Stochastic Nash Equilibrium Problems: Models, Analysis, and Algorithms

Abstract

Decision making under uncertainty has been studied extensively over the last 70 years, if not earlier. In the field of optimization, models for two-stage, stochastic, linear programming, presented by Dantzig <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> and Beale <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> , are often viewed as the basis for the subsequent development of the field of stochastic optimization. This subfield of optimization now encompasses a breadth of models that can accommodate both convexity and nonconvexity, probabilistic constraints, risk-aversion, discreteness, and multistage decision-making (compare <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> , <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[4]</xref> ). Similarly, stochastic control <xref ref-type="bibr" rid="ref5" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[5]</xref> has proven to be an enormously impactful subarea of control theory. When one extends the decision-making paradigm to multiple self-interested decision makers, then the resulting problem can be viewed as a noncooperative game that is rooted in the groundbreaking text by Von Neumann and Morgenstern <xref ref-type="bibr" rid="ref6" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[6]</xref> .