Stability likelihood of coalitions in a two-stage cartel game: An estimation method

Abstract

Existing formulations of a cartel game aim at finding stable coalitions, i.e. a coalition is labelled stable or not stable. Uncertainty about the underlying structure and/or parameter values gives rise to sensitivity or uncertainty analysis. In this paper we follow a probabilistic robustness concept: What is the probability a product, design or policy really fulfils the requirements or properties it is expected to. Following this idea, we introduce the concept of stability likelihood: What is the probability a coalition can be labelled as stable. Methods are described based on Monte Carlo Simulation and Directional Simulation to estimate such a probability and we illustrate the performance for several cases.

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.