Algorithm
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, other tasks; as an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, the Euclidean algorithm for finding the greatest common divisor of two numbers; the word algorithm itself is derived from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized Algoritmi.
A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem posed by David Hilbert in 1928. Formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, Alan Turing's Turing machines of 1936–37 and 1939. The word'algorithm' has its roots in Latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī was a Persian mathematician, astronomer and scholar in the House of Wisdom in Baghdad, whose name means'the native of Khwarazm', a region, part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, translated into Latin during the 12th century under the title Algoritmi de numero Indorum; this title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Al-Khwarizmi was the most read mathematician in Europe in the late Middle Ages through another of his books, the Algebra. In late medieval Latin, English'algorism', the corruption of his name meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός'number', the Latin word was altered to algorithmus, the corresponding English term'algorithm' is first attested in the 17th century. In English, it was first used in about 1230 and by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu, it begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five; the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that defines a sequence of operations". Which would include all computer programs, including programs that do not perform numeric calculations. A program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers. Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† to list all members of an enumerably infinite set by writing out their names, one after another, in some notation, but humans can do something useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human, capable of carrying out only elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large, thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of: Precise instructions for a fast, efficient, "good" process that specifies the "moves" of "the computer" to find and process arbitrary input integers/symbols m and n, symbols + and =... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format
John Forbes Nash Jr.
John Forbes Nash Jr. was an American mathematician who made fundamental contributions to game theory, differential geometry, the study of partial differential equations. Nash's work has provided insight into the factors that govern chance and decision-making inside complex systems found in everyday life, his theories are used in economics. Serving as a Senior Research Mathematician at Princeton University during the part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi. In 2015, he shared the Abel Prize with Louis Nirenberg for his work on nonlinear partial differential equations. John Nash is the only person to be awarded both the Nobel Memorial Prize in Economic Sciences and the Abel Prize. In 1959, Nash began showing clear signs of mental illness, spent several years at psychiatric hospitals being treated for paranoid schizophrenia. After 1970, his condition improved, allowing him to return to academic work by the mid-1980s.
His struggles with his illness and his recovery became the basis for Sylvia Nasar's biography, A Beautiful Mind, as well as a film of the same name starring Russell Crowe as Nash. On May 23, 2015, Nash and his wife Alicia were killed in a car crash while riding in a taxi on the New Jersey Turnpike, he is survived by John Charles Martin Nash and John Stier. Nash was born on June 1928, in Bluefield, West Virginia, his father, John Forbes Nash, was an electrical engineer for the Appalachian Electric Power Company. His mother, Margaret Virginia Nash, had been a schoolteacher, he was baptized in the Episcopal Church. He had Martha. Nash attended kindergarten and public school, he learned from books provided by his parents and grandparents. Nash's parents pursued opportunities to supplement their son's education, arranged for him to take advanced mathematics courses at a local community college during his final year of high school, he attended Carnegie Institute of Technology through a full benefit of the George Westinghouse Scholarship majoring in chemical engineering.
He switched to a chemistry major and at the advice of his teacher John Lighton Synge, to mathematics. After graduating in 1948 with both a B. S. and M. S. in mathematics, Nash accepted a scholarship to Princeton University, where he pursued further graduate studies in mathematics. Nash's adviser and former Carnegie professor Richard Duffin wrote a letter of recommendation for Nash's entrance to Princeton stating, "He is a mathematical genius." Nash was accepted at Harvard University. However, the chairman of the mathematics department at Princeton, Solomon Lefschetz, offered him the John S. Kennedy fellowship, convincing Nash that Princeton valued him more. Further, he considered Princeton more favorably because of its proximity to his family in Bluefield. At Princeton, he began work on his equilibrium theory known as the Nash equilibrium. Nash earned a Ph. D. degree in 1950 with a 28-page dissertation on non-cooperative games. The thesis, written under the supervision of doctoral advisor Albert W. Tucker, contained the definition and properties of the Nash equilibrium, a crucial concept in non-cooperative games.
