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本词条由Solitude初步翻译

本词条由Solitude初步翻译<br>

 

{{about|the theory of combinatorial games|the theory that includes games of chance and games of imperfect knowledge|Game theory}}

{{about|the theory of combinatorial games|the theory that includes games of chance and games of imperfect knowledge|Game theory}}

Combinatorial game theory (CGT) 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 in which the players take turns changing in defined ways or moves to achieve a defined winning condition. CGT 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.

Combinatorial game theory (CGT) 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 in which the players take turns changing in defined ways or moves to achieve a defined winning condition. CGT 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.

Combinatorial games include well-known games such as [[chess]], [[Draughts|checkers]], and [[Go (board game)|Go]], which are regarded as non-trivial, and [[tic-tac-toe]], which is considered as trivial in the sense of being "easy to solve". Some combinatorial games may also have an [[Bounded set|unbounded]] playing area, such as [[infinite chess]]. In CGT, the moves in these and other games are represented as a [[game tree]].   

Combinatorial games include well-known games such as [[chess]], [[Draughts|checkers]], and [[Go (board game)|Go]], which are regarded as non-trivial, and [[tic-tac-toe]], which is considered as trivial in the sense of being "easy to solve". Some combinatorial games may also have an [[Bounded set|unbounded]] playing area, such as [[infinite chess]]. In CGT, the moves in these and other games are represented as a [[game tree]].   

Combinatorial games include well-known games such as chess, checkers, and Go, which are regarded as non-trivial, and tic-tac-toe, which is considered as trivial in the sense of being "easy to solve". Some combinatorial games may also have an unbounded playing area, such as infinite chess. In CGT, the moves in these and other games are represented as a game tree.   

Combinatorial games include well-known games such as chess, checkers, and Go, which are regarded as ’’’<font color=“#32CD32”> non-trivial </font>’’’, and tic-tac-toe, which is considered as trivial in the sense of being "easy to solve". Some combinatorial games may also have an unbounded playing area, such as infinite chess. In CGT, the moves in these and other games are represented as a game tree.   

Combinatorial games also include one-player combinatorial puzzles such as Sudoku, and no-player automata, such as Conway's Game of Life, (although in the strictest definition, "games" can be said to require more than one participant, thus the designations of "puzzle" and "automata".)  

Combinatorial games also include one-player combinatorial puzzles such as Sudoku, and no-player automata, such as Conway's Game of Life, (although in the strictest definition, "games" can be said to require more than one participant, thus the designations of "puzzle" and "automata".)  

组合游戏还包括单人组合游戏，如数独游戏，以及无玩家自动机，如康威的生命游戏(虽然在最严格的定义中，“游戏”可以说需要不止一个参与者，因此命名为“拼图”和“自动机”。)

组合游戏还包括单人组合谜题，如数独游戏，以及无人自动机，如康威的生命游戏(虽然在最严格的定义中，“游戏”可以说需要一个以上参与者，因此命名为“拼图”和“自动机”。)

An important notion in CGT is that of the [[solved game]]. For example, [[tic-tac-toe]] is considered a solved game, as it can be proven that any game will result in a draw if both players play optimally. Deriving similar results for games with rich combinatorial structures is difficult. For instance, in 2007 it was announced that [[checkers]] has been [[Solved game#overview|weakly solved]]&mdash;optimal play by both sides also leads to a draw&mdash;but this result was a [[computer-assisted proof]].<ref>{{Cite journal| last1 = Schaeffer | first1 = J.| last2 = Burch | first2 = N.| last3 = Bjornsson | first3 = Y.| last4 = Kishimoto | first4 = A.| last5 = Muller | first5 = M.| last6 = Lake | first6 = R.| last7 = Lu | first7 = P.| last8 = Sutphen | first8 = S.| title = Checkers is solved| journal = [[Science (journal)|Science]]| volume = 317| issue = 5844| pages = 1518–1522| year = 2007| pmid = 17641166| doi = 10.1126/science.1144079| citeseerx = 10.1.1.95.5393| bibcode = 2007Sci...317.1518S}}</ref> Other real world games are mostly too complicated to allow complete analysis today, although the theory has had some recent successes in analyzing Go endgames. Applying CGT to a ''position'' attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is torturously difficult unless the game is very simple.

