# 组合博弈论

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Mathematicians playing Konane at a combinatorial game theory workshop

数学家们在组合博弈论研讨会上玩Konane

**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.^{[1]} However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing.^{[2]} 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.

组合博弈论是数学和理论计算机科学的一个分支，主要研究具有完全信息的序贯博弈。研究很大程度上局限于双人游戏，在这种游戏中，玩家以规定的方式或招数轮流变化，以达到规定的制胜条件。CGT传统上并不研究机会博弈，也不研究那些使用不完全或不完整信息的博弈，而是倾向于那些提供完全信息的博弈，在这种博弈中，双方都知道博弈的状态和可用的招数。然而，随着数学技术的进步，可以用数学方法分析的博弈类型不断扩展，因此该领域的边界也在不断变化。学者们通常会在论文的开头定义他们所说的 "博弈 "，这些定义经常随着他们所分析的特定博弈而变化，并不代表该领域的全部范围。

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 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.

组合游戏包括众所周知的游戏，如国际象棋、跳棋、围棋等，这些游戏被认为是 非琐碎的 ，而井字棋则被认为是 "容易解决 "的琐碎游戏。一些组合游戏也可能有一个无限制的游戏区域，如无限棋。在CGT中，这些游戏和其他游戏中的棋步被表示为博弈树。

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".^{[3]})

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”。)

Game theory in general includes games of chance, games of imperfect knowledge, and games in which players can move simultaneously, and they tend to represent real-life decision making situations.

Game theory in general includes games of chance, games of imperfect knowledge, and games in which players can move simultaneously, and they tend to represent real-life decision making situations.

博弈论通常包括机会博弈、不完全知识博弈和参与者可以同时移动的博弈，它们往往代表现实生活中的决策情况。

CGT has a different emphasis than "traditional" or "economic" game theory, which was initially developed to study games with simple combinatorial structure, but with elements of chance (although it also considers sequential moves, see extensive-form game). Essentially, CGT has contributed new methods for analyzing game trees, for example using surreal numbers, which are a subclass of all two-player perfect-information games.^{[3]} The type of games studied by CGT is also of interest in artificial intelligence, particularly for automated planning and scheduling. In CGT there has been less emphasis on refining practical search algorithms (such as the alpha–beta pruning heuristic included in most artificial intelligence textbooks), but more emphasis on descriptive theoretical results (such as measures of game complexity or proofs of optimal solution existence without necessarily specifying an algorithm, such as the strategy-stealing argument).

CGT has a different emphasis than "traditional" or "economic" game theory, which was initially developed to study games with simple combinatorial structure, but with elements of chance (although it also considers sequential moves, see extensive-form game). Essentially, CGT has contributed new methods for analyzing game trees, for example using surreal numbers, which are a subclass of all two-player perfect-information games. The type of games studied by CGT is also of interest in artificial intelligence, particularly for automated planning and scheduling. In CGT there has been less emphasis on refining practical search algorithms (such as the alpha–beta pruning heuristic included in most artificial intelligence textbooks), but more emphasis on descriptive theoretical results (such as measures of game complexity or proofs of optimal solution existence without necessarily specifying an algorithm, such as the strategy-stealing argument).

CGT与“传统”或“经济”博弈论的重点不同，后者最初是研究具有简单组合结构但具有机会元素的博弈的（尽管它也考虑了顺序移动，请参阅扩展形式的博弈)。本质上，CGT 提供了分析博弈树的新方法，例如使用超现实数字，它是所有两人完全信息博弈的一个子类。CGT研究的游戏类型在人工智能中也很受关注，特别是在自动计划和调度方面。在CGT中，很少强调改进实用的搜索算法(例如大多数人工智能教科书中包含的 alpha-beta剪枝算法) ，而更多强调描述性的理论结果(例如对策复杂性的度量或最优解存在性的证明，而无需指定算法，例如策略窃取论点)。

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—optimal play by both sides also leads to a draw—but this result was a computer-assisted proof.^{[4]} 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—optimal play by both sides also leads to a draw— ’’’ 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.

