Elsevier

Applied Mathematics and Computation

Volume 273, 15 January 2016, Pages 535-542
Applied Mathematics and Computation

The arising of cooperation in Cournot duopoly games

https://doi.org/10.1016/j.amc.2015.10.027Get rights and content

Abstract

In literature, several studies have been made to study conflict among duopolistic Cournot game with incomplete information. This paper attempts to investigate equilibrium stability of a nonlinear Cournot duopoly game on which cooperation may be existed among firms. Discrete time scales under the assumption of logarithmic price function and linear cost are used to build our models in the proposed games. We study here an adjustment dynamic strategy based on the so-called tit-for-tat strategy. For each model, the stability analysis of the fixed point is analyzed. Numerical simulations are carried out to show the complex behavior of the proposed models and to point out the impact of the models’ parameters on the cooperation.

Introduction

Game theory is characterized by its ability to consider interactions among firms. It is one of the most important theories that is used to describe and study such competition among competitors statically and dynamically. The dynamic case in which the equilibrium point (Nash equilibrium) is sought and its complex dynamic characteristics are of main interest have been studied in literature [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. In such games there are often several duopolistic firms in economic market where competition among them is controlled by the amount of commodities they produce, the demand scheme they adopt and the profit each firm wants to maximize. In competition, firms produce the same or homogenous goods and they must focus not only on the market size, but also on the actions their competitors do.

The Cournot duopolistic game has been studied intensively in literature. Agiza and Elsadany [14] have modeled a Cournot duopolistic game on which one of the competitors is heterogeneous. They have studied the proposed game in details and particularly when the game’s fixed point becomes unstable due to bifurcation occurred. In [28], a heterogeneous duopoly game with rational and adaptive competitors was examined. The authors have studied the impact of quadratic cost function on the complex behavior of the game and came up with the conclusion that introducing nonlinearity in cost generalizes the implications of heterogeneous Cournot duopoly with adjusting strategies. Other interesting works related to Cournot duopolistic games can be in literature [29], [30], [31].

In this paper, we focus and study the cooperation between firms in repeated Cournot duopoly games with a logarithmic price function. In Cournot duopoly games, Nash equilibrium or Cournot equilibrium is the basic solution in such games and reflects the rationality of the firms within games. Since, firm rationality contradicts with Pareto optimality (in cooperation case), then Nash equilibrium in duopoly game is not Pareto optimal. In other words, Pareto optimality in such games cannot be achieved by firm interest’s maximization. As reported in [15], [16], theoretical and experimental studies have leaded up to several ways by which the cooperative solution can be obtained. For instance, in the well-known short game of prisoner’s dilemma, the Nash equilibrium point is Pareto optimal as cooperation is obtainable. But for the repeated games, emergence of cooperation among competitors (firms) may be possible to achieve and then cooperation in iterated prisoner’s dilemma can be explained [15], [17]. In [18], it has been shown that the conditional cooperative strategy such as the so-called “tit-for-tat” may be used to achieve cooperation among firms in repeated games.

The emergence of cooperation has attracted much of interest for a long time and it would look like even pleonastic to report some of the recent and important papers in this field. In [19], setups based on discrete, continuous and mixed strategy have been proposed in the social dilemma games and their performance on networks populations has been shown. A useful source of information on the evolutionary games on multilayer networks and particularly in the evolution of cooperation is reported in details in [20]. An evolutionary dictator game model is introduced in [21] by which the evolution of altruism and fairness of populations has been studied. In this study, the influence of assignation on heterogeneous populations has been investigated. An important review of the universality of scaling for the dilemma strength in evolutionary games has been reported in [22]. The review has shown that social viscosity or spatial structure causes the existing scaling parameters to fail. In addition to the review has developed new parameters to resolve the paradox of cooperation benefits. Two-layer scale-free networks has been introduced in [23] to show evolution of cooperation. In [24], the authors have demonstrated that the influence of simple strategy-independent form may expand the scope of cooperation on structured populations. For more related works, readers are advised to have a look on some important papers [25], [26] and a more informational report [27].

