### Abstract

For arbitrary cycle-free directed graph games tree-type values are introduced axiomatically and their explicit formula representation is provided. These values may be considered as natural extensions of the tree and sink values as has been defined correspondingly for rooted and sink forest graph games. The main property for the tree value is that every player in the game receives the worth of this player together with his successors minus what these successors receive. It implies that every coalition of players consisting of one of the players with all his successors receives precisely its worth. Additionally their efficiency and stability are studied. Simple recursive algorithms to calculate the values are also provided. The application to the water distribution problem of a river with multiple sources, a delta and possibly islands is considered.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 25 |

Publication status | Published - Oct 2010 |

### Publication series

Name | Memorandum / Department of Applied Mathematics |
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Publisher | University of Twente, Department of Applied Mathematics |

No. | 1926 |

ISSN (Print) | 1874-4850 |

ISSN (Electronic) | 1874-4850 |

### Keywords

- MSC-91A43
- Cooperation structure
- Deletion link property
- METIS-277436
- IR-73838
- EWI-18640
- TU game
- Stability
- Efficiency
- MSC-91A12
- Myerson value

## Cite this

Khmelnitskaya, A. B., & Talman, D. (2010).

*Tree-type values for cycle-free directed graph games*. (Memorandum / Department of Applied Mathematics; No. 1926). Enschede: University of Twente, Department of Applied Mathematics.