It won Nash the Nobel Memorial Prize in Economic Sciences in 1994. Publications authored by Nash relating to the concept are in the following papers: Nash, John Forbes. "Equilibrium Points in N-person Games". Proceedings of the National Academy of Sciences of the United States of America. 36: 48–49. Doi:10.1073/pnas.36.1.48. MR 0031701. PMC 1063129. PMID 16588946. Nash, John Forbes. "The Bargaining Problem". Econometrica. Basel, Switzerland: MDPI. 18: 155–62. Doi:10.2307/1907266. JSTOR 1907266. MR 0035977. Nash, John Forbes. "Non-cooperative Games". Annals of Mathematics. Princeton, New Jersey: Princeton University. 54: 286–95. Doi:10.2307/1969529. JSTOR 1969529. MR 0043432. Nash, John Forbes. "Two-person Cooperative Games". Econometrica. Basel, Switzerland: MDPI. 21: 128–40. Doi:10.2307/1906951. MR 0053471. Archived from the original on March 29, 2017. Retrieved January 4, 2017. Nash did groundbreaking work in the area of real algebraic geometry: John Forbes. "Real algebraic manifolds". Annals of Mathematics. 56: 405–21.
Doi:10.2307/1969649. JSTOR 1969649. MR 0050928. See "Proc. Internat. Congr. Math". AMS. 1952: 516–17. His work in mathematics includes the Nash embedding theorem, which shows that every abstract Riemannian manifold can be isometrically realized as a submanifold of Euclidean space, he made significant contributions to the theory of nonlinear parabolic partial differential equations and to singularity theory. Mikhail Leonidovich Gromov writes about Nash's work: Nash was solving classical mathematical problems, difficult problems, something that nobody else was able to do, not to imagine how to do it.... But what Nash discovered in the course of his constructions of isometric embeddings is far from'classical' — it is something that brings about a dramatic alteration of our understanding of the basic logic of analysis and differential geometry. Judging from the classical perspective, what Nash has achieved in his papers is as impossible as the story of his life... is work on isometric immersions... opened a new world of mathematics that stretches in front of our eyes in yet unknown directions and still waits to be explored.
John Milnor gives a list of 21 publications. In the Nash biography A Beautiful Mind, author Sylvia Nasar explains that Nash was working on proving Hilbert's nineteenth problem, a theorem involving elliptic partial differential equ
Robot
A robot is a machine—especially one programmable by a computer— capable of carrying out a complex series of actions automatically. Robots can be guided by an external control device or the control may be embedded within. Robots may be constructed on the lines of human form, but most robots are machines designed to perform a task with no regard to how they look. Robots can be autonomous or semi-autonomous and range from humanoids such as Honda's Advanced Step in Innovative Mobility and TOSY's TOSY Ping Pong Playing Robot to industrial robots, medical operating robots, patient assist robots, dog therapy robots, collectively programmed swarm robots, UAV drones such as General Atomics MQ-1 Predator, microscopic nano robots. By mimicking a lifelike appearance or automating movements, a robot may convey a sense of intelligence or thought of its own. Autonomous things are expected to proliferate in the coming decade, with home robotics and the autonomous car as some of the main drivers; the branch of technology that deals with the design, construction and application of robots, as well as computer systems for their control, sensory feedback, information processing is robotics.
These technologies deal with automated machines that can take the place of humans in dangerous environments or manufacturing processes, or resemble humans in appearance, behavior, or cognition. Many of today's robots are inspired by nature contributing to the field of bio-inspired robotics; these robots have created a newer branch of robotics: soft robotics. From the time of ancient civilization there have been many accounts of user-configurable automated devices and automata resembling animals and humans, designed as entertainment; as mechanical techniques developed through the Industrial age, there appeared more practical applications such as automated machines, remote-control and wireless remote-control. The term comes from a Czech word, meaning "forced labor". U. R. by the Czech writer, Karel Čapek but it was Karel's brother Josef Čapek, the word's true inventor. Electronics evolved into the driving force of development with the advent of the first electronic autonomous robots created by William Grey Walter in Bristol, England in 1948, as well as Computer Numerical Control machine tools in the late 1940s by John T. Parsons and Frank L. Stulen.
