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# Topic: Minimax theorem

###### In the News (Wed 21 Aug 19)

 Minimax - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-05) Minimax (sometimes minmax) is a method in decision theory for minimizing the maximum possible loss. The performance of the naïve minimax algorithm may be improved dramatically, without affecting the result, by the use of alpha-beta pruning. John von Neumann proved the Minimax theorem in 1928, stating that such strategies always exist in two-person zero-sum games and can be found by solving a set of simultaneous equations. en.wikipedia.org /wiki/Minimax   (1213 words)

 Minimax   (Site not responding. Last check: 2007-11-05) Minimax is a method in decision theory for minimizing the expected maximum loss. The performance of the naive minimax algorithm may be improved dramatically, without affecting the result, by the use of alpha-beta pruning. For example in the prisoner's dilemma, the minimax strategy for each prisioner is to confess even though they would each do better if both denied their guilt. www.wapipedia.com /wikipedia/mobiletopic.aspx?cur_title=Minimax_algorithm   (911 words)

 Folk theorem (game theory) - Wikipedia, the free encyclopedia In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players' minimax conditions are satisfied. In mathematics, the term folk theorem refers generally to a theorem which is believed and discussed, but has not been published. A commonly referenced proof of a folk theorem was published in 1979 by Ariel Rubinstein. en.wikipedia.org /wiki/Folk_theorem_(game_theory)   (522 words)

 Minimax Article, Minimax Information   (Site not responding. Last check: 2007-11-05) Minimax is a method in decision theory forminimizing the expected maximum loss. John von Neumann proved the Minimax theorem in 1928, stating that such strategies always exist in two-person zero-sum games and can befound by solving a set of simultaneous equations. Minimax theory has been extended to decisions where there is no other player, but where the consequences of decisions dependof unknown facts. www.anoca.org /moves/algorithm/minimax.html   (973 words)

 Von_Neumann The Minimax Theorem says that the way to recapture rationality is to use strategies that randomize a player's choice over all possible actions. Von Neumann's Minimax Theorem makes the strong assertion that there always exists at least one mixed strategy for each player, such that the average payoff to each player is the same when he uses these strategies. Minimax Theorem: For every two-person, zero-sum game there exists a mixed strategy for each player such that the expected payoff for both is the same value V when the players use these strategies. hypatia.math.uri.edu /~kulenm/mth381pr/GAMETH/gametheory.html   (2866 words)

 Strategies of Play The minimax theorem was proven by John von Neumann in 1928. Minimax is a strategy of always minimizing the maximum possible loss which can result from a choice that a player makes. The Minimax Regret Principle is based on the Minimax Theorem advanced by John von Neumann, but is geared only towards one-person games. cse.stanford.edu /class/sophomore-college/projects-98/game-theory/Minimax.html   (1470 words)

 THE MINIMAX STRATEGY “The theorem states that in zero-sum games in which the players’ interests are strictly opposed (one’s gain is the other other’s loss), one player should attempt to minimize his opponent’s maximum payoff while his opponent attempts to maximize his own minimum payoff. “The general proof of the mini-max theorem is quite complicated, but the result is useful and worth remembering. “To put it in plain language, the minimax theorem says that there is always a rational solution to a precisely defined conflict between two people whose interests are completely opposite. www.charleswarner.us /MinimaxStrategy.htm   (900 words)

 Minimax Theorems and Their Proofs, by Stephen Simons   (Site not responding. Last check: 2007-11-05) The original motivation for the study of minimax theorems was, of course, Von Neumann's 1928 work on games of strategy. We discuss various minimax theorems in which X and Y are not assumed to be subsets of vector space. The recent unifying metaminimax theorems, theorems which imply simultaneously the minimax theorems of all the above three types and which depend on abstract generalizations of connectedness tend to show that the classification above is perhaps too rigid. at.yorku.ca /z/a/a/a/28.htm   (232 words)

 Minimax & Applications   (Site not responding. Last check: 2007-11-05) he minimax (or maximin) criterion is a widely used criterion for parameter optimization and more generally as a solution criterion for problems in many areas of pure and applied mathematics. They are also used to extend and unify the First Frobenius Theorem for positive matrices and Jentzsch’s Theorem for positive integral operators, while also providing a new short proof of the foundation existence result for Jentzsch’s Theorem. Finally, they can be used to provide a sharp lower bound of the second derivative of a function of one variable on a finite interval, and to prove the existence of a weight matrix in robust regression. www.irishscientist.ie /GITMF117.htm   (317 words)

