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Within a classical discrete-time Cournot oligopoly model with linear demand and quadratic cost functions, minimum and maximum production constraints are imposed in order to explore their effects on the dynamic of the system. Due to the presence of such constraints, the dynamic model assumes the form of a continuous piecewise linear map of the plane. The study of Nash equilibria of the oligopoly game, together with an analytical and numerical investigation of the different kinds of attractors of the dynamical system, shows how the presence of production constraints generates so called border collision bifurcations, a kind of global bifurcations recently introduced in the literature on non-smooth dynamical systems, which gives rise to a quite rich spectrum of dynamic scenarios, characterized by drastic changes in the qualitative dynamic properties of the system.

De Finetti’s commitment to mathematics education was always concrete and vigorous, as shown by his publications of treatises, textbooks, educational papers, and articles for non-specialists, as well by intense organisational activities.

In this chapter we present a unified approach for modelling the diffusion of alternative choices within a population of individuals in the presence of social externalities, starting from two particular discrete-time dynamic models – Galam’s model of rumors spreading [10] and a formalization of Schelling’s binary choices [7]. We describe some peculiar properties of discrete-time (or event-driven) dynamic processes and we show how some long-run (asymptotic) outcomes emerging from repeated short time decisions can be seen as emerging properties, sometimes unexpected, or difficult to be forecasted.

In this chapter we will introduce the classical Cournot model, which is also known as the single-product quantity setting oligopoly model without product differentiation. In the first section of the chapter the Cournot model will be discussed as an N-firm static game and the best responses of the firms and the equilibria will be determined in a series of examples, many of which will be built upon in developing the ideas in subsequent chapters. Section 1.2 introduces the dynamic adjustment processes via which we shall assume that firms adjust output over time. We will in particular discuss expectation formation processes and adaptive adjustments and gradient adjustments. The final section will illustrate by simple examples the complexity of the dynamics that can arise in these models due to certain nonlinear features to be described below. The fundamental techniques for the global analysis of the dynamics of such models will be explained in Sect. 1.3.

The previous chapters have introduced and analyzed the classical Cournot model under a number of assumptions. In this chapter we discuss some important modifications and extensions. We first introduce market share attraction games where the dynamics are driven by a generalization of the gradient adjustment process introduced in Chaps. 1 and 1. We carry out both a local and global analysis of the stability of this game. In Sect. 4.2 we consider labor-managed oligopolies with best response dynamics. We give a detailed discussion of the local stability in the discrete time case and via an example show the type of global dynamical behavior that is possible in this model type. The section concludes with a brief discussion of the local stability of a continuous time version of the labor-managed oligopoly. In Sect. 4.3 we introduce intertemporal demand interaction effects, brought about for example by habit formation, into dynamic oligopolies with best response dynamics. We give a local and global stability analysis of the model in discrete time. For the continuous time version we study the local stability of the dynamics, including also the case when there are information lags. In Sect. 4.4 we analyze oligopolies with production adjustment costs. For the case of best reply dynamics in discrete time we give local stability conditions. In the final section we consider oligopolies where there is partial cooperation amongst the firms of the industry. We show various properties of the best response function, give local stability for best reply dynamics in continuous time, and analyze the global dynamics of a particular example under discrete time best response dynamics.

The previous chapters have already dealt with the behavior of boundedly rational firms in an oligopoly. Although the firms know the true demand relationship, we have assumed that they do not know their competitors’ quantity choices. Instead they form expectations about these quantities and they base their own decisions on these beliefs. In particular, we have focused on several adjustment processes that firms might use to determine their quantity selections and we have investigated the circumstances under which such adjustment processes might lead to convergence to the Nash equilibrium of the static oligopoly game. However, the information that firms have about the environment may be incomplete on several accounts. For example, players may misspecify the true demand function or just misestimate the slope of the demand relationship, the reservation price, or the market saturation point. However, if firms base their decisions on such wrong estimates, they will realize that their beliefs are incorrect, since the market data they observe (for example, market prices or quantities) will be different from their predictions. Obviously, firms will try to update their beliefs on the demand relationship and this will give rise to an adjustment process. In other words, firms will try to learn the game they are playing. Following this line of thought, in this chapter we study oligopoly models under the assumption that firms either use misspecified price functions (Sect. 5.1) or do not know certain parameters of the market demand (Sect. 5.2). The main questions we want to answer are the following. If we understand an equilibrium in a game as a steady state of some non-equilibrium process of adjustment and “learning,” what happens if the players use an incorrect model of their environment? Does a reasonable adaptive process (for example, based on the best response) converge to anything? If so, to what does it converge? Is the limit that can be observed when the players play their perceived games (close to) an equilibrium of an equilibrium of the underlying true model? Is the observed situation consistent with the (limit) beliefs of the players?

