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 Schelling , T. C. ( 1973 ). Hockey helmets, concealed weapons, and daylight saving . Journal of Conflict Resolution , 17 ( 3 ), 381 – 428 .[Crossref], [Web of Science ®] , [Google Scholar], 1978 Schelling , T. C. ( 1978 ). Micromotives and Macrobehavior . New York : W. W. Norton . [Google Scholar]). 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 Schelling , T. C. ( 1978 ). Micromotives and Macrobehavior . New York : W. W. Norton . [Google Scholar]). 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.