Abstract
Consider a group of agents located along a polluted river where every agent must pay a certain cost for cleaning up the polluted river. Following the model of Ni and Wang (2007), we propose weaker versions of some of their axioms, and introduce two classes of cost sharing methods. We first show that the upstream equal sharing method (for short, UES method) is characterized by relaxing independence of upstream costs and no blind cost appearing in Ni and Wang (2007). After that, we propose the classes of equal upstream responsibility methods (for short, EUR methods) and weighted upstream sharing methods (for short, WUS methods), which generalizes the local responsibility sharing method (for short, LRS method) and the UES method. We provide two axiomatizations of the class of EUR methods by replacing upstream symmetry for the UES method with weak upstream symmetry. Meanwhile, we also provide two axiomatizations of the class of WUS methods by introducing two other weak versions of upstream symmetry. Finally, we define a pollution cost-sharing game, and show that each of the EUR methods and WUS methods is obtained as a Harsanyi solution for these pollution cost-sharing games. Moreover, we also show that the average of the LRS method and UES method, referred to as the compromise method, coincides with the Shapley value of this game.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We explicitly mention that there are no property rights with respect to ownership of an international river. This is reflected in water allocation principles from International Water Law, such as ATS and UTI that are applied in this paper, but also from ‘milder’ principles such as Territorial Intergation of all Basin States (TIBS). Some water allocation principles might be seen as reflecting property rights (such as ATS), but these are still different from property rights.
For the more general multiple spring rivers, Dong, Dong et al. (2012) also introduced the downstream equal sharing method (for short, DES method) that allocates the cost of a segment equally among this segment and each of its downstream segments. The DES method on \(\mathcal {P}^N\) is defined by \(DES_i(N,c)=\sum _{k=1}^i\frac{1}{n-k+1}c_k\) for all \((N,c)\in \mathcal {P}^N\) and \(i\in N\).
In van den Brink et al. (2018), it is shown that the UES method coincides with the permission value of a game with a permission structure where the game is the LRS game \(\langle N, v^L\rangle\) and the linear order of the players is determined by the flow of the river.
In the context of cooperative TU games, Casajus (2019) introduces a qualitative weaker version of symmetry, where payoffs of symmetric players are required to have the same sign instead of the usual stronger requirement that these payoffs should be equal.
We take the sum \(\sum _{i=n+1}^n ....\) to be equal to 0.
Given any rational \(r_k\), there must exist two integers \(a, b\in \mathbb {N}\), \(a \ne 0\), such that \(r_k=\frac{b}{a}\). Then by additivity, we have \(\psi (N,r_k c)=\psi (N,\frac{b}{a} c)=b\psi (N,\frac{1}{a} c)=\frac{b}{a}\cdot a\psi (N,\frac{1}{a} c)=\frac{b}{a}\psi (N,\frac{a}{a} c)=r_k\psi (N, c)\).
This follows since \(v^{cd}(S)=v^c(N)-v^c(N\backslash S)=\sum _{i \in N} c_i - \sum _{i \in N \setminus S} c_i - \sum _{i\in S}\frac{\vert \bar{P}_i(N \setminus S)\vert }{\vert \bar{P}_i(N)\vert }c_i = \sum _{i \in S} c_i - \sum _{i\in S}\frac{\vert \bar{P}_i(N)\vert - \vert \bar{P}_i(S)\vert }{\vert \bar{P}_i(N)\vert }c_i = \sum _{i \in S} (1-\frac{\vert \bar{P}_i(N)\vert - \vert \bar{P}_i(S)\vert }{\vert \bar{P}_i(N)\vert })c_i = \sum _{i\in S}\frac{\vert \bar{P}_i(S)\vert }{\vert \bar{P}_i(N)\vert }c_i\).
A solution \(\varphi\) satisfies self-duality, if for every \(\langle N,v\rangle \in \mathcal {G}^N\), it holds that \(\varphi (N,v)=\varphi (N,v^d)\) where \(\langle N,v^d\rangle\) is the dual game of \(\langle N,v\rangle\).