An important notion in CGT is that of the [[solved game]]. For example, [[tic-tac-toe]] is considered a solved game, as it can be proven that any game will result in a draw if both players play optimally. Deriving similar results for games with rich combinatorial structures is difficult. For instance, in 2007 it was announced that [[checkers]] has been [[Solved game#overview|weakly solved]]&mdash;optimal play by both sides also leads to a draw&mdash;but this result was a [[computer-assisted proof]].<ref>{{Cite journal| last1 = Schaeffer | first1 = J.| last2 = Burch | first2 = N.| last3 = Bjornsson | first3 = Y.| last4 = Kishimoto | first4 = A.| last5 = Muller | first5 = M.| last6 = Lake | first6 = R.| last7 = Lu | first7 = P.| last8 = Sutphen | first8 = S.| title = Checkers is solved| journal = [[Science (journal)|Science]]| volume = 317| issue = 5844| pages = 1518–1522| year = 2007| pmid = 17641166| doi = 10.1126/science.1144079| citeseerx = 10.1.1.95.5393| bibcode = 2007Sci...317.1518S}}</ref> Other real world games are mostly too complicated to allow complete analysis today, although the theory has had some recent successes in analyzing Go endgames. Applying CGT to a ''position'' attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is torturously difficult unless the game is very simple.

An important notion in CGT is that of the solved game. For example, tic-tac-toe is considered a solved game, as it can be proven that any game will result in a draw if both players play optimally. Deriving similar results for games with rich combinatorial structures is difficult. For instance, in 2007 it was announced that checkers has been weakly solved&mdash;optimal play by both sides also leads to a draw&mdash;but this result was a computer-assisted proof. Other real world games are mostly too complicated to allow complete analysis today, although the theory has had some recent successes in analyzing Go endgames. Applying CGT to a position attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is torturously difficult unless the game is very simple.

An important notion in CGT is that of the solved game. For example, tic-tac-toe is considered a solved game, as it can be proven that any game will result in a draw if both players play optimally. Deriving similar results for games with rich combinatorial structures is difficult. For instance, in 2007 it was announced that checkers has been weakly solved&mdash;optimal play by both sides also leads to a draw&mdash; ’’’<font color=“#32CD32”> but this result was a computer-assisted proof </font>’’’. Other real world games are mostly too complicated to allow complete analysis today, although the theory has had some recent successes in analyzing Go endgames. Applying CGT to a position attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is torturously difficult unless the game is very simple.

Cgt 中的一个重要概念是求解博弈。例如，井字棋被认为是一个已解决的游戏，因为它可以证明如果两个玩家都发挥最佳状态，那么任何游戏都将导致平局。对于具有丰富组合结构的游戏，获得相似的结果是困难的。例如，在2007年，有人宣布跳棋已经得到了弱解---- 双方的最佳玩法也会导致平局---- 但这是计算机辅助的证明。尽管该理论最近在分析围棋终局游戏方面取得了一些成功，但其他现实世界的游戏大多过于复杂，以至于今天无法进行全面分析。将 CGT 应用到一个位置，试图确定两个玩家的最佳移动顺序，直到游戏结束，并以此发现在任何位置的最佳移动。在实践中，除非游戏非常简单，否则这个过程非常困难。

Cgt 中的一个重要概念是求解博弈。例如，井字棋被认为是一个已解决的游戏，因为它可以证明如果两个玩家都发挥最佳状态，那么任何游戏都将导致平局。对于具有丰富组合结构的游戏，获得相似的结果是困难的。例如，在2007年，有人宣布跳棋已被弱解---- 双方的最佳玩法也会导致平局---- ’’’<font color=“#32CD32”> 但这个结果是计算机辅助证明 </font>’’’。尽管该理论最近在分析围棋终局游戏方面取得了一些成功，但其他现实世界的游戏大多过于复杂，以至于今天无法进行全面分析。将 CGT 应用到一个位置，试图确定两个玩家的最佳移动顺序，直到游戏结束，并以此发现在任何位置的最佳移动。在实践中，除非游戏非常简单，否则这个过程非常折磨人。

CGT arose in relation to the theory of impartial games, in which any play available to one player must be available to the other as well. One such game is nim, which can be solved completely. Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses.  In the 1930s, the Sprague–Grundy theorem showed that all impartial games are equivalent to heaps in nim, thus showing that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs.

CGT arose in relation to the theory of impartial games, in which any play available to one player must be available to the other as well. One such game is nim, which can be solved completely. Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses.  In the 1930s, the Sprague–Grundy theorem showed that all impartial games are equivalent to heaps in nim, thus showing that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs.

