Decision SupportStability likelihood of coalitions in a two-stage cartel game: An estimation method
Introduction
Game theory models the idea of a group of decision makers deciding to agree on co-operation, because it increases their benefit by a concept that is called coalition formation. Stable coalitions have been described in this perspective by among others Yi in [13]. A Coalition of players is considered stable, if no player inside the coalition has the incentive to leave the coalition and no player outside the coalition has the incentive to join the coalition. More recently, the idea has been applied to get a feeling for incentives to form international climate agreements on reduction of the emission of greenhouse gasses (the so-called Kyoto discussions) in some empirical studies. Eykmans and Finus [3] show a study where the world is divided into six regions and the tendency to co-operate is analysed using estimated payoff models. Their study is elaborated and extended by Finus et al. [4] using new estimates of the payoff function for 12 world regions and algorithms developed in co-operation with the authors of the current paper. The final outcome of such an analysis is whether a coalition is stable yes or no. The current paper extends this analysis by formulating the tendency of co-operation by a so-called stability likelihood, taking into account the uncertainty of the estimated model parameters in the underlying payoff functions. Specifically an algorithm (MCDS) is introduced that efficiently combines two methods for estimating stability likelihood: Monte Carlo Simulation (MC) and Directional Simulation (DS). Results of applying the suggested approach to the case described in [4], gave new insight to economic researchers about economic incentives for coalition formation with respect to CO2 emission reduction.
The stability likelihood concept is introduced mathematically after a formal description of the coalition formation game in Section 2 as a probability of the coalition to be stable. Methods to estimate this probability assuming underlying distribution functions of the uncertain parameters are elaborated in Section 3. In Section 4 we discuss the application of MCDS to the so-called STACO model. This is followed by the results in Section 5, the conclusions in Section 6 and recommendations in Section 7.
Section snippets
Cartel coalition stability
In this section the cartel coalition game, such as described in [13], [4], [3] is formalised in a mathematical way and we introduce the concept of stability likelihood.
Stability likelihood estimation algorithm
An algorithm, called MCDS, is designed for estimating stability likelihood of a coalition strategy c. This algorithm estimates the stability likelihood at an parameterised accuracy level: The estimate of P(c) will have a standard error of at most tarSE. The algorithm consists of two different methods for probability estimation, namely Monte Carlo Simulation and Directional Simulation. The algorithm can be called in three modes: MC, DS, or efficiently combine MC and DS.
The DS method is suitable
Test case
To test the MCDS algorithm, we made an implementation to experiment on the so-called STACO model, which is described in [4]. The STACO project investigates the formation and stability of international climate agreements. The basic structure of the STACO models consists of interacting regions that (i) choose to join an international climate agreement or not; and (ii) choose their optimal climate policy given the coalition formed. The regions are characterised by their abatement costs and damage
Results
In the paper of Finus et al. [4], a stable coalition between the European Community (EEC) and Japan was found in three different scenario’s, which we will call the “120%”, “200%” and “300%” scenario. These three scenario’s refer to three different assumptions related to the benefit term in the payoff model. For details about these scenario’s we refer to the paper [4]. Here, to illustrate the concept of stability likelihood, we base our experiments on the parameter settings related to these
Conclusions
An algorithm has been designed for estimating coalition stability likelihood, in the context of a two-stage cartel game. Coalition stability conclusions are based on the Nash Equilibrium condition. The algorithm computes an estimate for the coalition stability likelihood of a coalition c, at an accuracy level of tarSE, given that the payoff model and stochastic model of the model parameter are available.
For the general situation, where the stochastic model parameter values can have any type of
Recommendations
When implementing the concept of stability likelihood, we suggest to start with an MC implementation, since we experienced that the MC implementation is shorter, faster, and more versatile since MC allows for a wider range of probability distributions.
On the other hand, we are still convinced that the DS method has the potential to be more efficient. Not only more experimental cases can be considered, but also the specific case can be analysed. It is known that estimation of the probability
Acknowledgements
The authors thank the members of the STACO working group for their contribution to the paper and Prof. Dr. P. van Beek and the anonymous referees for their valuable comments on the manuscript.
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