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Fuzzy Game
In combinatorial game theory, a fuzzy game is a game which is ''incomparable'' with the zero game: it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right; nor equal to 0 which would be a win for the second player to move. It is therefore a first-player win. Classification of games In combinatorial game theory, there are four types of game. If we denote players as Left and Right, and G be a game with some value, we have the following types of game: 1. Left win: G > 0 :No matter which player goes first, Left wins. 2. Right win: G < 0 :No matter which player goes first, Right wins. 3. Second player win: G = 0 :The first player (Left or Right) has no moves, and thus loses. 4. First player win: G ║ 0 (G is fuzzy with 0) :The first player (Left or Right) wins. Using standard Dedekind-section game notation, , where L is the list of undominated move ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Game Theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players take turns changing in defined ways or ''moves'' to achieve a defined winning condition. Combinatorial game theory has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Game
In combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, under the normal play convention, the first player automatically loses, and it is a second-player win. The zero game has a Sprague–Grundy value of zero. The combinatorial notation of the zero game is: .. A zero game should be contrasted with the star game , which is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win. Examples Simple examples of zero games include Nim with no piles or a Hackenbush diagram with nothing drawn on it. Sprague-Grundy value The Sprague–Grundy theorem applies to impartial game In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference bet ...s (in which each move may be played by either pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surreal Number
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star (game Theory)
In combinatorial game theory, star, written as * or *1, is the value given to the game where both players have only the option of moving to the zero game. Star may also be denoted as the surreal form . This game is an unconditional first-player win. Star, as defined by John Conway in '' Winning Ways for your Mathematical Plays'', is a value, but not a number in the traditional sense. Star is not zero, but neither positive nor negative, and is therefore said to be ''fuzzy'' and ''confused with'' (a fourth alternative that means neither "less than", "equal to", nor "greater than") 0. It is less than all positive rational numbers, and greater than all negative rationals. Games other than may have value *. For example, the game *2 + *3, where the values are nimbers, has value * despite each player having more options than simply moving to 0. Why * ≠ 0 A combinatorial game has a positive and negative player; which player moves first is left ambiguous. The combinatorial game ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-player Win
In combinatorial game theory, a two-player deterministic perfect information turn-based game is a first-player-win if with perfect play the first player to move can always force a win. Similarly, a game is second-player-win if with perfect play the second player to move can always force a win. With perfect play, if neither side can force a win, the game is a draw. Some games with relatively small game trees have been proven to be first or second-player wins. For example, the game of nim with the classic 3–4–5 starting position is a first-player-win game. However, Nim with the 1-3-5-7 starting position is a second-player-win. The classic game of Connect Four has been mathematically proven to be first-player-win. With perfect play, checkers has been determined to be a draw; neither player can force a win. Another example of a game which leads to a draw with perfect play is tic-tac-toe, and this includes play from any opening move. Significant theory has been completed in the ef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blue-Red-Green Hackenbush
Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line. Gameplay The game starts with the players drawing a "ground" line (conventionally, but not necessarily, a horizontal line at the bottom of the paper or other playing area) and several line segments such that each line segment is connected to the ground, either directly at an endpoint, or indirectly, via a chain of other segments connected by endpoints. Any number of segments may meet at a point and thus there may be multiple paths to ground. On their turn, a player "cuts" (erases) any line segment of their choice. Every line segment no longer connected to the ground by any path "falls" (i.e., gets erased). According to the normal play convention of combinatorial game theory, the first player who is unable to move loses. Hackenbush boards can consist of finitely many (in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surreal Number
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
