WEBVTT 00:00:00.070 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high quality educational resources for free. 00:00:10.730 --> 00:00:13.340 To make a donation or view additional materials 00:00:13.340 --> 00:00:17.229 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.229 --> 00:00:17.854 at ocw.mit.edu. 00:00:21.066 --> 00:00:24.390 PROFESSOR: Today we're going to be talking about games. 00:00:24.390 --> 00:00:29.724 And I know you guys as well as I hope I do. 00:00:29.724 --> 00:00:32.140 The main thing that you guys want to talk about with games 00:00:32.140 --> 00:00:34.330 is how to do that alpha-beta thing. 00:00:34.330 --> 00:00:36.300 Because it's pretty confusing. 00:00:36.300 --> 00:00:40.490 And it's easy to get lost in a corner or something. 00:00:40.490 --> 00:00:43.820 Whereas doing the regular minimax, in my experience, 00:00:43.820 --> 00:00:45.590 most 6034 students can do that. 00:00:45.590 --> 00:00:48.450 And they do it right pretty much all the time. 00:00:48.450 --> 00:00:51.560 However, we're going to focus on all the different components 00:00:51.560 --> 00:00:52.940 of games. 00:00:52.940 --> 00:00:56.910 And I put up two provocative silver star ideas 00:00:56.910 --> 00:01:00.370 up on the board, which will come into play here. 00:01:00.370 --> 00:01:02.840 The Snow White principle is a new name. 00:01:02.840 --> 00:01:05.489 And it has never been revealed until today. 00:01:05.489 --> 00:01:08.050 Because I made up name recently. 00:01:08.050 --> 00:01:11.000 So you will be the first people to hear it and decide 00:01:11.000 --> 00:01:13.310 if it works better than the term "grandfather 00:01:13.310 --> 00:01:15.630 clause" for the thing that I'm trying to describe. 00:01:15.630 --> 00:01:19.420 Because most grandfathers don't eat their children. 00:01:19.420 --> 00:01:25.300 So here we've got a beautiful game tree. 00:01:25.300 --> 00:01:29.970 It has nodes from A through R. This 00:01:29.970 --> 00:01:33.250 is our standard game tree from 6034. 00:01:33.250 --> 00:01:35.922 We've got a maximizer up at the top 00:01:35.922 --> 00:01:37.880 who's trying to get the highest score possible. 00:01:37.880 --> 00:01:41.770 The minimizer is her opponent. 00:01:41.770 --> 00:01:44.530 And the minimizer is trying to get to the lowest score 00:01:44.530 --> 00:01:45.930 possible. 00:01:45.930 --> 00:01:49.264 And it's really unclear who wins or loses at each point. 00:01:49.264 --> 00:01:51.680 They're just trying to get it to the highest or the lowest 00:01:51.680 --> 00:01:54.140 score. 00:01:54.140 --> 00:01:56.660 All right, so let's do a refresher. 00:01:56.660 --> 00:01:59.590 Hopefully the quiz didn't put people into such panic modes 00:01:59.590 --> 00:02:01.180 that they forgot Monday's lecture. 00:02:01.180 --> 00:02:06.130 So let's make sure that we can do regular minimax 00:02:06.130 --> 00:02:09.639 algorithm on this tree and figure out the minimax 00:02:09.639 --> 00:02:11.270 value at A. 00:02:11.270 --> 00:02:13.900 So let's see how that works. 00:02:13.900 --> 00:02:19.010 All right, as you guys remember, the game search 00:02:19.010 --> 00:02:22.320 when using regular minimax is essentially 00:02:22.320 --> 00:02:24.550 a depth first search. 00:02:24.550 --> 00:02:28.080 And at each level, it chooses between all 00:02:28.080 --> 00:02:32.510 of the children whichever value that the parent wants. 00:02:32.510 --> 00:02:35.480 So here after it would choose the maximum of K and L, 00:02:35.480 --> 00:02:36.330 for instance. 00:02:36.330 --> 00:02:37.570 But that's getting ahead of ourselves. 00:02:37.570 --> 00:02:38.986 Because it's a depth first search. 00:02:38.986 --> 00:02:40.874 So we best start at the top. 00:02:40.874 --> 00:02:42.290 I'll help you guys up for a while. 00:02:42.290 --> 00:02:44.480 So we're doing A. We need the maximum of B, C, 00:02:44.480 --> 00:02:45.770 D, depth first search. 00:02:45.770 --> 00:02:49.760 We go to B. We're looking for the minimum of E and F. 00:02:49.760 --> 00:02:55.980 So having looked at E, our current minimum of E and F 00:02:55.980 --> 00:02:58.780 is just 2 for the moment. 00:02:58.780 --> 00:03:04.040 So this is going to be less than or equal to 2. 00:03:04.040 --> 00:03:06.690 All right, then we go down to F, which is a maximizer. 00:03:06.690 --> 00:03:09.850 And its children are K and L. So now I'm 00:03:09.850 --> 00:03:11.650 going to start making you guys do stuff. 00:03:11.650 --> 00:03:13.960 So what do you think? 00:03:13.960 --> 00:03:17.620 What is going to be the minimax value at F? 00:03:17.620 --> 00:03:20.659 The minimax value at F, what will that be? 00:03:20.659 --> 00:03:21.534 AUDIENCE: [INAUDIBLE] 00:03:25.150 --> 00:03:29.090 PROFESSOR: So that level is a maximizer-- max, min, max. 00:03:29.090 --> 00:03:30.280 F is a maximizer. 00:03:30.280 --> 00:03:32.420 K and L themselves are minimizers. 00:03:32.420 --> 00:03:34.200 But they're pretty impotent minimizers. 00:03:34.200 --> 00:03:35.590 Because they don't get to choose. 00:03:35.590 --> 00:03:39.950 They just have to do K or L. So the minimax value is three. 00:03:39.950 --> 00:03:42.660 And yeah, the path it would like to go is K. 00:03:42.660 --> 00:03:45.510 So we'll say that the minimax value here is 3. 00:03:45.510 --> 00:03:48.680 It's in fact exactly equal to 3. 00:03:48.680 --> 00:03:53.410 So if this is 3 and this is 2, then everyone, 00:03:53.410 --> 00:03:55.476 we know that the value of B is? 00:03:55.476 --> 00:03:57.710 AUDIENCE: [INAUDIBLE] 00:03:57.710 --> 00:03:58.960 PROFESSOR: I hear 3 and 2. 00:03:58.960 --> 00:04:00.281 Which one is it? 00:04:00.281 --> 00:04:00.780 AUDIENCE: 2. 00:04:00.780 --> 00:04:03.150 PROFESSOR: 2, that's right. 00:04:03.150 --> 00:04:04.600 So the value here is 2. 00:04:04.600 --> 00:04:06.790 Great, let's go down into this branch. 00:04:06.790 --> 00:04:09.427 So C is going to be the minimum of G and 6. 00:04:09.427 --> 00:04:10.510 But we don't see that yet. 00:04:10.510 --> 00:04:12.218 Because we're doing a depth first search. 00:04:12.218 --> 00:04:13.840 It's going to be the minimum of G. 00:04:13.840 --> 00:04:16.540 Now we need the maximum of M and N. 00:04:16.540 --> 00:04:18.160 We're going to need the minimum. 00:04:18.160 --> 00:04:23.530 M is the minimum of Q and R. So let's switch sides. 00:04:23.530 --> 00:04:27.100 The minimum of Q and R is? 00:04:27.100 --> 00:04:28.305 AUDIENCE: 1. 00:04:28.305 --> 00:04:29.180 PROFESSOR: Let's see. 00:04:29.180 --> 00:04:30.570 That's right, it's 1. 00:04:30.570 --> 00:04:32.170 So M has a value of 1. 00:04:32.170 --> 00:04:34.240 But I'm going to stay over here. 00:04:34.240 --> 00:04:36.140 Because M has a value of 1. 00:04:36.140 --> 00:04:39.100 Knowing that, then we know that G has a value of? 00:04:39.100 --> 00:04:39.600 AUDIENCE: 7 00:04:39.600 --> 00:04:40.600 PROFESSOR: That's right. 00:04:40.600 --> 00:04:41.930 7 is higher than 1. 00:04:41.930 --> 00:04:45.550 And since G is a 7, we now know going up to C 00:04:45.550 --> 00:04:47.850 that C has a value of? 00:04:47.850 --> 00:04:48.850 AUDIENCE: 6. 00:04:48.850 --> 00:04:50.830 PROFESSOR: Yes, C has a value of 6. 00:04:50.830 --> 00:04:52.710 That's the minimum 6 and 7. 00:04:52.710 --> 00:04:54.600 So now I'm going to go back down. 00:04:54.600 --> 00:04:57.370 Because we've done one of the other sub-trees. 00:04:57.370 --> 00:04:58.751 This is a 6. 00:04:58.751 --> 00:04:59.950 All right, great. 00:04:59.950 --> 00:05:03.900 Now we're going to go down to D. Hopefully it won't be too bad. 00:05:03.900 --> 00:05:05.760 These things usually aren't terrible. 00:05:05.760 --> 00:05:08.710 Because they're made to be pruned a lot in alpha-beta. 00:05:08.710 --> 00:05:12.350 So let's see, in D, we go down to I. And that's just a 1. 00:05:12.350 --> 00:05:17.040 We go down to J. And let's see, what's the minimax value of J? 00:05:17.040 --> 00:05:17.860 AUDIENCE: It's 20. 00:05:17.860 --> 00:05:18.860 PROFESSOR: That's right. 00:05:18.860 --> 00:05:21.120 20 is the maximum of 20 and 2. 00:05:21.120 --> 00:05:24.780 Great, so what's the minimax value of D? 00:05:24.780 --> 00:05:25.990 Everyone said it-- 1. 00:05:25.990 --> 00:05:27.946 All right, so what's the minimax value at A? 00:05:27.946 --> 00:05:29.574 AUDIENCE: 6. 00:05:29.574 --> 00:05:30.490 PROFESSOR: 6 is right. 00:05:30.490 --> 00:05:31.660 6 is higher than 2. 00:05:31.660 --> 00:05:32.740 It's higher than 1. 00:05:32.740 --> 00:05:34.020 Our value is 6. 00:05:34.020 --> 00:05:41.960 And our path is-- everyone-- A, C, H. That's it. 00:05:41.960 --> 00:05:46.329 Great, is everyone good with minimax? 00:05:46.329 --> 00:05:47.995 I know that usually a lot of people are. 00:05:47.995 --> 00:05:49.510 There's usually a few people who aren't. 00:05:49.510 --> 00:05:50.950 So if you're one of the people who 00:05:50.950 --> 00:05:54.150 would like some clarifications on minimax, raise your hand. 00:05:54.150 --> 00:05:58.680 There's probably a few other people who would like some too. 00:05:58.680 --> 00:06:00.536 OK. 00:06:00.536 --> 00:06:03.350 AUDIENCE: When you're doing this minimax, whatever values 00:06:03.350 --> 00:06:06.752 are not showing, you keep going down the tree 00:06:06.752 --> 00:06:08.452 and then just look at whether you're 00:06:08.452 --> 00:06:09.670 trying to find the minimax. 00:06:09.670 --> 00:06:12.380 And just whatever values you get you go back up one? 00:06:12.380 --> 00:06:14.710 PROFESSOR: Yes. 00:06:14.710 --> 00:06:20.340 The question was, when you go to do the minimax-- and let's 00:06:20.340 --> 00:06:21.724 say you got E was 2, and you know 00:06:21.724 --> 00:06:23.640 that B is going to be less than or equal to 2, 00:06:23.640 --> 00:06:25.160 but you don't know F yet. 00:06:25.160 --> 00:06:28.350 The question is, do you go down the tree, find the value at F, 00:06:28.350 --> 00:06:29.460 and then go back up? 00:06:29.460 --> 00:06:31.270 The answer is yes. 00:06:31.270 --> 00:06:33.300 By default, we use a depth first search. 