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Smoothed Complexity of 2-player Nash Equilibria [article]

Shant Boodaghians and Joshua Brakensiek and Samuel B. Hopkins and Aviad Rubinstein
2020 arXiv   pre-print
Our results therefore separate smoothed complexity and hardness of approximation for Nash equilibria in two-player games.  ...  In contrast to prior work proving PPAD-hardness after smoothing by noise of magnitude 1/poly(n) [CDT09], our smoothed complexity result is not proved via hardness of approximation for Nash equilibria.  ...  In this work, we resolve the complexity of two-player Nash equilibrium with constant-magnitude smoothing, proving that it is PPAD-complete (under randomized reductions).  ... 
arXiv:2007.10857v1 fatcat:i7tgbussgrcwxk3rva6vzwhrwm

Query complexity of approximate nash equilibria

Yakov Babichenko
2014 Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14  
Nash Equilibria We prove that the query complexity of end-of-a-simple-path is exp(n).Yakov BabichenkoQuery Complexity of Approximate Nash Equilibria We prove that the query complexity of end-of-a-simple-path  ...  Yakov Babichenko Query Complexity of Approximate Nash Equilibria The Main Theorem The well supported Nash problem, WSN(n, m, ε): INPUT: n-players, m-actions game.  ... 
doi:10.1145/2591796.2591829 dblp:conf/stoc/Babichenko14 fatcat:ijnxd5ep35chnbz7tfhntsezji

Query Complexity of Approximate Nash Equilibria

Yakov Babichenko
2016 Journal of the ACM  
Nash Equilibria We prove that the query complexity of end-of-a-simple-path is exp(n).Yakov BabichenkoQuery Complexity of Approximate Nash Equilibria We prove that the query complexity of end-of-a-simple-path  ...  Yakov Babichenko Query Complexity of Approximate Nash Equilibria The Main Theorem The well supported Nash problem, WSN(n, m, ε): INPUT: n-players, m-actions game.  ... 
doi:10.1145/2908734 fatcat:s2rshlll5bgx3ivkic6f7q2v6y

The price of anarchy in games of incomplete information

Tim Roughgarden
2012 ACM SIGecom Exchanges  
We outline a recently developed theory for bounding the inefficiency of equilibria in games of incomplete information, with applications to auctions and routing games.  ...  outcome of the game. (2) Invoke the Nash equilibrium hypothesis once per player, to derive that each player i's payoff in the Nash equilibrium a is at least as high as if it played a * i instead.  ...  equilibria in congestion games with incomplete information. (5) For Bayes-Nash equilibria in games with correlated player preferences, there is no general extension theorem for smooth games.  ... 
doi:10.1145/2325713.2325716 fatcat:2jdxa7z2i5ehxpylyorwhfwhyq

The price of anarchy in games of incomplete information

Tim Roughgarden
2012 Proceedings of the 13th ACM Conference on Electronic Commerce - EC '12  
We outline a recently developed theory for bounding the inefficiency of equilibria in games of incomplete information, with applications to auctions and routing games.  ...  outcome of the game. (2) Invoke the Nash equilibrium hypothesis once per player, to derive that each player i's payoff in the Nash equilibrium a is at least as high as if it played a * i instead.  ...  equilibria in congestion games with incomplete information. (5) For Bayes-Nash equilibria in games with correlated player preferences, there is no general extension theorem for smooth games.  ... 
doi:10.1145/2229012.2229078 dblp:conf/sigecom/Roughgarden12 fatcat:325pi75qrzfsthmvwh5em6jhqe

The Price of Anarchy in Auctions [article]

Tim Roughgarden, Vasilis Syrgkanis, Eva Tardos
2016 arXiv   pre-print
This survey outlines a general and modular theory for proving approximation guarantees for equilibria of auctions in complex settings.  ...  We highlight three user-friendly analytical tools: smoothness-type inequalities, which immediately yield approximation guarantees for many auction formats of interest in the special case of complete information  ...  Bayes-Nash equilibria are relatively well understood. 2. Bayes-Nash equilibria are fully efficient, with the item always allocated to the bidder with the highest valuation.  ... 
arXiv:1607.07684v1 fatcat:b5t3xgmk75bi7iymdoelblnnl4

The Price of Anarchy in Games of Incomplete Information

Tim Roughgarden
2015 ACM Transactions on Economics and Computation  
We outline a recently developed theory for bounding the inefficiency of equilibria in games of incomplete information, with applications to auctions and routing games.  ...  outcome of the game. (2) Invoke the Nash equilibrium hypothesis once per player, to derive that each player i's payoff in the Nash equilibrium a is at least as high as if it played a * i instead.  ...  equilibria in congestion games with incomplete information. (5) For Bayes-Nash equilibria in games with correlated player preferences, there is no general extension theorem for smooth games.  ... 
doi:10.1145/2737816 fatcat:jcqvxmvfizhgtmml3g2ve5b4sa

