Obtaining the Minimum Winning Coalition Through Fuzzy Subsets and Binary Integer Programming (BIP)
13 Pages Posted: 23 Aug 2010
Date Written: August 22, 2010
Abstract
This paper deals with the problem of obtaining the minimum winning coalition and its stability level. The construction of a coalition is modeled using fuzzy sets by taking into account the position of the members on a set of political subjects of interest. Fuzzy sets allow to consider the nuances and uncertainties that constitute political thinking, making it possible to calculate the distance from each member to all others. We present a binary integer programming (BIP) model to find the winning coalition that minimizes the sum of distances between all its members. Nonetheless, the average time needed to solve this BIP model grows exponentially with the size of the group. This is the reason a heuristic approach is presented to solve this problem. The method solves iteratively, several simplified BIP models based on the concept of alpha-cuts. Each iteration minimizes the aggregated distance between coalition members, while the stability value alpha is maximized. The algorithm stops when a minimum winning coalition is found with the highest possible stability. The solution is then improved by a local search procedure that reaches the optimal solution in close to 90% of the simulated cases. The experimental results over these cases show that the heuristic method gives an excellent balance between speed and quality, obtaining, in average, solutions within 1% from the optimum.
Keywords: Coalition formation, fuzzy sets, distance, binary integer programming (BIP), local search
JEL Classification: D71, D20, D72
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