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Berge Solution Concepts in the Graph Model for Conflict Resolution

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Abstract

In this work, we generalize Berge solution concepts in the graph model for conflict resolution for conflicts with 2 or more decision makers (DMs). These concepts are useful to the analysis of interactions among DMs with altruistic behaviors. Berge behavior can be observed in conflicts where DMs act altruistically expecting others to reciprocate so that in the end it is in their own self-interests to behave in this way. The Berge stabilities presented are inspired on commonly used stability notions in the GMCR, such as: generalized metarationality, symmetric metarationality, sequential and symmetric sequential stabilities for conflicts with 2 or more DMs. We investigate the relation among these proposed concepts and also between such concepts and the standard ones. We also establish a relationship between Berge stability and coalition Nash stability of a modified conflict. The chicken and stag hunt games are used as examples to illustrate applications of the Berge stabilities in conflicts. In particular, we show that in the stag hunt game and in a modified version of it, Berge stabilities may be used to select a more desired Nash equilibria.

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Fig. 1

Source: Adapted from Rêgo and Vieira (2017)

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Source: Adapted from Fang et al. (1993)

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Notes

  1. Although this assumption is not explicitly made in the GMCR literature, it relies on an inertia assumption that is usual in the GMCR literature, since no coalition would make efforts to return to the initial state in a cycle.

  2. Following a suggestion of an anonymous referee, we changed the names of the stabilities as given in Vieira and Rêgo (2018) as follows: Credible Berge stability is now Weak Berge stability.

  3. Again, following a suggestion of an anonymous referee, we changed the names of the stabilities as given in Vieira and Rêgo (2018) as follows: (i) metarational general Berge is now Meta-Berge, (ii) symmetric metarational Berge is now Symmetric Meta-Berge, (iii) Sequential Berge is now Weak Meta-Berge and (iv) Symmetric Sequential Berge is now Weak Symmetric Meta-Berge.

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Acknowledgements

Funding was provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grant Nos. 307556/2017-4 and 428325/2018-1) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (001).

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Authors and Affiliations

  1. Universidade Federal do Ceará, Crateús, CE, 63700-000, Brazil

    Giannini Italino Alves Vieira

  2. Graduate Program in Modelling and Quantitative Methods, Fortaleza, CE, 60455-760, Brazil

    Giannini Italino Alves Vieira

  3. Statistics and Applied Math Department, Universidade Federal do Ceará, Fortaleza, CE, 60455-760, Brazil

    Leandro Chaves Rêgo

  4. Graduate Programs in Statistics and Management Engineering, Universidade Federal de Pernambuco, Recife, PE, 50740-550, Brazil

    Leandro Chaves Rêgo

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  1. Giannini Italino Alves Vieira
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  2. Leandro Chaves Rêgo
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Correspondence to Leandro Chaves Rêgo.

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Vieira, G.I.A., Rêgo, L.C. Berge Solution Concepts in the Graph Model for Conflict Resolution. Group Decis Negot 29, 103–125 (2020). https://doi.org/10.1007/s10726-019-09650-5

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