# Twenty-five years of random asset exchange modeling

Max Greenberg  Electronic address<EMAIL_ADDRESS>Department of Economics,
University of Massachusetts Amherst, Amherst, MA 01002, USA  Cornell Systems
Engineering, Cornell University, Ithaca, NY 14850, USA  H. Oliver Gao
Electronic address<EMAIL_ADDRESS>Cornell Systems Engineering, Cornell
University, Ithaca, NY 14850, USA

###### Abstract

The last twenty-five years have seen the development of a significant
literature within the subfield of econophysics which attempts to model
economic inequality as an emergent property of stochastic interactions among
ensembles of agents. In this article, the literature surrounding this approach
to the study of wealth and income distributions, henceforth the ”random asset
exchange” literature following the terminology of Sinha (2003), is thoroughly
reviewed for the first time. The foundational papers of Drăgulescu and
Yakovenko (2000), Chakraborti and Chakrabarti (2000), and Bouchaud and Mézard
(2000) are discussed in detail, and principal canonical models within the
random asset exchange literature are established. The most common variations
upon these canonical models are enumerated, and significant papers within each
kind of modification are introduced. The successes of such models, as well as
the limitations of their underlying assumptions, are discussed, and it is
argued that the literature should move in the direction of more explicit
representations of economic structure and processes to acquire greater
explanatory power.

###### Contents

  1. I Introduction
    1. I.1 The universality of economic inequality
    2. I.2 Measuring inequality: the Gini coefficient
    3. I.3 Pareto, Gibrat, and the econophysicists
  2. II The Taxonomy of Random Asset Exchange Models
    1. II.1 Kinetic wealth exchange
      1. II.1.1 Theft, fraud, and yard sales
    2. II.2 Bouchaud-Mézard models
    3. II.3 Other formulations
  3. III Notable Trends in the Literature
    1. III.1 Non-conservation of wealth
    2. III.2 Networks and preferential attachment
    3. III.3 Goods and rationality
    4. III.4 Strategic behavior
    5. III.5 Class division
    6. III.6 Taxation and redistribution
    7. III.7 Miscellanea
  4. IV Discussion

##  I Introduction

Over the last fifteen years, the question of economic inequality has become
the epicenter of one of the most intense political debates in the United
States. Awareness of the growing gap between rich and poor has been growing
since the 2008 American bank bailouts and the 2010 Citizens United v. FEC
Supreme Court decision, but the inequality question was decisively pushed to
the forefront of American politics in September 2011 with the beginning of the
Occupy Wall Street protest movement, which introduced the dichotomy of ”the
1%” vs. ”the 99%” to public consciousness. Though the Occupy movement did not
immediately produce anything by way of practical politics, it nonetheless laid
the foundation for U.S. Senator Bernie Sanders’ two campaigns for president,
in which he used the language of Occupy to reframe economic inequality as the
result of policy choices which could be rectified through a social-democratic
”political revolution.”

But the idea that economic inequality is a problem which needs to be addressed
by way of policy is by no means an uncontroversial one. The majority of
American adults who electorally favor the Republican Party do not believe that
the current level of economic inequality in the United States is excessive
Horowitz  _et al._ (2020). The legacy of ”Reaganomics”—the economic policy
pursued by the Federal government of the United States under the tenure of
former President Ronald Reagan, which was characterized by cuts to tax rates
and other concessions to proponents of ”supply-side” economic theory—remains
contentious. And within the Republican delegation to the United States House
of Representatives, a proposal to eliminate the current bracketed income tax
system and to replace it with a much higher nationwide flat sales tax, in
order to dramatically lessen the tax burden on the wealthy, is gaining some
traction Dore (2023).

The controversy surrounding economic inequality is just as longstanding and
just as intense within the realm of academic economics. On one hand,
economists tend to be more skeptical than other social scientists of
government intervention in the economy, as a great deal of emphasis is placed
on the fact that, within the discipline’s canonical models, free markets have
no trouble arriving at socially optimal allocations of resources all on their
own. On the other hand, French economist Thomas Piketty’s 2013 magnum opus
Capital in the 21st Century, which proposes the imposition of a global,
progressive tax on wealth in order to rein in what he views as the excessive
amount of economic inequality in the world today, has become greatly
influential in the popular-academic debates concerning the issue Piketty
(2014). Needless to say, the intense political disputes around the question of
inequality show no sign of abating anytime soon.

###  I.1 The universality of economic inequality

What is indisputable, however, is that in nearly every single developed market
economy, the degree of stratification between the rich and everyone else is
not only staggering, but is also increasing. In the United States, the share
of household wealth owned by the top 1% of the population by net worth grew
from 29.9% in 1989 to 35.5% in 2013; meanwhile, the share of wealth owned by
the bottom 50% of the population shrunk from 3.0% to 1.1% during the same
period Killewald  _et al._ (2017). In Germany, individuals at the 90th
percentile of net assets own 13 times as much wealth as the median individual
and over a quarter of individuals have liabilities equal to or greater than
their assets, resulting in a negative net worth Grabka and Westermeier (2014).
While there are many countries where the degree of wealth inequality is not
this extreme—the United States has one of the most unequal distributions of
wealth in the world—the overall structure is strikingly similar in almost
every country Davies  _et al._ (2009). In every market economy for which data
exists, many possess very little wealth and a few possess much.

Great inequality also governs the distribution of incomes within market
economies. As reported by Horowitz et al. for the Pew Research Center, the
share of aggregate income possessed by high-income households, defined as
households with incomes greater than twice the national median, has grown from
29% in 1970 to 48% today. In the same period, the share of aggregate income
possessed by low-income households, defined as households with incomes less
than two-thirds the national median, fell from 10% to 9% over the same period.
In a similar vein, the incomes of those who are already in the top 5% of the
population in terms of earnings have consistently grown the faster than all
other earners over the past 40 years.

