How do you like your equilibrium selection problems? hard, or very hard? - Goldberg, Paul W. (Proceedings of the Third international conference on Algorithmic game theory, 2010)
The PPAD-completeness of Nash equilibrium computation is taken as evidence that the problem is computationally hard in the worst case. This evidence is necessarily rather weak, in the sense that PPAD is only know to lie "between P and NP", and there is not a strong prospect of showing it to be as hard as NP. Of course, the problem of finding an equilibrium that has certain sought-after properties should be at least as hard as finding an unrestricted one, thus we have for example the NP-hardness of finding equilibria that are socially optimal (or indeed that have various efficiently checkable properties), the results of Gilboa and Zemel [6], and Conitzer and Sandholm [3]. In the talk I will give an overview of this topic, and a summary of recent progress showing that the equilibria that are found by the Lemke-Howson algorithm, as well as related homotopy methods, are PSPACE-complete to compute. Thus we show that there are no short cuts to the Lemke-Howson solutions, subject only to the hardness of PSPACE. I mention some open problems.