Monotone Smooth Game

A Hilbert smooth game is monotone if, for all profiles $s,t\in S$, $$ \sum_{i\in I} \langle \nabla_i g_i(s)-\nabla_i g_i(t),\,s_i-t_i\rangle \le 0. $$ It is strictly monotone if the inequality is strict whenever $s\ne t$.

References

• [MFoGT, Def. 4.7.9] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Rosen monotonicity condition for Hilbert smooth games.