CGT的产生与公正博弈理论有关，在这个理论中，一个玩家可用的任何比赛必须对另一个球员也可用。尼姆就是这样一种游戏，它可以完全解决。尼姆是一款适用于两名玩家的公正游戏，受到正常游戏条件的限制，这意味着不能移动的玩家就是输家。在20世纪30年代，Sprague-Grundy 定理表明，所有公正的游戏都等价于尼姆游戏，这表明在组合层次上考虑的游戏可能具有重大的统一性，在这种情况下，详细的策略很重要，而不仅仅是收益。

CGT的产生与公正博弈理论有关，在这个理论中，一个玩家可用的任何比赛必须对另一个玩家也可用。尼姆就是这样一种游戏，它可以完全解决。尼姆是一款适用于两名玩家的公正游戏，受到正常游戏条件的限制，这意味着不能移动的玩家就是输家。在20世纪30年代，Sprague-Grundy 定理表明，所有公正的游戏都等价于尼姆游戏，这表明在组合层次上考虑的游戏可能具有重大的统一性，在这种情况下，详细的策略很重要，而不仅仅是收益。

In the 1960s, Elwyn R. Berlekamp, John H. Conway and Richard K. Guy jointly introduced the theory of a partisan game, in which the requirement that a play available to one player be available to both is relaxed. Their results were published in their book Winning Ways for your Mathematical Plays in 1982. However, the first work published on the subject was Conway's 1976 book On Numbers and Games, also known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was also a fruit of the collaboration between Berlekamp, Conway, and Guy.

In the 1960s, Elwyn R. Berlekamp, John H. Conway and Richard K. Guy jointly introduced the theory of a partisan game, in which the requirement that a play available to one player be available to both is relaxed. Their results were published in their book Winning Ways for your Mathematical Plays in 1982. However, the first work published on the subject was Conway's 1976 book On Numbers and Games, also known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was also a fruit of the collaboration between Berlekamp, Conway, and Guy.

在20世纪60年代，埃尔温·伯利坎普Elwyn R. Berlekamp，约翰·h·康威。 John h. Conway 和 理查德·盖伊 Richard k. Guy 共同提出了一个党派博弈理论，在这个理论中，放宽了一个游戏可供两个玩家使用的要求。他们的研究结果发表在1982年的《数学游戏的获胜方法》一书中。但是，关于这一主题的第一部著作是Conway于1976年出版的《数与游戏》（也称为ONAG），该书引入了超现实数的概念以及对游戏。《数字与游戏》也是Berlekamp，Conway和Guy之间合作的成果。

在20世纪60年代，埃尔温·伯利坎普Elwyn R. Berlekamp，约翰·h·康威。 约翰·h·康威 John h. Conway 和 理查德·盖伊 Richard k. Guy 共同提出了一个党派博弈理论，在这个理论中，放宽了一个游戏可供两个玩家使用的要求。他们的研究结果发表在1982年的《数学游戏的获胜方法》一书中。但是，关于这一主题的第一部著作是Conway于1976年出版的《数与游戏》（也称为ONAG），该书引入了超现实数的概念以及对游戏。《数字与游戏》也是Berlekamp，Conway和Guy之间合作的成果。

==Examples==

==Examples==<br>

The introductory text ''[[Winning Ways]]'' introduced a large number of games, but the following were used as motivating examples for the introductory theory:

The introductory text ''[[Winning Ways]]'' introduced a large number of games, but the following were used as motivating examples for the introductory theory:

The introductory text Winning Ways introduced a large number of games, but the following were used as motivating examples for the introductory theory:

The introductory text Winning Ways introduced a large number of games, but the following were used as motivating examples for the introductory theory:

获胜的方式介绍了大量的游戏，但以下是作为引导性理论的动机的例子:

《赢家之道》介绍了大量的游戏，但以下是作为引导性理论的激励例子:

* Blue–Red [[Hackenbush]] - At the finite level, this partisan combinatorial game allows constructions of games whose values are [[Dyadic rational|dyadic rational number]]s. At the infinite level, it allows one to construct all real values, as well as many infinite ones that fall within the class of [[surreal numbers]].

* Blue–Red [[Hackenbush]] - At the finite level, this partisan combinatorial game allows constructions of games whose values are [[Dyadic rational|dyadic rational number]]s. At the infinite level, it allows one to construct all real values, as well as many infinite ones that fall within the class of [[surreal numbers]].

* Blue–Red–Green Hackenbush - Allows for additional game values that are not numbers in the traditional sense, for example, [[star (game theory)|star]].

* Blue–Red–Green Hackenbush - Allows for additional game values that are not numbers in the traditional sense, for example, [[star (game theory)|star]].

* [[Toads and Frogs]] - Allows various game values. Unlike most other games, a position is easily represented by a short string of characters.

* [[Toads and Frogs]] - Allows various game values. Unlike most other games, a position is easily represented by a short string of characters.

* [[Domineering]] - Various interesting games, such as [[hot game]]s, appear as positions in Domineering, because there is sometimes an incentive to move, and sometimes not.  This allows discussion of a game's [[temperature (game theory)|temperature]].

* [[Domineering]] - Various interesting games, such as [[hot game]]s, appear as positions in Domineering, because there is sometimes an incentive to move, and sometimes not.  This allows discussion of a game's [[temperature (game theory)|temperature]].

