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## 2 A Cournot dynamic game on a graph 

Transportation of natural gas electricity and water allocation for different purposes is an important problem. However most of its studies are static while the problem is dynamic. The vertices of the graph representing the problem are pf three types markets m$_{i}$ , i = 1 , 2 , ..., k 1 firms (producers) f$_{j}$, j = 1 , 2 , ..., k 2 and transit countries through which the lines pass. The profit function of firm f$_{j}$ is given by 

<img src='content_image/316490.jpg'>

<img src='content_image/316491.jpg'>

where q$_{ij}$ are the production quantity from firm j to market i and α$_{i}$ , β$_{i}$ , γ$_{j}$ are positive constants. The sum over i is on all markets connected to firm j. Conversely the sum in c$_{i}$ is over all firms supplying market i. 

The dynamic Cournot game with bounded rationality is given by [3,4] 

<img src='content_image/316489.jpg'>

The parameters b$_{j}$ are proportionality parameters. They may be taken as functions of the production quantities but here we will take them as con- 