In Chap. 1 we introduced the classical Cournot model and after setting up the general framework we focused on a number of specific examples involving combinations of linear and hyperbolic price functions and linear and quadratic cost functions, also taking careful account of capacity constraints. These examples illustrated the variety of reaction functions that can occur and the various types of equilibria (possibly multiple) both in the interior of the domain of interest and on its boundaries. We then went on to introduce the various types of adjustment processes that underpin the dynamic processes, the study of the local and global dynamics of which has occupied much of the space in this book. In particular we considered discrete time and continuous time versions of partial adjustment towards the best response with naive expectations and adaptive expectations as well as the gradient adjustment process. We then introduced some of the basic tools for the analysis of global dynamics via some examples involving duopoly or symmetric and semi-symmetric oligopolies. We introduced the important concept of basins of attraction of different equilibria and the important tool of the critical curve and the concept of border collision bifurcations method. Already with the simple examples considered we see the types of complexity that can arise in oligopoly models under the type of dynamic adjustment processes we consider here.

Starting from a family of discontinuous piece-wise linear one-dimen- sional maps, recently introduced as a dynamic model in social sciences, we pro- pose a geometric method for flnding the analytic expression of the bifurcation curves, in the space of the parameters, that bound the regions characterized by the existence of stable periodic cycles of any period. The conditions for the creation and the destruction of periodic cycles, as well as the analytic expres- sions of the bifurcation conditions, are obtained by studying the occurrence of border-collision bifurcations. In this paper we consider the case of maps formed by three linear portions separated by two discontinuity points. After summarizing the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modifled by the intro- duction of a second discontinuity point. Finally we show how the considered map can be obtained as the limit case of a family of continuous maps as a pa- rameter is increased without bounds, and we show how the low period cycles, which are typical of the discontinuous map we consider, emerge from the more complex (i.e. chaotic) behaviors observed in the continuous maps when a pa- rameter value is large enough. From the point of view of the social application the increasing values of the parameter can be interpreted as higher degrees of impulsivity of the agents involved in binary decisions.

Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise linear functions, in one- or two- dimensional spaces. In this paper we study a one-dimensional map made up of three linear pieces which are separated by two discontinuity points, motivated by a dynamic model arising in social sciences. Starting from the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point, and we give the analytic expressions of the bifurcation curves of the principal tongues (or tongues of first degree) for the family of maps considered, which depends on five parameters.

In this article, single product Cournot oligopolies are considered, where the demand and cost functions are linear. While cost functions are completely known by all firms, they only partially know the demand function, as they misspecify the slope. At any stage of the repeated oligopoly game firms update the slope of their subjective demand functions on the basis of the discrepancy they observe between the expected price, computed according to believed demand, and the price they actually observe. This adjustment process has a unique steady state, where any subjective demand function coincides with the true demand function. If such steady state is stable, then the true slope of the demand function can be learned by all oligopolists, even if they start from misspecified initial guesses. Sufficient conditions for the stability of the steady state are given for n-firms oligopolies. In the particular case of a duopoly, an exact delimitation of the stability region in the parameters' space is given, and with the help of numerical simulations, the size and the shape of the basins of attraction is analysed, as well as the kinds of attracting sets that characterise the long-run dynamics of the learning process when the steady state is unstable.

