TY - GEN

T1 - How hard is it to approximate the best Nash equilibrium?

AU - Hazan, Elad

AU - Krauthgamer, Robert

PY - 2009

Y1 - 2009

N2 - The quest for a PTAS for Nash equilibrium in a two-player game seeks to circumvent the PPAD-completeness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibrium maximizing a certain objective, such as the social welfare. This optimization problem was shown to be NP-hard by Gilboa and Zemel [Games and Economic Behavior 1989]. However, this NP-hardness is unlikely to extend to finding an approximate equilibrium, since the latter admits a quasi-polynomial time algorithm, as proved by Lipton, Markakis and Mehta [Proc. of 4th EC, 2003]. We show that this optimization problem, namely, finding in a two-player game an approximate equilibrium achieving large social welfare is unlikely to have a polynomial time algorithm. One interpretation of our results is that the quest for a PTAS for Nash equilibrium should not extend to a PTAS for finding the best Nash equilibrium, which stands in contrast to certain algorithmic techniques used so far (e.g. sampling and enumeration). Technically, our result is a reduction from a 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 k = O(log n). For comparison, the currently known algorithms due to Alon, Krivelevich and Sudakov [Random Struct. & Algorithms, 1998], and Krauthgamer and Feige [Random Struct. & Algorithms, 2000], are effective for a much larger clique size k = Ω(√n).

AB - The quest for a PTAS for Nash equilibrium in a two-player game seeks to circumvent the PPAD-completeness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibrium maximizing a certain objective, such as the social welfare. This optimization problem was shown to be NP-hard by Gilboa and Zemel [Games and Economic Behavior 1989]. However, this NP-hardness is unlikely to extend to finding an approximate equilibrium, since the latter admits a quasi-polynomial time algorithm, as proved by Lipton, Markakis and Mehta [Proc. of 4th EC, 2003]. We show that this optimization problem, namely, finding in a two-player game an approximate equilibrium achieving large social welfare is unlikely to have a polynomial time algorithm. One interpretation of our results is that the quest for a PTAS for Nash equilibrium should not extend to a PTAS for finding the best Nash equilibrium, which stands in contrast to certain algorithmic techniques used so far (e.g. sampling and enumeration). Technically, our result is a reduction from a 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 k = O(log n). For comparison, the currently known algorithms due to Alon, Krivelevich and Sudakov [Random Struct. & Algorithms, 1998], and Krauthgamer and Feige [Random Struct. & Algorithms, 2000], are effective for a much larger clique size k = Ω(√n).

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U2 - 10.1137/1.9781611973068.79

DO - 10.1137/1.9781611973068.79

M3 - Conference contribution

AN - SCOPUS:70349144464

SN - 9780898716801

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

SP - 720

EP - 727

BT - Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms

PB - Association for Computing Machinery

T2 - 20th Annual ACM-SIAM Symposium on Discrete Algorithms

Y2 - 4 January 2009 through 6 January 2009

ER -