1.
Best response
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In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players strategies as given. Reaction correspondences, also known as best response correspondences, are used in the proof of the existence of mixed strategy Nash equilibria, one constructs a correspondence b, for each player from the set of opponent strategy profiles into the set of the players strategies. So, for any set of opponents strategies σ − i, b i represents player i s best responses to σ − i. Response correspondences for all 2x2 normal form games can be drawn with a line for each player in a unit square strategy space, figures 1 to 3 graphs the best response correspondences for the stag hunt game. The dotted line in Figure 1 shows the probability that player Y plays Stag. In Figure 2 the dotted line shows the probability that player X plays Stag. There are three distinctive reaction correspondence shapes, one for each of the three types of symmetric 2x2 games, coordination games, discoordination games and games with dominated strategies, any payoff symmetric 2x2 game will take one of these three forms. Games in which players score highest when both players choose the strategy, such as the stag hunt and battle of the sexes are called coordination games. Games such as the game of chicken and hawk-dove game in which players score highest when they choose opposite strategies, the third Nash equilibrium is a mixed strategy which lies along the diagonal from the bottom left to top right corners. If the players do not know one of them is which, then the mixed Nash is an evolutionarily stable strategy. Otherwise an uncorrelated asymmetry is said to exist, and the corner Nash equilibria are ESSes, Games with dominated strategies have reaction correspondences which only cross at one point, which will be in either the bottom left, or top right corner in payoff symmetric 2x2 games. For instance, in the prisoners dilemma, the Cooperate move is not optimal for any probability of opponent Cooperation. Figure 5 shows the correspondence for such a game, where the dimensions are Probability play Cooperate. A wider range of reaction correspondences shapes is possible in 2x2 games with payoff asymmetries, for each player there are five possible best response shapes, shown in Figure 6. From left to right these are, dominated strategy, dominated strategy, rising, falling, while there are only four possible types of payoff symmetric 2x2 games, the five different best response curves per player allow for a larger number of payoff asymmetric game types. Many of these are not truly different from each other, the dimensions may be redefined to produce symmetrical games which are logically identical. One well-known game with payoff asymmetries is the matching pennies game, player Ys reaction correspondence is that of a coordination game, while that of player X is a discoordination game. The only Nash equilibrium is the combination of mixed strategies where both players independently choose heads and tails with probability 0.5 each
2.
Cambridge University Press
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Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the worlds oldest publishing house and it also holds letters patent as the Queens Printer. The Presss mission is To further the Universitys mission by disseminating knowledge in the pursuit of education, learning, Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global presence, publishing hubs, and offices in more than 40 countries. Its publishing includes journals, monographs, reference works, textbooks. Cambridge University Press is an enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press and it originated from Letters Patent granted to the University of Cambridge by Henry VIII in 1534, and has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses, authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, and Stephen Hawking. In 1591, Thomass successor, John Legate, printed the first Cambridge Bible, the London Stationers objected strenuously, claiming that they had the monopoly on Bible printing. The universitys response was to point out the provision in its charter to print all manner of books. In July 1697 the Duke of Somerset made a loan of £200 to the university towards the house and presse and James Halman, Registrary of the University. It was in Bentleys time, in 1698, that a body of scholars was appointed to be responsible to the university for the Presss affairs. The Press Syndicates publishing committee still meets regularly, and its role still includes the review, John Baskerville became University Printer in the mid-eighteenth century. Baskervilles concern was the production of the finest possible books using his own type-design, a technological breakthrough was badly needed, and it came when Lord Stanhope perfected the making of stereotype plates. This involved making a mould of the surface of a page of type. The Press was the first to use this technique, and in 1805 produced the technically successful, under the stewardship of C. J. Clay, who was University Printer from 1854 to 1882, the Press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks, during Clays administration, the Press also undertook a sizable co-publishing venture with Oxford, the Revised Version of the Bible, which was begun in 1870 and completed in 1885. It was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories, the Cambridge Modern History was published between 1902 and 1912
3.
