Game theory
Game theory is the study of mathematical models of strategic interaction between rational decision-makers. It has applications in all fields of social science, as well as in computer science, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, is now an umbrella term for the science of logical decision making in humans and computers. Modern game theory began with the idea regarding the existence 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 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 book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.
Game theory was developed extensively in the 1950s by many scholars. It was 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; as of 2014, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. 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, mathematical game theory; the first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, uncle to James Waldegrave, a British diplomat, in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her, the problem is now known as Waldegrave problem. In his 1838 Recherches sur les principes mathématiques de la théorie des richesses, Antoine Augustin Cournot considered a duopoly and presents a solution, a restricted version of the Nash equilibrium.
In 1913, Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. It proved that the optimal chess strategy is determined; this paved the way for more general theorems. In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric. Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture, proved false. Game theory did not exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics, his paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern.
The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book; this foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During the following time period, work on game theory was focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies. In 1950, the first mathematical discussion of the prisoner's dilemma appeared, an experiment was undertaken by notable mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy. Around this same time, John Nash developed a criterion for mutual consistency of players' strategies, known as Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern.
Nash proved that every n-player, non-zero-sum non-cooperative game has what is now known as a Nash equilibrium. Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, the Shapley value were developed. In addition, the first applications of game theory to philosophy and political science occurred during this time. In 1979 Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was a simple "tit-for-tat" program that cooperates on the first step on subsequent steps just does whatever its opponent did on the previous step; the same winner was often obtained by natural selection. In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium. In 1994 Nash and Harsanyi became Economics Nobel Laureates for their contributi
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world, it holds letters patent as the Queen's Printer. The press mission is "to further the University's mission by disseminating knowledge in the pursuit of education and research at the highest international levels of excellence". Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global sales presence, publishing hubs, offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries, its publishing includes academic journals, reference works and English language teaching and learning publications. Cambridge University Press is a charitable 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.
It originated from letters patent granted to the University of Cambridge by Henry VIII in 1534, 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, Stephen Hawking. University printing began in Cambridge when the first practising University Printer, Thomas Thomas, set up a printing house on the site of what became the Senate House lawn – a few yards from where the press's bookshop now stands. In those days, the Stationers' Company in London jealously guarded its monopoly of printing, which explains the delay between the date of the university's letters patent and the printing of the first book. In 1591, Thomas's successor, John Legate, printed the first Cambridge Bible, an octavo edition of the popular Geneva Bible; the London Stationers objected strenuously. The university's response was to point out the provision in its charter to print "all manner of books".
Thus began the press's tradition of publishing the Bible, a tradition that has endured for over four centuries, beginning with the Geneva Bible, continuing with the Authorized Version, the Revised Version, the New English Bible and the Revised English Bible. The restrictions and compromises forced upon Cambridge by the dispute with the London Stationers did not come to an end until the scholar Richard Bentley was given the power to set up a'new-style press' in 1696. In July 1697 the Duke of Somerset made a loan of £200 to the university "towards the printing house and presse" and James Halman, Registrary of the University, lent £100 for the same purpose, it was in Bentley's time, in 1698, that a body of senior scholars was appointed to be responsible to the university for the press's affairs. The Press Syndicate's publishing committee still meets and its role still includes the review and approval of the press's planned output. John Baskerville became University Printer in the mid-eighteenth century.
Baskerville's concern was the production of the finest possible books using his own type-design and printing techniques. Baskerville wrote, "The importance of the work demands all my attention. Caxton would have found nothing to surprise him if he had walked into the press's printing house in the eighteenth century: all the type was still being set by hand. A technological breakthrough was badly needed, it came when Lord Stanhope perfected the making of stereotype plates; this involved making a mould of the whole surface of a page of type and casting plates from that mould. The press was the first to use this technique, in 1805 produced the technically successful and much-reprinted Cambridge Stereotype Bible. By the 1850s the press was using steam-powered machine presses, employing two to three hundred people, occupying several buildings in the Silver Street and Mill Lane area, including the one that the press still occupies, the Pitt Building, built for the press and in honour of William Pitt the Younger.
