Discrete-time deterministic and stochastic triopoly game with heterogeneous players and delay Mihaela Neamt ¸u, Nicoleta Sˆ ırghi, Carmen B˘ ab˘ ait ¸˘ a, Renata Antonie-Nit ¸u Abstract– In this paper the discrete-time triopoly game with heterogeneous players has been studied. We take into consideration the deterministic and stochastic cases. A study for the local stability of the fixed points is carried out. The bifurcation flip and its normal form are analyzed. Also, the case when the system contains delay is discussed. Numerical simulations are performed for the above models. Finally, some conclusions and future prospects are provided. Keywords– Triopoly game, Heterogeneous players, Flip bifurcation, Discrete-time dynamical system, Stochas- tic dynamical system, Strange attractor. I. I NTRODUCTION Economic models and economic-mathematical mo- deling practice constituted an excellent instrument for studying the economic games, stimulating research in this area. Currently a number of modeling methods of economic and mathematical theory were used to study the evolution of the social-economic status parameters. From this perspective the study, in a dynamic environment, of the oligopoly market mechanism is an extremely important issue. Based on these considerations it is possible to approach the microeconomic problems working with a modern instrument, namely game theory. The game theory has not changed the principle of rationality, but developed it by using strategically and informational complex, thus raising questions on the hy- pothesis of rational behavior for oligopoly type market structures [14]. The oligopoly market is an imperfect market structure that is found mostly in the actual economy, characterized by a limited number of company proceedings. The strate- gies of oligopoly companies are different and adapted to each actual situation on the market. Augustin Cournot studied the oligopoly markets oper- ation where each company acts knowing that the volume of production affects the market price [3]. He defined balance as a situation where each company chooses the Mihaela Neamt ¸u, West University of Timisoara, Pestalozzi Street, Timisoara, 300115, ROMANIA, e-mail: mihaela.neamtu@feaa.uvt.ro Nicoleta Sˆ ırghi, West University of Timisoara, Pestalozzi Street, Timisoara, 300115, ROMANIA, e-mail: nicoleta.sirghi@feaa.uvt.ro Carmen B˘ ab˘ ait ¸˘ a, West University of Timisoara, Pestalozzi Street, Timisoara, 300115, ROMANIA, e-mail: carmen.babaita@feaa.uvt.ro Renata Antonie-Nit ¸u, West University of Timisoara, Pestalozzi Street, Timisoara, 300115, ROMANIA, e-mail: renate.nitu@feaa.uvt.ro output which can maximize its profit but taking into ac- count the output forecast by the other companies, showing that such a balance leads to a price above the marginal productivity [4]. The oligopoly market structure showing the action of only three companies is called triopoly. This paper presents an oligopoly market analysis on the specific case of triopoly using game theory as a working instrument. The players choose simple expectations such as naive or complex as rational expectations. The players can use the same or different strategies. Based on [1,2,5,9,11], in the present paper we con- sider a triopoly game with heterogenous players, where each player thinks with different strategy to maximize his output. We consider the first player to be boundedly rational, the second one is an adaptive player and the third one is a naive player. They all produce the same or homogeneous goods which are perfect substitutes and over them at discrete-time periods n=0,1,2,.. on a common market. The paper is organized as follows. The discrete-time dynamical triopoly game with heterogenous players is described in Section 2. Section 3 provides the existence and the local stability of the fixed points, and the existence of the flip bifurcation and its normal form, as well. Section 4 presents the stochastic model. Section 5 present the deterministic and stochastic model with delay. Using Maple 12, some numerical simulations are carried out in Section 6. The strange attractor and Lyapunov exponent are measured numerically. Finally, some conclusions are offered. II. THE MODEL We consider a Cournot triopoly game, where q i , i = 1, 2, 3 denotes the quantity supplied by i th firm. Also, let P : IR + → IR + be a twice differentiable and non-increasing inverse demand function and C i : IR + → IR + , i =1, 2, 3 the twice differentiable increasing cost functions. The profit functions Π i : IR 3 + → IR are defined by: Π i (q 1 (n),q 2 (n),q 3 (n))= P (q 1 (n)+q 2 (n)+q 3 (n))q i - - C i (q i ), i =1, 2, 3. (1) If q i (n), i =1, 2, 3 are the outputs at the moment n ∈ IN , then in the moment n+1 the first player’s output q 1 (n+1) INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Issue 2, Volume 5, 2011 343