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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.

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 [

The issue of decision making under uncertainty as distinct from risk has recently been revisited, amongst others, by [

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., [

In this paper we study local and global dynamics of a Cournot duopoly model with strategic uncertainty as in [

The rest of the paper proceeds as follows. Section

We consider a Cournot duopoly for a single homogenous product with a liner negatively sloped inverse demand given by

Profits of firm

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

Let firm

From (

In addition, from (

We note that the values of price and profits corresponding to Nash equilibrium (

One of the first dynamic adjustment mechanisms studied in the literature on nonlinear oligopolies is the one proposed by [

Since the market demand is linear and average (and marginal) costs are constant, map (

The fixed point (

First of all, we focus on

Then, we can conclude that fixed point (

In addition, we note that in contrast with the standard Cournot game (

Starting from the red (resp., black) region, the first iterate is

This section studies the dynamics of the Cournot model with strategic uncertainty by using an adjustment mechanism of production introduced by [

By using (

The fixed points of map (

The study of local stability of equilibrium solutions is based on the study of the Jacobian matrix:

The fixed points

We have

The eigenvalues associated with (

The fixed point

The Jacobian matrix of map (

Now, since

To sum up,

With regard to the first condition in (

An important feature of map (

Since

From direct computation, we have that

Critical curves are represented for the parameter set:

In this section, we describe the properties of the basins of attraction of map (

It follows that the dynamics on axes

In this section we study the dynamic system

Parameter set:

By increasing the value of

(a) Portions of the basin of attraction of trajectories that converge to invariant axes for map

In this section, we assume that firms adjust production period by period by using different mechanisms. In particular, the pessimistic firm has limited information and modifies production depending on the value of its own marginal profits, while the optimistic firm has complete information and static expectations (“best reply” dynamics). Given this type of heterogeneity and using (

Map (

The study of local stability of equilibrium solutions is based on the study of the Jacobian matrix of the dynamic system. The Jacobian matrix of map

The following propositions hold.

The fixed point

The Jacobian matrix of map (

The fixed point

The Jacobian matrix of map (

This section develops the global analysis of map (

Different colours correspond to different behaviours of the model. In the yellow region,

Parameter set:

We note, however, that by starting from points that lie in the yellow region it is possible that the subsequent iteration continues to lie in the yellow region or, alternatively, it leads to either the red region or black region and the dynamics will definitely end up on the point

Analogously with map

In addition, in the model with heterogeneous adjustment mechanisms we note that coexistence of interior attractors can occur by slightly reducing the value of

Parameter set:

This paper developed a nonlinear Cournot duopoly to study the role of strategic uncertainty on the dynamics of the model economy. We characterised the local and global properties of a discrete two-dimensional map by considering that

The authors gratefully acknowledge that this work has been performed within the activity of the PRIN-2009 project “Local interactions and global dynamics in economics and finance: models and tools,” MIUR (Ministry of Education), Italy, and PRIN-2009 project “Structural change and growth,” MIUR, Italy. Numerical simulations have benefited from algorithms that can be found in