Under the stewardship of C. J. Clay, 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 Clay's administration, the press undertook a sizeable co-publishing venture with Oxford: the Revised Version of the Bible, begun in 1870 and completed in 1885, it was in this period as well that the Syndics of the press turned down what became the Oxford English Dictionary—a proposal for, brought to Cambridge by James Murray before he turned to Oxford. The appointment of R. T. Wright as Secretary of the Press Syndicate in 1892 marked the beginning of the press's development as a modern publishing business with a defined editorial policy and administrative structure, 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
John von Neumann
John von Neumann was a Hungarian-American mathematician, computer scientist, polymath. Von Neumann was regarded as the foremost mathematician of his time and said to be "the last representative of the great mathematicians", he made major contributions to a number of fields, including mathematics, economics and statistics. He was a pioneer of the application of operator theory to quantum mechanics in the development of functional analysis, a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer, he published over 150 papers in his life: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, the remainder on special mathematical subjects or non-mathematical ones. His last work, an unfinished manuscript written while in hospital, was published in book form as The Computer and the Brain, his analysis of the structure of self-replication preceded the discovery of the structure of DNA. In a short list of facts about his life he submitted to the National Academy of Sciences, he stated, "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, subsequently in Berlin in 1927–1929.
My work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939. During World War II, von Neumann worked on the Manhattan Project with theoretical physicist Edward Teller, mathematician Stanisław Ulam and others, problem solving key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb, he developed the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon, coined the term "kiloton", as a measure of the explosive force generated. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, consulted for a number of organizations, including the United States Air Force, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project, the Lawrence Livermore National Laboratory; as a Hungarian émigré, concerned that the Soviets would achieve nuclear superiority, he designed and promoted the policy of mutually assured destruction to limit the arms race.
Von Neumann was born Neumann János Lajos to a wealthy and non-observant Jewish family. After his arrival in the U. S. he had been baptized a Roman Catholic prior to the marriage to his Catholic first wife. Von Neumann was born in Budapest, Kingdom of Hungary, part of the Austro-Hungarian Empire, he was the eldest of three brothers. His father, Neumann Miksa was a banker, he had moved to Budapest from Pécs at the end of the 1880s. Miksa's father and grandfather were both born in Zemplén County, northern Hungary. John's mother was Kann Margit. Three generations of the Kann family lived in spacious apartments above the Kann-Heller offices in Budapest. On February 20, 1913, Emperor Franz Joseph elevated his father to the Hungarian nobility for his service to the Austro-Hungarian Empire; the Neumann family thus acquired the hereditary appellation Margittai. The family had no connection with the town. Neumann János became margittai Neumann János, which he changed to the German Johann von Neumann. Von Neumann was a child prodigy.
When he was 6 years old, he could divide two 8-digit numbers in his head and could converse in Ancient Greek. When the 6-year-old von Neumann caught his mother staring aimlessly, he asked her, "What are you calculating?"Children did not begin formal schooling in Hungary until they were ten years of age. Max believed that knowledge of languages in addition to Hungarian was essential, so the children were tutored in English, French and Italian. By the age of 8, von Neumann was familiar with differential and integral calculus, but he was interested in history, he read his way through Wilhelm Oncken's 46-volume Allgemeine Geschichte in Einzeldarstellungen. A copy was contained in a private library. One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor. Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1911. Eugene Wigner soon became his friend; this was one of the best schools in Budapest and was part of a brilliant education system designed for the elite.
Under the Hungarian system, children received all their education at the one gymnasium. The Hungarian school system produced a generation noted for intellectual achie
Simultaneous game
In game theory, a simultaneous game is a game where each player chooses his action without knowledge of the actions chosen by other players. Simultaneous games contrast with sequential games, which are played by the players taking turns. Normal form representations are used for simultaneous games. Rock-paper-scissors, a played hand game, is an example of a simultaneous game. Both players make a decision without knowledge of the opponent's decision, reveal their hands at the same time. There are two players in this game and each of them has three different strategies to make their decision. We will display Player 1's strategies as Player 2's strategies as columns. In the table, the numbers in red represent the payoff to Player 1, the numbers in blue represent the payoff to Player 2. Hence, the pay off for a 2 player game in rock-paper-scissors will look like this: The prisoner's dilemma is an example of a simultaneous game; some variants of chess that belong to this class of games include Synchronous chess and Parity chess.
Sequential game Simultaneous action selection Bibliography Pritchard, D. B.. Beasley, John, ed; the Classified Encyclopedia of Chess Variants. John Beasley. ISBN 978-0-9555168-0-1
Preference (economics)
In economics and other social sciences, preference is the ordering of alternatives based on their relative utility, a process which results in an optimal "choice". The character of the individual preferences is determined purely by taste factors, independent of considerations of prices, income, or availability of goods. With the help of the scientific method many practical decisions of life can be modelled, resulting in testable predictions about human behavior. Although economists are not interested in the underlying causes of the preferences in themselves, they are interested in the theory of choice because it serves as a background for empirical demand analysis. In 1926 Ragnar Frisch developed for the first time a mathematical model of preferences in the context of economic demand and utility functions. Up to economists had developed an elaborated theory of demand that omitted primitive characteristics of people; this omission ceased when, at the end of the 19th and the beginning of the 20th century, logical positivism predicated the need of theoretical concepts to be related with observables.
