The matching game is a cooperative profit game defined on an edge-weighted graph, where the players are the vertices and the profit of a coalition is the maximum weight of matchings in the subgraph induced by the coalition. A population monotonic allocation scheme is a collection of rules defining how to share the profit among players in each coalition such that every player is better off when the coalition expands. In this paper, we study matching games and provide a necessary and sufficient characterization for the existence of population monotonic allocation schemes. Our characterization implies that whether a matching game admits population monotonic allocation schemes can be determined efficiently.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Biró P, Kern W, Paulusma D (2012) Computing solutions for matching games. Int J Game Theory 41(1):75–90
Deng X, Ibaraki T, Nagamochi H (1999) Algorithmic aspects of the core of combinatorial optimization games. Math Oper Res 24(3):751–766
Deng X, Ibaraki T, Nagamochi H, Zang W (2000) Totally balanced combinatorial optimization games. Math Program 87(3):441–452
Eriksson K, Karlander J (2001) Stable outcomes of the roommate game with transferable utility. Int J Game Theory 29(4):555–569
Faigle U, Kern W, Fekete SP, Hochstättler W (1998) The nucleon of cooperative games and an algorithm for matching games. Math Program 83(1–3):195–211
Grossman JW, Harary F, Klawe M (1979) Generalized Ramsey theory for graphs, x: double stars. Discret Math 28(3):247–254
Immorlica N, Mahdian M, Mirrokni VS (2008) Limitations of cross-monotonic cost-sharing schemes. ACM Trans Algorithms 4(2):1–25
Kern W, Paulusma D (2003) Matching games: the least core and the nucleolus. Math Oper Res 28(2):294–308
Klaus B, Nichifor A (2010) Consistency in one-sided assignment problems. Soc Choice Welf 35(3):415–433
Könemann J, Pashkovich K, Toth J (2020) Computing the nucleolus of weighted cooperative matching games in polynomial time. Math Program 183(1–2):555–581
Kumabe S, Maehara T (2020) Convexity of \(b\)-matching games. In: Proceedings of the 29th international joint conference on artificial intelligence—IJCAI’20, Yokohama, Japan, pp 261–267
Kumabe S, Maehara T (2020) Convexity of hypergraph matching game. In: Proceedings of the 19th international conference on autonomous agents and multiagent systems—AAMAS’20, Auckland, New Zealand, pp 663–671
Llerena F, Núñez M, Rafels C (2015) An axiomatization of the nucleolus of assignment markets. Int J Game Theory 44(1):1–15
Moulin H (1999) Incremental cost sharing: characterization by coalition strategy-proofness. Soc Choice Welf 16(2):279–320
Moulin H, Shenker S (2001) Strategyproof sharing of submodular costs: budget balance versus efficiency. Econ Theor 18(3):511–533
Núñez M, Rafels C (2003) Characterization of the extreme core allocations of the assignment game. Games Econom Behav 44(2):311–331
Shapley LS, Shubik M (1971) The assignment game I: the core. Int J Game Theory 1(1):111–130
Solymosi T, Raghavan TES (1994) An algorithm for finding the nucleolus of assignment games. Int J Game Theory 23(2):119–143
Solymosi T, Raghavan TES (2001) Assignment games with stable core. Int J Game Theory 30(2):177–185
Sprumont Y (1990) Population monotonic allocation schemes for cooperative games with transferable utility. Games Econom Behav 2(4):378–394
Toda M (2005) Axiomatization of the core of assignment games. Games Econom Behav 53(2):248–261
Vazirani VV (2021) The general graph matching game: approximate core. arXiv: 2101.0739
We appreciate two anonymous referees for their helpful comments and valuable suggestions which greatly improved the presentation of this work.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported in part by the National Natural Science Foundation of China (Nos. 12001507, 11971447, 11871442) and the Natural Science Foundation of Shandong (No. ZR2020QA024).
About this article
Cite this article
Xiao, H., Fang, Q. Population monotonicity in matching games. J Comb Optim (2021). https://doi.org/10.1007/s10878-021-00804-3
- Cooperative game theory
- Matching game
- Population monotonic allocation scheme
Mathematics Subject Classification