1. An explicit dynamic model of segregation Gian-Italo Bischi Dipartimento di Economia e Metodi Quantitativi Università di Urbino "Carlo Bo" e-mail:gian.bischi@uniurb.it Ugo Merlone Dip. di Statistica e Matematica Applicata "Diego de Castro" Università di Torino e-mail:merlone@econ.unito.it

2. Schelling, T. (1969) "Models of Segregation", The American Economic Review, vol. 59, 488-493. Schelling, T. (1971) "Dynamic Models of Segregation." Journal of Mathematical Sociology 1: 143-186. Thomas Schelling, Micromotives and Macrobehavior, W. Norton, 1978 Chapter 4: Sorting and mixing: race and sex. Chapter 5: Sorting and mixing:age and income Peyton Young “Individual strategy and Social Structure”, Princeton Univ. Press, 1998 Akira Namatame “Adaptation and Evolution in Collective Systems”, World Scientific, 2006.

3. "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, 1971). Two models proposed by Schelling: 1)An agent based simulation model, a cellular automata migration model, where actors are not confined to a particular cell; 2) A 2-dim. dynamical system, even if no explicit expression is given. Only a qualitative-graphical dynamical analysis is proposed Schelling suggested that minor variations in nonrandom preferences can lead in the aggregate to distinct patterns of segregation. “In some cases, small incentives, almost imperceptible differentials, can lead to strikingly polarized results” (Schelling, 1971).

4. The dynamic model of Schelling Population of individuals partitioned in two classes C 1 and C 2 of numerosity N 1 and N 2 respectively. Let x i (t) be the number of C i individuals included in the system (district, society, political party etc.) The individuals of each group care about the color of the people in the system and can observe the ratio of individuals of the two types at any moment According to this information they can decide if move out (in) if they are dissatisfied (satisfied) with the observed proportion of opposite color agents to one's own color.

5. Individual preferences Following Schelling, we define for each class a cumulative Distribution of Tolerance R i = R i (x i ) maximum ratio R i = x j /x i of individuals of class C j to those of class C i which is tolerated by a fraction x i of the population C i. Simplest assumption: linear ii NiNi All can tolerate 0 different individuals R i =x j /x i xixi nobody can tolerate a ratio  i or more of different individuals 0<x i <N i can tolerate at most a ratio R i (x i ) of class j individuals  i = maximum tolerance of class C i

6. If R i (x i ) is the maximum tolerated ratio of C j individuals to C i ones, then x i R i (x i ) represents the absolute number of C j individuals tolerated by C i ones. from Schelling, 1971

7. From: Clark, W. A. V. (1991) "Residential Preferences and Neighborhood Racial Segregation: A Test of the Schelling Segregation Model" Demography, 28

8. A discrete-time explicit dynamic model Adaptive adjustment Two-dimensional dynamical system  i = speed of reaction low value denotes inertia, patience high value strong reactivity, fast decisions

9. With linear tolerance distribution Equilibria: x i (t+1) = x i (t) i=1,2 Boundary equilibria: E 0 =(0,0) E 1 = (N 1,0) E 2 =(0,N 2 ) Inner equilibria, solutions of a 3 ° degree algebraic equation

10. N 1 =1 N 2 =1  1 =0.5  2 =0.3  1 = 3  2 = 3.5 K 1 = 1 K 2 =1 E3E3 x2x2 x1x1 0 0 1 1 E1E1 E2E2 E0E0 E3E3 x2x2 x1x1 0 0 1 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =0.5  2 =0.3  1 = 3.8  2 = 3.5 K 1 = 1 K 2 =1 E4E4 E5E5

11. E3E3 x2x2 x1x1 0 0 1 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =0.5  2 =1  1 = 3.8  2 = 3.5 K 1 = 1 K 2 =1 E4E4 E5E5 E3E3 x2x2 x1x1 0 0 1 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1  2 =1  1 = 3.8  2 = 3.5 K 1 = 1 K 2 =1 E4E4 E5E5

12. E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1.2  2 =1.2  1 = 3  2 = 3.5 K 1 = 1 K 2 =1 E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1.2  2 =1.2  1 = 3.2  2 = 3.5 K 1 = 1 K 2 =1 c1c1 c2c2 1 1