The first commercial and programmable robot was built by George Devol in 1954 and was named the Unimate. It was sold to General Motors in 1961 where it was used to lift pieces of hot metal from die casting machines at the Inland Fisher Guide Plant in the West Trenton section of Ewing Township, New Jersey. Robots have replaced humans in performing repetitive and dangerous tasks which humans prefer not to do, or are unable to do because of size limitations, or which take place in extreme environments such as outer space or the bottom of the sea. There are concerns about the increasing use of their role in society. Robots are blamed for rising technological unemployment as they replace workers in increasing numbers of functions; the use of robots in military combat raises ethical concerns. The possibilities of robot autonomy and potential repercussions have been addressed in fiction and may be a realistic concern in the future; the word robot can refer to both physical robots and virtual software agents, but the latter are referred to as bots.
There is no consensus on which machines qualify as robots but there is general agreement among experts, the public, that robots tend to possess some or all of the following abilities and functions: accept electronic programming, process data or physical perceptions electronically, operate autonomously to some degree, move around, operate physical parts of itself or physical processes and manipulate their environment, exhibit intelligent behavior behavior which mimics humans or other animals. Related to the concept of a robot is the field of Synthetic Biology, which studies entities whose nature is more comparable to beings than to machines; the idea of automata originates in the mythologies of many cultures around the world. Engineers and inventors from ancient civilizations, including Ancient China, Ancient Greece, Ptolemaic Egypt, attempted to build self-operating machines, some resembling animals and humans. Early descriptions of automata include the artificial doves of Archytas, the artificial birds of Mozi and Lu Ban, a "speaking" automaton by Hero of Alexandria, a washstand automaton by Philo of Byzantium, a human automaton described in the Lie Zi.
Many ancient mythologies, most modern religions include artificial people, such as the mechanical servants built by the Greek god Hephaestus, the clay golems of Jewish legend and clay giants of Norse legend, Galatea, the mythical statue of Pygmalion that came to life. Since circa 400 BC, myths of Crete include Talos, a man of bronze who guarded the island from pirates. In ancient Greece, the Greek engineer Ctesibius "applied a knowledge of pneumatics and hydraulics to produce the first organ and water clocks with moving figures." In the 4th century BC, the Greek mathematician Archytas of Tarentum postulated a mechanical steam-operated bird he called "The Pigeon". Hero of Alexandria, a Greek mathematician and inventor, created numerous user-configurable automated devices, described machines powered by air pressure and water; the 11th century Lokapannatti tells of how the Buddha's relics were protected by mechanical robots, from the kingdom of Roma visaya. In ancient China, the
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
Extensive-form game
An extensive-form game is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, their payoffs for all possible game outcomes. Extensive-form games allow for the representation of incomplete information in the form of chance events modeled as "moves by nature"; some authors in introductory textbooks define the extensive-form game as being just a game tree with payoffs, add the other elements in subsequent chapters as refinements. Whereas the rest of this article follows this gentle approach with motivating examples, we present upfront the finite extensive-form games as constructed here; this general definition was introduced by Harold W. Kuhn in 1953, who extended an earlier definition of von Neumann from 1928. Following the presentation from Hart, an n-player extensive-form game thus consists of the following: A finite set of n players A rooted tree, called the game tree Each terminal node of the game tree has an n-tuple of payoffs, meaning there is one payoff for each player at the end of every possible play A partition of the non-terminal nodes of the game tree in n+1 subsets, one for each player, with a special subset for a fictitious player called Chance.
