CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES

We introduce the concept of a Lyapunov game as a subclass of strictly dominated games and potential games. The advantage of this approach is that every ergodic system (repeated game) can be represented by a Lyapunov-like function. A direct acyclic graph is associated with a game. The graph structure...

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Autor: Clepner Kerik, Julio Bernardo
Formato: artículo
Estado:Versión publicada
Fecha de publicación:2011
País:México
Recursos:Instituto Politécnico Nacional
Repositorio:Repositorio Digital del IPN
OAI Identifier:oai:www.repositoriodigital.ipn.mx:123456789/19985
Acesso em linha:http://www.repositoriodigital.ipn.mx/handle/123456789/19985
Access Level:acceso abierto
Palavra-chave:Lyapunov game
Lyapunov equilibrium point
forward decision process
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spelling CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMESClepner Kerik, Julio BernardoLyapunov gameLyapunov equilibrium pointforward decision processWe introduce the concept of a Lyapunov game as a subclass of strictly dominated games and potential games. The advantage of this approach is that every ergodic system (repeated game) can be represented by a Lyapunov-like function. A direct acyclic graph is associated with a game. The graph structure represents the dependencies existing between the strategy profiles. By definition, a Lyapunov-like function monotonically decreases and converges to a single Lyapunov equilibrium point identified by the sink of the game graph. It is important to note that in previous works this convergence has not been guaranteed even if the Nash equilibrium point exists. The best reply dynamics result in a natural implementation of the behavior of a Lyapunov-like function. Therefore, a Lyapunov game has also the benefit that it is common knowledge of the players that only best replies are chosen. By the natural evolution of a Lyapunov-like function, no matter what, a strategy played once is not played again. As a construction example, we show that, for repeated games with bounded nonnegative cost functions within the class of differentiable vector functions whose derivatives satisfy the Lipschitz condition, a complex vector-function can be built, where each component is a function of the corresponding cost value and satisfies the condition of the Lyapunov-like function. The resulting vector Lyapunov-like function is a monotonic function which can only decrease over time. Then, a repeated game can be represented by a one-shot game. The functionality of the suggested method is successfully demonstrated by a simulated experiment.Instituto Politécnico Nacional.CIECASFuncionespdfInternational Journal of applied mathematics and computer science, Vol. 21, No. 22014-09-10T18:44:10Z2014-09-10T18:44:10Z2011info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article10.2478/v10006-011-0026-xhttp://www.repositoriodigital.ipn.mx/handle/123456789/19985reponame:Repositorio Digital del IPNinstname:Instituto Politécnico Nacionalinstacron:IPNeninfo:eu-repo/semantics/openAccessoai:www.repositoriodigital.ipn.mx:123456789/199852026-02-18T16:47:37Z
dc.title.none.fl_str_mv CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES
title CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES
spellingShingle CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES
Clepner Kerik, Julio Bernardo
Lyapunov game
Lyapunov equilibrium point
forward decision process
title_short CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES
title_full CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES
title_fullStr CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES
title_full_unstemmed CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES
title_sort CONVERGENCE METHOD, PROPERTIES AND COMPUTATIONAL COMPLEXITY FOR LYAPUNOV GAMES
dc.creator.none.fl_str_mv Clepner Kerik, Julio Bernardo
author Clepner Kerik, Julio Bernardo
author_facet Clepner Kerik, Julio Bernardo
author_role author
dc.subject.none.fl_str_mv Lyapunov game
Lyapunov equilibrium point
forward decision process
topic Lyapunov game
Lyapunov equilibrium point
forward decision process
description We introduce the concept of a Lyapunov game as a subclass of strictly dominated games and potential games. The advantage of this approach is that every ergodic system (repeated game) can be represented by a Lyapunov-like function. A direct acyclic graph is associated with a game. The graph structure represents the dependencies existing between the strategy profiles. By definition, a Lyapunov-like function monotonically decreases and converges to a single Lyapunov equilibrium point identified by the sink of the game graph. It is important to note that in previous works this convergence has not been guaranteed even if the Nash equilibrium point exists. The best reply dynamics result in a natural implementation of the behavior of a Lyapunov-like function. Therefore, a Lyapunov game has also the benefit that it is common knowledge of the players that only best replies are chosen. By the natural evolution of a Lyapunov-like function, no matter what, a strategy played once is not played again. As a construction example, we show that, for repeated games with bounded nonnegative cost functions within the class of differentiable vector functions whose derivatives satisfy the Lipschitz condition, a complex vector-function can be built, where each component is a function of the corresponding cost value and satisfies the condition of the Lyapunov-like function. The resulting vector Lyapunov-like function is a monotonic function which can only decrease over time. Then, a repeated game can be represented by a one-shot game. The functionality of the suggested method is successfully demonstrated by a simulated experiment.
publishDate 2011
dc.date.none.fl_str_mv 2011
2014-09-10T18:44:10Z
2014-09-10T18:44:10Z
dc.type.none.fl_str_mv info:eu-repo/semantics/publishedVersion
info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv 10.2478/v10006-011-0026-x
http://www.repositoriodigital.ipn.mx/handle/123456789/19985
identifier_str_mv 10.2478/v10006-011-0026-x
url http://www.repositoriodigital.ipn.mx/handle/123456789/19985
dc.language.none.fl_str_mv en
language_invalid_str_mv en
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv International Journal of applied mathematics and computer science, Vol. 21, No. 2
publisher.none.fl_str_mv International Journal of applied mathematics and computer science, Vol. 21, No. 2
dc.source.none.fl_str_mv reponame:Repositorio Digital del IPN
instname:Instituto Politécnico Nacional
instacron:IPN
instname_str Instituto Politécnico Nacional
instacron_str IPN
institution IPN
reponame_str Repositorio Digital del IPN
collection Repositorio Digital del IPN
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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