TY - GEN

T1 - Approximating the best Nash Equilibrium in no (1ogn)-time breaks the exponential time hypothesis

AU - Braverman, Mark

AU - Ko, Young Kun

AU - Weinstein, Omri

PY - 2015/1/1

Y1 - 2015/1/1

N2 - The celebrated PPAD hardness result for finding an exact Nash equilibrium in a two-player game initiated a quest for finding approximate Nash equilibria efficiently, and is one of the major open questions in algorithmic game theory. We study the computational complexity of finding an ε-approximate Nash equilibrium with good social welfare. Hazan and Krauthgamer and subsequent improvements showed that finding an e-approximate Nash equilibrium with good social welfare in a two player game and many variants of this problem is at least as hard as finding a planted clique of size O (1ogn) in the random graph Q (n, 1/2). We show that any polynomial time algorithm that finds an ε-approximate Nash equilibrium with good social welfare refutes (the worst-case) Exponential Time Hypothesis by Impagliazzo and Paturi, confirming the recent conjecture by Aaronson, Impagliazzo and Moshkovitz. Specifically, it would imply a 2O (n1/2)algorithm for SAT. Our lower bound matches the quasi-polynomial time algorithm by Lipton, Markakis and Mehta for solving the problem. Our key tool is a reduction from the PCP machinery to finding Nash equilibrium via free games, the framework introduced in the recent work by Aaronson, Impagliazzo and Moshkovitz. Techniques developed in the process may be useful for replacing planted clique hardness with ETH-hardness in other applications.

AB - The celebrated PPAD hardness result for finding an exact Nash equilibrium in a two-player game initiated a quest for finding approximate Nash equilibria efficiently, and is one of the major open questions in algorithmic game theory. We study the computational complexity of finding an ε-approximate Nash equilibrium with good social welfare. Hazan and Krauthgamer and subsequent improvements showed that finding an e-approximate Nash equilibrium with good social welfare in a two player game and many variants of this problem is at least as hard as finding a planted clique of size O (1ogn) in the random graph Q (n, 1/2). We show that any polynomial time algorithm that finds an ε-approximate Nash equilibrium with good social welfare refutes (the worst-case) Exponential Time Hypothesis by Impagliazzo and Paturi, confirming the recent conjecture by Aaronson, Impagliazzo and Moshkovitz. Specifically, it would imply a 2O (n1/2)algorithm for SAT. Our lower bound matches the quasi-polynomial time algorithm by Lipton, Markakis and Mehta for solving the problem. Our key tool is a reduction from the PCP machinery to finding Nash equilibrium via free games, the framework introduced in the recent work by Aaronson, Impagliazzo and Moshkovitz. Techniques developed in the process may be useful for replacing planted clique hardness with ETH-hardness in other applications.

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M3 - Conference contribution

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 970

EP - 982

BT - Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015

PB - Association for Computing Machinery

T2 - 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015

Y2 - 4 January 2015 through 6 January 2015

ER -