* [[Nim]] - An [[impartial game]]. This allows for the construction of the [[nimber]]s.  (It can also be seen as a green-only special case of Blue-Red-Green Hackenbush.)

* [[Nim]] - An [[impartial game]]. This allows for the construction of the [[nimber]]s.  (It can also be seen as a green-only special case of Blue-Red-Green Hackenbush.)

The classic game Go was influential on the early combinatorial game theory, and Berlekamp and Wolfe subsequently developed an endgame and temperature theory for it (see references).  Armed with this they were able to construct plausible Go endgame positions from which they could give expert Go players a choice of sides and then defeat them either way.

The classic game Go was influential on the early combinatorial game theory, and Berlekamp and Wolfe subsequently developed an endgame and temperature theory for it (see references).  Armed with this they were able to construct plausible Go endgame positions from which they could give expert Go players a choice of sides and then defeat them either way.

经典的围棋游戏对早期的组合博弈论产生了影响，Berlekamp 和 Wolfe 随后为围棋发展出了一套终局和温度理论(见参考文献)。有了这些武器，他们就能够构建出合理的围棋终局位置，从中他们可以给围棋高手一个选择阵营的机会，然后以任何一种方式击败他们。

经典的围棋游戏对早期的组合博弈论影响很大，Berlekamp 和 Wolfe 随后为其发展出了一套终局和温度理论(见参考文献)。有了这些，他们就能够构建出合理的围棋终局局面，从中可以给围棋高手一个选择阵营的机会，然后以任何一种方式击败他们。

Another game studied in the context of combinatorial game theory is chess. In 1953 Alan Turing wrote of the game, "If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate."<ref>{{cite web

Another game studied in the context of combinatorial game theory is chess. In 1953 Alan Turing wrote of the game, "If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate."<ref>{{cite web

另一个在组合博弈论中研究的游戏是国际象棋。1953年，阿兰 · 图灵写道: “如果一个人能够用英语清楚地解释，如果需要的话，可以借助数学符号来解释如何进行计算，那么，只要存储容量足够，任何数字计算机都有可能进行计算。1.1.1.1.1.1.1.2.1.2.1.2.2.2.1.2.2.2.2.2.2.2.2.2.2.1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2

组合博弈论研究的另一种游戏是国际象棋。1953年，阿兰 · 图灵Alan Turing写道: “如果一个人能够用英语清楚地解释，如果需要的话，可以借助数学符号来解释如何进行计算，那么，只要存储容量足够，任何数字计算机都有可能进行计算。1.1.1.1.1.1.1.2.1.2.1.2.2.2.1.2.2.2.2.2.2.2.2.2.2.1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2

  | url=http://www.turingarchive.org/browse.php/B/7

  | url=http://www.turingarchive.org/browse.php/B/7

}}</ref> In a 1950 paper, Claude Shannon estimated the lower bound of the game-tree complexity of chess to be 10¹²⁰, and today this is referred to as the Shannon number. Chess remains unsolved, although extensive study, including work involving the use of supercomputers has created chess end-game tablebases, which shows the result of perfect play for all end-games with seven pieces or less. Infinite chess has an even greater combinatorial complexity than chess (unless only limited end-games, or composed positions with a small number of pieces are being studied).

}}</ref> In a 1950 paper, Claude Shannon estimated the lower bound of the game-tree complexity of chess to be 10¹²⁰, and today this is referred to as the Shannon number. Chess remains unsolved, although extensive study, including work involving the use of supercomputers has created chess end-game tablebases, which shows the result of perfect play for all end-games with seven pieces or less. Infinite chess has an even greater combinatorial complexity than chess (unless only limited end-games, or composed positions with a small number of pieces are being studied).

} / ref 在1950年的一篇论文中，Claude Shannon 估计国际象棋博弈树复杂度的下限是10 sup 120 / sup，今天这个下限被称为 Shannon 数。国际象棋仍未解决，尽管广泛的研究——包括使用超级计算机的研究——已经创造出了国际象棋最终游戏桌面基础，它显示了所有七个或更少棋子的最终游戏的完美结果。与国际象棋相比，无限大国际象棋具有更大的组合复杂性(除非只研究有限的最终对策，或者只研究有少量棋子的组合位置)。

} / ref 在1950年的一篇论文中，Claude Shannon 估计国际象棋博弈树复杂度的下限为10 sup 120 / sup，今天这被称为 Shannon 数。国际象棋仍然是无解的，尽管广泛的研究，包括涉及使用超级计算机的工作已经创建了国际象棋终局表库，它显示了所有七个或更少棋子的终局的完美结果。与国际象棋相比，无限棋比国际象棋有更大的组合复杂性(除非只研究有限的终局，或者只研究有少量棋子的组成局面)。

==Overview==

==Overview==

A game, in its simplest terms, is a list of possible "moves" that two players, called ''left'' and ''right'', can make.  The game position resulting from any move can be considered to be another game. This idea of viewing games in terms of their possible moves to other games leads to a [[recursion|recursive]] mathematical definition of games that is standard in combinatorial game theory. In this definition, each game has the notation '''{L|R}'''.  L is the [[set (mathematics)|set]] of game positions that the left player can move to, and R is the set of game positions that the right player can move to; each position in L and R is defined as a game using the same notation.

