Continuously Tunable Pareto Exponent in a Continuously Tunable Pareto Exponent in a Random Shuffling Money Exchange ModelRandom Shuffling Money Exchange Model

K. Bhattacharya, G. Mukherjee and S. S. Manna

Satyendra Nath Bose National Centre for Basic Sciencesmanna@bose.res.in

Econophys – Kolkata IEconophys – Kolkata I

Random pair wise conservative money shuffling:Random pair wise conservative money shuffling:A.A. Dragulescu and V. M. Yakovenko, Eur. Phys. J. B. 17 (2000) 723.

)]()())[1(1()1(

)]()()[1()1(

tmtmttm

tmtmttm

jij

jii

● N traders, each has money mi (i=1,N), ∑Ni=1mi=N, <m>=1

● Time t = number of pair wise money exchanges

● A pair i and j are selected {1 ≤i,j ≤ N, i ≠ j} with uniform probability who reshuffle their total money:

● Result: Wealth Distribution in the stationary state

)/exp(1

)(

mmm

mP

● Fixed Saving Propensity (Fixed Saving Propensity (λλ))A. Chakraborti and B. K. Chakrabarti, Eur. Phys. J. B 17 (2000) 167.

)}]()({1))[(1(1()()1(

)}]()({1)[(1()()1(

tmtmttmtm

tmtmttmtm

jij

jiii

j

● Result: Wealth Distribution in the stationary state

Gamma distribution: P(m) ~ ma exp(-bm)

Most probable valuemp=a/b

● Quenched Saving Propensities (Quenched Saving Propensities (λλii, i=1,N), i=1,N)A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica A, 335, 155 (2004)

)]t(m)1()t(m1))[(1t(1()t(m)1t(m

)]t(m)1()t(m1)[(1t()t(m)1t(m

jjijj

jjiii

ij

ii

● Result: Wealth Distribution in the stationary state

Pareto distribution:P(m) ~ m-(1+ν)

● Dynamics with a tagged traderDynamics with a tagged trader

● N-th trader is assigned λmax and others 0≤λ < λmax for 1 ≤ i ≤ N-1 ● λmax is tuned and <m(λmax)> are calculated for different λ● <m(λmax)> diverges like:

725.0

maxmax)1(N/)(m

[<m(λmax)>/N]N-0.125 ~ G[(1-λmax)N1.5]

where G[x]→x-δ as x→0 with δ ≈ 0.725

<m(λmax)>N-9/8 ~ (1-λmax)-3/4N-9/8 assuming 0.725 ≈ ¾

<m(λmax)> ~ (1-λmax)-3/4

For a system of N traders (1-λmax) ~ 1/N. Therefore

<m(λmax)> ~ N3/4

● Approaching the Stationary StateApproaching the Stationary State

● As λmax→1, the time tx required for the N-th trader to reach the stationary state diverges.● Scaling shows that: tx ~ (1-λmax)-1

Rule 1: Rule 1: Probability of selecting the i-th trader is: πi ~ mi

α

where α is a continuously varying tuning parameter

Rule 2:Rule 2: Trading is done by random pair wise conservative money exchange as before:

Weighted selection of Weighted selection of traders:traders:

PRESENT WORK

(t)]m(t)))[m1ε(t1()1(tm

(t)]m(t))[m1ε(t)1(tm

jij

jii

Results for Results for αα=2=2

P(m,N) follows a scaling form:

)/(),( )()( NmGNNmP

Where G(x) →x-(1+ν(α)) as x→0 G(x) →const. as x→1

Money Distribution in the Stationary StateMoney Distribution in the Stationary State

η(2)=1 and ζ(2)=2 giving ν(2)=1

Height of hor. part ~ 1/N2

Length of hor. part ~ NArea under hor. part ~ 1/N

Results for Results for αα=3/2=3/2

η(3/2)=3/2 and ζ(3/2)=1 giving ν(3/2)=1/2

Results for Results for αα=1=1

ν(1)=0

Conclusion

● There are complex inherent structures in the model with quenched random saving propensities which are disturbing. More detailed and extensive study are required.● Model with weighted selection of traders seems to be free from these problems.

Thank you.Thank you.