# Sharing a River with Downstream Externalities

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A River Sharing Model with Downstream Pollution Externalities

**Proposition**

**1**

**.**Given a river sharing problem $(N,e,c,d)$ it is a dominant strategy for all agents $i\in N$ not to abate at all, i.e., ${x}_{i}=0$.

**Proof.**

**Proposition**

**2**

**.**Given a river sharing problem $(N,e,c,d)$ there exists a unique vector ${x}^{\star}$, which is the solution to the following constrained minimization problem:

**Proof.**

## 3. Coalitions and Cost Upper Bounds

**Proposition**

**3.**

## 4. Cost Lower Bounds

**Proposition**

**4.**

## 5. The Downstream Incremental Distribution

**Theorem**

**1**

**.**The downstream incremental distribution (DID) ${z}^{\star}$ is the only river sharing agreement satisfying the non-cooperative core upper bounds $v(S)$ and the cost lower bounds $a(S)$ for any coalition S.

**Proof.**

## 6. Discussion and Extensions

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Proposition 3

## Appendix B. Proof of Proposition 4

## References

- Ambec, S.; Sprumont, Y. Sharing a River. J. Econ. Theory
**2002**, 107, 453–462. [Google Scholar] [CrossRef] - Van den Brink, R.; Van der Laan, G.; Vasil’ev, V. Component efficient solutions in line-graph games with applications. Econ. Theory
**2007**, 33, 349–364. [Google Scholar] [CrossRef] - Van den Brink, R.; Van der Laan, G.; Moes, N. Fair agreements for sharing international rivers with multiple springs and externalities. J. Environ. Econ. Manag.
**2012**, 63, 388–403. [Google Scholar] [CrossRef] [Green Version] - Ansink, E.; Weikard, H.P. Sequential sharing rules for river sharing problems. Soc. Choice Welf.
**2012**, 38, 187–210. [Google Scholar] [CrossRef] - Demange, G. On group stability in hierarchies and networks. J. Political Econ.
**2004**, 112, 754–778. [Google Scholar] [CrossRef] - Ni, D.; Wang, Y. Sharing a polluted river. Games Econ. Behav.
**2007**, 60, 176–186. [Google Scholar] [CrossRef] - Gengenbach, M.F.; Weikard, H.P.; Ansink, E. Cleaning a river: An analysis of voluntary joint action. Nat. Resour. Model.
**2010**, 23, 565–590. [Google Scholar] [CrossRef] - Van der Laan, G.; Moes, N. Collective decision making in an international river pollution model. Nat. Resour. Model.
**2016**, 29, 374–399. [Google Scholar] [CrossRef] - Ambec, S.; Ehlers, L. Sharing a river among satiable agents. Games Econ. Behav.
**2008**, 64, 35–50. [Google Scholar] [CrossRef] [Green Version] - Herings, P.; Van Der Laan, G.; Talman, D. The average tree solution for cycle-free graph games. Games Econ. Behav.
**2008**, 62, 77–92. [Google Scholar] [CrossRef] [Green Version] - Chander, P.; Tulkens, H. The core of an economy with multilateral environmental externalities. Int. J. Game Theory
**1997**, 26, 379–401. [Google Scholar] [CrossRef] [Green Version] - Aumann, R. The core of a cooperative game without side payments. Trans. Am. Math. Soc.
**1961**, 98, 539–552. [Google Scholar] [CrossRef] - Scarf, H. On the existence of a coopertive solution for a general class of N-person games. J. Econ. Theory
**1971**, 3, 169–181. [Google Scholar] [CrossRef] - Shapley, L. Cores of convex games. Int. J. Game Theory
**1971**, 1, 11–26. [Google Scholar] [CrossRef]

1. | As we characterize the game in cost space instead of utility space, one might rather speak of a “transferable cost” game. |

2. | In ref. [8] agents choose pollution levels to maximize the difference between benefits from individual pollution levels and damages depending on the vector of pollution levels over all agents, while in our model framework agents choose abatement levels to minimize the sum of the costs of individual abatement and the damage from aggregated pollution (see Section 2). Our model set-up could also be expressed in terms of pollution levels, yielding a model structure formally identical to [8]. |

3. | This concept of $\gamma $-core was first derived by [11]. Note that in our particular model set-up, in which emission abatement choices of ${x}_{i}=0$ are dominant strategies for all agents $i\notin S$, the $\gamma $-core is also identical to the $\alpha $-core (where agents outside the coalition are assumed to react to the strategy of the coalition such as to minimize the pay-off of the coalition S) and the $\beta $-core (where agents outside the coalition S are assumed to move first and choose emissions such as to minimize the maximum pay-off of coalition S), see [12,13]. |

4. | We gratefully acknowledge that we owe this idea to an anonymous reviewer. |

5. | In fact, the core of a convex game lies within a convex polyhedron, the vertices of which are determined by the set of marginal contribution vectors, see [14]. |

6. |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Steinmann, S.; Winkler, R.
Sharing a River with Downstream Externalities. *Games* **2019**, *10*, 23.
https://doi.org/10.3390/g10020023

**AMA Style**

Steinmann S, Winkler R.
Sharing a River with Downstream Externalities. *Games*. 2019; 10(2):23.
https://doi.org/10.3390/g10020023

**Chicago/Turabian Style**

Steinmann, Sarina, and Ralph Winkler.
2019. "Sharing a River with Downstream Externalities" *Games* 10, no. 2: 23.
https://doi.org/10.3390/g10020023