In greedy exchange, when two individuals meet, the richer person takes one unit of capital from the poorer person, as represented by the reaction scheme (j, k) → (j + 1, k − 1) for j ≥ k. In the rate equation approximation, the densities ck(t) now evolve according to
dck/dt = ck-1∑j=1k-1cj + ck+1∑j=k+1∞cj – ckN – c2k —– (1)
The first two terms account for the gain in ck(t) due to the interaction between pairs of individuals of capitals (j, k−1), with j k, respectively. The last two terms correspondingly account for the loss of ck(t). One can check that the wealth density M1 ≡ ∑∞k=1 k ck(t) is conserved, and that the population density obeys
dN/dt = -c1N —– (2)
Equation (1) are conceptually similar to the Smoluchowski equations for aggregation with a constant reaction rate. Mathematically, however, they appear to be more complex and we have been unable to solve them analytically. Fortunately, equation (1) is amenable to a scaling solution. For this purpose, we first re-write equation (1) as
dck/dt = -ck(ck + ck+1) + N(ck-1 – ck) + (ck+1 – ck-1)∑j=k∞cj —– (3)
Taking the continuum limit and substituting the scaling ansatz,
ck(t) ≅ N2C(x), with x = kN —– (4)
transforms equations (2) and (3) to
dN/dt = -C(0)N3 —– (5)
C(0)[2C + xC’] = 2C2 + C'[1 – 2∫x∞dyC(y)] —– (6)
where C ′ = dC/dx. Note also that the scaling function must obey the integral relations
∫x∞dxC(x) = 1 and ∫x∞dxxC(x) = 1 —– (7)
The former follows from the definition of density, N = ∑ck(t) ≅ N∫dx C(x), while the latter follows if we set, without loss of generality, the conserved wealth density equal to unity, ∑kkck(t) = 1.
Introducing B(x) = ∫0x dyC(y) recasts equation (6) into C(0)[2B′ + xB′′] = 2B′2 + B′′[2B − 1]. Integrating twice gives [C(0)x − B][B − 1] = 0, with solution B(x) = C(0)x for x < xf and B(x) = 1 for x ≥ xf, from which we conclude that the scaled wealth distribution C(x) = B′(x) coincides with the zero-temperature Fermi distribution;
C(x) = C(0), for x < xf
= 0, for x ≥ xf —– (8)
Hence the scaled profile has a sharp front at x = xf, with xf = 1/C(0) found by matching the two branches of the solution for B(x). Making use of the second integral relation, equation (7), gives C(0) = 1/2 and thereby closes the solution. Thus, the unscaled wealth distribution ck(t) reads,
ck(t) = 1/(2t), for k < 2√t
= 0, for k ≥ 2√t —– (9)
and the total density is N(t) = t-1/2
Figure: Simulation results for the wealth distribution in greedy additive exchange based on 2500 configurations for 106 traders. Shown are the scaled distributions C(x) versus x = kN for t = 1.5n, with n = 18, 24, 30, and 36; these steepen with increasing time. Each data set has been av- eraged over a range of ≈ 3% of the data points to reduce fluctuations.
These predictions by numerical simulations are shown in the figure. In the simulation, two individuals are randomly chosen to undergo greedy exchange and this process is repeated. When an individual reaches zero capital he is eliminated from the system, and the number of active traders is reduced by one. After each reaction, the time is incremented by the inverse of the number of active traders. While the mean-field predictions are substantially corroborated, the scaled wealth distribution for finite time actually resembles a finite-temperature Fermi distribution. As time increases, the wealth distribution becomes sharper and approaches equation (9). In analogy with the Fermi distribution, the relative width of the front may be viewed as an effective temperature. Thus the wealth distribution is characterized by two scales; one of order √t characterizes the typical wealth of active traders and a second, smaller scale which characterizes the width of the front.
To quantify the spreading of the front, let us include the next corrections in the continuum limit of the rate equations, equation (3). This gives,
∂c/∂t = 2∂/∂k [c∫k∞djc(j)] – c∂c/∂k – N∂c/∂k + N/2 ∂2c/∂k2 —– (10)
Here, the second and fourth terms on the RHS denote the second corrections. since, the convective third term determines the location of the front to be at kf = 2√t, it is natural to expect that the diffusive fourth term describes the spreading of the front. the term c∂c/∂k turns out to be negligible in comparison to the diffusive spreading term and is henceforth neglected. The dominant convective term can be removed by transforming to a frame of reference which moves with the front namely, k → K = k − 2√t. among the remaining terms in the transformed rate equation, the width of the front region W can now be determined by demanding that the diffusion term has the same order of magnitude as the reactive terms, i.e. N ∂2c/∂k2 ∼ c2. This implies W ∼ √(N/c). Combining this with N = t−1/2 and c ∼ t−1 gives W ∼ t1/4, or a relative width w = W/kf ∼ t−1/4. This suggests the appropriate scaling ansatz for the front region is
ck(t) = 1/t X(ξ), ξ = (k – 2√t)/ t1/4 —– (11)
Substituting this ansatz into equation (10) gives a non-linear single variable integro-differential equation for the scaling function X(ξ). Together with the appropriate boundary conditions, this represents, in principle, a more complete solution to the wealth distribution. However, the essential scaling behavior of the finite-time spreading of the front is already described by equation (11), so that solving for X(ξ) itself does not provide additional scaling information. Analysis gives w ∼ t−α with α ≅ 1/5. We attribute this discrepancy to the fact that w is obtained by differentiating C(x), an operation which generally leads to an increase in numerical errors.