###  I.2 Measuring inequality: the Gini coefficient

One of the most popular metrics used to quantify the degree of inequality
present in a given wealth or income distribution—or, indeed, any density
distribution over a non-negative domain—is the Gini coefficient, named for
Italian statistician Corrado Gini. The Gini coefficient of a distribution
$f(x)$ is defined by reference to the Lorenz curve, itself defined as the
function:

$L\left(F(x)\right)=\frac{1}{\mu}\int_{0}^{x}s\cdot p(s)ds$ (1)

where $F(x)=\int_{0}^{x}f(s)ds$ is the cumulative density distribution of
$f(x)$ and $\mu=\int_{0}^{\infty}s\cdot f(s)ds$ is the mean of $f(x)$
Gastwirth (1971). Intuitively, this integral represents the share of some
asset—say, income—held by the bottom $100x$% of the population, normalized by
the mean of the distribution. The Gini coefficient is then given by twice the
difference between the area under Lorenz curve of a perfectly egalitarian
distribution—a straight line with a slope of 1—and the Lorenz curve of the
distribution in question Dorfman (1979). Thus, the canonical formula used to
calculate the Gini coefficient is:

$G=1-2\int_{0}^{1}L(x)dx$ (2)

Thus, the Gini coefficient can take on any value between 0—perfect
equality—and 1—perfect inequality. To extend this statistic to describe
dispersion within a finite population $\\{x_{i}\\}_{i=1}^{N}$, however, it is
more convenient to leverage the alternative (but equivalent) definition of the
Gini coefficient, given by half the relative mean absolute difference of a
distribution:

$G=\frac{1}{2}\frac{\left(\sum_{i=1}^{N}\sum_{j=1}^{N}|x_{i}-x_{j}|\right)/N^{2}}{\mu}$
(3)

If the population $\\{x_{i}\\}_{i=1}^{N}$ is sorted such that $x_{j}>x_{k}$ if
and only if $j>k$, the Gini coefficient may be calculated using the
computationally much faster formula:

$G=\frac{N+1}{N}-\frac{2}{\mu N^{2}}\sum_{i=1}^{N}(N-i+1)x_{i}$ (4)

as demonstrated by Allison (1978). Note that the Gini coefficient over a
discrete population does not perfectly correspond to its continuous
counterpart, however, as the former has an upper bound of $1-1/N$ Allison
(1979).

The Gini coefficent is not a perfect measure of inequality. It has been
criticized on the basis that distributions with very different levels of
concentration in the right tail can produce identical indices; there is
therefore significant information lost when using it to represent an entire
distribution with a single scalar value Osberg (2017). Nonetheless, the Gini
coefficient serves as a useful and widely-used benchmark for summarizing the
degree of dispersion present in a given wealth or income distribution.

Gini Coefficients of Wealth and Income for Ten Countries
---
Country |  Gini Coefficient of Wealth |  Gini Coefficient of Income
United States |  0.801 |  0.401
France |  0.730 |  0.311
United Kingdom |  0.697 |  0.396
India |  0.669 |  0.344*
Germany |  0.667 |  0.289
Netherlands |  0.650 |  0.298*
Australia |  0.622 |  0.331*
Italy |  0.609 |  0.353
Spain |  0.570 |  0.343
China |  0.550 |  0.420*
Table 1: Gini coefficients of wealth and income inequality for ten countries,
all major world economies, based on data from the year 2000. Data for Gini
coefficients of wealth are taken from Davies et al. (2009), while data for
Gini coefficients of income are taken from the FRED database hosted by the
Federal Reserve Bank of St. Louis. Gini coefficients of income for India, the
Netherlands, and Australia are from 2004. Gini coefficient of income for China
is from 2002.

Underscoring the universality of steep economic inequality in both wealth and
income distributions, Table 1 displays the Gini coefficients for the wealth
and income distributions of ten countries. We observe that wealth
distributions are almost always ”more unequal” than income distributions: Gini
coefficients for wealth distributions tend to range between $0.5$ and $0.8$,
while Gini coefficients for income distributions tend to range from $0.25$ to
$0.45$. Furthermore, there is no obvious correlation between the Gini
coefficients for wealth distributions and for income distributions: some
countries, such as China, have coefficients relatively close in value, while
other countries, such as France, have Gini coefficients for wealth over twice
as high as the corresponding value for income.

###  I.3 Pareto, Gibrat, and the econophysicists

Regardless of whether one personally believes governments should play a role
in redistributing wealth from the rich to the poor, the universality of the
phenomenon of extreme inequality should raise eyebrows. Different countries
have dramatically different approaches to welfare programs, taxation, and all
other sorts of policy. Yet the distributions of wealth and income which emerge
in these countries are remarkably similar in form. It follows that there must
be some shared set of characteristics that account for this common structure
of wealth distribution. This line of questioning points us to an often
overlooked and still poorly understood aspect of economic inequality: its
origin.

The nature and origin of the distribution of wealth and income in market
economies has been an open problem in economics for more than a century. In
1897, the Italian civil engineer-turned-economist Vilfredo Pareto attempted to
provide an answer after noticing a striking pattern in data for land-ownership
rates in Italy. Specifically, Pareto posited that income in every society was
distributed according to a decreasing power law; namely:

$p(x)\propto x^{-1-\alpha}$ (5)

where $p(x)$ represents the probability density function of income and
$\alpha$ represents the ”Pareto index,” with smaller values producing fatter
tails and thus representing more unequal distributions. This observation has
come to be known as the ”weak Pareto law,” with its strong counterpart
including the additional claim that the Pareto index possesses a value in the
range $1.5\pm 0.5$ Pareto (1897). But not long thereafter it became apparent
that this law did not actually well characterize the entire income
distribution. Instead, when low- and middle-income strata were taken into
account, the data seemed to be much better fit by a right-skewed lognormal
distribution:

$p(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{(\ln(x)-\mu)^{2}}{2\sigma^{2}}\right)$
(6)

This fact was first noticed by Robert Gibrat Gibrat (1931). It is now well
established that, in fact, both Pareto and Gibrat were correct: a lognormal-
like distribution tends to characterize the bulk of incomes, while the Pareto
distribution tends to characterize the highest 2-3% of incomes Montroll and
Shlesinger (1982).

