Equilibrium Points of Bimatrix Games

Lemke, C. E.; Howson, Joseph T.

doi:10.1137/0112033

Cited by 994 publications

(502 citation statements)

References 4 publications

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“…Motivated by this work, we extend the algorithm of [4] by using Lemke's pivoting algorithm [7,14] to solve the successive sub-problems in the branch-and-bound tree. Unlike [4], we do not explicitly add the variable bound constraints, x j ≤ 0 and x j ≥ α i , thus the size of the subproblems never increases.…”

Section: Introductionmentioning

confidence: 99%

Algorithm for cardinality-constrained quadratic optimization

2007

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This paper describes an algorithm for cardinality-constrained quadratic optimization problems, which are convex quadratic programming problems with a limit on the number of non-zeros in the optimal solution. In particular, we consider problems of subset selection in regression and portfolio selection in asset management and propose branch-and-bound based algorithms that take advantage of the special structure of these problems. We compare our tailored methods against CPLEX's quadratic mixed-integer solver and conclude that the proposed algorithms have practical advantages for the special class of problems we consider.

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Section: Introductionmentioning

confidence: 99%

Algorithm for cardinality-constrained quadratic optimization

2007

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“…Locally, solving dyadic games with three players takes some technique of visualization of the cube of situations in pure strategies [6], [10], whereupon dyadic games with four players and more are solved purely in analytics, requiring more computational resources [10], [36]. Naturally, that finite noncooperative games with greater numbers of pure strategies at their players (three and more) are significantly hard to solve them [10], [37], [38]. Moreover, often an admissible player's action is described with a series of its continuous parameters, constituting thus an infinite (continuous) set of pure strategies [1], [6], [7], [12], [39], [40].…”

Section: Solving Noncooperative Gamesmentioning

confidence: 99%

Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values

Romanuke

2015

Electrical, Control and Communication Engineering

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-The paper suggests a method of obtaining an approximate solution of the infinite noncooperative game on the unit hypercube. The method is based on sampling uniformly the players' payoff functions with the constant step along each of the hypercube dimensions. The author states the conditions for a sufficiently accurate sampling and suggests the method of reshaping the multidimensional matrix of the player's payoff values, being the former player's payoff function before its sampling, into a matrix with minimally possible number of dimensions, where also maintenance of one-to-one indexing has been provided. Requirements for finite NE-strategy from NE (Nash equilibrium) solution of the finite game as the initial infinite game approximation are given as definitions of the approximate solution consistency. The approximate solution consistency ensures its relative independence upon the sampling step within its minimal neighborhood or the minimally decreased sampling step. The ultimate reshaping of multidimensional matrices of players' payoff values to the minimal number of dimensions, being equal to the number of players, stimulates shortened computations.

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“…algorithm [24]; see von Stengel [30,Chapter 3] for a careful exposition. This algorithm is a pathfollowing algorithm in the spirit of local search, but it is not guided by an objective or potential function and thus does not prove that computing an MNE of a bimatrix game is in P LS.…”

Section: Complexity Of Equilibrium Computationmentioning

confidence: 99%

Algorithmic Game Theory

Persiano¹

2011

Lecture Notes in Computer Science

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Equilibrium Points of Bimatrix Games

Cited by 994 publications

References 4 publications

Algorithm for cardinality-constrained quadratic optimization

Algorithm for cardinality-constrained quadratic optimization

Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values

Algorithmic Game Theory

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