$^{∗†}$Mathematics Department, Faculty of Science, Mansoura 35516, EGYPT E.Ahmed $^{∗}$, M.I.Shehata$^{†}$and H.A.El-Saka‡ arXiv:1208.4528v1 [cs.GT] 13 Jul 2012 Cournot dynamical game is studied on a graph. The stability of the system is studied. Prisoner’s dilemma game is used to model natural gas transmission. Key words: Dynamical Cournot game, graph; gas, water and elec- tricity transportation. Game theory [1] was first introduced by von Neumann and Morgenstern in 1944 as a mathematical model. It is the study of ways in which strategic $^{∗}$E-mail: magd45@yahoo.com $^{‡}$E-mail: halaelsaka@yahoo.com 1 Introduction Abstract 1 On Dynamical Cournot Game on a Graph