Recent development in computational complexity characterization of Nash equilibrium

Chen, Xi; Deng, Xiaotie

doi:10.1016/j.cosrev.2007.09.002

Cited by 3 publications

(4 citation statements)

References 53 publications

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“…In the case of the infinite support probability measure, the majority of the support pure strategies cannot be recurred even once as a number of the recurred ones constitute the zero-measure set [5], [6]. However, for most classes of infinite twoperson non-cooperative games their solutions are unknown or at least are non-effectively computable [1], [2], [7]. Then, the conversion of the infiniteness into finiteness is needed anyway.…”

Section: Infinite Two-person Non-cooperative Games Isomorphic To mentioning

confidence: 99%

“…Nevertheless, the satisfied requirements (12) are not the concluding step in converting the infinite game (3) into the bimatrix game. We should firstly arrange the finite game (6) by (7) into the bimatrix game. Secondly, acceptance of the drawn bimatrix game depends on features of its solution, approximating a solution of the infinite game (3).…”

Section: Requirements To the Samplingmentioning

confidence: 99%

“…be the r -th player's pure strategy, corresponding to its pure strategy in initial infinite game (3) with components in (15), whose indices are included into set (16). Hence, the first , has many solutions in forms of utility or equity equilibrium -Nash equilibrium, strong Nash equilibrium [7], [23], [31], [32], Pareto equilibrium [2], [6], [33], [34], Mertens-stable equilibrium [35], trembling hand perfect equilibrium [36], perfect Bayesian equilibrium [9], [37], [38], Markov perfect equilibrium [39], [40] and many others.…”

Section: Reshaping the Multidimensional Matricesmentioning

confidence: 99%

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Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game

Romanuke

Kamburg

2016

Applied Computer Systems

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Approximation in solving the infinite two-person non-cooperative games is studied in the paper. An approximation approach with conversion of infinite game into finite one is suggested. The conversion is fulfilled in three stages. Primarily the players’ payoff functions are sampled variously according to the stated requirements to the sampling. These functions are defined on unit hypercube of the appropriate Euclidean finite-dimensional space. The sampling step along each of hypercube dimensions is constant. At the second stage, the players’ payoff multidimensional matrices are reshaped into ordinary two-dimensional matrices, using the reversible index-to-index reshaping. Thus, a bimatrix game as an initial infinite game approximation is obtained. At the third stage of the conversion, the player’s finite equilibrium strategy support is checked out for its weak consistency, defined by five types of inequalities within minimal neighbourhood of every specified sampling step. If necessary, the weakly consistent solution of the bimatrix game is checked out for its consistency, strengthened in that the cardinality of every player’s equilibrium strategy support and their densities shall be non-decreasing within minimal neighbourhood of the sampling steps. Eventually, the consistent solution certifies the game approximation acceptability, letting solve even games without any equilibrium situations, including isomorphic ones to the unit hypercube game. A case of the consistency light check is stated for the completely mixed Nash equilibrium situation.

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Section: Infinite Two-person Non-cooperative Games Isomorphic To mentioning

confidence: 99%

Section: Requirements To the Samplingmentioning

confidence: 99%

Section: Reshaping the Multidimensional Matricesmentioning

confidence: 99%

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Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game

Romanuke

Kamburg

2016

Applied Computer Systems

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“…ON COMPACT ACTION SPACES Finding NE-solutions in even the finite noncooperative game is a computational difficulty [8], [34], [35]. 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].…”

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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A survey on algorithms for Nash equilibria in finite normal-form games

Li,

Huang,

Duan

et al. 2024

Computer Science Review

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Recent development in computational complexity characterization of Nash equilibrium

Cited by 3 publications

References 53 publications

Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game

Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game

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

A survey on algorithms for Nash equilibria in finite normal-form games

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