Skip to main content

Responsibility and sharing the cost of cleaning a polluted river

Mathematical Methods of Operations Research volume 89, pages 143–156 (2019)Cite this article

Abstract

Consider n firms (agents) located at a river, indexed by \(1, \dots , n\) from upstream to downstream. The pollution generated by these firms induce cleaning costs \(c_1, \dots , c_n\), where \(c_i\) is the cost for cleaning the water in region i (according to the local environmental standards). The corresponding cost allocation problem is highly interesting both in theory and practice. Among the most prominent allocation schemes are the so-called Local Responsibility and Upstream Equal Sharing. The first one allocates simply each local cost \(c_i\) to the corresponding firm i. The second distributes each \(c_i\) equally among firms \(1, \dots , i\). We propose and characterize a dynamic scheme which, given a particular order of arrival, allocates the current total cost among the firms that have arrived so far. The corresponding expected allocation (w.r.t. a random arrival order) turns out to be a convex combination of the two schemes above.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  • Alcalde-Unzu J, Góez-Rúa M, Molis E (2015) Sharing the cost of cleaning a river: the upstream responsibility rule. Games Econ Behav 90:134–150

    MathSciNet  Article  MATH  Google Scholar 

  • Ambec S, Sprumont Y (2002) Sharing a river. J Econ Theory 107:453–462

    MathSciNet  Article  MATH  Google Scholar 

  • Ambec S, Ehlers L (2008) Sharing a river among satiable agents. Games Econ Behav 64:35–50

    MathSciNet  Article  MATH  Google Scholar 

  • Barrett S (1994) Conflict and cooperation in managing international water resources. Working paper 1303. World Bank, Washington

  • Dong B, Ni D, Wang Y (2012) Sharing a polluted river network. Environ Resour Econ 53:367–387

    Article  Google Scholar 

  • Godana B (1985) Africa’s shared water resources. France Printer, London

    Google Scholar 

  • Khmelnitskaya AB (2010) Values for rooted-tree and sink-tree digraphs games and sharing a river. Theory Decis 69:657–669

    MathSciNet  Article  MATH  Google Scholar 

  • Kilgour M, Dinar A (1996) Are stable agreements for sharing international river waters now possible? Working paper 1474. World Bank, Washington

  • Malawski M (2013) “Procedural” values for cooperative games. Int Game Theory 42:305–324

    MathSciNet  Article  MATH  Google Scholar 

  • Ni D, Wang Y (2007) Sharing a polluted river. Games Econ Behav 60:176–186

    MathSciNet  Article  MATH  Google Scholar 

  • Shapley LS (1953) A value for n-person games. Ann Math Stud 28:307–317

    MathSciNet  MATH  Google Scholar 

  • Sun P, Hou D, Sun H, Driessen T (2017a) Optimization implementation and characterization of the equal allocation of noseparable costs value. J Optim Theory Appl 173:336–352

    MathSciNet  Article  MATH  Google Scholar 

  • Sun P, Hou D, Sun H, Zhang H (2017b) Process and optimization implementation of the alpha-ENSC value. Math Methods Oper Res 86:293–308

    MathSciNet  Article  MATH  Google Scholar 

  • van den Brink R, van der Laan G(2008) Comment on sharing a polluted river. Mimeo

  • van den Brink R, van der Laan G, Moes N (2012) Fair agreements for sharing international rivers with multiple springs and externalities. J Environ Econ Manag 3:388–403

    Article  Google Scholar 

  • van den Brink R, He S, Huang J (2018) Polluted river problems and games with a permission structure. Games Econ Behav 108:182–205

    MathSciNet  Article  MATH  Google Scholar 

  • Young HP (1985) Monotonic solutions of cooperative games. Int Game Theory 14:65–72

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, China

    Panfei Sun, Dongshuang Hou & Hao Sun

  2. Applied Mathematics, University of Twente, P.O.Box 217, 7500AE, Enschede, The Netherlands

    Panfei Sun

Authors
  1. Panfei Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dongshuang Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongshuang Hou.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by China Scholarship Council (No. 201706290159) and National Natural Science Foundation of China (Nos. 71571143, 71871180 and 71601156). And we also appreciate the help of Dr. W. Kern and the anonymous reviewers, who has given us some constructive suggestions.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sun, P., Hou, D. & Sun, H. Responsibility and sharing the cost of cleaning a polluted river. Math Meth Oper Res 89, 143–156 (2019). https://doi.org/10.1007/s00186-019-00658-w

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-019-00658-w

Keywords

  • Cost allocation
  • Local Responsibility Sharing
  • Upstream Equal Sharing
  • Axiomatization