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.
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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
Ambec S, Sprumont Y (2002) Sharing a river. J Econ Theory 107:453–462
Ambec S, Ehlers L (2008) Sharing a river among satiable agents. Games Econ Behav 64:35–50
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
Godana B (1985) Africa’s shared water resources. France Printer, London
Khmelnitskaya AB (2010) Values for rooted-tree and sink-tree digraphs games and sharing a river. Theory Decis 69:657–669
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
Ni D, Wang Y (2007) Sharing a polluted river. Games Econ Behav 60:176–186
Shapley LS (1953) A value for n-person games. Ann Math Stud 28:307–317
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
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
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
van den Brink R, He S, Huang J (2018) Polluted river problems and games with a permission structure. Games Econ Behav 108:182–205
Young HP (1985) Monotonic solutions of cooperative games. Int Game Theory 14:65–72
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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.
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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
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DOI: https://doi.org/10.1007/s00186-019-00658-w