Since these discoveries, mainstream economic theory has, broadly speaking,
shied away from further attempts to impose a universal form to these
distributions or to explain the processes responsible for their emergence.
There are both normative and methodological reasons for this gap in the
economics literature. The normative aversion, as voiced by Piketty, manifests
as skepticism that a universal law governing the right tails of income
distributions exists at all Piketty (2014). The methodological aversion, on
the other hand, stems from the fact that most macroeconomic models make use of
single, representative agents, which are ill-suited for describing
heterogeneity within a population. Meanwhile, more sophisticated tools capable
of addressing such questions, such as the Heterogeneous Agent New Keynesian
(HANK) class of models, are still relatively new to the scene Achdou  _et al._
(2022). Nonetheless, this aversion has meant that there has remained
comparatively little in the way of literature concerning one of the most
crucial questions in economics today. This gap drew the attention of
physicists interested in applying methods developed for the study of the
natural sciences to questions in the social sciences in the late 1990s.

The aim of these ”econophysicists” was to capture the characteristic features
of empirical wealth and income distributions, as made known by extensive
statistical analyses. There is now substantial evidence that the bulk of the
income distribution in all capitalist countries follows an exponential
distribution Tao  _et al._ (2019). The right tail of the income distribution
follows the aforementioned Pareto law and the left tail follows Gibrat’s law.
The exponential bulk and the log-normal left tail are sometimes unified in the
form of the closely related Gamma distribution:

$p(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$ (7)

where $\alpha$ is called the ”shape parameter” and $\beta$ the ”rate
parameter.” However, there remains insufficient data to conclude whether the
Gamma or log-normal distribution provides the better empirical fit Chakrabarti
_et al._ (2013).

Wealth distributions are unfortunately much less well understood due to a
dearth of publicly available data. Rough estimates of wealth distributions in
pre-capitalist societies, such as in the New Kingdom of Egypt and medieval
Hungary, provide some evidence that such societies exhibited power-law
distributions of wealth, but these results are far from conclusive Abul-Magd
(2002); Hegyi  _et al._ (2007). Drăgulescu and Yakovenko (2001b) used
inheritance tax data to study the wealth distribution in the modern United
Kingdom, which was found to have a similar structure to the UK’s income
distribution. Further supporting this conclusion, Sinha (2006), among others,
found evidence that the very wealthiest stratum of society, as measured by
published ”rich lists,” follows a power law distribution as well. These
features appear to emerge even in artificial economies, with Fuchs et al.
(2014) having observed an exponential bulk and power-law tail even in the
wealth distribution across players of a massively multiplayer online game with
inbuilt systems of production and trade. Thus, early exchange models in the
econophysics literature sought to generate distributions exhibiting both the
exponential bulk and power-law tail observed in data by means of symmetric
binary interactions.

The first paper in this lineage was Ispolatov et al. (1998), and shortly
thereafter two papers which would ultimately become the cornerstones of the
random asset exchange modeling literature—Drăgulescu and Yakovenko (2000) and
Bouchaud and Mézard (2000)—emerged. As it turned out, however, the
econophysicists were not the first to approach the question of inequality in
this way. The sociologist John Angle had actually published a series of papers
containing a model extremely similar to Ispolatov et al.’s more than a decade
earlier, though the literature had no knowledge of this fact until it was
pointed out by Lux (2005) Angle (1986, 1992, 1993). Likewise, it was noticed
by Patriarca et al. (2005) that Drăgulescu and Yakovenko’s model was
anticipated by a series of papers by Eleonora Bennati, which had been
published in ill-known Italian economics journals in the 1980s and which had
not been translated into English Bennati (1988, 1993).

Nonetheless, in the twenty-five years since Ispolatov et al.’s initial paper,
a sizeable literature on this subject has emerged, with countless variations
of the aforementioned models having been proposed and investigated. The
literature has also become much more diverse in that time: though this subject
was initially solely the domain of a subset of physicists interested in
exploring economic questions, they have since been joined by researchers with
backgrounds in mathematics, economics, systems science, and more.

This article provides, for the first time, a comprehensive and thoroughgoing
review of this literature, which, following the terminology of Sinha (2003),
will be referred to as the ”random asset exchange” literature. While many
excellent partial reviews do already exist (see Chatterjee and Chakrabarti
(2007), Yakovenko and Rosser (2009), Patriarca et al. (2010), and Patriarca
and Chakraborti (2013), just to name a few), all either have become dated or
have focused on only a delimited part of the literature. This review is the
first, to our knowledge, that not only discusses all significant econophysical
models of income inequality, but fully enumerates all the most common
variations upon the literature’s canonical models as well.

##  II The Taxonomy of Random Asset Exchange Models

Most random asset exchange models tend to fall into one of two classes. The
first of these is conventionally called the ”kinetic wealth exchange” (KWE)
class of model, which was popularized by Drăgulescu and Yakovenko (2000).
Named such because of their similarity to thermodynamic models from the
kinetic theory of gases, KWE models are typically—though not
always—characterized by the following properties:

  1. 1.

Pairwise exchange between agents is the primary system state transition
function;

  2. 2.

Total money present in the system is conserved; and

  3. 3.

Total money present between all pairs of agents engaged in exchange is
conserved.

These features are analogous to the role of particle collisions, conservation
of energy, and conservation of momentum in the kinetic theory of gases,
respectively.

The second prominent class of model, inspired by models of directed polymers
rather than ideal gases, is the Bouchaud-Mézard (BM) type model, first
introduced by Bouchaud and Mézard (2000). In contrast to KWE-style models, BM-
style models tend to be characterized instead by fixed wealth flow rates
between all ”adjacent” pairs of agents, as defined by an implicit adjacency
network, being the main mechanism of system evolution. Furthermore, each
agent’s wealth is subject to endogenous stochastic variation, leading to
systemic non-conservation of wealth.

While other formulations exist, models belonging to one of these two classes
represent the great majority of the literature. In this section, we review the
most significant variations of both classes—as well as a handful of minor but
nonetheless significant alternative model classes.

