Revisiting Schelling segregation dynamics€¦ · Revisiting Schelling segregation dynamics C...

Revisiting Schelling segregation dynamicsC Alais, R Mourgues, JP Bouchaud, M BenzaquenEcole Normale Supérieure

NAEC-X day at OECD

December 10th 2019

www.EconophysiX.com CFM Chair of Econophysics & Complex Systems

https://www.econophysix.com/

2/21 Table of contents

1. Schelling modelPresentationFree energy

2. SimulationsEvolution of a cityAnnealed dynamicsActivation Barrier

3. Market-Mediated Schelling ModelStationary StateWith Evolving Landlords



3/21 Schelling Model



4/21 Schelling model

A modification of the original Schelling model [1], defined by [2],[3].

Move with probability p(q −→ q′) = 11+e−β(u(q′)−u(q))



5/21 Schelling model

Allowing the system to evolve from a homogeneous initial state to a stationary state :



6/21 Free energy

This process does not optimize the sum of individual utilities,but a potential H which depends on the configuration of the city x .For fully rational agents H reads :

H(x) =Q∑

q=1

nxq∑n=1

u(n)

Or considering densities :

H(x) =Q∑

q=1

∫ ρq0

u(ρ)dρ



7/21 Phase separation

∫ ρq0 u(ρ)dρ as a function of ρq :

Here for fully rational agents.To maximizeH(x) =

∑Qq=1

∑nxqn=1 u(n), it is better

to have some neighbourhoods at 0 andsome at 0.7 rather than all at 0.5.



8/21 Phase separation

∫ ρq0 u(ρ)dρ as a function of ρq :

Here for irrational agents.In this case the maximization of H isachieved with a homogeneous state,which also maximizes individualutilities.



9/21 Phase diagram

Measure of segregation in the final state for different temperatures and global density(the higher the more segregated the city is)

Rationality

0.0

5

10

Density0.0 0.5 1



10/21 Evolution of a city

ZZZ

ZZZ

ZZ}

Increasing time

Each line is the trajectory of a system with initial state indicated by its color.Only the end of the trajectory is represented.



11/21 Evolution of a city

��>

Increasing time



12/21 Annealed dynamics

?

Increasing time and increasing rationality



13/21 Activation barrier

n non-empty neighbourhoods

n − 1

∆U‡

The agents of the neigh-bourhood to be emptied areevenly distributed into theother filled neighbourhoods.For instance for Q = 36,L = 100, N = 1800 andβ = 5 in average 1018 stepsare required !



14/21 Market-Mediated Schelling Model

One landlord per neighbourhood, he has an unknown price function p(ρ).Agents have a new utility function u(ρ)− αp(ρ)We look for a stationary state which maximizes :

H =Q∑

q=1

∫ ρq0

(u(ρ)− αp(ρ))dρ

with∑Q

q=1 ρq = Qρ̄. Under the hypothesis that:

- all landlords have the same price function as a function of density

- the stationary state is reached when all landlords have the same profit ρp(ρ)



15/21 Market-Mediated Schelling Model

Concave case : the stationary state is homogeneous.Convex case : there is segregation(ρ1, ρ2), the conservation of the number ofinhabitants and the stationary state condition give∫ ρ2

ρ1

(u(ρ)− αp(ρ))dρ = ρ2u(ρ2)− ρ1u(ρ1)

No natural p(ρ) gives a segregated state.



Embedded Animation 16/21 Market-Mediated Schelling Dynamics

A simulation where the landlords are blind, they change their prices randomly, keepingthe new value if they make a higher profit on the next step.



Conclusion

- A simple model which does not converge to a situation that maximizes individualutilities

- It is not obvious that a housing market helps avoiding segregation, or evenallows a stationary state

Thank you for your attention !



18/21 References

Thomas C. Schelling.Dynamic models of segregation.Journal of Mathematical Sociology, pages 143–186, 1971.

Sebastien Grauwin, Dominc Hunt, Eric Bertin, and Pablo Jensen.Effective free energy for individuals dynamics.Advances in Complex Systems, pages 529–536, 2011.

Pablo Jensen, Thomas Matreux, Jordan Cambe, Hernan Larralde, and Eric Bertin.

Giant catalytic effect of altruists in schelling’s segregation model.Physical Review Letter, page 208301, may 2018.



Evolution of a city



20/21 Market-Mediated Schelling Dynamics

Every million steps :



21/21 Market-Mediated Schelling Dynamics

Every 4 millions steps :



Schelling modelPresentationFree energy

SimulationsEvolution of a cityAnnealed dynamicsActivation Barrier

Market-Mediated Schelling ModelStationary StateWith Evolving Landlords

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