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---- ’’’ 但这个结果是计算机辅助证明 ’’’。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，否则这个过程非常折磨人。

It can be helpful to distinguish between combinatorial "mathgames" of interest primarily to mathematicians and scientists to ponder and solve, and combinatorial "playgames" of interest to the general population as a form of entertainment and competition.^{[5]} However, a number of games fall into both categories. Nim, for instance, is a playgame instrumental in the foundation of CGT, and one of the first computerized games.^{[6]} Tic-tac-toe is still used to teach basic principles of game AI design to computer science students.

It can be helpful to distinguish between combinatorial "mathgames" of interest primarily to mathematicians and scientists to ponder and solve, and combinatorial "playgames" of interest to the general population as a form of entertainment and competition. However, a number of games fall into both categories. Nim, for instance, is a playgame instrumental in the foundation of CGT, and one of the first computerized games. Tic-tac-toe is still used to teach basic principles of game AI design to computer science students.

它有助于区分主要是供数学家和科学家思考和解决的组合型”数学游戏”和作为一种娱乐和竞争形式的广大民众感兴趣的组合型”游戏”。然而，许多游戏都属于这两种类型。以尼姆为例，它是在 CGT 基础上的一种游戏，也是最早的电脑游戏之一。井字棋仍然用于向计算机科学专业的学生教授游戏AI设计的基本原理。

## History

历史

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。 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。

Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining. It is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the sum of two games, which is a game where each player may choose to move either in one game or the other at any point in the game, and a player wins when his opponent has no move in either game. This way of combining games leads to a rich and powerful mathematical structure.

Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining. It is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the sum of two games, which is a game where each player may choose to move either in one game or the other at any point in the game, and a player wins when his opponent has no move in either game. This way of combining games leads to a rich and powerful mathematical structure.

按照惯例，组合游戏通常是一种形式，即一个玩家在另一方没有剩余移动时获胜。将只有两个可能结果的任何有限博弈转换为适用此约定的等效博弈是很容易的。组合博弈理论中最重要的概念之一是两个博弈之和，即每个玩家可以选择在一个博弈中或另一个博弈中的任何一个时刻移动，当对手在其中任何一个博弈中没有移动时，玩家获胜。这种组合游戏的方式导致了丰富而强大的数学结构。

John Conway states in ONAG that the inspiration for the theory of partisan games was based on his observation of the play in go endgames, which can often be decomposed into sums of simpler endgames isolated from each other in different parts of the board.

John Conway states in ONAG that the inspiration for the theory of partisan games was based on his observation of the play in go endgames, which can often be decomposed into sums of simpler endgames isolated from each other in different parts of the board.

约翰 · 康威在 ONAG 中指出，党派博弈理论的灵感来源于他对围棋终局游戏的观察。

==Examples==

例子

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:

《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 numbers. 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.

蓝-红哈肯布什 - 在有限层面上，这种党派组合游戏允许构造其值是二元有理数的游戏。在无限水平上，它允许构造所有实值，以及许多属于超现实数类的无限值。

- Blue–Red–Green Hackenbush - Allows for additional game values that are not numbers in the traditional sense, for example, star.

- 蓝-红-绿 哈肯布什-允许附加的游戏值不是传统意义上的数字，例如星（博弈论）。

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

- [蟾蜍和青蛙]- 允许各种游戏值。与大多数其他游戏不同，一个位置很容易用一串短字符来表示。

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

- 霸气 - 各种有趣的棋局，比如热棋，都会出现在主宰局面中，因为有时有下棋的动机，有时没有。 这允许讨论一盘棋的温度（博弈论）。

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

- 尼姆 - 一个公正的游戏。这使得尼姆数的构建成为可能。 (它也可被视为蓝-红-绿 哈肯布什中仅有绿色的特例)。

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."引用错误：没有找到与`</ref>`

对应的`<ref>`

标签 In a 1950 paper, Claude Shannon estimated the lower bound of the game-tree complexity of chess to be 10^{120}, and today this is referred to as the Shannon number.^{[7]} 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 10^{120}, 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

总揽

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，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]\displaystyle{ \{(\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]\displaystyle{ \{(\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.

