A Note on Repeated Games with Vanishing Actions

    Research output: Contribution to journalArticleAcademicpeer-review

    2 Citations (Scopus)

    Abstract

    A two-person general-sum repeated game with vanishing actions is an infinitely repeated game where the players face the following restrictions. Each action must be used by player k ¿ {1,2} at least once in every rk ¿ ¿ consecutive stages, otherwise the action vanishes for the remaining play. We assume that the players wish to maximize their limiting average rewards over the entire time-horizon.A strategy-pair is jointly convergent if for each action pair a number exists to which the relative frequency with which this action pair is chosen, converges with probability one. A pair of feasible rewards is called individually rational if each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does given r1, r2 and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which are endogenous to the play.We prove that all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Furthermore, each convex combination of individually-rational jointly-convergent pure-strategy rewards, can be supported by an equilibrium for m × n-games provided r1 > m ¿ 2, r2 > n ¿ 2. Keywords: Stochastic games; vanishing actions; limiting Average rewards; endogenous threats
    Original languageUndefined
    Pages (from-to)107-115
    Number of pages8
    JournalInternational game theory review
    Volume7
    Issue number1
    Publication statusPublished - 2005

    Keywords

    • METIS-225221

    Cite this

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    title = "A Note on Repeated Games with Vanishing Actions",
    abstract = "A two-person general-sum repeated game with vanishing actions is an infinitely repeated game where the players face the following restrictions. Each action must be used by player k ¿ {1,2} at least once in every rk ¿ ¿ consecutive stages, otherwise the action vanishes for the remaining play. We assume that the players wish to maximize their limiting average rewards over the entire time-horizon.A strategy-pair is jointly convergent if for each action pair a number exists to which the relative frequency with which this action pair is chosen, converges with probability one. A pair of feasible rewards is called individually rational if each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does given r1, r2 and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which are endogenous to the play.We prove that all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Furthermore, each convex combination of individually-rational jointly-convergent pure-strategy rewards, can be supported by an equilibrium for m × n-games provided r1 > m ¿ 2, r2 > n ¿ 2. Keywords: Stochastic games; vanishing actions; limiting Average rewards; endogenous threats",
    keywords = "METIS-225221",
    author = "Joosten, {Reinoud A.M.G.}",
    year = "2005",
    language = "Undefined",
    volume = "7",
    pages = "107--115",
    journal = "International game theory review",
    issn = "0219-1989",
    publisher = "World Scientific Publishing Co. Pte Ltd",
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    }

    A Note on Repeated Games with Vanishing Actions. / Joosten, Reinoud A.M.G.

    In: International game theory review, Vol. 7, No. 1, 2005, p. 107-115.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - A Note on Repeated Games with Vanishing Actions

    AU - Joosten, Reinoud A.M.G.

    PY - 2005

    Y1 - 2005

    N2 - A two-person general-sum repeated game with vanishing actions is an infinitely repeated game where the players face the following restrictions. Each action must be used by player k ¿ {1,2} at least once in every rk ¿ ¿ consecutive stages, otherwise the action vanishes for the remaining play. We assume that the players wish to maximize their limiting average rewards over the entire time-horizon.A strategy-pair is jointly convergent if for each action pair a number exists to which the relative frequency with which this action pair is chosen, converges with probability one. A pair of feasible rewards is called individually rational if each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does given r1, r2 and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which are endogenous to the play.We prove that all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Furthermore, each convex combination of individually-rational jointly-convergent pure-strategy rewards, can be supported by an equilibrium for m × n-games provided r1 > m ¿ 2, r2 > n ¿ 2. Keywords: Stochastic games; vanishing actions; limiting Average rewards; endogenous threats

    AB - A two-person general-sum repeated game with vanishing actions is an infinitely repeated game where the players face the following restrictions. Each action must be used by player k ¿ {1,2} at least once in every rk ¿ ¿ consecutive stages, otherwise the action vanishes for the remaining play. We assume that the players wish to maximize their limiting average rewards over the entire time-horizon.A strategy-pair is jointly convergent if for each action pair a number exists to which the relative frequency with which this action pair is chosen, converges with probability one. A pair of feasible rewards is called individually rational if each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does given r1, r2 and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which are endogenous to the play.We prove that all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Furthermore, each convex combination of individually-rational jointly-convergent pure-strategy rewards, can be supported by an equilibrium for m × n-games provided r1 > m ¿ 2, r2 > n ¿ 2. Keywords: Stochastic games; vanishing actions; limiting Average rewards; endogenous threats

    KW - METIS-225221

    M3 - Article

    VL - 7

    SP - 107

    EP - 115

    JO - International game theory review

    JF - International game theory review

    SN - 0219-1989

    IS - 1

    ER -