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The Model Theoretical Analysis Simulation Results

Political Power and Socio-Economic Inequality

An Application of the Canonical Ensemble to Social Sciences

Daniel Kraft July 25th, 2012

The Model Theoretical Analysis Simulation Results

Overview

1

The Model

2

Theoretical Analysis

3

Simulation Results

The Model Theoretical Analysis Simulation Results

The Model

The Model Theoretical Analysis Simulation Results

Social Inequality

“In 2010, average real income per family [in the United States] grew by 2.3 % but the gains were very uneven. Top 1 % incomes grew by 11.6 % while bottom 99 % incomes grew only by 0.2 %. Hence, the top 1 % captured 93 % of the income gains in the first year of recovery.”

The Model Theoretical Analysis Simulation Results

Social Inequality

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Austria USA Turkey

The Model Theoretical Analysis Simulation Results

The Social Space

Individuals described by three dimensions:

The Model Theoretical Analysis Simulation Results

The Social Space

Individuals described by three dimensions: Labour a ∈ [0, 1] Income l ∈ [L, ∞) to model the economy.

The Model Theoretical Analysis Simulation Results

The Social Space

Individuals described by three dimensions: Power m ∈ [0, 1] to model possibly unfair political decisions, and Labour a ∈ [0, 1] Income l ∈ [L, ∞) to model the economy.

The Model Theoretical Analysis Simulation Results

The Social Space

Individuals described by three dimensions: Power m ∈ [0, 1] to model possibly unfair political decisions, and Labour a ∈ [0, 1] Income l ∈ [L, ∞) to model the economy. Definition My social space: U = [0, 1] × [0, 1] × [L, ∞) Individuals: x = (m, p) = (m, a, l) ∈ U

The Model Theoretical Analysis Simulation Results

Strain Functions

Individuals try to maximise their personal happiness, respectively minimise their strain:

The Model Theoretical Analysis Simulation Results

Strain Functions

Individuals try to maximise their personal happiness, respectively minimise their strain: Definition f : [0, 1] × [L, ∞) → R ∪ {∞} is a strictly convex strain function: f possesses certain regularity, f is strictly increasing in a and decreasing in l, and f is strictly convex.

The Model Theoretical Analysis Simulation Results

Strain Functions

A note on convexity, a. k. a. decreasing marginal utility:

The Model Theoretical Analysis Simulation Results

Strain Functions

A note on convexity, a. k. a. decreasing marginal utility:

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1 1.5 2 2.5 3 3.5 4 4.5 5 Strain f Labour l

The Model Theoretical Analysis Simulation Results

Strain Functions

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 Income l Labour a

Indifference curves for f(a, l) = ea - log l

The Model Theoretical Analysis Simulation Results

Coupling the Individuals

Of course, single individuals do not yet form a society!

The Model Theoretical Analysis Simulation Results

Coupling the Individuals

Of course, single individuals do not yet form a society! We require a closed economy: N

n=1 an = N n=1 ln

Normalisation of powers: N

n=1 mn = 1

The Model Theoretical Analysis Simulation Results

Coupling the Individuals

Of course, single individuals do not yet form a society! We require a closed economy: N

n=1 an = N n=1 ln

Normalisation of powers: N

n=1 mn = 1

Definition Ω ⊂ UN is the set of all configurations X = (x1, . . . , xN) that satisfy these conditions. xi are the individuals in my social space.

The Model Theoretical Analysis Simulation Results

“Dynamics” of the System

Definition We define the abstract energy H : Ω → R ∪ {∞}: H(X) =

N

• n=1

γ N + (1 − γ)mn

• f (an, ln),

where γ ∈ [0, 1].

The Model Theoretical Analysis Simulation Results

“Dynamics” of the System

Definition We define the abstract energy H : Ω → R ∪ {∞}: H(X) =

N

• n=1

γ N + (1 − γ)mn

• f (an, ln),

where γ ∈ [0, 1]. Assume that the system tries to minimise H over Ω.

The Model Theoretical Analysis Simulation Results

“Dynamics” of the System

For a temperature T > 0 (or equivalently β =

1 kT > 0) assume a

Boltzmann distribution (canonical ensemble):

The Model Theoretical Analysis Simulation Results

“Dynamics” of the System

For a temperature T > 0 (or equivalently β =

1 kT > 0) assume a

Boltzmann distribution (canonical ensemble): Definition For A ⊂ Ω, define its probability as πT(A) = 1 Z

• A

e−βH(X) dX, where Z =

• Ω

e−βH(X) dX.

The Model Theoretical Analysis Simulation Results

Theoretical Analysis

The Model Theoretical Analysis Simulation Results

Structure of the Minimum

Theorem Let γ = 1. Then X ∈ Ω is a global minimum of H over Ω iff an = ln = a∗, for all n = 1, . . . , N. a∗ is the minimum of a → f (a, a) over [L, 1].