00:06:33.300 --> 00:06:36.550 However, in non alpha-beta version, just regular minimax, 00:06:36.550 --> 00:06:40.900 it turns out it probably doesn't matter what you do. 00:06:40.900 --> 00:06:43.150 I suggested doing a depth first search to get yourself 00:06:43.150 --> 00:06:44.630 in the mindset of alpha-beta. 00:06:44.630 --> 00:06:49.290 Because order is very, very important in alpha-beta. 00:06:49.290 --> 00:06:52.150 But here, I don't know, you could do some weird bottom 00:06:52.150 --> 00:06:53.110 up search. 00:06:53.110 --> 00:06:56.010 Whatever you want, it's going to give you the right answer 00:06:56.010 --> 00:06:58.380 unless it asks what order they evaluated. 00:06:58.380 --> 00:06:59.280 But here's a hint. 00:06:59.280 --> 00:07:03.790 The order they're evaluated in is depth first search order. 00:07:03.790 --> 00:07:09.620 So without even doing anything, E, K, L, Q, R, N, H, I, O, P 00:07:09.620 --> 00:07:13.355 are the order of starting evaluation in this tree. 00:07:13.355 --> 00:07:14.230 AUDIENCE: [INAUDIBLE] 00:07:20.330 --> 00:07:23.950 PROFESSOR: So the question is, nodes like M and G 00:07:23.950 --> 00:07:26.130 we don't have to put values next to. 00:07:26.130 --> 00:07:28.420 Technically, if we were doing this very formally, 00:07:28.420 --> 00:07:30.560 and we couldn't remember, and I wasn't 00:07:30.560 --> 00:07:33.780 up there among the people, we would put 1 there. 00:07:33.780 --> 00:07:36.580 So at M, we would put a 1. 00:07:36.580 --> 00:07:37.829 But people remembered that. 00:07:37.829 --> 00:07:38.620 So we didn't do it. 00:07:38.620 --> 00:07:40.599 But then at G, we would put a 7. 00:07:40.599 --> 00:07:42.390 So if we were writing it out very formally, 00:07:42.390 --> 00:07:45.280 we would have a 1 and a 7. 00:07:45.280 --> 00:07:47.920 And at this D, we would have a 1. 00:07:47.920 --> 00:07:50.650 And then at the A, we would put a 6. 00:07:50.650 --> 00:07:53.140 And then that's the answer. 00:07:53.140 --> 00:07:55.510 Also, we've even put things like less than or greater 00:07:55.510 --> 00:07:57.070 than part way along the way. 00:07:57.070 --> 00:08:00.590 However, I believe that our alpha-beta search 00:08:00.590 --> 00:08:03.960 is going to definitely fulfill everyone's quota 00:08:03.960 --> 00:08:07.030 of pedantically putting lots of numbers next to nodes 00:08:07.030 --> 00:08:08.650 on a game tree. 00:08:08.650 --> 00:08:11.470 And so once you've done alpha-beta, 00:08:11.470 --> 00:08:13.870 if you can do it correctly, you'll 00:08:13.870 --> 00:08:16.260 think, oh, minimax, oh, those were the days. 00:08:16.260 --> 00:08:17.590 It's going to be easy. 00:08:17.590 --> 00:08:22.170 Because alpha-beta is a little bit more complicated. 00:08:22.170 --> 00:08:24.490 There's a lot of things that trip people up here. 00:08:24.490 --> 00:08:26.270 For alpha-beta, however, I will erase 00:08:26.270 --> 00:08:28.750 some of these numbers for the moment. 00:08:28.750 --> 00:08:30.660 They're still right. 00:08:30.660 --> 00:08:33.700 But we do it a little differently. 00:08:33.700 --> 00:08:38.470 So what do alpha-beta and beta add to this formula? 00:08:38.470 --> 00:08:40.710 Well, this is all sort of a winning formula, 00:08:40.710 --> 00:08:41.679 except for it's not. 00:08:41.679 --> 00:08:43.490 Because it takes too long. 00:08:43.490 --> 00:08:45.620 But it's a very nice formula. 00:08:45.620 --> 00:08:47.600 U is the maximizer, say. 00:08:47.600 --> 00:08:50.410 I would try to think, if I do this, what's he going to do? 00:08:50.410 --> 00:08:52.430 And then if he does that, what am I going to do? 00:08:52.430 --> 00:08:54.340 And then what is he going to do if I 00:08:54.340 --> 00:08:57.040 do that, et cetera, et cetera, all the way to the bottom. 00:08:57.040 --> 00:08:58.560 With alpha and beta, we add in what 00:08:58.560 --> 00:09:02.644 I like to call nuclear options. 00:09:02.644 --> 00:09:04.560 I'd like in this game of maximizer minimizer-- 00:09:04.560 --> 00:09:07.389 you can think of it as like the Cold War or the Peloponnesian 00:09:07.389 --> 00:09:09.680 War, except the Peloponnesian War didn't have nukes, so 00:09:09.680 --> 00:09:12.710 probably the Cold War. 00:09:12.710 --> 00:09:17.200 And in the Cold War, or any situation 00:09:17.200 --> 00:09:19.477 where you're up against an adversary who-- actually, 00:09:19.477 --> 00:09:21.560 this doesn't really work as well for the Cold War. 00:09:21.560 --> 00:09:22.935 But in any situation where you're 00:09:22.935 --> 00:09:24.920 up against an adversary whose only goal in life 00:09:24.920 --> 00:09:27.260 is to destroy you, you always want 00:09:27.260 --> 00:09:29.480 to find out what the best thing you can possibly do 00:09:29.480 --> 00:09:32.850 is if they hit that button and send nukes in from Cuba, 00:09:32.850 --> 00:09:36.270 or if they send fighter pilots, or whatever is going on. 00:09:36.270 --> 00:09:38.600 So the idea of alpha and beta is that they 00:09:38.600 --> 00:09:45.030 are numbers that represent the fail-safe, the worst case. 00:09:45.030 --> 00:09:47.590 Because obviously in the Cold War, 00:09:47.590 --> 00:09:50.820 sending nukes was not a good plan. 00:09:50.820 --> 00:09:55.230 But presumably, us sending nukes would 00:09:55.230 --> 00:09:57.880 be better than just being attacked and killed. 00:09:57.880 --> 00:10:04.150 So the alpha and beta represent the worst possible outcome 00:10:04.150 --> 00:10:07.470 you'd be willing to accept for your side. 00:10:07.470 --> 00:10:10.080 Because right now, you know you're 00:10:10.080 --> 00:10:15.400 guaranteed to be able to force the conflict to that point 00:10:15.400 --> 00:10:16.840 or better. 00:10:16.840 --> 00:10:19.780 So the alpha is the nuclear option, the fail-safe, 00:10:19.780 --> 00:10:22.420 of the maximizer. 00:10:22.420 --> 00:10:27.890 Nuclear options-- alpha is maximizer's nuclear option. 00:10:34.980 --> 00:10:40.430 And beta is the minimizer's nuclear option. 00:10:46.994 --> 00:10:49.410 So we ask ourselves-- and people who were paying attention 00:10:49.410 --> 00:10:52.240 at lecture or wrote stuff down know the answer already-- 00:10:52.240 --> 00:10:55.240 what could we possibly set to start off? 00:10:55.240 --> 00:10:57.230 Before we explore the tree and find anything, 00:10:57.230 --> 00:10:59.760 what will we set as our nuclear option, 00:10:59.760 --> 00:11:01.280 as our sort of fail-safe? 00:11:01.280 --> 00:11:04.490 We could always fall back on this number. 00:11:04.490 --> 00:11:06.630 So you could set 0. 00:11:06.630 --> 00:11:10.580 You could try to set some low number for the maximizer. 00:11:10.580 --> 00:11:12.960 Because if you set a high number for the maximizer 00:11:12.960 --> 00:11:16.150 as its fail-safe, it's going to be really snooty and just say, 00:11:16.150 --> 00:11:17.310 oh, I won't take this path. 00:11:17.310 --> 00:11:18.770 I already have a fail-safe that's 00:11:18.770 --> 00:11:19.970 better than all these paths. 00:11:19.970 --> 00:11:23.460 If you set, like, 100, you have no tree. 00:11:23.460 --> 00:11:26.990 Our default usually, in 6034, is to set negative infinity 00:11:26.990 --> 00:11:30.340 for alpha or negative some very large number if you're doing it 00:11:30.340 --> 00:11:32.770 in your lab. 00:11:32.770 --> 00:11:37.670 So if we set negative infinity as a default for alpha, 00:11:37.670 --> 00:11:41.600 that negative infinity is basically maximizer loses. 00:11:41.600 --> 00:11:43.020 So the maximizer goes in thinking, 00:11:43.020 --> 00:11:44.894 oh my god, if I don't look at this game tree, 00:11:44.894 --> 00:11:46.260 I automatically lose. 00:11:46.260 --> 00:11:51.070 He's willing to take the first path possibly presented. 00:11:51.070 --> 00:11:52.764 And that's why that negative infinity 00:11:52.764 --> 00:11:53.930 is a good default for alpha. 00:11:53.930 --> 00:11:56.096 Anyone have a good idea what a good default for beta 00:11:56.096 --> 00:11:57.710 is, or just remember? 00:11:57.710 --> 00:11:59.090 Positive infinity, that's right. 00:11:59.090 --> 00:12:01.423 Because the minimizer comes in, and she's like, oh crap, 00:12:01.423 --> 00:12:05.660 the maximizer automatically wins if I don't look at this node 00:12:05.660 --> 00:12:06.320 here. 00:12:06.320 --> 00:12:08.278 That makes sure the maximizer and the minimizer 00:12:08.278 --> 00:12:12.240 both are willing to look at the first path they see every time. 00:12:12.240 --> 00:12:13.635 Because look on this tree. 00:12:13.635 --> 00:12:16.560 If 10 was alpha, the maximizer would just 00:12:16.560 --> 00:12:19.020 reject out of hand everything except for P. 00:12:19.020 --> 00:12:20.440 And then we wouldn't have a tree. 00:12:20.440 --> 00:12:24.750 The maximizer would lose, because he would be like, hmm, 00:12:24.750 --> 00:12:26.790 this test game is very interesting. 00:12:26.790 --> 00:12:29.120 However, I have another option-- pft, 00:12:29.120 --> 00:12:31.380 and then you throw over the table. 00:12:31.380 --> 00:12:33.976 That's 10 for me, because you have to pick up the pieces. 00:12:33.976 --> 00:12:34.850 I don't own this set. 00:12:34.850 --> 00:12:36.990 I don't know. 00:12:36.990 --> 00:12:40.440 So that is why we set negative infinity and positive infinity 00:12:40.440 --> 00:12:42.770 as the default for alpha and beta. 00:12:42.770 --> 00:12:44.240 So how do alpha and beta propagate? 00:12:44.240 --> 00:12:45.870 And what do they do? 00:12:45.870 --> 00:12:49.110 The main purpose of alpha and beta is that, as we said, 00:12:49.110 --> 00:12:52.130 alpha-- let's say we have some chart of values. 00:12:52.130 --> 00:12:55.700 Alpha, which starts at negative infinity, 00:12:55.700 --> 00:12:58.935 is the worst that the maximizer is willing to accept. 00:12:58.935 --> 00:13:01.060 Because they know they can get that much or better. 00:13:01.060 --> 00:13:03.800 It starts out, that's the worst thing you can have. 00:13:03.800 --> 00:13:05.470 So it's not a problem. 