在标准的Cross-Cram游戏中，玩家交替轮换，但这种交替是由组合博弈论的定义隐性处理的，而不是被编码在游戏状态中。

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图片: 20x20square. png

- [math]\displaystyle{ \{(\mathrm{A}1,\mathrm{A}2) | (\mathrm{A}1,\mathrm{B}1)\} = \{ \{|\} | \{|\} \}. }[/math]

[math]\displaystyle{ \{(\mathrm{A}1,\mathrm{A}2) | (\mathrm{A}1,\mathrm{B}1)\} = \{ \{|\} | \{|\} \}. }[/math]

Math (mathrum { a }1，mathrum { a }2) | (mathrum { a }1，mathrum { b }1) | | math

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 {|} 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 {|} 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 {|} (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.

上图中的游戏类型也有一个简单的名字，它叫做星棋，也可以缩写为∗。 在星型棋局中，唯一有效的棋步是导致零棋，也就是说，在星型棋局中，谁的回合出现，谁就自动获胜。

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.

另一种类型的游戏，不存在于霸气中，是一种循环游戏，在这种游戏中，左或右的有效举动都是一种游戏，然后可以导致回到第一个游戏。 例如，跳棋，当其中一个棋子晋升时，就会变得很循环，因为它可以在两个或更多的格子之间无休止地循环。不具备这种棋步的棋局称为无环棋。

## Game abbreviations

游戏缩写

### Numbers

数字

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.

数字代表自由移动的次数，或者某个特定玩家的移动优势。按照惯例，正数代表左边的优势，而负数代表右边的优势。它们是递归定义的，0是基本情况。

- 0 = {|}

0 = {|}

0 = {|}

- 1 = {0|}, 2 = {1|}, 3 = {2|}

1 = {0|}, 2 = {1|}, 3 = {2|}

1 = {0|}, 2 = {1|}, 3 = {2|}

- −1 = {|0}, −2 = {|−1}, −3 = {|−2}

−1 = {|0}, −2 = {|−1}, −3 = {|−2}

−1 = {|0}, −2 = {|−1}, −3 = {|−2}

The zero game is a loss for the first player.

The zero game is a loss for the first player.

0局，则是先发制人的失利。

—- 意译

The sum of number games behaves like the integers, for example 3 + −2 = 1.

The sum of number games behaves like the integers, for example 3 + −2 = 1.

数字游戏之和的行为类似于整数，例如3 +-21。

### Star

星

*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.

∗ + ∗ = 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.

游戏∗既不是正数，也不是负数；它和所有其他最先获胜的游戏（无论玩家在哪一方）都被称为与0模糊或混淆的游戏；象征性地，我们写成∗ || 0。

### Up

上

*Up*, written as ↑, is a position in combinatorial game theory.^{[8]} In standard notation, ↑ = {0|∗}.

Up, written as ↑, is a position in combinatorial game theory. 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|∗}。

- −↑ = ↓ (
*down*)

−↑ = ↓ (down)

−↑ = ↓ (down)

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.

上 是严格的正数（↑ > 0），但却是无穷小的。上 的定义见《数学游戏的赢家》。

### Down

下

*Down*, written as ↓, is a position in combinatorial game theory.^{[8]} In standard notation, ↓ = {∗|0}.

Down, written as ↓, is a position in combinatorial game theory. 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}。

- −↓ = ↑ (
*up*)

−↓ = ↑ (up)

−↓ = ↑ (up)

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.

Down是严格意义上的负数（↓<0），但却是无穷小的。Down的定义见《数学游戏的赢家》。

### "Hot" games

热门游戏

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}.

