Settling the complexity of computing approximate two-player Nash equilibria

Rubinstein, Aviad

doi:10.1145/3055589.3055596

Cited by 4 publications

(5 citation statements)

References 49 publications

(65 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For two-player games, Lipton, Mehta, and Markakis gave a quasi-polynomial time algorithm to find a constant approximate Nash equilibrium Lipton et al [2003]. Recently, Rubinstein Rubinstein [2017] showed this to be the best possible assuming exponential time hypothesis for PPAD, and Kothari and Mehta Kothari and Mehta [2018] showed a matching unconditional hardness under the powerful algorithmic framework of Sum-of-Squares with oblivious rounding and enumeration. These results are complemented by communication Babichenko and Rubinstein [2017], Göös and Rubinstein [2018] and query complexity lower bounds Babichenko [2016], Goldberg and Roth [2016], Fearnley and Savani [2016].…”

Section: Related Workmentioning

confidence: 99%

“…Although it is well accepted that PPAD and PLS are unlikely to be in P Beame et al [1998], Bitansky et al [2015], Rubinstein [2017], problems in these classes admit respectively path-following style complementary pivot algorithms Lemke and Howson [1964], Govindan and Wilson [2003], Adsul et al [2011], and local-search-type algorithms Johnson et al [1988]. The natural local-search algorithms for PLS problems have been observed to be empirically fast Johnson et al [1988], Codenotti et al [2008], Deligkas et al [2016a].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Smoothed Efficient Algorithms and Reductions for Network Coordination Games

Boodaghians¹,

Kulkarni²,

Mehta³

2018

Preprint

View full text Add to dashboard Cite

Worst-case hardness results for most equilibrium computation problems have raised the need for beyond-worst-case analysis. To this end, we study the smoothed complexity of finding pure Nash equilibria in Network Coordination Games, a PLS-complete problem in the worst case. This is a potential game where the sequential-better-response algorithm is known to converge to a pure NE, albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial) smoothed complexity when the underlying game graph is a complete (resp. arbitrary) graph, and every player has constantly many strategies. We note that the complete graph case is reminiscent of perturbing all parameters, a common assumption in most known smoothed analysis results.Second, we define a notion of smoothness-preserving reduction among search problems, and obtain reductions from 2-strategy network coordination games to local-max-cut, and from kstrategy games (with arbitrary k) to local-max-cut up to two flips. The former together with the recent result of Bibak et al. [2018] gives an alternate O(n 8 )-time smoothed algorithm for the 2-strategy case. This notion of reduction allows for the extension of smoothed efficient algorithms from one problem to another.For the first set of results, we develop techniques to bound the probability that an (adversarial) better-response sequence makes slow improvements on the potential. Our approach combines and generalizes the local-max-cut approaches of Etscheid and Röglin [2017], Angel et al. [2017] to handle the multi-strategy case: it requires a careful definition of the matrix which captures the increase in potential, a tighter union bound on adversarial sequences, and balancing it with good enough rank bounds. We believe that the approach and notions developed herein could be of interest in addressing the smoothed complexity of other potential and/or congestion games.

show abstract

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Smoothed Efficient Algorithms and Reductions for Network Coordination Games

Boodaghians¹,

Kulkarni²,

Mehta³

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…This naturally leads to the consideration of ϵ-approximate Nash equilibria (ϵ-NE), where the players are permitted to gain at most ϵ by deviation. In view of computational complexity, a similar negative result is given by Rubinstein [46]. It shows that, under a certain moderate assumption, computing ϵ-NE cannot be polynomial time when ϵ < ϵ * , where ϵ * is some constant.…”

Section: Bound Computed By Our Programmentioning

confidence: 64%

“…A process of lower bound and upper bound results is presented in Figure 1. 0 1 ϵ * ?, [46] 1/3 + δ, [20] 0.3393 + δ, [48] 0.36, [4] 0.38, [4,13] 0.38 + δ, [16] 1/2, [17] 3/4, [32] Figure 1: The process on approximation NE in bimatrix games. The blue points are the upper bound results while the orange point is the lower bound result.…”

Section: Bound Computed By Our Programmentioning

confidence: 99%

The Transformation of China's Soes

Deng¹

2015

Peking University Law Journal

View full text Add to dashboard Cite

AI in Math deals with mathematics in a constructive manner so that reasoning becomes automated, less laborious, and less error-prone. For algorithms, the question becomes how to automate analyses for specific problems. For the first time, this work provides an automatic method for approximation analysis on a well-studied problem in theoretical computer science: computing approximate Nash equilibria in two-player games. We observe that such algorithms can be reformulated into a search-and-mix paradigm, which involves a search phase followed by a mixing phase. By doing so, we are able to fully automate the procedure of designing and analyzing the mixing phase. For example, we illustrate how to perform our method with a program to analyze the approximation bounds of all the algorithms in the literature. Same approximation bounds are computed without any hand-written proof. Our automatic method heavily relies on the LP-relaxation structure in approximate Nash equilibria. Since many approximation algorithms and online algorithms adopt the LP relaxation, our approach may be extended to automate the analysis of other algorithms.

show abstract

“…Even for large values of ε, despite considerable effort [20,19,33,23,14,26], polynomial-time algorithms for computing ε-approximate equilibria are known only for ε ≥ 0.3393 [33]. Recently, Rubinstein [32] showed that unless surprisingly fast algorithms exist for PPAD-complete problems (running in time 2Õ ( √ n) ), no general poly-time algorithm for computing ε-approximate equilibria, for a sufficiently small constant ε, exists. These results suggest a difficult computational landscape for equilibrium and approximate equilibrium computation on worst-case instances.…”

Section: Introductionmentioning

confidence: 99%