Conjuncted: Avarice

Greed followed by avarice….We consider the variation in which events occur at a rate equal to the difference in capital of the two traders. That is, an individual is more likely to take capital from a much poorer person rather than from someone of slightly less wealth. For this “avaricious” exchange, the corresponding rate equations are

dc_k/dt = c_k-1∑_j=1^k-1(k – 1 – j)c_j + c_k+1∑_j=k+1^∞(j – k – 1)c_j – c_k∑_j=1^∞|k – j|c_j —– (1)

while the total density obeys,

dN/dt = -c₁(1 – N) —– (2)

under the assumption that the total wealth density is set equal to one, ∑kc_k = 1

These equations can be solved by again applying scaling. For this purpose, it is first expedient to rewrite the rate equation as,

dc_k/dt = (c_k-1 – c_k)∑_j=1^k-1(k – j)c_j – c_k-1∑_j=1^k-1c_j + (c_k+1 – c_k)∑_j=k+1^∞(j – k)c_j – c_k+1∑_j=k+1^∞c_j —– (3)

taking the continuum limits

∂c/∂t = ∂c/∂k – N∂/∂k(kc) —— (3)

We now substitute the scaling ansatz,

c_k(t) ≅ N²C(x), with x = kN to yield

C(0)[2C + xC′] = (x − 1)C′ + C —– (4)

and

dN/dt = -C(0)N² —– (5)

Solving the above equations gives N ≅ [C(0)t]⁻¹ and

C(x) = (1 + μ)(1 + μx)^−2−1/μ —– (6)

with μ = C(0) − 1. The scaling approach has thus found a family of solutions which are parameterized by μ, and additional information is needed to determine which of these solutions is appropriate for our system. For this purpose, note that equation (6) exhibits different behaviors depending on the sign of μ. When μ > 0, there is an extended non-universal power-law distribution, while for μ = 0 the solution is the pure exponential, C(x) = e^−x. These solutions may be rejected because the wealth distribution cannot extend over an unbounded domain if the initial wealth extends over a finite range.

The accessible solutions therefore correspond to −1 < μ < 0, where the distribution is compact and finite, with C(x) ≡ 0 for x ≥ x_f = −μ⁻¹. To determine the true solution, let us re-examine the continuum form of the rate equation, equation (3). From naive power counting, the first two terms are asymptotically dominant and they give a propagating front with k_f exactly equal to t. Consequently, the scaled location of the front is given by x_f = Nk_f. Now the result N ≃ [C(0)t]⁻¹ gives x_f = 1/C(0). Comparing this expression with the corresponding value from the scaling approach, x_f = [1 − C(0)]⁻¹, selects the value C(0) = 1/2. Remarkably, this scaling solution coincides with the Fermi distribution that found for the case of constant interaction rate. Finally, in terms of the unscaled variables k and t, the wealth distribution is

c_k(t) = 4/t², k < t

= 0, k ≥ 0 —– (7)

This discontinuity is smoothed out by diffusive spreading. Another interesting feature is that if the interaction rate is sufficiently greedy, “gelation” occurs, whereby a finite fraction of the total capital is possessed by a single individual. For interaction rates, or kernels K(j, k) between individuals of capital j and k which do not give rise to gelation, the total density typically varies as a power law in time, while for gelling kernels N(t) goes to zero at some finite time. At the border between these regimes N(t) typically decays exponentially in time. We seek a similar transition in behavior for the capital exchange model by considering the rate equation for the density

dN/dt = -c₁∑_k=1^∞k(1, k)c_k —– (8)

For the family of kernels with K(1, k) ∼ k^ν as k → ∞, substitution of the scaling ansatz gives N ̇ ∼ −N^3−ν. Thus N(t) exhibits a power-law behavior N ∼ t^1/(2−ν) for ν < 2 and an exponential behavior for ν = 2. Thus gelation should arise for ν > 2.

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Greed

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 c_k(t) now evolve according to

dc_k/dt = c_k-1∑_j=1^k-1c_j + c_k+1∑_j=k+1^∞c_j – c_kN – c²_k —– (1)

The first two terms account for the gain in c_k(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 c_k(t). One can check that the wealth density M1 ≡ ∑^∞_k=1 k c_k(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

dc_k/dt = -c_k(c_k + c_k+1) + N(c_k-1 – c_k) + (c_k+1 – c_k-1)∑_j=k^∞c_j —– (3)

Taking the continuum limit and substituting the scaling ansatz,

c_k(t) ≅ N²C(x), with x = kN —– (4)

transforms equations (2) and (3) to

dN/dt = -C(0)N³ —– (5)

and

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 = ∑c_k(t) ≅ N∫dx C(x), while the latter follows if we set, without loss of generality, the conserved wealth density equal to unity, ∑_kkc_k(t) = 1.

Introducing B(x) = ∫₀^x 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 < x_f and B(x) = 1 for x ≥ x_f, 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 < x_f

= 0, for x ≥ x_f —– (8)

Hence the scaled profile has a sharp front at x = x_f, with x_f = 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 c_k(t) reads,

c_k(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^∞d_jc(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 k_f = 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 ∼ t^1/4, or a relative width w = W/k_f ∼ t^−1/4. This suggests the appropriate scaling ansatz for the front region is

c_k(t) = 1/t X(ξ), ξ = (k – 2√t)/ t^1/4 —– (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.