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Biased Positional Game
A biased positional game is a variant of a positional game. Like most positional games, it is described by a set of ''positions/points/elements'' (X) and a family of subsets (\mathcal), which are usually called the ''winning-sets''. It is played by two players who take turns picking elements until all elements are taken. While in the standard game each player picks one element per turn, in the biased game each player takes a different number of elements. More formally, for every two positive integers ''p'' and ''q'', a (p:q)-positional game is a game in which the first player picks ''p'' elements per turn and the second player picks ''q'' elements per turn. The main question of interest regarding biased positional games is what is their ''threshold bias'' - what is the bias in which the winning-power switches from one player to the other player. Example As an example, consider the ''triangle game''. In this game, the elements are all edges of a complete graph on ''n'' vertices, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positional Game
A positional game 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 winning strategy, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, 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. The term "collection" is used here because, 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 rather than a set. 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 power set of S and is denoted by \wp(S). The power set \wp(S) of a given set S is a family of sets over S. A subset of S having k elements is called a k-subset of S. The k-subsets S^ of a set S form a family of sets. Let S = \. An ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edges (a ... [...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] |
Acta Mathematica Hungarica
'' Acta Mathematica Hungarica'' is a peer-reviewed mathematics journal of the Hungarian Academy of Sciences, published by Akadémiai Kiadó and Springer Science+Business Media. The journal was established in 1950 and publishes articles on mathematics related to work by Hungarian mathematicians. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.39, and its 2015 impact factor was 0.469. The editor-in-chief is Imre Bárány, honorary editor is Ákos Császár, the editors are the mathematician members of the Hungarian Academy of Sciences. Abstracting and indexing According to the ''Journal Citation Reports'', the journal had a 2020 impact factor of 0.623. This journal is indexed by the following services: * Science Citation Index * Journal Citation Reports/Science Edition * Scopus * Mathematical Reviews * Zentralblatt Math zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Avoider-Enforcer Game
An Avoider-Enforcer game (also called Avoider-Forcer game or Antimaker-Antibreaker game) is a kind of positional game. Like most positional games, it is described by a set of ''positions/points/elements'' (X) and a family of subsets (\mathcal), which are called here the ''losing-sets''. It is played by two players, called Avoider and Enforcer, who take turns picking elements until all elements are taken. Avoider wins if he manages to avoid taking a losing set; Enforcer wins if he manages to make Avoider take a losing set. A classic example of such a game is ''Sim''. There, the positions are all the edges of the complete graph on 6 vertices. Players take turns to shade a line in their color, and lose when they form a full triangle of their own color: the losing sets are all the triangles. Comparison to Maker-Breaker games The winning condition of an Avoider-Enforcer game is exactly the opposite of the winning condition of the Maker-Breaker game on the same \mathcal. Thus, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box-making Game
A box-making game (often called just a box game) is a biased positional game where two players alternately pick elements from a family of pairwise-disjoint sets ("boxes"). The first player - called ''BoxMaker'' - tries to pick all elements of a single box. The second player - called ''BoxBreaker'' - tries to pick at least one element of all boxes. The box game was first presented by Paul Erdős and Václav Chvátal. It was solved later by Hamidoune and Las-Vergnas. Definition A box game is defined by: * A family of ''n'' pairwise-disjoint sets, A_1,\ldots,A_n, of different sizes. The sets are often called "boxes" and the elements are called "balls". * Two integers, ''p'' and ''q''. The first player, ''BoxMaker'', picks ''p'' balls (from the same or different boxes). Then the second player, ''BoxBreaker'', breaks ''q'' boxes. And so on. BoxMaker wins if he has managed to pick all balls in at least one box, before BoxBreaker managed to break this box. BoxBreaker wins if he has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
