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Partisan Game
In combinatorial game theory, a game is partisan (sometimes partizan) if it is not impartial. That is, some moves are available to one player and not to the other, or the payoffs are not symmetric. Most games are partisan. For example, in chess, only one player can move the white pieces. More strongly, when analyzed using combinatorial game theory, many chess positions have values that cannot be expressed as the value of an impartial game, for instance when one side has a number of extra tempos that can be used to put the other side into zugzwang. Partisan games are more difficult to analyze than impartial games, as the Sprague–Grundy theorem In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented ... does not apply. However, the application of combinatorial game theory to partisan games al ... [...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. Research in this field has primarily focused on two-player games in which a ''position'' evolves through alternating ''moves'', each governed by well-defined rules, with the aim of achieving a specific winning condition. Unlike game theory, economic game theory, combinatorial game theory generally avoids the study of games of chance or games involving imperfect information, preferring instead games in which the current state and the full set of available moves are always known to both players. However, as mathematical techniques develop, the scope of analyzable games expands, and the boundaries of the field continue to evolve. Authors typically define the term "game" at the outset of academic papers, with definitions tailored to the specific game under analysis rather than reflecting the field’s full scope. Combinatorics, Comb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 between player 1 and player 2 is that player 1 goes first. The game is played until a terminal position is reached. A terminal position is one from which no moves are possible. Then one of the players is declared the winner and the other the loser. Furthermore, impartial games are played with perfect information and no chance moves, meaning all information about the game and operations for both players are visible to both players. Impartial games include Nim, Sprouts, Kayles, Quarto, Cram, Chomp, Subtract a square, Notakto, and poset games. Go and chess are not impartial, as each player can only place or move pieces of their own color. Games such as poker, dice or dominos are not impartial games as they rely on chance. Impartial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chess
Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arranged in an 8×8 grid. The players, referred to as White and Black in chess, "White" and "Black", each control sixteen Chess piece, pieces: one king (chess), king, one queen (chess), queen, two rook (chess), rooks, two bishop (chess), bishops, two knight (chess), knights, and eight pawn (chess), pawns, with each type of piece having a different pattern of movement. An enemy piece may be captured (removed from the board) by moving one's own piece onto the square it occupies. The object of the game is to "checkmate" (threaten with inescapable capture) the enemy king. There are also several ways a game can end in a draw (chess), draw. The recorded history of chess goes back to at least the emergence of chaturanga—also thought to be an ancesto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zugzwang
Zugzwang (; ) is a situation found in chess and other turn-based games wherein one player is put at a disadvantage because of their obligation to make a move; a player is said to be "in zugzwang" when any legal move will worsen their position. Although the term is used less precisely in games such as chess, it is used specifically in combinatorial game theory to denote a move that directly changes the outcome of the game from a win to a loss. Putting the opponent in zugzwang is a common way to help the superior side win a game, and in some cases it is necessary in order to make the win possible. More generally, the term can also be used to describe a situation where none of the available options lead to a good outcome. The term ''zugzwang'' was used in German chess literature in 1858 or earlier, and the first known use of the term in English was by World Champion Emanuel Lasker in 1905. The concept of zugzwang was known to chess players many centuries before the term was coine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sprague–Grundy Theorem
In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization, or alternatively as a nimber, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim. The Grundy value or nim-value of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the nim-sequence of the game. The Sprague–Grundy theorem and its proof encapsulate the main results of a theory discovered independently b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nimber
In mathematics, the nimbers, also called Grundy numbers (not to be confused with Grundy chromatic numbers), are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with ''nimber addition'' and ''nimber multiplication'', which are distinct from ordinal addition and ordinal multiplication. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering. The nimber addition and multiplication operations are associative and commutative. Each nimber is its own additive inverse. In particular for some pairs of ordinals, their nimber sum is smaller than either addend. The minimum excludant operation is applied to sets of nimbers. Definition As a class, nimbers are indexed by ordinal numbers, and form a sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
