Abstract
We examine strategic cost sharing games with so-called arbitrary sharing based on various combinatorial optimization problems. These games have recently been popular in computer science to study cost sharing in the context of the Internet. We concentrate on the existence and computational complexity of strong equilibria (SE), in which no coalition can improve the cost of each of its members. Our main result reveals a connection to the core in coalitional cost sharing games studied in operations research. For set cover and facility location games this results in a tight characterization of the existence of SE using the integrality gap of suitable linear programming formulations. Furthermore, it allows to derive all existing results for SE in network design cost sharing games with arbitrary sharing via a unified approach. In addition, we show that in general there is no efficiency loss, i.e., the strong price of anarchy is always 1. Finally, we indicate how the LP-approach is useful for the computation of near-optimal and near-stable approximate SE.
Similar content being viewed by others
References
Albers S (2009) On the value of coordination in network design. SIAM J Comput 38(6): 2273–2302
Andelman N, Feldman M, Mansour Y (2009) Strong price of anarchy. Games Econ Behav 65(2): 289–317
Anshelevich E, Caskurlu B (2011a) Price of stability in survivable network design. Theory Comput Syst 49(1): 98–138
Anshelevich E, Caskurlu B (2011b) Exact and approximate equilibria for optimal group network formation. Theor Comput Sci 412(39): 5298–5314
Anshelevich E, Karagiozova A (2011) Terminal backup, 3D matching, and covering cubic graphs. SIAM J Comput 40(3): 678–708
Anshelevich E, Dasgupta A, Kleinberg J, Roughgarden T, Tardos É, Wexler T (2008a) The price of stability for network design with fair cost allocation. SIAM J Comput 38(4): 1602–1623
Anshelevich E, Dasgupta A, Tardos É, Wexler T (2008b) Near-optimal network design with selfish agents. Theory Comput 4: 77–109
Anshelevich E, Caskurlu B, Hate A (2010) Strategic multiway cut and multicut games. In: Proc. 8th intl. workshop approximation and online algorithms (WAOA), pp 1–12
Aumann R (1959) Acceptable points in general cooperative n-person games. In: Contributions to the theory of games IV, vol 40 of annals of mathematics study. Princeton University Press, Princeton, pp 287–324
Bird C (1976) On cost allocation for a spanning tree: a game theoretic approach. Networks 6: 335–350
Calinescu G, Karloff H, Rabani Y (2000) An improved approximation algorithm for multiway cut. J Comput Syst Sci 60(3): 564–574
Cardinal J, Hoefer M (2010) Non-cooperative facility location and covering games. Theor Comput Sci 411(16–18): 1855–1876
Chen H-L, Roughgarden T, Valiant G (2010) Designing network protocols for good equilibria. SIAM J Comput 39(5): 1799–1832
Deng X, Ibaraki T, Nagamochi H (1999) Algorithmic aspects of the core of combinatorial optimization games. Math Oper Res 24(3): 751–766
Epstein A, Feldman M, Mansour Y (2009) Strong equilibrium in cost sharing connection games. Games Econ Behav 67(1): 51–68
Faigle U, Fekete S, Hochstättler W, Kern W (1998) On approximately fair cost allocation in Euclidean TSP games. OR Spektrum 20(1): 29–37
Goemans M, Skutella M (2004) Cooperative facility location games. J Algorithms 50(2): 194–214
Goemans M, Williamson D (1995) A general approximation technique for constrained forest problems. SIAM J Comput 24(2): 296–317
Granot D (1986) A generalized linear production model: a unifying model. Math Prog 34: 212–222
Granot D, Huberman G (1981) On minimum cost spanning tree games. Math Prog 21: 1–18
Granot D, Maschler M (1998) Spanning network games. Int J Game Theory 27: 467–500
Hoefer M (2009) Non-cooperative tree creation. Algorithmica 53(1): 104–131
Hoefer M (2010) Strategic cooperation in cost sharing games. In: Proc. 6th intl workshop Internet and network economics (WINE), pp 258–269
Hoefer M (2011) Competitive cost sharing with economies of scale. Algorithmica 60(4): 743–765
Hoefer M, Krysta P (2005) Geometric network design with selfish agents. In: Proc. 11th conf. computing and combinatorics (COCOON), pp 167–178
Immorlica N, Mahdian M, Mirrokni V (2008) Limitations of cross-monotonic cost sharing schemes. ACM Trans Algorithms 4(2). doi:10.1145/1361192.1361201
Jain K, Mahdian M (2007) Cost sharing. In: Nisan N, Tardos É, Roughgarden T, Vazirani V (eds) Algorithmic game theory, chapter 15. Cambridge University Press, Cambridge
Jain K, Vazirani V (2001) Applications of approximation algorithms to cooperative games. In: Proc. 33rd symp. theory of computing (STOC), pp 364–372
Könemann J, Leonardi S, Schäfer G, van Zwam S (2008) A group-strategyproof cost sharing mechanism for the Steiner forest game. SIAM J Comput 37(5): 1319–1341
Leonardi S, Sankowski P (2007) Network formation games with local coalitions. In Proc. 26th symp. principles of distributed computing (PODC), pp 299–305
Megiddo N (1978) Cost allocation for Steiner trees. Networks 8(1): 1–6
Owen G (1975) On the core of linear production games. Math Prog 9: 358–370
Pál M, Tardos É (2003) Group strategyproof mechanisms via primal-dual algorithms. In: Proc. 44th symp. foundations of computer science (FOCS), pp 584–593
Prodon A, Libeling TM, Gröflin H (1985) Steiner’s problem on two-trees. Technical report, Départment de Mathemátiques, EPF Lausanne. Working paper RO 850315
Skorin-Karpov D (1995) On the core of the minimum cost Steiner tree game in networks. Ann Oper Res 57: 233–249
Tamir A (1991) On the core of network synthesis games. Math Prog 50: 123–135
Tamir A (1993) On the core of cost allocation games defined on location problems. Transp Sci 27: 81–86
Wong R (1984) A dual ascent approach for Steiner tree problems on a directed graph. Math Prog 28(3): 271–287
Young HP (1994) Cost allocation. In: Aumann R, Hart S (eds) Handbook of game theory with economic applications vol 2, Chap 34. North-Holland Science Publishers, Amsterdam, pp 1194–1235
Author information
Authors and Affiliations
Corresponding author
Additional information
An extended abstract of this work appeared in the Proceedings of the 6th Workshop on Internet and Network Economics (Hoefer 2010).
Rights and permissions
About this article
Cite this article
Hoefer, M. Strategic cooperation in cost sharing games. Int J Game Theory 42, 29–53 (2013). https://doi.org/10.1007/s00182-011-0312-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-011-0312-8