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 The Model 1 Theoretical Analysis 2 Simulation Results 3
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 1 Austria USA Turkey 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1
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 : 0.4 0.35 0.3 0.25 Strain f 0.2 0.15 0.1 0.05 0 1 1.5 2 2.5 3 3.5 4 4.5 5 Labour l
The Model Theoretical Analysis Simulation Results Strain Functions Indifference curves for f(a, l) = e a - log l 2 1.5 Income l 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Labour a
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 a n = � N n =1 l n Normalisation of powers: � N n =1 m n = 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 a n = � N n =1 l n Normalisation of powers: � N n =1 m n = 1 Definition Ω ⊂ U N is the set of all configurations X = ( x 1 , . . . , x N ) that satisfy these conditions. x i 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 ∪ {∞} : � γ N � � H ( X ) = N + (1 − γ ) m n f ( a n , l n ) , n =1 where γ ∈ [0 , 1].
The Model Theoretical Analysis Simulation Results “Dynamics” of the System Definition We define the abstract energy H : Ω → R ∪ {∞} : � γ N � � H ( X ) = N + (1 − γ ) m n f ( a n , l n ) , n =1 where γ ∈ [0 , 1]. Assume that the system tries to minimise H over Ω.
The Model Theoretical Analysis Simulation Results “Dynamics” of the System 1 For a temperature T > 0 (or equivalently β = kT > 0) assume a Boltzmann distribution (canonical ensemble):
The Model Theoretical Analysis Simulation Results “Dynamics” of the System 1 For a temperature T > 0 (or equivalently β = kT > 0) assume a Boltzmann distribution (canonical ensemble): Definition For A ⊂ Ω, define its probability as π T ( A ) = 1 � e − β H ( X ) dX , Z A where � e − β H ( X ) dX . Z = Ω
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 a n = l n = 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 a n = l n = 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 � γ a 0 , l 0 , a 1 , l 1 γ N − 1 � min f ( a 0 , l 0 ) + N + (1 − γ ) f ( a 1 , l 1 ) , N where a 0 , a 1 ∈ [0 , 1], l 0 , l 1 ≥ L and ( N − 1) a 0 + a 1 = ( N − 1) l 0 + l 1 .
The Model Theoretical Analysis Simulation Results A Simplified Problem � γ a 0 , l 0 , a 1 , l 1 γ N − 1 � min f ( a 0 , l 0 ) + N + (1 − γ ) f ( a 1 , l 1 ) , N where a 0 , a 1 ∈ [0 , 1], l 0 , l 1 ≥ L and ( N − 1) a 0 + a 1 = ( N − 1) l 0 + l 1 . 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 1 0.8 0.6 Income l 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Labour a
The Model Theoretical Analysis Simulation Results Further Results Consider the simplified problem. Theorem f ( a 1 , l 1 ) ≤ f ( a 0 , l 0 ) If γ < γ ′ , we also have f ( a 0 , l 0 ) ≥ f ( a ′ 0 , l ′ 0 ) and f ( a 1 , l 1 ) ≤ f ( a ′ 1 , l ′ 1 ) .
The Model Theoretical Analysis Simulation Results Further Results Consider the simplified problem. Theorem f ( a 1 , l 1 ) ≤ f ( a 0 , l 0 ) If γ < γ ′ , we also have f ( a 0 , l 0 ) ≥ f ( a ′ 0 , l ′ 0 ) and f ( a 1 , l 1 ) ≤ f ( a ′ 1 , l ′ 1 ) . Let f be everywhere finite. Theorem The minimiser ( a 0 , l 0 , a 1 , l 1 ) ∈ R 4 depends continuously on γ .
The Model Theoretical Analysis Simulation Results Further Results Consider the simplified problem. Theorem f ( a 1 , l 1 ) ≤ f ( a 0 , l 0 ) If γ < γ ′ , we also have f ( a 0 , l 0 ) ≥ f ( a ′ 0 , l ′ 0 ) and f ( a 1 , l 1 ) ≤ f ( a ′ 1 , l ′ 1 ) . Let f be everywhere finite. Theorem The minimiser ( a 0 , l 0 , a 1 , l 1 ) ∈ R 4 depends continuously on γ . Theorem If γ < 1 , we have a 0 > l 0 and a 1 < l 1 .
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 π T .
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 π T . 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 π T . 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 3 2 1 H 0 -1 1 0.8 -2 0.6 0 50 0.4 γ 100 0.2 150 β 0 200
The Model Theoretical Analysis Simulation Results A Phase Transition Energy Histogram for β = 16.3 60000 50000 40000 Histogram Count 30000 20000 10000 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Energy H
The Model Theoretical Analysis Simulation Results A Phase Transition 3.5 Left Phase Right Phase 3 2.5 2 Income l 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Power m
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