A systematic survey of Nash equilibrium variants and their solution methods and applications in multi-agent problems
Abstract
The literature on multi-agent systems lacks a unified framework connecting problem structure to the choice of equilibrium concept and solution method. This survey synthesizes work on Nash equilibrium and its variants, namely standard NE, Mixed/Approximate NE, Generalized Nash Equilibrium (GNE), and Stackelberg equilibrium, across a PRISMA-screened corpus of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = 333$$\end{document} papers with first-online years from 1987 to 2025, coded on a multi-axis taxonomy by two independent coders reconciled to consensus. Three patterns emerge. First, feasibility coupling (A3) tracks equilibrium-concept choice most closely, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{Cramer's }V(A3 \times B) = 0.794$$\end{document}; because GNE presupposes coupling, this is partly definitional and is reported as a construct-validity check, with GNE in 63.9% of coupled cases and NE in 46.2% of the rest. Second, player structure (A1) is the next closest correlate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(A1 \times B) = 0.566$$\end{document}: leader-follower systems take Stackelberg equilibrium in 86.8% of cases, and no peer-structured paper uses it. Third, strategy domain and equilibrium family associate only weakly with method, so solver choice rests on finer mathematical properties. Metaheuristics account for 75.1% of the corpus. The survey proposes a four-step guidance framework annotated with each step’s evidence status, and identifies priority gaps in evaluation, benchmarking, learning-based hierarchical models, and uncertainty-aware equilibrium computation.