Abstract
We study partial coalition formation and the strategic timing of membership of an IEA for environmental issues in the Coalitional Bargaining Game (CBG) of Gomes (Econometrica 73:1329–1350, 2005). We apply the CBG to a specific river sharing problem with two symmetric upstream agents, each at a tributary, and one downstream agent located at the junction of tributaries. We identify five regions in the parameter space of a discount factor and a productivity parameter for water. In one region the grand coalition always forms immediately. In two other regions, immediate formation of the grand coalition and gradual coalition formation both occur with positive probability. In another two regions only gradual coalition formation occurs. In one of these latter regions, a region with discount factors close to one, both upstream agents form a monopoly with positive probability. Formation of the monopoly persists in the limit as the discount factor goes to one.
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Notes
For technical simplicity, Gomes (2005; 2006) assumes that the proposing agent buys the ownership rights of his proposed coalition. In order to buy out all other agents in this coalition, it is implicit that the proposer is able to finance the amount needed. Alternatively, one might consider all agents in a coalition as shareholders and the shares determine the payoffs. For the context of IEAs with streams of payoffs over time, the alternative interpretation is more appropriate but also it is more elaborate. Therefore, we will closely follow Gomes (2005; 2006).
These benefits can be determined prior to the analysis and summarized as the partition function form. This form is taken as the primitive of many studies.
Here we do not allow \(e_{i}=0\) for \(i=1,2\). An upstream agent is irrelevant when it has zero water resources, because water flows from upstream to downstream. Furthermore, the water resource of agent 3 can only be utilized by agent 3, while the water of an upstream agent can be used by him and agent 3.
We write \(d_{\{i\}}\) as \(d_{i}\) and \(d_{\{i,j\}}\) as \(ij\) without abuse of notation. This short notation also follows later on for \(v_{ij}\), \(p_{ij}\), \( e^{0}(ij)\), etc.
In Gomes (2005; 2006), \(p_{C}\in \left( 0,1\right) \) is arbitrary. Also, \(p_{C}\) reflects the relative importance of coalition \(C\) in setting the agenda. Then, \(p_{C}<\sum _{k\in C}~p_{k}\) would mean that the relative importance of coalition \(C\) is smaller than the sum of those of its individual members, which is a special case elaborated in Gomes (2006) where \(p_{C}\) is taken proportional to the number of coalitions in \(\pi \), i.e. \( p_{C}=1/\left| \pi \right| \). In general, the choice \(<\), or alternatively \(>\) or \(=\) should be motivated by the application under consideration. We assume that the relative importance of individual members is inalienable and preserved by merging, or alternatively merging has a neutral effect on setting the agenda for (non)members.
The amount of financial transfer will be analyzed in detail in Sect. 3.
Note that it may be possible that coalition \(\{i,j\}\) is not able to pay the lump-sum transfer immediately. For instance, in our river problem, \(\{1,2\}\) realizes 0 payoff in the current period. In this situation, we consider the financial transfer as a promised present value. So we allow the possibility to pay the financial transfer in the future periods. For instance, agent 1 proposes to agent 2, and the financial transfer can be realized in the future but has a present value equal to \(t_2^0\).
To see this, \(t_{i}^{0}+t_{j}^{0}+t_{k}^{0}= \delta \left( v_{i}^{0}+v_{j}^{0}+v_{k}^{0}\right) \le \delta v_{N}^{N}<v_{N}^{N}.\)
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This research is financially supported by Netherlands Organization for Scientific Research NWO Grant 022.003.025. The authors thank Hans-Peter Weikard, Arantza Estévez-Fernández, Randolph Sloof, the anonymous referees and the participants of seminars at Wageningen University, HEC Montréal, University of Amsterdam, VU University Amsterdam and Tinbergen Institute Amsterdam. All derivations are given as an online appendix.
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Houba, H., van der Laan, G. & Zeng, Y. International Environmental Agreements for River Sharing Problems. Environ Resource Econ 62, 855–872 (2015). https://doi.org/10.1007/s10640-014-9862-0
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DOI: https://doi.org/10.1007/s10640-014-9862-0
Keywords
- International environmental agreements
- River sharing problem
- Negotiations
- Coalitional bargaining game
- Markov perfect equilibrium
- Efficiency
- Monopoly