The current paper is motivated by the work done by Ding and Shi [15]. We introduce a duopoly game based on a logarithmic price function. An adjustment dynamic strategy is introduced and studied to detect the cooperation condition that may be occurred based on this strategy. Under the proposed function, the tit-for-tat is used and a two dimension discrete map is introduced. The complex dynamic characteristics of this map are studied and the stability of the Nash equilibrium is investigated. The qualitative study of bifurcation is studied analytically and numerically. We conclude our study with a tit-for-tat strategy with team profit.

The structure of the paper is as follows: In Section 2 a description of a Cournot duopoly game based on a generalized inverse demand function is presented. In Section 3, the two-dimensional map whose iteration gives the time evolution based on a proposed dynamic adjustment is defined. The steady state point of the map, which is Nash equilibrium, is computed. Then the stability of this point is investigated and its complex dynamics is detected. Section 4 introduces a tit-for-tat Cournot duopoly game using the same function. As in Section 3, the dynamic characteristics of the game are investigated. In Section 5, the system studied in Section 4 is improved by adding a control strategy in this system and some discussion is illustrated. Finally, we end the paper with some conclusions to show the significance of our results.

Section snippets

Cooperative duopoly model

Suppose a market with two firms producing the same product or homogenous product. The decisions in this market are the quantities both firms sell in the market and are taken at discrete time scale, t=0,1,2,. The produced quantity by each firm at time t is denoted by qi,t(i=1,2). We assume that the cost of production is linear, Ci,t=ciqi,t, where c ≥ 0 is a marginal cost. Further we assume an inverse demand function as follows: pt=ablogQ

It is well-known that p indicates commodity price, while Q

Dynamic tit-for-tat behavior

For achieving the cooperation between the two firms, the tit-for-tat strategy is used. With this strategy, every firm is doing what its opponent has done in the previous move. This is an incomplete information scenario however the only things each firm knows are the output and the profit. In this situation each firm compares its profit πi, t with the cooperative profit πc that is Pareto optimal. If πi, t > πc this means that each firm will probably reduce its output to keep the cooperation

Dynamic tit-for-tat with control

A feedback control is added in the map (4) to improve it. This gives the following new map: qi,t+1=qi,t+νi(πcπi,t)ν(qi,tqc),i=1,2where νi>0,i=1,2 is an adjusting parameter, and ν > 0 and ν(qi,tqc),i=1,2 is the feedback control of the system. This feedback control shows that whether the firms cooperate or not. It is used to show that if qi, t > qc(i=1,2) the firm is going to reduce its output in the next period of time since, the current is over the cooperative output. On the other hand, if

Dynamic tit-for-tat team profit

Here, we assume that both firms construct a team with a team profit given in the following form: Πteam=ω[(acblogQ)q1,t]+(1ω)[(acblogQ)q2,t]where, ω is the weight parameter of the firm’s profit. The first-order derivative of (8) gives the cooperative output qc*=12e(2ω(ac)b2ωb)and then Πc=b4ωe(2ω(ac)b2ωb). Similar to the discussion in Section 2, the following system is obtained. q1,t+1=q1,t+ν1[Πc(acblogQ)q1,t]q2,t+1=q2,t+ν2[Πc(acblogQ)q2,t]

The system (9) has a unique positive fixed

Conclusion

In this paper, we have studied the cooperation that may be obtained among duopolistic firms in the economic market. Based on a logarithmic price function, three duopolistic Cournot models have been investigated. For each model, the fixed point has been computed and complete analytical and numerical studies of the stability conditions for the fixed point have been obtained. The analysis has shown that under the dynamic adjustment strategy and the tit-for-tat strategy, the cooperation may be

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Research group no. (RG-1435-054).

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