1 00:00:00,012 --> 00:00:04,925 >> Hey, welcome back. Good to see you, although I can't really 2 00:00:04,925 --> 00:00:10,174 see you, but I'm looking that way and it looks like I can see you. 3 00:00:10,174 --> 00:00:12,656 Okay. What are we doing now? 4 00:00:12,656 --> 00:00:18,661 Examples from up, down, and star. Now, and the general reversible move. 5 00:00:18,661 --> 00:00:25,797 This is probably one of the most subtle things we've done in the whole course. 6 00:00:25,797 --> 00:00:33,111 And it's not something that's easily converted into hand computations and 7 00:00:33,111 --> 00:00:37,552 examples. So if this seems like you have to go back 8 00:00:37,552 --> 00:00:43,984 and look at it over and over again that's okay it's, it's something that, that's, 9 00:00:43,984 --> 00:00:50,964 that's a little bit a little bit more Complicated than some of the other stuff 10 00:00:50,964 --> 00:00:55,224 we've done. What I want to do is look at the general 11 00:00:55,224 --> 00:01:02,030 reversible moves and then do an example. And the example is very, very small, so 12 00:01:02,030 --> 00:01:08,292 what's really going on in, in talking about reversible moves is this is And 13 00:01:08,292 --> 00:01:14,958 what's part of the general engine inside computer programs that analyze these 14 00:01:14,958 --> 00:01:20,055 things. And although we can compute small examples 15 00:01:20,055 --> 00:01:25,053 by hand, you can't really do much bigger examples by hand. 16 00:01:25,053 --> 00:01:29,322 So bear with me, so here's a whole lot of text. 17 00:01:29,322 --> 00:01:36,371 Just and this is, is a the definition of reversible moves in games without chance, 18 00:01:36,371 --> 00:01:43,103 so we start off with the game which says certain left options and moves and certain 19 00:01:43,103 --> 00:01:49,439 right options and moves then we say this game reverses through one of the left 20 00:01:49,439 --> 00:01:55,018 options Or left moves. If there is a right move of this left 21 00:01:55,018 --> 00:02:00,921 option, call that aR. With aR better for G, better for right 22 00:02:00,921 --> 00:02:06,996 than G is, so remember is a game is less it's better for right. 23 00:02:06,996 --> 00:02:14,465 So this means aR is better for right than G was to begin with. 24 00:02:14,465 --> 00:02:20,898 So think about it this way. Left moves to a then right immediately 25 00:02:20,898 --> 00:02:27,954 moves to one of the options inside a namely aR and right now is better off than 26 00:02:27,954 --> 00:02:32,613 when right started. So right may as well do that. 27 00:02:32,613 --> 00:02:39,962 All left moves to A right immediately is better off by moving to AR is better off 28 00:02:39,962 --> 00:02:44,342 than, than he or she was when the game started. 29 00:02:44,343 --> 00:02:51,450 In that case right, may as well do this all the time whenever left moves to a. 30 00:02:51,450 --> 00:02:58,701 And, and therefor left might as well move directly to the right opt, left options of 31 00:02:58,701 --> 00:03:03,827 they are. Rather then rather than to a, itself. 32 00:03:03,827 --> 00:03:11,024 Now, that's a little bit complicated. We'll look at an example, so don't panic 33 00:03:11,024 --> 00:03:12,801 yet. Alright. 34 00:03:12,801 --> 00:03:17,440 I, I read some book that had this in it, right? 35 00:03:17,440 --> 00:03:22,145 I've forgotten what the book was. Don't panic. 36 00:03:22,146 --> 00:03:28,080 Okay, it was a book of advice, and whatever you asked it, it said Don't 37 00:03:28,080 --> 00:03:30,991 panic. All right, so don't panic. 38 00:03:30,992 --> 00:03:37,351 Now, in games we always have to look at things from the point of view of left and 39 00:03:37,351 --> 00:03:42,442 the point of view of right. So, here is the analog for reversing 40 00:03:42,442 --> 00:03:46,387 through a right move. Now, so here's an example. 41 00:03:46,387 --> 00:03:50,862 And feel free to go back and look at these definitions again. 42 00:03:50,862 --> 00:03:56,051 Ask questions about them in the forum. And to re-look at this example. 43 00:03:56,051 --> 00:03:59,837 So here's the example. The example is up plus star. 44 00:03:59,837 --> 00:04:03,424 Now up is left move, only left moves to zero. 45 00:04:03,424 --> 00:04:08,738 The only right move is to star. Star is the only left move is to zero. 46 00:04:08,738 --> 00:04:12,134 The only right move is to[INAUDIBLE]. 0. 47 00:04:12,134 --> 00:04:16,469 So, up plus star is the same as this plus this. 48 00:04:16,469 --> 00:04:23,323 So, we have 2 games, we play them the, there sum by playing in one game or the 49 00:04:23,323 --> 00:04:27,219 other. So, let's look and see what happens. 