Skip to main content
SpringerLink
Log in

A value for communication situations with players having different bargaining abilities

  • S.I. : SING 14
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The aim of this paper is to extend the Myerson value (Myerson in Math Oper Res 2:225–229, 1977) to situations in which players in a TU-game, in addition to having cooperation possibilities restricted by a graph, also have different bargaining abilities. Then, we will associate to each player in a communication situation a weight in the interval [0, 1] that measures his bargaining ability. A unitary weight corresponds to a fully cooperative player whereas a null weight corresponds to a player that is not willing to cooperate in any way. Intermediate values modulate the bargaining ability. We modify the original TU-game to a new game which is, in turn, a modification of the Myerson’s graph-restricted game. We will assume that the reduction in the will to cooperate implies that players can not obtain the total dividend of the connected coalitions which must be discounted by an appropriate factor. Then, we propose as a solution for these situations the Shapley value (Shapley, in: Kuhn, Tucker (eds) Annals of mathematics studies, Princeton University Press, Princeton, 1953) of the modified game. This solution extends the Myerson value (and also the Shapley value). Moreover it satisfies monotonicity in the weights. Different characterizations of this rule can be obtained. They are based on properties as bargaining component efficiency, fairness, balanced contributions and balanced bargaining ability contributions, and thus they are parallel to those more prominent existing in the literature for the Myerson value.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Although Shapley (1953a) suggested the bargaining ability of each player as a possible interpretation of his weight, Owen (1968) showed the lack of weight monotonicity in the weighted Shapley values and provided another interpretation for weights that “can better be thought of as coefficients of slowness to reach a decision” (Owen 1968).

  2. The framework of this paper, graph-restricted games with players having weights, is similar to the one in Haeringer (1999).

  3. Aubin (1981) also defined cooperative fuzzy games in which each fuzzy coalition obtains a reward by means of a characteristic function which is assumed positively homogeneous.

  4. In the definition of fuzzy communication structure in Jiménez-Losada et al. (2010) also the graphs were restricted to being cycle-free.

  5. Details on Choquet integral for fuzzy measures can be found in Choquet (1953), Sugeno and Murofushi (1987), and Grabisch et al. (1992).

  6. In fact, \({\bar{\mu }}\) satisfies the following generalized version of the balanced bargaining ability contributions property: given \((N,v,\varvec{\lambda },\gamma ) \in {\mathcal {C}}{\mathcal {S}}_{\Lambda }^{N}\), and \(i,j \in N\)

    $$\begin{aligned} \bar{\mu _{i}}(N,v,\varvec{\lambda },\gamma ) - \bar{\mu _{i}}(N,v,\varvec{\lambda }^{-j,c},\gamma ) = \bar{\mu _{i}}(N,v,\varvec{\lambda },\gamma ) - \bar{\mu _{i}}(N,v,\varvec{\lambda }^{-i,c},\gamma ), \end{aligned}$$

    where for \(k=i,j\), \(\varvec{\lambda }^{-k,c} = (\lambda _1^{-k,c},\lambda _2^{-k,c},\ldots ,\lambda _n^{-k,c})\) is given by

    $$\begin{aligned} \lambda _l^{-k,c}=\left\{ \begin{array}{ll} \lambda _{l}, &{} \quad \mathrm{if }\, l \not = k\\ c<\lambda _{k}, &{} \quad \mathrm{if }\, l = k,\end{array}\right. \end{aligned}$$

    for \(l=1,\ldots ,n.\)

References

  • Algaba, E., Bilbao, J. M., Borm, P., & López, J. J. (2001). The Myerson value for union stable systems. Mathematical Methods of Operations Research, 54, 359–371.

    Article  Google Scholar 

  • Aubin, J. P. (1981). Cooperative fuzzy games. Mathematics of Operation Research, 6, 1–13.

    Article  Google Scholar 

  • Béal, S., Casajus, A., & Huettner, F. (2018). Efficient extensions of communication values. Annals of Operations Research, 264, 41–56.

    Article  Google Scholar 

  • Bergantiños, G., & Sánchez, E. (2001). Weighted Shapley values for games in generalized characteristic function form. TOP, 9(1), 55–67.

    Article  Google Scholar 

  • Bilbao, J. M., & Edelman, P. H. (2000). The Shapley value on convex geometries. Discrete Applied Mathematics, 103(1), 33–40.

