Nash Field

A Nash field is a continuous map, or an upper semicontinuous correspondence, $$ \Psi:G\times\Sigma\to\Sigma $$ such that for every finite game $g$, $$ NE(g)=\{\sigma\in\Sigma:\Psi(g,\sigma)=\sigma\}. $$

Each Nash field induces a dynamical system $$ \dot\sigma=\Psi(g,\sigma)-\sigma $$ whose rest points are exactly the Nash equilibria of $g$.

References

• [MFoGT, Chapter 5, Def. 5.4.1] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Nash fields whose fixed points are Nash equilibria.