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# The lattice of embedded subsets

## Author

Listed:
• Michel Grabisch

() (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

## Abstract

In cooperative game theory, games in partition function form are real-valued function on the set of so-called embedded coalitions, that is, pairs $(S,\pi)$ where $S$ is a subset (coalition) of the set $N$ of players, and $\pi$ is a partition of $N$ containing $S$. Despite the fact that many studies have been devoted to such games, surprisingly nobody clearly defined a structure (i.e., an order) on embedded coalitions, resulting in scattered and divergent works, lacking unification and proper analysis. The aim of the paper is to fill this gap, thus to study the structure of embedded coalitions (called here embedded subsets), and the properties of games in partition function form.

## Suggested Citation

• Michel Grabisch, 2010. "The lattice of embedded subsets," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00457827, HAL.
• Handle: RePEc:hal:cesptp:hal-00457827
DOI: 10.1016/j.dam.2009.10.015
Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00457827
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File URL: https://hal.archives-ouvertes.fr/hal-00457827/document

## Citations

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Cited by:

1. Caulier, Jean-François & Mauleon, Ana & Vannetelbosch, Vincent, 2015. "Allocation rules for coalitional network games," Mathematical Social Sciences, Elsevier, vol. 78(C), pages 80-88.
2. repec:spr:grdene:v:26:y:2017:i:6:d:10.1007_s10726-017-9542-x is not listed on IDEAS
3. repec:hal:journl:halshs-00690696 is not listed on IDEAS
4. José María Alonso-Meijide & Mikel Álvarez-Mozos & María Gloria Fiestras-Janeiro, 2015. "Power Indices and Minimal Winning Coalitions in Simple Games with Externalities Abstract: We propose a generalization of simple games to situations with coalitional externalities. The main novelty of ," UB Economics Working Papers 2015/328, Universitat de Barcelona, Facultat d'Economia i Empresa, UB Economics.
5. Grabisch, Michel & Funaki, Yukihiko, 2012. "A coalition formation value for games in partition function form," European Journal of Operational Research, Elsevier, vol. 221(1), pages 175-185.
6. José María Alonso-Meijide & Mikel Alvarez-Mozos & María Gloria Fiestras-Janeiro & Andrés Jiménez-Losada, 2016. "Some structural properties of a lattice of embedded coalitions," UB Economics Working Papers 2016/349, Universitat de Barcelona, Facultat d'Economia i Empresa, UB Economics.

### Keywords

Partition; Embedded subset; Game; Valuation; k-monotonicity;

### NEP fields

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