Chaos and complexity in economics

History of Political Economy, 2003

It is a well-recognized peculiarity of the so-called years of "high theory" that for the first time in the history of economics the mental variablesthat is, the expectations, conjectures, and beliefs-of the agents featuring in economic models gained the spotlight and became the explicit object of formal analysis. This happened as a consequence of the effort to extend the notion of static equilibrium to a multiperiod setup. Thus, elements such as an agent's knowledge, uncertainty, and foresight became absolutely central, so much so that, following G. L. S. Shackle's 1967 classic, it became commonplace to single them out as the leading topics during the period ranging from the mid-1920s to the early 1940s. Indeed, it was deemed impossible even to define the notion of economic equilibrium without relating it to the time dimension and to planning behavior, that is, without specifying how an agent's mental variables are formed and how economic interactions are framed in a multiperiod horizon. Crucial notions in contemporary economics such as temporary and intertemporal equilibrium or the ex ante-ex post distinction bear witness to the achievements of the "high theorists." In a single paper one cannot even briefly sketch the main events and characters leading to the rise of mental variables in interwar economics. Luckily, we already have some brilliant accounts of these developments

Studies in Nonlinear Dynamics & Econometrics, 2012

Chaos, Solitons & Fractals, 2000

A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map X xY y 3 r 1 yY r 2 x, is considered. Results on the existence of cycles and more complex attractors are given, based on the study of the one-dimensional map p x r 1 r 2 x. The property of multistability, i.e. the existence of many coexisting attractors (that may be cycles or cyclic chaotic sets), is proved to be a characteristic property of such games. The problem of the delimitation of the attractors and of their basins is studied. These general results are applied to the study of a particular duopoly game, proposed in M. Kopel [Chaos, Solitons & Fractals, 7 (12) (1996) 2031±2048] as a model of an economic system, in which the reaction functions r 1 and r 2 are logistic maps. Ó

Nonlinear Dynamics, 2015

In this paper we propose and compare three heterogeneous Cournotian duopolies, in which players adopt best response mechanisms based on different degrees of rationality. The economic setting we assume is described by an isoelastic demand function with constant marginal costs. In particular, we study the effect of the rationality degree on stability and convergence speed to the equilibrium output. We study conditions required to converge to the Nash equilibrium and the possible route to destabilization when such conditions are violated, showing that a more elevated degree of rationality of a single player does not always guarantee an improved stability. We show that the considered duopolies exhibit either a flip or a Neimark-Sacker bifurcation. In particular, in heterogeneous oligopolies models, the Neimark-Sacker bifurcation usually arises in the presence of a player adopting gradient-like decisional mechanisms, and not best response heuristic, as shown in the present case. Moreover, we show that the cost ratio crucially influences not only the size of the stability region, but also the speed of convergence toward the equilibrium.

Dynamic Games and Applications

We consider a model of evolutionary competition between adjustment processes in the Cournot oligopoly model and investigate the effect of increasing the number of firms. Our focus is on Nash play versus a general short-memory adaptive adjustment process. We find that, although Nash play has a stabilizing influence, a sufficient increase in the number of firms in the market tends to make the Cournot-Nash equilibrium unstable. This shows that the famous result by Theocharis (Rev Econ Stud 1960), that Cournot oligopoly markets are unstable for more than three firms, is robust, although the instability threshold increases in the presence of Nash firms. We establish that both the existence and the level of this threshold depend on the information costs associated with Nash play. Moreover, the interaction between adjustment processes naturally leads to the emergence of complicated endogenous fluctuations as the number of firms increases, even when demand and costs are linear. Keywords Stability of Cournot-Nash equilibrium • n-Player Cournot games • Evolutionary competition • Endogenous fluctuations JEL Classification C72 • C73 • D43 An earlier version of this paper circulated under the title "On the stability of the Cournot equilibrium: An evolutionary approach".

Central European Journal of Operations Research, 2021

In this paper, we analyse the company behaviour in duopoly taking into account the most common strategies, including dominant, reactive, cooperative and tit-for-tat strategies, since they account for most of the decisions made by companies. Dominant, reactive and cooperative strategies may lead to different outcomes, such as Stackelberg, Cournot, Cartel, Cartel with one cheater and Perfect competition equilibria, while the tit-for-tat strategy may lead to Cartel, Cournot or Perfect competition equilibria due to its retribution nature. However, we argue that these outcomes are mainly valid in the short-run since in the long-run the companies learn and adapt to the behavioural pattern of their peer, which leads them to evolve to a new way of thinking and strategic planning taking into account the long-run effects. Due to the long-term perspective, the outcome of the implementation of various strategies in duopoly may not be efficiently solved using simple game theory analysis. For that purpose, we propose the utilization of more complex analysis, an evolutionary game theory, which is based on the adaption of the company to the behaviour of other players in a duopoly. When the probability of choices is applied, new fitness equations are obtained which show the changing tendencies towards the companies' strategies that ensure payoffs above the average, in the long run, using a replicator dynamics concept. We analyse several scenarios, in which players choose among two, three or four strategies. Our results indicate that the long-run equilibrium and preferred options are significantly altered depending on the starting set of strategies.

