For competitive optimization, a change of coordinates of $(x,y) \in {\mathbb{R}}^{m +n}$ can correspond, for instance, to taking a decision variable of one player and giving it to the other player. This changes the underlying game significantly and thus we do *not* want our approximation to be invariant under this transformation. Instead, we want our local approximation to only be invariant to coordinate changes of $x \in {\mathbb{R}}^m$ and $y \in {\mathbb{R}}^{n}$ *in separation*, that is to block-diagonal coordinate changes on ${\mathbb{R}}^{m+n}$. *Mixed* order approximations (bilinear, biquadratic, etc.) have exactly this invariance property and thus are the natural approximation for two-player games.

While we are convinced that the right notion of first order competitive optimization is given by quadratically regularized bilinear approximations, we believe that the right notion of second order competitive optimization is given by *cubically* regularized *biquadratic* approximations, in the spirit of [@nesterov2006cubic].

Consensus, optimism, or competition? {#sec:comparison}
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We will now show that many of the convergent modifications of GDA correspond to different subsets of four common ingredients. *Consensus optimization* (ConOpt) [@mescheder2017numerics], penalises the players for non-convergence by adding the squared norm of the gradient at the next location, $\gamma \|\nabla_x f(x_{k+1},y_{k+1}), \nabla_x f(x_{k+1},y_{k+1})\|^2$ to both player’s 