International Journal of Game Theory

, Volume 42, Issue 1, pp 29–53 | Cite as

Strategic cooperation in cost sharing games

  • Martin Hoefer


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.


Cost sharing Strong equilibrium Integrality gap Combinatorial optimization 


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  1. Albers S (2009) On the value of coordination in network design. SIAM J Comput 38(6): 2273–2302CrossRefGoogle Scholar
  2. Andelman N, Feldman M, Mansour Y (2009) Strong price of anarchy. Games Econ Behav 65(2): 289–317CrossRefGoogle Scholar
  3. Anshelevich E, Caskurlu B (2011a) Price of stability in survivable network design. Theory Comput Syst 49(1): 98–138CrossRefGoogle Scholar
  4. Anshelevich E, Caskurlu B (2011b) Exact and approximate equilibria for optimal group network formation. Theor Comput Sci 412(39): 5298–5314CrossRefGoogle Scholar
  5. Anshelevich E, Karagiozova A (2011) Terminal backup, 3D matching, and covering cubic graphs. SIAM J Comput 40(3): 678–708CrossRefGoogle Scholar
  6. 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): 1602–1623CrossRefGoogle Scholar
  7. Anshelevich E, Dasgupta A, Tardos É, Wexler T (2008b) Near-optimal network design with selfish agents. Theory Comput 4: 77–109CrossRefGoogle Scholar
  8. Anshelevich E, Caskurlu B, Hate A (2010) Strategic multiway cut and multicut games. In: Proc. 8th intl. workshop approximation and online algorithms (WAOA), pp 1–12Google Scholar
  9. 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 287–324Google Scholar
  10. Bird C (1976) On cost allocation for a spanning tree: a game theoretic approach. Networks 6: 335–350CrossRefGoogle Scholar
  11. Calinescu G, Karloff H, Rabani Y (2000) An improved approximation algorithm for multiway cut. J Comput Syst Sci 60(3): 564–574CrossRefGoogle Scholar
  12. Cardinal J, Hoefer M (2010) Non-cooperative facility location and covering games. Theor Comput Sci 411(16–18): 1855–1876CrossRefGoogle Scholar
  13. Chen H-L, Roughgarden T, Valiant G (2010) Designing network protocols for good equilibria. SIAM J Comput 39(5): 1799–1832CrossRefGoogle Scholar
  14. Deng X, Ibaraki T, Nagamochi H (1999) Algorithmic aspects of the core of combinatorial optimization games. Math Oper Res 24(3): 751–766CrossRefGoogle Scholar
  15. Epstein A, Feldman M, Mansour Y (2009) Strong equilibrium in cost sharing connection games. Games Econ Behav 67(1): 51–68CrossRefGoogle Scholar
  16. Faigle U, Fekete S, Hochstättler W, Kern W (1998) On approximately fair cost allocation in Euclidean TSP games. OR Spektrum 20(1): 29–37CrossRefGoogle Scholar
  17. Goemans M, Skutella M (2004) Cooperative facility location games. J Algorithms 50(2): 194–214CrossRefGoogle Scholar
  18. Goemans M, Williamson D (1995) A general approximation technique for constrained forest problems. SIAM J Comput 24(2): 296–317CrossRefGoogle Scholar
  19. Granot D (1986) A generalized linear production model: a unifying model. Math Prog 34: 212–222CrossRefGoogle Scholar
  20. Granot D, Huberman G (1981) On minimum cost spanning tree games. Math Prog 21: 1–18CrossRefGoogle Scholar
  21. Granot D, Maschler M (1998) Spanning network games. Int J Game Theory 27: 467–500CrossRefGoogle Scholar
  22. Hoefer M (2009) Non-cooperative tree creation. Algorithmica 53(1): 104–131CrossRefGoogle Scholar
  23. Hoefer M (2010) Strategic cooperation in cost sharing games. In: Proc. 6th intl workshop Internet and network economics (WINE), pp 258–269Google Scholar
  24. Hoefer M (2011) Competitive cost sharing with economies of scale. Algorithmica 60(4): 743–765CrossRefGoogle Scholar
  25. Hoefer M, Krysta P (2005) Geometric network design with selfish agents. In: Proc. 11th conf. computing and combinatorics (COCOON), pp 167–178Google Scholar
  26. Immorlica N, Mahdian M, Mirrokni V (2008) Limitations of cross-monotonic cost sharing schemes. ACM Trans Algorithms 4(2). doi: 10.1145/1361192.1361201
  27. 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, CambridgeGoogle Scholar
  28. Jain K, Vazirani V (2001) Applications of approximation algorithms to cooperative games. In: Proc. 33rd symp. theory of computing (STOC), pp 364–372Google Scholar
  29. 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): 1319–1341CrossRefGoogle Scholar
  30. Leonardi S, Sankowski P (2007) Network formation games with local coalitions. In Proc. 26th symp. principles of distributed computing (PODC), pp 299–305Google Scholar
  31. Megiddo N (1978) Cost allocation for Steiner trees. Networks 8(1): 1–6CrossRefGoogle Scholar
  32. Owen G (1975) On the core of linear production games. Math Prog 9: 358–370CrossRefGoogle Scholar
  33. Pál M, Tardos É (2003) Group strategyproof mechanisms via primal-dual algorithms. In: Proc. 44th symp. foundations of computer science (FOCS), pp 584–593Google Scholar
  34. Prodon A, Libeling TM, Gröflin H (1985) Steiner’s problem on two-trees. Technical report, Départment de Mathemátiques, EPF Lausanne. Working paper RO 850315Google Scholar
  35. Skorin-Karpov D (1995) On the core of the minimum cost Steiner tree game in networks. Ann Oper Res 57: 233–249CrossRefGoogle Scholar
  36. Tamir A (1991) On the core of network synthesis games. Math Prog 50: 123–135CrossRefGoogle Scholar
  37. Tamir A (1993) On the core of cost allocation games defined on location problems. Transp Sci 27: 81–86CrossRefGoogle Scholar
  38. Wong R (1984) A dual ascent approach for Steiner tree problems on a directed graph. Math Prog 28(3): 271–287CrossRefGoogle Scholar
  39. 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 1194–1235Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.RWTH Aachen UniversityAachenGermany

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