Chaos, Solitons & Fractals, 2010

Applied Mathematics and Computation, 2014

Dynamic monopolies are investigated with discrete and continuous time scales by assuming general forms of the price and cost functions. The existence of the unique profit maximizing output level is proved. The discrete model is then constructed with gradient adjustment. It is shown that the steady state is locally asymptotically stable if the speed of adjustment is small enough and it goes to chaos through period-doubling cascade as the speed becomes larger. The non-negativity condition that prevents time trajectories from being negative is derived. The discrete model is converted into the continuous model augmented with time delay and inertia. It is then demonstrated that stability can be switched to instability and complex dynamics emerges as the length of the delay increases and that instability can be switched to stability as the inertia coefficient becomes larger. Therefore the delay has the destabilizing effect while the inertia has the stabilizing effect.

Mathematics and Computers in Simulation, 1999

The time evolution of a dynamic oligopoly game with three competing firms is modeled by a discrete dynamical system obtained by the iteration of a three-dimensional non-invertible map. For the symmetric case of identical players a complete analytical study of the stability conditions for the fixed points, which are Nash equilibria of the game, is given. For the situation of several coexisting stable Nash equilibria a numerical study of their basins of attraction is provided. This gives, evidence of the occurrence of global bifurcations at which the basins are transformed from simply connected sets into nonconnected sets, a basin structure which is peculiar of non-invertible maps. The presence of several coexisting attractors (or multistability) is observed even when complex attractors exist. Two different routes to complexity are presented: one related to the creation of more and more complex attractors; the other related to the creation of more and more complex structures of the basins. Starting from the benchmark case of identical players, the effects of heterogeneous behavior of the players, causing the loss of the symmetry properties of the dynamical system, are investigated through numerical explorations. # 1999 IMACS/ Elsevier Science B.V. All rights reserved.

This study with the help of non-linear Cournot Duopoly model shows that duopoly market can be chaotic because of non-linearity. It establishes conditions for the stability of chaotic market. The study considers the conditions for generating chaos and controlling chaos from the perspectives of both the firms. It explains how adaptive expectations can be used due to inconsistency of naïve expectations with the help of autocorrelation coefficient and then explains the method of controlling chaos.

International Game Theory Review, 2004

In this paper we study a model of a quantity-setting duopoly market where firms lack knowledge of the market demand. Using a misspecified demand function firms determine their profit-maximizing choices of their corresponding perceived market game. For illustrative purposes we assume that the (true) demand function is linear and that the reaction functions of the players are quadratic. We then investigate the global dynamics of this game and characterize the number of steady states and their welfare properties. We study the basins of attraction of these steady states and present situations in which global bifurcations of their basins occur when model parameters are varied. The economic significance of our result is to show that in situations where players choose their actions based on a misspecified model of the environment, additional self-confirming steady states may emerge, despite the fact that the Nash-equilibrium of the game under perfect knowledge is unique. As a consequence t...

Physica A: Statistical Mechanics and its Applications, 2003

We analyze a nonlinear discrete-time Cournot duopoly game, where players have heterogeneous expectations. Two types of players are considered: boundedly rational and naive expectations. In this study we show that the dynamics of the duopoly game with players whose beliefs are heterogeneous, may become complicated. The model gives more complex chaotic and unpredictable trajectories as a consequence of increasing the speed of adjustment of boundedly rational player. The equilibrium points and local stability of the duopoly game are investigated. As some parameters of the model are varied, the stability of the Nash equilibrium point is lost and the complex (periodic or chaotic) behavior occurs. Numerical simulations is presented to show that players with heterogeneous beliefs make the duopoly game behaves chaotically. Also we get the fractal dimension of the chaotic attractor of our map which is equivalent to the dimension of Henon map.

Mathematics

The current paper analyzes a competition of the Cournot duopoly game whose players (firms) are heterogeneous in a market with isoelastic demand functions and linear costs. The first firm adopts a rationally-based gradient mechanism while the second one chooses to share the market with certain profit in order to update its production. It trades off between profit and market share maximization. The equilibrium point of the proposed game is calculated and its stability conditions are investigated. Our studies show that the equilibrium point becomes unstable through period doubling and Neimark–Sacker bifurcation. Furthermore, the map describing the proposed game is nonlinear and noninvertible which lead to several stable attractors. As in literature, we have provided an analytical investigation of the map’s basins of attraction that includes lobes regions.

Chaos, Solitons & Fractals, 2000

Puu's economic dynamical model is extended to the following cases: Bounded rationality, dierentiated goods and bounded rationality with delay. It is shown that delay increases stability. Hence ®rms using bounded rationality with delay has a higher chance of reaching Nash equilibrium. Stability and instability (cycles and chaos) conditions for all these cases are determined.

Journal of the Egyptian Mathematical Society, 2016

In this paper, two different mechanisms are used to study a homogeneous Cournot duopoly in a market characterized by the downward sloping and concave price function. Two firms, which have constant marginal costs, use adaptive, low-rationality mechanisms to adjust their production levels toward equilibrium. In particular, the stability of the equilibrium for two different mechanisms is studied. However, complex dynamics arise, especially when the reaction coefficient increases. Finally, we compare the obtained results of the two models.