Extensive-form game
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Extensive-form games also allow representation of incomplete information in the form of chance events encoded as moves by nature. Whereas the rest of this article follows this approach with motivating examples. This general definition was introduced by Harold W. Kuhn in 1953, each players subset of nodes is referred to as the nodes of the player. Each node of the Chance player has a probability distribution over its outgoing edges, at any given non-terminal node belonging to Chance, an outgoing branch is chosen according to the probability distribution. A pure strategy for a player thus consists of a selection—choosing precisely one class of outgoing edges for every information set, in a game of perfect information, the information sets are singletons. Its less evident how payoffs should be interpreted in games with Chance nodes and these can be made precise using epistemic modal logic, see Shoham & Leyton-Brown for details. A perfect information two-player game over a tree can be represented as an extensive form game with outcomes. Examples of such games include tic-tac-toe, chess, and infinite chess, a game over an expectminimax tree, like that of backgammon, has no imperfect information but has moves of chance. For example, poker has both moves of chance, and imperfect information, the numbers by every non-terminal node indicate to which player that decision node belongs. The numbers by every terminal node represent the payoffs to the players, the labels by every edge of the graph are the name of the action that edge represents. The initial node belongs to player 1, indicating that player 1 moves first, play according to the tree is as follows, player 1 chooses between U and D, player 2 observes player 1s choice and then chooses between U and D. The payoffs are as specified in the tree, there are four outcomes represented by the four terminal nodes of the tree, and. The payoffs associated with each outcome respectively are as follows, if player 1 plays D, player 2 will play U to maximise his payoff and so player 1 will only receive 1. However, if player 1 plays U, player 2 maximises his payoff by playing D, player 1 prefers 2 to 1 and so will play U and player 2 will play D. This is the perfect equilibrium. An advantage of representing the game in this way is that it is clear what the order of play is, the tree shows clearly that player 1 moves first and player 2 observes this move. However, in some games play does not occur like this, one player does not always observe the choice of another. An information set is a set of decision nodes such that, in extensive form, an information set is indicated by a dotted line connecting all nodes in that set or sometimes by a loop drawn around all the nodes in that set
4.
Perfect information
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In economics, perfect information is a feature of perfect competition. Perfect information is importantly different from information, which implies common knowledge of each players utility functions, payoffs. Chess is an example of a game with perfect information as each player can see all of the pieces on the board at all times. Other examples of games include tic-tac-toe, Irensei, and Go. Card games where each players cards are hidden from other players, as in contract bridge, complete information Extensive form game Information asymmetry Partial knowledge Perfect competition Screening game Signaling game Fudenberg, D. and Tirole, J. Game Theory, MIT Press. A primer in theory, Harvester-Wheatsheaf
5.
MIT Press
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The MIT Press is a university press affiliated with the Massachusetts Institute of Technology in Cambridge, Massachusetts. Six years later, MITs publishing operations were first formally instituted by the creation of an imprint called Technology Press in 1932 and this imprint was founded by James R. Killian, Jr. at the time editor of MITs alumni magazine and later to become MIT president. Technology Press published eight titles independently, then in 1937 entered into an arrangement with John Wiley & Sons in which Wiley took over marketing, in 1962 the association with Wiley came to an end after a further 125 titles had been published. The press acquired its name after this separation, and has since functioned as an independent publishing house. A European marketing office was opened in 1969, and a Journals division was added in 1972, other areas, such as technology and design, have been added since. A recent addition is environmental science, in January 2010 the MIT Press published its 9000th title, and published about 200 books and 30 journals. In 2012 the Press celebrated its 50th anniversary, including publishing a booklet on paper. The MIT Press is a distributor for such publishers as Zone Books, in 2000, the MIT Press created CogNet, an online resource for the study of the brain and the cognitive sciences. In 1981 the MIT Press published its first book under the Bradford Books imprint, Brainstorms, Philosophical Essays on Mind and Psychology by Daniel C. The MIT Press also operates the MIT Press Bookstore showcasing both its front and backlist titles, along with a selection of complementary works from other academic. Once extensive construction around its location is completed, the Bookstore is planned to be returned to a site adjacent to the subway entrance. The Bookstore offers customized selections from the MIT Press at many conferences and symposia in the Boston area, the Press uses a colophon or logo designed by its longtime design director, Muriel Cooper, in 1962. It later served as an important reference point for the 2015 redesign of the MIT Media Lab logo by Pentagram, the Arts and Humanities Economics International Affairs, History, and Political Science Science and Technology Official Website MIT Press Journals Homepage The MIT PressLog
6.
