An H Theorem for Boltzmann’s Equation for the Yard-Sale Model of Asset Exchange

Boghosian, B.M., Johnson, M. & Marcq, J.A. An H Theorem for Boltzmann’s Equation for the Yard-Sale Model of Asset Exchange. J Stat Phys 161, 1339–1350 (2015). https://doi.org/10.1007/s10955-015-1316-8

0.1 Pareto’s observations in modern terms

A(w) is the cumulative distribution function (CDF) of economic agents, ordered by wealth. Let us denote the corresponding probability density function (PDF) of agents by P(w), but we shall adopt the convention of normalizing P to the total number of economic agents, rather than to unity, so that ∫badwP(w) is the total number of agents with wealth in [a, b], for 0≤a<b.

The total number of economic agents is given by the zeroth moment of the PDF of agents:

--> N = ' integrate( p( w), w, 0,∞) ;
(%o7) N = ∫ 0 ∞ p ⁡ ( w ) d w

and the total amount of wealth in the economy is given by the first moment,

--> W = ' integrate( p( w)· w, w, 0,∞) ;
(%o8) W = ∫ 0 ∞ w ⁢ p ⁡ ( w ) d w

so that A(w) is the following:

--> A( w) = ( 1 / N) · ' integrate( p( x), x, w,∞) ;
(%o9) A ⁡ ( w ) = ∫ w ∞ p ⁡ ( x ) d x N

From the Fundamental Theorem of Calculus, we see that the PDF of agents is related to the derivative of Pareto’s function

--> p( w) = diff( − N · A( w), w, 1) ;
(%o11) p ⁡ ( w ) = − N ⁢ ( d d ⁢ w ⁢ A ⁡ ( w ) )

0.2 The Lorenz Curve

Max O. Lorenz plotted the cumulative distribution of wealth versus the cumulative distribution of agents. That is, Lorenz made a parametric plot of the quantity

--> L( w) : = ( 1 / W)· ' integrate( p( x)· x, x, 0, w) ;
(%o14) L ⁡ ( w ) := 1 W ⁢ ∫ 0 w p ⁡ ( x ) ⁢ x d x

vs quantity

L(w) := (1/W)* 'integrate(p(x)*x,x,0,w);

--> F( w) : = ( 1 / N)· ' integrate( p( x), x, 0, w) ;
(%o15) F ⁡ ( w ) := 1 N ⁢ ∫ 0 w p ⁡ ( x ) d x

https://en.wikipedia.org/wiki/Lorenz_curve

0.3 Background to the Yard-Sale Model of Asset Exchange

Asset-Exchange Models (AEMs) and they typically involve binary transactions between agents involving some increment of wealth Δw, with rules for which agent gains it and which agent loses it.

The amount of wealth that will be transferred from agent 1 to agent 2 in this transaction is then Δw=β min (w1,w2), where β∈(−1,+1) is sampled from a symmetric PDF denoted by η(β). Note that, because η(β) is symmetric, the two agents both have even odds of winning and losing.

in which Δw is proportional to the wealth of the poorer agent

--> Δw = β· min ( w1, w2) ;
(%o4) Δw = min ⁡ ( w1 , w2 ) ⁢ β
--> β∈( − 1, + 1) ;
(%o5) β∈ ⁡ ( − 1 , 1 )

This is implemented in de.urszeidler.ascape.ysae.YardSaleModel

YardSaleAgent ysa1 = this;
YardSaleAgent ysa2 = (YardSaleAgent) agent;
double beta = factor * (randomIs() ? 1.0 : -1.0);
double dwealth = beta * Math.min(ysa1.wealth, ysa2.wealth);
ysa2.wealth = ysa2.wealth + dwealth;
ysa1.wealth = ysa1.wealth - dwealth;

0.4 Fokker–Planck Equation for the Yard-Sale Model

--> C( w) : = 1 − 1 / N · ' integrate ( P( x)·( 1 − x ^ 2 / w ^ 2), x, 0, w) ;
(%o1) C ⁡ ( w ) := 1 − 1 N ⁢ ∫ 0 w P ⁡ ( x ) ⁢ ( 1 − x 2 w 2 ) d x
--> ∂ P( w) /∂ t =∂ ^ 2 /∂ w ^ 2·( γ·( w ^ 2 / 2)· C( w)· P( w)) ;
(%o2) ∂P ⁡ ( w ) ∂t = w 2 ⁢ P ⁡ ( w ) ⁢ ( 1 − ∫ 0 w ( 1 − x 2 w 2 ) ⁢ P ⁡ ( x ) d x N ) ⁢ γ ⁢ ∂ 2 2 ⁢ ∂w 2

