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We investigate Nash equilibrium learning in a competitive Markov Game (MG)environment, where multiple agents compete, and multiple Nash equilibria canexist. In particular, for an oligopolistic dynamic pricing environment, exactNash equilibria are difficult to obtain due to the curse-of-dimensionality. Wedevelop a new model-free method to find approximate Nash equilibria.Gradient-free black box optimization is then applied to estimate $\epsilon$,the maximum reward advantage of an agent unilaterally deviating from any jointpolicy, and to also estimate the $\epsilon$-minimizing policy for any givenstate. The policy-$\epsilon$ correspondence and the state to$\epsilon$-minimizing policy are represented by neural networks, the latterbeing the Nash Policy Net. During batch update, we perform Nash Q learning onthe system, by adjusting the action probabilities using the Nash Policy Net. Wedemonstrate that an approximate Nash equilibrium can be learned, particularlyin the dynamic pricing domain where exact solutions are often intractable.

Approximate Nash Equilibrium Learning for n-Player Markov Games in Dynamic Pricing