On No-Regret Learning, Fictitious Play, and Nash Equilibrium
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No-regret algorithms have been proposed to control a wide variety of multi-agent systems. The appeal of no-regret algorithms is that they are easily implementable in large scale multi-agent systems because players make decisions using only retrospective or "regret based" information. Furthermore, there are existing results proving that the collective behavior will asymptotically converge to a set of points of "no-regret" in any game. We illustrate, through a simple example, that no-regret points need not reflect desirable operating conditions for a multi-agent system. Multi-agent systems often exhibit an additional structure (i.e. being "weakly acyclic") that has not been exploited in the context of no-regret algorithms. In this paper, we introduce a modification of the traditional no-regret algorithms by (i) exponentially discounting the memory and (ii) bringing in a notion of inertia in players' decision process. We show how these modifications can lead to an entire class of regret based algorithms that provide almost sure convergence to a pure Nash equilibrium in any weakly acyclic game.