The Price of Anarchy in Auctions

Tim Roughgarden, Vasilis Syrgkanis, Eva Tardos
2017 The Journal of Artificial Intelligence Research  
This survey outlines a general and modular theory for proving approximation guarantees for equilibria of auctions in complex settings.  ...  We highlight three user-friendly analytical tools: smoothness-type inequalities, which immediately yield approximation guarantees for many auction formats of interest in the special case of complete information  ...  Bayes-Nash equilibria are relatively well understood. 2. Bayes-Nash equilibria are fully efficient, with the item always allocated to the bidder with the highest valuation.  ... 
doi:10.1613/jair.5272 fatcat:7gcmpshgivh2vj4xu2enqe3nue

On the Approximation and Smoothed Complexity of Leontief Market Equilibria [article]

Li-Sha Huang, Shang-Hua Teng
2006 arXiv   pre-print
We show that the problem of finding an \epsilon-approximate Nash equilibrium of an n by n two-person games can be reduced to the computation of an (\epsilon/n)^2-approximate market equilibrium of a Leontief  ...  PPAD is not in RP, that the smoothed complexity of the Scarf's general fixed-point approximation algorithm (when applying to solve the approximate Leontief market exchange problem) or of any algorithm  ...  To relate the complexity of finding an approximate Nash equilibrium of two-person games with the smoothed complexity of Leontief economies, we examine the equilibria of perturbations of the reduction presented  ... 
arXiv:cs/0602090v1 fatcat:3ghfwx53lnhilmsqv5rl7mqkha

Settling the Complexity of Computing Two-Player Nash Equilibria [article]

Xi Chen, Xiaotie Deng, Shang-Hua Teng
2007 arXiv   pre-print
This is the first of a series of results concerning the complexity of Nash equilibria.  ...  We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD Polynomial Parity Argument, Directed version) introduced by Papadimitriou  ...  We would like to thank everyone who asked about the smoothed complexity of the Lemke-Howson algorithm, especially John Reif for being the first player to ask us this question.  ... 
arXiv:0704.1678v1 fatcat:bv5kllzvurg7fjlt2gq7d6nolu

Local Nash Equilibria are Isolated, Strict Local Nash Equilibria in `Almost All' Zero-Sum Continuous Games [article]

Eric Mazumdar, Lillian Ratliff
2020 arXiv   pre-print
That is, there exists an open-dense subset of zero-sum games for which local Nash equilibria are non-degenerate differential Nash equilibria.  ...  We further show that differential Nash equilibria of zero-sum games are structurally stable.  ...  Proof: Consider a two-player, zero-sum game (f, −f ) for some generic sufficiently smooth f ∈ C r (X, R). Then, by Theorem 2, a local Nash equilibria x is a differential Nash equilibria.  ... 
arXiv:2002.01007v1 fatcat:spoeiiv4jvd5lezooq2wcpj32y

Settling the complexity of computing two-player Nash equilibria

Xi Chen, Xiaotie Deng, Shang-Hua Teng
2009 Journal of the ACM  
It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria.  ...  Our result, building upon the work of Daskalakis, Goldberg, and Papadimitriou on the complexity of four-player Nash equilibria [21], settles a long standing open problem in algorithmic game theory.  ...  The Smoothed Complexity of Nash Equilibria The proof of Theorem 1.4 is then the simplest part of the paper.  ... 
doi:10.1145/1516512.1516516 fatcat:4csizuhga5h4jnbkrickxw6d2y

On the complexity of constrained Nash equilibria in graphical games

Gianluigi Greco, Francesco Scarcello
2009 Theoretical Computer Science  
Based on this framework, the complexity of deciding the existence and of computing constrained equilibria is then investigated, in the light of evidencing how the intrinsic difficulty of these tasks is  ...  A widely accepted rational behavior for non-cooperative players is based on the notion of Nash equilibrium.  ...  on the complexity issues related to Nash equilibria by considering the various classes of games defined in Section 2.  ... 
doi:10.1016/j.tcs.2009.05.030 fatcat:e5fwu7ypxrgpnb7d3umyagi7zm

Query Complexity of Approximate Equilibria in Anonymous Games [chapter]

Paul W. Goldberg, Stefano Turchetta
2015 Lecture Notes in Computer Science  
We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers.  ...  The general topic we consider is query complexity, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium.  ...  Related Work In the last decade, there has been interest in the complexity of computing approximate Nash equilibria.  ... 
doi:10.1007/978-3-662-48995-6_26 fatcat:7jcyuggwo5bc5lyz2w2v3zc2ou

Intrinsic Robustness of the Price of Anarchy

Tim Roughgarden
2015 Journal of the ACM  
First, we identify a "canonical sufficient condition" for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument.  ...  This measure implicitly assumes that players successfully reach some Nash equilibrium.  ...  ) Nash equilibria [2, 8, 9] .  ... 
doi:10.1145/2806883 fatcat:lx6nlwhv5jd2vdry57jigwnql4
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