    Article  Google Scholar 

  • Borm, P., Owen, G., & Tijs, S. (1992). On the position value for communication situations. SIAM Journal on Discrete Mathematics, 5(3), 305–320.

    Article  Google Scholar 

  • Calvo, E., Lasaga, J., & van den Nouweland, A. (1999). Values of games with probabilistic graphs. Mathematical Social Sciences, 37, 79–95.

    Article  Google Scholar 

  • Casajus, A. (2009). Networks and outside options. Social Choice and Welfare, 32, 1–13.

    Article  Google Scholar 

  • Choquet, G. (1953). Theory of capacities. Annales de I’institut Fourier, 5, 131–295.

    Article  Google Scholar 

  • del Pozo, M., Manuel, C., González-Arangüena, E., & Owen, G. (2011). Centrality in directed social networks. A game theoretic approach. Social Networks, 33, 191–200.

    Google Scholar 

  • Fernández, J. R., Algaba, E., Bilbao, J. M., Jiménez, A., Jiménez, N., & López, J. J. (2002). Generating functions for computing the Myerson value. Annals of Operations Research, 109, 143–158.

    Article  Google Scholar 

  • Gallardo, J. M., Jiménez, N., & Jiménez-Losada, A. (2018). A Shapley distance in graphs. Information Sciences, 432, 269–277.

    Article  Google Scholar 

  • Gilles, R. P., Owen, G., & van den Brink, R. (1992). Games with permission structures: The conjunctive approach. International Journal of Game Theory, 20(3), 277–293.

    Article  Google Scholar 

  • Gómez, D., González-Arangüena, E., Manuel, C., Owen, G., del Pozo, M., & Tejada, J. (2003). Centrality and power in social networks: A game theoretic approach. Mathematical Social Sciences, 46, 27–54.

    Article  Google Scholar 

  • Gómez, D., González-Arangüena, E., Manuel, C., Owen, G., & del Pozo, M. (2008). A value for probabilistic communication situations. European Journal of Operational Research, 190, 539–556.

    Article  Google Scholar 

  • González-Arangüena, E., Manuel, C., & del Pozo, M. (2015). Values of games with weighted graphs. European Journal of Operational Research, 243, 248–257.

    Article  Google Scholar 

  • González-Arangüena, E., Manuel, C., Owen, G., & del Pozo, M. (2017). The within groups and the between groups Myerson values. European Journal of Operational Research, 257, 586–600.

    Article  Google Scholar 

  • Grabisch, M., Murofushi, T., & Sugeno, M. (1992). Fuzzy measure of fuzzy events defined by fuzzy integrals. Fuzzy Sets and Systems, 50, 293–313.

    Article  Google Scholar 

  • Haeringer, G. (1999). Weighted Myerson value. International Game Theory Review, 1, 187–192.

    Article  Google Scholar 

  • Haeringer, G. (2006). A new weight scheme for the Shapley value. Mathematical Social Science, 52, 88–98.

    Article  Google Scholar 

  • Harsanyi, J. C. (1959). A bargaining model for cooperative n-person games. In A. W. Tucker & R. D. Luce (Eds.), Contributions to the theory of games IV (pp. 325–355). Princeton: Priceton University Press.

    Google Scholar 

  • Hart, S., & Mas-Colell, A. (1989). Potential, value, and consistency. Econometrica, 57, 589–614.

    Article  Google Scholar 

  • Heitzig, J., Donges, J. F., Zou, Y., Marwan, N., & Kurths, J. (2012). Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes. The European Physical Journal, 85, 38.

    Article  Google Scholar 

  • Jackson, M. O. (2005). Allocation rules for network games. Games and Economic Behavior, 51(1), 128–154.

    Article  Google Scholar 

  • Jackson, M. O., & Wolinsky, A. (1996). A strategic model of social and economic networks. Journal of Economic Theory, 71, 44–74.

    Article  Google Scholar 

  • Jiménez-Losada, A., Fernández, J. R., & Ordóñez, M. (2013). Myerson values for games with fuzzy communication structures. Fuzzy Sets and Systems, 213, 74–90.

    Article  Google Scholar 

  • Jiménez-Losada, A., Fernández, J. R., Ordóñez, M., & Grabisch, M. (2010). Games on fuzzy communication structures with Choquet players. European Journal of Operational Research, 207, 836–847.