00:13:05.470 --> 00:13:09.890 Infinity is the highest that the minimizer is willing to accept. 00:13:09.890 --> 00:13:11.430 That's beta. 00:13:11.430 --> 00:13:13.250 As you go along, though, the minimizer 00:13:13.250 --> 00:13:14.970 sees, oh, look at that. 00:13:14.970 --> 00:13:18.610 I can guarantee that at best the maximizer gets 100. 00:13:18.610 --> 00:13:20.700 Haha, beta is now 100. 00:13:20.700 --> 00:13:22.790 The maximizer says, oh yeah? 00:13:22.790 --> 00:13:27.490 Well I can guarantee that the lowest you can get me to go to 00:13:27.490 --> 00:13:28.280 is 0. 00:13:28.280 --> 00:13:31.330 So it's going to be 0. 00:13:31.330 --> 00:13:36.990 And this keeps going on until maybe at 6-- note, 00:13:36.990 --> 00:13:38.670 not drawn to scale. 00:13:38.670 --> 00:13:41.940 Maybe at 6, the maximizer said, haha, 00:13:41.940 --> 00:13:44.030 you can't make me go lower than 6. 00:13:44.030 --> 00:13:47.650 And the minimizer says, aha, you can't make me go higher than 6. 00:13:47.650 --> 00:13:50.120 And then 6 is the answer. 00:13:50.120 --> 00:13:56.950 If you ever get to a point where beta gets lower than alpha, 00:13:56.950 --> 00:14:02.000 or alpha gets lower than beta, then you just say, screw this. 00:14:02.000 --> 00:14:05.570 I'm not even going to look at the remaining stuff. 00:14:05.570 --> 00:14:09.870 I'm going to just prune now and go somewhere else that's 00:14:09.870 --> 00:14:11.780 less pointless than this. 00:14:11.780 --> 00:14:15.220 Because if the alpha gets higher than the beta, what that's 00:14:15.220 --> 00:14:18.515 saying is the maximizer says, oh man, look at this, minimizer. 00:14:18.515 --> 00:14:22.450 The lowest you can make me go is, say, 50. 00:14:22.450 --> 00:14:24.300 And the minimizer says, that's strange. 00:14:24.300 --> 00:14:28.760 Because the highest that you can make me go is 40. 00:14:28.760 --> 00:14:32.210 So something's generally amiss there. 00:14:32.210 --> 00:14:35.220 It usually means that one of the two of them 00:14:35.220 --> 00:14:37.890 doesn't even want to be exploring that branch at all. 00:14:37.890 --> 00:14:40.816 So you prune at that point. 00:14:40.816 --> 00:14:43.470 All right, so given that that's what we're looking for, 00:14:43.470 --> 00:14:46.650 how do we move the alphas and betas throughout the tree? 00:14:46.650 --> 00:14:48.400 There's a few different ways to draw them. 00:14:48.400 --> 00:14:52.830 And some of them I consider to be very busy. 00:14:52.830 --> 00:14:54.930 Probably in recitation and tutorial 00:14:54.930 --> 00:14:57.980 you will see a way that's busier and has more numbers. 00:14:57.980 --> 00:15:01.060 Technically, every node has both an alpha and a beta. 00:15:01.060 --> 00:15:04.560 However, the one that that node is paying attention to 00:15:04.560 --> 00:15:06.190 is the alpha, if it's a maximizer, 00:15:06.190 --> 00:15:08.360 and the beta if it's a minimizer. 00:15:08.360 --> 00:15:12.290 So I generally, for my purposes, only draw the alpha out 00:15:12.290 --> 00:15:16.330 for the maximizer and only draw the beta out for the minimizer. 00:15:16.330 --> 00:15:19.742 Very rarely, but it happens, they'll sometimes ask you, 00:15:19.742 --> 00:15:21.450 well, what's the beta of this node, which 00:15:21.450 --> 00:15:22.460 is a maximizer node? 00:15:22.460 --> 00:15:24.680 So it's good to know how it's derived. 00:15:24.680 --> 00:15:27.540 But I think that it wastes your time to write it out. 00:15:27.540 --> 00:15:28.520 That's my opinion. 00:15:28.520 --> 00:15:29.820 We'll see how it goes. 00:15:29.820 --> 00:15:32.720 So the way that it works, the way that alpha and beta works, 00:15:32.720 --> 00:15:33.970 is the Snow White principal. 00:15:33.970 --> 00:15:37.100 So does everyone know the story of Snow White? 00:15:37.100 --> 00:15:38.860 So there's a beautiful princess. 00:15:38.860 --> 00:15:40.696 There's an evil queen stepmother. 00:15:40.696 --> 00:15:43.070 Mirror mirror on the wall, who's the fairest of them all, 00:15:43.070 --> 00:15:46.620 finds out that it's the stepdaughter. 00:15:46.620 --> 00:15:49.130 So much like in the real world, in Snow White, 00:15:49.130 --> 00:15:52.440 the stepdaughter, Snow White, had the beauty of her parents. 00:15:52.440 --> 00:15:54.070 She inherited those. 00:15:54.070 --> 00:15:56.560 However, much like in the real world, maybe 00:15:56.560 --> 00:16:00.810 or perhaps not, the stepmother had an even better plan. 00:16:00.810 --> 00:16:04.980 She hired a hunter to sort of hunt Snow White, 00:16:04.980 --> 00:16:07.560 pull out Snow White's heart, and feed it 00:16:07.560 --> 00:16:09.710 to her so that she could gain Snow White's 00:16:09.710 --> 00:16:11.590 beauty for herself. 00:16:11.590 --> 00:16:14.900 How many people knew that version of the story? 00:16:14.900 --> 00:16:15.590 A few people. 00:16:15.590 --> 00:16:17.298 That's the original version of the story. 00:16:17.298 --> 00:16:18.570 Disney didn't put that in. 00:16:18.570 --> 00:16:19.944 The hunter then brought the heart 00:16:19.944 --> 00:16:22.250 of a deer, which I think in Disney the hunter did kill 00:16:22.250 --> 00:16:24.130 a deer arbitrarily, but it was not 00:16:24.130 --> 00:16:26.260 explained that that's why he was doing it. 00:16:26.260 --> 00:16:29.540 So in alpha-beta, it's just like that. 00:16:29.540 --> 00:16:32.760 By which I mean you start by inheriting the alpha 00:16:32.760 --> 00:16:34.400 and beta of your parents. 00:16:34.400 --> 00:16:38.360 But if you see something that you like amongst your children, 00:16:38.360 --> 00:16:42.810 you take it for yourself-- the Snow White principle. 00:16:42.810 --> 00:16:44.100 So let's see how that goes. 00:16:44.100 --> 00:16:48.200 Well, I told you guys that the default alpha was-- 00:16:48.200 --> 00:16:49.612 AUDIENCE: Negative infinity. 00:16:49.612 --> 00:16:50.820 PROFESSOR: Negative infinity. 00:16:50.820 --> 00:16:54.300 So here alpha is negative infinity. 00:16:54.300 --> 00:16:58.310 And I told you that the default beta was positive infinity. 00:16:58.310 --> 00:17:01.110 We're doing a depth first search here. 00:17:01.110 --> 00:17:03.590 All right, beta is infinity. 00:17:03.590 --> 00:17:09.099 All right, so we come here to E. Now, we could put an alpha. 00:17:09.099 --> 00:17:12.140 But I never put an alpha or a beta 00:17:12.140 --> 00:17:14.250 for one of the terminal nodes. 00:17:14.250 --> 00:17:17.329 Because it can't really do anything. 00:17:17.329 --> 00:17:19.060 It's just 2. 00:17:19.060 --> 00:17:22.352 So as we go down, we take the alpha and beta 00:17:22.352 --> 00:17:23.060 from our parents. 00:17:23.060 --> 00:17:25.830 But as we go up to a parent, if the parent likes 00:17:25.830 --> 00:17:28.450 what it sees in the child, it takes it instead. 00:17:28.450 --> 00:17:31.440 So I ask you all the question, would the minimizer 00:17:31.440 --> 00:17:35.490 prefer this 2 that it sees from its child or its own infinity 00:17:35.490 --> 00:17:36.630 for a beta? 00:17:36.630 --> 00:17:37.390 AUDIENCE: 2 00:17:37.390 --> 00:17:38.790 PROFESSOR: It likes the 2. 00:17:38.790 --> 00:17:40.190 That's absolutely right. 00:17:40.190 --> 00:17:44.060 So 2. 00:17:44.060 --> 00:17:48.575 All right, great, so now we go down to F. What is F's alpha? 00:17:51.854 --> 00:17:55.938 Who says negative infinity? 00:17:55.938 --> 00:17:58.220 Who says 2? 00:17:58.220 --> 00:18:01.690 No one-- oh, you guys are good. 00:18:01.690 --> 00:18:03.220 It's negative infinity. 00:18:03.220 --> 00:18:06.094 Technically, it also will have a beta of 2. 00:18:06.094 --> 00:18:07.260 But we're ignoring the beta. 00:18:07.260 --> 00:18:09.890 And the alphas that have been progressing downward 00:18:09.890 --> 00:18:11.962 from the parents-- negative infinity. 00:18:11.962 --> 00:18:14.170 That's why I called it the grandfather clause before. 00:18:14.170 --> 00:18:19.280 Because you would often look up to your grandparent to see what 00:18:19.280 --> 00:18:21.800 your default number is. 00:18:21.800 --> 00:18:23.830 So we get an alpha of negative infinity. 00:18:23.830 --> 00:18:26.134 We then go down to the K. It's a static evaluation. 00:18:26.134 --> 00:18:28.550 And now I'm going to start calling on people individually. 00:18:28.550 --> 00:18:30.450 So hopefully people paid attention 00:18:30.450 --> 00:18:34.010 to the mob, who were always correct. 00:18:34.010 --> 00:18:38.520 All right, so we go down to K. And we see a 3. 00:18:38.520 --> 00:18:40.520 F is a maximizer node. 00:18:40.520 --> 00:18:43.300 So what does F do now? 00:18:43.300 --> 00:18:45.450 AUDIENCE: Switches its alpha to 3. 00:18:45.450 --> 00:18:48.350 PROFESSOR: Yes, switches its alpha to 3, great. 00:18:55.430 --> 00:18:59.005 All right, so that's already quite good. 00:18:59.005 --> 00:19:00.075 It switches alpha to 3. 00:19:00.075 --> 00:19:01.330 It's very happy. 00:19:01.330 --> 00:19:03.526 It's got a 3 here. 00:19:03.526 --> 00:19:06.360 That's a nice value. 00:19:06.360 --> 00:19:16.736 So what does it do at L, the next node? 00:19:16.736 --> 00:19:18.865 It's gone to K, went back up to F. Depth 00:19:18.865 --> 00:19:20.740 first search, the next one would be L, right? 00:19:23.274 --> 00:19:24.149 AUDIENCE: [INAUDIBLE] 00:19:27.376 --> 00:19:32.420 PROFESSOR: Well, technically F could take L's value of 0 00:19:32.420 --> 00:19:34.690 if it liked it better than 3. 00:19:34.690 --> 00:19:35.590 But it's a maximizer. 00:19:35.590 --> 00:19:37.535 So does it want to take that? 00:19:37.535 --> 00:19:38.410 AUDIENCE: [INAUDIBLE] 00:19:38.410 --> 00:19:40.310 PROFESSOR: OK, that technically would be correct. 00:19:40.310 --> 00:19:40.893 But I'm sorry. 00:19:40.893 --> 00:19:43.830 I burdened you with a trick question. 00:19:43.830 --> 00:19:47.180 In fact, we don't look at L at all. 00:19:51.780 --> 00:19:53.510 Does everyone see that? 00:19:53.510 --> 00:19:54.740 I'll explain. 00:19:54.740 --> 00:19:57.190 The alpha at F has reached 3. 