13. E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1.2  2 =1.2  1 = 3.3  2 = 3.5 K 1 = 1 K 2 =1 E4E4 E5E5 E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1.2  2 =1.2  1 = 4  2 = 3.5 K 1 = 1 K 2 =1 E4E4 E5E5 c1c1 c2c2 1 1

14. E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1.2  2 =1.2  1 = 2  2 = 3 K 1 = 1 K 2 =1 E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1.2  2 =1.2  1 = 2.9  2 = 3 K 1 = 1 K 2 =1 1 1

15. E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1.2  2 =1.2  1 = 4  2 = 3 K 1 = 1 K 2 =1 E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1.2  2 =1.2  1 = 3.1  2 = 3 K 1 = 1 K 2 =1 1 1

16. N 1 =1 N 2 =0.5  1 =1  2 =1  1 = 3  2 = 3 K 1 = 1 K 2 =0.5 E3E3 x2x2 x1x1 0 0 0.5 E1E1 E2E2 E0E0 N 1 =1 N 2 =0.5  1 =1  2 =1  1 = 2  2 = 8 K 1 = 1 K 2 =0.5 E3E3 x2x2 x1x1 0 0 0.5 E1E1 E2E2 E0E0 E4E4 E5E5 1 1

17. N 1 =1 N 2 =0.5  1 =1  2 =1  1 = 2  2 = 10 K 1 = 1 K 2 =0.5 E3E3 x2x2 x1x1 0 0 0.5 E1E1 E2E2 E0E0 E4E4 E5E5 1

18. N 1 =1 N 2 =1  1 =1  2 =1  1 = 4  2 = 2 K 1 = 1 K 2 = 1 E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 N 1 =1 N 2 =1  1 =1  2 =1  1 = 4  2 = 2 K 1 = 0.6 K 2 = 1 E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 0.6 1 1 Constraints

19. N 1 =1 N 2 =1  1 =1  2 =1  1 = 4  2 = 2 K 1 = 0.4 K 2 = 1 E3E3 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 0.4 E4E4 E5E5 1 N 1 =1 N 2 =1  1 =1  2 =1  1 = 4  2 = 2 K 1 = 0.2 K 2 = 1 x2x2 0 1 E1E1 E2E2 E0E0 0.2 E4E4 1 x1x1 0

20. N 1 =1 N 2 =1  1 =0.4  2 =0.5  1 = 4  2 = 3 K 1 = 0.8 K 2 = 0.5 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 0.5 E3E3 1 0.8 N 1 =1 N 2 =1  1 =0.4  2 =0.5  1 = 4  2 = 3 K 1 = 0.8 K 2 = 0.5 x2x2 x1x1 0 0 1 E1E1 E2E2 E0E0 0.5 E3E3 1 0.4 E 4 =(K 1,K 2 )

21. N 1 =1 N 2 =1  1 =0.3  2 =1.2  1 = 4  2 = 2 K 1 = 0.4 K 2 = 1 E3E3 x2x2 x1x1 0 0 E1E1 E2E2 E0E0 0.4 E4E4 E5E5 1 N 1 =1 N 2 =1  1 =0.3  2 =0.4  1 = 4  2 = 2 K 1 = 0.4 K 2 = 1 E3E3 x2x2 0 E1E1 E2E2 E0E0 E4E4 E5E5 x1x1 0 0.4 1 The role of patience

22. Different distributions of tolerance 11 N1N1 R1R1 x1x1 N2N2 R2R2 x2x2 A fraction of the population C 2 always exists that tolerates any ratio of different colored individuals

23. Equilibria: E 0 =(0,0) E 1 = (N 1,0) E 2 =(0,N 2 ) and solutions of the 2°degree algebraic system 22 N1N1 x1x1  2 N 2 x2x2

24. N 1 =1 N 2 =0.8  1 =0.4  2 =0.5  1 = 2  2 = 1 K 1 = 1 K 2 = 0.8 x2x2 x1x1 0 0 0.8 E1E1 E2E2 E0E0 E3E3 1  2 N 2 N 1 =1 N 2 =0.8  1 =0.4  2 =0.5  1 = 3  2 = 2 K 1 = 1 K 2 = 0.8 x2x2 x1x1 0 0 0.8 E1E1 E2E2 E0E0 E3E3 1 E4E4