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From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory
2018
Entropy
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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
2016
Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science - ITCS '16
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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
2009
Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference
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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]
2014
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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
2018
Entropy
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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
2019
Scientific Reports
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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]
2019
arXiv
pre-print
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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
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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]
2016
arXiv
pre-print
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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]
2015
arXiv
pre-print
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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
2010
Mathematics of Operations Research
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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
2005
SIAM Journal of Control and Optimization
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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
2016
Automation and remote control
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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]
2018
arXiv
pre-print
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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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