00:19:57.190 --> 00:20:00.700 But the beta at B is 2. 00:20:00.700 --> 00:20:03.600 So B looks down and says, wait a minute. 00:20:03.600 --> 00:20:06.990 If I go down to F, my enemy's nuclear option, 00:20:06.990 --> 00:20:10.450 my enemy is the worst it can be for-- the best it 00:20:10.450 --> 00:20:11.960 can be for me is 3. 00:20:11.960 --> 00:20:14.320 F is trumpeting it around. 00:20:14.320 --> 00:20:16.436 I was thinking of eating his heart, or whatever, 00:20:16.436 --> 00:20:17.310 but I didn't want to. 00:20:17.310 --> 00:20:19.050 But it's going to be 3. 00:20:19.050 --> 00:20:24.146 It's going to be 3 or higher down there at F. 00:20:24.146 --> 00:20:25.270 There's no way I want that. 00:20:25.270 --> 00:20:29.310 I already have my own default escape plan. 00:20:29.310 --> 00:20:30.219 And that's 2. 00:20:30.219 --> 00:20:32.260 That's going to be better than whatever comes out 00:20:32.260 --> 00:20:35.871 of that horrible F. So screw it. 00:20:35.871 --> 00:20:39.535 And we never look at L. Does everyone get that? 00:20:39.535 --> 00:20:43.520 That is the main principle of alpha-beta pruning. 00:20:43.520 --> 00:20:47.100 If you see an alpha that's higher than the beta above it-- 00:20:47.100 --> 00:20:50.020 as I said, if alpha goes up above the beta-- 00:20:50.020 --> 00:20:55.360 or if you see a beta, like if there's a beta down here, 00:20:55.360 --> 00:20:59.450 and it's lower than the alpha above it, prune it. 00:20:59.450 --> 00:21:02.450 Stop doing that. 00:21:02.450 --> 00:21:04.270 And the question is, who prunes? 00:21:04.270 --> 00:21:06.840 Who decides that you don't look at L? 00:21:06.840 --> 00:21:09.560 The person who is thinking not to look at L 00:21:09.560 --> 00:21:13.460 is always up higher by at least two levels. 00:21:13.460 --> 00:21:15.670 So up here, B is saying, hmm, I don't 00:21:15.670 --> 00:21:19.130 want to look at L. Because F is already so terrible for me 00:21:19.130 --> 00:21:22.890 that it's just beyond belief. 00:21:22.890 --> 00:21:25.920 If this is 100, it might be 100. 00:21:25.920 --> 00:21:29.350 Even if it's lower, I'm still going to get a three. 00:21:29.350 --> 00:21:33.330 There's a sanity check that I've written that I sort of came 00:21:33.330 --> 00:21:36.180 up with just in case you're not sure that you can skip it. 00:21:36.180 --> 00:21:38.060 Because on a lot of these tests, we ask you, 00:21:38.060 --> 00:21:40.884 which one's do you evaluate, which ones do you skip, right? 00:21:40.884 --> 00:21:42.675 Or we just say, which ones do you evaluate, 00:21:42.675 --> 00:21:44.760 and you don't write the ones that you skip. 00:21:44.760 --> 00:21:48.860 Here's my sanity test to see if you can skip it. 00:21:48.860 --> 00:21:52.080 Ask yourself, if that node that I'm 00:21:52.080 --> 00:21:54.040 about to skip contained a negative infinity 00:21:54.040 --> 00:21:56.550 or some arbitrarily small number, negative infinity 00:21:56.550 --> 00:22:01.260 being the minimizer wins, would it change anything? 00:22:01.260 --> 00:22:02.920 Now that I've answered that, if it 00:22:02.920 --> 00:22:07.350 contained a positive infinity, would it change anything? 00:22:07.350 --> 00:22:11.820 If the answer is no both times, then you're 00:22:11.820 --> 00:22:13.760 definitely correct in pruning it. 00:22:13.760 --> 00:22:14.850 So look at that 0. 00:22:14.850 --> 00:22:16.810 If it was a negative infinity, minimizer wins, 00:22:16.810 --> 00:22:17.705 what would happen? 00:22:17.705 --> 00:22:20.320 The maximizer would say, I'm not touching that was a 10 foot 00:22:20.320 --> 00:22:22.370 pole, choosing 3. 00:22:22.370 --> 00:22:26.380 The minimizer would say, screw that, I'll take E. 00:22:26.380 --> 00:22:28.005 Let's say it was a positive infinity. 00:22:28.005 --> 00:22:31.075 The maximizer would say, eureka, holy grain, I win. 00:22:31.075 --> 00:22:33.080 The minimizer would say, yeah, if I'm a moron, 00:22:33.080 --> 00:22:36.450 and go down to F, and then would go to E and take 2. 00:22:36.450 --> 00:22:41.640 So no matter what was there, the minimizer would go to E. 00:22:41.640 --> 00:22:44.100 And you could say, well, what if it was exactly 2? 00:22:44.100 --> 00:22:46.130 But still the maximizer would choose 00:22:46.130 --> 00:22:48.520 K. The minimizer would go to E. So there's 00:22:48.520 --> 00:22:49.680 no reason to go down there. 00:22:49.680 --> 00:22:51.410 We can just prune it off right now. 00:22:51.410 --> 00:22:54.070 Does everyone agree, everyone see 00:22:54.070 --> 00:22:55.720 what I'm talking about here? 00:22:55.720 --> 00:23:00.225 Great, so we're now done with this branch. 00:23:00.225 --> 00:23:01.790 Because beta is 2. 00:23:01.790 --> 00:23:05.232 So now we're up at old grandpappy A. 00:23:05.232 --> 00:23:06.940 And he has an alpha of negative infinity. 00:23:06.940 --> 00:23:11.660 Everyone, what will he do? 00:23:11.660 --> 00:23:12.709 He'll take the 2. 00:23:12.709 --> 00:23:14.500 It's better than negative infinity for him. 00:23:14.500 --> 00:23:15.600 It's not wonderful. 00:23:15.600 --> 00:23:19.670 But certainly anything is better than an automatic loss. 00:23:19.670 --> 00:23:23.780 All right, now our highest node is a 2. 00:23:23.780 --> 00:23:27.380 So let's keep that in mind for our alpha. 00:23:27.380 --> 00:23:29.820 OK, so let's go over here. 00:23:29.820 --> 00:23:32.854 Let's see, so what will be the value at C? 00:23:32.854 --> 00:23:34.020 What will be the beta value? 00:23:37.366 --> 00:23:40.282 AUDIENCE: [INAUDIBLE] 00:23:42.712 --> 00:23:44.660 PROFESSOR: You go back to which one? 00:23:44.660 --> 00:23:47.090 To G. I'm not at G yet. 00:23:47.090 --> 00:23:49.800 I'm actually just starting the middle branch. 00:23:49.800 --> 00:23:52.280 So I'm going to C. And what's going to be its starting 00:23:52.280 --> 00:23:54.902 beta before I go down? 00:23:54.902 --> 00:23:56.170 AUDIENCE: Infinity. 00:23:56.170 --> 00:23:58.910 PROFESSOR: Infinity, that's right-- default value. 00:23:58.910 --> 00:24:00.570 It's easier than it seemed. 00:24:00.570 --> 00:24:03.690 All right, so yes, beta is equal to infinity. 00:24:06.370 --> 00:24:08.090 This should be better erased. 00:24:08.090 --> 00:24:09.570 I think it's confusing people. 00:24:09.570 --> 00:24:14.790 Great, OK, so beta is equal to infinity at C. 00:24:14.790 --> 00:24:17.110 Now we go down depth first search to G. What's 00:24:17.110 --> 00:24:18.460 going to be our alpha at G? 00:24:18.460 --> 00:24:19.620 AUDIENCE: Minus infinity. 00:24:19.620 --> 00:24:21.560 PROFESSOR: Ahh, it would seem so. 00:24:21.560 --> 00:24:26.110 However, take a look up at the great-grandpappy A. 00:24:26.110 --> 00:24:28.490 It seems to have changed to 2. 00:24:28.490 --> 00:24:29.930 So this time it's 2. 00:24:29.930 --> 00:24:31.680 Why is it 2 instead of negative infinity? 00:24:31.680 --> 00:24:34.873 Why can we let A be so noxious and not start with saying, 00:24:34.873 --> 00:24:36.710 oh, I automatically lose? 00:24:36.710 --> 00:24:40.040 Well, A knows that no matter how awful things get 00:24:40.040 --> 00:24:42.680 in that middle branch, he can just say, 00:24:42.680 --> 00:24:43.930 screw the whole middle branch. 00:24:43.930 --> 00:24:45.110 I'm going to B. 00:24:45.110 --> 00:24:47.796 That's something that the minimizer can't do. 00:24:47.796 --> 00:24:49.920 And we have to start at infinity for the minimizer. 00:24:49.920 --> 00:24:51.140 But the maximizer can. 00:24:51.140 --> 00:24:52.790 Because he has the choice at the top. 00:24:52.790 --> 00:24:53.850 Does everyone see that? 00:24:53.850 --> 00:24:56.440 He can just say, oh, I'm not even going to C. Yeah, 00:24:56.440 --> 00:24:57.330 shows you. 00:24:57.330 --> 00:24:59.320 I'm going to A and taking the 2. 00:24:59.320 --> 00:25:03.640 So therefore alpha is actually 2 at G. 00:25:03.640 --> 00:25:06.500 All right, great, so we've got an alpha that's 00:25:06.500 --> 00:25:10.790 2 at G. We're going to go down to M. It's a minimizer. 00:25:10.790 --> 00:25:14.407 All right, what's going to be our beta value at M? 00:25:14.407 --> 00:25:15.282 AUDIENCE: [INAUDIBLE] 00:25:19.230 --> 00:25:21.325 PROFESSOR: Or which is the beta default, minus 00:25:21.325 --> 00:25:22.200 or positive infinity? 00:25:22.200 --> 00:25:24.554 What would be the minimizer? 00:25:24.554 --> 00:25:25.345 AUDIENCE: Positive. 00:25:25.345 --> 00:25:27.180 PROFESSOR: Positive infinity, that's right. 00:25:27.180 --> 00:25:29.250 M is going to be a positive infinity for beta. 00:25:29.250 --> 00:25:34.670 Again, it picks it up from C. Great, 00:25:34.670 --> 00:25:39.030 now we get to some actual values. 00:25:39.030 --> 00:25:41.390 So we're at some actual values. 00:25:41.390 --> 00:25:45.520 We are at Q. So what's going to happen 00:25:45.520 --> 00:25:47.230 at M when M sees that Q is 1? 00:25:53.431 --> 00:25:55.040 AUDIENCE: [INAUDIBLE] 00:25:55.040 --> 00:25:56.040 PROFESSOR: What is beta? 00:25:56.040 --> 00:25:56.371 It says infinity. 00:25:56.371 --> 00:25:57.890 I'm sorry, it's hard to read. 00:25:57.890 --> 00:25:59.486 Beta is infinity at M. 00:25:59.486 --> 00:26:01.470 AUDIENCE: OK, so it's going to minimize, right? 00:26:01.470 --> 00:26:05.001 So it's going to be like, OK, [INAUDIBLE]. 00:26:05.001 --> 00:26:06.000 PROFESSOR: That's right. 00:26:06.000 --> 00:26:08.270 So they're going to put beta to 1. 00:26:08.270 --> 00:26:16.250 Because it sees Q. Great, so my next question is, 00:26:16.250 --> 00:26:19.124 what's going to happen at R? 00:26:19.124 --> 00:26:22.052 AUDIENCE: [INAUDIBLE] 00:26:28.030 --> 00:26:30.010 PROFESSOR: Very smart. 00:26:30.010 --> 00:26:31.397 You've detected my trap. 00:26:31.397 --> 00:26:32.855 The question is, does it look at R? 00:26:32.855 --> 00:26:35.142 The answer is, no. 00:26:35.142 --> 00:26:37.100 It doesn't look at R. Why doesn't it look at R? 00:26:37.100 --> 00:26:39.020 Does everyone see? 00:26:39.020 --> 00:26:42.330 Yeah, alpha is now greater than the beta below it. 00:26:42.330 --> 00:26:43.917 Beta has gotten lower than alpha. 