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Probably Approximately Correct Nash Equilibrium Learning

We consider a multiagent noncooperative game with agents’ objective functions being affected by uncertainty. Following a data driven paradigm, we represent uncertainty by means of scenarios and seek a robust Nash equilibrium solution. We treat the Nash equilibrium computation problem within the realm of probably approximately correct learning. Building upon recent developments in scenario-based optimization, we accompany the computed Nash equilibrium with a priori and a posteriori probabilistic robustness certificates, providing confidence that the computed equilibrium remains unaffected (in probabilistic terms) when a new uncertainty realization is encountered. For a wide class of games, we also show that the computation of the so called compression set—which is at the core of scenario-based optimization—can be directly obtained as a byproduct of the proposed methodology. We demonstrate the efficacy of our approach on an electric vehicle charging control problem.