Expected complexity of graph partitioning problems
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On limited nondeterminism and the complexity of the V-C dimension
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Finding a large hidden clique in a random graph
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Finding and certifying a large hidden clique in a semirandom graph
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Hiding Cliques for Cryptographic Security
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Relations between average case complexity and approximation complexity
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The Probable Value of the Lovász-Schrijver Relaxations for Maximum Independent Set
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Playing large games using simple strategies
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On Limited versus Polynomial Nondeterminism
On Limited versus Polynomial Nondeterminism
Spectral Partitioning of Random Graphs
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Testing k-wise and almost k-wise independence
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Computing good nash equilibria in graphical games
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Approximating nash equilibria using small-support strategies
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Progress in approximate nash equilibria
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The complexity of computing a Nash equilibrium
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Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games
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The Search for Equilibrium Concepts
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Approximate Equilibria for Strategic Two Person Games
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How hard is it to approximate the best Nash equilibrium?
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Settling the complexity of computing two-player Nash equilibria
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Random Tensors and Planted Cliques
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Small Clique Detection and Approximate Nash Equilibria
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Complexity results about Nash equilibria
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The Complexity of Computing a Nash Equilibrium
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New algorithms for approximate Nash equilibria in bimatrix games
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An optimization approach for approximate Nash equilibria
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
Graphical models for game theory
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Polynomial algorithms for approximating nash equilibria of bimatrix games
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A note on approximate nash equilibria
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HAPLOFREQ: estimating haplotype frequencies efficiently
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Inapproximability of NP-complete variants of nash equilibrium
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Statistical algorithms and a lower bound for detecting planted cliques
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The quest for a polynomial-time approximation scheme (PTAS) for Nash equilibrium in a two-player game, which emerged as a major open question in algorithmic game theory, seeks to circumvent the PPAD-completeness of finding an (exact) Nash equilibrium by finding an approximate equilibrium. The closely related problem of finding an equilibrium maximizing a certain objective, such as social welfare, was shown to be NP-hard [Gilboa and Zemel, Games Econom. Behav., 1 (1989), pp. 80-93]. However, this NP-hardness is unlikely to extend to approximate equilibria, since the latter admits a quasi-polynomial time algorithm [Lipton, Markakis, and Mehta, in Proceedings of the 4th ACM Conference on Electronic Commerce, ACM, New York, 2003, pp. 36-41]. We show that this optimization problem, namely, finding in a two-player game an approximate equilibrium achieving a large social welfare, is unlikely to have a polynomial-time algorithm. One interpretation of our results is that a PTAS for Nash equilibrium (if it exists) should not extend to a PTAS for finding the best Nash equilibrium. Technically, our result is a reduction from the notoriously difficult problem in modern combinatorics, of finding a planted (but hidden) clique in a random graph $G(n,1/2)$. Our reduction starts from an instance with planted clique size $O(\log n)$. For comparison, the currently known algorithms are effective only for a much larger clique size $\Omega(\sqrt{n})$.