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Goofspiel
Goofspiel (also known as The Game of Pure Strategy, GOPS or Psychological Jujitsu) is a card game for two or more players. It was invented by Merrill Flood while at Princeton University in the 1930s, and Alex Randolph describes a similar game as having been popular with the 5th Indian Army during the Second World War. The game is simple to learn and play, but has some degree of strategic depth. It is commonly used as an example of multi-stage simultaneous move game in game theory and artificial intelligence. Game play Goofspiel is played using cards from a standard deck of cards, and is typically a two-player game, although more players are possible."GOPS" iThe Very Best Two Player Card Games ''PlayingCardDecks'', 5 October 2021. Retrieved 13 October 2021. Each suit is ranked A (low), 2, ..., 10, J, Q, K (high). One suit is singled out as the "prizes"; each of the remaining suits becomes a hand for one player, with one suit discarded if there are only two players, or taken ...
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Card Game
A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specific. Countless card games exist, including families of related games (such as poker). A small number of card games played with traditional decks have formally standardized rules with international tournaments being held, but most are folk games whose rules vary by region, culture, and person. Traditional card games are played with a ''deck'' or ''pack'' of playing cards which are identical in size and shape. Each card has two sides, the ''face'' and the ''back''. Normally the backs of the cards are indistinguishable. The faces of the cards may all be unique, or there can be duplicates. The composition of a deck is known to each player. In some cases several decks are shuffled together to form a single ''pack'' or ''shoe''. Modern card games usually have bespoke decks, often with a vast amount of cards, and can include number or action cards. This ...
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Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ...
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Card Games Introduced In The 1930s
Card or The Card may refer to: * Various types of plastic cards: **By type ***Magnetic stripe card ***Chip card ***Digital card **By function ***Payment card ****Credit card ****Debit card ****EC-card ****Identity card ****European Health Insurance Card ****Driver's license * Playing card, a card used in games * Printed circuit board * Punched card, a piece of stiff paper that holds digital data represented by the presence or absence of holes in predefined positions. *In communications ** Postcard ** Greeting card, an illustrated piece of card stock featuring an expression of friendship or other sentiment * \operatorname, in mathematical notation, a function that returns the cardinality of a set * Card, a tool for carding, the cleaning and aligning of fibers * Sports terms ** Card (sports), the lineup of the matches in an event ** Penalty card As a proper name People with the name * Card (surname) Companies * Cards Corp, a South Korean internet company Arts and entertainment * " ...
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Pagat
The trull is a trio of three special trump cards used in tarock games in Austria and other countries that have a much higher card value than the other trumps. The individual cards are known as trull cards (''Trullstücke''). The word ''trull'' is derived from the French ''tous les trois'' which means "all three". In spite of its French roots the term is not common in the game of French tarot, where the trull cards are called ''les bouts'' ("butts", "ends") or, in earlier times, ''les oudlers'', which has no other meaning. Introduction The games of the tarot (French) or tarock (German) family are distinguished mainly in that, in addition to the suit cards, their decks have a series of 21 classical, permanent trumps, most of which are numbered with Roman or Arabic numerals. In games of German-language origin the trumps are also called ''tarocks''. The special role of the 'fool' (''Narren'') is described below. Tarock games are trick-taking card games, in which the cards have ...
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Dynamic Programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have ''optimal substructure''. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.Cormen, T. H.; Leiserson, C. E.; Rives ...
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Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ...
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Mixed Strategies
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 ...
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Nash Equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs. 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 their's 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 ...
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Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ...
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Merrill M
Merrill may refer to: Places in the United States *Merrill Field, Anchorage, Alaska *Merrill, Iowa * Merrill, Maine *Merrill, Michigan * Merrill, Mississippi, an unincorporated community near Lucedale in George County *Merrill, Oregon *Merrill, Wisconsin * Merrill (town), Wisconsin *Merrill Township, Michigan * Merrill Township, North Dakota *Merrill College at the University of California, Santa Cruz People * Merrill Moses (born 1977), Olympic water polo player *Merrill (surname) *Merrill Cook, Utah politician *Merrill Garbus, musician behind the experimental indie project Tune-yards *Merrill Ashley (born 1950), American ballet dancer and ''répétiteur'' Other uses *Merrill (company), a division of Bank of America *Skidmore, Owings and Merrill, architectural firm * USS ''Merrill'' (DD-976) *Nine men's morris, a strategy board game also called ''Merrills'' * Merrill (crater) Merrill is a lunar impact crater. It is located in the high northern latitudes, on the far side. Less th ...
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First-price Sealed-bid Auction
A first-price sealed-bid auction (FPSBA) is a common type of auction. It is also known as blind auction. In this type of auction, all bidders simultaneously submit sealed bids so that no bidder knows the bid of any other participant. The highest bidder pays the price that was submitted. Strategic analysis In a FPSBA, each bidder is characterized by their monetary valuation of the item for sale. Suppose Alice is a bidder and her valuation is a. Then, if Alice is rational: *She will never bid more than a, because bidding more than a can only make her lose net value. *If she bids exactly a, then she will not lose but also not gain any positive value. *If she bids less than a, then she ''may'' have some positive gain, but the exact gain depends on the bids of the others. Alice would like to bid the smallest amount that can make her win the item, as long as this amount is less than a. For example, if there is another bidder Bob and he bids y and y, then Alice would like to ...
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Suit (cards)
In playing cards, a suit is one of the categories into which the cards of a deck are divided. Most often, each card bears one of several pips (symbols) showing to which suit it belongs; the suit may alternatively or additionally be indicated by the color printed on the card. The rank for each card is determined by the number of pips on it, except on face cards. Ranking indicates which cards within a suit are better, higher or more valuable than others, whereas there is no order between the suits unless defined in the rules of a specific card game. In a single deck, there is exactly one card of any given rank in any given suit. A deck may include special cards that belong to no suit, often called jokers. History Modern Western playing cards are generally divided into two or three general suit-systems. The older Latin suits are subdivided into the Italian and Spanish suit-systems. The younger Germanic suits are subdivided into the German and Swiss suit-systems. The French suits a ...
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