We propose a repeated oligopoly game where quantity setting firms have incomplete knowledge of the demand function of the market in which they operate. At each time step they solve a profit maximization problem by using a subjective approximation of the demand function based on a local estimate its partial derivative, computed at the current values of prices and outputs, obtained through market experiments. At each time step they extrapolate such local approximation by assuming a linear demand function and ignoring the effects of the competitors outputs. Despite a so rough approximation, that we call "Local Monopolistic Approximation" (LMA), the repeated game may converge to a Nash equilibrium of the true oligopoly game, i.e. the game played under the assumption of full information. An explicit form of the dynamical system that describes the time evolution of oligopoly games with LMA is given for arbitrary differentiable demand functions, provided that the cost functions are linear or quadratic. Sufficient conditions for the local stability of Nash Equilibria are given. In the particular case of an isoelastic demand function, we show that the repeatead game based on LMA always converges to a Nash equilibrium, both with linear and quadratic cost functions. This stability result is compared with "best reply" dynamics, obtained under the assumption of isoelastic demand (fully known by the players) and linear costs.

In this lesson we consider discrete time dynamical systems with coexisting attractors, and we analyze the problem of the structure of the boundaries that separate their basins of attraction. This problem may become particularly challenging when the discrete dynamical system is represented by the iteration of a noninvertible map, because in this case nonconnected basins can be obtained, formed by several (even infinitely many) disjoint portions. Measure theoretic attractors, known as Milnor attractors, are also described, together with riddled basins, an extreme form of complex basin’s structure that can be observed in the presence of such attractors. Some tools for the study of global bifurcations that lead to the creation of complex structures of the basins are described, as well as some applications in discrete time models taken from economic dynamics.

We propose a dynamic model for studying the time evolution of fish stocks in an environment divided into two adjacent zones with different fishing policies. We analyze two particular harvesting methods: constant fishing effort and profit maximization. In this case, some agents engage in competition based on maximization of individual profit whereas others cooperate. The asymptotic behavior of the system pinpoints that, even if the presence of a reserve area leads to higher levels of sustainability in exploiting fish stocks, attention should be paid in properly regulating the harvesting activity in order to avoid severe depletion of the resource.

In this paper we analyze a recurrence , where is a weighted power mean of ,…., . Such an iteration scheme has been proposed to model a class of non-linear forward-looking economic models ( the state today is affected by tomorrow’ s expectation ) under bounded rationality; the agents employ a recursive learning rule to update beliefs using weighted power means of the past states. A proposition on the convergence of the dynamical system with memory, proven with a general weighted power mean, generalizes some results given in the literature, where only the arithmetic mean is considered. A power weighted mean with exponentially decreasing weights decreasing is proposed to simulate a fading memory. In this case the iteration scheme with memory is reduced to an equivalent two-dimensional autonomous map whose possible kinds of asymptotic behaviors are the same as those of a one-dimensional map. By this general technique it is proved, for a function f which maps a compact interval into itself, that the presence of a long memory has a stabilizing effect, in the sense that with a sufficiently strong memory convergence to a steady state is obtained even for an otherwise oscillating, or chaotic, dynamical system. In the appendix is considered an economic example from an overlapping generation models which leads to a harmonic mean.

It is well known that models of nonlinear oscillators applied to the study of the business cycle can be formulated both as continuos or discrete time dynamic models (see e.g. [23], [33], [34]). However, economic time is often discontinuous (discrete) because decisions in economics cannot be continuously revised. For this reason discrete-time dynamical systems, represented by difference equations or, more properly, by the iterated application of maps, are often a more suitable tool for modelling dynamic economic processes. So, it is useful to study the peculiarities of discrete dynamical systems and their possible applications to the study of self sustained oscillations. This is the main goal of this Chapter, where we describe, on the light of some recent results about local and global properties of iterated maps of the plane, some particular routes to the creation/destruction of closed invariant curves, along which self sustained oscillations occur.

