|
Zermelo's Theorem (game Theory)
In game theory, Zermelo's theorem is a theorem about finite two-person games of perfect information in which the players move alternately and in which chance does not affect the decision making process. It says that if the game cannot end in a draw, then one of the two players must have a winning strategy (i.e. can force a win). An alternate statement is that for a game meeting all of these conditions except the condition that a draw is now possible, then either the first-player can force a win, or the second-player can force a win, or both players can at least force a draw. The theorem is named after Ernst Zermelo, a German mathematician and logician, who proved the theorem for the example game of chess in 1913. Example Zermelo's theorem can be applied to all finite-stage two-player games with complete information and alternating moves. The game must satisfy the following criteria: there are two players in the game; the game is of perfect information; the board game is finite; ... [...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] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...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] |
László Kalmár
László Kalmár ( ; 27 March 1905, Edde – 2 August 1976, Mátraháza) was a Hungarian mathematician and Professor at the University of Szeged. Kalmár is considered the founder of mathematical logic and theoretical computer science in Hungary. Biography Kalmár was of Jewish ancestry. His early life mixed promise and tragedy. His father died when he was young, and his mother died when he was 17, the year he entered the University of Budapest, making him essentially an orphan. Kalmár's brilliance manifested itself while in Budapest schools. At the University of Budapest, his teachers included Kürschák and Fejér. His fellow students included the future logician Rózsa Politzer, from 1934 on Rózsa Péter. Kalmár graduated in 1927. He discovered mathematical logic, his chosen field, while visiting Göttingen in 1929. Upon completing his doctorate at Budapest, he took up a position at the University of Szeged. That university was mostly made up of staff from the form ... [...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] |
Threefold Repetition
In chess, the threefold repetition rule states that a player may claim a draw if the same position occurs three times during the game. The rule is also known as repetition of position and, in the USCF rules, as triple occurrence of position.Article 14K.2 in Two positions are by definition "the same" if pieces of the same type and color occupy the same squares, the same player has the move, the remaining castling rights are the same and the possibility to capture ''en passant'' is the same. The repeated positions need not occur in succession. The game is not automatically drawn if a position occurs for the third time – one of the players, on their turn, must claim the draw with the arbiter. The claim must be made either before making the move which will produce the third repetition, or after the opponent has made a move producing a third repetition. By contrast, the fivefold repetition rule requires the arbiter to intervene and declare the game drawn if the same position occurs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fifty Move Rule
The fifty-move rule in chess states that a player can claim a draw if no has been made and no pawn has been moved in the last fifty moves (where a "move" consists of a player completing a turn followed by the opponent completing a turn). The purpose of this rule is to prevent a player with no chance of winning from obstinately continuing to play indefinitely or seeking to win by tiring the opponent. Chess positions with only a few pieces can be " solved", that is, the outcome of best play for both sides can be determined by exhaustive analysis; if the outcome is a win for one side or the other (rather than a draw), it is of interest to know whether the defending side can hold out long enough to invoke the fifty-move rule. The simplest common endings, called the basic checkmates, such as king and queen versus king, can all be won in well under 50 moves. However, in the 20th century it was discovered that certain endgame positions are winnable but require more than 50 moves (with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dénes Kőnig
Dénes Kőnig (September 21, 1884 – October 19, 1944) was a Hungarian mathematician of Hungarian Jewish heritage who worked in and wrote the first textbook on the field of graph theory. Biography Kőnig was born in Budapest, the son of mathematician Gyula Kőnig. In 1907, he received his doctorate Translated by Richard McCoart; with commentary by W.T. Tutte. at, and joined the faculty of the Royal Joseph University in Budapest (today Budapest University of Technology and Economics). His classes were visited by Paul Erdős, who, as a first year student, solved one of his problems. Kőnig became a full professor there in 1935. To honor his fathers' death in 1913, Kőnig and his brother György created the Gyula Kőnig prize in 1918. This prize was meant to be an endowment for young mathematicians, however was later devaluated. But the prize remained as a medal of high scientific recognition. In 1899, he published his first work while still attending High School in a journal ''Matem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pigeonhole Principle
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, of three gloves, at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into. This seemingly obvious statement, a type of combinatorics, counting argument, can be used to demonstrate possibly unexpected results. For example, given that the Demographics of London, population of London is more than one unit greater than the maximum number of hairs that can be on a human's head, the principle requires that there must be at least two people in London who have the same number of hairs on their heads. Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Pet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Contradiction
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, and ''reductio ad impossibile''. A mathematical proof employing proof by contradiction usually proceeds as follows: #The proposition to be proved is ''P''. #We assume ''P'' to be false, i.e., we assume ''¬P''. #It is then shown that ''¬P'' implies falsehood. This is typically accomplished by deriving two mutually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Congress Of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before 2022 as the Nevanlinna Prize), the Carl Friedrich Gauss Prize, Gauss Prize, and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being List of International Congresses of Mathematicians Plenary and Invited Speakers, invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame". History German mathematicians Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.A. John Coleman"Mathematics without borders": a book review. ''CMS Notes'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Information
Perfect information is a concept in game theory and economics that describes a situation where all players in a game or all participants in a market have knowledge of all relevant information in the system. This is different than complete information, which implies Common knowledge (logic), common knowledge of each agent's utility functions, payoffs, strategies and "types". A system with perfect information may or may not have complete information. In economics this is sometimes described as "no hidden information" and is a feature of perfect competition. In a market with perfect information all consumers and producers would have complete and instantaneous knowledge of all market prices, their own utility and cost functions. In game theory, a sequential game has perfect information if each player, when making any decision, is perfectly informed of all the events that have previously occurred, including the "initialization event" of the game (e.g. the starting hands of each player ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
