### Abstract

We introduce the two-step average tree value for transferable utility games with

restricted cooperation represented by undirected communication graphs or hypergraphs. The solution can be considered as an alternative for both the average tree solution for graph games and the average tree value for hypergraph games.

Instead of averaging players' marginal contributions corresponding to all admis-sible rooted spanning trees of the underlying (hyper)graph, which determines

the average tree solution or value, we consider a two-step averaging procedure,

in which in the first step for each player the average of players' marginal contributions corresponding to all admissible rooted spanning trees that have this

player as the root is calculated, and in the second step the average over all players of all the payoffs obtained in the first step is computed. In general these two approaches lead to different solution concepts. When each component in the

underlying communication structure is cycle-free, a linear cactus with cycles, or

the complete graph, the two-step average tree value coincides with the average

tree value. A comparative analysis of both solution concepts is done and an

axiomatization of the the two-step average tree value on the subclass of TU games with semi-cycle-free hypergraph communication structure, which is more general than that given by a cycle-free hypergraph, is obtained.

restricted cooperation represented by undirected communication graphs or hypergraphs. The solution can be considered as an alternative for both the average tree solution for graph games and the average tree value for hypergraph games.

Instead of averaging players' marginal contributions corresponding to all admis-sible rooted spanning trees of the underlying (hyper)graph, which determines

the average tree solution or value, we consider a two-step averaging procedure,

in which in the first step for each player the average of players' marginal contributions corresponding to all admissible rooted spanning trees that have this

player as the root is calculated, and in the second step the average over all players of all the payoffs obtained in the first step is computed. In general these two approaches lead to different solution concepts. When each component in the

underlying communication structure is cycle-free, a linear cactus with cycles, or

the complete graph, the two-step average tree value coincides with the average

tree value. A comparative analysis of both solution concepts is done and an

axiomatization of the the two-step average tree value on the subclass of TU games with semi-cycle-free hypergraph communication structure, which is more general than that given by a cycle-free hypergraph, is obtained.

Original language | English |
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Place of Publication | Tilburg |

Publisher | CentER, Center for Economic Research |

Number of pages | 21 |

Volume | 2020-018 |

Publication status | Published - 9 Jul 2020 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2020-018 |

### Keywords

- TU game
- hypergraph communication structure
- average tree value
- component fairness

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## Cite this

Kang, L., Khmelnitskaya, A., Shan, E., Talman, A. J. J., & Zhang, G. (2020).

*The Two-Step Average Tree Value for Graph and Hypergraph Games*. (CentER Discussion Paper; Vol. 2020-018). CentER, Center for Economic Research.