1 Oligarchy as a Phase Transition: The effect of wealth-attained advantage in a
Fokker-Planck description of asset exchange

Bruce M. Boghosian, 1 Adrian Devitt-Lee, 1 Merek Johnson, 1 Jeremy A. Marcq, 1 and Hongyan Wang 1
1
Department of Mathematics, Tufts University, Medford, Massachusetts 02155, USA
(Dated: October 8, 2018)

1.1 Pareto potentials.

--> A( w, t) : = ( 1 / N) · ' integrate( P( x, t), x, w,∞) ;
(%o17) A ⁡ ( w , t ) := 1 N ⁢ ∫ w ∞ P ⁡ ( x , t ) d x
--> F( w, t) : = ( 1 / N) · ' integrate( P( x, t), x, 0, w) ;
(%o18) F ⁡ ( w , t ) := 1 N ⁢ ∫ 0 w P ⁡ ( x , t ) d x
--> L( w, t) : = ( 1 / W) · ' integrate( P( x, t)· x, x, 0, w) ;
(%o19) L ⁡ ( w , t ) := 1 W ⁢ ∫ 0 w P ⁡ ( x , t ) ⁢ x d x
--> B( w, t) : = ( 1 / N) · ' integrate( P( x, t)·( x² / 2), x, 0, w) ;
(%o21) B ⁡ ( w , t ) := 1 N ⁢ ∫ 0 w P ⁡ ( x , t ) ⁢ x 2 2 d x
--> A( w, t) + F ( w, t) = 1 ;
(%o22) ∫ w ∞ P ⁡ ( x , t ) d x N + ∫ 0 w P ⁡ ( x , t ) d x N = 1

1.2 B.
Transactional exchange

--> w = w1 + ∆ w ;
(%o23) w = ∆w + w1
--> ∆ w = sqr( γ·∆ t) · min( t1, t2)· η ;
(%o24) ∆w = min ⁡ ( t1 , t2 ) ⁢ η ⁢ sqr ⁡ ( γ ⁢ ∆t )
--> ∆ w = √( γ·∆ t) · min( t1, t2)· η ;
(%o25) ∆w = min ⁡ ( t1 , t2 ) ⁢ η ⁢ γ ⁢ ∆t
--> η : { − 1, + 1} ;
(η) { − 1 , 1 }

Here we have taken β = √γ∆t, where ∆t is the characteristic time associated with a transaction, for reasons
which will become clear shortly

This in implemented by de.urszeidler.ascape.ysae1.YardSaleModel

YardSaleAgent ysa1 = this;
YardSaleAgent ysa2 = (YardSaleAgent) agent;
double beta = Math.sqrt(deltaT * gamma) * (randomIs() ? 1.0 : -1.0);
double dwealth = beta * Math.min(ysa1.wealth, ysa2.wealth);
ysa2.wealth = ysa2.wealth + dwealth;
ysa1.wealth = ysa1.wealth - dwealth;

--> E[ η] = 0 ;
(%o27) E { − 1 , 1 } = 0
--> E[ η ^ 2 ] = 1 ;
(%o28) E { − 1 , 1 } 2 = 1
--> E[ f] = ( 1 / N) · ' integrate( P( x, t) · E[ f( η, x)], x, 0,∞) ;
(%o29) E f = ∫ 0 ∞ E f ⁡ ( { − 1 , 1 } , x ) ⁢ P ⁡ ( x , t ) d x N

REDISTRIBUTION IN THE YARD-SALE MODEL
Derivation of the Fokker-Planck equation for the redistributional YSM

Redistribution funktion:

--> Δw = ( τ · Δt) · (( W / N) − w) + √( γ· Δt) · min( w, z) ·  η ;
(%o1) Δw = ( W N − w ) ⁢ Δt ⁢ τ + min ⁡ ( w , z ) ⁢  η ⁢ Δt ⁢ γ

implemented by de.urszeidler.ascape.ysae_distubution.YardSaleModel

YardSaleAgent ysa1 = this;
YardSaleAgent ysa2 = (YardSaleAgent) agent;
double beta = Math.sqrt(deltaT * gamma) * (randomIs() ? 1.0 : -1.0);
double dwealth_static = beta * Math.min(ysa1.wealth, ysa2.wealth);
double redist1 = tau * deltaT * (wealthPropotion - ysa1.getWealth());
double redist2 = tau * deltaT * (wealthPropotion - ysa2.getWealth());

ysa2.wealth = ysa2.wealth + dwealth_static + redist2;
ysa1.wealth = ysa1.wealth - dwealth_static + redist1;

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