 History of Game Theory Waldegrave's solution is a minimax mixed strategy equilibrium, but he made no extension of his result to other games, and expressed concern that a mixed strategy "does not seem to be in the usual rules of play" of games of chance. This theorem was published by E. Zermelo in his paper Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels and hence is referred to as Zermelo's Theorem. In their paper A Limit Theorem on the Core of an Economy G. Debreu and H. Scarf generalised Edgeworth, in the context of a NTU game, by allowing an arbitrary number of commodities and an arbitrary but finite number of types of traders. free.prohosting.com /cepr/data/001gamehis.html   (5828 words)

 Von Neumann's Minimax Theorem The minimax theorem states that one can assign every two person finite zero-sum game a value V that is the average payoff each player can expect to win from the other while playing sensibly. With it and the mixed strategies, we can treat all two person zero-sum games as games with equilibrium points, which provides the assurance that in a given state of things, it is impossible to do better as long as the opponent is as good as you are. It is important to know however that minimax only predict security, which, like a complete description of all chess moves, is hard to come by. library.thinkquest.org /26408/math/minimax.shtml   (526 words)

 Extended minimax theorems and such a result is called a minimax theorem. The proofs are based on the extended Riesz representation theorem and the separation theorems. Theorems 1 and 2 generalize several previously known results on minimax, in particular, van Noumann theorem and equilibrium theorems in game theory (mixed strategies in matrix games). www.math.uu.nl /hpopt/abstracts/protassov   (227 words)

 A Versatile Theorem The applet illustrates a theorem that can be used to establish the Minimax Theorem for two-person zero-sum games. The theorem is just a particular case of Pappus' theorem: if (there) AbaB, then, since Ab·aB, Bc·bC, and Ca·cA are collinear, the line through Bc·bC and Ca·cA is parallel to both Ab and aB. This is exactly the statement of the theorem at hand, where D and E replace B and C, while B, C, F replace a, b, and c. www.cut-the-knot.org /Curriculum/Geometry/MMParallel.shtml   (235 words)

 Minimax Theorem The maximin solution for the allied forces is to search north, and the minimax solution for the Japanese forces is to sail north. In the proof of the minimax theorem, you will encounter theorem 1 which states that if an equilibrium exists than the minimax value equals the maximin value. When we are done with the theorem, we will have developed a rigorous proof based on the Brower Fixed Point Theorem. students.cs.byu.edu /~cs670ta/Lectures/Minimax2.html   (1784 words)

 Minimax Theory And Applications - Bokus bokhandel   (Site not responding. Last check: 2007-11-05) These proceedings deal with classical minimax theory and look at the role of connectedness, which replaces that of convexity appearing in most classical results. Intersection Theorems, Minimax Theorems and Abstract Connectedness; J. Kindler. On a Topological Minimax Theorem and its Applications; B. Ricceri. www.bokus.com /b/0792350642.html   (262 words)

 The Utility Theory The whole existence of the minimax theorem is based on the assumption that opponents will play to maximize winnings at all times. It is a simple fact of life that players will not always act by the best strategy, and has little to do with the content of the theory. Needless to say, where wants are involved, context and experience matter, but in many cases a generalized baseline can be calculated for a large group, which allows us to use minimax on solid footing on the utility values, which produce zero sums in games involving "wants" regardless of the players final goals. library.thinkquest.org /26408/math/utility.shtml   (371 words)

 On Classes of Generalized Convex Functions, Farkas-Type (SMEALSearch) -   (Site not responding. Last check: 2007-11-05) Moreover, for some of those function classes a Farkas-type theorem is proved. As such this paper unifies and extends results existing in the literature and shows how these results can be used to verify Farkas-type theorems and strong Lagrangian duality results in finite dimensional optimization. 1 A Generalization of a Minimax Theorem of Fan Via a Theorem o.. smealsearch2.psu.edu /2925.html   (411 words)

 Backgammon Theory A _mixed strategy_ is a general class of strategy including those where a player makes decisions probabilistically (an example of why this may be necessary is the game "paper, scissors, rock" where the best strategy is to pick a move at random -- always picking "paper" is not the best strategy). The practical upshot is that if you believe the Minimax Theorem (and you can read a proof elsewhere :-) then yes, there are optimal strategies for backgammon. Yes -- as it is written in the Gospel according to Saint John :-) (John von Neumann proved the Minimax Theorem in 1928). www.bkgm.com /rgb/rgb.cgi?view+551   (936 words)

 On Zero-Sum Two Person Games [Jaky Joseph] These games can be solved using the minimax theorem. The minimax theorem was proven by John von Neumann in 1928 in his article "Zur Theorie der Gesellschaftsspiele" using topology and functional calculus. This theorem has proven to be extremely useful in analyzing and finding the Nash equilibrium outcome of two-person zero sum games and has acted as spring board in analyzing more complex games with contant-sum payoffs. www.swarthmore.edu /NatSci/math_stat/webspot/Joseph,Jaky/Games/index.html   (180 words)