Game theory
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Game theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory is used in economics, political science, and psychology, as well as logic, computer science. Originally, it addressed zero-sum games, in one persons gains result in losses for the other participants. Today, game theory applies to a range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals. Modern game theory began with the idea regarding the existence of equilibria in two-person zero-sum games. Von Neumanns original proof used Brouwer fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this provided an axiomatic theory of expected utility. This theory was developed extensively in the 1950s by many scholars, Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2014, John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Early discussions of examples of two-person games occurred long before the rise of modern, the first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, and uncle to James Waldegrave, a British diplomat, in 1713. In this letter, Waldegrave provides a mixed strategy solution to a two-person version of the card game le Her. James Madison made what we now recognize as an analysis of the ways states can be expected to behave under different systems of taxation. In 1913 Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels and it proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems, the Danish mathematician Zeuthen proved that the mathematical model had a winning strategy by using Brouwers fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false. Game theory did not really exist as a field until John von Neumann published a paper in 1928. Von Neumanns original proof used Brouwers fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern
7.
Bayes' theorem
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In probability theory and statistics, Bayes’ theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. One of the applications of Bayes’ theorem is Bayesian inference. When applied, the involved in Bayes’ theorem may have different probability interpretations. With the Bayesian probability interpretation the theorem expresses how a subjective degree of belief should rationally change to account for availability of related evidence, Bayesian inference is fundamental to Bayesian statistics. Bayes’ theorem is named after Rev. Thomas Bayes, who first provided an equation that allows new evidence to update beliefs. It was further developed by Pierre-Simon Laplace, who first published the modern formulation in his 1812 “Théorie analytique des probabilités. ”Sir Harold Jeffreys put Bayes’ algorithm and Laplaces formulation on an axiomatic basis. Jeffreys wrote that Bayes’ theorem “is to the theory of probability what the Pythagorean theorem is to geometry. ”Bayes theorem is stated mathematically as the equation, P = P P P. P and P are the probabilities of observing A and B without regard to each other, P, a conditional probability, is the probability of observing event A given that B is true. P is the probability of observing event B given that A is true, Bayes’ theorem was named after the Reverend Thomas Bayes, who studied how to compute a distribution for the probability parameter of a binomial distribution. Bayes’ unpublished manuscript was edited by Richard Price before it was posthumously read at the Royal Society. Price edited Bayes’ major work “An Essay towards solving a Problem in the Doctrine of Chances”, Price wrote an introduction to the paper which provides some of the philosophical basis of Bayesian statistics. In 1765 he was elected a Fellow of the Royal Society in recognition of his work on the legacy of Bayes, the French mathematician Pierre-Simon Laplace reproduced and extended Bayes’ results in 1774, apparently quite unaware of Bayes’ work. The Bayesian interpretation of probability was developed mainly by Laplace, stephen Stigler suggested in 1983 that Bayes’ theorem was discovered by Nicholas Saunderson, a blind English mathematician, some time before Bayes, that interpretation, however, has been disputed. Martyn Hooper and Sharon McGrayne have argued that Richard Prices contribution was substantial, By modern standards, Price discovered Bayes’ work, recognized its importance, corrected it, contributed to the article, and found a use for it. The modern convention of employing Bayes’ name alone is unfair but so entrenched that anything else makes little sense, suppose a drug test is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users, suppose that 0. 5% of people are users of the drug. If a randomly selected individual tests positive, what is the probability that he is a user and this surprising result arises because the number of non-users is very large compared to the number of users, thus the number of false positives outweighs the number of true positives. To use concrete numbers, if 1000 individuals are tested, there are expected to be 995 non-users and 5 users, from the 995 non-users,0.01 ×995 ≃10 false positives are expected
8.