A game, in its simplest terms, is a list of possible "moves" that two players, called ''left'' and ''right'', can make.  The game position resulting from any move can be considered to be another game. This idea of viewing games in terms of their possible moves to other games leads to a [[recursion|recursive]] mathematical definition of games that is standard in combinatorial game theory. In this definition, each game has the notation '''{L|R}'''.  L is the [[set (mathematics)|set]] of game positions that the left player can move to, and R is the set of game positions that the right player can move to; each position in L and R is defined as a game using the same notation.

A game, in its simplest terms, is a list of possible "moves" that two players, called left and right, can make.  The game position resulting from any move can be considered to be another game. This idea of viewing games in terms of their possible moves to other games leads to a recursive mathematical definition of games that is standard in combinatorial game theory. In this definition, each game has the notation {L|R}.  L is the set of game positions that the left player can move to, and R is the set of game positions that the right player can move to; each position in L and R is defined as a game using the same notation.

A game, in its simplest terms, is a list of possible "moves" that two players, called left and right, can make.  The game position resulting from any move can be considered to be another game. This idea of viewing games in terms of their possible moves to other games leads to a recursive mathematical definition of games that is standard in combinatorial game theory. In this definition, each game has the notation {L|R}.  L is the set of game positions that the left player can move to, and R is the set of game positions that the right player can move to; each position in L and R is defined as a game using the same notation.

一个博弈，用最简单的术语来说，就是两个参与者(左边和右边)可以完成的一系列可能的“动作”。游戏的位置产生的任何移动可以被认为是另一个游戏。这种根据游戏可能转移到其他游戏的观点导致了游戏的递归数学定义，这是21组合博弈论的标准。在这个定义中，每个游戏都有一个符号{ l | r }。是左边玩家可以移动到的游戏位置的集合，r 是右边玩家可以移动到的游戏位置的集合; l 和 r 中的每个位置都被定义为使用相同符号的游戏。

一个游戏，用最简单的说法，就是两个参与者(左边和右边)可以完成的一系列可能的“动作”。游戏的位置产生的任何移动可以被认为是另一个游戏。这种根据游戏可能转移到其他游戏的观点导致了组合博弈论中标准的递归数学定义。在这个定义中，每个游戏都有{ l | r }的符号。L是左边玩家可以移动到的游戏位置的集合，R 是右边玩家可以移动到的游戏位置的集合; L和 R 中的每个位置都用同样的记号定义为一个游戏。

Using Domineering as an example, label each of the sixteen boxes of the four-by-four board by A1 for the upper leftmost square, C2 for the third box from the left on the second row from the top, and so on. We use e.g. (D3, D4) to stand for the game position in which a vertical domino has been placed in the bottom right corner. Then, the initial position can be described in combinatorial game theory notation as

Using Domineering as an example, label each of the sixteen boxes of the four-by-four board by A1 for the upper leftmost square, C2 for the third box from the left on the second row from the top, and so on. We use e.g. (D3, D4) to stand for the game position in which a vertical domino has been placed in the bottom right corner. Then, the initial position can be described in combinatorial game theory notation as

以霸气为例，用 A1表示最左边的方格，用 C2表示第二排左边的第三个方格，以此类推。我们使用例如。(D3，D4)代表游戏位置，即一块垂直的多米诺骨牌被放置在游戏的右下角。然后，初始位置可以用组合博弈论表示法描述为

以霸气为例，在四乘四棋盘的十六个棋格中，分别用A1代表最左上角的棋格，C2代表第二行从上到下左边的第三个棋格，以此类推。我们用如(D3，D4)代表在右下角放置了一张竖直的多米诺骨牌的游戏位置。那么，初始位置可以用组合博弈论的符号描述为

 

<math>\{(\mathrm{A}1,\mathrm{A}2),(\mathrm{B}1,\mathrm{B}2),\dots|(\mathrm{A}1,\mathrm{B}1), (\mathrm{A}2,\mathrm{B}2),\dots\}.</math>

<math>\{(\mathrm{A}1,\mathrm{A}2),(\mathrm{B}1,\mathrm{B}2),\dots|(\mathrm{A}1,\mathrm{B}1), (\mathrm{A}2,\mathrm{B}2),\dots\}.</math>

In standard Cross-Cram play, the players alternate turns, but this alternation is handled implicitly by the definitions of combinatorial game theory rather than being encoded within the game states.

