### Abstract

A cooperative game with non-transferable utility (NTU-game) consists of a collection of payoffsets for the subsets of a nite set of players, for which it has to be determined how much payof each player must receive. The core of an NTU-game consists of all payoffvectors that are in the payoff set of the coalition of all players and cannot be improved upon by any coalition of players. For cooperative games with transferable utility (TU-games) the notion of convexity was introduced to guarantee that the Shapley value, being the average of all marginal vectors of the game, is an element of the core. Convexity of a TU-game is equivalent to supermodularity of the characteristic function underlying the

game. In this paper we introduce the concept of supermodularity for NTU-games. Super-modularity for NTU-games is weaker than other existing types of convexity. Under super-modularity of an NTU-game it is shown that all appropriately dened marginal vectors of the game are elements of the core. As solution concept for NTU-games we propose a set of solutions that is determined by the average of all marginal vectors of the game. For TU-games the solution set coincides with the Shapley value of the game. Also conditions

are stated under which the solution set is a subset of the core and is the set of bargaining solutions of a corresponding bargaining problem.

game. In this paper we introduce the concept of supermodularity for NTU-games. Super-modularity for NTU-games is weaker than other existing types of convexity. Under super-modularity of an NTU-game it is shown that all appropriately dened marginal vectors of the game are elements of the core. As solution concept for NTU-games we propose a set of solutions that is determined by the average of all marginal vectors of the game. For TU-games the solution set coincides with the Shapley value of the game. Also conditions

are stated under which the solution set is a subset of the core and is the set of bargaining solutions of a corresponding bargaining problem.

Original language | English |
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Place of Publication | Tilburg |

Publisher | Operations research |

Number of pages | 16 |

Volume | 2014-067 |

Publication status | Published - 10 Nov 2014 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2014-067 |

### Keywords

- core
- shapley value
- convexity
- supermodularity
- marginal vector

### Cite this

Koshevoy, G. A., Suzuki, T., & Talman, A. J. J. (2014).

*Supermodular NTU-games*. (CentER Discussion Paper; Vol. 2014-067). Tilburg: Operations research.