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

尼姆数

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 nimbers. The Sprague–Grundy theorem states that every impartial game is equivalent to a nimber.

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。

## See also

另见

- Alpha–beta pruning, an optimised algorithm for searching the game tree

Alpha-beta剪枝算法，一种用于搜索博弈树的优化算法

- Backward induction, reasoning backwards from a final situation

逆向归纳，从最终情况反向推理

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

制冷和制热（组合博弈论），博弈的各种转变使它们更适合该理论

- Connection game, a type of game where players attempt to establish connections

连接游戏，玩家尝试建立连接的一种游戏

- Endgame tablebase, a database saying how to play endgames

端游表库，一个数据库，说明如何玩残局

- Expectiminimax tree, an adaptation of a minimax game tree to games with an element of chance

Expectiminimax树，最小化博弈树的改编，以适应有偶然因素的博弈。

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

广泛形式的游戏，丰富了收益和玩家可用信息的博弈树

- Game classification, an article discussing ways of classifying games

游戏分类，有关讨论游戏分类方法的文章

- Game complexity, an article describing ways of measuring the complexity of games

游戏复杂度，一篇描述衡量游戏复杂度的方法的文章

- Grundy's game, a mathematical game in which heaps of objects are split

格伦迪游戏，一种数学游戏，其中拆分成堆的对象

- Multi-agent system, a type of computer system for tackling complex problems

多主体系统，一种用于解决复杂问题的计算机系统

解决国际象棋

- Sylver coinage, a mathematical game of choosing positive integers that are not the sum of non-negative multiples of previously chosen integers

Sylver coinage，一种选择正整数的数学游戏，该整数不是先前选择的整数的非负倍数的总和

- Wythoff's game, a mathematical game of taking objects from one or two piles

Wythoff的游戏，一种从一堆或两堆中取出物体的数学游戏

- Topological game, a type of mathematical game played in a topological space

拓扑游戏，一种在拓扑空间中玩的数学游戏

- Zugzwang, being obliged to play when this is disadvantageous

强制被动，不利时唯一可行的走法

## Notes

- ↑ Lessons in Play, p. 3
- ↑ Thomas S. Fergusson's analysis of poker is an example of CGT expanding into games that include elements of chance. Research into Three Player NIM is an example of study expanding beyond two player games. Conway, Guy and Berlekamp's analysis of partisan games is perhaps the most famous expansion of the scope of CGT, taking the field beyond the study of impartial games.
- ↑
^{3.0}^{3.1}http://erikdemaine.org/papers/AlgGameTheory_GONC3/paper.pdf - ↑ Schaeffer, J.; Burch, N.; Bjornsson, Y.; Kishimoto, A.; Muller, M.; Lake, R.; Lu, P.; Sutphen, S. (2007). "Checkers is solved".
*Science*.**317**(5844): 1518–1522. Bibcode:2007Sci...317.1518S. CiteSeerX 10.1.1.95.5393. doi:10.1126/science.1144079. PMID 17641166. - ↑ Fraenkel, Aviezri (2009). "Combinatorial Games: selected bibliography with a succinct gourmet introduction".
*Games of No Chance 3*.**56**: 492. - ↑ Grant, Eugene F.; Lardner, Rex (2 August 1952). "The Talk of the Town - It".
*The New Yorker*. - ↑ Claude Shannon (1950). "Programming a Computer for Playing Chess" (PDF).
*Philosophical Magazine*.**41**(314): 4. Archived from the original (PDF) on 2010-07-06. - ↑
^{8.0}^{8.1}E. Berlekamp; J. H. Conway; R. Guy (1982).*Winning Ways for your Mathematical Plays*.**I**. Academic Press. ISBN 0-12-091101-9.

E. Berlekamp; J. H. Conway; R. Guy (1982).*Winning Ways for your Mathematical Plays*.**II**. Academic Press. ISBN 0-12-091102-7.

## References

- [[Michael H. Albert

作者: 迈克尔 · h · 艾伯特 |Albert, Michael H.]]; Nowakowski, Richard J.

