41:44.530 --> 00:41:46.710 align:middle line:84% then the students who submit the essay are also 1/2 passionate, 00:41:46.710 --> 00:41:47.502 align:middle line:90% 1/2 not passionate. 00:41:47.502 --> 00:41:48.390 align:middle line:90% Nothing has changed. 00:41:48.390 --> 00:41:51.722 align:middle line:90% 00:41:51.722 --> 00:41:53.930 align:middle line:84% But what does the school believe if they see that you 00:41:53.930 --> 00:41:55.070 align:middle line:90% didn't submit the essay? 00:41:55.070 --> 00:41:57.177 align:middle line:90% 00:41:57.177 --> 00:41:58.510 align:middle line:90% I don't know, what do you think. 00:41:58.510 --> 00:41:59.468 align:middle line:90% This is a bit trickier. 00:41:59.468 --> 00:42:00.390 align:middle line:90% Should it be 1/2, 1/2? 00:42:00.390 --> 00:42:01.090 align:middle line:90% Why should it be 1/2, 1/2? 00:42:01.090 --> 00:42:02.070 align:middle line:90% What do you think? 00:42:02.070 --> 00:42:04.928 align:middle line:90% 00:42:04.928 --> 00:42:06.470 align:middle line:84% There's not one right answer to this. 00:42:06.470 --> 00:42:09.130 align:middle line:84% So I think it's unclear, but, Amy, you said 1/2, 1/2, why? 00:42:09.130 --> 00:42:13.010 align:middle line:84% AUDIENCE: I guess the idea is symmetric to the other one where 00:42:13.010 --> 00:42:14.250 align:middle line:90% everyone's-- 00:42:14.250 --> 00:42:17.370 align:middle line:90% no one [INAUDIBLE] essay, but-- 00:42:17.370 --> 00:42:18.250 align:middle line:90% wait. 00:42:18.250 --> 00:42:20.505 align:middle line:84% So if everyone submits essay [INAUDIBLE] 1/2, 1/2. 00:42:20.505 --> 00:42:23.130 align:middle line:84% And then that's like, you have a student that you don't receive 00:42:23.130 --> 00:42:27.970 align:middle line:84% an essay from would also be 1/2, 1/2 because, basically, 00:42:27.970 --> 00:42:29.470 align:middle line:84% the way that Bayes' rule plays out, 00:42:29.470 --> 00:42:33.143 align:middle line:84% they're all the same per person for other cases. 00:42:33.143 --> 00:42:34.810 align:middle line:84% IAN BALL: OK, so let's maybe write out-- 00:42:34.810 --> 00:42:36.410 align:middle line:84% let's get another thought, and then let's write out Bayes' rule 00:42:36.410 --> 00:42:37.493 align:middle line:90% and see what happens here. 00:42:37.493 --> 00:42:40.250 align:middle line:84% Were there any other thoughts on what we do here? 00:42:40.250 --> 00:42:40.890 align:middle line:90% Yeah. 00:42:40.890 --> 00:42:42.690 align:middle line:84% AUDIENCE: I mean, just like qualitatively, then you 00:42:42.690 --> 00:42:45.080 align:middle line:84% assume that there's a greater chance that they're not 00:42:45.080 --> 00:42:46.320 align:middle line:90% passionate about [INAUDIBLE]. 00:42:46.320 --> 00:42:47.820 align:middle line:84% IAN BALL: That also seems reasonable. 00:42:47.820 --> 00:42:49.987 align:middle line:84% I think there's a lot of reasonable answers to this, 00:42:49.987 --> 00:42:52.800 align:middle line:84% but let's see how Bayes' rule works. 00:42:52.800 --> 00:42:56.020 align:middle line:90% So to apply Bayes' rule-- 00:42:56.020 --> 00:43:00.312 align:middle line:90% 00:43:00.312 --> 00:43:02.682 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:43:02.682 --> 00:43:03.860 align:middle line:90% 00:43:03.860 --> 00:43:06.360 align:middle line:84% IAN BALL: Yeah, so I think we're going to have an issue when 00:43:06.360 --> 00:43:07.235 align:middle line:90% we apply Bayes' rule. 00:43:07.235 --> 00:43:09.560 align:middle line:90% So let's see what happens. 00:43:09.560 --> 00:43:14.783 align:middle line:84% So remember, just a quick review of Bayes' rule. 00:43:14.783 --> 00:43:16.200 align:middle line:84% Bayes' rule allows us to calculate 00:43:16.200 --> 00:43:17.540 align:middle line:90% conditional probabilities. 00:43:17.540 --> 00:43:21.100 align:middle line:84% So it allows us to say something like if we have two events, A 00:43:21.100 --> 00:43:24.680 align:middle line:84% and B, what is the probability that event A occurs 00:43:24.680 --> 00:43:27.332 align:middle line:90% conditional on B occurring. 00:43:27.332 --> 00:43:29.040 align:middle line:84% And the rule for Bayes' rule is that this 00:43:29.040 --> 00:43:34.600 align:middle line:84% is the probability of A and B occurring 00:43:34.600 --> 00:43:39.320 align:middle line:84% divided by the probability of B occurring. 00:43:39.320 --> 00:43:44.050 align:middle line:84% And this basic picture that we give, this is A, 00:43:44.050 --> 00:43:48.350 align:middle line:84% and this is B. Well, what's the probability of A occurring given 00:43:48.350 --> 00:43:48.850 align:middle line:90% B? 00:43:48.850 --> 00:43:50.390 align:middle line:84% Well, if we know that B occurred, 00:43:50.390 --> 00:43:53.310 align:middle line:84% we know we're somewhere in this circle. 00:43:53.310 --> 00:43:55.870 align:middle line:84% The probability that A occurs given B 00:43:55.870 --> 00:43:57.667 align:middle line:84% is saying what's given that we're 00:43:57.667 --> 00:43:59.750 align:middle line:84% in this circle B, what's the likelihood that we're 00:43:59.750 --> 00:44:01.230 align:middle line:90% in the shaded region. 00:44:01.230 --> 00:44:03.350 align:middle line:84% And that's exactly what's being pictured here, 00:44:03.350 --> 00:44:05.870 align:middle line:84% P of A intersect B is the probability 00:44:05.870 --> 00:44:07.830 align:middle line:84% that we're in this shaded region. 00:44:07.830 --> 00:44:10.070 align:middle line:84% And probability B is just the probability 00:44:10.070 --> 00:44:14.770 align:middle line:84% that we're in this outer circle here. 00:44:14.770 --> 00:44:17.513 align:middle line:84% So let's try to apply Bayes' rule 00:44:17.513 --> 00:44:19.430 align:middle line:84% when we see that the student writes the essay. 00:44:19.430 --> 00:44:27.750 align:middle line:84% So what we're interested in is, given essay, 00:44:27.750 --> 00:44:31.550 align:middle line:84% we want to say, what is the probability that the student is 00:44:31.550 --> 00:44:34.410 align:middle line:84% passionate, tP, given that they write the essay? 00:44:34.410 --> 00:44:39.030 align:middle line:90% 00:44:39.030 --> 00:44:41.997 align:middle line:84% And Bayes' rule tells us this is equal to the probability 00:44:41.997 --> 00:44:43.580 align:middle line:84% that they're passionate and they write 00:44:43.580 --> 00:44:45.517 align:middle line:84% the essay divided by the probability 00:44:45.517 --> 00:44:46.600 align:middle line:90% that they write the essay. 00:44:46.600 --> 00:45:00.420 align:middle line:90% 00:45:00.420 --> 00:45:02.710 align:middle line:84% Now, the probability that they write the essay 00:45:02.710 --> 00:45:03.960 align:middle line:90% is a little hard to calculate. 00:45:03.960 --> 00:45:06.620 align:middle line:90% Let's break it down. 00:45:06.620 --> 00:45:08.027 align:middle line:90% So write this one more time-- 00:45:08.027 --> 00:45:10.360 align:middle line:84% maybe I'll write it over here to give myself more space. 00:45:10.360 --> 00:45:12.540 align:middle line:84% This is the probability that they're passionate 00:45:12.540 --> 00:45:17.820 align:middle line:84% and they write the essay divided by-- well, 00:45:17.820 --> 00:45:19.660 align:middle line:84% there's two ways they could write the essay. 00:45:19.660 --> 00:45:21.780 align:middle line:84% They could be passionate and then write the essay, 00:45:21.780 --> 00:45:23.560 align:middle line:84% or they could be not passionate and write the essay. 00:45:23.560 --> 00:45:25.000 align:middle line:84% So let me break it down like that. 00:45:25.000 --> 00:45:30.685 align:middle line:84% We have the probability of tP and essay 00:45:30.685 --> 00:45:33.520 align:middle line:90% plus the probability of tN. 00:45:33.520 --> 00:45:46.200 align:middle line:90% 00:45:46.200 --> 00:45:48.420 align:middle line:84% So let's see if we can fill this in. 00:45:48.420 --> 00:45:50.420 align:middle line:84% What is the probability that they're passionate, 00:45:50.420 --> 00:45:51.900 align:middle line:90% and they write the essay? 00:45:51.900 --> 00:45:53.995 align:middle line:84% Well, they always write the essay. 00:45:53.995 --> 00:45:56.120 align:middle line:84% So this is just the probability they're passionate, 00:45:56.120 --> 00:45:56.745 align:middle line:90% and that's 1/2. 