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Schelling’s Segregation Model

Outline

In 1969, Thomas C. Schelling developed a simple but striking model of racial segregation [Sch69]

His model studies the dynamics of racially mixed neighborhoods

Like much of Schelling’s work, the model shows how local interactions can lead to surprising aggregate structure

In particular, it shows that relatively mild preference for neighbors of similar race can lead in aggregate to the collapse of mixed neighborhoods, and high levels of segregation

In recognition of this and other research, Schelling was awarded the 2005 Nobel Prize in Economic Sciences (joint with Robert Aumann)

In this lecture we (in fact you) will build and run a version of Schelling’s model

The Model

We will cover a variation of Schelling’s model that is easy to program and captures the main idea

Set Up

Suppose we have two types of people: orange people and green people

For the purpose of this lecture, we will assume there are 250 of each type

These agents all live on a single unit square

The location of an agent is just a point \((x, y)\), where \(0 < x, y < 1\)

Preferences

We will say that an agent is happy if half or more of her 10 nearest neighbors are of the same type

Here ‘nearest’ is in terms of Euclidean distance

An agent who is not happy is called unhappy

An important point here is that agents are not averse to living in mixed areas

They are perfectly happy if half their neighbors are of the other color

Behavior

Initially, agents are mixed together (integrated)

In particular, the initial location of each agent is an independent draw from a bivariate uniform distribution on \(S = (0, 1)^2\)

Now, cycling through the set of all agents, each agent is now given the chance to stay or move

We assume that each agent will stay put if they are happy and move if unhappy

The algorithm for moving is as follows

  1. Draw a random location in \(S\)
  2. If happy at new location, move there
  3. Else, go to step 1

In this way, we cycle continuously through the agents, moving as required

We continue to cycle until no one wishes to move

Results

Let’s have a look at the results we got when we coded and ran this model

As discussed above, agents are initially mixed randomly together

../_images/schelling_fig1.png

But after several cycles they become segregated into distinct regions

../_images/schelling_fig2.png
../_images/schelling_fig3.png
../_images/schelling_fig4.png

In this instance, the program terminated after 4 cycles through the set of agents, indicating that all agents had reached a state of happiness

What is striking about the pictures is how rapidly racial integration breaks down

This is despite the fact that people in the model don’t actually mind living mixed with the other type

Even with these preferences, the outcome is a high degree of segregation

Exercises

Exercise 1

Implement and run this simulation for yourself

Use 250 agents of each type

Solutions

Exercise 1

Here’s one solution that does the job we want. If you feel like a further exercise you can probably speed up some of the computations and then increase the number of agents.

using Plots
pyplot()
srand(42)  # set seed for random numbers. Reproducible output
mutable struct Agent{TI<:Integer, TF<:AbstractFloat}
    kind::TI
    location::Vector{TF}
end


# constructor
Agent(k::Integer) = Agent(k, rand(2))


function draw_location!(a::Agent)
    a.location = rand(2)
    nothing
end

# distance is just 2 norm: uses our subtraction function
get_distance(a::Agent, o::Agent) = norm(a.location - o.location)

function is_happy(a::Agent, others::Vector{Agent})
    "True if sufficient number of nearest neighbors are of the same type."
    # distances is a list of pairs (d, agent), where d is distance from
    # agent to self
    distances = Any[]

    for agent in others
        if a != agent
            dist = get_distance(a, agent)
            push!(distances, (dist, agent))
        end
    end

    # == Sort from smallest to largest, according to distance == #
    sort!(distances)

    # == Extract the neighboring agents == #
    neighbors = [agent for (d, agent) in distances[1:num_neighbors]]

    # == Count how many neighbors have the same type as self == #
    num_same_type = sum([a.kind == other.kind for other in neighbors])

    return num_same_type >= require_same_type
end

function update!(a::Agent, others::Vector{Agent})
    "If not happy, then randomly choose new locations until happy."
    while !is_happy(a, others)
        draw_location!(a)
    end
    return nothing
end


function plot_distribution(agents::Vector{Agent}, cycle_num)
    x_vals_0, y_vals_0 = Float64[], Float64[]
    x_vals_1, y_vals_1 = Float64[], Float64[]

    # == Obtain locations of each type == #
    for agent in agents
        x, y = agent.location
        if agent.kind == 0
            push!(x_vals_0, x)
            push!(y_vals_0, y)
        else
            push!(x_vals_1, x)
            push!(y_vals_1, y)
        end
    end

    p = scatter(x_vals_0, y_vals_0, color=:orange, markersize=8, alpha=0.6)
    scatter!(x_vals_1, y_vals_1, color=:green, markersize=8, alpha=0.6)
    plot!(title="Cycle $(cycle_num)", legend=:none)

    return p
end;
# == Main == #

num_of_type_0 = 250
num_of_type_1 = 250
num_neighbors = 10      # Number of agents regarded as neighbors
require_same_type = 5   # Want at least this many neighbors to be same type

# == Create a list of agents == #
agents = Agent[Agent(0) for i in 1:num_of_type_0]
push!(agents, [Agent(1) for i in 1:num_of_type_1]...)

count = 1

# ==  Loop until none wishes to move == #
while true
    println("Entering loop $count")
    p = plot_distribution(agents, count)
    display(p)
    count += 1
    no_one_moved = true
    movers = 0
    for agent in agents
        old_location = agent.location
        update!(agent, agents)
        if !isapprox(0.0, maximum(old_location - agent.location))
            no_one_moved = false
        end
    end
    if no_one_moved
        break
    end
end

println("Converged, terminating")
Entering loop 1
../_images/schelling_solutions_1_jl.png
Entering loop 2
../_images/schelling_solutions_2_jl.png
Entering loop 3
../_images/schelling_solutions_3_jl.png
Entering loop 4
Converged, terminating
../_images/schelling_solutions_4_jl.png