50 00:04:27,219 --> 00:04:30,452 Left can move to this 0. Make that move. 51 00:04:30,452 --> 00:04:34,681 In which case what's left is 0 plus star. So that's a star. 52 00:04:34,681 --> 00:04:40,163 Or left can move over here to this 0. In which case what's left is 0 over here 53 00:04:40,163 --> 00:04:44,517 and up over here. So the two possible left moves leaves star 54 00:04:44,517 --> 00:04:48,211 and up. And now let's look at the right move. 55 00:04:48,211 --> 00:04:53,487 The right can move to star, in which case you have star plus star. 56 00:04:53,487 --> 00:05:00,773 You can check it out, star plus star is 0 or right can move to this 0 over here then 57 00:05:00,773 --> 00:05:06,793 you have 0 plus op and that's therefore, the other right option. 58 00:05:06,793 --> 00:05:14,467 Now, let's look at this. So, the first thing we want to notice is 59 00:05:14,467 --> 00:05:19,082 the following. I'm going to recopy some of this. 60 00:05:19,082 --> 00:05:24,921 So, up plus star is, by our computation. Star up 0 up. 61 00:05:24,921 --> 00:05:33,614 Now we know that 0 is less than up. From a few slides ago remember? 62 00:05:33,614 --> 00:05:41,185 0 is less than up, is less than 1 over 2 to the n, for any n. 63 00:05:41,185 --> 00:05:48,295 If you look back, this, this was actually done in, in one of the earlier weeks. 64 00:05:48,295 --> 00:05:53,486 Although maybe we, we didn't actually write it that way. 65 00:05:53,486 --> 00:05:57,396 So, so that says this op is always worse for. 66 00:05:57,396 --> 00:06:05,136 Right then this 0. 0 is always better for right than up. 67 00:06:05,136 --> 00:06:11,315 So this move is dominates so we can leave it. 68 00:06:11,315 --> 00:06:19,367 Okay, that's the easy part or the easier part, all right. 69 00:06:21,477 --> 00:06:33,680 So now, let's look at this up. Up, that up is 0 star. 70 00:06:33,680 --> 00:06:47,030 And so it has a right option of star. Right move of star. 71 00:06:47,031 --> 00:06:54,604 Now, star is less than up plus star, because just subtract star, just add star 72 00:06:54,604 --> 00:07:02,510 to both sides, star plus star is 0, and over here we'll have, we'll have up plus 73 00:07:02,510 --> 00:07:09,159 star plus star which will be up. So, so since 0 is less than up we have 74 00:07:09,159 --> 00:07:14,308 this. Which says that there's a right option of 75 00:07:14,308 --> 00:07:21,315 up, this up here, that's better for right than the original gain. 76 00:07:21,315 --> 00:07:30,909 So, right will move there automatically once left moves the star, and then left is 77 00:07:30,909 --> 00:07:41,343 left with the if right moves >> If left moves to up, right moves to star, and then 78 00:07:41,343 --> 00:07:48,721 since star is 0, 0, the only left option from there is to move to 0. 79 00:07:48,721 --> 00:07:57,143 Go back. Rewind the video, look at the definitions 80 00:07:57,143 --> 00:08:07,781 that says this reverses to 0. 0 is the left options of the right option 81 00:08:07,781 --> 00:08:17,721 of up divided which is better for right then you would know gain. 82 00:08:17,721 --> 00:08:26,390 I think I said that right. So, so that says opp plus star is actually 83 00:08:26,390 --> 00:08:32,286 zero star zero. Now, where have we seen that before? 84 00:08:32,286 --> 00:08:56,827 Take a look at this position here, Hackenbush. 85 00:08:56,827 --> 00:09:02,122 Green edge down here, blue edge up here. Blue can cut the, cut the blue edge. 86 00:09:02,122 --> 00:09:06,697 Left can cut the blue edge. In which case, what's left is star or left 87 00:09:06,697 --> 00:09:10,612 can cut the green edge. In which case, what's left is zero. 88 00:09:10,612 --> 00:09:16,183 Right can only cut the green edge. In which case, what is left is 0. 89 00:09:16,183 --> 00:09:22,951 So, these are equal games and they're both equal to up plus star, which is sometimes 90 00:09:22,951 --> 00:09:31,118 abbreviated in literature as up star. So it turns out that we've seen this thing 91 00:09:31,118 --> 00:09:39,902 before and now viewing this game is up plus star with for instance now what 92 00:09:39,902 --> 00:09:48,974 happens when we added to itself, so up star plus up star is up plus up plus star 93 00:09:48,974 --> 00:09:58,046 plus star But star plus star is 0, so that's up plus up, which is sometimes 94 00:09:58,046 --> 00:10:07,949 written in the literature double up. So double up to be sure and we'll see you 95 00:10:07,949 --> 00:10:09,679 next module.