Advertisement

Review of Economic Design

, Volume 22, Issue 1–2, pp 25–53 | Cite as

Stable cost sharing in production allocation games

  • Eric Bahel
  • Christian Trudeau
Original Paper

Abstract

Suppose that a group of agents have demands for some good. Every agent owns a technology which allows them to produce the good, with these technologies varying in their effectiveness. If all technologies exhibit increasing returns to scale (IRS) then it is always efficient to centralize production of the good, whereas efficiency in the case of decreasing returns to scale (DRS) typically requires to spread production. We search for stable cost allocations while differentiating allocations with homogeneous prices, in which all units produced are traded at the same price, from allocations with heterogeneous prices. For the respective cases of IRS or DRS, it is shown that there always exist stable cost sharing rules with homogeneous prices. Finally, in the general framework (under which there may exist no stable allocation at all) we provide a sufficient condition for the existence of stable allocations with homogeneous prices. This condition is shown to be both necessary and sufficient in problems with unitary demands.

Keywords

Cost sharing Stability Production allocation Returns to scale Homogeneous prices 

JEL Classification

C71 D63 

Supplementary material

References

  1. Anderson RM (1992) The core in perfectly competitive economies. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol 1. Elsevier, New York, pp 413–457CrossRefGoogle Scholar
  2. Anshelevich E, Dasgupta A, Kleinberg J, Tardos E, Wexler T, Roughgarden T (2008) The price of stability for network design with fair cost allocation. SIAM J Comput 38:1602–1623CrossRefGoogle Scholar
  3. Bahel E, Trudeau C (2013) A discrete cost sharing model with technological cooperation. Int J Game Theory 42:439–460CrossRefGoogle Scholar
  4. Bahel E, Trudeau C (2014) Stable lexicographic rules for shortest path games. Econ Lett 125:266–269CrossRefGoogle Scholar
  5. Bergantinos G, Vidal-Puga J (2007) A fair rule in minimum cost spanning tree problems. J Econ Theory 137:326–352CrossRefGoogle Scholar
  6. Bird CJ (1976) On cost allocation for a spanning tree: a game theoretic approach. Networks 6:335–350CrossRefGoogle Scholar
  7. Camiña E (2006) A generalized assignment game. Math Soc Sci 52:152–161CrossRefGoogle Scholar
  8. Crawford VP, Knoer EM (1981) Job matching with heterogeneous firms and workers. Econometrica 49:437–450CrossRefGoogle Scholar
  9. Gillies DB (1953) Some theorems on n-person games. Ph.D. Thesis, Department of Mathematics, Princeton UniversityGoogle Scholar
  10. Jaume D, Massó J, Neme A (2016) The multiple-partners assignment game with heterogeneous sells and multi-unit demands: competitive equilibria. Polar Biol 39:2189–2205CrossRefGoogle Scholar
  11. Kaneko M (1976) On the core and competitive equilibria of a market with indivisible goods. Naval Res Logist 21:321–337CrossRefGoogle Scholar
  12. Moulin H (2013) Cost sharing in networks: some open questions. Int Game Theory Rev 15:1340001CrossRefGoogle Scholar
  13. Moulin H, Sprumont Y (2007) Fair allocation of production externalities: recent results. Rev d’Econ Politique 117:7–36Google Scholar
  14. Núñez M, Rafels C (2002) The assignment game: the \(\tau \)-value. Int J Game Theory 31:411–422CrossRefGoogle Scholar
  15. Núñez M, Rafels C (2017) A survey on assignment markets. J Dyn Games 2:227–256Google Scholar
  16. Quant M, Borm P, Reijnierse H (2006) Congestion network problems and related games. Eur J Oper Res 172:919–930CrossRefGoogle Scholar
  17. Quinzii M (1984) Core and competitive equilibria with indivisibilities. Int J Game Theory 13:41–60CrossRefGoogle Scholar
  18. Rosenthal EC (2013) Shortest path games. Eur J Oper Res 224(1):132–140CrossRefGoogle Scholar
  19. Sanchez-Soriano J, Lopez MA, Garcia-Jurado I (2001) On the core of transportation games. Math Soc Sci 41:215–225CrossRefGoogle Scholar
  20. Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26CrossRefGoogle Scholar
  21. Shapley LS, Shubik M (1971) The assignment game I: the core. Int J Game Theory 1:9–25Google Scholar
  22. Sharkey WW (1995) Network models in economics. In: Ball MO, Magnanti TL, Nonma CL, Nemhauser GL (eds) Network routing. Handbooks in operation research and management science, vol 8. Elsevier, New York, pp 713–765Google Scholar
  23. Sotomayor M (1992) The multiple partners game. In: Majumdar M (ed) Equilibrium and dynamics: essays in honor to David Gale. Springer, Berlin, pp 269–283Google Scholar
  24. Sotomayor M (2002) A labor market with heterogeneous firms and workers. Int J Game Theory 31:269–283CrossRefGoogle Scholar
  25. Sotomayor M (2007) Connecting the cooperative and competitive structures of the multiplepartners assignment game. J Econ Theory 134:155–174CrossRefGoogle Scholar
  26. Sprumont Y (2005) On the discrete version of the Aumann–Shapley cost-sharing method. Econometrica 73:1693–1712CrossRefGoogle Scholar
  27. Thompson GL (1981) Auctions and market games. In: Aumann R (ed) Essays in game theory and mathematical economics in Honor of Oskar Morgenstern. Princeton University Press, PrincetonGoogle Scholar
  28. Trudeau C (2009a) Cost sharing with multiple technologies. Games Econ Behav 67:695–707CrossRefGoogle Scholar
  29. Trudeau C (2009b) Network flow problems and permutationally concave games. Math Soc Sci 58:121–131CrossRefGoogle Scholar
  30. Trudeau C (2012) A new stable and more responsive cost sharing solution for minimum cost spanning tree problems. Games Econ Behav 75:402–412CrossRefGoogle Scholar
  31. Yokote K (2016) Core and competitive equilibria: an approach from discrete convex analysis. J Math Econ 66:1–13CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of EconomicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of EconomicsUniversity of WindsorWindsorCanada

Personalised recommendations