1 00:00:00,012 --> 00:00:03,604 >> Hello, this is games without dice or cards. 2 00:00:03,604 --> 00:00:07,245 Combinatoriai Game Theory. I'm Tom Morley. 3 00:00:07,245 --> 00:00:13,343 So let's start by jumping into the water head first, and let's play a game. 4 00:00:13,343 --> 00:00:17,127 Let me first introduce you to our two players. 5 00:00:17,127 --> 00:00:23,269 Our two players are called left and right. These two players alternate. 6 00:00:23,269 --> 00:00:29,841 One goes, what the other one makes a move, then the first one makes a move, then the 7 00:00:29,841 --> 00:00:35,571 other one makes a move. Typically, left is associated with the 8 00:00:35,571 --> 00:00:42,816 color blue and you can remember that because there's an L is blue and right is 9 00:00:42,816 --> 00:00:50,200 associated with red, and you can remember that, because there's an r in red. 10 00:00:50,200 --> 00:01:01,315 Now, these players are going to alternate, alternate, play and the 1st 1 that can't 11 00:01:01,315 --> 00:01:05,129 move loses. So there's no ties. 12 00:01:05,129 --> 00:01:13,975 At each stage of the game there will be various moves available to one player or 13 00:01:13,975 --> 00:01:21,925 the other player or perhaps both, and whoever runs out of moves loses. 14 00:01:21,925 --> 00:01:29,185 So let's take a look at something like this, and So let's start with a little 15 00:01:29,185 --> 00:01:30,536 horsey. Okay. 16 00:01:30,536 --> 00:01:36,907 So this is a game called Hackenbush, well, it's a particular position in the game 17 00:01:36,907 --> 00:01:41,674 called Hackenbush. And you can start in various ways but we 18 00:01:41,674 --> 00:01:47,781 start with this little horsey. The two players, red and blue, left and 19 00:01:47,781 --> 00:01:55,111 right, take turns, and what they do is cut one of their edges of their own color. 20 00:01:55,111 --> 00:02:02,064 So if blue starts, he could cut this edge, and that edge is no longer there. 21 00:02:02,064 --> 00:02:08,570 Now, these, these edges have helium in them, so and they're attached to the 22 00:02:08,570 --> 00:02:14,897 ground here, which is green. So if something gets disconnected from the 23 00:02:14,897 --> 00:02:19,936 ground, it floats off and is not available to either player. 24 00:02:19,936 --> 00:02:28,866 So, for instance maybe, right goes here maybe left goes here. 25 00:02:28,866 --> 00:02:41,366 Now, at this point, this whole piece here is still connected to the ground. 26 00:02:41,366 --> 00:02:50,927 But once right moves, it all floats up, and now left moves, and there's no moves 27 00:02:50,927 --> 00:02:57,040 left for left. Okay, so, so whenever the two players 28 00:02:57,040 --> 00:03:03,445 alternate, left cuts blue edges, right cuts red edges. 29 00:03:03,445 --> 00:03:08,969 Any edge that's either cut or no longer connected to the ground to where the 30 00:03:08,969 --> 00:03:13,366 nutrients are, the nitrogen and the water and whatever. 31 00:03:13,367 --> 00:03:18,892 Any edges no longer connected to the ground, floats off and is not available to 32 00:03:18,892 --> 00:03:22,710 either player. When a player has no edges to cut, the 33 00:03:22,710 --> 00:03:28,713 game ends, that player loses, okay? We'll see a lot of examples in terms of 34 00:03:28,713 --> 00:03:34,691 Hackenbush as, as the game goes on. Let's take a look at some very simple 35 00:03:34,691 --> 00:03:38,636 positions in Hackenbush. Here's the ground. 36 00:03:38,636 --> 00:03:42,649 It's green. And now we go, I don't know. 37 00:03:42,649 --> 00:03:47,198 Blue, and blue, and maybe a red in between. 