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 opponentsʹ actions without knowledge of their opponentsʹ 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/twomove 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 twoplayer game in which only one player has two-moves and use properties of planar dynamical systems to establish convergence.