All right. Let's talk about extensive-form games. In the previous video when I introduced game theory, we had two types of non-cooperative games: extensive-form and normal-form. In the extensive-form game, we want to think of an example of an extensive-form game. Extensive-form game is kind of like a tree diagram. So the example we're going to use here is we're going to think about chess, modeling the game of chess. This is game theory, so let's model this out. The game of chess has got some fairly concrete rules. Here's the first player. By convention, white plays first on a chessboard. Now we've build a tree diagram. How many different possible moves does white have? Stop and think about it. Let's see. How many different possible moves does white have? Well, that's the number of branches of the tree that we have to actually write out. What are the possible moves that white has? It turns out there are 20 different moves for the opening move of a chess game. Think about it. You have eight pawns in a row. You can move each one of those eight pawns either one piece forward or two hops forward. So there's 16 possible moves with a pawn, eight pawns. Each pawn has two possible positions they can move to. Plus you have king's knight can go out right, king's knight can go out left, king queen's knight can go out right, queen's knight can go out left. So there are four possible moves for the knights, and that means that the white has 20 possible positions of the board after white's first move. Each one of these nodes, and I'm not going to draw all 20 of them. There's 20 after just one move. Now black's turn. Now black comes in. How many possible moves for black on this? Well, figure that out. It's still the same 20. The game is really in an infancy, and black can also move any one of the pawns out one space or two spaces at 16 possible outcomes of the pawn, plus black could move king's knight out right or left, queen's knight out right or left. So for each one of the possible positions of the board when white got done, black has 20 possible outcomes then it can put there. So I'm not going to put them all down there. But you can see on each one of these nodes, black now has 20 possible that black could choose. Now it's white's turn again. Now how many of them are very? Well, now there's 400 because there were 20 nodes, 20 possible outcomes of the board after white's first move. Now black has 20 possible moves that it could make off of any of those initial both moves from white. So the 20 times 20 is there's 400 nodes already, just after one play. White plays and then black plays and you've already got 400 possible branches of the tree. Now lets go back to what white has. Well, white did pass all these notes to 400 of them and one of them is where the game board is; and from that one, white has how many moves? Well, again white can move some pawns out more or less but maybe the first move, the white moved a pawn out so that the queen can come out one or queen can two or queen cross at an angle three, queen four. So some of these nodes white may have as many as 25, 26 moves. Some of them less if they hadn't broken any of the moves out in front of the queens or the bishops or the rooks. It might be a situation where they have less developed. But you can see what's happening. These nodes are exploding exponentially. So it only have to be a few moves down and you can see the large number of outcomes of this possible game. If you've ever played these computer games on your computer. When you play these computer games on your computer or at least when I do, if you play at level 1, I can win all the time. Game is real fast. If you set it up, say, level 2 the game is a little bit slower. I could still win. Level 3, yeah, I can hold my [inaudible]. But by my time you move to level 4, the computer is a little bit slower. In fact, sometimes noticeably slower, and I never win. The computer beats me all the time in these games. See what's happening is when you're setting the different level when you're playing these games, you're telling the computer how many branches the computer can go down and evaluate the outcomes of game to say which one of those is best for me. I can see if the computer could see three moves ahead, the computer could see millions more outcomes, but the computer, of course, that's what they're good at is evaluating millions of things in nanoseconds, all the clicks in the CPU. So if you tell the computer that it has a lot of time, it can go a long way down this game. It can think way down here and maybe be five, six moves ahead of you if you give it all that time. In competitive matches, they can't do that because they've time limits on that. It's really hard. There was a competition after the mathematicians first talked about this extensive-form game of chess. People wondered is there a finite number of possible moves, is there a finite number of board positions or is it just too big. It turns out; they put up rise up. The prize money was a lot of money back in those days. We're talking about in the '50s and '60s, it was like $100,000 which was several times more than what faculty salaries were in those days. So it was pretty big price. In fact, it was proved that there is a finite number of moves in a chess game. Once you think about that, there's a finite number of moves than presumably we can always find to win. If there's a finite number, let's just evaluate them all. Well, it turns out it's a finite number but it's so big that even if you took the fastest supercomputers we know right now, it would take decades to figure this process out. So it's a finite number, but it's a huge finite number. So the point is even though if the computer was so fast they could do it all then the computer would never lose IBM has this computer called Deep Blue. Deep Blue is a computer that has been able to beat grandmasters in chess. Deep Blue is a computer. that only does chess. That's all it does. Deep Blue does this type of chess. It just races through the nodes. It races ahead, and by the time you get in the middle of a chess game, the number of possible nodes to look at is huge. It was estimated that probably the best grand champion ever in terms of chess skills and ability was this guy named Bobby Fischer. People claimed that Bobby Fischer could see seven moves ahead which is more than anybody ever. Deep Blue cannot see seven moves ahead. So how could Bobby Fischer be better than the Deep Blue? Well, the difference is Deep Blue is a pure number cruncher. Deep Blue just is a node searcher. Bobby Fischer did something called pattern recognition where Bobby Fischer would just throw up fast parts of the entire thing. Bobby Fischer say this stuff is all irrelevant, I'm only going to look at the the nodes that come down from here. So Bobby Fischer, through the human brain, could basically recognize that some of these things are just dumb and I don't and I don't need to waste time evaluating all those nodes. Computers don't have that right now yet. Artificial Intelligence, AI, is working hard on trying to incorporate pattern recognition into Deep Blue. So it's not just a pure, let me solve all these nodes and go down to the next level. Let me solve all these nodes, which will make Deep Blue even further. So that's an example of a game in extensive-form. Let's think about this with respect to economics. Thanks.