Each player's 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; each set of nodes of a rational player is further partitioned in information sets, which make certain choices indistinguishable for the player when making a move, in the sense that: there is a one-to-one correspondence between outgoing edges of any two nodes of the same information set—thus the set of all outgoing edges of an information set is partitioned in equivalence classes, each class representing a possible choice for a player's move at some point—, every path in the tree from the root to a terminal node can cross each information set at most once the complete description of the game specified by the above parameters is common knowledge among the playersA play is thus a path through the tree from the root to a terminal node. At any given non-terminal node belonging to Chance, an outgoing branch is chosen according to the probability distribution.
At any rational player's node, the player must choose one of the equivalence classes for the edges, which determines one outgoing edge except the player doesn't know which one is being followed. A pure strategy for a player thus consists of a selection—choosing one class of outgoing edges for every information set. In a game of perfect information, the information sets are singletons. It's less evident, it is assumed that each player has a von Neumann–Morgenstern utility function defined for every game outcome. The above presentation, while defining the mathematical structure over which the game is played, elides however the more technical discussion of formalizing statements about how the game is played like "a player cannot distinguish between nodes in the same information set when making a decision"; these can be made precise using epistemic modal logic. A perfect information two-player game over a game tree can be represented as an extensive form game with outcomes. Examples of such games include tic-tac-toe 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 imperfect information. A complete extensive-form representation specifies: the players of a game for every player every opportunity they have to move what each player can do at each of their moves what each player knows for every move the payoffs received by every player for every possible combination of moves The game on the right has two players: 1 and 2; the numbers by every non-terminal node indicate. 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. The initial node belongs to player 1. Play according to the tree is as follows: player 1 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 are as follows, and. If player 1 plays D, player 2 will play U' to maximise their payoff and so player 1 will only receive 1.
However, if player 1 plays U, player 2 maximises their payoff by playing D' and player 1 receives 2. Player 1 prefers 2 to 1 and s
Rock–paper–scissors
Rock–paper–scissors is a hand game played between two people, in which each player forms one of three shapes with an outstretched hand. These shapes are "rock", "paper", "scissors". "Scissors" is identical to the two-fingered V sign except that it is pointed horizontally instead of being held upright in the air. A simultaneous, zero-sum game, it has only two possible outcomes: a draw, or a win for one player and a loss for the other. A player who decides to play rock will beat another player who has chosen scissors, but will lose to one who has played paper. If both players choose the same shape, the game is tied and is immediately replayed to break the tie; the type of game originated in China and spread with increased contact with East Asia, while developing different variants in signs over time. Other names for the game in the English-speaking world include roshambo and other orderings of the three items, with "rock" sometimes being called "stone". Rock–paper–scissors is used as a fair choosing method between two people, similar to coin flipping, drawing straws, or throwing dice in order to settle a dispute or make an unbiased group decision.
Unlike random selection methods, rock–paper–scissors can be played with a degree of skill by recognizing and exploiting non-random behavior in opponents. The players count aloud to three, or speak the name of the game, each time either raising one hand in a fist and swinging it down on the count or holding it behind, they "throw" by extending it towards their opponent. Variations include a version where players use only three counts before throwing their gesture, or a version where they shake their hands three times before "throwing"; the first known mention of the game was in the book Wuzazu by the Chinese Ming-dynasty writer Xie Zhaozhi, who wrote that the game dated back to the time of the Chinese Han dynasty. In the book, the game was called shoushiling. Li Rihua's book Note of Liuyanzhai mentions this game, calling it shoushiling, huozhitou, or huoquan. Throughout Japanese history there are frequent references to sansukumi-ken, meaning ken games where "the three who are afraid of one another".
