# American Institute of Mathematical Sciences

April  2017, 4(2): 87-96. doi: 10.3934/jdg.2017006

## A simple family of solutions for forest games

 1 CIMAT, A. C., Jalisco S/N, Valenciana, C.P. 36240, Guanajuato, Gto, México 2 UASLP, School of Economics, Av. Pintores S/N, Burócratas del Estado, C.P. 78213, San Luis Potosí, SLP, México

* Corresponding author: Tel. +52 473 73 27155, Ext 4991

Received  April 2016 Revised  December 2016 Published  March 2017

In this paper we study TU-games where the cooperation structure among the players is modeled by a forest. Using the classical component efficiency axiom and a generalized version of the component fairness axiom we obtain a family of solutions. We show that every solution in this family is based on a process of transfers among the players, and the average tree solution belongs to the family. Finally, we obtain a solution based on the degree of the nodes and we study a set of properties satisfied by this family.

Citation: Oliver Juarez-Romero, William Olvera-Lopez, Francisco Sanchez-Sanchez. A simple family of solutions for forest games. Journal of Dynamics and Games, 2017, 4 (2) : 87-96. doi: 10.3934/jdg.2017006
##### References:
 [1] R. J. Aumann and M. Maschler, Game theoretic analysis of a bankruptcy problem from the Talmud, J Econ theory, 36 (1985), 195-213.  doi: 10.1016/0022-0531(85)90102-4. [2] R. Baron, S. Béal, E. Rémila and P. Solal, Average Tree solutions and the distribution of Harsanyi dividends, Int. J. Game Theory, 40 (2001), 331-349.  doi: 10.1007/s00182-010-0245-7. [3] S. Béal, A. Lardon, E. Rémila and P. Solal, The Average Tree solution for multi-choice forest games, Annals of Operations Research, 196 (2012a), 27-51.  doi: 10.1007/s10479-012-1150-1. [4] S. Béal, E. Rémila and P. Solal, Weighted component fairness for forest games, Math Social Sci, 64 (2012b), 144-151.  doi: 10.1016/j.mathsocsci.2012.03.004. [5] K. Binmore, A. Rubinstein and A. Wolinsky, The Nash bargaining solution in economic modelling, The RAND Journal of Economics, (1986), 176-188.  doi: 10.2307/2555382. [6] P. Herings, G. van der Laan and D. Talman, The Average Tree solution for cycle-free graph games, Game Econ Behav, 62 (2008), 77-92.  doi: 10.1016/j.geb.2007.03.007. [7] P. Herings, G. van der Laan, D. Talman and Z. Yang, The average tree solution for cooperative games with limited communication structure, Game Econ Behav, 68 (2010), 626-633.  doi: 10.1016/j.geb.2009.10.002. [8] M. O. Jackson, Allocation rules for network games, Game Econ Behav, 51 (2005), 128-154.  doi: 10.1016/j.geb.2004.04.009. [9] R. B. Myerson, Graphs and cooperation on games, Math Operations Res, 2 (1977), 225-229.  doi: 10.1287/moor.2.3.225. [10] L. S. Shapley, A value for n-person game Ⅱ, Annals of Mathematics Studies, (eds. Kuhn, H.W. and Tucker, A.W.), Princeton University Press, 28 (1953), 307-317. [11] A. van den Nouweland, Games and graphs in economic situations Ph. D. thesis, Tilburg University, The Netherlands, 1993.

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##### References:
 [1] R. J. Aumann and M. Maschler, Game theoretic analysis of a bankruptcy problem from the Talmud, J Econ theory, 36 (1985), 195-213.  doi: 10.1016/0022-0531(85)90102-4. [2] R. Baron, S. Béal, E. Rémila and P. Solal, Average Tree solutions and the distribution of Harsanyi dividends, Int. J. Game Theory, 40 (2001), 331-349.  doi: 10.1007/s00182-010-0245-7. [3] S. Béal, A. Lardon, E. Rémila and P. Solal, The Average Tree solution for multi-choice forest games, Annals of Operations Research, 196 (2012a), 27-51.  doi: 10.1007/s10479-012-1150-1. [4] S. Béal, E. Rémila and P. Solal, Weighted component fairness for forest games, Math Social Sci, 64 (2012b), 144-151.  doi: 10.1016/j.mathsocsci.2012.03.004. [5] K. Binmore, A. Rubinstein and A. Wolinsky, The Nash bargaining solution in economic modelling, The RAND Journal of Economics, (1986), 176-188.  doi: 10.2307/2555382. [6] P. Herings, G. van der Laan and D. Talman, The Average Tree solution for cycle-free graph games, Game Econ Behav, 62 (2008), 77-92.  doi: 10.1016/j.geb.2007.03.007. [7] P. Herings, G. van der Laan, D. Talman and Z. Yang, The average tree solution for cooperative games with limited communication structure, Game Econ Behav, 68 (2010), 626-633.  doi: 10.1016/j.geb.2009.10.002. [8] M. O. Jackson, Allocation rules for network games, Game Econ Behav, 51 (2005), 128-154.  doi: 10.1016/j.geb.2004.04.009. [9] R. B. Myerson, Graphs and cooperation on games, Math Operations Res, 2 (1977), 225-229.  doi: 10.1287/moor.2.3.225. [10] L. S. Shapley, A value for n-person game Ⅱ, Annals of Mathematics Studies, (eds. Kuhn, H.W. and Tucker, A.W.), Princeton University Press, 28 (1953), 307-317. [11] A. van den Nouweland, Games and graphs in economic situations Ph. D. thesis, Tilburg University, The Netherlands, 1993.
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