Nonlinear Dynamics in a Cournot Duopoly with Different Attitudes towards Strategic Uncertainty

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 323290, 11 pages http://dx.doi.org/10.1155/2013/323290 Research Article Nonlinear Dynamics in a Cournot Duopoly with Different Attitudes towards Strategic Uncertainty Luciano Fanti,1 Luca Gori,2 and Mauro Sodini1 1 2 Department of Economics and Management, University of Pisa, Via Cosimo Ridolfi 10, 56124 Pisa, Italy Department of Law, University of Genoa, Via Balbi 30/19, 16126 Genoa, Italy Correspondence should be addressed to Luca Gori; Received 2 September 2013; Accepted 12 November 2013 Academic Editor: Luca Guerrini Copyright © 2013 Luciano Fanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper analyses the dynamics of a duopoly with quantity-setting firms and different attitudes towards strategic uncertainty. By following the recent literature on decision making under uncertainty, where the Choquet expected utility theory is adopted to allow firms to plan their strategies, we investigate the effects of the interaction between pessimistic and optimistic firms on economic dynamics described by a two-dimensional map. In particular, the study of the local and global behaviour of the map is performed under three assumptions: (1) both firms have complete information on the market demand and adjust production over time depending on past behaviours (static expectations—“best reply” dynamics); (2) both firms have incomplete information and production is adjusted over time by following a mechanism based on marginal profits; and (3) one firm has incomplete information on the market demand and production decisions are based on the behaviour of marginal profits, and the rival has complete information on the market demand and static expectations. In cases 2 and 3 it is shown that complex dynamics and coexistence of attractors may arise. The analysis is carried forward through numerical simulations and the critical lines technique. 1. Introduction In this paper, we analyse the dynamics of a Cournot duopoly under strategic uncertainty with pessimistic and optimistic firms within the framework of a nonlinear dynamic oligopoly as those developed by a recent burgeoning literature (see [1] and the papers cited therein). The issue of decision making under uncertainty as distinct from risk has recently been revisited, amongst others, by [2–6]. In these papers, strategic uncertainty is represented by means of the Choquet expected utility (CEU) theory [7] where agents exhibit different attitudes towards uncertainty, that is, pessimism or optimism, overweighing less or more uncertain events. This theory has also been adopted by [8] to represent strategic behaviour à la Cournot with different firms’ attitude towards uncertainty (CEU theory has been applied to other economic contexts, where optimism and pessimism can explain the paradox of people buying insurance and gambling, the equity premium puzzle, and the small stock puzzle [6]). In a strategic context such as a duopoly game, it is crucial to forecast the behaviour of the competitor in order to make a decision and to specify the information set available to each player. In the literature on nonlinear oligopolies, two distinct assumptions with regard to available information are usually made: players have a complete knowledge of the market demand and use some form of expectations about the rival’s strategic variable decision (e.g., naive, rational, or adaptive expectations or, alternatively, some weighted sum of previous rules) to set the price or the quantity in the future period (e.g., [9, 10]); players have limited information about the market demand and use some forms of estimation of their own current marginal profits (e.g., [11–14]) or other adjustment mechanisms such as the local monopolistic approximation to determine the price or quantity in the future period [15]. This is because, under the hypothesis of limited information, players are unable to solve the optimisation problem by accounting for expectations about the value of the strategic variable that the competitor will choose for the next period, but they are able to get either a correct estimate of the slope 2 Abstract and Applied Analysis of their own profit function in the current period, that is, the partial derivative of the profit function computed at the current state of production, or use a linear approximation of the demand function by market experiments without any guess about the influence of the competitors (i.e., monopolistic approximation). In this paper we study local and global dynamics of a Cournot duopoly model with strategic uncertainty as in [8] by considering different information sets of players. In particular, (1) both firms have complete information on the market demand and use the “best reply” adjustment mechanism (static expectations) to vary production period by period; (2) both firms have incomplete information on the market demand and adjust production by following a mechanism based on marginal profits; (3) one firm has incomplete information and production decisions are based on the behaviour of marginal profits, and the rival has complete information and static expectations. These assumptions make the topological structure of the map different and then comparing the local and global properties of the three dynamic systems is relevant. In particular, in cases 2 and 3 we find that complex dynamics as well as coexistence of attractors may occur. These phenomena depend on the relative value of the parameter that weights the strategic uncertainty of firms. The analysis is performed by applying the critical lines techniques as well as through numerical simulations. The rest of the paper proceeds as follows. Section 2 develops the Cournot model with strategic uncertainty. Section 3 studies the dynamics of the model under complete information and static expectations of both firms. Section 4 introduces the adjustment mechanism of production based on marginal profits for both firms and performs the local and global dynamics. Section 5 analyses the mixed case. Section 6 outlines the conclusions. attitude, that is, it is either pessimistic or optimistic, towards uncertainty. In particular, we assume that the pessimistic firm assigns a positive probability of being in the worst case (which is realised when the market price equals zero) (the firm considers it possible that the supply of the other firm is large enough to get the price to zero), while the optimistic firm assigns a positive probability of being in the best case, (which is realised when it behaves as a monopolist in the market). By using CEU theory, it is assumed that each firm maximises its own CEU function which is given by a weighted average (with the parameter 𝛾 ∈ [0, 1]) of its expected profits and the profits (...truncated)