This type of game originated in China before being imported to Japan and subsequently becoming popular among the Japanese. The earliest Japanese sansukumi-ken game was known as mushi-ken, imported directly from China. In mushi-ken the "frog" is superseded by the "slug", which, in turn is superseded by the "snake", superseded by the "frog". Although this game was imported from China the Japanese version differs in the animals represented. In adopting the game, the original Chinese characters for the poisonous centipede were confused with the characters for the slug; the most popular sansukumi-ken game in Japan was kitsune-ken. In the game, a supernatural fox called a kitsune defeats the village head, the village head defeats the hunter, the hunter defeats the fox. Kitsune-ken, unlike mushi-ken or rock–paper–scissors, is played by making gestures with both hands. Today, the best-known sansukumi-ken is called jan-ken, a variation of the Chinese games introduced in the 17th century. Jan-ken uses the rock and scissors signs and is the game that the modern version of rock–paper–scissors derives from directly.
Hand-games using gestures to represent the three conflicting elements of rock and scissors have been most common since the modern version of the game was created in the late 19th century, between the Edo and Meiji periods. By the early 20th century, rock–paper–scissors had spread beyond Asia through increased Japanese contact with the west, its English-language name is therefore taken from a translation of the names of the three Japanese hand-gestures for rock and scissors: elsewhere in Asia the open-palm gesture represents "cloth" rather than "paper". The shape of the scissors is adopted from the Japanese style. In Britain in 1924 it was described in a letter to The Times as a hand game of Mediterranean origin, called "zhot". A reader wrote in to say that the game "zhot" referred to was evidently Jan-ken-pon, which she had seen played throughout Japan. Although at this date the game appears to have been new enough to British readers to need explaining, the appearance by 1927 of a popular thriller with the title Scissors Cut Paper, followed by Stone Blunts Scissors, suggests it became popular.
In 1927 La Vie au patronage, a children's magazine in France, described it in detail, referring to it as a "jeu japonais". Its French name, "Chi-fou-mi", is based on the Old Japanese words for "one, three". A 1932 New York Times article on the Tokyo rush hour describes the rules of the game for the benefit of American readers, suggesting it was not at that time known in the U. S; the 1933 edition of the Compton's Pictured Encyclopedia described it as a common method of settling disputes between children in its article on Japan.
Ariel Rubinstein
Ariel Rubinstein is an Israeli economist who works in Economic Theory, Game Theory and Bounded Rationality. Ariel Rubinstein is a professor of economics at the School of Economics at Tel Aviv University and the Department of Economics at New York University, he studied mathematics and economics at the Hebrew University of Jerusalem, 1972–1979. In 1982, he published "Perfect equilibrium in a bargaining model", an important contribution to the theory of bargaining; the model is known as a Rubinstein bargaining model. It describes two-person bargaining as an extensive game with perfect information in which the players alternate offers. A key assumption is; the main result gives conditions under which the game has a unique subgame perfect equilibrium and characterizes this equilibrium. He co-wrote A Course in Game Theory with Martin J. Osborne, a textbook, cited in excess of 7,790 times as of Jan 2018. Rubinstein was elected a member of the Israel Academy of Sciences and Humanities, a Foreign Honorary Member of the American Academy of Arts and Sciences in and the American Economic Association.
In 1985 he was elected a fellow of the Econometric Society, served as its president in 2004. In 2002, he was awarded an honorary doctorate by the Tilburg University, he has received the Bruno Prize, the Israel Prize for economics, the Nemmers Prize in Economics, the EMET Prize. and the Rothschild Prize. Bargaining and Markets, with Martin J. Osborne, Academic Press 1990 A Course in Game Theory, with Martin J. Osborne, MIT Press, 1994. Modeling Bounded Rationality, MIT Press, 1998. Economics and Language, Cambridge University Press, 2000. Lecture Notes in Microeconomic Theory: The Economic Agent, Princeton University Press, 2006. Economic Fables, Open Book Publishers, 2012. AGADOT HAKALKALA, Zmora, Bitan, 2009. List of Israel Prize recipients Personal Web site Nash lecture Russ. "Rubinstein on Game Theory and Behavioral Economics". EconTalk. Library of Economics and Liberty