Jean Tirole
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Jean Tirole is a French professor of economics. He focuses on industrial organization, game theory, banking and finance, in 2014 he was awarded the Nobel Memorial Prize in Economic Sciences for his analysis of market power and regulation. Tirole received engineering degrees from the École Polytechnique in Paris in 1976 and he graduated as a member of the elite Corps of Bridges, Waters and Forests. Tirole pursued graduate studies at the Paris Dauphine University and was awarded a DEA degree in 1976, in 1981, he received a Ph. D. in economics from the Massachusetts Institute of Technology for his thesis titled Essays in economic theory, under the supervision of Eric Maskin. After receiving his doctorate from MIT in 1981, he worked as a researcher at the École nationale des ponts et chaussées until 1984, from 1984–1991, he worked as Professor of Economics at MIT. From 1994 to 1996 he was a Professor of Economics at the École Polytechnique, Tirole was involved with Jean-Jacques Laffont in the project of creating a new School of Economics in Toulouse. He was president of the Econometric Society in 1998 and of the European Economic Association in 2001, Tirole has been a member of the Académie des Sciences morales et politiques since 2011, the Conseil danalyse économique since 2008 and the Conseil stratégique de la recherché since 2013. In the early 2010s, he showed that generally tend to take short-term risks. Tirole was awarded the Nobel Memorial Prize in Economic Sciences in 2014 for his analysis of market power and he is a foreign honorary member of the American Academy of Arts and Sciences and of the American Economic Association. He has also been a Sloan Fellow and a Guggenheim Fellow and he was a fellow of the Econometric Society in 1986 and an Economic Theory Fellow in 2011. In 2007 he was awarded the highest award of the French CNRS and he is among the most influential economists in the world according to IDEAS/RePEc. His research covers industrial organization, regulation, game theory, banking and finance, psychology and economics, international finance, the Theory of Industrial Organization, MIT Press,1988. Dynamic Models of Oligopoly (avec Drew Fudenberg, Harwood Academic Publishers GMbH,1986, the Theory of Industrial Organization, MIT Press,1988. A Theory of Incentives in Regulation and Procurement, MIT Press,1993, the Prudential Regulation of Banks, MIT Press,1994. Competition in Telecommunications, MIT Press,1999, financial Crises, Liquidity and the International Monetary System, Princeton University Press,2002. The Theory of Corporate Finance, Princeton University Press,2005, Association of American Publishers 2006 Award for Excellence. Balancing the Banks, Princeton University Press,2010, inside and Outside Liquidity, MIT Press,2011
9.
