### Abstract

In our previous papers (2002, 2017), we derived conditions for the existence of a strong Nash equilibrium in multistage nonzero-sum games under additional constraints on the possible deviations of coalitions from their agreed-upon strategies. These constraints allowed only one-time simultaneous deviations of all the players in a coalition. However, it is clear that in real-world problems the deviations of different members of a coalition may occur at different times (at different stages of the game), which makes the punishment strategy approach proposed by the authors earlier inapplicable in the general case. The fundamental difficulty is that in the general case the players who must punish the deviating coalition know neither the members of this coalition nor the times when each player performs the deviation. In this paper, we propose a new punishment strategy, which does not require the full information about the deviating coalition but uses only the fact of deviation of at least one player of the coalition. Of course, this punishment strategy can be realized only under some additional constraints on stage games. Under these additional constraints, it was proved that the punishment of the deviating coalition can be effectively realized. As a result, the existence of a strong Nash equilibrium in the game was established.

Original language | English |
---|---|

Pages (from-to) | S140-S149 |

Number of pages | 10 |

Journal | Proceedings of the Steklov Institute of Mathematics |

Volume | 305 |

Issue number | Suppl 1 |

Early online date | 18 Oct 2019 |

DOIs | |

Publication status | Published - 2019 |

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### Scopus subject areas

- Mathematics (miscellaneous)

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*Proceedings of the Steklov Institute of Mathematics*, vol. 305, no. Suppl 1, pp. S140-S149. https://doi.org/10.1134/S008154381904014X

**Construction of a Strong Nash Equilibrium in a Class of Infinite Nonzero-Sum Games.** / Petrosyan, L. A.; Pankratov, Ya.B.

Research output

TY - JOUR

T1 - Construction of a Strong Nash Equilibrium in a Class of Infinite Nonzero-Sum Games

AU - Petrosyan, L. A.

AU - Pankratov, Ya.B.

N1 - Petrosyan, L.A. & Pankratova, Y.B. Proc. Steklov Inst. Math. (2019) 305(Suppl 1): S140. https://doi.org/10.1134/S008154381904014X

PY - 2019

Y1 - 2019

N2 - In our previous papers (2002, 2017), we derived conditions for the existence of a strong Nash equilibrium in multistage nonzero-sum games under additional constraints on the possible deviations of coalitions from their agreed-upon strategies. These constraints allowed only one-time simultaneous deviations of all the players in a coalition. However, it is clear that in real-world problems the deviations of different members of a coalition may occur at different times (at different stages of the game), which makes the punishment strategy approach proposed by the authors earlier inapplicable in the general case. The fundamental difficulty is that in the general case the players who must punish the deviating coalition know neither the members of this coalition nor the times when each player performs the deviation. In this paper, we propose a new punishment strategy, which does not require the full information about the deviating coalition but uses only the fact of deviation of at least one player of the coalition. Of course, this punishment strategy can be realized only under some additional constraints on stage games. Under these additional constraints, it was proved that the punishment of the deviating coalition can be effectively realized. As a result, the existence of a strong Nash equilibrium in the game was established.

AB - In our previous papers (2002, 2017), we derived conditions for the existence of a strong Nash equilibrium in multistage nonzero-sum games under additional constraints on the possible deviations of coalitions from their agreed-upon strategies. These constraints allowed only one-time simultaneous deviations of all the players in a coalition. However, it is clear that in real-world problems the deviations of different members of a coalition may occur at different times (at different stages of the game), which makes the punishment strategy approach proposed by the authors earlier inapplicable in the general case. The fundamental difficulty is that in the general case the players who must punish the deviating coalition know neither the members of this coalition nor the times when each player performs the deviation. In this paper, we propose a new punishment strategy, which does not require the full information about the deviating coalition but uses only the fact of deviation of at least one player of the coalition. Of course, this punishment strategy can be realized only under some additional constraints on stage games. Under these additional constraints, it was proved that the punishment of the deviating coalition can be effectively realized. As a result, the existence of a strong Nash equilibrium in the game was established.

KW - characteristic function

KW - core

KW - imputation

KW - multistage game

KW - repeated game

KW - strong Nash equilibrium

UR - http://www.scopus.com/inward/record.url?scp=85073570016&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/construction-strong-nash-equilibrium-class-infinite-nonzerosum-games

U2 - 10.1134/S008154381904014X

DO - 10.1134/S008154381904014X

M3 - статья

AN - SCOPUS:85073570016

VL - 305

SP - S140-S149

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

SN - 0081-5438

IS - Suppl 1

ER -