00:45:56.745 --> 00:46:00.420 align:middle line:90% 00:46:00.420 --> 00:46:05.180 align:middle line:84% And on the bottom, again, we have the 1/2 plus we're going 00:46:05.180 --> 00:46:07.690 align:middle line:84% to get a second 1/2 because the probability that they're not 00:46:07.690 --> 00:46:09.440 align:middle line:84% passionate and they write the essay, well, 00:46:09.440 --> 00:46:11.820 align:middle line:84% if they're not passionate, they always write the essay. 00:46:11.820 --> 00:46:14.320 align:middle line:84% So this is just the probability that they're not passionate, 00:46:14.320 --> 00:46:15.620 align:middle line:90% which is also 1/2. 00:46:15.620 --> 00:46:17.735 align:middle line:84% So we get 1/2 divided by 1/2 plus 1/2, 00:46:17.735 --> 00:46:18.860 align:middle line:90% and that's going to be 1/2. 00:46:18.860 --> 00:46:22.020 align:middle line:90% 00:46:22.020 --> 00:46:24.640 align:middle line:84% I think it's good to go over this carefully once. 00:46:24.640 --> 00:46:26.480 align:middle line:84% And then I think people get the hang of it. 00:46:26.480 --> 00:46:29.180 align:middle line:90% Any questions on this? 00:46:29.180 --> 00:46:30.280 align:middle line:90% Two questions, OK, great. 00:46:30.280 --> 00:46:31.080 align:middle line:90% Yeah. 00:46:31.080 --> 00:46:32.100 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:46:32.100 --> 00:46:34.780 align:middle line:84% IAN BALL: This is just confirming this. 00:46:34.780 --> 00:46:36.548 align:middle line:90% Yeah, yes. 00:46:36.548 --> 00:46:38.090 align:middle line:84% AUDIENCE: So then in the other case-- 00:46:38.090 --> 00:46:41.930 align:middle line:84% IAN BALL: So let's go to the other case, yeah, yeah. 00:46:41.930 --> 00:46:46.330 align:middle line:84% So maybe I'll erase our little review and go to the other case. 00:46:46.330 --> 00:46:49.142 align:middle line:84% So now we want to say, given that we see a student-- 00:46:49.142 --> 00:46:50.350 align:middle line:90% we didn't expect to see this. 00:46:50.350 --> 00:46:51.790 align:middle line:84% We thought everyone was supposed to write an essay. 00:46:51.790 --> 00:46:53.330 align:middle line:84% That's what they're supposed to do under their equilibrium 00:46:53.330 --> 00:46:54.150 align:middle line:90% strategy. 00:46:54.150 --> 00:46:56.490 align:middle line:84% But now we see that someone submits 00:46:56.490 --> 00:46:59.305 align:middle line:90% an application with no essay. 00:46:59.305 --> 00:47:00.930 align:middle line:84% And we say, well, what belief should we 00:47:00.930 --> 00:47:02.090 align:middle line:90% form about this student? 00:47:02.090 --> 00:47:04.123 align:middle line:90% Well, let's apply Bayes' rule. 00:47:04.123 --> 00:47:06.290 align:middle line:84% We say, what's the probability that they're actually 00:47:06.290 --> 00:47:09.230 align:middle line:84% passionate, that I want to accept them, given no essay? 00:47:09.230 --> 00:47:13.490 align:middle line:90% 00:47:13.490 --> 00:47:16.690 align:middle line:84% And that's exactly equal to the probability 00:47:16.690 --> 00:47:25.016 align:middle line:84% that they're passionate and no essay 00:47:25.016 --> 00:47:27.190 align:middle line:84% divided by the probability of no essay. 00:47:27.190 --> 00:47:32.010 align:middle line:90% 00:47:32.010 --> 00:47:32.950 align:middle line:90% And what is this? 00:47:32.950 --> 00:47:38.000 align:middle line:90% 00:47:38.000 --> 00:47:38.840 align:middle line:90% Yeah. 00:47:38.840 --> 00:47:39.740 align:middle line:90% AUDIENCE: 0 over 0. 00:47:39.740 --> 00:47:41.200 align:middle line:90% IAN BALL: This is 0 over 0. 00:47:41.200 --> 00:47:44.320 align:middle line:90% 00:47:44.320 --> 00:47:48.000 align:middle line:84% Because I'm never-- no one is writing no essay. 00:47:48.000 --> 00:47:49.208 align:middle line:90% Everyone is writing an essay. 00:47:49.208 --> 00:47:50.917 align:middle line:84% So the probability that you're passionate 00:47:50.917 --> 00:47:53.740 align:middle line:84% and you don't write an essay is 0 because if you're passionate, 00:47:53.740 --> 00:47:54.982 align:middle line:90% you always write an essay. 00:47:54.982 --> 00:47:56.940 align:middle line:84% The probability that you don't submit an essay, 00:47:56.940 --> 00:47:59.440 align:middle line:84% well, if you're passionate, you always submit an essay. 00:47:59.440 --> 00:48:01.648 align:middle line:84% If you're not passionate, you always submit an essay. 00:48:01.648 --> 00:48:04.240 align:middle line:90% So we get a 0 here as well. 00:48:04.240 --> 00:48:06.600 align:middle line:84% So what we see is that there's a fundamental problem 00:48:06.600 --> 00:48:08.800 align:middle line:84% with Bayes' rule, that Bayes' rule only 00:48:08.800 --> 00:48:11.160 align:middle line:84% allows us to define conditional probabilities 00:48:11.160 --> 00:48:14.200 align:middle line:84% conditional on events that have positive probability. 00:48:14.200 --> 00:48:16.240 align:middle line:84% When we try to define conditional probabilities 00:48:16.240 --> 00:48:18.200 align:middle line:84% conditioned on 0 probability events, 00:48:18.200 --> 00:48:20.960 align:middle line:84% Bayes' rule doesn't really apply, and we run into problems. 00:48:20.960 --> 00:48:23.340 align:middle line:90% So I'll say not well defined. 00:48:23.340 --> 00:48:28.040 align:middle line:90% 00:48:28.040 --> 00:48:30.800 align:middle line:84% And I think this is exactly why we got a few different answers 00:48:30.800 --> 00:48:31.400 align:middle line:90% here. 00:48:31.400 --> 00:48:33.690 align:middle line:84% What should we conclude when someone doesn't write an essay? 00:48:33.690 --> 00:48:35.470 align:middle line:84% I think there's a few reasonable choices. 00:48:35.470 --> 00:48:38.130 align:middle line:84% You might say, well, this isn't supposed to happen. 00:48:38.130 --> 00:48:40.990 align:middle line:84% So let me just go back to my prior and say it's 1/2, 1/2. 00:48:40.990 --> 00:48:42.710 align:middle line:84% You could say, well, the fact that they 00:48:42.710 --> 00:48:44.350 align:middle line:84% don't want to write an essay, that 00:48:44.350 --> 00:48:46.250 align:middle line:84% must mean they're not really passionate about the program. 00:48:46.250 --> 00:48:46.790 align:middle line:90% I'm going to assume-- 00:48:46.790 --> 00:48:48.290 align:middle line:84% I'm going to say that they're not passionate. 00:48:48.290 --> 00:48:50.582 align:middle line:84% Or maybe you think they are passionate, and they just-- 00:48:50.582 --> 00:48:52.810 align:middle line:84% I don't know, you could tell a lot of stories. 00:48:52.810 --> 00:48:57.630 align:middle line:84% But basically, we don't really pin down what the belief is. 00:48:57.630 --> 00:49:01.050 align:middle line:84% But now this creates a problem because now the question is-- 00:49:01.050 --> 00:49:04.310 align:middle line:90% 00:49:04.310 --> 00:49:07.130 align:middle line:84% so this equilibrium seems to fit. 00:49:07.130 --> 00:49:08.750 align:middle line:90% Let's see if this makes sense. 00:49:08.750 --> 00:49:11.910 align:middle line:84% If I'm the student, if I believe that the school is going 00:49:11.910 --> 00:49:14.870 align:middle line:84% to accept me if I write an essay and not accept me 00:49:14.870 --> 00:49:17.870 align:middle line:84% if I don't write the essay, then it's optimal for me 00:49:17.870 --> 00:49:20.030 align:middle line:90% to write the essay. 00:49:20.030 --> 00:49:21.500 align:middle line:90% That makes sense. 00:49:21.500 --> 00:49:23.750 align:middle line:84% And given that all the students are writing the essay, 00:49:23.750 --> 00:49:26.310 align:middle line:84% well, the school is happy to accept you 00:49:26.310 --> 00:49:29.030 align:middle line:84% because the school gets a payoff of 3 00:49:29.030 --> 00:49:31.850 align:middle line:84% from accepting a passionate student, 00:49:31.850 --> 00:49:34.190 align:middle line:84% negative 2 from accepting a not passionate student, 00:49:34.190 --> 00:49:37.110 align:middle line:84% And the average of 3 and negative 2 is strictly positive. 00:49:37.110 --> 00:49:38.530 align:middle line:90% So everything looks good here. 00:49:38.530 --> 00:49:41.530 align:middle line:84% The only question is, is this a credible threat? 00:49:41.530 --> 00:49:49.643 align:middle line:90% 00:49:49.643 --> 00:49:50.810 align:middle line:90% But there's a question here. 00:49:50.810 --> 00:49:51.550 align:middle line:90% Yeah. 00:49:51.550 --> 00:49:53.070 align:middle line:90% AUDIENCE: I was just curious. 