Definition (Nash equilibirum)

For a given $N-$person stochastic team with a fixed information structure, $\{ J;\Gamma^{i},i\in\mathcal{N} \}$, and $N-$tuple of strategies $\underline{\gamma}^{*}:=(\gamma^{1*},\dots,\gamma^{N*})$ constitutes a Nash equilibrium (or person-by-person optimal) if the following inequalities hold: $$J^{*}:=J(\underline{\gamma}^{*})\le J(\underline{\gamma}^{-i*},\beta),\quad\forall \beta\in\Gamma^{i},\forall i\in\mathcal{N}$$where $$(\underline{\gamma}^{-i*},\beta):=(\gamma^{1,*},\dots,\gamma^{i-1,*},\beta,\gamma^{i+1,*},\dots,\gamma^{N,*})$$