Continuously Tunable Pareto Exponent in a Random Shuffling Money Exchange Model K. Bhattacharya, G. Mukherjee and S. S. Manna Satyendra Nath Bose National Centre for Basic Sciences Econophys – Kolkata I

Random pair wise conservative money shuffling: A.A. Dragulescu and V. M. Yakovenko, Eur. Phys. J. B. 17 (2000) 723. ● N traders, each has money m i (i=1,N), ∑ N i=1 m i =N, =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

Fixed Saving Propensity (λ) ● Fixed Saving Propensity (λ) A. Chakraborti and B. K. Chakrabarti, Eur. Phys. J. B 17 (2000) 167. ● Result: Wealth Distribution in the stationary state Gamma distribution: P(m) ~ m a exp(-bm) Most probable value m p =a/b

Quenched Saving Propensities (λ i, i=1,N) ● Quenched Saving Propensities (λ i, i=1,N) A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica A, 335, 155 (2004) ● Result: Wealth Distribution in the stationary state Pareto distribution: P(m) ~ m -(1+ν)

Dynamics with a tagged trader ● Dynamics with a tagged trader ● N-th trader is assigned λ max and others 0≤λ < λ max for 1 ≤ i ≤ N-1 ● λ max is tuned and are calculated for different λ ● diverges like:

[ /N]N ~ G[(1-λ max )N 1.5 ] where G[x]→x -δ as x→0 with δ ≈ N -9/8 ~ (1-λ max ) -3/4 N -9/8 assuming ≈ ¾ ~ (1-λ max ) -3/4 For a system of N traders (1-λ max ) ~ 1/N. Therefore ~ N 3/4

Approaching the Stationary State ● Approaching the Stationary State ● As λ max →1, the time t x required for the N-th trader to reach the stationary state diverges. ● Scaling shows that: t x ~ (1-λ max ) -1

Rule 1: Probability of selecting the i-th trader is: π i ~ m i α where α is a continuously varying tuning parameter Rule 2: Trading is done by random pair wise conservative money exchange as before: Weighted selection of traders: PRESENT WORK

Results for α=2 P(m,N) follows a scaling form: Where G(x) →x -(1+ν(α)) as x→0 G(x) →const. as x→1 Money Distribution in the Stationary State η(2)=1 and ζ(2)=2 giving ν(2)=1 Height of hor. part ~ 1/N 2 Length of hor. part ~ N Area under hor. part ~ 1/N

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

Results for α=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.