An explicit dynamic model of segregation

Transcription

1 An explicit dynamic model of segregation Gian-Italo Bischi Dipartimento di Economia e Metodi Quantitativi Universita di Urbino "Carlo Bo" gian.bischi@uniurb.it Ugo Merlone Dip. di Statistica e Matematica Applicata "Diego de Castro" Universita di Torino merlone@econ.unito.it Introduction In a celebrated seminal paper, Schelling (969, see also the enlarged version of 97) proposed two models for the description of residential separation of a population formed by two kinds of inhabitants, diering e.g. for racial or religious or cultural features. The separation into exclusive districts is explained by Schelling in terms of individual preferences (or tolerance, following Schelling's terminology) on coexistence with neighborhoods of the opposite kind. Schelling analyzed the eects of local decisions on global behaviour, and showed that even if agents have mild preferences for same-type neighbors the system can evolve towards a complete separation in the long run, even if this is not the outcome preferred by the individuals. In his papers Schelling refers to blacks and whites that have to decide if they want to stay in a given city district or leave it, and denotes the phenomenon of separation as segregation, to stress the dramatic problem of formation of ghettos. However, similar intergroup discrimination pheneomena can be osserved in dierent situations when individuals are partioned either according to some apparently irrelevant criteria (see e.g. Tajfel et al., 97) or when they have to decide if joining or not a given club or enter an organization, or a political party or an academic group. Indeed, Schelling himself begins his paper with the sentence "People get separated along many lines and in many ways. There is segregation by sex, age, income, language, religion, color, taste... and the accidents of historical location" (Schelling, 97). The rst model proposed by Schelling has been extensively modeled as an agent-based simulation model. In fact, it can be considered as a migration model, i.e., a cellular automata where actors are not conned to a particular cell, see Gilbert and Troitzsch (2005). By contrast, the second one is formulated in terms of a 2-dim dynamical system modelling, even if no explicit expression is given, as a qualitative-graphical dynamical analysis is proposed.

2 While the rst model has given rise to a ourishing stream of literature, and many researchers have developed extensions, renements, computer and graphical implementations of Schelling's agent-based simulation model, the second approach, based on the qualitative theory of nonlinear dynamical systems, has been rather neglected. In this paper we try to ll this gap, and propose an explicit analytic formulation of the model described, by words and graphical analysis, in Schelling (969, 97). This explicit formulation will allow us to emphasize how the parameters that represent individual preferences of the two populations, aect the stability of the equilibria and other kinds of long run global behaviours. Moreover, when the system may evolve towards several dierent attractors, an analysis of the extension and the shapes of their basins of attraction is possible. This will allow us to give a more accurate description of the role of initial conditions on the system evolution, and the inuence of their changes as consequences of small variations, as those induced by particular laws or policies. In fact, as already stressed by Schelling, "In some cases, small incentives, almost imperceptible dierentials, can lead to strikingly polarized results" (Schelling 97, p. 46). Finally, we shall study the eects of constraints on local and global stability analysis, as well as the creation of attracting sets, which are more complicated than stable equilibria, such as periodic or chaotic phenomena. These attractors may be created through the usual bifurcations occurring in smooth dynamical systems or by border collision bifurcations, caused by the presence of points of non dierentiability due to the existence of constraints. Starting from the model proposed in this paper, further research can be devoted to the study of the inuence of dierent functions that give the distribution of tolerance inside the populations, such as those arising from experimental data, see Clark (99) on this point. 2 The discrete time modelization Following Schelling (97) we assume that individuals are partitioned in two classes C and C 2 (denoted as color and color 2 in the following) of respective numerosity N and, and we postulate that the individuals of each group care about the color of the people living in the district they live (or in the association they belong, or political party and so on, according to the dierent interpretations attached to the modelled system); individuals of color i, i = ; 2, can observe the ratio of individuals of the two types at any moment, and they can decide if move out (in) if they are dissatised (satised) with the observed proportion of opposite color agents to one's own color. As in Schelling, this can be expressed through the denition, for each population, of a cumulative Distribution of Tolerance R i = R i (x i ), that represents a cumulative density function that gives the maximum ratio R i of individuals of population

