Heterogeneous Society in Binary Choices with Externalities

We study a two-strategy model with negative externalities proposed by Schelling, in a dynamical setting where a society consists of two interacting populations with different behaviors derived from experiments with human participants. The resulting dynamics is a three-dimensional piecewise smooth map with one discontinuity, which inherits some of the characteristics of each homogeneous population dynamics, while others are lost and new ones emerge. We propose a technique to represent the dynamics on a bidimensional space and prove that the heterogeneous society dynamics can be obtained as a linear combination of the dynamics of the two homogeneous populations. As expected, complexity arises with respect to some aspects. Firstly, the number of equilibria expands to infinity and we were able to determine possible focal equilibria in the sense of Schelling. Secondly, when heterogeneity is introduced, the period adding structure of cycles is replaced by a period incrementing structure. Thirdly, the phenomenon of overreaction and cyclic oscillations can be mitigated even if it never completely disappears. We also derive the orbits of cycles of period two and provide numerical evidences of coexistence of cycles with different periods. It is worth noticing that with the heterogeneous society, the dynamics does not depend on the society aggregate choices only, rather on each population choice; neglecting it will make impossible to determine the future evolution of the system. The implications are important as heterogeneity makes the system path-dependent and a policy maker, considering aggregate society choice only, would be unable to make the proper decisions, unless further information is considered.