It will also be instructive to compare the algorithms to linearized competitive gradient descent (LCGD), which is obtained by skipping the matrix inverse in CGD (which corresponds to taking only the leading order term in the limit $\eta D_{xy}^2f \rightarrow 0$) and also coincides with first order LOLA [@foerster2018learning]. As illustrated in Figure \[fig:ingredients\], these six algorithms amount to different subsets of the following four terms.

$$\begin{aligned}
       & \text{GDA: } &\Delta x =  &&&- \nabla_x f&\\
       & \text{LCGD: } &\Delta x =  &&&- \nabla_x f& &-\eta D_{xy}^2 f \nabla_y f&\\
       & \text{SGA: } &\Delta x =  &&&- \nabla_x f& &- \gamma D_{xy}^2 f \nabla_y f&  & & \\
       & \text{ConOpt: } &\Delta x =  &&&- \nabla_x f& &- \gamma D_{xy}^2 f \nabla_y f&  &- \gamma D_{xx}^2 f \nabla_x f& \\
       & \text{OGDA: } &\Delta x \approx &&&- \nabla_x f& &-\eta D_{xy}^2 f \nabla_y f&  &+\eta D_{xx}^2 f \nabla_x f& \\
       & \text{CGD: } &\Delta x = &\left({\operatorname{Id}}+ \eta^2 D_{xy}^2 f D_{yx}^2 f\right)^{-1}&\bigl( &- \nabla_x f&  &-\eta D_{xy}^2 f \nabla_y f& & & \bigr)
     \end{aligned}$$

1.  \[item:grad\] The *gradient term* $-\nabla_{x}f$, $\nabla_{y}f$ which corresponds to the most immediate way in which the players can improve their cost.

2.  \[item:comp\] The *competitive 