Computing the nucleolus of cooperative games

Abstract

The computation of the nucleolus, one of the most widespread single valued solution concept of cooperative game theory has developed greatly, starting from an immensely complex lexicographical optimisation problem to solving a sequence of LPs. Focusing on the general case, in this thesis we provide an extensive review of the various existing computational methods in the literature, also providing a categorisation of these methods in a new aspect, mainly based on the process of updating in between LPs, that influences the required number of iterations. The disparity in the number of iterations between primal and dual methods lead to the classical primal-dual trade-off: one method requires minimal number of iterations (primal), while solving the large LPs in the other is easier (dual). We introduce a hybrid variant, that inherits good qualities from each method, dissolving the classical trade-off. After showing how to reduce the number of iterations for the dual sequence, we introduce a conceptually new, practical approach to one of the main tasks involved in the sequential LP framework, that is finding all coalitions in the span of some coalitions. We argue that it could be beneficial in practice not to do this, possibly leading to a decrease in computational time despite the increasing number of iterations, pivots and subroutine iterations. Through analysing the numerical results, it becomes clear, that the literature not yet considered every aspect of the primal-dual trade-off: namely warm starting, favouring primal over the dual. Balancedness is related to the nucleolus early on through the Kohlberg criterion verifying whether a solution is the nucleolus or not. Our results on balancedness lay the foundations for the main contribution of the thesis, a new constructive approach to compute the nucleolus. A primal based active-set method with a dual subroutine, that benefits from every advantage of primal methods, including efficient warm starting, however it does not suffer from it’s disadvantage, by solving large LPs without explicitly formulating them. Results on balancedness further allow us to improve on the original Kohlberg criterion. The new active-set method outperforms other classical sequential LP models and the dual counterpart of the proposed method, based on extensive amount of numerical testing. Another important contribution is providing open-source codes for all algorithms and instances involved in creating those numerical results.

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ORCID for Joerg Fliege: orcid.org/0000-0002-4459-5419

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