Whereas economists in the 18th and 19th centuries felt comfortable theorizing about utility, with the advent of logical positivism in the 20th century, they felt that it needed more of an empirical structure. Because binary choices are directly observable, it appealed to economists; the search for observables in microeconomics is taken further by revealed preference theory. Since the pioneer efforts of Frisch in the 1920s, one of the major issues which has pervaded the theory of preferences is the representability of a preference structure with a real-valued function; this has been achieved by mapping it to the mathematical index called utility. Von Neumann and Morgenstern 1944 book "Games and Economic Behaviour" treated preferences as a formal relation whose properties can be stated axiomatically; these type of axiomatic handling of preferences soon began to influence other economists: Marschak adopted it by 1950, Houthakker employed it in a 1950 paper, Kenneth Arrow perfected it in his 1951 book "Social Choice and Individual Values".
Gérard Debreu, influenced by the ideas of the Bourbaki group, championed the axiomatization of consumer theory in the 1950s, the tools he borrowed from the mathematical field of binary relations have become mainstream since then. Though the economics of choice can be examined either at the level of utility functions or at the level of preferences, to move from one to the other can be useful. For example, shifting the conceptual basis from an abstract preference relation to an abstract utility scale results in a new mathematical framework, allowing new kinds of conditions on the structure of preference to be formulated and investigated. Another historical turnpoint can be traced back to 1895, when Georg Cantor proved in a theorem that if a binary relation is linearly ordered it is isomorphically embeddable in the ordered real numbers; this notion would become influential for the theory of preferences in economics: by the 1940s prominent authors such as Paul Samuelson, would theorize about people having weakly ordered preferences.
Suppose the set of all states of the world is X and an agent has a preference relation on X. It is common to mark the weak preference relation by ⪯, so that x ⪯ y means "the agent wants y at least as much as x" or "the agent weakly prefers y to x"; the symbol ∼ is used as a shorthand to the indifference relation: x ∼ y ⟺, which reads "the agent is indifferent between y and x". The symbol ≺ is used as a shorthand to the strong preference relation: x ≺ y ⟺, which reads "the agent prefers y to x". In everyday speech, the statement "x is preferred to y" is understood to mean that someone chooses x over y. However, decision theory rests on more precise definitions of preferences given that there are many experimental conditions influencing people's choices in many directions. Suppose a person is confronted with a mental experiment that she must solve with the aid of introspection, she is offered apples and oranges, is asked to verbally choose one of the two. A decision scientist observing this single event would be inclined to say that whichever is chosen is the preferred alternative.
Under several repetitions of this experiment, if the scientist observes that apples are chosen 51% of the time it would mean that x ≻ y. If half of the time oranges are chosen x ∼ y. If 51% of the time she chooses oranges it means that y ≻ x. Preference is here being identified with a greater frequency of choice; this experiment implicitly assumes. Otherwise, out of 100 repetitions, some of them will give as a result that neither apples, oranges or ties are chosen; these few cases of uncertainty will ruin any preference information resulting from the frequency attributes of the other valid cases. However, this example was used
Perfect information
In economics, perfect information is a feature of perfect competition. With perfect information in a market, all consumers and producers have perfect and instantaneous knowledge of all market prices, their own utility, own cost functions. In game theory, a sequential game has perfect information if each player, when making any decision, is informed of all the events that have occurred, including the "initialization event" of the game. Chess is an example of a game with perfect information as each player can see all the pieces on the board at all times. Other examples of games with perfect information include tic-tac-toe, infinite chess, Go. Card games where each player's cards are hidden from other players such as poker and bridge are examples of games with imperfect information. Academic literature has not produced consensus on a standard definition of perfect information which defines whether games with chance, but no secret information, games with simultaneous moves are games of perfect information.
