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Game Theory
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Difficulty: Beginner
Keywords: Problem
Game Theory is a system for describing "games" and practical problems that can sometimes lead to the best (most "rational") solution for a player.
(This is very simplified. Go As game theory applies to go, I think there are some very valuable concepts that can be used to analyze a position. Go is a zero-sum game. One person loses and one person wins. It is not possible for both to win, because the 1/2 point komi does not allow ties. But are individual battles zero-sum? At first glance, they do not appear to be. Both players can come out better off (such as in a joseki). In zero-sum situations, all outcomes are matched, ie if black gains 10 points, white loses 10 points. But everywhere on this site, I read remarks like "this piece trades off territory for influence" or vice versa. This, to me, shows that battles are zero-sum. I think this needs a bit more explanation. In games like economics, the aim is to maximize your profits. It does not matter how well other players do because no matter how much money they make, you can still make as much as possible. However, in Go, the "real" score (in my opinion) is the difference between scores. Since both players are competing for the same resources and there is a finite amount of it, every point white takes is a point black cannot take. Since the difference of the score is important, that is what a player should try to maximize. (Eg maximize the lead) So if we look at it this way, battles can be seen as zero-sum. If black takes 15 points of territory, black is 15 points ahead of white (and conversely white is 15 points behind black, so the sum is 0). If white take 8 points of territory and black gains 8 points of influence, white is 0 points ahead, black is 0 points behind, so the sum is 0. More later. -- SifuEric TakeNGive (10k): Game theory is interesting to me, but i know little about it. Q: (for Sifu, or whoever knows) How would zero-sum theory quantify the value of a reducing attack -- that is, a play by Black that turns some of what should have been White's territory into dame? A: Let's say white was ahead in territory, so Black was behind. ahead+behind=0. After the reducing attack, white is only a bit ahead so black is a bit behind. a bit ahead+a bit behind=0. The idea of Zero-sum game does not apply to Go for it is not a economic game, as that class of games are known as. Go belongs to the class of games called Combinatoric Games whose mathematical treatment did not come fully until John Conway discovered Go. For more info check out the book: On Numbers and Games by John Conway. Also Mathematical Go: Chilling Gets the Last Point by Elwyn Berlekamp and David Wolfe. BillSpight: Go is a 2-person zero-sum game with perfect information. In the end game it tends to decompose into more or less independent regions, each of which is a game. Combinatorial games add and subtract. Games with kos do not add and subtract, and are not, strictly speaking, combinatorial games. For instance, a zero game may be a ko threat. Adding a zero game to a combinatorial game does not change it, just as adding zero to a score does not change it. But adding a ko threat to a ko may change it, even if the threat is a zero game. However, much of combinatorial game theory can be applied to go, particularly thermography, a method that determines the count and size of plays in a position. Thermography has been extended to cover kos by Professor Berlekamp and myself. A related concept, temperature, is useful in thinking about go. Path: CGTPath · Prev: CGTReferences · Next: This is a copy of the living page "Game Theory" at Sensei's Library. (C) the Authors, published under the OpenContent License V1.0. |