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Values for level structures with polynomial-time algorithms, relevant coalition functions, and general considerations

Besner, Manfred (2020): Values for level structures with polynomial-time algorithms, relevant coalition functions, and general considerations.

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Abstract

Exponential runtimes of algorithms for TU-values like the Shapley value are one of the biggest obstacles in the practical application of otherwise axiomatically convincing solution concepts of cooperative game theory. We discuss how the hierarchical structure of a level structure improves the runtimes compared to an unstructured set of players. As examples, we examine the Shapley levels value, the nested Shapley levels value, and, as a new LS-value, the nested Owen levels value. Polynomial-time algorithms for these values (under ordinary conditions) are provided. Furthermore, we introduce relevant coalition functions where all coalitions which are not relevant for the payoff calculation have a Harsanyi dividend of zero. By these coalition functions, our results shed new light on the computation of values of the Harsanyi set and many values from extensions of this set.

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