A systematic survey of Nash equilibrium variants and their solution methods and applications in multi-agent problems

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 $$N = 333$$ 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, $$\text{Cramer's }V(A3 \times B) = 0.794$$ ; 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, $$V(A1 \times B) = 0.566$$ : 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.