these methods can be considered variations of the *extragradient method* [@korpelevich1977extragradient](see also [@facchinei2003finite]\[Chapter 12\]). Finally, some methods directly modify the gradient dynamics, either by promoting convergence through gradient penalties [@mescheder2017numerics], or by attempting to disentangle convergent *potential* parts from rotational *Hamiltonian* parts of the vector field [@balduzzi2018mechanics; @letcher2019differentiable; @gemp2018global].\
**Our contributions:** Our main *conceptual* objection to most existing methods is that they lack a clear game-theoretic motivation, but instead rely on the ad-hoc introduction of additional assumptions, modifications, and model parameters.\
Their main *practical* shortcoming is that to avoid divergence the stepsize has to be chosen inversely proportional to the magnitude of the interaction of the two players (as measured by $D_{xy}^2 f$, $D_{xy}^2 g$).\
On the one hand, the small stepsize results in slow convergence. On the other hand, a stepsize small enough to prevent divergence will not be known in advance in most problems. Instead it has to be discovered through tedious trial and error, which is further aggravated by the lack of a good diagnostic for improvement in multi-agent optimization (which is given by the objective function in single agent optimization).\
We alleviate the above mentioned problems by introducing a novel algorithm, *competitive gradient descent* (CGD) that 