###  II.1 Kinetic wealth exchange

While their work was anticipated by Angle (1986), Bennati (1988, 1993), and
Ispolatov et al. (1998), it is Drăgulescu and Yakovenko (2000) who are
credited with first formalizing and thoroughly studying the KWE model. In
their initial formulation, a system of $N\gg 0$ agents with $M\gg N$ units of
wealth between them is posited. Agents then engage in random pairwise
exchanges, with a winner and loser being randomly selected in each pair and a
transfer of wealth occurring, following some exchange rule:

$\begin{bmatrix}w_{i}\\\
w_{j}\end{bmatrix}\rightarrow\begin{bmatrix}w_{i}+\Delta w\\\ w_{j}-\Delta
w\end{bmatrix}$ (8)

with $\Delta w>0$ if agent $i$ is the winner of the exchange, and $\Delta w<0$
if instead $j$ is victorious. Since KWE models almost always feature
exclusively linear, pairwise exchanges, it is often convenient to represent
the model’s exchange rule as a $2\times 2$ matrix $M$, such that:

$\begin{split}\begin{bmatrix}w_{i}(t+1)\\\
w_{j}(t+1)\end{bmatrix}&=M\begin{bmatrix}w_{i}(t)\\\
w_{j}(t)\end{bmatrix}\end{split}$ (9)

Drăgulescu and Yakovenko showed that, so long as $\Delta w$ is chosen so that
the exchange process was time-reversal symmetric, then the distribution of
money among agents converges to the entropy-maximizing exponential
distribution:

$p(w)=\frac{1}{T}\exp\left(-\frac{w}{T}\right)$ (10)

where $T=\langle w\rangle$ represents the average wealth held by
agents—analogous to temperature in the equivalent thermodynamic system
Dragulescu and Yakovenko (2000). This result proves to be extremely robust,
not varying with one’s choice of time-reversal symmetric exchange rule or
underlying adjacency network Lanchier (2017).

The differences between Drăgulescu and Yakovenko’s model and those of Angle,
Ispolatov et al., and Bennati are subtle. In both Angle’s initial model (the
”one-parameter inequality process,” or OPIP) and Ispolatov et al.’s
”multiplicative-random” exchange model, $\Delta w=\varepsilon w_{loser}$, such
that exchanges are of the form:

$\begin{bmatrix}w_{i}\\\
w_{j}\end{bmatrix}\rightarrow\begin{bmatrix}w_{i}+\varepsilon w_{j}\\\
(1-\varepsilon)w_{j}\end{bmatrix}$ (11)

if agent $i$ wins the exchange. The sole difference between these two
formulations is that Angle (1986) draws $\varepsilon$ from a uniform
distribution before each exchange, whereas Ispolatov et al (1998) define
$\varepsilon$ as a fixed, global parameter. In contrast to the exchange rules
investigated by Drăgulescu and Yakovenko (2000), both of these models break
time symmetry and produce identical distributions which are very well-
approximated by, but not exactly given by, Gamma distributions Angle (1993).

In both Ispolatov et al.’s additive-random exchange model and Bennati’s model,
on the other hand, agents exchange constant, quantized amounts of wealth,
equivalent under rescaling to $\Delta w=1$. In Ispolatov et al. (1998), agents
with 0 wealth are removed from the system entirely, causing all the wealth in
the system to eventually be accumulated by a single agent (a phenomenon termed
”condensation”). In Bennati (1988), however, agents with 0 wealth are
permitted to win, but not to lose, exchanges, identical to the provision in
the constant exchange rule discussed by Drăgulescu and Yakovenko. For that
reason, the KWE model with time reversal-symmetric exchange rule is sometimes
referred to as the Bennati-Drăgulescu-Yakovenko (BDY) model of wealth exchange
Yakovenko and Rosser (2009).

An extension of Drăgulescu and Yakovenko’s initial model proposed
contemporaneously with its initial publication was investigated by Chakraborti
and Chakrabarti (2000), which introduced a ”saving propensity” parameter
$\lambda$. Called the CC model (or, more rarely, the ”saved wealth” model),
its system dynamics are characterized by the fact that, for $\lambda\in[0,1)$,
every agent engages in multiplicative exchange with only a fraction
$1-\lambda$ of their total wealth. The exchange rule in such models can thus
be defined by:

$M=\begin{bmatrix}\lambda+\varepsilon(1-\lambda)&\varepsilon(1-\lambda)\\\
(1-\varepsilon)(1-\lambda)&\lambda+(1-\varepsilon)(1-\lambda)\end{bmatrix}$
(12)

where $\varepsilon$ is drawn from a uniform distribution on $[0,1]$ at every
exchange.

Curiously, this slight modification dramatically changes the equilibrium
distribution of money among agents within the system as the mode of the
distribution (the ”most likely agent wealth”) becomes non-zero, approaching
$T=\frac{M}{N}$ (an egalitarian distribution) as $\lambda$ approaches 1. Gupta
(2006) observes that this departure from the entropy-maximizing distribution
is a consequence of the fact that the introduction of the saving propensity
parameter $\lambda$ results in the system transition matrix becoming non-
singular Gupta (2006). Patriarca et al. (2004a, b) demonstrate that the
resultant distribution is extremely well fit by a scaled Gamma distribution:

$\begin{split}p(w)=\frac{1}{\Gamma(n)}\left(n\cdot\frac{w}{T}\right)^{n-1}\exp\left(-n\cdot\frac{w}{T}\right)\end{split}$
(13)

where $n=1+3\lambda/(1-\lambda)$. The fit is not exact, however, as the
distributions differ in their fourth moments Lallouache  _et al._ (2010).

The CC model is extremely influential in the random asset exchange literature,
and it itself has two major variations which must be mentioned. The first,
introduced by Chatterjee et al. (2003), defines the saving propensity
parameter to be heterogenously distributed throughout the population; instead
of having identical saving propensities, each agent $i$ has their own
individual saving propensity $\lambda_{i}\in[0,1)$ drawn from the uniform
distribution during model initialization. The exchange rule of this model,
called the CCM model, is thus:

$M=\begin{bmatrix}\lambda_{i}+\varepsilon(1-\lambda_{i})&\varepsilon(1-\lambda_{j})\\\
(1-\varepsilon)(1-\lambda_{i})&\lambda_{j}+(1-\varepsilon)(1-\lambda_{j})\end{bmatrix}$
(14)

The steady-state distribution then exhibits a Gamma-like bulk, as in the CC
model, as well as a right tail well fit by a power law with Pareto parameter
$\alpha=1$; this power law is robust for any distribution of saving propensity
of the form $\rho(\lambda)\approx|\lambda_{0}-\lambda|^{\alpha}$, or for
uniform distributions within a restricted range
$\lambda_{i}\in[a,b]\subset[0,1)$ Chatterjee  _et al._ (2004).