00:26:43.917 --> 00:26:46.000 This is the same thing I was talking about before, 00:26:46.000 --> 00:26:48.090 when we figured out that the alpha here is 2. 00:26:48.090 --> 00:26:52.470 The maximizer says, wait a minute. 00:26:52.470 --> 00:26:55.740 The maximizer G says, if I go to M, the best I'm getting out 00:26:55.740 --> 00:26:56.759 of this is 1. 00:26:56.759 --> 00:26:58.300 Because if this is negative infinity, 00:26:58.300 --> 00:26:59.680 the minimizer will choose it. 00:26:59.680 --> 00:27:01.555 If this is positive infinity, he'll choose 1. 00:27:01.555 --> 00:27:04.237 The best I'm going to get out of here is 1. 00:27:04.237 --> 00:27:06.320 If that's the case, I might as well have just gone 00:27:06.320 --> 00:27:09.600 to B and not even gone to C. So I'm not going to go to M. 00:27:09.600 --> 00:27:10.440 I'll go to N, maybe. 00:27:10.440 --> 00:27:13.370 Maybe N is better. 00:27:13.370 --> 00:27:15.300 Does everyone see that? 00:27:15.300 --> 00:27:20.270 Great, so let's say that the maximizer does go to N. 00:27:20.270 --> 00:27:23.379 So what's going to happen with this alpha? 00:27:23.379 --> 00:27:24.800 AUDIENCE: [INAUDIBLE] 00:27:24.800 --> 00:27:26.640 PROFESSOR: That's right, it's going to be 7. 00:27:26.640 --> 00:27:28.080 7 is better than 2. 00:27:28.080 --> 00:27:30.960 And the maximizer has control to get to that seven, 00:27:30.960 --> 00:27:36.940 at least if it gets to G. All right, now the minimizer at C-- 00:27:36.940 --> 00:27:38.580 we'll do everyone this time. 00:27:38.580 --> 00:27:43.798 The minimizer at C, seeing that 7, what does the minimizer do? 00:27:43.798 --> 00:27:44.298 Anyone? 00:27:48.200 --> 00:27:49.210 So it sees the 7. 00:27:49.210 --> 00:27:50.680 What does it do to its beta? 00:27:50.680 --> 00:27:55.330 It takes the 7-- better than infinity, anyway. 00:27:55.330 --> 00:27:58.290 And yeah, then it checks H. And everybody, again, what 00:27:58.290 --> 00:28:00.100 happens at H? 00:28:00.100 --> 00:28:00.970 It takes the 6. 00:28:00.970 --> 00:28:01.865 It's lower than 7. 00:28:04.730 --> 00:28:07.747 All right, now we'll go back to having 00:28:07.747 --> 00:28:08.830 people do it on their own. 00:28:08.830 --> 00:28:10.246 Well, all the way back to the top, 00:28:10.246 --> 00:28:13.216 what does A do when it sees the 6 coming out of C? 00:28:13.216 --> 00:28:15.150 AUDIENCE: Changes to 6. 00:28:15.150 --> 00:28:17.330 PROFESSOR: Changes to 6, that's right. 00:28:17.330 --> 00:28:18.910 Alpha equals 6. 00:28:18.910 --> 00:28:24.620 Great-- homestretch, people, homestretch. 00:28:24.620 --> 00:28:31.650 So the minimizer, everyone, has a beta of infinity. 00:28:31.650 --> 00:28:35.070 And if I wasn't a static node, it would have an alpha of 6. 00:28:35.070 --> 00:28:36.090 But it is a static node. 00:28:36.090 --> 00:28:38.570 So it just has a value of 1. 00:28:38.570 --> 00:28:43.380 So since it has a value of 1, everyone, the beta becomes 1. 00:28:43.380 --> 00:28:45.264 And what next, everyone? 00:28:45.264 --> 00:28:45.930 AUDIENCE: Prune. 00:28:45.930 --> 00:28:49.960 PROFESSOR: Prune, that's right. 00:28:49.960 --> 00:28:51.110 Why prune? 00:28:51.110 --> 00:28:54.650 Well, this time it's A himself who can prune. 00:28:54.650 --> 00:28:56.723 A says, well darn, if I go to D, I'm 00:28:56.723 --> 00:28:59.360 going to get 1 or something even worse than 1. 00:28:59.360 --> 00:29:02.550 I might as well take my 6 while I have it, 00:29:02.550 --> 00:29:04.875 prune all the rest all the way down. 00:29:09.210 --> 00:29:10.620 Everyone see that? 00:29:10.620 --> 00:29:12.710 Everyone cool with that? 00:29:12.710 --> 00:29:16.060 It's not too bad if you take it one step at a time. 00:29:16.060 --> 00:29:18.410 We did it. 00:29:18.410 --> 00:29:21.966 Our question is, which nodes are evaluated in order? 00:29:21.966 --> 00:29:35.170 Our answer is, everyone-- E, K, Q, N, H, I. OK, not so obvious, 00:29:35.170 --> 00:29:35.670 I guess. 00:29:35.670 --> 00:29:36.980 A few people followed me. 00:29:36.980 --> 00:29:40.590 But it is E, K, Q, N, H, I. It's just depth first order. 00:29:40.590 --> 00:29:43.220 And we pruned some of them away. 00:29:43.220 --> 00:29:47.260 Great, so that is alpha-beta. 00:29:47.260 --> 00:29:49.470 Any questions about that before I give some questions 00:29:49.470 --> 00:29:52.022 about progressive deepening? 00:29:52.022 --> 00:29:53.230 All right, we've got a bunch. 00:29:53.230 --> 00:29:55.182 So first question. 00:29:55.182 --> 00:30:02.480 AUDIENCE: [INAUDIBLE] nodes like F, B, C, and D? 00:30:02.480 --> 00:30:04.190 PROFESSOR: The question is, when asked 00:30:04.190 --> 00:30:07.900 for the order of evaluation, are we excluding F, B, C, and D? 00:30:07.900 --> 00:30:11.250 The answer is we're talking about here static evaluation. 00:30:11.250 --> 00:30:14.720 The static evaluator is a very important and interesting 00:30:14.720 --> 00:30:15.400 function. 00:30:15.400 --> 00:30:18.230 And I'll get back to something a few students have asked me 00:30:18.230 --> 00:30:21.484 about the static evaluator later and try to explain what it is. 00:30:21.484 --> 00:30:23.650 It's basically the thing that pops out those numbers 00:30:23.650 --> 00:30:25.015 at the bottom of the leaves. 00:30:25.015 --> 00:30:26.765 So when we ask, what is the order of nodes 00:30:26.765 --> 00:30:31.510 that were statically evaluated, we mean leaves only. 00:30:31.510 --> 00:30:32.580 That's a good question. 00:30:32.580 --> 00:30:33.620 Any other questions? 00:30:33.620 --> 00:30:35.430 Let's see, there was one up here before. 00:30:35.430 --> 00:30:36.170 But it's gone. 00:30:36.170 --> 00:30:37.885 It might have been the same one. 00:30:37.885 --> 00:30:38.384 Question? 00:30:38.384 --> 00:30:39.717 AUDIENCE: So a similar question. 00:30:39.717 --> 00:30:42.578 When you say, static nodes, that just means the leaf nodes? 00:30:42.578 --> 00:30:43.750 PROFESSOR: Means the leaf nodes, that's right. 00:30:43.750 --> 00:30:46.041 The question is, does static nodes mean the leaf nodes. 00:30:46.041 --> 00:30:46.903 The answer is yes. 00:30:46.903 --> 00:30:48.319 AUDIENCE: And so static evaluation 00:30:48.319 --> 00:30:51.737 is when you compare the value of a static node to something? 00:30:51.737 --> 00:30:53.570 PROFESSOR: Static evaluation is when you get 00:30:53.570 --> 00:30:55.180 that number, the static node. 00:30:55.180 --> 00:30:56.480 Let me explain. 00:30:56.480 --> 00:30:59.140 Unless someone else has another question about alpha-beta, 00:30:59.140 --> 00:31:00.500 let me explain static values. 00:31:00.500 --> 00:31:02.030 Because I was about to do that. 00:31:02.030 --> 00:31:03.571 There is a question about alpha-beta. 00:31:03.571 --> 00:31:06.326 I'll come back to both of yours after I answer this. 00:31:06.326 --> 00:31:09.437 AUDIENCE: You were mentioning [INAUDIBLE]. 00:31:09.437 --> 00:31:11.385 And I'm a little bit confused. 00:31:11.385 --> 00:31:13.576 If you're looking at one node, and you're 00:31:13.576 --> 00:31:16.255 seeing either grab the value from the grandparent 00:31:16.255 --> 00:31:18.203 or grab it from the-- 00:31:18.203 --> 00:31:21.134 PROFESSOR: So it always starts-- the question is, what 00:31:21.134 --> 00:31:22.300 is the Snow White principle? 00:31:22.300 --> 00:31:23.380 How does it work? 00:31:23.380 --> 00:31:26.760 Every node always starts off with taking 00:31:26.760 --> 00:31:29.865 the value of the same type, alpha or beta, 00:31:29.865 --> 00:31:30.740 from its grandparent. 00:31:30.740 --> 00:31:33.010 It always starts that way. 00:31:33.010 --> 00:31:34.427 Now, you say, why the grandparent? 00:31:34.427 --> 00:31:35.926 Wouldn't it take it from the parent? 00:31:35.926 --> 00:31:37.050 It actually does. 00:31:37.050 --> 00:31:39.430 But I'm not drawing out the alphas at all the minimizer 00:31:39.430 --> 00:31:39.700 levels. 00:31:39.700 --> 00:31:40.991 Because they don't do anything. 00:31:40.991 --> 00:31:44.200 They're only even there to pass them down. 00:31:44.200 --> 00:31:48.110 So all of the values pass down, down, down, down, down 00:31:48.110 --> 00:31:49.050 to begin. 00:31:49.050 --> 00:31:55.390 Every node, in fact, starts off with its grandparents 00:31:55.390 --> 00:31:57.920 with its parents' values, OK? 00:31:57.920 --> 00:32:00.730 But then when the node sees a child, 00:32:00.730 --> 00:32:02.780 it's completely done evaluating. 00:32:02.780 --> 00:32:05.360 It's finished. 00:32:05.360 --> 00:32:08.380 It can't be in the process. 00:32:08.380 --> 00:32:13.550 Let's say C. When C sees that G is completely 00:32:13.550 --> 00:32:15.780 done with all of its sub-branches 00:32:15.780 --> 00:32:18.760 and is ready to return a value, or if it's just 00:32:18.760 --> 00:32:22.120 a static evaluation, then it's automatically completely done. 00:32:22.120 --> 00:32:25.020 Because it has no children. 00:32:25.020 --> 00:32:29.550 A static value like K of 3 is automatically completely done. 00:32:29.550 --> 00:32:30.830 It's got a 3. 00:32:30.830 --> 00:32:33.360 Similarly, when we came back to G after going to N, 00:32:33.360 --> 00:32:36.140 and we knew that the value was 7, that was completely done. 00:32:36.140 --> 00:32:37.730 The value was definitely 7. 00:32:37.730 --> 00:32:39.480 There was no other possibilities. 00:32:39.480 --> 00:32:40.830 AUDIENCE: That's after looking at the children, right? 00:32:40.830 --> 00:32:41.500 PROFESSOR: Yes. 00:32:41.500 --> 00:32:44.630 So once you're done with all the children of G, 00:32:44.630 --> 00:32:46.250 then G comes up and says, guess what? 00:32:46.250 --> 00:32:47.409 Guess what, guys? 00:32:47.409 --> 00:32:48.950 So technically before that, you would 00:32:48.950 --> 00:32:53.210 have said that G's alpha is greater than or equal to 1 00:32:53.210 --> 00:32:55.950 when we looked at Q. And then we looked at M. We'd say, 00:32:55.950 --> 00:32:57.640 it's equal exactly to 7. 