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Waldegrave Problem
In probability and game theory, the Waldegrave problem refers to a problem first described in the second edition of Pierre Raymond de Montmort`s '' Essay d'analyse sur les jeux de hazard''. This problem is remarkable in that it is the first appearance to a mixed strategy solution in game theory. Montmort originally called Waldegrave's Problem the ''Problème de la Poulle'' or the Problem of the Pool. He provides a minimax mixed strategy solution to a two-person version of the card game le Her Le Her (or ''le Hère'') is a French card game that dates back to the 16th century. It is quoted by the French poet Marc Papillon de Lasphrise in 1597. Under the name ''coucou'' it is mentioned in Rabelais' long list of games (in Gargantua, 1534). .... It was Isaac Todhunter who called it Waldegrave's Problem. The general description of the problem is as follows: Suppose there are n+1 players with each player putting one unit into the pot or pool. The first two players play each other and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game 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 mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Raymond De Montmort
Pierre Remond de Montmort was a French mathematician. He was born in Paris on 27 October 1678 and died there on 7 October 1719. His name was originally just Pierre Remond. His father pressured him to study law, but he rebelled and travelled to England and Germany, returning to France in 1699 when, upon receiving a large inheritance from his father, he bought an estate and took the name de Montmort. He was friendly with several other notable mathematicians, and especially Nicholas Bernoulli, who collaborated with him while visiting his estate. He was elected a fellow of the Royal Society in 1715, while traveling again to England, and became a member of the French Academy of Sciences in 1716. De Montmort is known for his book on probability and games of chance, Essay d'analyse sur les jeux de hazard, which was also the first to introduce the combinatorial study of derangements. He is also known for naming Pascal's triangle after Blaise Pascal, calling it "Table de M. Pascal pour ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essay D'analyse Sur Les Jeux De Hazard
''Essay d'analyse sur les jeux de hazard'' ("Analysis of games of chance") is a book on combinatorics and mathematical probability written by Pierre Remond de Montmort and published in 1708. The work applied ideas from combinatorics and probability to analyse various games of chances popular during the time. This book was mainly influenced by Huygens' treatise ''De ratiociniis in ludo aleae'' and the knowledge of the fact that Jakob Bernoulli had written an unfinished work in probability. The work was intended to re-create the yet unpublished work of Jakob Bernoulli called Ars Conjectandi. The work greatly influenced the thinking of Montmort's contemporary, Abraham De Moivre. Montmort collaborated with Nicolaus(I) Bernoulli in a fascinating correspondence which began in 1710. They discussed many topics, particularly the probability questions that arose from Montmort's book. A second edition of the book was published in 1714, a year after the publication of Ars Conjectandi, published ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimax
Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. Originally formulated for several-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty. Game theory In general games The maximin value is the highest value that the player can be sure to get without knowing the actions of the other players; equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Its formal definition is: :\underline = \max_ \min_ Where: * is the index of the player of interest. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Strategy
In game theory, a player's strategy is any of the options which they choose in a setting where the 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. A player's strategy will determine the action which the player will take at any stage of the game. In studying game theory, economists enlist a more rational lens in analyzing decisions rather than the psychological or sociological perspectives taken when analyzing relationships between decisions of two or more parties in different disciplines. The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Le Her
Le Her (or ''le Hère'') is a French card game that dates back to the 16th century. It is quoted by the French poet Marc Papillon de Lasphrise in 1597. Under the name ''coucou'' it is mentioned in Rabelais' long list of games (in Gargantua, 1534). Le Her belongs to the family of Ranter-Go-Round games. It is played with a standard deck of 52 cards by two people, designated the dealer and the receiver. King is ranked high and ace low. To play, the dealer gives one card to the receiver and one to the dealer. The receiver may choose to exchange cards with the dealer, unless the dealer has a king, in which case no exchange occurs. Then, the dealer may choose to exchange with the top card of the deck, unless the top card is a king, in which case no exchange occurs. In the case of the dealer and receiver having same ranked cards, the dealer is the winner. Le Her played a role in the development of the mathematical theory of probability with solutions being sought by Bernoulli Bernoulli c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolaus I Bernoulli
Nicolaus Bernoulli (also spelled Nicolas or Nikolas; 21 October 1687, Basel – 29 November 1759, Basel) was a Swiss people, Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. Biography He was the son of Nicolaus Bernoulli, painter and Alderman of Basel. In 1704 he graduated from the University of Basel under Jakob Bernoulli and obtained his PhD five years later (in 1709) with a work on probability theory in law. His thesis was titled ''Dissertatio Inauguralis Mathematico-Juridica de Usu Artis Conjectandi in Jure''. In 1716 he obtained the Galileo-chair at the University of Padua, where he worked on differential equations and geometry. In 1722 he returned to Switzerland and obtained a chair in Logics at the University of Basel. He was elected a Fellow of the Royal Society of London in March, 1714. His most important contributions can be found in his letters, in particular to Pierre Rémond de Montmort. In these letters, he introduced in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game 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 mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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