Analysis on the dynamics of a Cournot investment game with bounded rationality
Introduction
Cournot (1838) introduced earliest the mathematical model which describes production competitions in an oligopolistic market. In a classical Cournot model each participant uses a naïve expectation to guess that opponents' output remains at the same level as in the previous period and adopts an output strategy which solves the corresponding profit maximization problem. Since then, a great number of literatures have been devoted to enrich and expand the Cournot oligopoly game theory. Much work has paid attention to the stability and the complex phenomena in a dynamical Cournot game with this kind of naïve expectation [e.g., (Teocharis, 1960, Puu, 1991, Puu, 1996, Puu, 1998, Agiza, 1998, Kopel, 1996, Ahmed and Agiza, 1998, Agliari et al., 2000, Rosser, 2002)]. As a more sophisticated kind of learning rule with respect to naïve expectations, adaptive expectations or adaptive adjustments have been proposed in other dynamical models [e.g., (Okuguchi, 1970, Bischi and Kopel, 2001, Agiza et al., 1999, Rassenti et al., 2000, Szidarovszky and Okuguchi, 1988)]. In recent years, many researchers have paid attention to a kind of bounded rationality, with which a player (without complete information of the demand function) uses local knowledge to update output by the marginal profit. Bischi and Naimzada (1999) gave a general formula of the dynamical Cournot model with this form of bounded rationality, assuming that producers behave as local profit maximizers in a local adjustment process, “where each firm increases its output if it perceives a positive marginal profit and decreases its production if the perceived marginal profit is negative (Bischi & Naimzada, 1999)”. Much work has been done on the dynamical Cournot games performed by players with this kind of marginal profit method. The models with homogeneous players (all players are boundedly rational players and use the marginal profit method to adjust strategies) are considered in Agiza et al. (2001), Agiza et al. (2002), Ahmed et al. (2000), and Bischi and Naimzada (1999). Some other work has focused on modeling the system with heterogeneous expectations. Agiza and Elsadany (2003) and Zhang et al. (2007) studied the dynamics of a Cournot duopoly game with one bounded rationality player and one naïve player. Agiza and Elsadany (2004) and Dubiel-Teleszynski (2011) considered a duopoly model in which one player has bounded rationality and the other has adaptive expectation. In the model by Fan et al. (2012), there is one player using the marginal profit method and one player adjusting production in terms of the market price in the previous period. Ding et al. (2009) studied the dynamics of a two-team Cournot game with heterogeneous players.
In these models for dynamical Cournot game, output is a key variable and each player is able to take any needed output updating for the purpose of local profit maximization; thus it is based on an implicit assumption that all players could provide sufficient quantity of products on the market. However, this implicit assumption may be impractical in an economy market where investment plays the most important role. For instance, in an emerging industry with immature development (e.g., a new energy market), it is unlikely for a firm to hold productivity large enough due to its lack of investment accumulation. As a strategic behavior in these economic activities, investment accumulation plays a significant role in achieving a good production level. Moreover, we know that even in a mature industry, the production capacity of a firm is greatly dependent on its large-scale investment stock. Only when the investment comes up to a certain level can a firm provide as much goods as the market demands. Therefore, during the developing period of an infant industry, the competition among producers lies mainly in their investment strategy. To obtain a competitive market share and get superiority over opponents, producers must consider their investment strategies in successive periods.
The main purpose of our work is to formulate a novel model, which puts investment decision as a substitute for output adjustment into the dynamical Cournot game. In our model, all producers are also assumed to have bounded rationality and make their investment decisions in line with the marginal profit in the previous period. That is to say, each firm will increase its investment if it perceives a positive marginal profit and decrease its investment if the perceived marginal profit is negative. It is analyzed as to how this novel dynamical Cournot game, in a local adjustment process, evolves to equilibrium or exhibits complicated dynamic behaviors.
This article is organized as the following. In Section 2, we model the dynamical investment game played by players with bounded rationality. In Section 3, we discuss the existence and local stability of the equilibrium points for the system. In Section 4, we show the dynamic features of this system with numerical simulations, including bifurcation diagram, phase portrait, stable region and sensitive dependence on initial conditions. In Section 5, time-delayed feedback control is used to stabilize the chaotic behaviors of the system.
Section snippets
The model
Our work focuses on firms' investment competition rather than their output competition. We pay attention to a duopoly investment game, where producers' investment choices are substituted for their output decisions discussed in classic Cournot games.
We consider a competition between two firms (players), labeled by i = 1,2, producing homogeneous goods. The strategy of each firm is to choose an investment in every period. Both players make their decisions in discrete periods t = 0,1,2 ⋯. We write Ki(t −
Equilibrium points and stability
Let xi(t + 1) = xi(t) and Ii(t + 1) = Ii(t) (i = 1,2) in system (10), then we get
Solving equations in Eq. (11), we obtain four equilibrium states of dynamics (Eq. (10)), which are listed as follows:where
Numerical simulation
In this section, we show by numerical simulations how the system evolves under different levels of parameters, especially of the capital residual rate θ and the adjustment speed α. In all the numerical simulations, the other parameters are fixed: a = 5, b = 1, c1 = 0.3, c2 = 0.5, B1 = 0.6 and B2 = 0.8.
For three cases of the capital residual rate θ, Fig. 1 is about bifurcation diagrams of system (10) with respect to the adjustment rate α1 while the other one is fixed as α2 = 2.2. Fig. 1(A) shows the
Chaos control
From the numerical simulations above, we see that the adjustment rate and the capital residual rate have great influence on the stability of system (10). If the model parameters fail to locate into the stable region required, the behaviors of the dynamics will be much complicated. In a real economic system, chaos is not desirable and will be not expected, and it is needed to be avoided or controlled so that the dynamic system would work better. In this section, we use the time-delayed feedback
Conclusion
In this work we have taken into consideration firms' investment decisions as substitute for the output choices considered by the existing work on classic Cournot games. We have formulated a novel Cournot form of investment game played by two players with bounded rationality. The main idea in our model is that each firm's decision is to choose its investment in each period according to the marginal profit observed from the previous period. We have established a corresponding dynamics of players'
Acknowledgements
We are very grateful to the anonymous referees for the valuable comments and suggestions that greatly help us to improve the paper.
Financial support by the National Natural Science Foundation of China (Nos. 71171098 and 51306072) and by the Jiangsu Provincial Fund of Philosophy and Social Sciences for Universities (No. 2010-2-8) is gratefully acknowledged.
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