This paper proposes the concept of interval-valued average tree solution (“AT solution” for short) of graph cooperative games with interval-valued payoffs, and develop an effective and a direct simplified method for solving a subclass of interval-valued graph cooperative games. In this method, the interval-valued AT solution is proved to be a monotonic and non-decreasing function of coalitions’ values under specific condition. Hence, the lower and upper bounds of interval-valued AT solutions of graph cooperative games can be obtained directly by using the lower and upper bounds of the interval-valued coalitions’ payoffs, respectively. The proposed method gives better results than general interval subtraction and the partial subtraction operator methods. In addition, some important properties of the interval-valued AT solutions of interval-valued graph cooperative games are discussed. At last, the applicability and superiority of the proposed approach is demonstrated by comparing with other methods.
BranzeiR., BranzeiO., Alparslan GökS.Z. and TijsS., Cooperative interval games: A survey, Central European Journal of Operations Research18(3) (2010), 397–411.
2.
HanW.B., SunH. and XuG.J., A new approach of cooperative interval games: The interval core and Shapley value revisited, Operations Research Letters40(6) (2012), 462–468.
3.
Alparslan GökS.Z., Some results on cooperative interval games, Optimization63 (2014), 7–13.
4.
MengF.Y., ChenX. and TanC., Cooperative fuzzy games with interval characteristic functions, Operations Research16 (2016), 1–24.
5.
HongF.X. and LiD.F., Nonlinear programming method for interval-valued n-person cooperative games, Operations Research17 (2017), 479–497.
6.
LiX., SunH. and HouD., On the position value for communication situations with fuzzy coalition, Journal of Intelligent & Fuzzy Systems33 (2017), 113–124.
7.
Alparslan GökS.Z., BranzeiR. and TijsS.The interval Shapley value: An axiomatization, Central European Journal of Operations Research18(2) (2010), 131–140.
HukuharaM., Integration des applications measurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj10 (1967), 205–223.
10.
MooreR., Methods and Applications of Interval Analysis. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1979).
11.
LiD.F., Linear programming approach to solve interval-valued matrix games, Omega39 (2011), 655–666.
12.
LiD.F. and YeY.F., Interval-valued least square prenucleolus of interval-valued cooperative games and a simplified method, Operational Research18 (2016), 205–220.
13.
MyersonR.B., Graphs and cooperation in games, Mathematics of Operations Research2(3) (1977), 225–229.
14.
Jiménez-LosadaA., FernándezJ.R. and OrdóñezM., Myerson values for games with fuzzy communication structure, Fuzzy Sets System213 (2013), 74–90.
15.
FernándezJ.R., GallegoI., Jiménez-LosadaA. and OrdóñezM., Cooperation among agents with a proximity relation, European Journal of Operational Research250(2) (2016), 555–565.
16.
HeringsP.J.J., LaanG.V.D. and TalmanD., The average tree solution for cycle-free graph games, Games and Economic Behavior62(1) (2008), 77–92.
17.
TalmanA.J.J. and YamamotoY., Average tree solution and subcore for acyclic graph games, Journal of the operational research society51 (2008), 203–212.
18.
HeringsP.J.J., LaanG.V.D., TalmanA.J.J. and YangZ., The average tree solution for cooperative games with communication structure, Games and Economic Behavior68(2) (2010), 626–633.
19.
BéalS., RémilaE. and SolalP., Rooted-tree solutions for tree games, European Journal of Operational Research203 (2010), 404–408.
20.
Van den BrinkRené, HeringsJ.J., LaanG.V.D. and Talman D.J.J., The average tree permission value for games with a permission tree, Economic Theory58(1) (2013), 99–123.
21.
NieC.P. and Zhang.Q., Fuzzy Average Tree Solution for Graph Games with Fuzzy Coalitions, In Fuzzy Information & Engineering and Operations Research & Management. Springer, Berlin, Heidelberg (2014).
22.
XuG., LiX., SunH. and SuJ., The Myerson value for cooperative games on communication structure with fuzzy coalition, Journal of Intelligent & Fuzzy Systems33 (2017), 27–39.
23.
MishraD. and TalmanD., A characterization of the average tree solution for tree games, International Journal of Game Theory39 (2010), 105–111.
24.
StefaniniL., A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets System161 (2010), 1564–1584.
25.
YagerR.R., OWA aggregation over a continuous interval argument with applications to decision making, IEEE Transactions on Systems Man & Cybernetics Part B Cybernetics A Publication of the IEEE Systems Man & Cybernetics Society34(5) (2004), 1952–1963.
26.
LiB.Z. and LuoX.F., Study on profit allocation of enterprise’s original innovation with an industry-university-research cooperative mode based on the Shapley value, Operations Research and Management Science22 (2013), 220–224. (in Chinese).
27.
YuX. and ZhangQ., An extension of cooperative fuzzy games, Fuzzy Sets System161 (2010), 1614–1634.
