The measurement of social polarization

1
The measurement of social polarization

• Iñaki Permanyer
• Institut dAnàlisi Econòmica (IAE-CSIC)
• E-mail inaki.permanyer_at_uab.es

2
Summary

• Income Polarization vs. Social Polarization
• A model for social polarization
• The symmetric case
• Axioms
• Characterization results
• The asymmetric case
• Axioms
• Characterization results

3
Income Polarization vs. Social Polarization

• Income/wealth distribution is not the only cause
  of social tension.
• Religion
• Ethnicity
• Nationalism
• Main goal Extend the notion of polarization to
  these contexts.
• Focus on measurement.

4
A model for social polarization (I)

• Measures of bi-polarization?
• Limited use.
• The identification-alienation (IA) framework.

• T(?,a) is increasing in ? and a.
• T(0,a)0, T(?,0)0.
• ?(i) is assumed to be equal to the size of the
  group to which i belongs.

5
A model for social polarization (II)

• Properties of the distance function d(i,j)?.
• In Esteban and Ray (1994), d(i,j)yi-yj, where
  yi,yj
• are i and js (log of) incomes.
• However, d(i,j) does not need to be a metric.
• Symmetry d(i,j)d(j,i). Why?
• 2) Triangle inequality d(i,k) d(i,j)d(j,k).
• Why?
• Consider i,k great enemies and j a
  mediator or common friend.

6
A model for social polarization (III)

• Notation
• MTotal population mass
• Population splitted in N (exogenously given)
  groups G1,,GNG.
• MiSize of group Gi
• Assumption For any A,B e Gi, C,D e Gj, then
• d(A,C) d(B,D).
• Hence, we will write d(Gi,Gj) to denote the
  degree of alienation felt by the members of Gi
  towards the members of Gj.

7
A model for social polarization (IV)

• We postulate that social polarization is
  proportional to
• Main goal of the paper Characterize P(G, d)
  using different sets of axioms. This will be done
  both for symmetric and asymmetric d(i,j).

8
The symmetric case. Axioms (I)

• Axiom 1. Consider a three group distribution
• G1, G2, G3 of respective masses m,n,n, where
  mgtn. There exists egt0 and kgt0 such that if
• d(G2, G3) lt e, and nltkm, then the merging of
  groups G2,G3 into a new group G with
• d(G,G1)(d(G2, G1)d(G3, G1))/2 does increase
  polarization.
• Intuition When two minority groups join their
  forces and fuse as a single group against the
  majority group, polarization should increase.

9
The symmetric case. Axioms (II)

• Axiom 2. Consider a three group distribution G1,
  G2, G3 of respective masses m,n,k, where mgtk and
  d(G2, G3) lt d(G2, G1). There exists egt0 such
  that if d(G2, G3) and d(G2, G1) are decreased and
  increased respectively by an amount not exceeding
  e, then polarization does increase.
• Intuition When two minority groups approach
  positions against a majority group, polarization
  should increase.

10
The symmetric case. Axioms (III)

• Axiom 3. If P(G1, d) P(G2, d) and ?gt0, then
• P(?G1, d) P(?G2, d), where ?G1, ?G2 represent
  population scalings of G1,G2 respectively.
• Intuition Polarization rankings should not
  change when populations are scaled up or down.

11
The symmetric case. Axioms (IV)

• Axiom 4. Consider a population splitted in two
  groups of equal mass. Then, if we shift mass from
  one of the groups to the other, polarization must
  decrease.
• Intuition Social tension is maximal when two
  opposing groups have the same size.

12
The symmetric case. Results (I)

• Theorem 1. A social polarization index as defined
  in this context satisfies axioms 1, 2, 3 and 4 if
  and only if it is proportional to
• with 0 lt a 2.
• Proposition 1. When axiom 4 is imposed to the
  continuous income polarization index defined in
  Duclos, Esteban and Ray (2004), one obtains the
  restriction a 1.

13
The symmetric case. Axioms (V)

• Axiom 5. For any population of fixed mass
  consider a symmetric distribution G1,,GN where
  all the groups have the same population mass and
  d(Gi,Gj)dgt0 for all i?j. Then, increasing the
  values of N will decrease polarization.
• Intuition Social tension is dilluted as the
  number of opposing groups increase.

14
The symmetric case. Axioms (VI)

• Axiom 6. Consider a three group distribution G1,
  G2, G3 of respective masses m,n,n, where mgtngt0
  and d(Gi, Gj) d gt 0 for all i?j. Then, a
  population mass transfer from G1 to G2 and G3 by
  the same amount without altering the size rank of
  the groups will increase polarization.
• Intuition As the three opposing groups become
  gradually similar, social tension increases.

15
The symmetric case. Results (II)

• Theorem 2. A social polarization index as defined
  in this context satisfies axioms 1, 2, 3, 5 and 6
  if and only if it is proportional to

16
The asymmetric case.

• Let us now consider the case in which d(Gi,Gj) is
  not necessarily equal to d(Gj,Gi).
• Some axioms (1,2 and 3) do make sense in the
  context of asymmetric alienation.
• Axiom 4 can not be stated clearly in the
  asymmetric context.
• Axioms 5 and 6 should be restated for the
  asymmetric case.

17
The asymmetric case. Axioms (I)

• Axiom 5A. For any population of fixed mass
  consider a symmetric distribution G1,,GN where
  all the groups have the same population mass and
  d(Gi,Gj), d(Gj,Gi)d1,d2 for some fixed d1?d2
  0. Then, increasing the values of N will
  decrease polarization.
• Intuition Social tension is dilluted as the
  number of opposing groups increase.

18
The asymmetric case. Axioms (II)

• Axiom 6A. Consider a three group distribution
  G1, G2, G3 of respective masses m,n,n, where
  mgtngt0.Assume that
• d(G1,G2) d(G2,G3) d(G3,G1) d gt 0
• d(G1,G3) d(G3,G2) d(G2,G1) 0.
• Then, a population mass transfer from G1 to G2
  and G3 by the same amount without altering the
  size rank of the groups will increase
  polarization.

G1
G1
G2
G3
G2
G3
19
The asymmetric case. Results

• Theorem 3. A social polarization index as defined
  in this context satisfies axioms 1, 2, 3, 5A and
  6A if and only if it is proportional to
• Proposition 2. Take a population of size M. For
  any two group distribution G1,G2 (of sizes M1,M2)
  with d(G1,G2)dgt0 and d(G2,G1)0, the
  corresponding mass distribution that maximizes
  polarization is M12M/3, M2M/3.

20
Conclusions and way forward

• P(G, d) can be seen as an axiomatically
  characterized generalization of the RQ index
  (RQSiSjM2iMj).
• P(G, d) is a conceptually simple measure that can
  be useful for empirical social polarization
  studies.
• Future research Define an explicit distance
  function d(i,j) that depends on certain
  attributes (income, religion, ethnic group and so
  on).

21
The measurement of social polarization

• Iñaki Permanyer
• Institut dAnàlisi Econòmica (IAE-CSIC)
• E-mail inaki.permanyer_at_uab.es