an explicit dynamic model of segregation gian-italo bischi dipartimento di economia e metodi...
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An explicit dynamic model of segregation
Gian-Italo BischiDipartimento di Economia e Metodi QuantitativiUniversità di Urbino "Carlo Bo"e-mail:[email protected]
Ugo MerloneDip. di Statistica e Matematica Applicata "Diego de Castro"Università di Torinoe-mail:[email protected]
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, 1978Chapter 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.
"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).
The dynamic model of Schelling
Population of individuals partitioned in two classes C1 and C2 of numerosity N1 and N2 respectively.
Let xi(t) be the number of Ci 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.
Individual preferences
Following Schelling, we define for each class a cumulative Distribution of Tolerance Ri = Ri(xi)maximum ratio Ri = xj /xi of individuals of class Cj to those of class Ci
which is tolerated by a fraction xi of the population Ci .
Simplest assumption: linear 1 , 1, 2ii i
i
xR i
N
i
Ni
All can tolerate0 different individuals
Ri=xj /xi
xi
nobody can tolerate a ratio i
or more of different individuals0<xi<Ni can tolerate at most a ratio Ri(xi) of class j individuals
i = maximum tolerance of class Ci
If Ri(xi) is the maximum tolerated ratio of Cj individuals to Ci ones, then xiRi(xi) represents the absolute number of Cj individuals tolerated by Ci ones.
from Schelling, 1971
From: Clark, W. A. V. (1991) "Residential Preferences and Neighborhood Racial Segregation: A Test of the Schelling Segregation Model" Demography, 28
A discrete-time explicit dynamic model
1
.i ii i i i j
i
x t x tx t R x t x t
x t
Adaptive adjustment
1 1 1 1 1 1 2 1
2 2 2 2 2 2 1 2
1 min 1 ( ( )) ,
1 min 1 ( ( )) ,
x t x t x t R x t x t K
x t x t x t R x t x t K
Two-dimensional dynamical system
ii NK
i = speed of reaction low value denotes inertia, patience high value strong reactivity, fast decisions
With linear tolerance distribution
11 1 1 1 1 2 1
1
22 2 2 2 2 1 2
2
( )1 min 1 1 ,
( )1 min 1 1 ,
x tx t x t x t x t K
N
x tx t x t x t x t K
N
Equilibria: xi (t+1) = xi (t) i=1,2
Boundary equilibria:E0=(0,0) E1= (N1,0) E2=(0,N2)
Inner equilibria, solutions of a 3°degree algebraic equation
12 1 1
1
21 2 2
2
1
1
xx x
N
xx x
N
N1=1 N2=1 1=0.5 2=0.3 1 = 3 2 = 3.5 K1= 1 K2=1
E3
x2
x100
1
1
E1
E2
E0
E3
x2
x100
1
1
E1
E2
E0
N1=1 N2=1 1=0.5 2=0.3 1 = 3.8 2 = 3.5 K1= 1 K2=1
E4
E5
E3
x2
x100
1
1
E1
E2
E0
N1=1 N2=1 1=0.5 2=1 1 = 3.8 2 = 3.5 K1= 1 K2=1
E4
E5
E3
x2
x100
1
1
E1
E2
E0
N1=1 N2=1 1=1 2=1 1 = 3.8 2 = 3.5 K1= 1 K2=1
E4
E5
E3
x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1.2 2=1.2 1 = 3 2 = 3.5 K1= 1 K2=1
E3
x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1.2 2=1.2 1 = 3.2 2 = 3.5 K1= 1 K2=1
c1
c2
11
E3
x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1.2 2=1.2 1 = 3.3 2 = 3.5 K1= 1 K2=1
E4 E5
E3
x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1.2 2=1.2 1 = 4 2 = 3.5 K1= 1 K2=1
E4
E5
c1
c2
1 1
E3
x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1.2 2=1.2 1 = 2 2 = 3 K1= 1 K2=1
E3
x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1.2 2=1.2 1 = 2.9 2 = 3 K1= 1 K2=1
1 1
E3x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1.2 2=1.2 1 = 4 2 = 3 K1= 1 K2=1
E3
x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1.2 2=1.2 1 = 3.1 2 = 3 K1= 1 K2=1
1 1
N1=1 N2=0.5 1=1 2=1 1 = 3 2 = 3 K1= 1 K2=0.5
E3x2
x100
0.5
E1
E2
E0
N1=1 N2=0.5 1=1 2=1 1 = 2 2 = 8 K1= 1 K2=0.5
E3x2
x100
0.5
E1
E2
E0
E4
E5
1 1
N1=1 N2=0.5 1=1 2=1 1 = 2 2 = 10K1= 1 K2=0.5
E3x2
x100
0.5
E1
E2
E0
E4
E5
1
N1=1 N2=1 1=1 2=1 1 = 4 2 = 2 K1= 1 K2= 1
E3x2
x100
1
E1
E2
E0
N1=1 N2=1 1=1 2=1 1 = 4 2 = 2 K1= 0.6 K2= 1
E3
x2
x100
1
E1
E2
E0
0.61 1
Constraints
N1=1 N2=1 1=1 2=1 1 = 4 2 = 2 K1= 0.4 K2= 1
E3
x2
x100
1
E1
E2
E0
0.4
E4
E5
1
N1=1 N2=1 1=1 2=1 1 = 4 2 = 2 K1= 0.2 K2= 1
x2
0
1
E1
E2
E0
0.2
E4
1x10
N1=1 N2=1 1=0.4 2=0.5 1 = 4 2 = 3 K1= 0.8 K2= 0.5
x2
x100
1
E1
E2
E0
0.5E3
10.8
N1=1 N2=1 1=0.4 2=0.5 1 = 4 2 = 3 K1= 0.8 K2= 0.5
x2
x100
1
E1
E2
E0
0.5E3
10.4
E4=(K1,K2)
N1=1 N2=1 1=0.3 2=1.2 1 = 4 2 = 2 K1= 0.4 K2= 1
E3
x2
x100
E1
E2
E0
0.4
E4
E5
1
N1=1 N2=1 1=0.3 2=0.4 1 = 4 2 = 2 K1= 0.4 K2= 1
E3
x2
0E1
E2
E0
E4
E5
x10 0.4 1
The role of patience
Different distributions of tolerance
1
N1
R1
x1 N2
R2
x2
A fraction of the population C2 always exists that tolerates any ratio of different colored individuals
1
1111 1)(
N
xxR
2 2 21 1
2
( )N x
R xx
11 1 1 1 1 2 1
1
2 2 2 2 2 2 1 2
( )1 min 1 1 ,
1 min 1 ( ) ,
x tx t x t x t x t K
N
x t x t N x t x t K
Equilibria:
E0=(0,0) E1= (N1,0) E2=(0,N2)
and solutions of the 2°degree algebraic system
12 1 1 1 1 1
1
1 2 1 2 2 2 2
1x
x x R x xN
x x R x N x
2
N1x12 N2
x2
N1=1 N2=0.8 1=0.4 2=0.5 1 = 2 2 = 1 K1= 1 K2= 0.8
x2
x100
0.8
E1
E2
E0
E3
12 N2
N1=1 N2=0.8 1=0.4 2=0.5 1 = 3 2 = 2 K1= 1 K2= 0.8
x2
x100
0.8
E1
E2
E0
E3
1
E4