On the Complexity of Approximating a Nash Equilibrium

Daskalakis, Constantinos

doi:10.1137/1.9781611973082.117

Cited by 16 publications

(23 citation statements)

References 22 publications

(58 reference statements)

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“…The -Gcircuit problem has already proven useful in several works in recent years (e.g [CDT09,Das13,CPY13,OPR14]). We believe that Theorem 2 will lead to stronger hardness results in many applications in algorithmic game theory and economics.…”

Section: Theorem 2 (Generalized Circuit)mentioning

confidence: 99%

Inapproximability of Nash Equilibrium

Rubinstein

2015

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

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We prove that finding an -approximate Nash equilibrium is PPADcomplete for constant and a particularly simple class of games: polymatrix, degree 3 graphical games, in which each player has only two actions.As corollaries, we also prove similar inapproximability results for Bayesian Nash equilibrium in a two-player incomplete information game with a constant number of actions, for relative -Well Supported Nash Equilibrium in a two-player game, for market equilibrium in a non-monotone market, for the generalized circuit problem defined by Chen et al [CDT09], and for approximate competitive equilibrium from equal incomes with indivisible goods.

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Section: Theorem 2 (Generalized Circuit)mentioning

confidence: 99%

Inapproximability of Nash Equilibrium

Rubinstein

2015

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

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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 practice, ISPs normally do not disclose private information, e.g., available resources and routing strategies, and therefore, this "common knowledge" assumption might not be valid. Moreover, the strategy space of the game increases exponentially with the system scale, and solving Nash equilibrium has been shown to be computationally expensive [4], [7] even under two players. This makes the Nash equilibrium computationally intractable for large-scale networks like the Internet.…”

Section: A Second-stage Simultaneous-move Game Of Cpsmentioning

confidence: 99%

Regulating Monopolistic ISPS without Neutrality

Tang

Richard

2014

2014 IEEE 22nd International Conference on Network Protocols

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Net neutrality has been heavily debated as a potential Internet regulation. Advocates have expressed concerns about the pricing power of ISPs, which might be used to discriminate Content Providers (CPs), and consequently destroy innovations at the edge of the Internet and hurt the user welfare. However, without service differentiation, ISPs do not have incentives to expand infrastructure capacities and provide quality of services, which will eventually impair the future Internet.Although competition among ISPs would alleviate the problem and reduce the need for regulations, the problem is more severe in monopolistic markets. We study the service differentiation offered by a monopolistic ISP and find that its profit-optimal strategy makes an ordinary service "damaged good", which hurts the welfare of CPs. Instead of imposing net neutrality regulations, we propose a flexible and lenient policy framework that generalizes net neutrality regulations. We find that a stringent regulation is needed when 1) the ISP's capacity is abundant, 2) the profit distribution of CPs is concentrated, or 3) the utility of CPs and their users are not positively correlated. We believe that by allowing the ISPs to differentiate services under a welldesigned policy constraint, the utility of the Internet ecosystem could be greatly improved.

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On the Complexity of Approximating a Nash Equilibrium

Cited by 16 publications

References 22 publications

Inapproximability of Nash Equilibrium

Inapproximability of Nash Equilibrium

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

Regulating Monopolistic ISPS without Neutrality

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