In standard Cross-Cram play, the players alternate turns, but this alternation is handled implicitly by the definitions of combinatorial game theory rather than being encoded within the game states.

The above game describes a scenario in which there is only one move left for either player, and if either player makes that move, that player wins. (An irrelevant open square at C3 has been omitted from the diagram.) The <nowiki>{|}</nowiki> in each player's move list (corresponding to the single leftover square after the move) is called the zero game, and can actually be abbreviated 0.  In the zero game, neither player has any valid moves; thus, the player whose turn it is when the zero game comes up automatically loses.

The above game describes a scenario in which there is only one move left for either player, and if either player makes that move, that player wins. (An irrelevant open square at C3 has been omitted from the diagram.) The <nowiki>{|}</nowiki> in each player's move list (corresponding to the single leftover square after the move) is called the zero game, and can actually be abbreviated 0.  In the zero game, neither player has any valid moves; thus, the player whose turn it is when the zero game comes up automatically loses.

上面的游戏描述了一个场景，在这个场景中，任何一个玩家只剩下一步棋，如果任何一个玩家走了那一步棋，那个玩家就赢了。(图中省略了 C3处一个不相关的开平方)每个玩家的移动列表中的 nowiki { | } / nowiki (对应于移动后剩下的单个方块)称为零游戏，实际上可以缩写为0。在零游戏中，两个玩家都没有任何有效的移动，因此，当零游戏出现时，轮到的玩家自动失败。

上面的游戏描述了这样一种情形：任何一方只剩下一步棋，如果任何一方下了这一步棋，该一方就获胜。(图中省略了C3处无关的空位。)每个棋手棋谱中的  <nowiki>{|}</nowiki> (对应于下棋后剩余的单个方格)称为零棋局，实际上可以缩写为0。在零棋局中，双方都没有任何有效棋步；因此，当零棋局出现时，轮到自己的玩家自动输掉。

The type of game in the diagram above also has a simple name; it is called the star game, which can also be abbreviated ∗.  In the star game, the only valid move leads to the zero game, which means that whoever's turn comes up during the star game automatically wins.

The type of game in the diagram above also has a simple name; it is called the star game, which can also be abbreviated ∗.  In the star game, the only valid move leads to the zero game, which means that whoever's turn comes up during the star game automatically wins.

An additional type of game, not found in Domineering, is a loopy game, in which a valid move of either left or right is a game that can then lead back to the first game.  Checkers, for example, becomes loopy when one of the pieces promotes, as then it can cycle endlessly between two or more squares. A game that does not possess such moves is called loopfree.

An additional type of game, not found in Domineering, is a loopy game, in which a valid move of either left or right is a game that can then lead back to the first game.  Checkers, for example, becomes loopy when one of the pieces promotes, as then it can cycle endlessly between two or more squares. A game that does not possess such moves is called loopfree.

 

==Game abbreviations==

==Game abbreviations==

===Numbers===

===Numbers===

Numbers represent the number of free moves, or the move advantage of a particular player. By convention positive numbers represent an advantage for Left, while negative numbers represent an advantage for Right. They are defined recursively with 0 being the base case.

Numbers represent the number of free moves, or the move advantage of a particular player. By convention positive numbers represent an advantage for Left, while negative numbers represent an advantage for Right. They are defined recursively with 0 being the base case.

The zero game is a loss for the first player.

The zero game is a loss for the first player.

===Star===

===Star===

''[[star (game theory)|Star]]'', written as ∗ or {0|0}, is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.

''[[star (game theory)|Star]]'', written as ∗ or {0|0}, is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.

Star, written as ∗ or {0|0}, is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.

Star, written as ∗ or {0|0}, is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.

星，写作 x 或{0 | 0} ，是第一个玩家的胜利，因为任何一个玩家必须(如果第一个在游戏中移动)移动到零游戏，因此赢。

星，写作 x 或{0 | 0} ，是第一个玩家的胜利，因为任何一个玩家必须(如果第一个在游戏中移动)移动到零局，因此先手赢。

  ∗ + ∗ = 0, because the first player must turn one copy of ∗ to a 0, and then the other player will have to turn the other copy of ∗ to a 0 as well; at this point, the first player would lose, since 0 + 0 admits no moves.

  ∗ + ∗ = 0, because the first player must turn one copy of ∗ to a 0, and then the other player will have to turn the other copy of ∗ to a 0 as well; at this point, the first player would lose, since 0 + 0 admits no moves.

