Global Dynamics in Binary Choice Models with Social Influence

Abstract

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.

Keywords:

• binary games
• discrete dynamical systems
• global bifurcations
• overshooting
• social externalities

ACKNOWLEDGMENTS

The authors are grateful to two anonymous referees whose helpful suggestions greatly improved the quality of this article.

Notes

See in Schelling (1973) the discussion on the differential payoff on page 391 and the discussion about coalitions starting on page 393.

As the function f is not differentiable in its fixed points, when the derivative is below −1 only on one side, the equilibrium may be stable or unstable according to the global properties of the function, in the sense that the equilibrium may be locally repelling from one side but globally asymptotically stable if the long run dynamics is governed by the stable branch.

Notice that also in this case is globally asymptotically stable even if .