Ugo Merlone

Global Dynamics in Binary Choice Models with Social Influence

A discrete-time dynamical system is proposed to model a class of binary choice games with externalities as those described by Schelling (1973, 1978). In order to analyze some oscillatory time patterns and problems of equilibrium selection that were not considered in the qualitative analysis given by Schelling, we introduce an explicit adjustment mechanism. We perform a global dynamic analysis that allows us to explain the transition toward nonconnected basins of attraction when several coexisting attractors are present. This gives a formal explanation of some overshooting effects in social systems and of the consequent cyclic behaviors qualitatively described in Schelling (1978). Moreover, we show how the occurrence of a global bifurcation may lead to the explanation of situations of path dependence and the creation of thresholds observed in real life situations of collective choices, leading to extreme forms of irreversible departure from an equilibrium and uncertainty about the long run evolution of the some social systems.

Impulsivity in Binary Choices and the Emergence of Periodicity

Binary choice games with externalities, as those described by Schelling (1973, 1978), have been recently modelled as discrete dynamical systems (Bischi and Merlone, 2009). In this paper we discuss the dynamic behavior in the case in which agents are impulsive; that is; they decide to switch their choices even when the difference between payoffs is extremely small. This particular case can be seen as a limiting case of the original model and can be formalized as a piecewise linear discontinuous map. We analyze the dynamic behavior of this map, characterized by the presence of stable periodic cycles of any period that appear and disappear through border-collision bifurcations. After a numerical exploration, we study the conditions for the creation and the destruction of periodic cycles, as well as the analytic expressions of the bifurcation curves.

Some notes on applying the Herfindahl–Hirschman Index

The Herfindahl–Hirschman Index is one of the most commonly used indicators to detect anticompetitive behaviour in industries. In fact, an increase in the value of the index is usually interpreted as an indicator of actions which may lessen competition or even create a monopoly. In this article we show that this is not always the case, since in certain instances it is impossible to detect cooperation. We also show an example when competition even has a decreasing effect on the value of the index.

Periodic cycles and bifurcation curves for one-dimensional maps with two discontinuities", Journal of Dynamical Systems and Geometric Theory

Abstract Starting from a family of discontinuous piece-wise linear one-dimensional maps, recently introduced as a dynamic model in social sciences, we propose a geometric method for finding 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 expressions 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 modified by the introduction 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 parameter 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 parameter 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.

Original article: Border-collision bifurcations in a model of Braess paradox

In Braess paradox adding an extra resource, and therefore an extra available choice, enriches the complexity of the game from a dynamic perspective. The analysis of the cycles and the bifurcations helps to visualize how this complexity changes, in a quite new way with respect to what is provided by the so far literature. We derive the conditions for the creation and the destruction of periodic cycles, as well as the analytical expressions of the bifurcation conditions, by studying the occurrence of border-collision bifurcations. We are also able to give a proof of the relation between the cost of the new resource and the existence of cycles of any given period, and also of the coexistence of equilibria, adding the path dependence to the problem.

Global dynamics in adaptive models of collective choice with social influence

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.

Cartelizing Groups in Dynamic Linear Oligopoly with Antitrust Threshold

The effects of partially cooperating firms are examined in N-firm oligopolies. The Herfindahl-Hirschmann Index is assumed to detect the violation of the antitrust regulation by the firms, and based on this assumption a piece-wise differentiable dynamic system can be developed. The firms stop cooperating if this index becomes larger than a certain threshold and they restart or continue cooperating otherwise. The equilibria of the dynamic system are first determined. Local and global asymptotic stability of the equilibria are then investigated showing the complexity of the dynamic behavior of the system.

Replicating human interaction in Braess paradox

The Braess Paradox shows how adding a new road to a traffic network may actually increase the total travel time. It has recently found new interest in research. Researchers conducted new experiments with human participants in order to observe the outcomes with an increasing number of people, with private or public monitoring. A small number of papers were devoted to the observation of different behaviors, and intuitively suggested some theoretical hypotheses about the heterogeneity of the participants. Analyzing the data gathered from the observation of an experiment with human participants, and coding artificial behaviors emerged by mean of Grounded Theory, we used ABM simulations to confirm or disprove possible behaviors and composition of the population that was so far suggested only theoretically.

Dynamics of a 2D Piecewise Linear Braess Paradox Model: Effect of the Third Partition

In this work, we investigate the dynamics of a piecewise linear 2D discontinuous map modeling a simple network showing the Braess paradox. This paradox represents an example in which adding a new route to a specific congested transportation network makes all the travelers worse off in terms of their individual travel time. In the particular case in which the modeled network corresponds to a binary choice situation, the map is defined on two partitions and its dynamics has already been described. In the general case corresponding to a ternary choice, a third partition appears leading to significantly more complex bifurcation structures formed by border collision bifurcations of stable cycles with points located in all three partitions. Considering a map taking a constant value on one of the partitions, we provide a first systematic description of possible dynamics for this case.

SYSTEMATIC APPROACH TO N-PERSON SOCIAL DILEMMA GAMES: CLASSIFICATION AND ANALYSIS

This paper presents a new systematic review of N-person social dilemma games using a new approach based on dynamic properties of the corresponding system. Traditionally N-person social dilemma games are classified by relative orders of magnitude of payoff parameters. Without border-line cases 24 are identified. The new approach introduced in this paper categorizes the social dilemma games in cases with different number and asymptotic properties of the equilibria. In these cases the solution structure or the trajectory of the percentage of cooperators is readily apparent. These cases also provide the modeler with additional information concerning the impacts of the model parameters on the game outcomes. The example of a simple cartel illustrates this methodology.