Impartial Games
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In combinatorial game theory, an impartial game is a
game A game is a structured form of play (activity), play, usually undertaken for enjoyment, entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator s ...
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 Nim is a mathematical two player game. Nim or NIM may also refer to: * Nim (programming language) * Nim Chimpsky, a signing chimpanzee Acronyms * Network Installation Manager, an IBM framework * Nuclear Instrumentation Module * Negative index met ...
, Sprouts, Kayles, Quarto,
Cram Cram may refer to: * Cram (surname), a surname, and list of notable persons having the surname * Cram.com, a website for creating and sharing flashcards * Cram (Australian game show), a television show * ''Cram'' (game show), a TV game show that ...
,
Chomp Chomp is a two-player strategy game played on a rectangular grid made up of smaller square cells, which can be thought of as the blocks of a chocolate bar. The players take it in turns to choose one block and "eat it" (remove from the board), tog ...
,
Subtract a square Subtract-a-square (also referred to as take-a-square) is a two-player mathematical subtraction game. It is played by two people with a pile of coins (or other tokens) between them. The players take turns removing coins from the pile, always removin ...
,
Notakto Notakto is a tic-tac-toe variant, also known as neutral or impartial tic-tac-toe. The game is a combination of the games tic-tac-toe and Nim, played across one or several boards with both of the players playing the same piece (an "X" or cross). T ...
, 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 Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game w ...
,
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing g ...
or
dominos Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also ca ...
are not impartial games as they rely on chance. Impartial games can be analyzed using the Sprague–Grundy theorem, stating that every impartial game under the normal play convention is equivalent to a nimber. The representation of this nimber can change from game to game, but every possible state of any variation of an impartial game board should be able to have some nimber value. For example, several nim heaps in the game nim can be calculated, then summed using nimber addition, to give a nimber value for the game. A game that is not impartial is called a partisan game, though some partisan games can still be evaluated using nimbers such as Domineering. Domineering would not be classified as an impartial game as players use differently acting pieces, one player with vertical dominoes, one with horizontal ones, thereby breaking the rule that each player must be able to act using the same operations.


Requirements

All impartial games must meet the following conditions: * Two players must alternate turns until a final state is reached. * A winner is chosen when one player may no longer change position or make any operation. * There must be a finite number of operations and positions for both players. For example, in Nim, players must take away a subset of a stack that is currently in play. As there is a finite number of coins in any stack, a player may only remove a finite number of coins. * All operations must be able to be done by both sides. In all Impartial games, the players are making actions to some game board whether in the form of stacks for Nim or rows and columns Cram. Both players are acting on the board till the board can no longer change in some way. * No action in the game may be reliant on chance. Any inclusion of chance would mean there is not perfect information about the game, furthermore actions could not be minmaxed ruling out any form inductive strategy.


References


Further reading

*; ; *; ; ; ; {{cite book, title=vol. 4, isbn=1-56881-144-6, last1=Berlekamp, first1=Elwyn R., last2=Conway, first2=John Horton, last3=Guy, first3=Richard K., date=15 June 2004 Combinatorial game theory