衰 x + 0，因为第一个参与者必须把一个 x 分数的副本变成0，然后另一个参与者也必须把另一个 x 分数的副本变成0; 这时，第一个参与者会输，因为0 + 0不允许移动。

∗ + ∗ = 0，因为第一个玩家必须把一个 ∗ 的一个棋子变成0，然后另一个玩家也必须把∗的另一个棋子变成0; 这时，第一个玩家会输，因为0 + 0不允许移动。

The game ∗ is neither positive nor negative; it and all other games in which the first player wins (regardless of which side the player is on) are said to be fuzzy with or confused with 0; symbolically, we write ∗ || 0.

The game ∗ is neither positive nor negative; it and all other games in which the first player wins (regardless of which side the player is on) are said to be fuzzy with or confused with 0; symbolically, we write ∗ || 0.

 

===Up===

===Up===

''Up'', written as ↑, is a position in combinatorial game theory.<ref name=winningways>{{cite book |author1=E. Berlekamp |author2=J. H. Conway |author3=R. Guy | title=[[Winning Ways for your Mathematical Plays]] | volume=I | publisher=Academic Press | year=1982 | isbn=0-12-091101-9}}<br/>{{cite book |author1=E. Berlekamp |author2=J. H. Conway |author3=R. Guy | title=Winning Ways for your Mathematical Plays | volume=II | publisher=Academic Press | year=1982 | isbn=0-12-091102-7}}</ref> In standard notation, ↑ = {0|∗}.

''Up'', written as ↑, is a position in combinatorial game theory.<ref name=winningways>{{cite book |author1=E. Berlekamp |author2=J. H. Conway |author3=R. Guy | title=[[Winning Ways for your Mathematical Plays]] | volume=I | publisher=Academic Press | year=1982 | isbn=0-12-091101-9}}<br/>{{cite book |author1=E. Berlekamp |author2=J. H. Conway |author3=R. Guy | title=Winning Ways for your Mathematical Plays | volume=II | publisher=Academic Press | year=1982 | isbn=0-12-091102-7}}</ref> In standard notation, ↑ = {0|∗}.

Up, written as ↑, is a position in combinatorial game theory. In standard notation, ↑ = {0|∗}.

Up, written as ↑, is a position in combinatorial game theory. In standard notation, ↑ = {0|∗}.

上面，注明↑ ，是组合博弈论中的一个位置。In standard notation, ↑ = {0|∗}.

上，写成↑，是组合博弈论中的一个位置。在标准符号中，↑={0|∗}。

Up is strictly positive (↑ > 0), but is infinitesimal. Up is defined in Winning Ways for your Mathematical Plays.

Up is strictly positive (↑ > 0), but is infinitesimal. Up is defined in Winning Ways for your Mathematical Plays.

===Down===

===Down===

''Down'', written as ↓, is a position in combinatorial game theory.<ref name=winningways /> In standard notation, ↓ = {∗|0}.

''Down'', written as ↓, is a position in combinatorial game theory.<ref name=winningways /> In standard notation, ↓ = {∗|0}.

Down, written as ↓, is a position in combinatorial game theory. In standard notation, ↓ = {∗|0}.

Down, written as ↓, is a position in combinatorial game theory. In standard notation, ↓ = {∗|0}.

是组合博弈论的一个职位↓。In standard notation, ↓ = {∗|0}.

下，写成↓，是组合博弈论中的一个位置。在标准符号中，↓={∗|0}。

 

 

 

Down is strictly negative (↓ < 0), but is infinitesimal. Down is defined in Winning Ways for your Mathematical Plays.

Down is strictly negative (↓ < 0), but is infinitesimal. Down is defined in Winning Ways for your Mathematical Plays.

 

==="Hot" games===

==="Hot" games===

Consider the game {1|−1}. Both moves in this game are an advantage for the player who makes them; so the game is said to be "hot;" it is greater than any number less than −1, less than any number greater than 1, and fuzzy with any number in between. It is written as ±1. It can be added to numbers, or multiplied by positive ones, in the expected fashion; for example, 4 ± 1 = {5|3}.

Consider the game {1|−1}. Both moves in this game are an advantage for the player who makes them; so the game is said to be "hot;" it is greater than any number less than −1, less than any number greater than 1, and fuzzy with any number in between. It is written as ±1. It can be added to numbers, or multiplied by positive ones, in the expected fashion; for example, 4 ± 1 = {5|3}.

Consider the game {1|−1}. Both moves in this game are an advantage for the player who makes them; so the game is said to be "hot;" it is greater than any number less than −1, less than any number greater than 1, and fuzzy with any number in between. It is written as ±1. It can be added to numbers, or multiplied by positive ones, in the expected fashion; for example, 4 ± 1 = {5|3}.

Consider the game {1|−1}. Both moves in this game are an advantage for the player who makes them; so the game is said to be "hot;" it is greater than any number less than −1, less than any number greater than 1, and fuzzy with any number in between. It is written as ±1. It can be added to numbers, or multiplied by positive ones, in the expected fashion; for example, 4 ± 1 = {5|3}.