One well-known (and, arguably, unrealistic) aspect of the CCM model is that
average agent wealth is highly correlated with their saving parameter, such
that the agents who save nearly all of their money in every transaction
invariably become the wealthiest. This remains the case even if a significant
bias in favor of poorer agents is introduced, because thrifty agents in the
CCM model always stand to gain much more than they lose from every transaction
Neñer and Laguna (2021a). This aspect of the model also explains the
surprising appearance of the Pareto tail, which is actually somewhat illusory:
the right tail of the equilibrium distribution of the CCM model is constituted
by the overlapping exponential tails of the exponential distributions
corresponding to the subpopulations with the highest saving parameters
Patriarca  _et al._ (2005).

Another significant drawback of the CCM model is that, while the right tails
of the steady state distributions change from approximately Pareto with index
1 to exponential as the distribution of $\lambda_{i}$ narrows, the empirical
value of $\alpha\approx 1.5$ is never reached Repetowicz  _et al._ (2005).
However, there are a number of ways to modify the CCM model and recover such a
regime: Repetowicz et al. (2006), for instance, note that introducing modified
wealth parameters with memory—$\hat{w}_{i}(t)=w_{i}(t)+\gamma w_{i}(q)$, with
$\gamma\in(0,1)$ and $q<t$—before each transaction and applying the CC
exchange rule thereto does permit Pareto tails with indices $\alpha>1$ to be
obtained. Likewise, Bisi (2017) demonstrates that replacing the saving
propensity parameter with a bounded, global function of an agent’s wealth
$\gamma(w_{i})$ also permits superunitary Pareto indices.

Figure 1: Stationary distributions produced by the BDY (left), CC (middle),
and CCM (right) models, with best-fit gamma curves shown. All simulations were
performed with the parameterization $N=5000$ and $\langle w\rangle=100$ over
$10^{5}$ iterations.

The second variation, introduced by Cordier et al. (2005) and often called the
CPT model, investigates the CC model with an additional stochastic growth
term:

$\begin{split}M&=\begin{bmatrix}(1-\lambda)+\eta_{i}&\lambda\\\
\lambda&(1-\lambda)+\eta_{j}\end{bmatrix}\end{split}$ (15)

Where $\eta_{i}$ and $\eta_{j}$ are independent and identically distributed
variables with mean 0 and variance $\sigma^{2}$. As in the BDY, CC, and CCM
models, debts are not permitted, so a transaction only takes place so long as
neither agent is reduced to a negative level of wealth. Because $\eta_{i}$ and
$\eta_{j}$ are uncorrelated, total wealth is now only preserved in the mean.
The CPT model has an inverse-Gamma equilibrium, with shape parameter
$\alpha=1+\frac{2\lambda}{\sigma^{2}}$ and scale parameter $\beta=\alpha-1$:

$\begin{split}p(w)&=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\cdot
w^{-1-\alpha}\cdot\exp\left(-\frac{\beta}{w}\right)\end{split}$ (16)

In this case, the shape parameter $\alpha$ may be interpreted as the ”Pareto
index” of the approximately power-law right tail.

The CPT model, like the CCM model, is quite flexible and has been studied in a
variety of other contexts. Düring and Toscani (2008) employ a CPT model with
quenched saving propensities to study international transactions, representing
countries as subpopulations with different saving propensities. Bisi and Spiga
(2010) consider a variation on the CPT wherein the amount of wealth an agent
receives from his trading partner is also subject to stochastic fluctuations.
More recently, Zhou et al. (2021) investigate the effect of introducing a non-
Maxwellian (i.e. wealth-varying) collision kernel in the CPT model.

####  II.1.1 Theft, fraud, and yard sales

Following the terminology of Hayes (2002), binary exchange models in which the
transfer amount is proportional to the wealth of the loser are commonly
referred to as ”theft and fraud” (TF) models, while those in which the
transfer amount is proportional to the wealth of the poorer agent are referred
to as ”yard sale” (YS) models. That is, the YS model posits an exchanged
quantity $\Delta w$ of the form:

$\begin{split}\Delta w&\propto\min\\{w_{i},w_{j}\\}\end{split}$ (17)

The advantage of the YS model is that, from a strategic perspective, agents
are not disincentivized from engaging in trade, as the expected value of an
exchange is always 0. This is in contrast to the TF model, which is so-named
precisely because the expected value of an exchange is always negative for the
richer agent. If agents were allowed to choose whether or not to engage in a
given exchange, a TF economy would immediately freeze as soon as a wealth
differential appeared. The principal drawback of the YS model, however, is
that it is now well-known that the unmodified YS model always exhibits
condensation, though a non-degenerate equilibrium can be recovered if the
probability of winning a given exchange is biased in favor of the less wealthy
agent, if a mechanism for redistributing wealth from richer agents to poorer
ones is introduced, or if extremal dynamics are coupled to the system
Boghosian  _et al._ (2015); Sinha (2003); Cardoso  _et al._ (2021); Bagatella-
Flores  _et al._ (2015).

Moukarzel et al. (2007) demonstrated that, in the case of the YS model where
the proportion of the poorer agent’s wealth at stake in each transaction is a
fixed constant $f$, a sufficient bias of the probability $p$ towards the
poorer agent alone was sufficient to avoid condensation. In particular, the
critical probability $p^{*}$ above which the system does not condense was
found to be:

$p^{*}=\frac{\log\left(\frac{1}{1-f}\right)}{\log\left(\frac{1+f}{1-f}\right)}$
(18)

Based on this result, Bustos-Guajardo and Moukarzel (2012) studied an
extension of the YS model on an adjacency network, such that exchanges may
only take place between adjacent agents. They found that the value of the
critical probability remains the same regardless of the choice of network. In
fact, most system dynamics in the stable phase of the system are independent
of the choice of network. However, certain dynamical aspects of the system
(such as time required for the system to fully condense) do differ from the
fully-connected case in the unstable (i.e. condensing) region. This is not
entirely surprising, seeing as the number of agents to whom the wealth will
condense is directly determined by the underlying network; instead of one
agent accumulating all the money, the distribution condenses to a set of
”locally rich agents,” sometimes termed the ”oligarchy.”

