corresponds to the optimal strategy if the other player thinks that the other player’s strategy stays constant (LCGD, see Figure \[fig:ingredients\]). The third partial sum ($N = 2$) corresponds to the optimal strategy if the other player thinks that the other player thinks that the other player’s strategy stays constant, and so forth, until the Nash equilibrium is recovered in the limit. For small enough $\eta$, we could use the above series expansion to solve for $(\Delta x, \Delta y)$, which is known as Richardson iteration and would recover high order LOLA [@foerster2018learning]. However, expressing it as a matrix inverse will allow us to use optimal Krylov subspace methods to obtain far more accurate solutions with fewer gradient evaluations.\
**Rigorous results on convergence and local stability:** We will now show some basic convergence results for CGD, the proofs of which we defer to the appendix. Our results are restricted to the case of a zero-sum game ($f = -g$), but we expect that they can be extended to games that are dominated by competition. To simplify notation, we define $$\bar{D} {\coloneqq}({\operatorname{Id}}+ \eta^2 D_{xy}^2f D_{yx}^2f)^{-1} \eta^2 D_{xy}^2f D_{yx}^2f, \quad
    \tilde{D} {\coloneqq}({\operatorname{Id}}+ \eta^2 D_{yx}^2f D_{xy}^2f)^{-1} \eta^2 D_{yx}^2f D_{xy}^2f.$$ We furthermore define the 