2; Wolfe, David

3 David (2007). *Lessons in Play: An Introduction to Combinatorial Game Theory*

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[国际标准图书馆编号978-1-56881-277-9]|978-1-56881-277-9

[国际标准图书馆编号978-1-56881-277-9]]].

| year = 2007}}

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*Combinatorial games: tic-tac-toe theory*

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出版商剑桥大学出版社. ISBN [[Special:BookSources/978-0-521-46100-9

[国际标准图书编号978-0-521-46100-9]|978-0-521-46100-9

[国际标准图书编号978-0-521-46100-9]]].

| year = 2008}}

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- [[Elwyn Berlekamp

1 |Berlekamp, E.]]; [[John Horton Conway

2 Conway |Conway, J. H.]]; [[Richard K. Guy

作者: 理查德 · 盖伊 |Guy, R.]] (1982). *Winning Ways for your Mathematical Plays: Games in general*. Academic Press

出版商: 学术出版社. ISBN 0-12-091101-9. 2nd ed., A K Peters Ltd (2001–2004),

,

| year = 1982}} 2nd ed., A K Peters Ltd (2001–2004), ,

| 1982年}}第二版，a k Peters Ltd (2001-2004) ，,

- Berlekamp, E.

1; Conway, J. H.

2 Conway; [[Richard K. Guy

作者: 理查德 · 盖伊 |Guy, R.]] (1982). *Winning Ways for your Mathematical Plays: Games in particular*. Academic Press

出版商: 学术出版社. ISBN 0-12-091102-7. https://archive.org/details/winningwaysforyo02berl. 2nd ed., A K Peters Ltd (2001–2004),

,

.

| url = https://archive.org/details/winningwaysforyo02berl | url-access = registration | year = 1982}} 2nd ed., A K Peters Ltd (2001–2004), , .

Https://archive.org/details/winningwaysforyo02berl : a k Peters Ltd (2001-2004) ，1982年。

- [[Elwyn Berlekamp

1 Elwyn |Berlekamp, Elwyn]]; [[David Wolfe (mathematician)

大卫 · 沃尔夫(数学家) |Wolfe, David]] (1997). [https://archive.org/details/mathematicalgoch0000berl *Mathematical Go: Chilling Gets the Last Point*

数学行动: 冷却得到最后一分*]. A K Peters Ltd. ISBN [[Special:BookSources/1-56881-032-6*

1-56881-032-6|1-56881-032-6

1-56881-032-6]]. https://archive.org/details/mathematicalgoch0000berl.

| url = https://archive.org/details/mathematicalgoch0000berl | url-access = registration | year = 1997}}

| https://archive.org/details/mathematicalgoch0000berl | url-access registration | year 1997}

- Bewersdorff, Jörg (2004).
*Luck, Logic and White Lies: The Mathematics of Games*

运气，逻辑和善意的谎言: 游戏的数学*. A K Peters Ltd. ISBN [[Special:BookSources/1-56881-210-8*

1-56881-210-8|1-56881-210-8

1-56881-210-8]]. See especially sections 21–26.

| year = 2004}} See especially sections 21–26.

特别是21-26节。

- [[John Horton Conway

最后一个康威 |Conway, John Horton]] (1976). *On Numbers and Games*

关于数字和游戏*. Academic Press*

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.

| year = 1976}} 2nd ed., A K Peters Ltd (2001), .

| 1976年}}第二版，a k Peters Ltd (2001) ，。

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*Games, Puzzles, and Computation*. A K Peters, Ltd.. ISBN 978-1-56881-322-6.

## External links

- List of combinatorial game theory links at the homepage of David Eppstein

- An Introduction to Conway's games and numbers by Dierk Schleicher and Michael Stoll

- Combinational Game Theory terms summary by Bill Spight

This page was moved from wikipedia:en:Combinatorial game theory. Its edit history can be viewed at 组合博弈论/edithistory