Bounds on the cost of stabilizing a cooperative game

Yoram Bachrach, Edith Elkind, Enrico Malizia, Reshef Meir, Dmitrii Pasechnik, Jeffrey S. Rosenschein, Jörg Rothe, Michael Zuckerman

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


A key issue in cooperative game theory is coalitional stability, usually captured by the notion of the core-the set of outcomes that are resistant to group deviations. However, some coalitional games have empty cores, and any outcome in such a game is unstable. We investigate the possibility of stabilizing a coalitional game by using subsidies. We consider scenarios where an external party that is interested in having the players work together offers a supplemental payment to the grand coalition, or, more generally, a particular coalition structure. This payment is conditional on players not deviating from this coalition structure, and may be divided among the players in any way they wish. We define the cost of stability as the minimum external payment that stabilizes the game. We provide tight bounds on the cost of stability, both for games where the coalitional values are nonnegative (profit-sharing games) and for games where the coalitional values are nonpositive (cost-sharing games), under natural assumptions on the characteristic function, such as superadditivity, anonymity, or both. We also investigate the relationship between the cost of stability and several variants of the least core. Finally, we study the computational complexity of problems related to the cost of stability, with a focus on weighted voting games.

Original languageAmerican English
Pages (from-to)987-1023
Number of pages37
JournalJournal of Artificial Intelligence Research
StatePublished - Dec 2018

Bibliographical note

Funding Information:
This work unifies and extends conference papers that appeared in the proceedings of the Eighth International Joint Conference on Autonomous Agents and Multiagent Systems (AA-MAS 2009), in the proceedings of the Second and the Third International Symposium on Algorithmic Game Theory (SAGT 2009 and SAGT 2010), and in the proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (IJCAI 2011), see (Bachrach et al., 2009b, 2009a; Meir et al., 2010, 2011). We thank Dov Monderer and David Parkes, as well as the AAMAS’09, SAGT’09, SAGT’10, IJCAI’11, and JAIR reviewers for their helpful comments. This work was supported in part by the following parties and grants: DFG grants RO-1202/{11-1, 12-1, 14-1, 14-2}; the European Science Foundation’s EUROCORES program LogICCC; NRF (Singapore) grant RF2009-08; ERC Starting Grant ACCORD (Grant Agreement 639945); Horizon 2020 European Research Infrastructure project OpenDreamKit (Project ID 676541); Israel Science Foundation grants #1227/12 and #1340/18; Israel Ministry of Science and Technology grant #3-6797; the Israel Ministry of Science and Technology—Knowledge Center in Machine Learning and Artificial Intelligence; the Google Inter-University Center for Electronic Markets and Auctions; the European Commission through the European Social Fund; the Calabria Region. This work was done in part while the third author was visiting University of Oxford, and while the seventh author was visiting The Hebrew University of Jerusalem and Stanford University.

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