Abstract
In recent work of Hazan and Krauthgamer (SICOMP 2011), it was shown that finding an $\eps$-approximate Nash equilibrium with near-optimal value in a two-player game is as hard as finding a hidden clique of size $O(\log n)$ in the random graph $G(n,1/2)$. This raises the question of whether a similar intractability holds for approximate Nash equilibrium without such constraints. We give evidence that the constraint of near-optimal value makes the problem distinctly harder: a simple algorithm finds an optimal 1/2-approximate equilibrium, while finding strictly better than 1/2-approximate equilibria is as hard as the Hidden Clique problem. This is in contrast to the unconstrained problem where more sophisticated algorithms, achieving better approximations, are known. Unlike general Nash equilibrium, which is in PPAD, optimal (maximum value) Nash equilibrium is NP-hard. We proceed to show that optimal Nash equilibrium is just one of several known NP-hard problems related to Nash equilibrium, all of which have approximate variants which are as hard as finding a planted clique. In particular, we show this for approximate variants of the following problems: finding a Nash equilibrium with value greater than $\eta$ (for any $\eta>0$, even when the best Nash equilibrium has value $1-\eta$), finding a second Nash equilibrium, and finding a Nash equilibrium with small support. Finally, we consider the complexity of approximate pure Bayes Nash equilibria in two-player games. Here we show that for general Bayesian games the problem is NP-hard. For the special case where the distribution over types is uniform, we give a quasi-polynomial time algorithm matched by a hardness result based on the Hidden Clique problem.
Highlights
The classical notion of Nash equilibrium is the most fundamental concept in the theory of non-cooperative games
A series of hardness results culminated in the work of Chen, Deng, and Teng [5], who showed that even for two-player games, the problem of computing a Nash equilibrium is PPAD-complete, and unlikely to be solvable in polynomial time
A notion which has emerged as the focus of several works is that of an ε-approximate Nash equilibrium, or ε-equilibrium for short, where neither player can gain more than ε by defecting to a different strategy
Summary
The classical notion of Nash equilibrium is the most fundamental concept in the theory of non-cooperative games. Hazan and Krauthgamer [16] have attempted to provide evidence for the optimality of the QPTAS of Lipton et al [18], by showing a reduction from a well-studied and seemingly intractable problem (which can be solved in quasi-polynomial time) to the related problem of finding an εequilibrium with near maximum value (the value of an equilibrium is the average of the payoffs of the two players). The problem they reduce from is the k-Hidden Clique Problem: Definition 1.1 (k-Hidden Clique Problem).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
