Dynamic fictitious play, dynamic gradient play, and distributed convergence to Nash equilibria

We consider a continuous-time form of repeated matrix games in which player strategies evolve in reaction to opponent actions. Players observe each other's actions, but do not have access to other player utilities. Strategy evolution may be of the best response sort, as in fictitious play, or a gradient update. Such mechanisms are known to not necessarily converge. We introduce a form of "dynamic" fictitious and gradient play strategy update mechanisms. These mechanisms use derivative action in processing opponent actions and, in some cases, can lead to behavior converging to Nash equilibria in previously nonconvergent situations. We analyze convergence in the case of exact and approximate derivative measurements of the dynamic update mechanisms. In the ideal case of exact derivative measurements, we show that convergence to Nash equilibrium can always be achieved. In the case of approximate derivative measurements, we derive a characterization of local convergence that shows how the dynamic update mechanisms can converge if the traditional static counterparts do not. We primarily discuss two player games, but also outline extensions to multiplayer games. We illustrate these methods with convergent simulations of the well known Shapley and Jordan counterexamples.