Redistribution in the YS model was examined by Boghosian (2014a), in which a
mechanism by which, at each time step, $\chi$ percent of each agent’s wealth
was confiscated and subsequently redistributed uniformly among the population.
Introducing this mechanism not only prevented condensation, but also produced
a gamma-like steady state distribution with a Pareto-like tail. The dynamics
of this mechanism were studied in more detail by Boghosian et al. (2017) and
Devitt-Lee et al. (2018), in which it was combined with an bias in exchanges
in favor of the wealthy, called the ”wealth-attained advantage.” In this
variation, termed the ”extended yard sale” (EYS) model, the wealthier agent
wins a given exchange with probability $p=rT(w_{i}-w_{j})$, where $T$ is the
average wealth of the system and $w_{i}$ is the wealth of the richer agent.
The wealth-attained advantage formally acts as a net tax on the non-oligarchy,
while the redistribution acts as a net tax on the oligarchy; once the poor-to-
rich flux of the redistributive mechanism was eclipsed by the rich-to-poor
flux of the wealthy agents’ advantage, the system moves from a subcritical to
a supercritical state and the inequality of the resulting wealth distribution,
as measured by the Gini coefficient, begins increasing rapidly. This acts as a
second-order phase transition within the system. One additional variation of
the EYS model, the ”affine wealth” (AW) model, was introduced by Li et al.
(2019). The AW model permits negative wealth by defining a debt limit
$\Delta$, adding $\Delta$ to the wealths of both agents before each exchange,
and subtracting $\Delta$ once the exchange is complete. The AW model provides
a remarkably good fit to the U.S. wealth distribution, as reported by the U.S.
Survey of Consumer Finances.

Figure 2: Stationary distribution produced by the IGAV model with quenched
savings propensities and a bias function per Eq. (12). Simulation was
performed with the parameterization $N=5000$ and $\langle w\rangle=100$ over
$10^{6}$ iterations.

It is also worth mentioning the variation on the YS model first formulated by
Iglesias et al. (2004). This model, sometimes referred to as the IGAV model,
sees agents with wealths $w_{i}$ and $w_{j}$ and saving parameters
$\lambda_{i}$ and $\lambda_{j}$ exchange quantities $\Delta
w_{ij}=\min\left\\{(1-\lambda_{i})w_{i},(1-\lambda_{j})w_{j}\right\\}$, where
the bias in favor of the poorer agent is defined as per Scafetta et al.
(2002):

$p=\frac{1}{2}+f\cdot\frac{w_{1}-w_{2}}{w_{1}+w_{2}}$ (19)

The asymmetry flux index $f\in[0,1/2]$ essentially defines the degree of
systemic social protection offered to the poor. A number of similar models
which modify Iglesias et al.’s exchange rule were studied by Caon et al.
(2007). Recently, Neñer and Laguna (2021a) showed that, in the IGAV model, the
richest agents are not necessarily the thriftiest. Instead, the saving
propensity $\lambda^{*}_{i}$ that maximizes equilibrium average wealth lies in
the interval $(0,1)$, increasing with $f$.

Heinsalu and Patriarca (2014) introduce a variation of the BDY model meant to
more explicitly model the dynamics of barter economies. While Drăgulescu and
Yakovenko examined, among others, the TF exchange rule:

$M=\begin{bmatrix}\varepsilon&\varepsilon\\\
1-\varepsilon&1-\varepsilon\end{bmatrix}$ (20)

where $\varepsilon$ is a uniform random variable with mean $0.5$, Heinsalu and
Patriarca consider the rule:

$M=\begin{bmatrix}1-\varepsilon_{i}&\varepsilon_{j}\\\
\varepsilon_{i}&1-\varepsilon_{j}\end{bmatrix}$ (21)

where $\varepsilon_{i}$ and $\varepsilon_{j}$ are i.i.d. uniform random
variables with mean $0.5$. This modification, called the ”immediate exchange”
(IE) model, changes the system from a pure TF one, where wealth flows
unidirectionally and, on average, from richer to poorer agents, to one in
which wealth flows bidirectionally. In the general IE model, transactions have
some probability $\mu$ of occurring unidirectionally in the manner of Angle
(1986). In the pure IE model with $\mu=0$ has a steady state distribution
$p(w)$ which is an exact Gamma distribution with a shape parameter of 2,
meaning $\lim_{x\rightarrow 0}p(x)=0$, $p(x)$ has a non-zero mode, and the
right tail is well-approximated by a Pareto distribution with $\alpha=1$
Katriel (2014).

###  II.2 Bouchaud-Mézard models

Unlike the KWE model, the BM model does not make use of agents pairing up and
engaging in binary transactions with a winner and a loser; rather, the rate of
exchange between agents are defined by a fixed adjacency matrix $\mathbf{J}$,
each entry of which $J_{ij}$ represents the ”cash flow rate” from agent $j$ to
agent$i$. In Bouchaud and Mézard’s original paper, each agent in the
population of size $N$ has two sources of income—stochastic returns from
investments and sales of a product to other agents—and one source of
expenses—purchases of products from other agents. Thus, the income of agent
$i$ is given by:

$\diff{w_{i}}{t}=\eta_{i}(t)w_{i}(t)+\sum_{j\neq i}J_{ij}w_{j}(t)-\sum_{j\neq
i}J_{ji}w_{i}(t)$ (22)

where $\eta_{i}$ is a Gaussian random variable with variance $2\sigma^{2}$
Bouchaud and Mézard (2000). Notably, the BM model has no restriction on total
wealth being conserved. The simplest case, in which all rates of exchange are
equalized such that $J_{ij}=\frac{J}{N}$, lends itself well to a mean-field
approximation, which produces an inverse-Gamma equilibrium distribution with
shape parameter $\alpha=1+\frac{J}{\sigma^{2}}$ and scale parameter
$\beta=\alpha-1$—strikingly similar in form to the equilibrium distribution of
the CPT model. However, the mean-field approximation is time-limited; for any
finite number of agents, the BM model on a complete graph will eventually
exhibit wealth condensation and the probability that a given agent will have
wealth less than any finite fraction of total wealth grows to 1 Medo (2009).