00:49:53.070 --> 00:49:56.050 align:middle line:84% If you were to get someone who submits an essay, 00:49:56.050 --> 00:49:59.150 align:middle line:84% could that also imply that maybe there's a third type of student 00:49:59.150 --> 00:50:00.360 align:middle line:90% that you don't know of? 00:50:00.360 --> 00:50:01.610 align:middle line:90% IAN BALL: That's a good point. 00:50:01.610 --> 00:50:03.777 align:middle line:84% So maybe you want to revise your model of the world. 00:50:03.777 --> 00:50:06.430 align:middle line:84% And maybe our types, we've only represented types 00:50:06.430 --> 00:50:08.370 align:middle line:84% by how excited they are about the program, 00:50:08.370 --> 00:50:09.953 align:middle line:84% but maybe there's a laziness component 00:50:09.953 --> 00:50:11.450 align:middle line:90% that we also need to model. 00:50:11.450 --> 00:50:13.590 align:middle line:84% Or-- and we'll talk about this later-- 00:50:13.590 --> 00:50:17.070 align:middle line:84% it could be that this cost c varies in the population. 00:50:17.070 --> 00:50:18.490 align:middle line:84% Again, talking about essay costs, 00:50:18.490 --> 00:50:20.290 align:middle line:84% I feel like it's 0 for everyone in the world of GPT, 00:50:20.290 --> 00:50:22.053 align:middle line:84% but in the old days, there was a cost. 00:50:22.053 --> 00:50:24.470 align:middle line:84% And some people were better at writing essays and the cost 00:50:24.470 --> 00:50:27.012 align:middle line:84% for them was lower, and we're going to talk about that later. 00:50:27.012 --> 00:50:28.430 align:middle line:90% Yeah, exactly. 00:50:28.430 --> 00:50:30.540 align:middle line:84% So I guess the basic question here 00:50:30.540 --> 00:50:32.640 align:middle line:90% is, is this a credible threat? 00:50:32.640 --> 00:50:34.940 align:middle line:84% Is it credible for the school to say, 00:50:34.940 --> 00:50:37.837 align:middle line:84% as they do in this equilibrium, if you don't submit an essay, 00:50:37.837 --> 00:50:38.920 align:middle line:90% we're going to reject you? 00:50:38.920 --> 00:50:41.840 align:middle line:90% 00:50:41.840 --> 00:50:42.340 align:middle line:90% Yeah. 00:50:42.340 --> 00:50:43.460 align:middle line:84% AUDIENCE: Well, isn't it kind of murky 00:50:43.460 --> 00:50:45.160 align:middle line:84% because, as you were just talking about, 00:50:45.160 --> 00:50:46.880 align:middle line:84% there are some ways you could just say, 00:50:46.880 --> 00:50:48.530 align:middle line:84% if you don't write an essay that you're not passionate, 00:50:48.530 --> 00:50:49.600 align:middle line:90% so we don't want you. 00:50:49.600 --> 00:50:51.700 align:middle line:84% But what if the school has a really different way 00:50:51.700 --> 00:50:53.280 align:middle line:84% of thinking like, oh, he didn't write the essay. 00:50:53.280 --> 00:50:55.947 align:middle line:84% That's because you were spending more time studying for an exam. 00:50:55.947 --> 00:50:58.040 align:middle line:84% So then we're actually more excited, 00:50:58.040 --> 00:50:59.420 align:middle line:84% so I think because you don't know 00:50:59.420 --> 00:51:02.900 align:middle line:84% exactly what the school will conclude, the threat is-- 00:51:02.900 --> 00:51:04.100 align:middle line:90% it's not necessarily-- 00:51:04.100 --> 00:51:05.000 align:middle line:84% IAN BALL: I think that's a great answer. 00:51:05.000 --> 00:51:06.000 align:middle line:90% So it's definitely murky. 00:51:06.000 --> 00:51:06.940 align:middle line:84% And that's exactly what I'm trying 00:51:06.940 --> 00:51:08.880 align:middle line:84% to convey with this example, that it is murky. 00:51:08.880 --> 00:51:11.020 align:middle line:84% But I think the key point, which you hinted at, 00:51:11.020 --> 00:51:15.060 align:middle line:84% was whether this threat is credible depends 00:51:15.060 --> 00:51:18.500 align:middle line:84% on what the school believes when they don't see an essay. 00:51:18.500 --> 00:51:19.900 align:middle line:90% So is this threat credible? 00:51:19.900 --> 00:51:21.560 align:middle line:90% Well, it depends on the belief. 00:51:21.560 --> 00:51:27.970 align:middle line:90% 00:51:27.970 --> 00:51:29.530 align:middle line:84% If the school believes that everyone 00:51:29.530 --> 00:51:31.645 align:middle line:84% who doesn't submit an essay is passionate, 00:51:31.645 --> 00:51:33.270 align:middle line:84% then it's not credible for them to say, 00:51:33.270 --> 00:51:35.687 align:middle line:84% we're going to reject those students because they'd rather 00:51:35.687 --> 00:51:37.370 align:middle line:84% accept passionate students than not. 00:51:37.370 --> 00:51:39.770 align:middle line:84% But if their belief, after not seeing an essay, 00:51:39.770 --> 00:51:42.725 align:middle line:84% is that the student is not passionate, 00:51:42.725 --> 00:51:44.850 align:middle line:84% then it is credible for them to reject that student 00:51:44.850 --> 00:51:46.750 align:middle line:84% because that's exactly what's optimal for them, 00:51:46.750 --> 00:51:49.370 align:middle line:84% if they see that a student-- if they believe that a student is 00:51:49.370 --> 00:51:51.210 align:middle line:90% not passionate. 00:51:51.210 --> 00:51:52.770 align:middle line:90% So how do we get around this? 00:51:52.770 --> 00:51:55.530 align:middle line:84% Well, beliefs aren't well defined, 00:51:55.530 --> 00:51:58.410 align:middle line:84% but whether something is credible depends on the belief. 00:51:58.410 --> 00:52:01.610 align:middle line:84% The way we get at this is our equilibrium notion 00:52:01.610 --> 00:52:04.010 align:middle line:84% is going to need to explicitly specify 00:52:04.010 --> 00:52:09.210 align:middle line:84% what beliefs the school forms, even at this information set. 00:52:09.210 --> 00:52:12.630 align:middle line:84% So as a result, beliefs must be part of the equilibrium. 00:52:12.630 --> 00:52:22.690 align:middle line:90% 00:52:22.690 --> 00:52:23.890 align:middle line:90% Why? 00:52:23.890 --> 00:52:27.730 align:middle line:84% Well, specifying strategies alone 00:52:27.730 --> 00:52:29.792 align:middle line:84% is not enough to tell us what people believe. 00:52:29.792 --> 00:52:31.250 align:middle line:84% But our whole equilibrium notion is 00:52:31.250 --> 00:52:33.370 align:middle line:84% supposed to capture whether these threats are credible. 00:52:33.370 --> 00:52:35.703 align:middle line:84% And therefore, we're going to need to explicitly specify 00:52:35.703 --> 00:52:37.210 align:middle line:90% what the belief is. 00:52:37.210 --> 00:52:40.650 align:middle line:84% And what we'll see here is that, here, there's 00:52:40.650 --> 00:52:43.350 align:middle line:84% no flexibility in what belief we specify here. 00:52:43.350 --> 00:52:45.670 align:middle line:84% In this case, the belief is pinned down by Bayes' rule. 00:52:45.670 --> 00:52:48.290 align:middle line:84% Bayes' rule tells us what the belief has to be. 00:52:48.290 --> 00:52:51.150 align:middle line:84% Here, we can choose whatever belief we want. 00:52:51.150 --> 00:52:53.970 align:middle line:84% There's no further restriction imposed by Bayes' rule 00:52:53.970 --> 00:52:56.010 align:middle line:84% on what the school's belief is here. 00:52:56.010 --> 00:52:58.090 align:middle line:84% Now, I said before, there's all this big debate 00:52:58.090 --> 00:53:00.173 align:middle line:84% in the literature about how to define it. 00:53:00.173 --> 00:53:02.090 align:middle line:84% People came along and made all these arguments 00:53:02.090 --> 00:53:04.702 align:middle line:84% about what belief should be at this kind of information set. 00:53:04.702 --> 00:53:05.910 align:middle line:90% What should you believe here? 00:53:05.910 --> 00:53:08.035 align:middle line:84% For now, we're not going to impose any restrictions 00:53:08.035 --> 00:53:09.210 align:middle line:90% on what those beliefs are. 00:53:09.210 --> 00:53:14.290 align:middle line:84% And now I'm going to go into the formal definition. 00:53:14.290 --> 00:53:15.910 align:middle line:90% Let's go back over here. 00:53:15.910 --> 00:53:30.560 align:middle line:90% 00:53:30.560 --> 00:53:38.020 align:middle line:84% Let's define perfect Bayesian equilibrium. 00:53:38.020 --> 00:53:43.280 align:middle line:90% 00:53:43.280 --> 00:53:45.080 align:middle line:84% So the first key point is that one thing 00:53:45.080 --> 00:53:47.020 align:middle line:84% I've harped on throughout the semester is no longer true. 00:53:47.020 --> 00:53:48.603 align:middle line:84% So throughout the semester, I've said, 00:53:48.603 --> 00:53:50.180 align:middle line:84% an equilibrium is a strategy profile. 