minimax theorem

The Man From the Future: The Visionary Life of John Von Neumann

by Ananyo Bhattacharya  · 6 Oct 2021  · 476pp  · 121,460 words

a ‘science of contest’ would be taken towards the end of the following year. On 7 December 1926, von Neumann unveiled his proof of the minimax theorem to mathematicians at Göttingen. Published in 1928, the paper expounding his proof, On the Theory of Parlour Games,10 would firmly establish game theory as

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. If he had, the effect would have been ‘primarily discouraging’.17 ‘Throughout the period in question I thought there was nothing worth publishing until the “minimax theorem” was proved,’ he added. ‘As far as I can see, there could be no theory of games on these bases without that theorem. By surmising

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laid down the axioms of game theory and given formal definitions of terms such as ‘game’ and ‘strategy’. He also offered a proof of his minimax theorem, which was much more elementary than the original of 1928. In 1938, Jean Ville, a student of Émile Borel, had published a simple algebraic proof

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sum games set out in Theory of Games entails finding tricks to reduce them to the zero-sum two-person case. Since von Neumann’s minimax theorem guarantees that every such game has a rational solution, he thought that analysing these ‘fake’ two-player games would reveal the best strategy for each

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Morgenstern hoped. The two-person zero-sum game, perhaps the most elegant and immediately applicable part of the book, was rooted in von Neumann’s minimax theorem, first developed nearly twenty years earlier. Non-zero-sum games with an arbitrary number of players were a work in progress. However, von Neumann’s

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analog of the one we have developed for games.’22 Von Neumann had instantly recognized that Dantzig’s optimization problem was mathematically related to his minimax theorem for two-person zero-sum games. The insight helped determine the conditions under which logistical problems of the type Dantzig was interested in could or

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world. 14. On von Neumann’s 1928 proof and the priority dispute with Borel see Tinne Hoff Kjeldsen, ‘John von Neumann’s Conception of the Minimax Theorem: A Journey Through Different Mathematical Contexts’, Archive for History of Exact Sciences, 56 (2001), pp. 39–6. 15. Maurice Fréchet, ‘Emile Borel, Initiator of the

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-Dove game 180 impact 175–82, 188–92, 203, 212–6, 218–20, 220–1, 223–4 loose ends 176 Matching Pennies 147, 164, 164 minimax theorem proof 143–8, 169, 176–7, 192 mixed strategies 147–148 multi-player (n-person) games 169–752, 176 Nash equilibria 201–3 Nash’s

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, 137, 139 Michigan, University of 105, 181, 258 Logic of Computers Group 243–4, 258, 265 Milgrom, Paul 177–8 military worth 188, 189, 191 minimax theorem proof 143–8, 154, 169, 176–7, 192 Minority Report (film) 231 Minsky, Marvin 274, 276 Minta/Model gimnázium 7–8, 9 Mises, Ludwig von

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problem 43–5, 57–8, 296n43 Medal of Freedom 277, 277 Medal for Merit 83 meets Klára 74–5 and Morgenstern 154, 155, 156–9 minimax theorem proof 143–8, 169, 176–7, 192 misanthropy 203 Monte Carlo bomb simulations 133–8 and Nash 199–203 nervous energy 77 on nuclear deterrence

Theory of Games and Economic Behavior: 60th Anniversary Commemorative Edition (Princeton Classic Editions)

by John von Neumann and Oskar Morgenstern  · 19 Mar 2007

. This was to become known as the “KMT model,” which also established firmly the relationship of the model to game theory. In both, the fundamental minimax theorem is of the essence; KMT showed why this had to be the case and—unexpectedly—that game theory can also be used as a mathematical

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chance can influence the direction of scientific work. At the time when we were about to write down a new proof for Johnny’s famous minimax theorem, originally developed in 1928, I went out for a walk on a brilliant, snowy cold winter day. I went towards the Institute for Advanced Study

Prisoner's Dilemma: John Von Neumann, Game Theory, and the Puzzle of the Bomb

by William Poundstone  · 2 Jan 1993  · 323pp  · 100,772 words

Behavior Cake Division Rational Players Games as Trees Games as Tables Zero-Sum Games Minimax and Cake Mixed Strategies Curve Balls and Deadly Genes The Minimax Theorem N-Person Games 4 THE BOMB Von Neumann at Los Alamos Game Theory in Wartime Bertrand Russell World Government Operation Crossroads The Computer Preventive War

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a rational course of action for games of two players, provided their interests are completely opposed. This proof is called the “minimax theorem.” The class of games covered by the minimax theorem includes many recreational games, ranging from such trivial contests as ticktacktoe to the sophistications of chess. It applies to any two-person

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exactly, an “optimal” way to play such games. Were this all there was to the minimax theorem, it would qualify as a clever contribution to recreational mathematics. Von Neumann saw more profound implications. He intended the minimax theorem to be the cornerstone of a game theory that would eventually encompass other types of games

