To enable an efficient interaction of non-communicating mobile robots in collision avoidance scenarios, we present a novel combined trajectory planning and prediction algorithm. Inverse optimal control is used to estimate unknown goal states of all robots based on observed past trajectories. Each robot also takes the perspective of other robots in considering self-prediction and solves a joint prediction problem using the estimated goal states. The resulting predictions are then considered for planning. Simulation results of scenarios with 2-8 robots show that the median of the durations until all vehicles reach their goals is 9.8 % faster compared to planning with constant acceleration based estimated goal states. Moreover, the proposed approach never leads to the solver being unable to find a solution to the planning or prediction problem.