The theory of dynamical systems has been one of the most developed mathematical areas in the last 50years with a wide spectrum of applications ranging from physics to biology to economics. During the last decades, economic theory in particular has witnessed an important shift in methodology. The classical approach that describes economic outcomes as resulting from the choices of fully rational and identical economic agents has failed to explain many important features of economic complexity in the real world. Fueled by its inability to predict sudden changes and (financial and economic) crises, it has been criticized on a number of grounds. As a consequence, a growing interest in alternative approaches based on concepts like bounded rationality and heterogeneity, social interaction, and learning has emerged. Following this new paradigm, economics and finance are viewed as complex evolving systems characterized by emerging non-equilibrium situations. The corresponding mathematical modeling approaches require a wide range of analytical and numerical techniques, some of which are new and innovative. These techniques consist of a mix of theories, among them the qualitative theory of nonlinear dynamical systems, optimal control theory, game theory, and the theory of stochastic processes. The current status and the upcoming trends of these new modeling techniques and numerical toolswere presented at the InternationalConferenceonNonlinearEconomic

This paper is devoted to the study of the global dynamical properties of a two-dimensional noninvertible map, with a denominator which can vanish, obtained by applying Bairstow's method to a cubic polynomial. It is shown that the complicated structure of the basins of attraction of the fixed points is due to the existence of singularities such as sets of nondefinition, focal points, and prefocal curves, which are specific to maps with a vanishing denominator, and have been recently introduced in the literature. Some global bifurcations that change the qualitative structure of the basin boundaries, are explained in terms of contacts among these singularities. The techniques used in this paper put in evidence some new dynamic behaviors and bifurcations, which are peculiar of maps with denominator; hence they can be applied to the analysis of other classes of maps coming from iterative algorithms (based on Newton's method, or others). (c) 1999 American Institute of Physics.

This paper studies the evolution of two clusters of firms competing on a common market. Firms exit and enter a cluster based on the perceived chances for profits inside and outside the cluster. Information about profits are diffused by direct communication between firms. Internal and external spillover effects reduce the overall costs of firms in the clusters depending on the number of firms in the own and the competing cluster. A discrete time deterministic dynamical system describing the evolution of cluster sizes is derived. An analysis of the long run attractors of the system and their basins of attraction is used to compare the effects of advantages of a cluster with respect to the size of internal and external spillover effects, respectively. Furthermore, the implications of slow and fast exit and entry behavior of firms for the long run survival and the size of the clusters are studied.

The time evolution of an oligopoly system is often described as an n-players game which is played repeatedly, in the sense that at each discrete time period t = 0, 1,..., n producers choose their actions, x 1(t), ... ,x n (t), by solving an optimization problem based on the knowledge of the actions observed in the past. For example, the classical Cournot adjustment (see Cournot, 1838, Teocharis, 1960, and many others) is obtained by assuming that at each period any player chooses its own production strategy which is a best response to the choices of the competitors in the previous period.

Discrete-time dynamic oligopoly games which exhibit chaotic time patters of the competitors’ strategic choices are at the center of a flourishing literature, including the seminal paper by Rand (1978) and several papers after it (see e.g. Dana and Montrucchio, 1986, Puu, 1991, Kopel, 1996), where simple microeconomic situations have been proposed which lead to duopoly games with chaotic dynamics. The main concern, in this stream of literature, is to emphasize the features of the duopoly games which are responsible for the transition from regular to chaotic dynamics and to analyze the bifurcations which cause the loss of stability of the equilibria and the appearance of more complex attractors (see e.g. Bischi et al., 2000).

