  A dynamical system is defined in terms of the gradient of a payoff function.
Dynamical variables are of two types, ascent and descent. The ascent variables
move in the direction of the gradient, while the descent variables move in the
opposite direction. Dynamical systems of this form or very similar forms have
been studied in diverse fields such as game theory, optimization, neural
networks, and population biology. Gradient descent-ascent is approximated as a
Newtonian dynamical system that conserves total energy, defined as the sum of
the kinetic energy and a potential energy that is proportional to the payoff
function. The error of the approximation is a residual force that violates
energy conservation. If the residual force is purely dissipative, then the
energy serves as a Lyapunov function, and convergence of bounded trajectories
to steady states is guaranteed. A previous convergence theorem due to Kose and
Uzawa required the payoff function to be convex in the descent variables, and
concave in the ascent variables. Here the assumption is relaxed, so that the
payoff function need only be globally `less convex' or `more concave' in the
ascent variables than in the descent variables. Such relative convexity
conditions allow the existence of multiple steady states, unlike the
convex-concave assumption. When combined with sufficient conditions that imply
the existence of a minimax equilibrium, boundedness of trajectories is also
assured.