Games which are sequential and which have chance events but no secret information, are sometimes considered games of perfect information. This includes games such as Monopoly, but there are some academic papers which do not regard such games as games of perfect information because the results of chance themselves are unknown prior to them occurring. Games with simultaneous moves are not considered games of perfect information; this is because each of the players holds information, secret, must play a move without knowing the opponent's secret information. Some such games are symmetrical, fair. An example of a game in this category includes rock–paper–scissors. Complete information Extensive form game Information asymmetry Partial knowledge Perfect competition Screening game Signaling game Fudenberg, D. and Tirole, J. Game Theory, MIT Press. Gibbons, R. A primer in game theory, Harvester-Wheatsheaf. Luce, R. D. and Raiffa, H. Games and Decisions: Introduction and Critical Survey, Wiley & Sons The Economics of Groundhog Day by economist D.
W. MacKenzie, using the 1993 film Groundhog Day to argue that perfect information, therefore perfect competition, is impossible
Jean Tirole
Jean Tirole is a French professor of economics. He focuses on industrial organization, game theory and finance, economics and psychology. 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, from the École nationale des ponts et chaussées in 1978, he graduated as a member of the elite Corps of Bridges and Forests. Tirole pursued graduate studies at the Paris Dauphine University and was awarded a DEA degree in 1976 and a Doctorat de troisième cycle in decision mathematics in 1978. 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. Tirole is chairman of the board of the Jean-Jacques Laffont Foundation at the Toulouse School of Economics, scientific director of the Industrial Economics Institute at Toulouse 1 University Capitole.
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, his work by 1988 helped to define modern industrial organization theory by organising and synthesising the main results of the game-theory revolution vis-à-vis understanding of non-competitive markets. 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 is Engineer General of the Corps of Bridges and Forest, serving as Chair of the Board of the Toulouse School of Economics, Visiting Professor at MIT and Professor "cumulant" at the École des hautes études en sciences sociales since 1995. He was president of the Econometric Society in 1998 and of the European Economic Association in 2001. Around this time, he was able to determine a way to calculate the optimal prices for the regulation of natural monopolies and wrote a number of articles about the regulation of capital markets—with a focus on the differential of control between decentralised lenders and the centralised control of bank management.
Tirole has been a member of the Académie des Sciences morales et politiques since 2011, the Conseil d'analyse économique since 2008 and the Conseil stratégique de la recherché since 2013. In the early 2010s, he showed that banks tend to take short-term risks and recommended a change in quantitative easing towards a more quality-based market stimulation policy. Tirole authored the famous textbook The Theory of Industrial Organization, which synthesised modern models of oligopolistic competition, analysing various cases where industries consist of a small number of firms, with significant market power, he and Oliver Hart published a paper showing the conditions in which a vertical merger can result in foreclosure. Rochet and Tirole analysed the implications of 2-sided markets for competition policy. Fundenberg and Tirole created a taxonomy of strategic effects in oligopolistic competition models. Tirole was awarded the Nobel Memorial Prize in Economic Sciences in 2014 for his analysis of market power and the regulation of natural monopolies.
Tirole received doctorates honoris causa from the Université libre de Bruxelles in 1989, the London Business School and the University of Montreal in 2007, the University of Mannheim in 2011, the Athens University of Economics and Business and the University of Rome Tor Vergata in 2012 as well as the University of Lausanne in 2013. Tirole received the inaugural BBVA Foundation Frontiers of Knowledge Award in the Economics and Management category in 2008, the Public Utility Research Center Distinguished Service Award in 1997, the Yrjö Jahnsson Award of the Yrjö Jahnsson Foundation and the European Economic Association in 1993, he is a foreign honorary member of the American Academy of Arts and Sciences and of the American Economic Association. He has been a Sloan Fellow and a Guggenheim Fellow, he was a fellow of the Econometric Society in 1986 and an Economic Theory Fellow in 2011. In 2013 Tirole was elected an Honorary Fellow of the Royal Society of Edinburgh. In 2007 he was awarded the highest award of the French CNRS.
In 2008, he received the Prix du Cercle d'Oc. He is among the most influential economists in the world according to IDEAS/RePEc. Besides his numerous academic distinctions, he was the recipient of the Gold Medal of the city of Toulouse in 2007, a Chevalier de la Légion d'honneur since 2007 and an Officer in the Ordre national du Mérite since 2010. Tirole has published about 200 professional articles in economics and finance, as well as 10 books, including The Theory of Industrial Organization, Game Theory, A Theory of Incentives in Procurement and Regulation, The Prudential Regulation of Banks, Competition in Telecommunications, Financial Crises and the International Monetary System, The Theory of Corporate Finance, his research covers industrial organization, game theory, public economics and finance, psychology and economics, international finance and macroeconomics. Dy