In this chapter oligopoly theory and population dynamics are combined to describe the exploitation of a living resource under imperfect competition. This requires an interdisciplinary approach, because profit maximization arguments must be combined with the biological laws which regulate the natural growth of living resources in order to determine the long-run behavior of the natural system. The results obtained are often characterized by complex behavior and bifurcation phenomena, because the interaction of human economic decision making with ecological dynamics are highly non linear (see e.g. Rosser Jr., 2001). Indeed, resource economics is a very important field of application of dynamic analysis. In the literature on the economics of renewable resources, a plethora of questions has been studied using the tools from optimal control theory, dynamic programming and the theory of nonlinear dynamical systems. Renewable resources, e.g. grass, trees or fish, have the capacity for reproduction and growth over time and their stock is diminished by the harvesting activities of a sole owner or several individuals. Among the problems which have been investigated extensively are: How do (optimal) harvesting paths look like? Under which conditions is it more likely to observe conservation or extinction of the resource? What is the influence of the market structure? In the last 50 years considerable progress has been made, in particular, in fishery economics (for an overview, see Conrad 1995). To appreciate the breadth of topics which are the focus in the economics of fisheries and for a presentation of the main economic insights, the interested reader is advised to consult the book by Clarke (1990). Furthermore, the texts of Conrad and Clarke (1987) and Conrad (1999) can be recommended. In this chapter, we will introduce a bioeconomic model of commercial fishing to study the evolution of the stock of a fish population which is subject to harvesting over time. We will focus on three points: (i) the influence of the market structure, in particular, the role of strategic effects; (ii) the problem of extinction or conservation of the resource and the role of harvesting costs; (iii) the influence of errors in predicting the fish stock when determining future harvesting activities.

This special issue of Computational Economics contains a collection of refereed papers presented at the sixth MDEF Modelli Dinamici in Economia e Finanza (Dynamic Models in Economics and Finance) International Workshop held at the Department of Economics and Quantitative Methods of the University of Urbino, Italy, on the 23rd–25th of September, 2010. Scientific meetings certainly derive their value from the scientific results presented during the formal sessions, the discussions after the conferences often leading to a comparison of different methods and results. However equally important is the atmosphere created amongst the participants during the breaks, dinners and other social events that afford them the opportunity to catch up with old acquaintances and to make new ones. During the six editions of MDEF, which has taken place every two years in Urbino since 2000, many collaborations have been established among the participants and several papers and books written as a result of such fruitful collaborations have been planned and developed thanks to these meetings, including some of the papers published in this special issue. The sixth MDEF workshop was held two years after a severe global economic crisis, one turning out to be the worst since the 1930s, as has been widely commented. The crisis has not only had widespread influence on social and economic life in many

We consider a Kaldor-type discrete-time nonlinear business cycle model in income and capital, where investment is assumed to depend both on the difference between normal and current levels of capital stock, and on the difference between the current income and its normal level, through a nonlinear S-shaped increasing function. As usual in Kaldor business cycle models, one or three steady states exist, and the standard analysis of the local stability and bifurcations suggests that endogenous oscillations occur in the presence of only one unstable equilibrium, whereas the coexistence of three equilibria is characterized by bi-stability, the central equilibrium being on the boundary which separates the basins of the two stable ones. However, a deeper analysis of the global dynamic properties of the model in the parameter ranges where three steady states exist, reveals the existence of an attracting limit cycle surrounding the three steady states, leading to a situation of multistability, with a rich and complex dynamic structure.

In this paper we study a class of dynamic promotional competition models, in which firms compete for market share by expending marketing effort. We investigate two main issues. First, we answer the question if it is possible to give a global characterization of the stability of the steady state effort allocation. We show that by using the concept of critical curves and an invariance property of the coordinate axes a characterization of the set of feasible points (points that generate positive trajectories converging to the steady state allocation) and its changes can be given. Second, we deal with the assumption of homogeneous firms, which is often made in the literature. We demonstrate that the symmetric model which derives from this assumption exhibits, in many situations, non-generic dynamical behavior. New phenomena, like Milnor attractors and synchronization of trajectories, arising in the homogeneous case are illustrated. The introduction of small heterogeneities into the model invalidates many of the conclusions derived under the hypothesis of homogeneous firms.

A dynamic Cournot duopoly game, characterized by firms with bounded rationality, is represented by a discrete-time dynamical system of the plane. Conditions ensuring the local stability of a Nash equilibrium, under a local (or myopic) adjustment process, are given, and the influence of marginal costs and speeds of adjustment of the two firms on stability is studied. The stability loss of the Nash equilibrium, as some parameter of the model is varied, gives rise to more complex (periodic or chaotic) attractors. The main result of this paper is given by the exact determination of the basin of attraction of the locally stable Nash equilibrium (or other more complex bounded attractors around it), and the study of the global bifurcations that change the structure of the basin from a simple to a very complex one, with consequent loss of predictability, as some parameters of the model are allowed to vary. These bifurcations are studied by the use of critical curves, a relatively new and powerful method for the study of noninvertible two-dimensional maps.

