Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints

Authors:
Ming Hu;Masao Fukushima
Affiliations:
School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang, China 212003 and Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto Un ...;Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan 606-8501
Venue:
Computational Optimization and Applications
Year:
2012

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Abstract

The equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalization of Nash equilibrium problem (NEP) and the mathematical program with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a special class of EPECs where a common parametric P-matrix linear complementarity system is contained in all players' strategy sets. After reformulating the EPEC as an equivalent nonsmooth NEP, we use a smoothing method to construct a sequence of smoothed NEPs that approximate the original problem. We consider two solution concepts, global Nash equilibrium and stationary Nash equilibrium, and establish some results about the convergence of approximate Nash equilibria. Moreover we show some illustrative numerical examples.