00:32:57.640 --> 00:32:58.890 We're done here. 00:32:58.890 --> 00:33:02.170 And then at that point, when it's fresh and ripe 00:33:02.170 --> 00:33:04.890 and has all of its highest value or its best value, 00:33:04.890 --> 00:33:08.100 that's when the parent can eat its heart and gain that value 00:33:08.100 --> 00:33:08.980 itself. 00:33:08.980 --> 00:33:12.340 So that's when C says, for instance, oh man, 00:33:12.340 --> 00:33:14.550 I have an infinity. 00:33:14.550 --> 00:33:16.060 I really like that 7 better. 00:33:16.060 --> 00:33:17.390 And it takes the 7. 00:33:17.390 --> 00:33:19.840 But then it saw H. And it said, oh man, that's a 6. 00:33:19.840 --> 00:33:20.980 That's even better than 7. 00:33:20.980 --> 00:33:21.765 So it took the 6. 00:33:21.765 --> 00:33:24.510 AUDIENCE: So shouldn't the alpha take 7 then? 00:33:24.510 --> 00:33:25.780 PROFESSOR: So alpha takes 6. 00:33:25.780 --> 00:33:27.300 Because C is a minimizer. 00:33:27.300 --> 00:33:29.960 C took the 7 from G, but then right 00:33:29.960 --> 00:33:32.810 after that C saw H and took the 6. 00:33:32.810 --> 00:33:35.880 Because 6 is even lower than 7. 00:33:35.880 --> 00:33:38.160 And then alpha took the 6. 00:33:38.160 --> 00:33:40.260 Because 6 was higher than 2. 00:33:40.260 --> 00:33:42.824 AUDIENCE: So it's not going to look below the branch? 00:33:42.824 --> 00:33:45.240 PROFESSOR: Yeah, the problem is that the maximizer doesn't 00:33:45.240 --> 00:33:46.490 have control there. 00:33:46.490 --> 00:33:48.464 The minimizer has got control at C. 00:33:48.464 --> 00:33:49.880 And the minimizer is going to make 00:33:49.880 --> 00:33:52.020 sure it's as low as possible. 00:33:52.020 --> 00:33:56.470 The maximizer at A, his only control, or her only control, 00:33:56.470 --> 00:33:59.460 is the ability to send either way to B or C 00:33:59.460 --> 00:34:02.346 or D. And then at that point, at C, 00:34:02.346 --> 00:34:04.345 the minimizer gets to choose if we go to G or H. 00:34:04.345 --> 00:34:07.300 And it's never going to choose G. Because G is higher than H. 00:34:07.300 --> 00:34:09.829 All right, awesome, was there another question? 00:34:09.829 --> 00:34:12.830 All right, let's go back to static evaluations. 00:34:12.830 --> 00:34:16.139 When I first took this class, I had some weird thoughts 00:34:16.139 --> 00:34:17.139 about static evolutions. 00:34:17.139 --> 00:34:18.739 I heard some students ask me this. 00:34:18.739 --> 00:34:21.475 I almost got a question about it onto one of the tests, 00:34:21.475 --> 00:34:23.600 but it was edited to some other weird question that 00:34:23.600 --> 00:34:25.451 was m to the b to the d minus 1 or something 00:34:25.451 --> 00:34:26.659 like that at the last minute. 00:34:26.659 --> 00:34:28.909 So I'm going to pose you guys the actual question that 00:34:28.909 --> 00:34:31.520 would have been on one of the older test, which 00:34:31.520 --> 00:34:32.790 is the following. 00:34:32.790 --> 00:34:34.873 I had a student who came to me and said, you know, 00:34:34.873 --> 00:34:37.489 [INAUDIBLE], when we do this alpha-beta pruning, and all 00:34:37.489 --> 00:34:40.560 this other stuff, we're trying to assume that we're really 00:34:40.560 --> 00:34:42.870 saving that much time by getting rid 00:34:42.870 --> 00:34:44.361 of a few static evaluations. 00:34:44.361 --> 00:34:46.110 In fact, when we do progressive deepening, 00:34:46.110 --> 00:34:48.401 we're always just counting, how many static evaluations 00:34:48.401 --> 00:34:49.290 do we have to do? 00:34:49.290 --> 00:34:51.586 And he said, I look at these static evaluations. 00:34:51.586 --> 00:34:52.710 And there's just a 3 there. 00:34:52.710 --> 00:34:56.520 It takes no time to do the static evaluation. 00:34:56.520 --> 00:34:57.890 It's on the board. 00:34:57.890 --> 00:34:59.920 It takes much longer to do the alpha-beta. 00:34:59.920 --> 00:35:02.770 It's faster by far to not do alpha-beta. 00:35:02.770 --> 00:35:04.680 So I then tried to explain to that student. 00:35:04.680 --> 00:35:07.050 I said, OK, we need to be clear about what 00:35:07.050 --> 00:35:08.340 static evaluations are. 00:35:08.340 --> 00:35:09.590 You guys get it easy. 00:35:09.590 --> 00:35:11.657 We put these numbers on the board. 00:35:11.657 --> 00:35:13.240 A static evaluation-- let's say you're 00:35:13.240 --> 00:35:17.010 playing a game like chess. 00:35:17.010 --> 00:35:19.050 Static evaluation takes a long time. 00:35:19.050 --> 00:35:21.910 When I was in 6170, Java [INAUDIBLE], 00:35:21.910 --> 00:35:24.150 the class that used to exist, we had a program 00:35:24.150 --> 00:35:28.440 called Anti-Chess where I used my 6034 skills to write the AI. 00:35:28.440 --> 00:35:31.890 And the static evaluator took a long time. 00:35:31.890 --> 00:35:32.970 And we were timed. 00:35:32.970 --> 00:35:34.970 So getting the static evaluator faster, 00:35:34.970 --> 00:35:36.602 that was the most important thing. 00:35:36.602 --> 00:35:37.810 Why does it take a long time? 00:35:37.810 --> 00:35:39.875 Well, the static evaluator is an evaluation 00:35:39.875 --> 00:35:43.730 of the board position, the state of the game, 00:35:43.730 --> 00:35:45.519 at a snapshot of time. 00:35:45.519 --> 00:35:48.060 And that's not as easy as just saying, oh, here's the answer. 00:35:48.060 --> 00:35:51.500 Because in chess, first of all, not only 00:35:51.500 --> 00:35:53.780 did I have to look at how many pieces I had, 00:35:53.780 --> 00:35:56.520 what areas that I controlled. 00:35:56.520 --> 00:35:58.154 Also-- well, it was anti-chess. 00:35:58.154 --> 00:35:59.320 But that's not withstanding. 00:35:59.320 --> 00:36:01.110 Let's pretend it's regular chess. 00:36:01.110 --> 00:36:03.900 I also had to look, if it was in regular chess-- 00:36:03.900 --> 00:36:05.650 and I still had to do this in anti-chess-- 00:36:05.650 --> 00:36:06.850 if my king was in check. 00:36:06.850 --> 00:36:08.350 And what that meant is I had to look 00:36:08.350 --> 00:36:11.040 at all of my opponent's moves, possible moves, 00:36:11.040 --> 00:36:13.120 to see if anyone of them could take my king. 00:36:13.120 --> 00:36:15.921 Because in regular chess, it's illegal to put your king 00:36:15.921 --> 00:36:16.420 into check. 00:36:16.420 --> 00:36:18.830 So you better not even allow that move. 00:36:18.830 --> 00:36:20.500 And regardless, getting into checkmate 00:36:20.500 --> 00:36:22.670 is negative infinity for you. 00:36:22.670 --> 00:36:27.100 So it takes a really long time to do static evaluations, 00:36:27.100 --> 00:36:29.380 at least good ones, usually. 00:36:29.380 --> 00:36:30.400 You want to avoid them. 00:36:30.400 --> 00:36:32.740 Because they're not just some number on the page. 00:36:32.740 --> 00:36:34.910 They are some function you wrote that does 00:36:34.910 --> 00:36:38.230 a very careful analysis of the state of the game and says, 00:36:38.230 --> 00:36:41.770 I'm good to heuristically guess that my value is 00:36:41.770 --> 00:36:45.580 pi, or some other number, and then rates 00:36:45.580 --> 00:36:47.060 that compared to other states. 00:36:47.060 --> 00:36:49.530 Does that make sense to everyone? 00:36:49.530 --> 00:36:51.790 So the answer to the hypothetical question 00:36:51.790 --> 00:36:53.628 that might have been on the old test, when 00:36:53.628 --> 00:36:56.086 the person said, I've got this great idea where you do tons 00:36:56.086 --> 00:36:57.544 of static evaluation, and you don't 00:36:57.544 --> 00:37:00.620 have to do this long alpha-beta, is, don't do that. 00:37:00.620 --> 00:37:04.805 The static evaluations actually take a long time. 00:37:04.805 --> 00:37:06.680 Does that clear it up for people who asked me 00:37:06.680 --> 00:37:11.130 before about what is a static evaluation, why are the leaf 00:37:11.130 --> 00:37:12.030 nodes called static? 00:37:14.730 --> 00:37:17.550 And you might ask, why are some of these static just 00:37:17.550 --> 00:37:19.140 arbitrarily? 00:37:19.140 --> 00:37:23.030 The answer is, when you're running out of time to expand 00:37:23.030 --> 00:37:26.270 deeper, and you just need to stop that stage of the game-- 00:37:26.270 --> 00:37:27.744 maybe it's just getting too hairy, 00:37:27.744 --> 00:37:29.160 maybe it's spreading out too much, 00:37:29.160 --> 00:37:30.760 you have some heuristic that says, 00:37:30.760 --> 00:37:33.500 this is where I stop for now-- it's 00:37:33.500 --> 00:37:35.740 a heuristic guess of the value. 00:37:35.740 --> 00:37:37.990 It's kind of like those heuristic values in the search 00:37:37.990 --> 00:37:38.330 tree. 00:37:38.330 --> 00:37:39.704 It's a guess of how much work you 00:37:39.704 --> 00:37:41.340 have left to get to the goal. 00:37:41.340 --> 00:37:43.720 Here, you say, well, I wish I could go deeper. 00:37:43.720 --> 00:37:45.100 But I just don't have the time. 00:37:45.100 --> 00:37:48.250 So here's how I think I'm doing at this level. 00:37:48.250 --> 00:37:49.359 It's not always right. 00:37:49.359 --> 00:37:51.150 And that's going to lead us into the answer 00:37:51.150 --> 00:37:53.210 to one of the questions about progressive deepening. 00:37:53.210 --> 00:37:54.876 So I'll put up the progressive deepening 00:37:54.876 --> 00:37:56.050 question really quickly. 00:37:56.050 --> 00:38:00.610 So the question is this. 00:38:00.610 --> 00:38:02.610 Let me see, this is a maximizer-- yes. 00:38:02.610 --> 00:38:10.515 Suppose that we do progressive deepening on the tree that 00:38:10.515 --> 00:38:11.600 is only two levels deep. 00:38:11.600 --> 00:38:13.620 What is progressive deepening in a nutshell if you don't 00:38:13.620 --> 00:38:14.930 remember from the lecture? 00:38:14.930 --> 00:38:17.890 The idea is this. 00:38:17.890 --> 00:38:20.080 In this tree, it doesn't work. 00:38:20.080 --> 00:38:22.550 But in trees that actually branch like 2 to the n, 00:38:22.550 --> 00:38:26.280 it doesn't take that much time to do some of the top levels 00:38:26.280 --> 00:38:29.240 first and then move on to the bottom levels. 