a deterministic strategy. We can then replace the ${\mathbb{E}}[X], {\mathbb{E}}[Y]$ with $x,y$, set the derivative of the first expression with respect to $x$ and of the second expression with respect to $y$ to zero, and solve the resulting system of two equations for the Nash equilibrium $(x,y)$.

According to Theorem \[thm:uniqueNash\], the Game  has exactly one optimal pair of strategies, which is deterministic. Thus, we can use these strategies as an update rule, generalizing the idea of local optimality from the single– to the multi agent setting and obtaining Algorithm \[alg:CGD\]. **What I think that they think that I think ... that they do**: Another game-theoretic interpretation of CGD follows from the observation that its update rule can be written as $$\label{eqn:whatIthink}
    \begin{pmatrix}
         \Delta x\\
         \Delta y
    \end{pmatrix}
    =  
    -
    \begin{pmatrix}
        {\operatorname{Id}}& \eta D_{xy}^2 f \\
        \eta D_{yx}^2 g & {\operatorname{Id}}\end{pmatrix}^{-1}
    \begin{pmatrix}
         \nabla_{x} f\\
         \nabla_{y} g
    \end{pmatrix}.$$ Applying the expansion $ \lambda_{\max} (A) < 1 \Rightarrow \left( {\operatorname{Id}}- A \right)^{-1} = \lim_{N \rightarrow \infty} \sum_{k=0}^{N} A^k$ to the above equation, we observe that the first partial sum ($N = 0$) corresponds to the optimal strategy if the other player’s strategy stays constant (GDA). The second partial sum ($N = 1$) 