 [No title]   (Site not responding. Last check: 2007-11-05) We introduce a generalized form of the Hahn--Banach theorem, which we will use to prove various classical results on the existence of linear functionals, and also to prove a minimax theorem. We will also explain why one cannot generalize the minimax theorem too much, in the sense that any reasonable attempt to generalize the minimax theorem to a pair of noncompact sets is probably doomed to failure. We will, however, mention an unreasonable generalization that is true, hard and related to R. James's "sup theorem''. oldweb.cecm.sfu.ca /events/WCOM01/simons.html   (85 words)

 A new minimax theorem and a perturbed James's theorem (ResearchIndex)   (Site not responding. Last check: 2007-11-05) A new minimax theorem and a perturbed James's theorem (ResearchIndex) A new minimax theorem and a perturbed James's theorem The main functional--analytic result is Theorem 14, which contains a su#cient condition for the minimax relation to hold for the canonical bilinear form on X Y, where X is a nonempty convex subset of a real locally convex space, E, and Y is a nonempty convex subset of its dual, E #. citeseer.ist.psu.edu /621583.html   (337 words)

 GameTheory (via CobWeb/3.1 planetlab2.cs.virginia.edu)   (Site not responding. Last check: 2007-11-05) The Minimax Theorem discovered by von Neumann in 1928 asserts that every finite, zero-sum, two-player game has a minimax value if mixed strategies are allowed. The Minimax Theorem does't apply to nonzero-sum games or games with more than two players. John Nash showed in 1950 that such games do have a weaker solution, a noncooperative equilibrium in which no player, acting on the assumption that the other players' strategies are fixed, can gain anything by changing his or her own strategy. eluzions.com.cob-web.org:8888 /Games/Theory   (1032 words)

 TheExperiment | Articles => John von Neumann and Game Theory: a Brief Study of   (Site not responding. Last check: 2007-11-05) In 1928 the Hungarian-American mathematician John von Neumann, published an article entitled "Zur Theorie der Gesellschaftspiele" ("Theory of Parlour Games")proving the "minimax theorem" which established the mathematical framework for all subsequent theoretical developments. This theorem says that there is always a rational solution to a precisely defined conflict between two people whose interests are completely opposite. In this way, greed ensures fair division because the first child cannot object because she divided the cake herself, and the second child was given the choice of pieces. www.theexperiment.org /articles.php?news_id=792   (1656 words)

 vu02-w04   (Site not responding. Last check: 2007-11-05) A player's effective minimax value crucially depends on the asynchronous move structure in the repeated game, but not on the player's minimax or effective minimax value in the stage game. We establish a folk theorem: when players are sufficiently patient, any feasible payoff vector where every player receives more than his effective minimax value can be approximated by a perfect equilibrium in the repeated game with asynchronous moves. This folk theorem integrates Fudenberg and Maskin's (1986) folk theorem for standard repeated games, Lagunoff and Matsui's (1997) anti-folk theorem for repeated pure coordination game with asynchronous moves, and Wen's (2002) folk theorem for repeated sequential games. www.vanderbilt.edu /Econ/wparchive/abstracts/vu02-w04.html   (170 words)

 CSC 2411H, Spring 2005: Home Page This is possibly the most fundamental theorem in the theory of optimization, and provides an extremely efficient tool to many optimization problems. We will discuss few of the applications of the theorem, Von Neumann minmax principle in game-theory, Yao's minmax principle and the "primal-dual" algorithmic paradigm. Week 7: von Neumann minimax theorem, Yao's minimax Principle, Ellipsoid Algorithm Lecture notes by Xuming He. www.cs.toronto.edu /~avner/teaching/2411   (518 words)

 David Blackwell page2, his research The Blackwell Renewal Theorem (see [1] and [2]) really defines renewal theory, and introduced the Lusin spaces now known as Blackwell spaces.Three of his favorite papers of his own work are: Blackwell, David; Freedman, David The tail $\sigma$-field of a Markov chain and a theorem of Orey. Blackwell, David An analog of the minimax theorem for vector payoffs. www.math.buffalo.edu /mad/PEEPS/blackwell_david2.html   (1069 words)

 PS 444 Formal Theory From these assumptions, Von Neumann, in conjunction with Oskar Morganstern, was able to develop the minimax theorem as a proof. According to Poundstone, "The minimax theorem proves that every finite, two-person, zero-sum game has a rational solution in the form of a pure or mixed strategy." (Poundstone p.61) Essentially, this theorem has several important components. However, minimax does not hold true if one or more of the players are not rational. www.missouri.edu /~polsjwe/ps444ja.html   (1305 words)

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