00:38:29.240 --> 00:38:31.250 Just do them one at a time. 00:38:31.250 --> 00:38:33.840 So let's say we only did it up through J. We only did 00:38:33.840 --> 00:38:35.890 the top two levels of the tree. 00:38:35.890 --> 00:38:38.690 We'd like to reorder the tree so that alpha-beta 00:38:38.690 --> 00:38:43.250 can prune as much as it possibly can, at least we hope. 00:38:43.250 --> 00:38:47.240 So let's pretend that we had a psychic awesome genius 00:38:47.240 --> 00:38:50.520 friend who told us that the static values when we went up 00:38:50.520 --> 00:38:52.680 to two levels-- remember, when we go to two levels, 00:38:52.680 --> 00:38:55.440 F, G, and J have to get a static value, right? 00:38:55.440 --> 00:38:56.750 Because we're not going down. 00:38:56.750 --> 00:38:58.180 We do a static evaluation. 00:38:58.180 --> 00:39:02.410 They get the exact correct numbers-- 3, 7, and 20. 00:39:02.410 --> 00:39:03.573 Genius, brilliant. 00:39:03.573 --> 00:39:07.500 All right, so if that happens, what 00:39:07.500 --> 00:39:10.200 is the best way that we could reorder that tree? 00:39:10.200 --> 00:39:18.880 Oh yeah, so it's A, B, C, D with values of 2, 3, 7, 6, 1, 20. 00:39:18.880 --> 00:39:21.880 I'll draw that. 00:39:21.880 --> 00:39:23.390 This is the non-reordered tree. 00:39:29.890 --> 00:39:41.410 Let's see, so it's 2, 3, 7, 6, 1, 20. 00:39:41.410 --> 00:39:43.584 So what's the best way to reorder? 00:39:43.584 --> 00:39:45.250 Well, first of all, does anyone remember 00:39:45.250 --> 00:39:49.300 what Patrick said when he talked about progressive deepening? 00:39:49.300 --> 00:39:53.800 Usually no one does, so don't worry about it. 00:39:53.800 --> 00:39:55.880 Because at that time you guys didn't think, 00:39:55.880 --> 00:39:57.390 oh, I have to do this for the quiz. 00:39:57.390 --> 00:39:58.987 You were just thinking, oh man, we've 00:39:58.987 --> 00:40:01.070 already heard alpha-beta and all this other stuff. 00:40:01.070 --> 00:40:02.570 And this is just a small fact. 00:40:02.570 --> 00:40:04.650 But it's a very important fact. 00:40:04.650 --> 00:40:06.720 And now you know you have to do it for the quiz. 00:40:06.720 --> 00:40:08.600 So you're probably going to remember it. 00:40:08.600 --> 00:40:13.510 The way you do it is you try to guess, and you say, 00:40:13.510 --> 00:40:16.030 which one of these is going to be a winner? 00:40:16.030 --> 00:40:19.530 Whichever one I think is going to be a winner at that level, 00:40:19.530 --> 00:40:21.360 I put first. 00:40:21.360 --> 00:40:22.500 Why is that the case? 00:40:22.500 --> 00:40:27.110 Well, something interesting you may have noticed here-- 00:40:27.110 --> 00:40:30.590 whenever you have a winner, like the middle node, 00:40:30.590 --> 00:40:34.060 or whenever you have whatever is the current best 00:40:34.060 --> 00:40:36.790 for your alpha, you sort of have to explore out 00:40:36.790 --> 00:40:39.160 a lot of that area. 00:40:39.160 --> 00:40:42.390 Like for instance, the left node was our current best at 2. 00:40:42.390 --> 00:40:44.250 The middle branch was our current best, 00:40:44.250 --> 00:40:45.130 at that time was 6. 00:40:45.130 --> 00:40:46.382 It was the total best. 00:40:46.382 --> 00:40:48.090 We had to explore a good number of nodes. 00:40:48.090 --> 00:40:50.370 But on the right, we just saw, oh, there's 1. 00:40:50.370 --> 00:40:50.930 We're done. 00:40:50.930 --> 00:40:53.070 We cut everything off. 00:40:53.070 --> 00:40:55.150 In other words, the branch that turns out 00:40:55.150 --> 00:40:57.052 to be the one that you take, you have 00:40:57.052 --> 00:40:58.760 to do a pretty good amount of exploration 00:40:58.760 --> 00:41:00.554 to prove that it's the right one. 00:41:00.554 --> 00:41:01.970 Whereas if it's the wrong one, you 00:41:01.970 --> 00:41:05.320 can sometimes with just one node say, this is wrong, done. 00:41:05.320 --> 00:41:07.100 So therefore, if the one that turns out 00:41:07.100 --> 00:41:10.510 to be the eventual winner is first of all, 00:41:10.510 --> 00:41:13.290 then it's really easy to reject all the other branches. 00:41:13.290 --> 00:41:16.490 Do people see that sort of conceptually a little bit, 00:41:16.490 --> 00:41:19.250 that if you get the best node right away, 00:41:19.250 --> 00:41:22.390 you can just reject all the wrong ones pretty quickly? 00:41:22.390 --> 00:41:23.560 That's our goal. 00:41:23.560 --> 00:41:25.970 So how can we, quote, "get the right one," the best one 00:41:25.970 --> 00:41:27.040 right away? 00:41:27.040 --> 00:41:30.050 Well, here's how we do it. 00:41:30.050 --> 00:41:33.590 Let's say we're at B. Which one is the minimizer likely to pick 00:41:33.590 --> 00:41:35.800 assuming that our heuristic is good 00:41:35.800 --> 00:41:38.920 and that these guesses are pretty much close to the truth? 00:41:38.920 --> 00:41:41.261 It turns out they're perfect, so this is going to work. 00:41:41.261 --> 00:41:42.760 So which one will the minimizer pick 00:41:42.760 --> 00:41:44.801 if it has to choose between E and F, do we think? 00:41:44.801 --> 00:41:45.534 AUDIENCE: E. 00:41:45.534 --> 00:41:46.450 PROFESSOR: E, perfect. 00:41:46.450 --> 00:41:48.981 Which one will it pick between G and H? 00:41:48.981 --> 00:41:49.481 AUDIENCE: H. 00:41:49.481 --> 00:41:51.689 PROFESSOR: H. Which one will it pick between I and J? 00:41:51.689 --> 00:41:52.470 AUDIENCE: I. 00:41:52.470 --> 00:41:54.660 PROFESSOR: OK, so what we're saying 00:41:54.660 --> 00:41:56.790 is we think it's going to pick E. 00:41:56.790 --> 00:41:59.576 We think it's going to pick H. We think it's going to pick I. 00:41:59.576 --> 00:42:02.830 So first of all, we should put E before F, H before G, 00:42:02.830 --> 00:42:06.490 and I before J. Because we think it's going to pick those first. 00:42:06.490 --> 00:42:10.190 Those are probably our best ones to invalidate a poor branch. 00:42:10.190 --> 00:42:12.360 So now between 2, 6, and 1, which 00:42:12.360 --> 00:42:15.110 is what we think we're going to get, which one do we think 00:42:15.110 --> 00:42:16.691 the maximizer is going to take? 00:42:16.691 --> 00:42:17.190 AUDIENCE: 6. 00:42:17.190 --> 00:42:18.050 PROFESSOR: 6. 00:42:18.050 --> 00:42:21.940 Then if it couldn't take 6, what would be its next best choice? 00:42:21.940 --> 00:42:23.690 2, then 1. 00:42:23.690 --> 00:42:27.120 That's just our order-- simple as that. 00:42:27.120 --> 00:42:28.630 It couldn't be anything easier that 00:42:28.630 --> 00:42:31.130 evolves really complex trees, a huge number of numbers, 00:42:31.130 --> 00:42:33.200 and reordering those trees. 00:42:33.200 --> 00:42:42.905 So C-- you guys told me C, B, D. You told me C, B, D, I think? 00:42:42.905 --> 00:42:44.780 Yeah, those are the ones the maximizer likes. 00:42:44.780 --> 00:42:49.740 And then the ones the minimizer likes you told me was H, 00:42:49.740 --> 00:42:53.150 and before G. Because H is smaller than G. 00:42:53.150 --> 00:43:03.710 You guys told me E before F. And you guys told me I before J. 00:43:03.710 --> 00:43:07.120 And you guys would be correct in all regards. 00:43:07.120 --> 00:43:12.490 We have 6, 7, 2, 3, 1, 20. 00:43:12.490 --> 00:43:15.450 All the minimizers choose from smallest to highest. 00:43:15.450 --> 00:43:19.840 The maximizer chooses from highest to lowest of the ones 00:43:19.840 --> 00:43:21.690 that the minimizers will take. 00:43:21.690 --> 00:43:24.320 And if we did that, you can see we would probably 00:43:24.320 --> 00:43:26.160 save some time. 00:43:26.160 --> 00:43:27.640 Let's see how much time. 00:43:27.640 --> 00:43:30.770 Let's say we looked at H first. 00:43:30.770 --> 00:43:34.010 Well, if we looked at H first, we 00:43:34.010 --> 00:43:39.060 would still have actually had to look at Q and N. However, 00:43:39.060 --> 00:43:43.340 we would not have had to look at K. Do people see why? 00:43:43.340 --> 00:43:44.970 If we already knew this branch was 6, 00:43:44.970 --> 00:43:48.160 as soon as we saw 2 for the beta here-- 2 is less than 6-- 00:43:48.160 --> 00:43:49.752 we could have pruned. 00:43:49.752 --> 00:43:51.710 We still would have had to look at I over here. 00:43:51.710 --> 00:43:53.480 Because you have to look at at least one 00:43:53.480 --> 00:43:55.760 thing in the new sub-branch. 00:43:55.760 --> 00:44:00.500 And it actually only would have saved us one node-- oops. 00:44:00.500 --> 00:44:04.740 So it winds up that in total, how many nodes would 00:44:04.740 --> 00:44:10.350 we have evaluated if we did that little scheme of reordering? 00:44:10.350 --> 00:44:17.050 Well, we normally had to do six-- E, K, Q, N, H, I. 00:44:17.050 --> 00:44:20.391 How many do we evaluate if we do this progressive deepening 00:44:20.391 --> 00:44:20.890 scheme? 00:44:20.890 --> 00:44:22.845 How many times do we run the static evaluator, 00:44:22.845 --> 00:44:24.970 which of course you know the static evaluator takes 00:44:24.970 --> 00:44:26.620 a long time? 00:44:26.620 --> 00:44:29.400 Anyone have a guess? 00:44:29.400 --> 00:44:36.170 I told you the only one we don't evaluate is K. Raise your hand. 00:44:36.170 --> 00:44:40.460 I won't make anyone give this one. 00:44:40.460 --> 00:44:43.410 So I said the only one we save on is K. 00:44:43.410 --> 00:44:49.180 So we still do E, Q, N, H, and I over here. 00:44:49.180 --> 00:44:51.180 There's two possible answers that I will accept. 00:44:51.180 --> 00:44:53.171 So you have a higher chance of guessing it. 00:44:53.171 --> 00:44:53.670 Anyway? 00:44:57.200 --> 00:44:59.405 Does everyone agree that we did six before? 00:44:59.405 --> 00:45:01.210 If we didn't do any progressive deepening, 00:45:01.210 --> 00:45:08.932 we just did E, K, Q, N, H, I. And now we're not doing K. OK, 00:45:08.932 --> 00:45:09.890 people are saying five. 00:45:09.890 --> 00:45:11.074 All right, good. 00:45:11.074 --> 00:45:12.240 That's not the right answer. 