### Abstract

In this paper, we consider a general class of cooperative multistage games with random time horizon and discuss the problem of implementing a cooperative solution. It is known that in many cases a cooperative solution can be time-inconsistent and hence not realizable. To solve this problem, the imputation distribution procedure was proposed. However, the computed payment distribution scheme may result in negative payments which are not feasible. In this case, one has to carry out a regularization procedure as described in the paper. We describe a general regularization scheme and apply it both to the core and to the Shapley value. It is shown that for the mentioned two cases the regularization can be carried out in two alternative ways thus providing a basis for developing efficient numerical schemes. For the Shapley value the regularization procedure was elaborated and described in the form of an algorithm. The obtained results are illustrated with two numerical examples.

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

Pages (from-to) | 40-55 |

Number of pages | 16 |

Journal | Discrete Applied Mathematics |

Volume | 255 |

Early online date | 2018 |

DOIs | |

Publication status | Published - 28 Feb 2019 |

### Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

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**On the regularization of a cooperative solution in a multistage game with random time horizon.** / Gromova, E. V.; Plekhanova, T. M.

Research output

TY - JOUR

T1 - On the regularization of a cooperative solution in a multistage game with random time horizon

AU - Gromova, E. V.

AU - Plekhanova, T. M.

PY - 2019/2/28

Y1 - 2019/2/28

N2 - In this paper, we consider a general class of cooperative multistage games with random time horizon and discuss the problem of implementing a cooperative solution. It is known that in many cases a cooperative solution can be time-inconsistent and hence not realizable. To solve this problem, the imputation distribution procedure was proposed. However, the computed payment distribution scheme may result in negative payments which are not feasible. In this case, one has to carry out a regularization procedure as described in the paper. We describe a general regularization scheme and apply it both to the core and to the Shapley value. It is shown that for the mentioned two cases the regularization can be carried out in two alternative ways thus providing a basis for developing efficient numerical schemes. For the Shapley value the regularization procedure was elaborated and described in the form of an algorithm. The obtained results are illustrated with two numerical examples.

AB - In this paper, we consider a general class of cooperative multistage games with random time horizon and discuss the problem of implementing a cooperative solution. It is known that in many cases a cooperative solution can be time-inconsistent and hence not realizable. To solve this problem, the imputation distribution procedure was proposed. However, the computed payment distribution scheme may result in negative payments which are not feasible. In this case, one has to carry out a regularization procedure as described in the paper. We describe a general regularization scheme and apply it both to the core and to the Shapley value. It is shown that for the mentioned two cases the regularization can be carried out in two alternative ways thus providing a basis for developing efficient numerical schemes. For the Shapley value the regularization procedure was elaborated and described in the form of an algorithm. The obtained results are illustrated with two numerical examples.

KW - Game theory

KW - Multistage games

KW - Dynamic games

KW - Decision making under uncertainty

KW - Random duration

KW - Cooperation

KW - CONSISTENT SHAPLEY VALUE

KW - DIFFERENTIAL-GAMES

KW - SUSTAINABILITY

UR - http://www.scopus.com/inward/record.url?scp=85053737087&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/regularization-cooperative-solution-multistage-game-random-time-horizon

U2 - 10.1016/j.dam.2018.08.008

DO - 10.1016/j.dam.2018.08.008

M3 - статья

VL - 255

SP - 40

EP - 55

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -