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Shapley Value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject. The setup is as follows: a coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of generated surplus among the players should arise in any particular game? Or phrased differently: how important is each player to the overall cooperation, and what payoff can he or she reasonably expect? The Shapley va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lloyd Shapley 2 2012
Lloyd, Lloyd's, or Lloyds may refer to: People * Lloyd (name), a variation of the Welsh word ' or ', which means "grey" or "brown" ** List of people with given name Lloyd ** List of people with surname Lloyd * Lloyd (singer) (born 1986), American singer Places United States * Lloyd, Florida * Lloyd, Kentucky * Lloyd, Montana * Lloyd, New York * Lloyd, Ohio * Lloyds, Alabama * Lloyds, Maryland * Lloyds, Virginia Elsewhere * Lloydminster, or "Lloyd", straddling the provincial border between Alberta and Saskatchewan, Canada Companies and businesses Derived from Lloyd's Coffee House *Lloyd's Coffee House, a London meeting place for merchants and shipowners between about 1688 and 1774 * Lloyd's of London, a British insurance market ** ''Lloyd's of London'' (film), a 1936 film about the insurance market ** Lloyd's building, its headquarters ** Lloyd's Agency Network * ''Lloyd's List'', a website and 275-year-old daily newspaper on shipping and global trade ** ''Lloyd's List ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Aumann
Robert John Aumann (Hebrew name: , Yisrael Aumann; born June 8, 1930) is an Israeli-American mathematician, and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel. He also holds a visiting position at Stony Brook University, and is one of the founding members of the Stony Brook Center for Game Theory. Aumann received the Nobel Memorial Prize in Economic Sciences in 2005 for his work on conflict and cooperation through game theory analysis. He shared the prize with Thomas Schelling. Early years Aumann was born in Frankfurt am Main, Germany, and fled to the United States with his family in 1938, two weeks before the Kristallnacht pogrom. He attended the Rabbi Jacob Joseph School, a yeshiva high school in New York City. Academic career Aumann graduated from the City College of New York in 1950 with a B.Sc. in mathematics. He received his M.Sc. in 1952, and his Ph. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cooperative Games
Cooperative game may refer to: * Cooperative board game, board games in which players work together to achieve a common goal * Cooperative game theory In game theory, a cooperative game (or coalitional game) is a game with competition between groups of players ("coalitions") due to the possibility of external enforcement of cooperative behavior (e.g. through contract law). Those are opposed to ..., in game theory, a game with competition between groups of players and the possibility of cooperative behavior * Cooperative video game, a video game that allows players to work together as teammates {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game 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 mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shapley–Shubik Power Index
The Shapley–Shubik power index was formulated by Lloyd Shapley and Martin Shubik in 1954 to measure the powers of players in a voting game. The index often reveals surprising power distribution that is not obvious on the surface. The constituents of a voting system, such as legislative bodies, executives, shareholders, individual legislators, and so forth, can be viewed as players in an ''n''-player game. Players with the same preferences form coalitions. Any coalition that has enough votes to pass a bill or elect a candidate is called winning, and the others are called losing. Based on Shapley value, Shapley and Shubik concluded that the power of a coalition was not simply proportional to its size. The power of a coalition (or a player) is measured by the fraction of the possible voting sequences in which that coalition casts the deciding vote, that is, the vote that first guarantees passage or failure. The power index is normalized between 0 and 1. A power of 0 means that a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banzhaf Power Index
The Banzhaf power index, named after John F. Banzhaf III (originally invented by Lionel Penrose in 1946 and sometimes called Penrose–Banzhaf index; also known as the Banzhaf–Coleman index after James Samuel Coleman), is a power index defined by the probability of changing an outcome of a vote where voting rights are not necessarily equally divided among the voters or shareholders. To calculate the power of a voter using the Banzhaf index, list all the winning coalitions, then count the critical voters. A ''critical voter'' is a voter who, if he changed his vote from yes to no, would cause the measure to fail. A voter's power is measured as the fraction of all swing votes that he could cast. There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods. Examples Voting game Simple voting game A simple voting game, taken from ''Game Theory and Strategy'' by Philip D. Straffin: ; 4, 3, 2, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airport Problem
In mathematics and especially game theory, the airport problem is a type of fair division problem in which it is decided how to distribute the cost of an airport runway among different players who need runways of different lengths. The problem was introduced by S. C. Littlechild and G. Owen in 1973. Their proposed solution is: # Divide the cost of providing the minimum level of required facility for the smallest type of aircraft equally among the number of landings of all aircraft # Divide the incremental cost of providing the minimum level of required facility for the second smallest type of aircraft (above the cost of the smallest type) equally among the number of landings of all but the smallest type of aircraft. Continue thus until finally the incremental cost of the largest type of aircraft is divided equally among the number of landings made by the largest aircraft type. The authors note that the resulting set of landing charges is the Shapley value for an appropriately def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data, known as training data, in order to make predictions or decisions without being explicitly programmed to do so. Machine learning algorithms are used in a wide variety of applications, such as in medicine, email filtering, speech recognition, agriculture, and computer vision, where it is difficult or unfeasible to develop conventional algorithms to perform the needed tasks.Hu, J.; Niu, H.; Carrasco, J.; Lennox, B.; Arvin, F.,Voronoi-Based Multi-Robot Autonomous Exploration in Unknown Environments via Deep Reinforcement Learning IEEE Transactions on Vehicular Technology, 2020. A subset of machine learning is closely related to computational statistics, which focuses on making pred ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham Neyman
Abraham Neyman (born June 14, 1949, Israel) is an Israeli mathematician and game theorist, Professor of Mathematics at the Federmann Center for the Study of Rationality and the Einstein Institute of Mathematics at the Hebrew University of Jerusalem in Israel. He served as president of the Israeli Chapter of the Game Theory Society (2014–2018). Biography Neyman received his BSc in mathematics in 1970 and his MSc in mathematics in 1972 from the Hebrew University. His MSc thesis was on the subject of “The Range of a Vector Measure” and was supervised by Joram Lindenstrauss. His PhD thesis, "Values of Games with a Continuum of Players," was completed under Robert Aumann in 1977. Neyman has been professor of mathematics at the Hebrew University since 1982, including serving as the chairman of the institute of mathematics 1992–1994, as well as holding a professorship in economics, 1982–1990. He has been a member of the Center for the Study of Rationality at the Hebrew Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-François Mertens
Jean-François Mertens (11 March 1946 – 17 July 2012) was a Belgian game theorist and mathematical economist. Mertens contributed to economic theory in regards to order-book of market games, cooperative games, noncooperative games, repeated games, epistemic models of strategic behavior, and refinements of Nash equilibrium (see solution concept). In cooperative game theory he contributed to the solution concepts called the core and the Shapley value. Regarding repeated games and stochastic games, Mertens 1982 and 1986 survey articles, and his 1994 survey co-authored with Sylvain Sorin and Shmuel Zamir, are compendiums of results on this topic, including his own contributions. Mertens also made contributions to probability theory and published articles on elementary topology. Epistemic models Mertens and Zamir implemented John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a privately known type that describes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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