00:45:12.240 --> 00:45:15.680 But it at least shows that you can do taking away the one. 00:45:15.680 --> 00:45:17.489 We did at least five over here. 00:45:17.489 --> 00:45:19.030 There's two possible answers, though. 00:45:19.030 --> 00:45:20.412 Because look over there. 00:45:20.412 --> 00:45:22.120 In order to do the progressive deepening, 00:45:22.120 --> 00:45:27.850 we had to do those static evaluations, right? 00:45:27.850 --> 00:45:34.190 So we either did all those static evaluations 00:45:34.190 --> 00:45:39.470 and these five-- E, K, Q, N, H, I-- static evaluations. 00:45:39.470 --> 00:45:42.590 Because we didn't do the K. 00:45:42.590 --> 00:45:46.010 Or we might have saved ourselves. 00:45:46.010 --> 00:45:47.970 Because maybe we were smart and decided 00:45:47.970 --> 00:45:51.899 to cache the static values when we were going down the tree. 00:45:51.899 --> 00:45:54.440 It's an implementation detail that on this test when we asked 00:45:54.440 --> 00:45:56.210 that question we didn't say. 00:45:56.210 --> 00:45:58.400 What I mean by cache is when we did it here and saw 00:45:58.400 --> 00:46:01.540 that E was a 2, and then here-- oh, we 00:46:01.540 --> 00:46:03.990 have to do the static value at E. If we were smart, 00:46:03.990 --> 00:46:06.520 we might have made a little hash table or something 00:46:06.520 --> 00:46:09.985 and put down 2 so we didn't have to do a static evaluation at E. 00:46:09.985 --> 00:46:12.602 And if that happened, well, we save E, H, and I, 00:46:12.602 --> 00:46:15.270 and we do three fewer. 00:46:15.270 --> 00:46:16.660 Does everyone see that? 00:46:16.660 --> 00:46:18.177 However, that's still more than six. 00:46:18.177 --> 00:46:19.260 So it didn't save us time. 00:46:19.260 --> 00:46:21.051 So you might say, oh, progressive deepening 00:46:21.051 --> 00:46:22.250 is a waste of time. 00:46:22.250 --> 00:46:23.670 But it's not. 00:46:23.670 --> 00:46:28.050 Because this is a very, very small, not very branchy tree 00:46:28.050 --> 00:46:30.750 that was made so that you guys could easily 00:46:30.750 --> 00:46:34.060 do alpha-beta and take the quiz, and it wouldn't be bad. 00:46:34.060 --> 00:46:40.120 If this was actually branching even double at each level, 00:46:40.120 --> 00:46:44.310 it would have, what, 16 nodes down here at the bottom. 00:46:44.310 --> 00:46:47.390 Then you would want to be doing that progressive deepening. 00:46:47.390 --> 00:46:51.360 So now I ask you a conceptual riddle question. 00:46:51.360 --> 00:46:53.180 It's not really that much of a riddle. 00:46:53.180 --> 00:46:54.846 But we'll see if anyone wants to answer. 00:46:54.846 --> 00:46:57.670 Again, I won't call on you for this. 00:46:57.670 --> 00:46:59.730 According to this test, a student 00:46:59.730 --> 00:47:01.730 named Steve says, OK, I know I have 00:47:01.730 --> 00:47:05.320 to pay to do the progressive deepening here. 00:47:05.320 --> 00:47:06.490 But let's ignore that. 00:47:06.490 --> 00:47:09.320 Because it's small in a large tree, right? 00:47:09.320 --> 00:47:11.320 It's not going to take that much. 00:47:11.320 --> 00:47:13.720 Let's ignore the costs of the progressive deepening 00:47:13.720 --> 00:47:16.450 and only look at how much we do here. 00:47:16.450 --> 00:47:18.920 He says, when it comes to performing the alpha-beta 00:47:18.920 --> 00:47:21.220 on the final level, I'm guaranteed 00:47:21.220 --> 00:47:23.600 to always prune at least as well or better 00:47:23.600 --> 00:47:25.970 if I rearrange the nodes based on the best result 00:47:25.970 --> 00:47:27.830 from progressive deepening. 00:47:27.830 --> 00:47:28.759 Do you agree? 00:47:31.876 --> 00:47:32.751 AUDIENCE: [INAUDIBLE] 00:47:32.751 --> 00:47:34.250 PROFESSOR: Can I repeat it? 00:47:34.250 --> 00:47:36.850 OK, the question is, ignoring the cost 00:47:36.850 --> 00:47:39.170 that we pay progressively deepening here-- just 00:47:39.170 --> 00:47:42.295 forget about it-- at the final step, at the final iteration, 00:47:42.295 --> 00:47:45.590 the question is, am I guaranteed to do at least as 00:47:45.590 --> 00:47:47.530 well or better in my alpha-beta pruning 00:47:47.530 --> 00:47:51.137 when I reorder based on the best order 00:47:51.137 --> 00:47:52.220 for progressive deepening? 00:47:52.220 --> 00:47:53.935 Here certainly we did. 00:47:53.935 --> 00:47:57.020 But the question is, is Steve guaranteed? 00:47:57.020 --> 00:47:58.526 Answer? 00:47:58.526 --> 00:47:59.675 AUDIENCE: [INAUDIBLE] 00:47:59.675 --> 00:48:00.841 PROFESSOR: What did you say? 00:48:00.841 --> 00:48:02.162 AUDIENCE: [INAUDIBLE] 00:48:02.162 --> 00:48:03.870 PROFESSOR: That's the answer and the why, 00:48:03.870 --> 00:48:05.150 which we asked to explain. 00:48:05.150 --> 00:48:08.160 The answer we got is, doesn't that depend on the heuristic? 00:48:08.160 --> 00:48:09.025 Perfectly correct. 00:48:09.025 --> 00:48:11.130 The answer is, no, we're not guaranteed, 00:48:11.130 --> 00:48:12.780 and it depends on the heuristic. 00:48:12.780 --> 00:48:14.920 So if we were guaranteed, that would 00:48:14.920 --> 00:48:17.310 be our heuristic was godlike, like this heuristic. 00:48:17.310 --> 00:48:18.960 If your heuristic already tells you 00:48:18.960 --> 00:48:22.180 the correct answer no matter what, don't do game search. 00:48:22.180 --> 00:48:25.340 Just go to the empty chess board, 00:48:25.340 --> 00:48:28.170 put all the pieces in the front rows, 00:48:28.170 --> 00:48:29.990 and run static evaluator on that. 00:48:29.990 --> 00:48:33.110 And it'll say, oh, it looks like with this game not 00:48:33.110 --> 00:48:36.466 started that white is stupid, so black will win in 15 turns. 00:48:36.466 --> 00:48:37.340 And then you're done. 00:48:37.340 --> 00:48:38.690 And you don't do a search. 00:48:38.690 --> 00:48:41.290 We know that our heuristic is flawed in some way. 00:48:41.290 --> 00:48:42.790 It could be very flawed. 00:48:42.790 --> 00:48:44.850 If it's flawed so badly that it tells us 00:48:44.850 --> 00:48:47.320 a very bad result of what's actually going to happen, 00:48:47.320 --> 00:48:49.646 even though we think the minimizer is going to go to H, 00:48:49.646 --> 00:48:52.070 maybe it's wrong by a lot and it goes to G. 00:48:52.070 --> 00:48:54.590 It could take us up an even worse path 00:48:54.590 --> 00:48:56.180 and make us take longer. 00:48:56.180 --> 00:48:56.972 Question? 00:48:56.972 --> 00:48:58.506 AUDIENCE: If it's the heuristic, how 00:48:58.506 --> 00:48:59.804 could you cache the values so you didn't 00:48:59.804 --> 00:49:01.096 have to recalculate them later? 00:49:01.096 --> 00:49:02.720 PROFESSOR: The question is, how can you 00:49:02.720 --> 00:49:04.780 cache the values if it's a heuristic so you don't 00:49:04.780 --> 00:49:06.390 have to recalculate them later? 00:49:06.390 --> 00:49:09.770 The answer is, it wouldn't help if there weren't 00:49:09.770 --> 00:49:11.780 these weird multi-level things where 00:49:11.780 --> 00:49:14.860 we stop at E for some reason, even though it goes down 00:49:14.860 --> 00:49:16.270 to five levels. 00:49:16.270 --> 00:49:19.430 The way you could cache it is it is a heuristic. 00:49:19.430 --> 00:49:20.635 But it's consistent. 00:49:20.635 --> 00:49:22.960 And I don't mean consistent from search. 00:49:22.960 --> 00:49:25.890 I mean it's a consistent heuristic in-- the game state 00:49:25.890 --> 00:49:29.190 E is, let's say that's the state where I moved out 00:49:29.190 --> 00:49:31.530 my knight as the maximizer, and the minimizer said, 00:49:31.530 --> 00:49:33.270 you're doing the knight opening, really, 00:49:33.270 --> 00:49:35.820 and then did a counterattack. 00:49:35.820 --> 00:49:37.740 No matter how we get to E, or where 00:49:37.740 --> 00:49:40.980 we go to get to E, that's always going to be state E. 00:49:40.980 --> 00:49:43.430 It's always going to have the same heuristic value. 00:49:43.430 --> 00:49:46.730 It's not like some guy who goes around and just randomly pulls 00:49:46.730 --> 00:49:49.007 a number out of a hat. 00:49:49.007 --> 00:49:50.840 We're going to have some value that gives us 00:49:50.840 --> 00:49:52.510 points based on state E. And it's 00:49:52.510 --> 00:49:54.670 going to be the same any time we go to state 00:49:54.670 --> 00:49:57.020 E. Does that make sense? 00:49:57.020 --> 00:49:57.980 It is a heuristic. 00:49:57.980 --> 00:50:01.250 But it's always going to give the same value at E no matter 00:50:01.250 --> 00:50:04.500 how you got to E. 00:50:04.500 --> 00:50:05.705 But it could be really bad. 00:50:05.705 --> 00:50:07.330 In fact, you might consider a heuristic 00:50:07.330 --> 00:50:09.830 that's the opposite of correct and always tells us the worst 00:50:09.830 --> 00:50:11.112 move and claims it's the best. 00:50:11.112 --> 00:50:13.070 That's the heuristic that the minimizer program 00:50:13.070 --> 00:50:14.860 did to our computer, perhaps. 00:50:14.860 --> 00:50:17.450 In that case, when we do progressive deepening and we 00:50:17.450 --> 00:50:22.100 reorder, we'll probably get the worst pruning possible. 00:50:22.100 --> 00:50:22.880 We might not. 00:50:22.880 --> 00:50:23.830 But we may. 00:50:23.830 --> 00:50:26.560 So in that case, you're not guaranteed. 00:50:26.560 --> 00:50:28.450 I hope that's given a few clues. 00:50:28.450 --> 00:50:30.090 In tutorial, you guys are going to see 00:50:30.090 --> 00:50:32.500 some more interesting problems that 00:50:32.500 --> 00:50:33.870 go into a few other details. 00:50:33.870 --> 00:50:36.160 I at least plan on doing [INAUDIBLE] interesting game 00:50:36.160 --> 00:50:41.649 problem from last year, which asked a bunch of varied things 00:50:41.649 --> 00:50:43.440 that are a little bit different from these. 00:50:43.440 --> 00:50:47.360 So it should be a lot of fun, hopefully, or at least useful, 00:50:47.360 --> 00:50:49.910 to do the next quiz. 00:50:49.910 --> 00:50:51.550 So have a great weekend. 00:50:51.550 --> 00:50:54.711 Don't stress out too much about the quiz.