Transcript An explicit dynamic model of segregation

An explicit dynamic model of segregation
Gian-Italo Bischi
Dipartimento di Economia e Metodi Quantitativi
Università di Urbino "Carlo Bo"
e-mail:[email protected]
Ugo Merlone
Dip. di Statistica e Matematica Applicata "Diego de Castro"
Università di Torino
e-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, 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.
"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).
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).
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
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
i

xi 
Ri   i 1   ,
 Ni 
i  1, 2
i = maximum tolerance of class Ci
Ri=xj /xi
xi
Ni
nobody can tolerate a ratio i
0<xi<Ni can tolerate at most
or more of different individuals a ratio R (x ) of class j individuals All can tolerate
i i
0 different individuals
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
Adaptive adjustment
xi  t  1  xi  t 
xi  t 
  i  xi  t  Ri  xi  t    x j  t   .
i = speed of reaction
low value denotes inertia, patience
high value strong reactivity, fast decisions
Two-dimensional dynamical system


 x  t  1  min x  t 
1
 1

 x2  t  1  min x2  t 

Ki  Ni

 x t  R ( x (t ))  x t   , K 
1   1  x1  t  R1 ( x1 (t ))  x2  t    , K1


1   2

2
2
2
1
2
With linear tolerance distribution


 x1  t  1  min  x1  t 





 x2  t  1  min  x2  t 






 x1 (t ) 
1   1  1 x1  t  1 
  x2  t    , K1 
N1 


 






 x2 (t ) 
1   2  2 x2  t  1 
  x1  t    , K 2 
N2 


 


Equilibria: xi (t+1) = xi (t) i=1,2


x1 
 x2   1 x1 1  

 N1 

 x   x 1  x2 

2 2
 1
N
2 


Boundary equilibria:
E0=(0,0) E1= (N1,0) E2=(0,N2)
Inner equilibria, solutions
of a 3°degree algebraic equation
N1=1 N2=1 1=0.5 2=0.3 1 = 3 2 = 3.5
K1= 1 K2=1
N1=1 N2=1 1=0.5 2=0.3 1 = 3.8 2 = 3.5
K1= 1 K2=1
1 E
2
1 E
2
x2
x2
E4
E5
E3
E3
E0
0
0
E1
x1
1
E0
0
0
E1
x1
1
N1=1 N2=1 1=0.5 2=1 1 = 3.8 2 = 3.5
K1= 1 K2=1
1 E
2
x2
N1=1 N2=1 1=1
K1= 1 K2=1
2=1 1 = 3.8 2 = 3.5
1 E
2
x2
E4
E4
E5
E5
E3
E0
0
0
E3
E1
x1
1
E0
0
0
E1
x1
1
N1=1 N2=1 1=1.2 2=1.2 1 = 3 2 = 3.5
K1= 1 K2=1
N1=1 N2=1 1=1.2 2=1.2 1 = 3.2 2 = 3.5
K1= 1 K2=1
1 E
2
1 E
2
x2
x2
c2
c1
E3
E0
0
0
x1
E1
1
E3
E0
0
0
E1
x1
1
N1=1 N2=1 1=1.2 2=1.2 1 = 3.3 2 = 3.5
K1= 1 K2=1
1 E
2
N1=1 N2=1 1=1.2 2=1.2 1 = 4 2 = 3.5
K1= 1 K2=1
1 E
2
x2
E4
x2
c2
E4
E5
c1
E5
E3
E3
E0
0
0
E1
x1
1
E0
0
0
E1
x1
1
N1=1 N2=1 1=1.2 2=1.2 1 = 2 2 = 3
K1= 1 K2=1
N1=1 N2=1 1=1.2 2=1.2 1 = 2.9 2 = 3
K1= 1 K2=1
1 E
2
1 E
2
x2
x2
E3
E3
E0
0
0
E1
x1
E0
0
1
0
E1
x1
1
N1=1 N2=1 1=1.2 2=1.2 1 = 3.1 2 = 3
K1= 1 K2=1
N1=1 N2=1 1=1.2 2=1.2 1 = 4 2 = 3
K1= 1 K2=1
1 E
2
1 E
2
x2
x2
E3
E0
0
0
E1
x1
1
E3
E0
0
0
E1
x1
1
N1=1 N2=0.5 1=1 2=1
K1= 1 K2=0.5
0.5
1 = 3 2 = 3
N1=1 N2=0.5 1=1 2=1
K1= 1 K2=0.5
0.5
E2
E3
x2
1 = 2 2 = 8
E2
x2
E3
E5
E4
E0
0
0
E1
x1
1
E0
0
0
E1
x1
1
N1=1 N2=0.5 1=1 2=1 1 = 2 2 = 10
K1= 1 K2=0.5
0.5
E2
x2
E3
E5
E4
E0
0
0
E1
x1
1
Constraints
N1=1 N2=1 1=1 2=1 1 = 4 2 = 2
K1= 0.6 K2= 1
N1=1 N2=1 1=1 2=1 1 = 4 2 = 2
K1 = 1 K 2 = 1
1
1
E2
x2
x2
E3
E0
0
0
E2
E1
x1
E3
E0
0
1
0
E1
0.6
x1
1
N1=1 N2=1 1=1 2=1
K1= 0.4 K2= 1
1
1 = 4 2 = 2
N1=1 N2=1 1=1 2=1 1 = 4 2 = 2
K1= 0.2 K2= 1
1
E2
x2
E2
x2
E3
E5
E4
E1
E0
0
0
0.4
x1
1
E0
0
0
E4
E1
0.2
x1
1
N1=1 N2=1 1=0.4 2=0.5 1 = 4 2 = 3
K1= 0.8 K2= 0.5
N1=1 N2=1 1=0.4 2=0.5 1 = 4 2 = 3
K1= 0.8 K2= 0.5
1
1
x2
x2
0.5
E2
0.5
E3
E0
0
0
E1
0.8
x1 1
E2
E3
E0
0
0
E4=(K1,K2)
E1
0.4
x1 1
The role of patience
N1=1 N2=1 1=0.3 2=0.4 1 = 4 2 = 2
K1= 0.4 K2= 1
N1=1 N2=1 1=0.3 2=1.2 1 = 4 2 = 2
K1= 0.4 K2= 1
E2
x2
E2
x2
E3
E3
E5
E5
E4
E1
E0
0
0
E4
0.4
x1
1
E1
E0
0
0
0.4
x1
1
Different distributions of tolerance
R1
1
R2

x 
R1 ( x1 )   1 1  1 
 N1 
N1 x1
A fraction of the population C2
always exists that tolerates any ratio
of different colored individuals
R1 ( x1 ) 
N2
 2  N 2  x2 
x2
x2






 x1 (t ) 
 x1  t  1  min  x1  t  1   1  1 x1  t  1 

x
t
,
K





2
1



N
1 



 




 x2  t  1  min x2  t  1   2  2  N 2  x2 (t )   x1  t    , K 2


Equilibria:
E0=(0,0) E1= (N1,0) E2=(0,N2)
and solutions of the 2°degree algebraic system


x1 
 x2  x1 R1  x1    1 x1 1  

 N1 
x  x R x   N  x
2 1 2
2 2
2
 1
x2
N2
N1
2 N2
x1
N1=1 N2=0.8 1=0.4 2=0.5 1 = 2 2 = 1
K1= 1 K2= 0.8
0.8
E2
N1=1 N2=0.8 1=0.4 2=0.5 1 = 3 2 = 2
K1= 1 K2= 0.8
0.8 E
2
x2
x2
E3
E3
E4
E0
0
0
E1
2 N2 x1 1
E1
E0
0
0
x1
1