3 C j to individuals of population C i which is tolerated by a fraction x i =N i of population C i. As suggested by Schelling, the simplest assumption is a linear cumulative distribution x i R i = i, i = ; 2 () N i where i is a parameter giving the higher ratio of tolerance for individuals of color C i. The dynamic model is obtained by assuming that at each time period each individual of population C i has perfect information about the ratio R i : if the number of individuals of the opposite color, x j, j 6= i, is below a given threshold, then some individuals of population C i will enter the system, whereas if it is above the threshold some C i individuals will leave the system. An explicit dynamic modelization of this adaptive mechanism is the following: assume that at time t, x i (t) class C i individuals are present in the system. They can tolerate at most x i (t) R i (x i (t)) class C j individuals, j 6= i; if this number is larger than the class C j individuals in the system at time t we assume that class C i individuals will increase, vice-versa, if the opposite inequality holds, class C i individuals will decrease, nally if no change in the class C population occurs when the number of class C j individuals is x i (t) R i ( (t)). Further we assume that this dierence times i, the speed of adjustment, is the relative variation of class C i individuals, formally: x i (t + ) x i (t) x i (t) = i [x i (t) R i (x i (t)) x j (t)] : where we adopted a discrete time scale in order to get a comparison with the agent based simulation model proposed by Schelling (97). Putting together the two populations dynamics, we obtain (t + ) = (t) h + x (t) (t + ) = (t) h + 2 x2 (t) 2 (t) N (t) (t) i (t) i (2) Obviously constraints 0 x i (t) N i are imposed, but even more restrictive constraints may be considered, such as 0 x i (t) K i < N i, i = ; 2, or x i 0 and (t) + (t) K < N +. The presence of these constraints gives rise to a piece-wise dierentiable dynamical system, i.e., the phase space of the dynamical system can be divided into disjoint regions where the dynamical system is smooth. In the presence If R i is the maximum tolerated ratio of C j individuals to C i ones, then x i R i represents the absolute number of C j individuals tolerated by C i ones.

4 of piecewise smooth dynamical systems the adjustment process may reveal the occurrence of so-called border-collision bifurcations, which are related to the crossing of invariant sets through the boundaries that separate these regions. These bifurcations may cause sudden stability switches and/or the appearance/disappearance of periodic cycles or chaotic attractors (see e.g. Banerjee et al. 2000, Zhanybai et al. 2003). If we neglect, for a while, the presence of constraints, some general statements on the existence and stability of equilibria con be given. First of all we notice that the coordinate axes x i = 0, i = ; 2, are invariant lines because from x i (t) = 0 it follows x i (t + ) = 0. The corresponding dynamics trapped inside the invariant x j axis is governed by the one-dimensional map x j (t + ) = x j (t) " + j x j (t) j!!# x j (t) N j a unimodal map characterized by the two xed points x 0 j = 0 (always unstable) and j = N j. For the two-dimensional dynamical system (2) the equilibria are obtained by solving the sixth degree algebraic system h x2 2 h x N from which we have three boundary equilibria i = 0 i = 0 O (0; 0) ; E (N ; 0) ; E 2 (0; ) and interior equilibria given by the real and positive solutions of the system = N = 2 (3) It can be shown that there are up to three interior xed points, that may be found by Cardano's formulas. A computation of the jacobian matrix at the equilibria give useful information on the stability of the equilibria. Of course, the equilibria E and E 2 indicate complete segregation, i.e. an outcome of the dynamic system evolution characterized by a unique colored population, whereas the interior equilibria represent situations of coexistence of dierent colored populations in the long run. The coexistence of several stable equilibria, both interior and on the coordinate axis, each with its own basin of attraction, is observed for certain

5 parameters' constellations. In this case the study of the extension and the shape of the basins is crucial to predict the outcome of the dynamic process. Finally, the creation of more complex attractors, characterized by persistent oscillations that may be periodic or chaotic, is another interesting occurrence. For the model proposed in this paper, these kinds of dynamic behaviours have been mainly observed in the presence of constraints that limit the number of individuals that can enter the system. References Banerjee, S., P. Ranjan, C. Grebogi (2000) "Bifurcations in Two-Dimensional Piecewise Smooth Maps - Theory and Applications in Switching Circuits", IEEE Trans. Circuits Syst.-I: Fund. Theory Appl. 47(5), Clark, W. A. V. (99) "Residential Preferences and Neighborhood Racial Segregation: A Test of the Schelling Segregation Model" Demography, 2 ()., pp. -9. Gilbert, N. and Troitzsch, K. G. (2005). Simulation for the social scientist (Second ed.). Milton Keynes: Open University Press. Schelling, T. (97) "Dynamic Models of Segregation." Journal of Mathematical Sociology : Schelling, T. (969) "Models od Segregation", The American Economic Review, vol. 59, Tajfel H., Billig M.G., Bundy R.P. and Flament C. (97) \Social categorization and intergroup behaviour", European Journal of Social Psychology, Vol.. no.2,49-7 Zhanybai, T., Zhusubaliyev and E. Mosekilde (2003) Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems, World Scientic, Singapore.