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Minimax
Minimax (sometimes Minmax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, combinatorial game theory, statistics, and philosophy for ''minimizing'' the possible loss function, loss for a Worst-case scenario, 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_ W ...
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Alpha–beta Pruning
Alpha–beta pruning is a search algorithm that seeks to decrease the number of nodes that are evaluated by the Minimax#Minimax algorithm with alternate moves, minimax algorithm in its game tree, search tree. It is an adversarial search algorithm used commonly for machine playing of two-player Combinatorial game theory, combinatorial games (Tic-tac-toe, Chess, Connect 4, etc.). It stops evaluating a move when at least one possibility has been found that proves the move to be worse than a previously examined move. Such moves need not be evaluated further. When applied to a standard minimax tree, it returns the same move as minimax would, but prunes away branches that cannot possibly influence the final decision. History John McCarthy during the Dartmouth workshop, Dartmouth Workshop met Alex Bernstein of IBM, who was writing a chess program. McCarthy invented alpha–beta search and recommended it to him, but Bernstein was "unconvinced". Allen Newell and Herbert A. Simon who us ...
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Minimax Theorem
In the mathematical area of game theory and of convex optimization, a minimax theorem is a theorem that claims that : \max_ \min_ f(x,y) = \min_ \max_f(x,y) under certain conditions on the sets X and Y and on the function f. It is always true that the left-hand side is at most the right-hand side ( max–min inequality) but equality only holds under certain conditions identified by minimax theorems. The first theorem in this sense is von Neumann's minimax theorem about two-player zero-sum games published in 1928, which is considered the starting point of game theory. Von Neumann is quoted as saying "''As far as I can see, there could be no theory of games ... without that theorem ... I thought there was nothing worth publishing until the Minimax Theorem was proved''". Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature. Bilinear functions and zero-sum games Von Neumann's original theorem was motivated by ...
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Evaluation Function
An evaluation function, also known as a heuristic evaluation function or static evaluation function, is a function used by game-playing computer programs to estimate the value or goodness of a position (usually at a leaf or terminal node) in a game tree. Most of the time, the value is either a real number or a quantized integer, often in ''n''ths of the value of a playing piece such as a stone in go or a pawn in chess, where ''n'' may be tenths, hundredths or other convenient fraction, but sometimes, the value is an array of three values in the unit interval, representing the win, draw, and loss percentages of the position. There do not exist analytical or theoretical models for evaluation functions for unsolved games, nor are such functions entirely ad-hoc. The composition of evaluation functions is determined empirically by inserting a candidate function into an automaton and evaluating its subsequent performance. A significant body of evidence now exists for several games ...
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Ply (chess)
In two-or-more-player sequential games, a ply is one turn taken by one of the players. The word is used to clarify what is meant when one might otherwise say "turn". The word "turn" can be a problem since it means different things in different traditions. For example, in standard chess terminology, one ''move'' consists of a turn by each player; therefore a ply in chess is a ''half-move''. Thus, after 20 moves in a chess game, 40 plies have been completed—20 by white and 20 by black. In the game of Go, by contrast, a ply is the normal unit of counting moves; so for example to say that a game is ''250 moves long'' is to imply 250 plies. In poker with ''n'' players the word "street" is used for a full betting round consisting of ''n'' plies; each dealt card may sometimes also be called a "street". For instance, in heads up Texas hold'em, a street consists of 2 plies, with possible plays being check/raise/call/fold: the first by the player at the big blind, and the second by th ...
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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 ...
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Loss Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Pierre-Simon Laplace, Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economi ...
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Worst-case Scenario
A worst-case scenario is a concept in risk management wherein the planner, in planning for potential disasters, considers the most severe possible outcome that can reasonably be projected to occur in a given situation. Conceiving of worst-case scenarios is a common form of strategic planning, specifically scenario planning, to prepare for and minimize contingencies that could result in accidents, quality problems, or other issues. Development and use The worst-case scenario is " e of the most commonly used alternative scenarios".Charles Yoe, ''Principles of Risk Analysis: Decision Making Under Uncertainty'' (2011), p. 429-30. A risk manager may request "a conservative risk estimate representing a worst-case scenario" in order to determine the latitude they may exercise in planning steps to reduce risks.Nicholas P. Cheremisinoff, ''Environmental Technologies Handbook'' (2005), page 73-74. Generally, a worst-case scenario "is settled upon by agreeing that a given worst case is bad e ...
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Repeated Games
In game theory, a repeated game (or iterated game) is an extensive form game that consists of a number of repetitions of some base game (called a stage game). The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of their current action on the future actions of other players; this impact is sometimes called their reputation. Single stage game or single shot game are names for non-repeated games. Example Consider two gas stations that are adjacent to one another. They compete by publicly posting pricing, and have the same and constant marginal cost ''c'' (the wholesale price of gasoline). Assume that when they both charge , their joint profit is maximized, resulting in a high profit for everyone. Despite the fact that this is the best outcome for them, they are motivated to deviate. By modestly lowering the price, either can steal all of their competitors' customers, nearly doub ...
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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 ...
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A Modern Approach
A, or a, is the first letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is '' a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, '' a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest known ancestor of A is ''aleph''—the first letter of the Phoenician ...
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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 ...
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Gresham College
Gresham College is an institution of higher learning located at Barnard's Inn Hall off Holborn in Central London, England that does not accept students or award degrees. It was founded in 1597 under the Will (law), will of Sir Thomas Gresham, and hosts over 140 free public lectures every year. Since 2001, all lectures have been made available online. , the Acting Provost is Sarah B. Hart, Professor Sarah Hart. History First four centuries Sir Thomas Gresham, founder of the Royal Exchange (London), Royal Exchange, left his estate jointly to the City of London Corporation and the Mercers' Company, which today support the college through the Joint Grand Gresham Committee under the presidency of the Lord Mayor of London. Gresham's will provided for the setting up of the college – in Gresham's mansion in Bishopsgate, on the site now occupied by Tower 42, the former NatWest Tower – and endowed it with the rental income from shops sited around the Royal Exchange. The early succes ...
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