00:53:50.180 --> 00:53:51.960 align:middle line:84% And I said this over and over again. 00:53:51.960 --> 00:53:54.320 align:middle line:84% Now, that's no longer true because equilibrium 00:53:54.320 --> 00:53:56.937 align:middle line:84% is going to be a strategy profile together with beliefs. 00:53:56.937 --> 00:53:59.520 align:middle line:84% So we're going to have to have this new object, which is going 00:53:59.520 --> 00:54:01.020 align:middle line:90% to be called an assessment. 00:54:01.020 --> 00:54:03.497 align:middle line:90% 00:54:03.497 --> 00:54:05.080 align:middle line:84% I think this is a silly word, but this 00:54:05.080 --> 00:54:06.400 align:middle line:90% is the word that's used. 00:54:06.400 --> 00:54:08.520 align:middle line:90% And that specifies two things. 00:54:08.520 --> 00:54:10.540 align:middle line:90% It specifies a strategy profile. 00:54:10.540 --> 00:54:14.880 align:middle line:90% 00:54:14.880 --> 00:54:18.020 align:middle line:84% And also beliefs, sometimes it's called a belief system. 00:54:18.020 --> 00:54:25.310 align:middle line:90% 00:54:25.310 --> 00:54:28.410 align:middle line:84% So strategy profile, we already know what that is. 00:54:28.410 --> 00:54:30.210 align:middle line:84% It specifies a strategy for each player. 00:54:30.210 --> 00:54:32.910 align:middle line:84% And a strategy in this game and in any game 00:54:32.910 --> 00:54:35.810 align:middle line:84% is a mapping from information sets to actions. 00:54:35.810 --> 00:54:37.310 align:middle line:84% It's a complete contingent plan that 00:54:37.310 --> 00:54:39.560 align:middle line:84% says that my information set and any of my information 00:54:39.560 --> 00:54:41.150 align:middle line:90% sets, what am I going to do? 00:54:41.150 --> 00:54:45.270 align:middle line:84% So in this game, it's exactly what we wrote down over here. 00:54:45.270 --> 00:54:49.310 align:middle line:84% The student has two information sets tP and tN. 00:54:49.310 --> 00:54:52.430 align:middle line:84% So a strategy for the student specifies what they do at tP 00:54:52.430 --> 00:54:54.190 align:middle line:90% and what they do at tN. 00:54:54.190 --> 00:54:56.327 align:middle line:84% Similarly, the school has two information sets. 00:54:56.327 --> 00:54:57.910 align:middle line:84% The information set where they observe 00:54:57.910 --> 00:54:59.827 align:middle line:84% the student wrote an essay and the information 00:54:59.827 --> 00:55:02.243 align:middle line:84% set where they observed the student didn't write an essay, 00:55:02.243 --> 00:55:03.910 align:middle line:84% and a strategy specifies what action 00:55:03.910 --> 00:55:08.030 align:middle line:84% they take at each of those informations. 00:55:08.030 --> 00:55:09.310 align:middle line:90% So I said that in words. 00:55:09.310 --> 00:55:13.550 align:middle line:84% Now, a belief system is going to specify-- 00:55:13.550 --> 00:55:22.172 align:middle line:90% 00:55:22.172 --> 00:55:34.980 align:middle line:84% specifies a belief at every non-singleton information set. 00:55:34.980 --> 00:55:37.340 align:middle line:84% So a lot of the information sets are singletons. 00:55:37.340 --> 00:55:38.960 align:middle line:84% I don't need to specify a belief here. 00:55:38.960 --> 00:55:41.820 align:middle line:84% I only need to specify a belief at an information set that 00:55:41.820 --> 00:55:43.700 align:middle line:90% contains multiple nodes. 00:55:43.700 --> 00:55:44.918 align:middle line:90% And what is a belief? 00:55:44.918 --> 00:55:46.460 align:middle line:84% A belief is going to be a probability 00:55:46.460 --> 00:55:49.700 align:middle line:84% distribution over the nodes in that information set. 00:55:49.700 --> 00:55:51.220 align:middle line:84% And that's exactly what we did here. 00:55:51.220 --> 00:55:55.540 align:middle line:84% We said if the student submitted an essay, 00:55:55.540 --> 00:55:57.520 align:middle line:84% we had two nodes in that information set. 00:55:57.520 --> 00:55:59.100 align:middle line:84% And we had to form belief about what 00:55:59.100 --> 00:56:01.380 align:middle line:84% was the relative likelihood of those two nodes. 00:56:01.380 --> 00:56:03.860 align:middle line:84% And similarly, at this information set, no essay-- 00:56:03.860 --> 00:56:05.080 align:middle line:90% or maybe I got it backwards. 00:56:05.080 --> 00:56:06.820 align:middle line:84% At the essay information set, we have 00:56:06.820 --> 00:56:09.140 align:middle line:84% to form a belief about the two nodes 00:56:09.140 --> 00:56:10.640 align:middle line:90% within that information set. 00:56:10.640 --> 00:56:13.000 align:middle line:84% So a belief at every non-singleton information set. 00:56:13.000 --> 00:56:14.460 align:middle line:90% What is a belief? 00:56:14.460 --> 00:56:28.000 align:middle line:84% i.e., a probability distribution over-- well, 00:56:28.000 --> 00:56:29.125 align:middle line:90% it can't be over all nodes. 00:56:29.125 --> 00:56:31.000 align:middle line:84% I certainly can't assign positive probability 00:56:31.000 --> 00:56:32.920 align:middle line:84% to nodes that are not in that information set. 00:56:32.920 --> 00:56:34.660 align:middle line:84% It's only a distribution over the nodes that are 00:56:34.660 --> 00:56:35.960 align:middle line:90% within that information set. 00:56:35.960 --> 00:56:37.377 align:middle line:84% So it's a probability distribution 00:56:37.377 --> 00:56:40.640 align:middle line:84% over nodes in the information set. 00:56:40.640 --> 00:56:50.700 align:middle line:90% 00:56:50.700 --> 00:56:52.160 align:middle line:84% So now we've defined an assessment. 00:56:52.160 --> 00:56:55.420 align:middle line:84% Let's formally define a perfect Bayesian equilibrium. 00:56:55.420 --> 00:57:04.460 align:middle line:84% So definition, an assessment is a PBE 00:57:04.460 --> 00:57:06.400 align:middle line:90% if it satisfies two conditions. 00:57:06.400 --> 00:57:11.180 align:middle line:90% 00:57:11.180 --> 00:57:14.120 align:middle line:84% The first condition is sequential rationality. 00:57:14.120 --> 00:57:21.010 align:middle line:90% 00:57:21.010 --> 00:57:22.950 align:middle line:84% And the second is belief consistency. 00:57:22.950 --> 00:57:31.850 align:middle line:90% 00:57:31.850 --> 00:57:33.350 align:middle line:84% Let's start with belief consistency. 00:57:33.350 --> 00:57:34.810 align:middle line:84% This is what we're getting at here. 00:57:34.810 --> 00:57:39.210 align:middle line:84% We want to say that the school's belief here must be 1/2, 1/2. 00:57:39.210 --> 00:57:42.050 align:middle line:84% But the school's belief here is totally flexible. 00:57:42.050 --> 00:57:44.830 align:middle line:84% So what belief consistency says is, 00:57:44.830 --> 00:57:46.690 align:middle line:84% well, the beliefs are consistent. 00:57:46.690 --> 00:57:51.070 align:middle line:84% The beliefs are consistent with Bayes' rule. 00:57:51.070 --> 00:57:58.770 align:middle line:90% 00:57:58.770 --> 00:58:02.030 align:middle line:84% Well, they can't be consistent with Bayes' rule everywhere. 00:58:02.030 --> 00:58:05.150 align:middle line:84% Sometimes we use this vague phrase wherever possible. 00:58:05.150 --> 00:58:12.890 align:middle line:90% 00:58:12.890 --> 00:58:15.560 align:middle line:84% There's often different ways of defining exactly what wherever 00:58:15.560 --> 00:58:16.460 align:middle line:90% possible means. 00:58:16.460 --> 00:58:19.520 align:middle line:84% What I mean by wherever possible is at any information set that's 00:58:19.520 --> 00:58:25.340 align:middle line:84% reached with positive probability, so i.e. 00:58:25.340 --> 00:58:33.560 align:middle line:90% 00:58:33.560 --> 00:58:35.640 align:middle line:84% Now, let's be careful-- the probability 00:58:35.640 --> 00:58:39.220 align:middle line:84% that we reach an information set depends on the strategy profile. 00:58:39.220 --> 00:58:42.760 align:middle line:84% So what we're saying is we have an assessment, looking 00:58:42.760 --> 00:58:45.080 align:middle line:84% at the strategy profile that's part of that assessment, 00:58:45.080 --> 00:58:46.540 align:middle line:90% I can go every information set. 00:58:46.540 --> 00:58:48.160 align:middle line:84% And I can compute the probability 00:58:48.160 --> 00:58:49.785 align:middle line:84% that we actually reach that information 00:58:49.785 --> 00:58:51.360 align:middle line:90% set under the strategy profile. 