Further investigation into this class of model demonstrated that the resulting
distribution is also sensitive to the nature of the underlying network
defining the non-zero entries of the transaction matrix $\mathbf{J}$. Souma et
al. (2001) demonstrated through simulation that defining $\mathbf{J}$ on a
small-world network—where each agent neighbors only $0.1\%$ of the
population—leads to distributions which are best fit by a combination of log-
normal and power-law distributions. Garlaschelli and Loffredo (2004, 2008)
likewise showed that it is possible to retrieve a realistic mixed log-normal-
power law distribution by simulating the model on a simple heterogeneous
network with a small number of ”hub” agents, and that the BM model on a
homogeneous network is able to reproduce either a log-normal or a power law
distribution—but not both—depending on the average number of adjacencies per
agent. Ma et al. (2013) simulated the BM model on a partially connected
network and found the generalized inverse Gamma (GIGa) distribution provided
the best fit to the steady state.

Though the original BM model is a continuous-time model, a number of authors
have studied similar models in discrete time as well. Di Matteo et al. (2003),
for example, considers the variation:

$\begin{split}\Delta w_{i}(t)&=A_{i}(t)+B_{i}(t)w_{t}\\\ &+\sum_{j\neq
i}Q_{j\rightarrow i}(t)w_{j}(t)-\sum_{j\neq i}Q_{i\rightarrow
j}(t)w_{i}(t)\end{split}$ (23)

For the purposes of their analysis, additive noise $A_{i}(t)$ is assumed to be
Gaussian with mean zero, and multiplicative noise $B_{i}(t)=0$. Additionally,
each agent $i$ is assumed to split a fixed share $q_{0}$ of their wealth
evenly with all of their neighbors $j\in\mathcal{I}_{i}$, where
$|\mathcal{i}|=z_{i}$. Thus, $Q_{i\rightarrow j}(t)=\frac{q_{0}}{z_{i}}$ if
$j\in\mathcal{I}_{i}$ and 0 otherwise. Their restricted system dynamics thus
become:

$\begin{split}w_{i}(t+1)-w_{i}(t)&=A_{i}(t)-q_{0}w_{i}(t)+\sum_{j\in\mathcal{I}_{i}}\frac{q_{0}}{z_{j}}w_{j}(t)\end{split}$
(24)

Simulating this time series produces results dependent on the choice of
adjacency network, the most notable being that scale-free networks produce
power-law distributions. In this case, the equilibrium wealth level of a given
node is nearly perfectly correlated with the number of neighbors it has in the
specified network, as shown in Figure 4c.

Figure 3: Stationary distribution produced by the discrete-time BM model on a
Barabási–Albert scale-free network, as described by Di Matteo et al. (2003).
Simulation was performed with the parameterization $N=5000$, $\langle
w\rangle=100$, $q_{0}=0.1$, and $E\left[A_{i}(t)^{2}\right]=1$ over $10^{5}$
iterations.

Scafetta et al. (2004) propose another discrete-time variation of Bouchaud and
Mézard’s model, motivated by a dissatisfaction with the formulation of wealth
transfer via exchange as it appears in their original paper, which sees a
constant wealth flux from rich to poor. This is not necessarily realistic as
wealth should only be transferred in exchange if an agent buys an asset for a
price different than its value; such a model cannot explain wealth inequality
under the assumption of perfect pricing. Thus, Scafetta et al. propose a model
in which the wealth of agent $i$ is given by:

$w_{i}(t+1)=w_{i}(t)+r_{i}\xi(t)w_{i}(t)+\sum_{j\neq i}w_{i\rightarrow j}(t)$
(25)

where $r_{i}=V\Pi_{i}>0$ is the ”individual investment index,” given as the
product of the global investment index and the proportion of wealth actually
invested by agent $i$, $\xi(t)$ is a Gaussian random variable representing
return on investment, and $w_{i\rightarrow j}(t)$ represents the flow of
wealth from agent $j$ to agent $i$ in period $t$, which is assumed to be
Gaussian with mean
$\mu=fh\frac{w_{i}-w_{j}}{w_{i}+w_{j}}\min\\{w_{i},w_{j}\\}$ and standard
deviation $\sigma=h\min\\{w_{i},w_{j}\\}$.

Varying $f$, $h$, and $r$ then allowed the authors to tune the strengths of
different system dynamics. If $h>0$ and $f=r=0$ (the symmetric trade-only
model), wealth condensation occurs. If $f,h>0$ and $r=0$ (the asymmetric
trade-only model), a Gamma-like distribution is observed. Finally, if
$f,h,r>0$ (the asymmetric trade-investment model), a Gamma-like distribution
with a power-law tail is observed.

Various other modifications to the BM model have been studied as well. Huang
(2004) extends the BM model to negative wealth levels, and Torregrossa and
Toscani (2017) prove analytically that a unique steady state with support on
the entire real number line exists. Johnston et al. (2005) imposes the
additional restriction of conservation of wealth, finding that wealth
condensation still occurs for high values of $\mu$. Finally Ichinomiya (2012a,
b) relaxes Bouchaud and Mézard’s mean field assumption to adiabatic and
independent assumptions, drawn from quantum mechanics. The power law-like tail
is reproduced and condensation is seen to take place at a higher $J$ than the
mean-field case would indicate, though the Pareto index obtained is smaller
than those empirically observed Ichinomiya (2013).

###  II.3 Other formulations

While the two principal model classes of the random asset exchange literature
are KWE model and the BM model, a variety of less influential formulations
also exist.

A simple model which has nonetheless been significant in the economics
literature is the multiplicative stochastic process (MSP), which was studied
by the economists Robert Gibrat and D.G. Champernowne Garlaschelli and
Loffredo (2008). It would however be a stretch to say that the MSP class of
models is truly a type of random asset exchange model as it is primarily
characterized by the lack of exchange or any other sort of interaction between
agents. Such models essentially represent agents’ levels of wealth in terms of
independent random walks, but nonetheless are able to capture some essential
characteristics of observed distributions. The simplest model in this vein is
the pure MSP $w(t+1)=\lambda(t)w(t)$, where $\lambda(t)$ is a Gaussian random
variable. It is straightforward to show that the distribution of wealth among
an ensemble of agents whose wealth evolution is governed by a pure MSP will
follow a log-normal distribution, though variations which include additive
noise (as in the Kesten process from the biological sciences) and ”minimum
wage”-style boundary constraints can also reproduce power law tails Souma
(2002). For examples of such models, see Biham et al. (1998), Huang and
Solomon (2001), Souma and Nirei (2005), and Basu and Mohanty (2008).