John Harsanyi
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John Charles Harsanyi was a Hungarian-American economist and Nobel Memorial Prize in Economic Sciences winner. He also made important contributions to the use of game theory, for his work, he was a co-recipient along with John Nash and Reinhard Selten of the 1994 Nobel Memorial Prize in Economics. Harsanyi was born on May 29,1920 in Budapest, Hungary, the son of Alice and Charles Harsanyi and his parents converted from Judaism to Catholicism a year before he was born. He attended high school at the Lutheran Gymnasium in Budapest, in high school, he became one of the best problem solvers of the KöMaL, the Mathematical and Physical Monthly for Secondary Schools. Founded in 1893, this periodical is generally credited with a share of Hungarian students success in mathematics. He also won the first prize in the Eötvös mathematics competition for school students. Although he wanted to study mathematics and philosophy, his father sent him to France in 1939 to enroll in engineering at the University of Lyon. However, because of the start of World War II, Harsanyi returned to Hungary to study pharmacology at the University of Budapest, as a pharmacology student, Harsanyi escaped conscription into the Hungarian Army which, as a person of Jewish descent, would have meant forced labor. However, in 1944 his military deferment was cancelled and he was compelled to join a labor unit on the Eastern Front. After the end of the war, Harsanyi returned to the University of Budapest for graduate studies in philosophy and sociology, then a devout Catholic, he simultaneously studied theology, also joining lay ranks of the Dominican Order. He later abandoned Catholicism, becoming an atheist for the rest of his life, Harsanyi spent the academic year 1947–1948 on the faculty of the Institute of Sociology of the University of Budapest, where he met Anne Klauber, his future wife. He was forced to resign the faculty because of openly expressing his anti-Marxist opinions, Harsanyi remained in Hungary for the following two years attempting to sell his familys pharmacy without losing it to the authorities. The two did not marry until they arrived in Australia because Klaubers immigration papers would need to be changed to reflect her married name, the two arrived with her parents on December 30,1950 and they looked to marry immediately. Harsanyi and Klauber were married on January 2,1951, neither spoke much English and understood little of what they were told to say to each other. Harsanyi later explained to his new wife that she had promised to better food than she usually did. Harsanyis Hungarian degrees were not recognized in Australia, but they earned him credit at the University of Sydney for a masters degree. Harsanyi worked in a factory during the day and studied economics in the evening at the University of Sydney, while studying in Sydney, he started publishing research papers in economic journals, including the Journal of Political Economy and the Review of Economic Studies. The degree allowed him to take a position in 1954 at the University of Queensland in Brisbane
10.
John Maynard Smith
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John Maynard Smith FRS was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a degree in genetics under the well-known biologist J. B. S. Haldane. Maynard Smith was instrumental in the application of theory to evolution and theorised on other problems such as the evolution of sex. John Maynard Smith was born in London, the son of the surgeon Sidney Maynard Smith, but following his fathers death in 1928, the moved to Exmoor. B. S. Haldane, whose books were in the schools library despite the bad reputation Haldane had at Eton for his communism and he became an atheist at age 14. On leaving school, Maynard Smith joined the Communist Party of Great Britain, when the Second World War broke out in 1939, he defied his partys line and volunteered for service. He was rejected, however, because of eyesight and was told to finish his engineering degree. He later quipped that under the circumstances, my poor eyesight was a selective advantage—it stopped me getting shot, the year of his graduation, he married Sheila Matthew, and they later had two sons and one daughter. Between 1942 and 1947, he applied his degree to military aircraft design, Maynard Smith, having decided that aircraft were “noisy and old-fashioned”, then took a change of career, entering University College London to study fruit fly genetics under Haldane. After graduating he became a lecturer in Zoology at UCL between 1952 and 1965, where he directed the Drosophila lab and conducted research on population genetics and he published a popular Penguin book, The Theory of Evolution, in 1958. In 1962 he was one of the members of the University of Sussex and was a Dean between 1965–85. He subsequently became a professor emeritus, prior to his death the building housing much of Life Sciences at Sussex was renamed the John Maynard Smith Building, in his honour. In 1973 Maynard Smith formalised a central concept in evolutionary theory called the evolutionarily stable strategy. This area of research culminated in his 1982 book Evolution and the Theory of Games, the Hawk-Dove game is arguably his single most influential game theoretical model. He was elected a Fellow of the Royal Society in 1977, in 1986 he was awarded the Darwin Medal. Maynard Smith published a book entitled The Evolution of Sex which explored in mathematical terms, during the late 1980s he also became interested in the other major evolutionary transitions with the evolutionary biologist Eörs Szathmáry. Together they wrote an influential 1995 book The Major Transitions in Evolution, a popular science version of the book, entitled The Origins of Life, From the birth of life to the origin of language was published in 1999. In 1995 he was awarded the Linnean Medal by The Linnean Society and in 1999 he was awarded the Crafoord Prize jointly with Ernst Mayr, in 2001 he was awarded the Kyoto Prize