00:58:51.360 --> 00:58:53.503 align:middle line:84% And I can split those information sets in two. 00:58:53.503 --> 00:58:54.920 align:middle line:84% I can say there's some information 00:58:54.920 --> 00:58:56.860 align:middle line:84% sets that are reached with positive probability. 00:58:56.860 --> 00:58:58.640 align:middle line:84% And there are some that are never reached. 00:58:58.640 --> 00:59:00.820 align:middle line:84% And the ones that are reached with positive probability, 00:59:00.820 --> 00:59:02.520 align:middle line:84% the beliefs that those information sets 00:59:02.520 --> 00:59:04.425 align:middle line:90% must satisfy Bayes' rule. 00:59:04.425 --> 00:59:05.800 align:middle line:84% But at the other information sets 00:59:05.800 --> 00:59:08.040 align:middle line:84% that are not reached with positive probability, 00:59:08.040 --> 00:59:11.880 align:middle line:84% I have total flexibility over those beliefs. 00:59:11.880 --> 00:59:14.390 align:middle line:84% There's no restriction on the beliefs there. 00:59:14.390 --> 00:59:17.710 align:middle line:90% So now sequential rationality. 00:59:17.710 --> 00:59:18.750 align:middle line:90% We're close. 00:59:18.750 --> 00:59:22.430 align:middle line:84% Well, the whole point of introducing beliefs 00:59:22.430 --> 00:59:25.630 align:middle line:84% was to determine whether these threats were credible. 00:59:25.630 --> 00:59:28.810 align:middle line:84% And what credible meant, precisely, was optimal. 00:59:28.810 --> 00:59:32.070 align:middle line:84% Optimal given my beliefs and given the strategy of others. 00:59:32.070 --> 00:59:39.710 align:middle line:84% So sequential rationality means, at every information set, 00:59:39.710 --> 00:59:46.910 align:middle line:84% the action under the strategy profile is optimal, 00:59:46.910 --> 00:59:56.970 align:middle line:84% given beliefs and the rest of the strategy profile. 00:59:56.970 --> 01:00:00.470 align:middle line:90% 01:00:00.470 --> 01:00:02.970 align:middle line:90% And let me say the rest. 01:00:02.970 --> 01:00:12.710 align:middle line:90% 01:00:12.710 --> 01:00:14.870 align:middle line:90% Put this in quotes. 01:00:14.870 --> 01:00:16.750 align:middle line:90% So what do I mean here? 01:00:16.750 --> 01:00:18.950 align:middle line:84% Whenever we get to an information set, 01:00:18.950 --> 01:00:21.030 align:middle line:84% a strategy is going to specify what 01:00:21.030 --> 01:00:25.023 align:middle line:84% action the player is supposed to take at that information set. 01:00:25.023 --> 01:00:27.190 align:middle line:84% And that action that they're supposed to take better 01:00:27.190 --> 01:00:28.142 align:middle line:90% be optimal. 01:00:28.142 --> 01:00:29.850 align:middle line:84% Well, how do we define that it's optimal? 01:00:29.850 --> 01:00:33.230 align:middle line:84% Well, we have to check that it's optimal given the beliefs that 01:00:33.230 --> 01:00:37.430 align:middle line:84% the player has there and how everyone else is going to play, 01:00:37.430 --> 01:00:39.990 align:middle line:84% which is the "rest of the strategy profile." 01:00:39.990 --> 01:00:41.830 align:middle line:84% So what's optimal for me at this node 01:00:41.830 --> 01:00:44.390 align:middle line:84% depends on-- or at this information set 01:00:44.390 --> 01:00:46.030 align:middle line:84% depends on my beliefs over which node 01:00:46.030 --> 01:00:49.150 align:middle line:84% I'm actually at and also, how people are going to subsequently 01:00:49.150 --> 01:00:50.190 align:middle line:90% play in the game. 01:00:50.190 --> 01:00:53.190 align:middle line:84% And the way they subsequently play is pinned down by the "rest 01:00:53.190 --> 01:00:54.548 align:middle line:90% of the strategy profile." 01:00:54.548 --> 01:00:57.090 align:middle line:84% I could write all this in math, and that's what the notes do, 01:00:57.090 --> 01:00:58.070 align:middle line:84% but I think that maybe just makes 01:00:58.070 --> 01:00:59.210 align:middle line:90% things a little more confusing. 01:00:59.210 --> 01:01:00.668 align:middle line:84% So I'm trying to write it in words, 01:01:00.668 --> 01:01:02.610 align:middle line:84% but if anything's unclear, let me know. 01:01:02.610 --> 01:01:06.830 align:middle line:90% 01:01:06.830 --> 01:01:09.238 align:middle line:90% Questions? 01:01:09.238 --> 01:01:10.780 align:middle line:84% So let's go back to this game and see 01:01:10.780 --> 01:01:17.500 align:middle line:90% if we can find some more PBE. 01:01:17.500 --> 01:01:20.320 align:middle line:90% So let's look here. 01:01:20.320 --> 01:01:36.620 align:middle line:90% 01:01:36.620 --> 01:01:38.700 align:middle line:84% So let's try to finish off our construction 01:01:38.700 --> 01:01:40.620 align:middle line:84% of a perfect Bayesian equilibrium over here. 01:01:40.620 --> 01:01:43.740 align:middle line:84% We already have a strategy for the student. 01:01:43.740 --> 01:01:45.900 align:middle line:84% We already have a strategy for the school. 01:01:45.900 --> 01:01:48.820 align:middle line:84% The next step is we have to specify beliefs. 01:01:48.820 --> 01:01:51.140 align:middle line:84% We've already specified the 1/2, 1/2 belief. 01:01:51.140 --> 01:01:55.420 align:middle line:84% But we have flexibility over the p1 minus p belief. 01:01:55.420 --> 01:01:58.300 align:middle line:84% So let's emphasize here that this is 01:01:58.300 --> 01:01:59.920 align:middle line:90% unconstrained by Bayes' rule. 01:01:59.920 --> 01:02:08.090 align:middle line:90% 01:02:08.090 --> 01:02:10.730 align:middle line:84% Now one confusion I often see on problem sets or exams 01:02:10.730 --> 01:02:13.090 align:middle line:84% is people say, oh, here's my equilibrium. 01:02:13.090 --> 01:02:16.370 align:middle line:84% You can take p whatever you want. 01:02:16.370 --> 01:02:18.890 align:middle line:84% Just because it's unconstrained by Bayes' rule, 01:02:18.890 --> 01:02:20.850 align:middle line:84% to write down an equilibrium, we still 01:02:20.850 --> 01:02:22.730 align:middle line:84% have to make a choice about what this is. 01:02:22.730 --> 01:02:24.710 align:middle line:84% We have a lot of flexibility in what that choice is, 01:02:24.710 --> 01:02:26.585 align:middle line:84% but you still have to make a concrete choice. 01:02:26.585 --> 01:02:28.930 align:middle line:84% You can't say the equilibrium is whatever p you want. 01:02:28.930 --> 01:02:30.685 align:middle line:84% You have to choose a particular P. 01:02:30.685 --> 01:02:32.310 align:middle line:84% And that's what we're going to do here. 01:02:32.310 --> 01:02:35.010 align:middle line:84% So let's see what p we can choose. 01:02:35.010 --> 01:02:38.050 align:middle line:84% So this p is unconstrained by Bayes' rule, 01:02:38.050 --> 01:02:40.130 align:middle line:84% but it is constrained by something else. 01:02:40.130 --> 01:02:42.350 align:middle line:84% It is constrained by Sequential Rationality. 01:02:42.350 --> 01:02:48.930 align:middle line:90% 01:02:48.930 --> 01:02:49.810 align:middle line:90% Say Sr. 01:02:49.810 --> 01:02:51.050 align:middle line:90% Why? 01:02:51.050 --> 01:02:53.830 align:middle line:84% Why am I not allowed to choose whatever p I want here? 01:02:53.830 --> 01:02:56.170 align:middle line:84% So I want to make this construct here 01:02:56.170 --> 01:02:58.048 align:middle line:90% a perfect Bayesian equilibrium. 01:02:58.048 --> 01:03:00.590 align:middle line:84% But you might think, oh, I can just choose whatever P I want, 01:03:00.590 --> 01:03:01.870 align:middle line:90% but I think there's an issue. 01:03:01.870 --> 01:03:02.730 align:middle line:90% Yeah. 01:03:02.730 --> 01:03:05.210 align:middle line:84% AUDIENCE: I mean, if you were to accept-- 01:03:05.210 --> 01:03:10.160 align:middle line:84% or to assume that it was more passionate and not passionate 01:03:10.160 --> 01:03:15.200 align:middle line:84% [INAUDIBLE] since you would be rejecting more [INAUDIBLE] 01:03:15.200 --> 01:03:17.560 align:middle line:84% likely would've been a higher utility [INAUDIBLE]. 01:03:17.560 --> 01:03:18.560 align:middle line:90% IAN BALL: Exactly right. 01:03:18.560 --> 01:03:21.600 align:middle line:84% So let's check whether rejecting is sequentially rational given 01:03:21.600 --> 01:03:23.200 align:middle line:90% this belief. 01:03:23.200 --> 01:03:27.400 align:middle line:84% So my payoff as the school is always 0 if I reject. 