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Fuzzy Coalitions and Fuzziness of Games

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Book cover Models for Cooperative Games with Fuzzy Relations among the Agents

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 355))

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Abstract

In this chapter we review basic concepts about fuzzy sets. We introduce the concept of fuzzy coalition and also of game with fuzzy coalitions. Although our study is interrelated to these games we only need some superficial notes about them. Two generalizations of the Shapley value for games with fuzzy coalitions are defined. Given a crisp cooperative game there are several ways to extend it to a game with fuzzy coalitions. We analyze three very known. The multilinear extension is a probabilistic version using a fuzzy coalition as a set of probability distributions for the players and then calculating an expected worth for each coalition. The proportional extension and the Choquet one use the fuzzy coalition as participation levels for the players in two different ways depending on the players’ behavior.

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Notes

  1. 1.

    This T-norm is usually named Lukasiewicz norm.

  2. 2.

    Product is understood as the usual probabilistic intersection.

  3. 3.

    The notion of fuzzy game is also used for cooperative games with fuzzy payoffs, see Mares [12]. The concept of game on \([0,1]^N\) can be studied in a different way as game with overlapping coalitions, see Chalkiadakis et al. [8].

  4. 4.

    Aubin named it pareto optimality.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada II, Universidad de Sevilla, Seville, Spain

    Andrés Jiménez-Losada

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  1. Andrés Jiménez-Losada
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Correspondence to Andrés Jiménez-Losada .

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Jiménez-Losada, A. (2017). Fuzzy Coalitions and Fuzziness of Games. In: Models for Cooperative Games with Fuzzy Relations among the Agents. Studies in Fuzziness and Soft Computing, vol 355. Springer, Cham. https://doi.org/10.1007/978-3-319-56472-2_2

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  • DOI: https://doi.org/10.1007/978-3-319-56472-2_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-56471-5

  • Online ISBN: 978-3-319-56472-2

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