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# Finding Robust Nash equilibria

*Proceedings of the 31st International Conference on Algorithmic Learning Theory*, PMLR 117:725-751, 2020.

#### Abstract

When agents or decision maker have uncertainties of underlying parameters, they may want to take “robust” decisions that are optimal against the worst possible values of those parameters, leading to some max-min optimization problems. With several agents in competition, in game theory, uncertainties are even more important and robust games - or game with non-unique prior - have gained a lot of interest recently, notably in auctions. The existence of robust equilibria in those games is guaranteed using standard fixed point theorems as in classical finite games, so we focus on the problem of finding and characterizing them. Under some linear assumption on the structure of the uncertainties, we provide a polynomial reduction of the robust Nash problem to a standard Nash problem (on some auxiliary different game). This is possible by proving the existence of a lifting transforming robust linear programs into standard linear programs. In the general case, the above direct reduction is not always possible. However, we prove how to adapt the Lemke-Howson algorithm to find robust Nash equilibria in non-degenerate games.