Abstract
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.
Highlights
In many situations the consequences of the choices of an actor are affected by the actions of other actors, that is, the population of agents that form the social system as a whole
Discrete Dynamics in Nature and Society the time evolution of the fraction of agents that make a binary choice, and provides arguments about existence and stability of equilibrium values. This implicit dynamic adjustment fails in describing some important phenomena observed in many real situations, such as oscillations caused by overshooting or overreaction of the actors involved in choices repeated over time, as well as problems of equilibrium selection when nonmonotonic payoff curves lead to the presence of several stable equilibria
In this paper we numerically show the gradual changes induced by increasing values of λ, and in the limiting case we show that the asymptotic dynamics is characterized by the existence of periodic cycles of any period
Summary
In many situations the consequences of the choices of an actor are affected by the actions of other actors, that is, the population of agents that form the social system as a whole. Bischi and Merlone 4 presented an explicit discrete-time dynamic model which is based on the qualitative properties described by Schelling 1 and simulates an adaptive adjustment process of repeated binary choices of boundedly rational agents with social externalities. This permitted them to study the effects on the dynamic behavior of different kinds of payoff functions as well as the qualitative changes of the asymptotic dynamics induced by variations of the main parameters of the model.
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