Consider the game {1|−1}.在这个游戏中，两个动作都是制造它们的玩家的优势; 所以这个游戏被称为“热” ; 它比任何小于 -1的数字都要大，小于大于1的数字都要小，而且在两者之间的任何数字都是模糊的。它被写成了1。它可以以预期的方式加到数字上，或者乘以正数; 例如，41{5 | 3}。

考虑游戏{1|−1}.在这盘棋中，两步棋都对下棋者有利，所以说这盘棋是 "热棋"；它大于小于-1的任何数字，小于大于1的任何数字，而模糊的是中间的任何数字。它的写法是±1.它可以与数字相加，也可以与正数相乘，按预期的方式进行；例如，4±1={5|3}。

==Nimbers==

==Nimbers==

An [[impartial game]] is one where, at every position of the game, the same moves are available to both players. For instance, [[Nim]] is impartial, as any set of objects that can be removed by one player can be removed by the other. However, [[domineering]] is not impartial, because one player places horizontal dominoes and the other places vertical ones. Likewise Checkers is not impartial, since the players own different colored pieces. For any [[ordinal number]], one can define an impartial game generalizing Nim in which, on each move, either player may replace the number with any smaller ordinal number; the games defined in this way are known as [[nimber]]s. The [[Sprague–Grundy theorem]] states that every impartial game is equivalent to a nimber.

An [[impartial game]] is one where, at every position of the game, the same moves are available to both players. For instance, [[Nim]] is impartial, as any set of objects that can be removed by one player can be removed by the other. However, [[domineering]] is not impartial, because one player places horizontal dominoes and the other places vertical ones. Likewise Checkers is not impartial, since the players own different colored pieces. For any [[ordinal number]], one can define an impartial game generalizing Nim in which, on each move, either player may replace the number with any smaller ordinal number; the games defined in this way are known as [[nimber]]s. The [[Sprague–Grundy theorem]] states that every impartial game is equivalent to a nimber.

一个公正的游戏是这样的，在游戏的每个位置，两个玩家都可以使用相同的动作。例如，尼姆是公正的，因为任何一组可以被一个玩家移除的物体都可以被另一个玩家移除。然而，霸道并不公平，因为一个玩家放置水平的多米诺骨牌，另一个放置垂直的。同样，西洋跳棋也是不公平的，因为玩家拥有不同颜色的棋子。对于任何序数，我们都可以定义一个公正的游戏来概括 Nim，在这个游戏中，任何一个玩家在每次移动时都可以用任何较小的序数来替换这个数字; 这样定义的游戏被称为 nimbers。斯普拉格-格兰迪定理指出，每一个公正的博弈都等价于一个智者。

 

公正的游戏是指在游戏中的每个位置上，双方都可以走同样的移动路线。例如，尼姆是公正的，因为一个玩家可以移走的任何一组物体都可以被另一个玩家移走。然而，霸气不是公正的，因为一个玩家放置水平的骨牌，而另一个玩家放置垂直的骨牌。同样跳棋也不是公正的，因为玩家拥有不同颜色的棋子。对于任何序数，我们可以定义一个公平的广义尼姆游戏。在这个游戏中，每一个玩家都可以用任何更小的序数来代替这个数字；以这种方式定义的游戏被称为尼姆数。Sprague-Grundy定理指出，每一个公正的游戏都等同于尼姆游戏。

The "smallest" nimbers – the simplest and least under the usual ordering of the ordinals – are 0 and ∗.

The "smallest" nimbers – the simplest and least under the usual ordering of the ordinals – are 0 and ∗.

“最小”的邻位数是0和 * ，即按照通常的顺序排列最简单和最小的邻位数。

“最小”的尼姆数是0和 * ，即按照通常的顺序排列最简单和最小的尼姆数。

* [[Alpha–beta pruning]], an optimised algorithm for searching the game tree

* [[Alpha–beta pruning]], an optimised algorithm for searching the game tree

* [[Backward induction]], reasoning backwards from a final situation

* [[Backward induction]], reasoning backwards from a final situation

* [[Cooling and heating (combinatorial game theory)]], various transformations of games making them more amenable to the theory

* [[Cooling and heating (combinatorial game theory)]], various transformations of games making them more amenable to the theory

* [[Endgame tablebase]], a database saying how to play endgames

* [[Endgame tablebase]], a database saying how to play endgames

* [[Expectiminimax tree]], an adaptation of a minimax game tree to games with an element of chance

* [[Expectiminimax tree]], an adaptation of a minimax game tree to games with an element of chance

* [[Extensive-form game]], a game tree enriched with payoffs and information available to players

* [[Extensive-form game]], a game tree enriched with payoffs and information available to players