A simple ecosystem with limiting nutrient cycling is modeled by chemostat equations with an integral term describing the continuous time lag involved in the process of nutrient regeneration from organic sediments. The same model has already been proposed in a previous paper, where conditions for boundedness of the solutions and stability of the equilibria were given. This paper is concerned with the relationships between resilience, that is, the speed with which the system returns to a stable equilibrium following a perturbation, and the time lag in the nutrient recycling process. Simple algorithms are given for the numerical calculation of the characteristic return time toward the stable equilibrium following a small perturbation. These methods also allow us to distinguish the case of monotone convergence from that of oscillatory convergence toward equilibrium The numerical results obtained show that the presence of the time lag causes both qualitative and quantitative modifications in the dependence of equilibriumresilience on some relevant ecological parameters, such as the input nutrient concentration and the recycling extent. Analytical results for “quasi-closed” ecosystems are given that show that such stable systems are characterized by a very low resilience.

Mathematical models are often used to solve decision problems. A typical decision problem in clinical practice is that of optimal dosage of drugs, i.e. the choice of a dosage regimen in order to maintain a therapeutic level for a long time without producing serious side effects (see e.g.l). If a drug is encapsulated within a given volume of erythrocytes for treatment, the values of two parameters must be determined in order to obtain a desired result: the volume of erythrocytes prepared and the amount of the drug which is loaded inside them. In this paper an optimal control problem is formulated on the basis of a mathematical model which has been recently proposed in2 and3 to describe the administration of a drug after having encapsulated it inside erythrocytes as a non-diffusible prodrug. This model allows us to obtain an estimate of the maximum concentration of the drug in the plasma, and from this information we can recognize the set of allowable decisions (or admissible controls) which is the set of values of the control parameters which lead to therapeutic and non-toxic drug concentration in plasma. The solution of the optimal control problem indicates that in the set of admissible control parameters a unique point exists which gives the maximum time of therapeutic effect without reaching a toxic concentration. This point represents the optimal dosage strategy.

The growth of a species feeding on a limiting nutrient supplied at a constant rate is modelled by chemostat-type equations with a general nutrient uptake function and delayed nutrient recycling. Conditions for boundedness of the solutions and the existence of non-negative equilibria are given for the integrodifferential equations with distributed time lags. When the time lags are neglected conditions for the global stability of the positive equilibrium and for the extinction of the species are provided. The positive equilibrium continues to be locally stable when the time lag in recycling is considered and this is proved for a wide class of memory functions. Computer simulations suggest that even in this case the region of stability is very large, but the solutions tend to the equilibrium through oscillations.

We study a system of two integrodifierential equations which models the evolution of a biotic species feeding on an abiotic resource. We also consider nutrient recycling with time delay. By Hopf bifurcation theory we prove the existence of stable oscillations for a range of values of the input of nutrients.

Muovendo dall’idea che esista un isomorfismo tra i recenti sviluppi di matematica e letteratura, il libro percorre il Novecento esplorando una rete di scambi fatta non solo di prestiti e citazioni (la tematizzazione letteraria della matematica), ma di parallelismi strutturali e di premesse condivise nel rapporto col reale. La postfazione dialoga con il volume, ripercorrendo alcune tappe del cammino verso una crescente consapevolezza della complessità, nel reale, nel dominio concettuale e nella rappresentazione letteraria.

What She Left (2015) di T.R. Richmond è un thriller in cui ci confrotniamo con la morte di una ragazza attraverso un mosaico di testimonianze, spesso legate al mondo della rete e presentate secondo una cronologia scomposta derivante da un percorso investigativo generatore di suspense. L'analisi del romanzo mostra come il libro non rientri semplicemente nelle coordinate di un genere popolare, ma problematizzi la costruzione dell'identità nelle società digitali e il rapporto della scrittura con la sua origine in culture transmediali sempre più caratterizzate dal predominio del simulacro.