# Probably Approximately Correct Nash Equilibrium Learning

## TL;DR

This paper introduces a data-driven, probabilistically robust method for computing Nash equilibria in uncertain multi-agent games, with theoretical guarantees and decentralized computation, demonstrated on electric vehicle charging.

## Contribution

It develops a PAC learning framework for Nash equilibrium computation with robustness certificates and a decentralized solution approach for scenario-based games.

## Key findings

- Provides probabilistic robustness guarantees for Nash equilibria.
- Enables decentralized equilibrium computation.
- Validates approach on electric vehicle charging problem.

## Abstract

We consider a multi-agent 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 (PAC) 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 - a key concept in scenario-based optimization - can be directly obtained as a byproduct of the proposed solution methodology. Finally, we illustrate how to overcome differentiability issues, arising due to the introduction of scenarios, and compute a Nash equilibrium solution in a decentralized manner. We demonstrate the efficacy of the proposed approach on an electric vehicle charging control problem.