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without doubt von Neumann’s 1928 article, “Zur Theorie der Gesellschaftspiele” (“Theory of Parlor Games”). In this he proved (as Borel had not) the famous “minimax theorem.” This important result immediately gave the field mathematical respectability. THEORY OF GAMES AND ECONOMIC BEHAVIOR Von Neumann wanted game theory to reach a larger audience

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as possible or as genes mindlessly reproducing as much as natural selection permits. We’ll hear more about biological interpretations of game theory later. THE MINIMAX THEOREM 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. Von Neumann

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’s position as founder of game theory rests mainly with his proof of the minimax theorem by 1926. Von Neumann considered the theorem crucial. In 1953 he wrote, “As far as I can see, there could be no theory of games

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on these bases without that theorem.... Throughout the period in question I thought there was nothing worth publishing until the ‘minimax theorem’ was proved.” 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

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, von Neumann answered no. Such questions lingered. What is game theory good for? If not to play games, then what? Von Neumann himself saw the minimax theorem as the first cornerstone of an exact science of economics. Toward this end, much of von Neumann and Morgenstern’s book treats games with three

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coalitions can be stable. Then it is difficult or impossible to predict what will happen. Von Neumann hoped to use the minimax theorem to tackle games of ever more players. The minimax theorem gives a rational solution to any two-person zero-sum game. A three-person game can be dissected into sub-games

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player C, then the resulting game (coalition of A and B vs. C) is effectively a two-person game with a solution guaranteed by the minimax theorem. By figuring out the results of all the potential coalitions, the players A, B, and C would be able to decide which coalitions were most

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player’s gain is another’s loss, there is no point in forming a coalition. That case, however, was already covered by von Neumann’s minimax theorem. Nash’s work was primarily concerned with non-zero-sum games and games of three or more players. With the

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minimax theorem, von Neumann struck a great blow for rationality. He demonstrated that any two rational beings who find their interests completely opposed can settle on a

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solution.” Nash proved that every two-person finite game has at least one equilibrium point. This result is an important extension of von Neumann’s minimax theorem. The minimax solutions of zero-sum games qualify as equilibrium points, but Nash’s proof says that non-zero-sum games have equilibrium points, too

The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling

by Adam Kucharski  · 23 Feb 2016  · 360pp  · 85,321 words

(2003): 395–415. 147Von Neumann completed his solution: Details of the dispute were given in: Kjedldsen, T. H. “John von Neumann’s Conception of the Minimax Theorem: A Journey Through Different Mathematical Contexts.” Archive for History of Exact Science 56 (2001). 149While earning his master’s degree in 2003: Follek, Robert. “Soar

The Pentagon's Brain: An Uncensored History of DARPA, America's Top-Secret Military Research Agency

by Annie Jacobsen  · 14 Sep 2015  · 558pp  · 164,627 words

“Theory of Parlor Games.” The paper, which examined game playing from a mathematical point of view, contained a soon-to-be famous proof, called the minimax theorem. Von Neumann wrote that when two players are involved in a zero-sum game—a game in which one player’s losses equal the other

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. It placed game theory on the world stage, and after the war it caught the attention of the Pentagon. By the 1950s, von Neumann’s minimax theorem was legendary at RAND, and to engage von Neumann in a discussion about game theory was like drinking from the Holy Grail. It became a

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amount of time, the police say. Von Neumann could not “solve” the Prisoner’s Dilemma. It is an unsolvable paradox. It does not fit the minimax theorem. There is no answer; the outcome of the dilemma game differs from player to player. Dresher and Flood posed the Prisoner’s Dilemma to dozens

Turing's Cathedral

by George Dyson  · 6 Mar 2012

Europe were scarce.35 He published twenty-five papers in the next three years, including a 1928 paper on the theory of games (with its minimax theorem proving the existence of good strategies, for a wide class of competitions, at the saddle point between convex sets) as well as the book Mathematical

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, 8.1, 14.1, 15.1, 17.1 Miller, Bernetta (1884–1972), 6.1, 6.2, 7.1, 8.1, 8.2 Mineville, New York minimax theorem (von Neumann) Minitotal (Alfvén) Minsky, Marvin “Model of General Economic Equilibrium” (von Neumann, 1932), 15.1, 15.2 Molotov-Ribbentrop Pact Monadology (Leibniz, 1714) Monte

Darwin Among the Machines

by George Dyson  · 28 Mar 2012  · 463pp  · 118,936 words

power of generalization,” wrote Jacob Marschak in the Journal of Political Economy in 1946.5 Von Neumann’s central insight was his proof of the “minimax” theorem on the existence of good strategies, demonstrating for a wide class of games that a determinable strategy exists that minimizes the expected loss to a

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, 222–24, 225 origins and evolution, 81–82, 172, 211, 219, 222–25 robustness of, 176 unpredictability of, 9, 35, 45, 72–73, 109–110 minimax theorem, defined, 154 Minsky, Marvin, 7, 72, 111 MTT (Massachusetts Institute of Technology), 80, 98, 144, 179, 180 Mivart, St. George Jackson (1827–1900), 18 “Model