01:03:27.400 --> 01:03:34.060 align:middle line:84% So my payoff from rejecting is 0. 01:03:34.060 --> 01:03:36.760 align:middle line:90% 01:03:36.760 --> 01:03:40.170 align:middle line:84% What's my expected payoff from accepting given this belief? 01:03:40.170 --> 01:03:42.420 align:middle line:84% My expected payoff of accepting depends on the belief. 01:03:42.420 --> 01:03:43.800 align:middle line:90% So let's write it down. 01:03:43.800 --> 01:03:46.800 align:middle line:84% So we have-- well, p is the probability 01:03:46.800 --> 01:03:49.440 align:middle line:90% that the student is passionate. 01:03:49.440 --> 01:03:51.240 align:middle line:84% And if they're passionate, I think 01:03:51.240 --> 01:03:54.240 align:middle line:90% we said I get a payoff of 3. 01:03:54.240 --> 01:03:57.080 align:middle line:90% 01:03:57.080 --> 01:04:03.520 align:middle line:84% And if they're not passionate, I get a payoff of negative 2. 01:04:03.520 --> 01:04:06.880 align:middle line:84% So in order for rejecting to be sequentially 01:04:06.880 --> 01:04:08.960 align:middle line:84% rational at this node, it must be 01:04:08.960 --> 01:04:10.640 align:middle line:84% that the expected payoff from rejecting 01:04:10.640 --> 01:04:14.560 align:middle line:84% is weakly higher than the expected payoff from accepting. 01:04:14.560 --> 01:04:18.280 align:middle line:84% So I need 0 to be greater than or equal to this. 01:04:18.280 --> 01:04:22.520 align:middle line:84% And now let's simplify this a little bit. 01:04:22.520 --> 01:04:34.400 align:middle line:84% We get 3p, and then we get 5p minus 2. 01:04:34.400 --> 01:04:35.440 align:middle line:90% That's right. 01:04:35.440 --> 01:04:38.240 align:middle line:84% So we need 0 greater than or equal to 5p minus 2. 01:04:38.240 --> 01:04:42.480 align:middle line:84% Let's put the 2 over here and then divide by 5. 01:04:42.480 --> 01:04:48.260 align:middle line:84% So we must get p less than or equal to 2/5. 01:04:48.260 --> 01:04:50.880 align:middle line:90% 01:04:50.880 --> 01:04:52.840 align:middle line:84% So we've actually constructed a family 01:04:52.840 --> 01:04:54.520 align:middle line:90% of perfect Bayesian equilibrium. 01:04:54.520 --> 01:04:57.360 align:middle line:84% For any p that you give me, as long as that p is smaller than 01:04:57.360 --> 01:05:00.960 align:middle line:84% 2/5, we have a perfect Bayesian equilibria where the student 01:05:00.960 --> 01:05:04.230 align:middle line:84% always submits the essay, the school accepts if you submit 01:05:04.230 --> 01:05:06.590 align:middle line:84% an essay, they reject if you don't submit an essay, 01:05:06.590 --> 01:05:08.590 align:middle line:84% and if they see that you don't submit an essay, 01:05:08.590 --> 01:05:11.950 align:middle line:84% they assign belief p to you being a passionate student. 01:05:11.950 --> 01:05:13.830 align:middle line:84% And that belief p must be small enough 01:05:13.830 --> 01:05:16.690 align:middle line:84% that it's credible for them to actually reject. 01:05:16.690 --> 01:05:17.215 align:middle line:90% Yes. 01:05:17.215 --> 01:05:18.590 align:middle line:84% AUDIENCE: So does that mean that, 01:05:18.590 --> 01:05:22.750 align:middle line:84% given different students, as long as they can assign 01:05:22.750 --> 01:05:25.150 align:middle line:90% one person 3 and one person-- 01:05:25.150 --> 01:05:27.927 align:middle line:84% like, different students who don't submit essays 01:05:27.927 --> 01:05:29.510 align:middle line:84% can have different values of p as long 01:05:29.510 --> 01:05:31.447 align:middle line:90% as it just doesn't make sense. 01:05:31.447 --> 01:05:33.030 align:middle line:84% IAN BALL: So I think you're describing 01:05:33.030 --> 01:05:35.023 align:middle line:90% a slightly different game. 01:05:35.023 --> 01:05:37.190 align:middle line:84% Yeah, here, we've kind of made a simplification here 01:05:37.190 --> 01:05:40.630 align:middle line:84% where we've basically imagined that there's really 01:05:40.630 --> 01:05:41.990 align:middle line:90% only one student. 01:05:41.990 --> 01:05:44.750 align:middle line:84% So here what we're imagining is a student 01:05:44.750 --> 01:05:49.630 align:middle line:84% is drawn from the population and applies. 01:05:49.630 --> 01:05:52.210 align:middle line:84% I think this is a reasonable model if-- in reality, 01:05:52.210 --> 01:05:53.770 align:middle line:84% what happens is many people apply, 01:05:53.770 --> 01:05:55.810 align:middle line:84% but the school adopts a uniform rule, 01:05:55.810 --> 01:05:58.962 align:middle line:84% they either accept students with essays or they don't, and they 01:05:58.962 --> 01:06:01.420 align:middle line:84% either accept students who don't have essays or they don't. 01:06:01.420 --> 01:06:03.740 align:middle line:84% If they wanted to do something more complicated where 01:06:03.740 --> 01:06:04.980 align:middle line:84% there's five applications and they 01:06:04.980 --> 01:06:06.480 align:middle line:84% do different things to different applications, 01:06:06.480 --> 01:06:07.897 align:middle line:84% then we'd have to explicitly model 01:06:07.897 --> 01:06:09.160 align:middle line:90% the number of applications. 01:06:09.160 --> 01:06:11.640 align:middle line:84% And here, look at the information set. 01:06:11.640 --> 01:06:14.700 align:middle line:84% All I see is I see 1 applicant, and it's either an essay or not 01:06:14.700 --> 01:06:15.380 align:middle line:90% an essay. 01:06:15.380 --> 01:06:16.980 align:middle line:84% And if we want to capture what you're describing, which 01:06:16.980 --> 01:06:18.563 align:middle line:84% I think is important for applications, 01:06:18.563 --> 01:06:20.660 align:middle line:90% we need to enrich the game. 01:06:20.660 --> 01:06:21.820 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:06:21.820 --> 01:06:23.653 align:middle line:84% IAN BALL: We're just looking at one student. 01:06:23.653 --> 01:06:26.620 align:middle line:84% Or the interpretation is a uniform policy that 01:06:26.620 --> 01:06:27.840 align:middle line:90% treats all students equally. 01:06:27.840 --> 01:06:29.200 align:middle line:84% If we wanted to treat students differently, 01:06:29.200 --> 01:06:30.617 align:middle line:84% then we'd have to enrich the game. 01:06:30.617 --> 01:06:33.660 align:middle line:90% 01:06:33.660 --> 01:06:37.460 align:middle line:84% Great, so this is what's called a-- 01:06:37.460 --> 01:06:39.060 align:middle line:90% let's put this in words. 01:06:39.060 --> 01:06:41.480 align:middle line:84% This is what's called a pooling equilibrium. 01:06:41.480 --> 01:06:47.780 align:middle line:90% 01:06:47.780 --> 01:06:49.260 align:middle line:84% It's called a pooling equilibrium 01:06:49.260 --> 01:06:51.700 align:middle line:84% because both types of students, the passionate 01:06:51.700 --> 01:06:54.900 align:middle line:84% and the non-passionate students, choose the same action. 01:06:54.900 --> 01:06:57.740 align:middle line:84% They pool together, and all submit essays. 01:06:57.740 --> 01:07:00.580 align:middle line:84% And as a result, their choice about whether to submit an essay 01:07:00.580 --> 01:07:03.020 align:middle line:84% doesn't reveal that much information about 01:07:03.020 --> 01:07:05.140 align:middle line:84% whether they're passionate or not. 01:07:05.140 --> 01:07:07.300 align:middle line:84% So next, let's see if we can compute 01:07:07.300 --> 01:07:10.220 align:middle line:84% what's called a separating equilibrium where 01:07:10.220 --> 01:07:13.360 align:middle line:84% different types of students make different choices. 01:07:13.360 --> 01:07:18.860 align:middle line:90% 01:07:18.860 --> 01:07:20.260 align:middle line:90% Let's go down here. 01:07:20.260 --> 01:07:21.760 align:middle line:84% This should be the last thing we do. 01:07:21.760 --> 01:07:36.620 align:middle line:90% 01:07:36.620 --> 01:07:38.440 align:middle line:84% So earlier, I had assumed that c-- 01:07:38.440 --> 01:07:40.460 align:middle line:84% remember c is the cost of submitting an essay-- 01:07:40.460 --> 01:07:41.680 align:middle line:90% was less than 1. 01:07:41.680 --> 01:07:43.040 align:middle line:90% Let's now relax that assumption. 01:07:43.040 --> 01:07:44.980 align:middle line:90% And let's let c be arbitrary. 01:07:44.980 --> 01:07:48.480 align:middle line:90% So let c be arbitrary. 01:07:48.480 --> 01:07:53.500 align:middle line:90% 01:07:53.500 --> 01:07:57.850 align:middle line:84% And now what we'd like to do is see, for which values of c, 01:07:57.850 --> 01:08:00.250 align:middle line:84% can we find a separating equilibrium? 