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Lemke–Howson Algorithm
The Lemke–Howson algorithm is an algorithm that computes a Nash equilibrium of a bimatrix game, named after its inventors, Carlton E. Lemke and J. T. Howson. It is said to be "the best known among the combinatorial algorithms for finding a Nash equilibrium", although more recently the Porter-Nudelman-Shoham algorithm has outperformed on a number of benchmarks. Description The input to the algorithm is a 2-player game ''G''. ''G'' is represented by two ''m'' × ''n'' game matrices A and B, containing the payoffs for players 1 and 2 respectively, who have ''m'' and ''n'' pure strategies respectively. In the following we assume that all payoffs are positive. (By rescaling, any game can be transformed to a strategically equivalent game with positive payoffs.) ''G'' has two corresponding polytopes (called the ''best-response polytopes'') P1 and P2, in ''m'' dimensions and ''n'' dimensions respectively, defined as follows. P1 is in R''m''; let denote the coordinates. P1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs. If each player has chosen a strategy an action plan based on what has happened so far in the game and no one can increase one's own expected payoff by changing one's strategy while the other players keep their's unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bimatrix Game
In game theory, a bimatrix game is a simultaneous game for two players in which each player has a finite number of possible actions. The name comes from the fact that the normal form of such a game can be described by two matrices - matrix A describing the payoffs of player 1 and matrix B describing the payoffs of player 2. Player 1 is often called the "row player" and player 2 the "column player". If player 1 has m possible actions and player 2 has n possible actions, then each of the two matrices has m rows by n columns. When the row player selects the i-th action and the column player selects the j-th action, the payoff to the row player is A ,j/math> and the payoff to the column player is B ,j/math>. The players can also play mixed strategies. A mixed strategy for the row player is a non-negative vector x of length m such that: \sum_^m x_i = 1. Similarly, a mixed strategy for the column player is a non-negative vector y of length n such that: \sum_^n y_j = 1. When the players ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlton E
Carlton may refer to: People * Carlton (name), a list of those with the given name or surname * Carlton (singer), English soul singer Carlton McCarthy * Carlton, a pen name used by Joseph Caldwell (1773–1835), American educator, Presbyterian minister, mathematician and astronomer Places Australia * Carlton, New South Wales, a suburb of Sydney * Carlton, Tasmania, a locality in Tasmania * Carlton, Victoria, a suburb of Melbourne Canada * Carlton, Edmonton, Alberta, a neighbourhood * Carlton, Saskatchewan, a hamlet * Fort Carlton, a Hudson's Bay Company fur trading post built in 1810, near present-day Carlton, Saskatchewan * Carlton Trail, a historic trail near Fort Carlton * Carlton Street, Toronto, Ontario England * Carlton, Bedfordshire, a village * Carlton, Cambridgeshire, a village * Carlton, County Durham, a village and civil parish * Carlton, Leicestershire, a village * Carlton, Nottinghamshire, a suburb to the east of Nottingham ** The Carlton Academy ** ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE-complete
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (polynomial space) and if every other problem that can be solved in polynomial space can be transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. Problems known to be PSPACE-complete include determining properties of regular expressions and context-sensitive grammars, determining the truth of quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization problems, and many puzzles and games. Theory A problem is defined to be PSPACE-complete if it can be solved using a polynomial amount of memory (it belongs to PSPACE) and every problem in PSPACE can be tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christos Papadimitriou
Christos Charilaos Papadimitriou ( el, Χρήστος Χαρίλαος "Χρίστος" Παπαδημητρίου; born August 16, 1949) is a Greek theoretical computer scientist and the Donovan Family Professor of Computer Science at Columbia University. Education Papadimitriou studied at the National Technical University of Athens, where in 1972 he received his Bachelor of Arts degree in electrical engineering. He then pursued graduate studies at Princeton University, where he received his Ph.D. in electrical engineering and computer science in 1976 after completing a doctoral dissertation titled "The complexity of combinatorial optimization problems." Career Papadimitriou has taught at Harvard, MIT, the National Technical University of Athens, Stanford, UCSD, University of California, Berkeley and is currently the Donovan Family Professor of Computer Science at Columbia University. Papadimitriou co-authored a paper on pancake sorting with Bill Gates, then a Harvard undergra ... [...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 mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