    Article  Google Scholar 

  • Kalai, E., & Samet, D. (1987). On weighted Shapley values. International Journal of Game Theory, 16, 205–222.

    Article  Google Scholar 

  • Khmelnitskaya, A., Selçuk, Ö., & Talman, D. (2016). The Shapley value for directed graph games. Operations Research Letters, 44(1), 143–147.

    Article  Google Scholar 

  • Manuel, C., & Martín, D. (2020). A monotonic weighted Shapley value. Group Decision and Negotiation, 29, 627–654.

    Article  Google Scholar 

  • Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2, 225–229.

    Article  Google Scholar 

  • Myerson, R. B. (1980). Conference structures and fair allocation rules. International Journal of Game Theory, 9, 169–182.

    Article  Google Scholar 

  • Navarro, F. (2018). Necessary players, Myerson fairness and the equal treatment of equals. Annals of Operations Research, 280, 111–119.

    Article  Google Scholar 

  • Owen, G. (1968). A note on the Shapley value. Management Science, 14, 731–732.

    Article  Google Scholar 

  • Owen, G. (1986). Values of graph-restricted games. SIAM Journal on Algebraic and Discrete Methods, 7, 210–220.

    Article  Google Scholar 

  • Shapley, L. S. (1953a). Additive and nom-additive set functions. Ph.D. thesis. Princeton University.

  • Shapley, L. S. (1953b). A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Annals of mathematics studies (Vol. 28, pp. 307–317). Princeton: Princeton University Press.

    Google Scholar 

  • Sugeno, M., & Murofushi, T. (1987). Choquet’s integral as an integral form for a general class of fuzzy measures. Preprints of Second IFSA Congress, 1, 408–411.

    Google Scholar 

  • Tsurumi, M., Tanino, T., & Inuiguchi, M. (2001). A Shapley function on a class of cooperative fuzzy games. European Journal of Operational Research, 129, 596–618.

    Article  Google Scholar 

  • van den Brink, R., & Gilles, R. P. (1996). Axiomatizations of the conjunctive permission value for games with permission structures. Games and Economic Behaviour, 12, 112–126.

    Google Scholar 

  • van den Brink, R., González-Arangüena, E., Manuel, C., & del Pozo, M. (2014). Order monotonic solutions for generalized characteristic functions. European Journal of Operational Research, 238(3), 786–796.

    Article  Google Scholar 

  • van den Nouweland, A., Borm, P., & Tijs, S. (1992). Allocation rules for hypergraph communication situations. International Journal of Game Theory, 20, 255–268.

    Article  Google Scholar 

  • Winter, E. (1992). The consistency and potential for values of games with coalition structure. Games and Economic Behavior, 4, 132–144.

    Article  Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy Sets. Information and Control, 8, 338–353.

    Article  Google Scholar 

  • Zemp, D. C., Wiedermann, M., Kurths, J., Rammig, A., & Donges, J. F. (2014). Node-weighted measures for complex networks with directed and weighted edges for studying continental moisture recycling. A Letters Journal Exploring The Frontiers of Physics, 107, 58005.

    Google Scholar 

  • Zou, Z., Zhang, Q., Borkotokey, S., & Yu, X. (2017). The extended Shapley value for generalized cooperative games under precedence constraints. Operational Research, 20, 899–925.

    Article  Google Scholar 

Download references

Acknowledgements

This research has been partially supported by the “Plan Nacional de I\(+\)D\(+\)i” of the Spanish Government under the project MTM2015-70550-P. D. Martín also wants to thank University Complutense of Madrid and Bank of Santander for his pre-doctoral contract. The authors wish to thank two anonymous referees and the editor for their useful suggestions and comments.

Author information

Authors and Affiliations

  1. Departamento de Estadística y Ciencia de los Datos, Facultad de Estudios Estadísticos, Universidad Complutense de Madrid, Av. Puerta de Hierro s/n, 28040, Madrid, Spain

    C. Manuel & D. Martín

Authors
  1. C. Manuel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Martín
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Manuel.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Manuel, C., Martín, D. A value for communication situations with players having different bargaining abilities. Ann Oper Res 301, 161–182 (2021). https://doi.org/10.1007/s10479-020-03825-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-020-03825-z

Keywords

Search

Navigation