01:08:00.250 --> 01:08:03.490 align:middle line:84% And this is actually an exam question I often like to ask, 01:08:03.490 --> 01:08:06.050 align:middle line:84% where you try to find the set of parameter values 01:08:06.050 --> 01:08:08.462 align:middle line:84% such that something is in equilibrium. 01:08:08.462 --> 01:08:10.170 align:middle line:84% So let's see if we can work through this. 01:08:10.170 --> 01:08:17.970 align:middle line:84% So we want to look for an equilibrium 01:08:17.970 --> 01:08:21.253 align:middle line:84% where the essay distinguishes the types of students. 01:08:21.253 --> 01:08:23.670 align:middle line:84% I think this is exactly why schools offer optional essays. 01:08:23.670 --> 01:08:25.370 align:middle line:84% What they're hoping is that people who really want to get in 01:08:25.370 --> 01:08:27.490 align:middle line:84% and are excited write the essay, and the people 01:08:27.490 --> 01:08:29.189 align:middle line:84% who don't really want to get in don't write the essay. 01:08:29.189 --> 01:08:30.609 align:middle line:84% I think it often doesn't work because we end up 01:08:30.609 --> 01:08:31.830 align:middle line:84% in the other equilibrium I described 01:08:31.830 --> 01:08:33.750 align:middle line:84% where everyone submits the essay even if they don't want to go. 01:08:33.750 --> 01:08:34.630 align:middle line:90% But this is the idea. 01:08:34.630 --> 01:08:37.569 align:middle line:84% And if essays are costly enough, this is what might work. 01:08:37.569 --> 01:08:39.090 align:middle line:84% So we're looking for an equilibrium 01:08:39.090 --> 01:08:42.290 align:middle line:90% where what is the student doing? 01:08:42.290 --> 01:08:49.010 align:middle line:84% Well, if they're passionate, they submit the essay. 01:08:49.010 --> 01:08:56.359 align:middle line:84% And if they're not passionate, they don't submit the essay. 01:08:56.359 --> 01:08:59.671 align:middle line:84% And then what does the school do? 01:08:59.671 --> 01:09:08.540 align:middle line:84% Well, let's first specify the school's beliefs. 01:09:08.540 --> 01:09:12.319 align:middle line:84% In this case, both information sets 01:09:12.319 --> 01:09:14.180 align:middle line:84% are reached with positive probability. 01:09:14.180 --> 01:09:15.640 align:middle line:84% There's a positive probability that a student 01:09:15.640 --> 01:09:17.457 align:middle line:84% submits an essay and a positive probability 01:09:17.457 --> 01:09:19.040 align:middle line:84% that student does not submit an essay. 01:09:19.040 --> 01:09:20.960 align:middle line:84% So in this case, we can apply Bayes' rule 01:09:20.960 --> 01:09:22.140 align:middle line:90% at both information sets. 01:09:22.140 --> 01:09:24.200 align:middle line:84% We don't run into this issue of dividing by 0. 01:09:24.200 --> 01:09:26.158 align:middle line:84% And these beliefs are going to be well defined. 01:09:26.158 --> 01:09:28.840 align:middle line:84% So if I see that a student submits an essay given 01:09:28.840 --> 01:09:30.946 align:middle line:84% that this is the strategy, what does Bayes' rule 01:09:30.946 --> 01:09:31.779 align:middle line:90% tell me as a school? 01:09:31.779 --> 01:09:33.140 align:middle line:84% What is the probability that they're passionate? 01:09:33.140 --> 01:09:35.859 align:middle line:84% And what is the probability that they're not passionate? 01:09:35.859 --> 01:09:36.359 align:middle line:90% Yeah. 01:09:36.359 --> 01:09:37.160 align:middle line:90% AUDIENCE: 1? 01:09:37.160 --> 01:09:38.327 align:middle line:90% IAN BALL: Yeah, it's just 1. 01:09:38.327 --> 01:09:40.840 align:middle line:84% And I think the way I'm writing it, let's write this 1, 0. 01:09:40.840 --> 01:09:42.298 align:middle line:84% So I say there's a probability of 1 01:09:42.298 --> 01:09:44.960 align:middle line:84% that they're passionate and 0 that they're not passionate. 01:09:44.960 --> 01:09:46.620 align:middle line:84% What if I see that there's no essay? 01:09:46.620 --> 01:09:47.852 align:middle line:90% Well, now it's the opposite. 01:09:47.852 --> 01:09:49.560 align:middle line:84% The only students who don't submit essays 01:09:49.560 --> 01:09:51.060 align:middle line:84% are the ones who are not passionate. 01:09:51.060 --> 01:09:53.319 align:middle line:84% So I'm certain that they must be not passionate. 01:09:53.319 --> 01:09:54.690 align:middle line:90% And I get 0, 1. 01:09:54.690 --> 01:09:58.790 align:middle line:90% 01:09:58.790 --> 01:09:59.670 align:middle line:90% OK. 01:09:59.670 --> 01:10:01.313 align:middle line:84% Well, now it's pretty clear, if I'm 01:10:01.313 --> 01:10:03.730 align:middle line:84% certain that students who submit the essay are passionate, 01:10:03.730 --> 01:10:06.652 align:middle line:90% I certainly want to accept them. 01:10:06.652 --> 01:10:09.110 align:middle line:84% And if I'm certain that students who don't submit the essay 01:10:09.110 --> 01:10:11.810 align:middle line:84% are not passionate, then I certainly want to reject them. 01:10:11.810 --> 01:10:19.530 align:middle line:90% 01:10:19.530 --> 01:10:23.310 align:middle line:84% So here's our candidate equilibrium. 01:10:23.310 --> 01:10:27.230 align:middle line:84% And let's just go through the conditions. 01:10:27.230 --> 01:10:29.450 align:middle line:84% And I think this is always a good way to check. 01:10:29.450 --> 01:10:42.870 align:middle line:90% 01:10:42.870 --> 01:10:46.390 align:middle line:84% So does this equilibrium satisfy belief consistency? 01:10:46.390 --> 01:10:47.210 align:middle line:90% Yes. 01:10:47.210 --> 01:10:50.830 align:middle line:84% We just checked Bayes' rule, so we're good here. 01:10:50.830 --> 01:10:52.890 align:middle line:84% And just to remind you, when I say equilibrium, 01:10:52.890 --> 01:10:55.130 align:middle line:84% notice I've not just specified the strategies. 01:10:55.130 --> 01:10:57.310 align:middle line:90% I've also specified the beliefs. 01:10:57.310 --> 01:10:59.470 align:middle line:84% And that's because this is an assessment, not just 01:10:59.470 --> 01:11:01.910 align:middle line:90% a strategy profile. 01:11:01.910 --> 01:11:03.510 align:middle line:84% And so I have belief consistency. 01:11:03.510 --> 01:11:05.303 align:middle line:84% Now we move to sequential rationality. 01:11:05.303 --> 01:11:07.970 align:middle line:84% Well, let me separately check it for the student and the school. 01:11:07.970 --> 01:11:15.430 align:middle line:90% 01:11:15.430 --> 01:11:17.730 align:middle line:84% Sequential rationality for the school is pretty clear. 01:11:17.730 --> 01:11:18.570 align:middle line:90% This is what I already said. 01:11:18.570 --> 01:11:19.670 align:middle line:84% They're accepting students who they 01:11:19.670 --> 01:11:21.090 align:middle line:84% think are likely to be passionate, 01:11:21.090 --> 01:11:22.548 align:middle line:84% and they're rejecting students they 01:11:22.548 --> 01:11:24.470 align:middle line:84% think are likely to be not passionate. 01:11:24.470 --> 01:11:26.880 align:middle line:90% So we're good here. 01:11:26.880 --> 01:11:29.130 align:middle line:84% But what about sequential rationality for the student, 01:11:29.130 --> 01:11:30.450 align:middle line:84% this is going to be a little trickier. 01:11:30.450 --> 01:11:31.825 align:middle line:84% And this is going to depend on c. 01:11:31.825 --> 01:11:32.390 align:middle line:90% Yeah. 01:11:32.390 --> 01:11:34.307 align:middle line:84% AUDIENCE: Yeah, [INAUDIBLE] the expected value 01:11:34.307 --> 01:11:39.652 align:middle line:84% for passionate students have to be like, negative [INAUDIBLE]. 01:11:39.652 --> 01:11:41.610 align:middle line:84% IAN BALL: Right, so let's exactly check it out. 01:11:41.610 --> 01:11:43.030 align:middle line:84% So there's two things we have to check. 01:11:43.030 --> 01:11:45.230 align:middle line:84% We have to check that if I'm a passionate student, 01:11:45.230 --> 01:11:47.330 align:middle line:84% I prefer writing an essay to not. 01:11:47.330 --> 01:11:48.830 align:middle line:84% And if I'm a not-passionate student, 01:11:48.830 --> 01:11:51.020 align:middle line:84% I prefer not writing an essay to writing an essay. 01:11:51.020 --> 01:11:53.360 align:middle line:84% So let's just write that down in math, but in words, 01:11:53.360 --> 01:11:55.220 align:middle line:90% that's exactly what we want. 01:11:55.220 --> 01:11:58.780 align:middle line:84% So let's suppose I'm a student who's passionate. 