Finding and verifying the nucleolus of cooperative games
Finding and verifying the nucleolus of cooperative games
The nucleolus offers a desirable payoff-sharing solution in cooperative games, thanks to its attractive properties—it always exists and lies in the core (if the core is non-empty), and it is unique. The nucleolus is considered as the most ‘stable’ solution in the sense that it lexicographically minimizes the dissatisfactions among all coalitions. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e. with the number of players n≤ 15). This approach, however, becomes more challenging for larger games because of the need to form and check a criterion involving possibly exponentially large collections of coalitions, with each collection potentially of an exponentially large size. The aim of this work is twofold. First, we develop an improved version of the Kohlberg criterion that involves checking the ‘balancedness’ of at most (n- 1) sets of coalitions. Second, we exploit these results and introduce a novel descent-based constructive algorithm to find the nucleolus efficiently. We demonstrate the performance of the new algorithms by comparing them with existing methods over different types of games. Our contribution also includes the first open-source code for computing the nucleolus for games of moderately large sizes.
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Benedek, Marton
72bc97bc-a373-4b88-9e9f-145d62551214
Fliege, Joerg
54978787-a271-4f70-8494-3c701c893d98
Nguyen, Tri-Dung
a6aa7081-6bf7-488a-b72f-510328958a8e
Benedek, Marton
72bc97bc-a373-4b88-9e9f-145d62551214
Fliege, Joerg
54978787-a271-4f70-8494-3c701c893d98
Nguyen, Tri-Dung
a6aa7081-6bf7-488a-b72f-510328958a8e
Benedek, Marton, Fliege, Joerg and Nguyen, Tri-Dung
(2020)
Finding and verifying the nucleolus of cooperative games.
Mathematical Programming, 0, .
(doi:10.1007/s10107-020-01527-9).
Abstract
The nucleolus offers a desirable payoff-sharing solution in cooperative games, thanks to its attractive properties—it always exists and lies in the core (if the core is non-empty), and it is unique. The nucleolus is considered as the most ‘stable’ solution in the sense that it lexicographically minimizes the dissatisfactions among all coalitions. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e. with the number of players n≤ 15). This approach, however, becomes more challenging for larger games because of the need to form and check a criterion involving possibly exponentially large collections of coalitions, with each collection potentially of an exponentially large size. The aim of this work is twofold. First, we develop an improved version of the Kohlberg criterion that involves checking the ‘balancedness’ of at most (n- 1) sets of coalitions. Second, we exploit these results and introduce a novel descent-based constructive algorithm to find the nucleolus efficiently. We demonstrate the performance of the new algorithms by comparing them with existing methods over different types of games. Our contribution also includes the first open-source code for computing the nucleolus for games of moderately large sizes.
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In preparation date: 2018
Accepted/In Press date: 14 May 2020
e-pub ahead of print date: 6 June 2020
Additional Information:
Funding Information:
Open access funding provided by Centre for Economic and Regional Studies. The authors would like to thank the Editor and the two anonymous reviewers for their valuable comments and detailed suggestions on how to improve the manuscript. The first author acknowledges that the research reported in this paper has been supported by the National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program). The third author acknowledges the funding support from the Engineering and Physical Sciences Research Council (Grant Nos. EP/P021042/1, EP/M50662X/1).
Publisher Copyright:
© 2020, The Author(s).
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Local EPrints ID: 426473
URI: http://eprints.soton.ac.uk/id/eprint/426473
ISSN: 0025-5610
PURE UUID: 8d6dab97-274f-4203-a983-9c5616401d14
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Date deposited: 28 Nov 2018 17:30
Last modified: 17 Mar 2024 06:04
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Author:
Marton Benedek
Author:
Tri-Dung Nguyen
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