The Model Theoretical Analysis Simulation Results

Structure of the Minimum

Theorem Let γ = 1. Then X ∈ Ω is a global minimum of H over Ω iff an = ln = a∗, for all n = 1, . . . , N. a∗ is the minimum of a → f (a, a) over [L, 1]. Theorem Let γ < 1, then X ∗ ∈ Ω of the form X ∗ = ((1, a∗

1, l∗ 1), (0, a∗ 0, l∗ 0), . . . , (0, a∗ 0, l∗ 0))

minimises H over Ω. a∗

0, a∗ 1 ∈ [0, 1] and l∗ 0, l∗ 1 ≥ L depend on f and

the parameters. This minimum is unique up to permutation of the individuals.

The Model Theoretical Analysis Simulation Results

A Simplified Problem

min

a0,l0,a1,l1 γ N − 1

N f (a0, l0) + γ N + (1 − γ)

• f (a1, l1),

where a0, a1 ∈ [0, 1], l0, l1 ≥ L and (N − 1)a0 + a1 = (N − 1)l0 + l1.

The Model Theoretical Analysis Simulation Results

A Simplified Problem

min

a0,l0,a1,l1 γ N − 1

N f (a0, l0) + γ N + (1 − γ)

• f (a1, l1),

where a0, a1 ∈ [0, 1], l0, l1 ≥ L and (N − 1)a0 + a1 = (N − 1)l0 + l1. Can be solved for instance by: Gradient projection techniques, or Newton’s method applied to the Lagrangian.

The Model Theoretical Analysis Simulation Results

A Simplified Problem

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Income l Labour a

The Model Theoretical Analysis Simulation Results

Further Results

Consider the simplified problem. Theorem f (a1, l1) ≤ f (a0, l0) If γ < γ′, we also have f (a0, l0) ≥ f (a′

0, l′ 0) and f (a1, l1) ≤ f (a′ 1, l′ 1).

The Model Theoretical Analysis Simulation Results

Further Results

Consider the simplified problem. Theorem f (a1, l1) ≤ f (a0, l0) If γ < γ′, we also have f (a0, l0) ≥ f (a′

0, l′ 0) and f (a1, l1) ≤ f (a′ 1, l′ 1).

Let f be everywhere finite. Theorem The minimiser (a0, l0, a1, l1) ∈ R4 depends continuously on γ.

The Model Theoretical Analysis Simulation Results

Further Results

Consider the simplified problem. Theorem f (a1, l1) ≤ f (a0, l0) If γ < γ′, we also have f (a0, l0) ≥ f (a′

0, l′ 0) and f (a1, l1) ≤ f (a′ 1, l′ 1).

Let f be everywhere finite. Theorem The minimiser (a0, l0, a1, l1) ∈ R4 depends continuously on γ. Theorem If γ < 1, we have a0 > l0 and a1 < l1.

The Model Theoretical Analysis Simulation Results

Simulation Results

The Model Theoretical Analysis Simulation Results

Metropolis Algorithm

Calculation of Z and expectation values intractable! → Numerical simulation, Monte-Carlo method

The Model Theoretical Analysis Simulation Results

Metropolis Algorithm

Calculation of Z and expectation values intractable! → Numerical simulation, Monte-Carlo method Custom Metropolis algorithm: Generate configurations sampled by πT.

The Model Theoretical Analysis Simulation Results

Metropolis Algorithm

Calculation of Z and expectation values intractable! → Numerical simulation, Monte-Carlo method Custom Metropolis algorithm: Generate configurations sampled by πT. Markov process, updating “current” configuration. We need P(X ′)

P(X) , but not P (X) directly.

→ Z drops out!

The Model Theoretical Analysis Simulation Results

Metropolis Algorithm

Calculation of Z and expectation values intractable! → Numerical simulation, Monte-Carlo method Custom Metropolis algorithm: Generate configurations sampled by πT. Markov process, updating “current” configuration. We need P(X ′)

P(X) , but not P (X) directly.

→ Z drops out! This generates a “time series”, but does not imply anything about real time evolution!

The Model Theoretical Analysis Simulation Results

Energy Expectation

50 100 150 200 β 0.2 0.4 0.6 0.8 1 γ

• 2
• 1

1 2 3 H

The Model Theoretical Analysis Simulation Results

A Phase Transition

10000 20000 30000 40000 50000 60000

• 1
• 0.5

0.5 1 1.5 2 2.5 3 Histogram Count Energy H

Energy Histogram for β = 16.3

The Model Theoretical Analysis Simulation Results

A Phase Transition

0.5 1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 0.8 1 Income l Power m Left Phase Right Phase

The Model Theoretical Analysis Simulation Results

Infinite Volume Limit

• 6
• 4
• 2

2 4 0.5 1 1.5 2 2.5 3 Energy H τ = β / N N = 5 N = 10 N = 20 N = 50 N = 100 N = 200 N = 500

The Model Theoretical Analysis Simulation Results

Summary

We set up a model describing individuals in a social space. It is crucial to model the power distribution!

The Model Theoretical Analysis Simulation Results

Summary

We set up a model describing individuals in a social space. It is crucial to model the power distribution! This model inherently shows social inequality. Transition happens as a first-order phase transition, breaking permutation symmetry spontaneously.

The Model Theoretical Analysis Simulation Results

Summary

We set up a model describing individuals in a social space. It is crucial to model the power distribution! This model inherently shows social inequality. Transition happens as a first-order phase transition, breaking permutation symmetry spontaneously.

Thanks for your attention!