  Cooperatively planning for multiple agents has been proposed as a promising
method for strategic and motion planning for automated vehicles. By taking into
account the intent of every agent, the ego agent can incorporate future
interactions with human-driven vehicles into its planning. The problem is often
formulated as a multi-agent game and solved using iterative algorithms
operating on a discretized action or state space. Even if converging to a Nash
equilibrium, the result will often be only sub-optimal. In this paper, we
define a linear differential game for a set of interacting agents and solve it
to optimality using mixed-integer programming. A disjunctive formulation of the
orientation allows us to formulate linear constraints to prevent agent-to-agent
collision while preserving the non-holonomic motion properties of the vehicle
model. Soft constraints account for prediction errors. We then define a joint
cost function, where a cooperation factor can adapt between altruistic,
cooperative, and egoistic behavior. We study the influence of the cooperation
factor to solve scenarios, where interaction between the agents is necessary to
solve them successfully. The approach is then evaluated in a racing scenario,
where we show the applicability of the formulation in a closed-loop receding
horizon replanning fashion. By accounting for inaccuracies in the cooperative
assumption and the actual behavior, we can indeed successfully plan an optimal
control strategy interacting closely with other agents.
