---
abstract: |
    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 electricity transportation.*
author:
- |
    E.Ahmed[^1] , M.I.Shehata and H.A.El-Saka[^2]\
    Mathematics Department, Faculty of Science, Mansoura 35516, EGYPT\
    Damietta Faculty of Science, Mansoura University, 34517, New Damietta, Egypt\
title: On Dynamical Cournot Game on a Graph
---

Introduction
============

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 interactions among rational players produce outcomes with respect to the preferences of the players. Each player in a game faces a choice among two or more possible strategies. A strategy is a predetermined program of play that tells the$i$ player which action to take in response to every possible strategy other players may use . Transportation of natural gas electricity and water allocation for different purposes was studied recently by Ilkihc \[2\]. He modeled it as static Cournot game on a graph. We study the dynamic game. The dynamic equations are derived. The stability of the equilibrium solution is studied. Prisoner’s dilemma game is used to model the source countries-transit countries- markets interaction on a graph.The problem studied in this letter can be explained by the following example: Assume Russia wants to export gas to two EU countries say EU1, EU2. Assume Azerbaijan wants to export gas to EU2 only. Thus Cournot game exists between Russia and Azerbaijan in EU2. The Russian gas pipelines pass through Belarus hence prisoner’s dilemma game (PD) exists between Russia and Belarus. Similarly Azeri gas pipelines pass through Turkey hence PD game exists between Azerbaijan and Turkey.

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$_{i},i=1,2,...,k1$ firms (producers) $f_{j},j=1,2,...,k2$ and transit countries through which the lines pass. The profit function of firm $f_{j}$ is given by

$$\begin{aligned}
\Pi _{j} &=&\sum_{i}\alpha _{i}q_{ij}-\gamma _{j}s_{j}^{2}/2-\sum_{i}\beta
_{i}q_{ij}c_{i} \\
s_{j} &=&\sum_{l}q_{lj},c_{i}=\sum_{l}q_{il}\end{aligned}$$

where $q_{ij}$ are the production quantity from firm j to market i and $\alpha _{i}$, $\beta _{i}$, $\gamma _{j}$ 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 \[3,4\]

$$dq_{ij}/dt=b_{j}(\partial \Pi _{j}/\partial q_{ij})$$

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

For general graph Cournot bounded rationality game the dynamic equations are

$$dq_{ij}/dt=b_{j}[\alpha _{i}-\gamma _{j}\sum_{l}q_{lj}-\beta
_{i}q_{ij}-\beta _{i}\sum_{k}q_{ik}]$$

Here we will consider a simple graph consisting of two firms and two markets. The first firm supplies both markets while second firm supply only second market. In this case the above system takes the form

$$\begin{aligned}
dq_{11}/dt &=&\alpha _{1}-\gamma _{1}(q_{11}+q_{21})-2\beta _{1}q_{11} \\
dq_{21}/dt &=&\alpha _{2}-\gamma _{1}(q_{11}+q_{21})-\beta
_{2}(2q_{21}+q_{22})  \nonumber \\
dq_{22}/dt &=&\alpha _{2}-\gamma _{2}q_{22}-\beta _{2}(2q_{22}+q_{21}) 
\nonumber\end{aligned}$$

after rescaling the system becomes:

$$\begin{aligned}
dq_{11}/dt &=&1-r_{1}q_{11}-q_{21} \\
dq_{22}/dt &=&1-r_{2}q_{22}-q_{21}  \nonumber \\
dq_{21}/dt &=&1-r_{3}q_{21}-r_{4}q_{11}-r_{5}q_{22}  \nonumber\end{aligned}$$

Applying Routh-Hurwitz conditions \[5\] the unique equilibrium of the above system is locally asymptotically stable if the following conditions are satisfied:

$$\begin{aligned}
a_{1} &>&0,a_{3}>0,a_{1}a_{2}>a_{3} \\
a_{1} &=&r_{1}+r_{2}+r_{3}  \nonumber \\
a_{2} &=&r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}-r_{4}-r_{5}  \nonumber \\
a_{3} &=&r_{1}r_{2}r_{3}-r_{1}r_{4}-r_{2}r_{5}  \nonumber\end{aligned}$$

For the special case $r_{1}=r_{2}$the above results simplifies significantly. The unique equilibrium solution become

$$\begin{aligned}
q_{11} &=&q_{22}=(1-q_{21})/r_{1} \\
q_{21} &=&(r_{1}-r_{4}-r_{5})/(r_{1}r_{3}-r_{4}-r_{5})  \nonumber\end{aligned}$$

The existence and stability conditions are

$$\begin{aligned}
r_{1} &>&r_{4}+r_{5} \\
r_{3} &>&1  \nonumber\end{aligned}$$

The approximate solutions for system (6):

If we take $\,q_{11}(0)=0.1,q_{22}(0)=0.2,q_{21}(0)=0.3,r_{1}=0.01,r_{2}=0.1,r_{3}=1.1,r_{4}=-0.3, 
$and $r_{5}=0.4.$

We find that the interior equilibrium point is unstable where the conditions (2.7) is not satisfied.

If we take $\,q_{11}(0)=0.1,q_{22}(0)=0.2,q_{21}(0)=0.3,r_{1}=0.2,r_{2}=0.5,r_{3}=1.5,r_{4}=-0.3, 
$and $r_{5}=0.4.$

We find that the interior equilibrium point $(1.13636,0.454545,0.772727)$ is locally asymptotically stable where the conditions (2.7) is satisfied.

Cooperation on graphs
=====================

In natural gas, electricity and water transport a new situation exists namely transit countries. Those where transmission lines pass e.g. Belarus in the line Russia-Belarus- EU gas line. In this case a Prisoner’s Dilemma (PD) game \[6\] exists between both producers and end users from one side and transit countries on the other. In PD game there are two strategies namely to cooperate or to defect. Typically the payoff matrix for such a game is given by

$$\left[ 
\begin{array}{cc}
R & S \\ 
T & U\end{array}\right] ,T>R>U>S$$

 The standard dominant strategy is to defect hence both sides lose a lot. A way to solve this problems is through side payments paid by both producers and end users to transit countries.

Games on graphs has been studied by May and Sigmund \[7\]. They have shown that in the case of Prisoner’s dilemma game cooperation can exist easier than the standard game.

Conclusions
===========

Dynamical games representing natural gas, electricity and water are studied using Cournot game on a graph. Stability of the unique equilibrium solution is investigated. Prisoner’s dilemma game is used to model the producers- transit countries-end users interactions. The existence of graphs will improve the possibility of cooperation between these countries.

[9]{} J. Webb, Game theory, Springer (2007).

R.Ilkihc, “Cournot competition on a network of markets and firms” On Line. (2010).

T. Puu, Structural Change in Flow-Based Spatial Economic Models-A Survey, Socio-Spatial Dynamics 2 (1991)1.

G.I. Bischi and A. Naimzada, “Global Analysis of a Duopoly Game with Bounded Rationality”, Advances in Dynamic Games and Applications, vol.5, Birkhauser (1999) pp. 361-385.

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge Univ. press. U.K. (1998).

I. Dombrowsky, Institutionalization in the management of international waters. (On line) 2005.

May R.M. and Nowak M.(1993), The spatial dilemmas of evolution, Int.J.Bifurc.Chaos 3, 35.

[^1]: E-mail: magd45@yahoo.com

[^2]: E-mail: halaelsaka@yahoo.com
