Approximating the best Nash Equilibrium in no(log n)-time breaks the Exponential Time Hypothesis

Braverman, Mark; Kun-Ko, Young; Weinstein, Omri

doi:10.1137/1.9781611973730.66

Cited by 48 publications

(45 citation statements)

References 20 publications

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“…Our result bypasses this hardness result, and obtains a PTAS for a problem for which noise-stability does not hold. Finally, we show that a PTAS for the maximum-prior problem yields a PTAS for the dual signaling problem, and we rule out the latter via a simple, clean reduction from the bestNash problem and the recent result of Braverman et al [BKW15]. This result also strengthens the hardness result from [CCD + 15] mentioned above, by showing that in the absence of noise-stability, a QPTAS is the best-possible approximation for the mixture selection problem, assuming the ETH.…”

Section: Introductionsupporting

confidence: 54%

Hardness Results for Signaling in Bayesian Zero-Sum and Network Routing Games

Bhaskar

Cheng

et al. 2016

Proceedings of the 2016 ACM Conference on Economics and Computation

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We study the optimization problem faced by a perfectly informed principal in a Bayesian game, who reveals information to the players about the state of nature to obtain a desirable equilibrium. This signaling problem is the natural design question motivated by uncertainty in games and has attracted much recent attention. We present new hardness results for signaling problems in (a) Bayesian two-player zero-sum games, and (b) Bayesian network routing games.For Bayesian zero-sum games, when the principal seeks to maximize the equilibrium utility of a player, we show that it is NP-hard to obtain an additive FPTAS. Our hardness proof exploits duality and the equivalence of separation and optimization in a novel way. Further, we rule out an additive PTAS assuming planted clique hardness, which states that no polynomial time algorithm can recover a planted clique from an Erdős-Rényi random graph. Complementing these, we obtain a PTAS for a structured class of zero-sum games (where obtaining an FPTAS is still NP-hard) when the payoff matrices obey a Lipschitz condition. Previous results ruled out an FPTAS assuming planted-clique hardness, and a PTAS only for implicit games with quasi-polynomial-size strategy sets.For Bayesian network routing games, wherein the principal seeks to minimize the average latency of the Nash flow, we show that it is NP-hard to obtain a (multiplicative) 4 3 − ǫ -approximation, even for linear latency functions. This is the optimal inapproximability result for linear latencies, since we show that full revelation achieves a 4 3 -approximation for linear latencies.

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Section: Introductionsupporting

confidence: 54%

Hardness Results for Signaling in Bayesian Zero-Sum and Network Routing Games

Bhaskar

Cheng

et al. 2016

Proceedings of the 2016 ACM Conference on Economics and Computation

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“…Nash equilibria is known to be computationally hard [12,14], and in light of these findings, a considerable effort has been directed towards understanding the complexity of approximate Nash equilibrium. Results in this direction include both upper bounds [25,22,15,21,16,23,18,7,33,34,3,2] and lower bounds [20,13,9]. In particular, it is known that for a general bimatrix game an approximate Nash equilibrium can be computed in quasi-polynomial time [25].…”

Section: Related Workmentioning

confidence: 99%

Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Caratheodory's Theorem

Barman

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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We present algorithmic applications of an approximate version of Carathéodory's theorem. The theorem states that given a set of vectors X in R d , for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with n × n payoff matrices A, B, if the number of non-zero entries in any column of A + B is at most s then an ε-Nash equilibrium of the game can be computed in time n O( log s ε 2 ) . This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games-since s can be at most n-the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003).The approximate Carathéodory's theorem also leads to an additive approximation algorithm for the normalized densest k-subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a subgraph with exactly k vertices with normalized density within ε (in the additive sense) of the optimal in time n O( log d ε 2 ) . Additionally, we show that a similar approximation result can be achieved for the problem of finding a k × k-bipartite subgraph of maximum normalized density.

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“…In Section 5 we will show that the problem of approximating the best social welfare achievable by an approximate Nash equilibrium in a two-player normal form game can be written down as a constrained ǫ-ETR formula where α, γ, d, and m are constant. It has been shown that, assuming the exponential time hypothesis, this problem cannot be solved faster than quasi-polynomial time [11,19], so this also implies that constrained ǫ-ETR where α, γ, d, and m are constant cannot be solved faster than quasi-polynomial time unless the exponential time hypothesis is false.…”

Section: The Existential Theory Of the Realsmentioning

confidence: 99%

Approximating the Existential Theory of the Reals

Deligkas

Fearnley

Melissourgos

et al. 2018

Web and Internet Economics

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The existential theory of the reals (ETR) consists of existentially quantified boolean formulas over equalities and inequalities of real-valued polynomials. We propose the approximate existential theory of the reals (ǫ-ETR), in which the constraints only need to be satisfied approximately. We first show that unconstrained ǫ-ETR = ETR, and then study the ǫ-ETR problem when the solution is constrained to lie in a given convex set. Our main theorem is a sampling theorem, similar to those that have been proved for approximate equilibria in normal form games. It states that if an ETR problem has an exact solution, then it has a k-uniform approximate solution, where k depends on various properties of the formula. A consequence of our theorem is that we obtain a quasi-polynomial time approximation scheme (QPTAS) for a fragment of constrained ǫ-ETR. We use our theorem to create several new PTAS and QPTAS algorithms for problems from a variety of fields.

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Approximating the best Nash Equilibrium in n^o^{(log n)}-time breaks the Exponential Time Hypothesis

Cited by 48 publications

References 20 publications

Hardness Results for Signaling in Bayesian Zero-Sum and Network Routing Games

Hardness Results for Signaling in Bayesian Zero-Sum and Network Routing Games

Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Caratheodory's Theorem

Approximating the Existential Theory of the Reals

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