Mixed complementarity problems for robust optimization equilibrium in bimatrix game.
type: Item
instance of: scholarly article
title: Mixed complementarity problems for robust optimization equilibrium in bimatrix game.
zbMATH Open document ID: 1265.91003
author: Gui-Mei Luo
published in: Applications of Mathematics
publication date: 2013-01-02T00:00:00Z
full work available at URL: http://hdl.handle.net/10338.dmlcz/142913
review text: The author studies the problem of finding the robust optimization equilibrium of a bimatrix game in which each player attempts to minimize his own cost with either each player's cost matrix or his opponent's uncertain strategies. Let \(Y\), \(Z\) denote the sets of mixed strategies of Player one and Player two, \(D_A\), \(D_B\) be bounded sets of matrices and \(Y^U\), \(Z^U\) bounded sets of the opponents' strategies. Player one solves the problem \[ \min _{y \in Y}\max _{\tilde {A} \in D_A, \tilde {z} \in Z^U}(y^T\tilde {A}\tilde {z}). \tag{1} \] Player two solves the problem \[ \min _{z \in Z}\max _{\tilde {B} \in D_B, \tilde {y} \in Y^U}(\tilde {y}^T\tilde {B}z). \tag{2} \] A pair of strategies (\(\hat {y}, \hat {z}\)) which solve problems (1) and (2), respectively, is called a robust equilibrium for the players. The author proposes a method for finding strategies \(\hat {y}\), \(\hat {z}\) based on linear programming and solving a mixed complementarity problem, which can be solved by methods known from the literature. The proposed procedure reduces the computational complexity as compared with other methods known from the literature, which lead to solving the second-order cone complementarity problem. The last section of the paper contains numerical examples illustrating the theoretical results of the preceding sections.
reviewed by: Karel Zimmermann
Mathematics Subject Classification ID: 91A05
Mathematics Subject Classification ID: 90C05
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 90C46
zbMATH DE Number: 6121198
zbMATH Keywords: robust optimization equilibrium
zbMATH Keywords: bimatrix game
zbMATH Keywords: \(l_1\cap l_\infty \)-norm
zbMATH Keywords: mixed complementarity problem
MaRDI profile type: Publication
OpenAlex ID: W1974781235
cites work: Robust game theory
cites work: Robust convex optimization
cites work: Robust solutions of uncertain linear programs
cites work: Robust solutions of linear programming problems contaminated with uncertain data
cites work: Robust linear optimization under general norms.
cites work: The Price of Robustness
cites work: Tractable approximations to robust conic optimization problems
cites work: A Robust Optimization Perspective on Stochastic Programming
cites work: Robust Solutions to Least-Squares Problems with Uncertain Data
cites work: Robust Solutions to Uncertain Semidefinite Programs
cites work: Finite-Dimensional Variational Inequalities and Complementarity Problems
cites work: A Combined Smoothing and Regularization Method for Monotone Second-Order Cone Complementarity Problems
cites work: Q5694955
cites work: Games with Incomplete Information Played by “Bayesian” Players Part II. Bayesian Equilibrium Points
cites work: Efficient and Durable Decision Rules with Incomplete Information
cites work: Q3183650
cites work: Formulation of Bayesian analysis for games with incomplete information
cites work: Equilibrium points in n -person games
cites work: Non-cooperative games
cites work: Technical Note—Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming
DOI: 10.1007/S10492-012-0029-4
description: scientific article; zbMATH DE number 6121198