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From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory

Christos Papadimitriou, Georgios Piliouras
2018 Entropy  
The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent  ...  In this paper, we propose a new class of universal non-equilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash.  ...  Since the diffeomorphism maps zero measure sets to zero measure sets, all but a zero-measure set of points of S n are recurrent and the theorem follows.  ... 
doi:10.3390/e20100782 pmid:33265870 fatcat:jjwfwqxgcjb53bvn37cwgzspta

From Nash Equilibria to Chain Recurrent Sets

Christos Papadimitriou, Georgios Piliouras
2016 Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science - ITCS '16  
Nash equilibria are guaranteed to be fixed points of such dynamics; however, the system behavior is captured by a more general object that is known in dynamical systems theory as chain recurrent set.  ...  Here, we shift focus to universal non-equilibrium solution concepts that arise from an important theorem in the topology of dynamical systems that was unavailable to Nash.  ...  In game theoretic terms, every game is a "potential" game, if only we change our solution concept from equilibria to chain recurrent sets. Theorem 2.  ... 
doi:10.1145/2840728.2840757 dblp:conf/innovations/PapadimitriouP16 fatcat:qtriky5uureh3iisnn76tzyn4u

Distributed coverage games for mobile visual sensors (I): Reaching the set of Nash equilibria

Minghui Zhu, Sonia Martinez
2009 Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference  
The algorithm is proven to be convergent in probability to the set of (restricted) Nash equilibria from which none has incentive to unilaterally deviate.  ...  We formulate a coverage optimization problem for mobile visual sensor networks as a repeated multi-player game.  ...  This algorithm is shown to be convergent in probability to the set of (restricted) Nash equilibria from which no agent is willing to unilaterally deviate.  ... 
doi:10.1109/cdc.2009.5399545 dblp:conf/cdc/ZhuM09 fatcat:uacuslslkfddjo6nwsctjshfgu

Game-theoretical control with continuous action sets [article]

Steven Perkins and Panayotis Mertikopoulos and David S. Leslie
2014 arXiv   pre-print
with continuous action sets, and we propose an actor-critic reinforcement learning algorithm that provably converges to equilibrium in this class of problems.  ...  To do so, we extend the theory of finite-dimensional two-timescale stochastic approximation to an infinite-dimensional, Banach space setting, and we prove that the continuous dynamics of the process converge  ...  This allows results from the theory of learning in games to be applied directly, resulting in learning algorithms that converge to the set of equilibria -and hence system optima.  ... 
arXiv:1412.0543v1 fatcat:mvmwybqrh5cgpmsdkcardhct24

Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics [article]

Jason Milionis, Christos Papadimitriou, Georgios Piliouras, Kelly Spendlove
2022 arXiv   pre-print
We also prove a stronger result for ϵ-approximate Nash equilibria: there are games such that no game dynamics can converge (in an appropriate sense) to ϵ-Nash equilibria, and in fact the set of such games  ...  We apply this theory, and in particular the concepts of chain recurrence, attractors, and Conley index, to prove a general impossibility result: there exist games for which any dynamics is bound to have  ...  K.S. would like to thank Rob Vandervorst for numerous enlightening discussions regarding Conley theory.  ... 
arXiv:2203.14129v1 fatcat:v3uzmg4gpfdfre2geyok3kybee

Information Theory in Game Theory

MHR Khouzani, Pasquale Malacaria
2018 Entropy  
Information theory, as the mathematics of communication and storage of information, and game theory, as the mathematics of adversarial and cooperative strategic behaviour, are each successful fields of  ...  arising from an important theorem in the topology of dynamical systems: Chain recurrent sets.  ...  Starting from recent algorithmic results establishing that Nash equilibria are computationally equivalent to fixed points, the authors propose a new class of universal non-equilibrium solution concepts  ... 
doi:10.3390/e20110817 pmid:33266541 fatcat:mhgcfqfgkvbzhmapfppad6duby

α-Rank: Multi-Agent Evaluation by Evolution

Shayegan Omidshafiei, Christos Papadimitriou, Georgios Piliouras, Karl Tuyls, Mark Rowland, Jean-Baptiste Lespiau, Wojciech M. Czarnecki, Marc Lanctot, Julien Perolat, Remi Munos
2019 Scientific Reports  
In contrast to the Nash equilibrium, which is a static solution concept based solely on fixed points, MCCs are a dynamical solution concept based on the Markov chain formalism, Conley's Fundamental Theorem  ...  Current models are fundamentally limited in one or more of these dimensions, and are not guaranteed to converge to the desired game-theoretic solution concept (typically the Nash equilibrium). α-Rank automatically  ...  ), and Nash (which is a static solution concept that does not capture dynamic behavior).  ... 
doi:10.1038/s41598-019-45619-9 pmid:31289288 pmcid:PMC6617105 fatcat:25smi37gu5ey3lhzwfcgvda4wq

α-Rank: Multi-Agent Evaluation by Evolution [article]

