g &+ y^{\top} D_{yx}^2 g x + x^{\top} \nabla_x g + \frac{1}{2\eta} y^{\top} y.
    \end{split}\end{aligned}$$ Indeed, the (unique) Nash equilibrium of the Game  can be computed in closed form.

\[thm:uniqueNash\] Among all (possibly randomized) strategies with finite first moment, the only Nash equilibrium of the Game  is given by $$\begin{aligned}
    \label{eqn:nash}
    &x = -\eta \left( {\operatorname{Id}}- \eta^2 D_{xy}^2f D_{yx}^2 g \right)^{-1}  
                \left( \nabla_{x} f - \eta D_{xy}^2f  \nabla_{y} g \right) \\
    &y = -\eta \left( {\operatorname{Id}}- \eta^2 D_{yx}^2g D_{xy}^2 f \right)^{-1}  
                \left( \nabla_{y} g - \eta D_{yx}^2g  \nabla_{x} f \right),
    \end{aligned}$$ given that the matrix inverses in the above expression exist. [^3]

Let $X,Y$ be randomized strategies. By subtracting and adding ${\mathbb{E}}[X]^2/(2\eta), {\mathbb{E}}[Y]^2/(2\eta)$, and taking expectations, we can rewrite the game as $$\begin{aligned}
    &\min_{{\mathbb{E}}[X] \in {\mathbb{R}}^m}  {\mathbb{E}}[X]^{\top} \nabla_x f + {\mathbb{E}}[X]^{\top} D_{xy}^2 f {\mathbb{E}}[Y] + 
    {\mathbb{E}}[Y]^{\top} \nabla_y f + \frac{1}{2\eta} {\mathbb{E}}[X]^{\top} {\mathbb{E}}[X] + \frac{1}{2\eta} {\operatorname{Var}}[X]\\
    &\min_{{\mathbb{E}}[Y] \in {\mathbb{R}}^n}  {\mathbb{E}}[Y]^{\top} \nabla_y g + {\mathbb{E}}[Y]^{\top} D_{yx}^2 g {\mathbb{E}}[X] + {\mathbb{E}}[X]^{\top} \nabla_x g + \frac{1}{2\eta} {\mathbb{E}}[Y]^{\top} {\mathbb{E}}[Y] + \frac{1}{2\eta} {\operatorname{Var}}[Y].\end{aligned}$$ Thus, the objective value for both players can always be improved by decreasing the variance while keeping the expectation the same, meaning that the optimal value will always (and only) be achieved by 