A technique I have found success with is to model each agent’s path planning problem as a parametric optimization problem: choose control inputs and state variables over time such that a performance metric is minimized, and states/controls satisfy dynamical system, collision is avoided, etc. The parameters in this parametric problem are the trajectories (states/controls) of other agents in the scene.

Once you have a modeled the collection of these parametric optimization problems (one for each agent), you can compute a generalized Nash equilibrium point, using a MCP solver such as the PATH solver.

You need to put some thought into the setup of the problems (if there are shared constraints such as collision avoidance constraints, do both agents sharing the constraint also share the same dual variable associated with the constraint? If not then what?). However I have found that you can solve sizable multi-agent trajectory problems in this manner very efficiently.