38 00:03:47,198 --> 00:03:51,958 And maybe a red. And then, say, a blue on top. 39 00:03:51,958 --> 00:03:57,489 Up, now up. It's okay that these aren't connected, but 40 00:03:57,489 --> 00:04:02,567 they're connected to the ground, so that's okay. 41 00:04:02,568 --> 00:04:12,220 So, they're there maybe, maybe red goes first and red says aha, I'm going to cut 42 00:04:12,220 --> 00:04:21,716 this and now these float off and that top blue end is no longer available for blue. 43 00:04:21,716 --> 00:04:28,462 Blue say, what's the best move for blue? I don't know offhand, but blue might cut 44 00:04:28,462 --> 00:04:34,117 this, and then red cuts this, and then blue cuts this, and now red loses. 45 00:04:34,117 --> 00:04:39,151 So that's how that works. And we'll, we'll look at various 46 00:04:39,151 --> 00:04:46,120 variations of this where we start with, with, with more complicated pictures and, 47 00:04:46,120 --> 00:04:52,584 and try to analyze in terms of, of who, who wins, if left goes first, who wins if 48 00:04:52,584 --> 00:04:54,673 right goes first? Okay. 49 00:04:54,673 --> 00:04:59,279 Let's take a look at another class of games. 50 00:04:59,280 --> 00:05:05,198 And this is a class of games all played with coins. 51 00:05:05,198 --> 00:05:12,449 In this case the coins are pennies and dimes, but and maybe right is pennies, 52 00:05:12,449 --> 00:05:19,589 because the copper in pennies, although, there's very little copper actually in 53 00:05:19,589 --> 00:05:26,308 pennies looks kind of red. And then the dimes, say are, are left. 54 00:05:26,308 --> 00:05:33,082 Now both players push coins to the left in the direction of, of that and the three 55 00:05:33,082 --> 00:05:38,152 games push, shove, and run over have slightly different rules. 56 00:05:38,152 --> 00:05:44,457 Now you can, you can, you can push a coin off the edge, off the end, and if it's off 57 00:05:44,457 --> 00:05:50,984 the end, it's no longer in play. So the two players alternate pushing one 58 00:05:50,984 --> 00:05:57,762 coin to the left one space. And let's look at let's look at, say, this 59 00:05:57,762 --> 00:06:05,543 position here in push, the first game. If left pushes this coin over this, this 60 00:06:05,543 --> 00:06:11,994 coin over one, then that pushes also that, the, the coin next to it over. 61 00:06:11,994 --> 00:06:17,951 So that's, that's called push. In, in, in shove, when you push a coin 62 00:06:17,951 --> 00:06:22,805 over, everything to the left moves over one space also. 63 00:06:22,805 --> 00:06:29,416 So with shove, this, this, this coin over here gets thrown off the cliff. 64 00:06:29,416 --> 00:06:35,994 In run over, what happens is when you push, say this coin over here one to the 65 00:06:35,994 --> 00:06:42,896 left, you run over that person and that, that coin disappear from playing. 66 00:06:42,896 --> 00:06:50,906 So you can look at longer positions to start with, different relations of coins 67 00:06:50,906 --> 00:06:55,013 on the thing. I think it's not too hard to show. 68 00:06:55,013 --> 00:06:58,609 Whoever has the right most coin will win the game. 69 00:06:58,609 --> 00:07:04,357 But, but if we have several of these going at once, then, then things get a little 70 00:07:04,357 --> 00:07:08,950 bit more complicated. And still what we want to do, even if we 71 00:07:08,950 --> 00:07:15,220 know who, who's, who wins the game what, what's the best play maybe to, to delay, 72 00:07:15,220 --> 00:07:22,193 going off the cliff as long as possible. So, so here's four games to play with and 73 00:07:22,193 --> 00:07:26,573 I urge you to start with pictures like this. 74 00:07:26,574 --> 00:07:33,406 Find someone to play, find, figure out whether we want to be left or right 75 00:07:33,406 --> 00:07:38,185 doesn't matter. Draw a picture like this and go play some 76 00:07:38,185 --> 00:07:42,196 games get some coins out, draw pictures like this. 77 00:07:42,196 --> 00:07:47,944 Play several of these at once and we'll explain as the course goes by how to do 78 00:07:47,944 --> 00:07:53,947 that more carefully and, and see what see what looks like good moves and see what 79 00:07:53,947 --> 00:08:04,022 looks like bad moves. So there's some games to start with and, 80 00:08:04,022 --> 00:08:09,550 let's, continue next time.