term* $-D_{xy}f \nabla_yf$, $D_{yx}f \nabla_x f$ which can be interpreted either as anticipating the other player to use the naive (GDA) strategy, or as decreasing the other players influence (by decreasing their gradient).

3.  \[item:consensus\] The *consensus term* $ \pm D_{xx}^2 \nabla_x f$, $\mp D_{yy}^2 \nabla_y f$ that determines whether the players prefer to decrease their gradient ($\pm = +$) or to increase it ($\pm = -$). The former corresponds the players seeking consensus, whereas the latter can be seen as the opposite of consensus.\
    (It also corresponds to an approximate Newton’s method. [^4])

4.  \[item:equilibrium\] The *equilibrium term* $({\operatorname{Id}}+ \eta^2 D_{xy}^2 D_{yx}^2 f)^{-1}$, $({\operatorname{Id}}+ \eta^2 D_{yx}^2 D_{xy}^2 f)^{-1}$, which arises from the players solving for the Nash equilibrium. This term lets each player prefer strategies that are less vulnerable to the actions of the other player.

Each of these is responsible for a different feature of the corresponding algorithm, which we can illustrate by applying the algorithms to three prototypical test cases considered in previous works.

-   We first consider the bilinear problem $f(x,y) = \alpha xy$ (see Figure \[fig:bilinear\_strong\]). It is well known that GDA will fail on this problem, for any value of $\eta$. For $\alpha = 