### Abstract

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

Place of Publication | Tilburg |

Publisher | Econometrics |

Number of pages | 34 |

Volume | 2013-012 |

Publication status | Published - 2013 |

### Publication series

Name | CentER Discussion Paper |
---|---|

Volume | 2013-012 |

### Fingerprint

### Keywords

- Set system
- nested set
- rooted tree
- chain
- core
- convexity
- marginal vector
- Shapley value

### Cite this

*Solutions For Games With General Coalitional Structure And Choice Sets*. (CentER Discussion Paper; Vol. 2013-012). Tilburg: Econometrics.

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**Solutions For Games With General Coalitional Structure And Choice Sets.** / Koshevoy, G.A.; Suzuki, T.; Talman, A.J.J.

Research output: Working paper › Discussion paper › Other research output

TY - UNPB

T1 - Solutions For Games With General Coalitional Structure And Choice Sets

AU - Koshevoy, G.A.

AU - Suzuki, T.

AU - Talman, A.J.J.

N1 - Pagination: 34

PY - 2013

Y1 - 2013

N2 - In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payo his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some speci cations of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently de ned for this class. For graph games it therefore diers from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.

AB - In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payo his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some speci cations of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently de ned for this class. For graph games it therefore diers from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.

KW - Set system

KW - nested set

KW - rooted tree

KW - chain

KW - core

KW - convexity

KW - marginal vector

KW - Shapley value

M3 - Discussion paper

VL - 2013-012

T3 - CentER Discussion Paper

BT - Solutions For Games With General Coalitional Structure And Choice Sets

PB - Econometrics

CY - Tilburg

ER -