01:11:58.780 --> 01:12:03.918 align:middle line:84% And if I write an essay, what is my payoff? 01:12:03.918 --> 01:12:05.960 align:middle line:84% And if I don't write an essay, what is my payoff? 01:12:05.960 --> 01:12:09.940 align:middle line:90% 01:12:09.940 --> 01:12:13.640 align:middle line:84% Well, if I'm passionate and I write an essay, what happens? 01:12:13.640 --> 01:12:14.680 align:middle line:90% What's my payoff here? 01:12:14.680 --> 01:12:18.580 align:middle line:90% 01:12:18.580 --> 01:12:20.200 align:middle line:90% 2 minus c. 01:12:20.200 --> 01:12:23.348 align:middle line:90% 01:12:23.348 --> 01:12:24.140 align:middle line:90% OK. 01:12:24.140 --> 01:12:28.580 align:middle line:84% And if I don't write an essay, well, I'm not accepted, 01:12:28.580 --> 01:12:31.040 align:middle line:84% and I don't write the essay, so my payoff is just 0. 01:12:31.040 --> 01:12:33.920 align:middle line:90% 01:12:33.920 --> 01:12:37.660 align:middle line:84% Now, notice a really crucial point here. 01:12:37.660 --> 01:12:41.340 align:middle line:84% The school's strategy is that if you don't submit an essay, 01:12:41.340 --> 01:12:43.380 align:middle line:90% you're rejected. 01:12:43.380 --> 01:12:46.660 align:middle line:84% When I, as a passionate student, contemplate deviating and not 01:12:46.660 --> 01:12:49.490 align:middle line:84% submitting an essay, when I don't submit an essay, 01:12:49.490 --> 01:12:52.182 align:middle line:84% the school's strategy is fixed, and therefore, they reject me. 01:12:52.182 --> 01:12:54.390 align:middle line:84% People often get confused here because you might say, 01:12:54.390 --> 01:12:55.870 align:middle line:84% well, the school's belief is wrong. 01:12:55.870 --> 01:12:58.690 align:middle line:84% Their belief is that only non-passionate students 01:12:58.690 --> 01:13:00.130 align:middle line:90% don't submit essays. 01:13:00.130 --> 01:13:02.462 align:middle line:84% And now a passionate student's submitting an essay. 01:13:02.462 --> 01:13:04.170 align:middle line:84% But the school can't tell the difference. 01:13:04.170 --> 01:13:07.270 align:middle line:84% If the school's belief is that if you don't submit an essay, 01:13:07.270 --> 01:13:08.690 align:middle line:90% you can't be passionate. 01:13:08.690 --> 01:13:11.130 align:middle line:84% And if a passionate student deviates and doesn't 01:13:11.130 --> 01:13:13.410 align:middle line:84% send an essay, they're not able to change the beliefs 01:13:13.410 --> 01:13:14.487 align:middle line:90% that the school has. 01:13:14.487 --> 01:13:16.570 align:middle line:84% The school has their beliefs fixed, and therefore, 01:13:16.570 --> 01:13:18.090 align:middle line:84% what happens to them when they don't 01:13:18.090 --> 01:13:21.000 align:middle line:84% write an essay is determined by the strategy of the school. 01:13:21.000 --> 01:13:21.750 align:middle line:90% It doesn't change. 01:13:21.750 --> 01:13:23.510 align:middle line:84% Some people are tempted to say, oh, they're passionate, 01:13:23.510 --> 01:13:25.218 align:middle line:84% so now the school's going to accept them. 01:13:25.218 --> 01:13:26.770 align:middle line:90% No, it's a unilateral deviation. 01:13:26.770 --> 01:13:28.390 align:middle line:90% The school strategy is fixed. 01:13:28.390 --> 01:13:29.807 align:middle line:84% And we're considering what happens 01:13:29.807 --> 01:13:32.290 align:middle line:84% when a passionate student deviates by not submitting this. 01:13:32.290 --> 01:13:35.090 align:middle line:84% OK, now let's consider a non-passionate student. 01:13:35.090 --> 01:13:37.890 align:middle line:90% They're not submitting an essay. 01:13:37.890 --> 01:13:40.570 align:middle line:84% And again, they're getting a payoff of 0. 01:13:40.570 --> 01:13:42.070 align:middle line:90% What if they submitted an essay? 01:13:42.070 --> 01:13:44.970 align:middle line:90% 01:13:44.970 --> 01:13:47.510 align:middle line:90% What would their payoff be? 01:13:47.510 --> 01:13:48.920 align:middle line:90% Yeah. 01:13:48.920 --> 01:13:53.538 align:middle line:84% 1 minus C. Again, they're able to trick the school. 01:13:53.538 --> 01:13:56.080 align:middle line:84% If they submit the essay, they successfully trick the school, 01:13:56.080 --> 01:13:57.620 align:middle line:84% and the school's going to accept them 01:13:57.620 --> 01:13:58.520 align:middle line:84% even though the school doesn't want 01:13:58.520 --> 01:14:00.640 align:middle line:84% to accept non-passionate people, but they have 01:14:00.640 --> 01:14:02.280 align:middle line:90% to pay the essay cost of c. 01:14:02.280 --> 01:14:04.920 align:middle line:84% So now let's check the equilibrium conditions. 01:14:04.920 --> 01:14:07.400 align:middle line:84% A passionate student is supposed to submit the essay, 01:14:07.400 --> 01:14:11.680 align:middle line:84% so we need 2 minus c to be greater than or equal to 0. 01:14:11.680 --> 01:14:13.400 align:middle line:84% But a non-passionate student is not 01:14:13.400 --> 01:14:14.960 align:middle line:90% supposed to submit the essay. 01:14:14.960 --> 01:14:20.280 align:middle line:84% So we need 0 greater than or equal to 1 minus c. 01:14:20.280 --> 01:14:24.440 align:middle line:90% So what's our solution? 01:14:24.440 --> 01:14:26.840 align:middle line:84% Well, this is going to be an equilibrium 01:14:26.840 --> 01:14:30.760 align:middle line:84% if both of these inequalities are satisfied. 01:14:30.760 --> 01:14:35.080 align:middle line:84% And that's going to be true if 1 is less than or equal to c 01:14:35.080 --> 01:14:37.465 align:middle line:90% less than or equal to 2. 01:14:37.465 --> 01:14:39.340 align:middle line:84% And let's try to understand the role of this. 01:14:39.340 --> 01:14:40.960 align:middle line:84% Why do we have these two assumptions? 01:14:40.960 --> 01:14:44.160 align:middle line:84% Well, we want the essay to separate the good students 01:14:44.160 --> 01:14:47.000 align:middle line:90% from the bad students. 01:14:47.000 --> 01:14:49.060 align:middle line:84% What would go wrong if c was too big? 01:14:49.060 --> 01:14:50.680 align:middle line:90% If c was greater than 2? 01:14:50.680 --> 01:14:51.200 align:middle line:90% Yeah. 01:14:51.200 --> 01:14:52.835 align:middle line:84% AUDIENCE: Even the good students-- 01:14:52.835 --> 01:14:54.460 align:middle line:84% IAN BALL: Yeah, it's so costly-- if you 01:14:54.460 --> 01:14:55.660 align:middle line:84% say you have to write 100 essays, well, 01:14:55.660 --> 01:14:57.327 align:middle line:84% no one's going to apply to your school-- 01:14:57.327 --> 01:14:58.520 align:middle line:90% again, in the pre-GPT world. 01:14:58.520 --> 01:15:00.312 align:middle line:84% So we're not able to separate the students. 01:15:00.312 --> 01:15:03.120 align:middle line:84% If c is too low, what goes wrong? 01:15:03.120 --> 01:15:06.180 align:middle line:84% If c is less than 1, well, even the non-passionate students say, 01:15:06.180 --> 01:15:08.480 align:middle line:84% well, it's a one-paragraph essay, I can do that. 01:15:08.480 --> 01:15:10.080 align:middle line:90% It's worth it for me to do that. 01:15:10.080 --> 01:15:12.080 align:middle line:84% So if you think of the school as-- this 01:15:12.080 --> 01:15:13.622 align:middle line:84% is a broader question about the game, 01:15:13.622 --> 01:15:15.247 align:middle line:84% you could think of the design question, 01:15:15.247 --> 01:15:17.800 align:middle line:84% if the school can choose c by choosing how hard the essay is, 01:15:17.800 --> 01:15:19.258 align:middle line:84% they might want to choose the essay 01:15:19.258 --> 01:15:20.660 align:middle line:90% to be a moderate difficulty. 01:15:20.660 --> 01:15:23.480 align:middle line:84% It's hard enough to deter the uninspired students 01:15:23.480 --> 01:15:26.520 align:middle line:84% from applying, but easy enough that the inspired students would 01:15:26.520 --> 01:15:27.280 align:middle line:90% still apply. 01:15:27.280 --> 01:15:29.920 align:middle line:84% And I think that's exactly what admissions offices 01:15:29.920 --> 01:15:32.840 align:middle line:90% and jobs think about. 01:15:32.840 --> 01:15:36.200 align:middle line:84% OK, let me stop there unless there's any questions. 01:15:36.200 --> 01:15:39.070 align:middle line:90% And I'll see everyone next week. 01:15:39.070 --> 01:15:44.000 align:middle line:90%