References
Alcalde-Unzu J, Gómez-Rúa M, Molis E (2015) Sharing the costs of cleaning a river: the Upstream Responsibility rule. Games and Economic Behavior 90:134–150
Ambec S, Ehlers L (2008) Sharing a river among satiable agents. Games and Economic Behavior 64(1):35–50
Ambec S, Sprumont Y (2002) Sharing a river. Journal of Economic Theory 107:453–462
Barrett S (1994) Conflict and cooperation in managing international water resources. Working paper 1303. World Bank, Washington
Béal S, Ghintran A, Rémila É, Solal P (2013) The river sharing problem: A survey. International Game Theory Review 15(03):1340016
Brânzei R, Fragnelli V, Tijs S (2002) Tree-connected peer group situations and peer group games. Mathematical Methods of Operations Research 55:93–106
Beard R (2011) The river sharing problem: A review of the technical literature for policy economists. MPRA Paper No. 34382
Casajus A (2019) Relaxations of symmetry and the weighted Shapley values. Economics Letters 176:75–78
Chun Y, Hokari T (2007) On the coincidence of the Shapley value and the nucleolus in queueing problems. Seoul Journal of Economics 20(2):223–237
Chun Y, Lee J (2012) Sequential contributions rules for minimum cost spanning tree problems. Mathematical Social Sciences 64(2):136–143
Deng X, Papadimitriou CH (1994) On the complexity of cooperative solution concepts. Mathematics of Operations Research 19(2):257–266
Dong B, Ni D, Wang Y (2012) Sharing a polluted river network. Environmental and Resource Economics 53(3):367–387
Godana B (1985) Africa’s shared water resources. France Printer, London
Gómez-Rúa M (2013) Sharing a polluted river through environmental taxes. SERIEs 4(2):137–153
Gudmundsson J, Leth Hougaard J, Ko C Y (2018) River sharing: Implementation of efficient water consumption. SSRN 3128889. Available at SSRN: https://doi.org/10.2139/ssrn.3128889
Harsanyi JC (1959) A bargaining model for the cooperative \(n\)-person game. In: Contributions to the Theory of Game. Ed. by H Kuhn and A Tucked. Vol. 4. Princeton University Press, 1959, pp. 325-355
Hou D, Driessen T, Sun H (2015) The Shapley value and the nucleolus of service cost savings games as an application of 1-convexity. The IMA Journal of Applied Mathematics 80(6):1799–1807
Hou D, Lardon A, Sun P, Xu G (2019) Sharing a polluted river under waste flow control. Economie, Gestion (GREDEG CNRS), University of Nice Sophia Antipolis, Groupe de REcherche en Droit
Hu CC, Tsay MH, Yeh CH (2018) A study of the nucleolus in the nested cost-sharing problem: Axiomatic and strategic perspectives. Games and Economic Behavior 109:82–98
Kalai E, Samet D (1987) On weighted Shapley values. International Journal of Game Theory 16(3):205–222
Kilgour D M, Dinar A (1995) Are stable agreements for sharing international river waters now possible? Policy Research Working Paper 1474. World Bank, Washington
Ni D, Wang Y (2007) Sharing a polluted river. Games and Economic Behavior 60(1):176–186
Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM Journal on Applied Mathematics 17(6):1163–1170
Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the Theory of Games II. Princeton University Press, pp 307–317
Steinmann S, Winkler R (2019) Sharing a river with downstream externalities. Games 10(2):23
Sun P, Hou D, Sun H (2019) Responsibility and sharing the cost of cleaning a polluted river. Mathematical Methods of Operations Research 89(1):143–156
Tijs SH (1981) Bounds for the core and the \(\tau\)-value. In: Moeschlin O, Pallaschke D (eds) Game Theory and Mathematical Economics. North-Holland Publishing Company, Amsterdam, pp 123–132
van den Brink R, He S, Huang JP (2018) Polluted river problems and games with a permission structure. Games and Economic Behavior 108:182–205
van den Nouweland A, Borm P, van Golstein Brouwers W, Groot Bruinderink R (1996) A game theoretic approach to problems in telecommunication. Management Science 42:294–303
Vasil’ev VA (1978) Support function of the core of a convex game. Optimizacija Vyp 21:30–35
Acknowledgements
The authors thank an associate editor and two anonymous referees for very helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 72071159), National Key R &D Program of China (Grant No. 2021YFA1000400) and China Scholarship Council (Grant Nos. 202006290157).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, W., Xu, G. & van den Brink, R. Two new classes of methods to share the cost of cleaning up a polluted river. Soc Choice Welf 61, 35–59 (2023). https://doi.org/10.1007/s00355-022-01439-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00355-022-01439-x