Unified convergence proofs of continuous-time fictitious play

We consider a continuous-time version of fictitious play (FP), in which interacting players evolve their strategies in reaction to their opponents´ actions without knowledge of their opponents´ utilities. It is known that FP need not converge, but that convergence is possible in certain special cases including zero-sum games, identical interest games, and two-player/two-move games. We provide a unified proof of convergence in all of these cases by showing that a Lyapunov function previously introduced for zero-sum games also can establish stability in the other special cases. We go on to consider a two-player game in which only one player has two-moves and use properties of planar dynamical systems to establish convergence.