Shayegan Omidshafiei, Christos Papadimitriou, Georgios Piliouras, Karl Tuyls, Mark Rowland, Jean-Baptiste Lespiau, Wojciech M. Czarnecki, Marc Lanctot, Julien Perolat, Remi Munos
2019 arXiv   pre-print
profiles, whereas computing a Nash equilibrium for a general-sum game is known to be intractable.  ...  In contrast to the Nash equilibrium, which is a static concept based on fixed points, MCCs are a dynamical solution concept based on the Markov chain formalism, Conley's Fundamental Theorem of Dynamical  ...  In game-theoretic terms, every game is a "potential" game, if only we change our solution concept from equilibria to chain recurrent sets.  ... 
arXiv:1903.01373v4 fatcat:p7qcpsvqsrctlhnocr5shaxfse

Page 1451 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
The purpose of the analysis is to classify the set of initial states into winning sets and draw sets for each player, and to prescribe strategies able to Game theory 2002b:9 1022 implement the kind of  ...  Under the condition that the ensuing Markov chain has a path from each state to a pure strategy Nash equilibrium, con- vergence to a pure strategy Nash equilibrium with probability one is established.  ... 

Jump-starting coordination in a stag hunt: Motivation, mechanisms, and their analysis [article]

Ioannis Avramopoulos
2016 arXiv   pre-print
Our analysis is based on the solution concepts of Nash equilibrium, dominance solvability, as well as a formalization of the notion of "incremental deployability," which is shown to be keenly relevant  ...  The stag hunt (or assurance game) is a simple game that has been used as a prototype of a variety of social coordination problems (ranging from the social contract to the adoption of technical standards  ...  I would, finally, like to thank an anonymous reviewer for simplifying my proof that stag hunts are weakly acyclic games.  ... 
arXiv:1601.03162v1 fatcat:vxnys2xxcjg33p5u7fp4osbnuq

Stochastic Coalitional Better-response Dynamics and Strong Nash Equilibrium [article]

Konstantin Avrachenkov
2015 arXiv   pre-print
Similar statement holds for strict strong Nash equilibrium.  ...  We consider coalition formation among players in an n-player finite strategic game over infinite horizon.  ...  Introduction Nash equilibrium is the most desirable solution concept in non-cooperative game theory.  ... 
arXiv:1506.03311v1 fatcat:otfwhkgtxnd73inosjudlecidm

Stochastic Approximations of Set-Valued Dynamical Systems: Convergence with Positive Probability to an Attractor

Mathieu Faure, Gregory Roth
2010 Mathematics of Operations Research  
We pursue the analogy by extending to this setting the result of convergence with positive probability to an attractor.  ...  Bena\"im et al. (2005) generalised this approach for stochastic approximation algorithms whose average behavior is related to a differential inclusion instead.  ...  Acknowledgements: the authors would like to thank Michel Benaïm for useful advices and discussions.  ... 
doi:10.1287/moor.1100.0455 fatcat:gh2ewr2xc5ccdn4d7p7cnlgjou

Individual Q-Learning in Normal Form Games

David S. Leslie, E. J. Collins
2005 SIAM Journal of Control and Optimization  
Playerdependent learning rates are then considered, and it is shown that this extension converges in some games for which many algorithms, including the basic algorithm initially considered, fail to converge  ...  We introduce a particular value-based learning algorithm, individual Q-learning, and use stochastic approximation to study the asymptotic behaviour, showing that strategies will converge to Nash distribution  ...  The authors thank three anonymous referees for helpful comments, and Prof. Josef Hofbauer and Dr. Andy Wright for helpful discussions in the course of the research leading to this paper.  ... 
doi:10.1137/s0363012903437976 fatcat:rmhpgsfdfjfoxejatrgatmhzgy

Stochastic coalitional better-response dynamics and stable equilibrium

K. Avrachenkov, V. V. Singh
2016 Automation and remote control  
We consider coalition formation among players in an n-player finite strategic game over infinite horizon.  ...  Similar statement holds for strict K-stable equilibrium. We apply the CBR dynamics to study the dynamic formation of the networks in the presence of mutations.  ...  Introduction Nash equilibrium is most desirable solution concept in non-cooperative game theory.  ... 
doi:10.1134/s0005117916120110 fatcat:5stef3tyhbhnzj3by63vr34rve

On incremental deployability [article]

Ioannis Avramopoulos
2018 arXiv   pre-print
For example, we show that incremental deployability is at least as general a concept as the Nash equilibrium (in that the latter can be derived from the former).  ...  To that end, we draw on an elementary concept in Internet systems engineering, namely, that of incremental deployability, which we study mathematically and computationally.  ...  Acknowledgments My account on YouTube has kept me company, through recommendations (for example, on music to listen to or on technical material to study) and through positive "clockwork orange" experiments  ... 
arXiv:1805.10115v1 fatcat:2ovf2na3jnbz3dwmrtiphdp6oy
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