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Discrepancy Game
A discrepancy game is a kind of positional game A positional game in game theory is a kind of a combinatorial game for two players. It is described by: *Xa finite set of elements. Often ''X'' is called the ''board'' and its elements are called ''positions''. *\mathcala family of subsets of .... Like most positional games, it is described by its set of ''positions/points/elements'' (X) and a family of ''sets'' (\mathcal- a family of subsets of X). It is played by two players, called ''Balancer'' and ''Unbalancer''. Each player in turn picks an element. The goal of Balancer is to ensure that every set in \mathcal is balanced, i.e., the elements in each set are distributed roughly equally between the players. The goal of Unbalancer is to ensure that at least one set is unbalanced. Formally, the goal of balancer is defined by a vector (b_1,\ldots,b_n) where ''n'' is the number of sets in \mathcal. Balancer wins if in every set ''i'', the difference between the number of elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positional Game
A positional game in game theory is a kind of a combinatorial game for two players. It is described by: *Xa finite set of elements. Often ''X'' is called the ''board'' and its elements are called ''positions''. *\mathcala family of subsets of X. These subsets are usually called the ''winning sets''. * A criterion for winning the game. During the game, players alternately claim previously-unclaimed positions, until one of the players wins. If all positions in X are taken while no player wins, the game is considered a draw. The classic example of a positional game is tic-tac-toe. In it, X contains the 9 squares of the game-board, \mathcal contains the 8 lines that determine a victory (3 horizontal, 3 vertical and 2 diagonal), and the winning criterion is: the first player who holds an entire winning-set wins. Other examples of positional games are Hex and the Shannon switching game. For every positional game there are exactly three options: either the first player has a winn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I, known as the index set, to F, in which case the sets of the family are indexed by members of I. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class. A finite family of subsets of a finite set S is also called a '' hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maker-Breaker Game
A Maker-Breaker game is a kind of positional game. Like most positional games, it is described by its set of ''positions/points/elements'' (X) and its family of ''winning-sets'' (\mathcal- a family of subsets of X). It is played by two players, called Maker and Breaker, who alternately take previously untaken elements. In a Maker-Breaker game, Maker wins if he manages to hold all the elements of a winning-set, while Breaker wins if he manages to prevent this, i.e. to hold at least one element in each winning-set. Draws are not possible. In each Maker-Breaker game, either Maker or Breaker has a winning strategy. The main research question about these games is which of these two options holds. Examples A classic Maker-Breaker game is Hex. There, the winning-sets are all paths from the left side of the board to the right side. Maker wins by owning a connected path; Breaker wins by owning a connected path from top to bottom, since it blocks all connected paths from left to right. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positional Games
Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred (however, the values may be modified when combined). In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string. The Babylonian numeral system, base 60, was the first positional system to be developed, and its influence is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
