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Strictly Determined Game
In game theory, a strictly determined game is a two-player zero-sum game that has at least one Nash equilibrium with both players using pure strategies. The value of a strictly determined game is equal to the value of the equilibrium outcome. Most finite combinatorial games, like tic-tac-toe, chess, draughts, and go, are strictly determined games. Notes The study and classification of strictly determined games is distinct from the study of Determinacy, which is a subfield of set theory. See also * Solved game A solved game is a game whose outcome (win, lose or tie (draw), draw) can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied to abstract strategy games, and especially to games with ... References Game theory game classes {{Mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 1950s, it was extended to the study of non zero-sum games, and was eventually applied to a wide range of Human behavior, behavioral relations. It is now an umbrella term for the science of rational Decision-making, decision making in humans, animals, and computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-player Game
A two-player game is a multiplayer game that is played by precisely two players. This is distinct from a solitaire game, which is played by only one player. Examples The following are some examples of two-player games. This list is not intended to be exhaustive. * Board games: ** Chess ** Checkers ** Go ** Some wargames, such as '' Hammer of the Scots'' * Card games: ** Cribbage ** Whist ** Rummy ** 66 ** Pinochle ** '' Magic: The Gathering'', a collectible card game in which players duel * Sports: ** Cue sports, a family of games that use cue sticks and billiard balls ** Many athletic games, such as tennis ( singles) * Video games: **''Pong'' ** A Way Out See also * List of types of games * Zero-sum game Zero-sum game is a Mathematical model, mathematical representation in game theory and economic theory of a situation that involves two competition, competing entities, where the result is an advantage for one side and an equivalent loss for the o ... References {{Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-sum
Zero-sum game is a mathematical representation in game theory and economic theory of a situation that involves two competing entities, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is equivalent to player two's loss, with the result that the net improvement in benefit of the game is zero. If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally. Other examples of zero-sum games in daily life include games like poker, chess, sport and bridge where one person gains and another person loses, which results in a zero-net benefit for every player. In the markets and financial instruments, futures contracts and options are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Equilibrium
In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed). The idea of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly. If each player has chosen a strategy an action plan based on what has happened so far in the game and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Strategy
In game theory, a move, action, or play is any one of the options which a player can choose in a setting where the optimal outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. The term strategy is typically used to mean a complete algorithm for playing a game, telling a player what to do for every possible situation. A player's strategy determines the action the player will take at any stage of the game. However, the idea of a strategy is often confused or conflated with that of a move or action, because of the correspondence between moves and pure strategies in most games: for any move ''X'', "always play move ''X''" is an example of a valid strategy, and as a result every move can also be considered to be a strategy. Other authors treat strate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...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] |
Tic-tac-toe
Tic-tac-toe (American English), noughts and crosses (English in the Commonwealth of Nations, Commonwealth English), or Xs and Os (Canadian English, Canadian or Hiberno-English, Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid, one with Xs and the other with Os. A player wins when they mark all three spaces of a row, column, or diagonal of the grid, whereupon they traditionally draw a line through those three marks to indicate the win. It is a solved game, with a forced draw assuming Best response, best play from both players. Names In American English, the game is known as "tic-tac-toe". It may also be spelled "tick-tack-toe", "tick-tat-toe", or "tit-tat-toe". In Commonwealth English (particularly British English, British, South African English, South African, Indian English, Indian, Australian English, Australian, and New Zealand English), the game is known as "noughts and crosses", alternatively spelled ... [...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] |
Draughts
Checkers (American English), also known as draughts (; Commonwealth English), is a group of strategy board games for two players which involve forward movements of uniform game pieces and mandatory captures by jumping over opponent pieces. Checkers is developed from alquerque. The term "checkers" derives from the checkered board which the game is played on, whereas "draughts" derives from the verb "to draw" or "to move". The most popular forms of checkers in Anglophone countries are American checkers (also called English draughts), which is played on an 8×8 checkerboard; Russian draughts, Turkish draughts and Armenian draughts, all of them on an 8×8 board; and international draughts, played on a 10×10 board – with the latter widely played in many countries worldwide. There are many other variants played on 8×8 boards. Canadian checkers and Malaysian/Singaporean checkers (also locally known as dam) are played on a 12×12 board. American checkers was weakly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Go (game)
# Go is an abstract strategy game, abstract strategy board game for two players in which the aim is to fence off more territory than the opponent. The game was invented in China more than 2,500 years ago and is believed to be the oldest board game continuously played to the present day. A 2016 survey by the International Go Federation's 75 member nations found that there are over 46 million people worldwide who know how to play Go, and over 20 million current players, the majority of whom live in East Asia. The Game piece (board game), playing pieces are called ''Go equipment#Stones, stones''. One player uses the white stones and the other black stones. The players take turns placing their stones on the vacant intersections (''points'') on the #Boards, board. Once placed, stones may not be moved, but ''captured stones'' are immediately removed from the board. A single stone (or connected group of stones) is ''captured'' when surrounded by the opponent's stones on all Orthogona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinacy
Determinacy is a subfield of game theory and set theory that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists. Determinacy was introduced by Gale and Stewart in 1950, under the name determinateness. The games studied in set theory are usually Gale–Stewart games—two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws. The field of game theory studies more general kinds of games, including games with draws such as tic-tac-toe, chess, or infinite chess, or games with imperfect information such as poker. Basic notions Games The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers. These games are often called Gale–St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
