### Abstract

Original language | Undefined |
---|---|

Pages (from-to) | 107-115 |

Number of pages | 8 |

Journal | International game theory review |

Volume | 7 |

Issue number | 1 |

Publication status | Published - 2005 |

### Keywords

- METIS-225221

### Cite this

*International game theory review*,

*7*(1), 107-115.

}

*International game theory review*, vol. 7, no. 1, pp. 107-115.

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

Research output: Contribution to journal › Article › Academic › peer-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 -