Another class of model which studied in the early days of the RAE literature
in particular is the Generalized Lotka-Volterra (GLV) model, which also has
its origins in the biological sciences. The original Lotka-Volterra process,
studied by Biham et al. (1998), is given by:

$w_{i}(t+1)=\lambda(t)w_{i}(t)+a\bar{w}(t)-bw_{i}(t)\bar{w}(t)$ (26)

where $\lambda$ is a time-dependent random variable and $\bar{w}(t)$ is the
average wealth in the system. The inclusion of the $\bar{w}(t)$ terms
represents a form of indirect interaction between agents: much like in the
mean-field approximation of the BM model, instead of including specific
interaction terms $b_{ij}w_{i}(t)w_{j}(t$, all interactions are assumed to be
symmetrical: $b_{ij}=b/N$.

The generalized form of this model was introduced by Solomon and Richmond
(2001, 2002), and follows:

$\begin{split}\Delta
w_{i}(t)&=\left(\varepsilon_{i}(t)\sigma_{i}+c_{i}(w_{1},w_{2},\dots,w_{N},t)\right)w_{i}(t)\\\
&+a_{i}\sum_{i}b_{j}w_{j}(t)\end{split}$ (27)

where $\epsilon_{i}$ is a stochastic variable such that $E[\epsilon_{i}]=0$
and $E[\epsilon_{i}^{2}]=1$, $c_{i}$ represents endogenous and exogenous
dynamics in returns, and $a_{i}$ and $b_{i}$ represent arbitrary
redistributions of wealth among agents. The restrictions on $\epsilon$ can be
made without loss of generality thanks to the $c_{i}$ term. Under certain
assumptions, this model also produced mixed exponential-Pareto distributions.
However, this model ultimately faded in popularity due to the difficulty it
has accurately representing the left tail of income distributions, as well as
the lack of economic justification for some of its terms Repetowicz  _et al._
(2005).

##  III Notable Trends in the Literature

While the papers discussed above serve as the foundation for the random asset
exchange literature, the flexibility of the underlying modeling framework have
allowed a vast number of featural variations upon these canonical models to
have proliferated. In this section, we provide an overview of the most
significant of these trends is provided and summarize a number of key papers
in each category.

###  III.1 Non-conservation of wealth

One of the primary criticisms leveled against the original KWE models is that
the assumption of total conservation of wealth, made by analogy with the
conservation of energy in ideal gas models, is highly unrealistic. In real
economies, wealth is constantly being created and destroyed—not just by means
of production and consumption, but even by the constant issuing and repaying
of loans. Thus, a number of modifications to the conservative KWE model have
attempted to represent this fact. Most such models can be classified into one
of two types: models which, like the CPT model, conserve wealth in the mean,
and models which tie the global wealth level to a fixed influx rate.

Bisi et al. (2009) and Bassetti and Toscani (2010) both consider models of the
first type. The latter considers the non-conservative exchange rule:

$M=\begin{bmatrix}\varepsilon_{i}&\varepsilon_{i}\\\
\varepsilon_{j}&\varepsilon_{j}\end{bmatrix}$ (28)

where $\varepsilon_{i}$ and $\varepsilon_{i}$ are i.i.d. and
$E[\varepsilon_{i}+\varepsilon_{j}]=1$. Bassetti et al. (2014) considers a
class of similar lotteries and demonstrates they tend to produce inverse-Gamma
steady states.

Slanina (2004) was the first to consider a non-conservative model of the
second type, in which a constant inflow of wealth from outside the system of
interacting agents is permitted. As in other formulations, the model sees
pairs of agents $i$ and $j$ chosen at random to engage in a transfer of
wealth, defined by the dynamics:

$\begin{split}M=\begin{bmatrix}1-\lambda+\epsilon&\lambda\\\
\lambda&1-\lambda+\epsilon\end{bmatrix}\end{split}$ (29)

where $\lambda\in[0,1]$ represents the wealth exchanged between two
interacting agents and $\epsilon>0$ represents the rate at which exogenous
wealth flows into the system. Slanina’s model produces a Gamma-like
equilibrium distribution with a Pareto tail with an index $\alpha\sim
1+\frac{2\lambda}{\epsilon^{2}}$. Coelho et al. (2008) extended this model,
redefining $\lambda(w_{i})$ as a piecewise function taking on two different
values depending on which side of a pre-specified wealth threshold
$n\bar{w}(t)$ an agent’s wealth $w_{i}(t)$ fell:

$M=\begin{bmatrix}1-\lambda(w_{i}(t))+\epsilon&\lambda(w_{j}(t))\\\
\lambda(w_{i}(t))&1-\lambda(w_{j}(t))+\epsilon\end{bmatrix}$ (30)

This modification reproduced a double power-law regime, a phenomenon observed
when comparing the right tail of income from tax data to estimates for the
capital gains of a country’s very wealthiest individuals.

A number of non-conservative models have dynamics which attempt to more
directly model the process of money creation through borrowing. For example,
Chen et al. (2013) consider a random exchange model in which agents who would
otherwise reach zero wealth are permitted to borrow money from a central bank,
which in turn can issue loans with no interest up to a certain global debt
limit. This process of money creation (issue of loans) and annihilation
(paying back of loans) leads to a system in which the money supply grows
logarithmically. Schmitt et al. (2014) introduces a similar system of money
creation and analyzes the non-local effect that issuing credit has on the rest
of the system; though the recipient of the loan clearly benefits, the effects
of the increase in the money supply quickly propagate and all agents suffer
the resultant inflationary effects.

Recently, Liu et al. (2021) and Klein et al. (2021) introduced a
generalization of the unaltered YS model which permits growth in the money
supply over time, which they call the ”Growth, Exchange, and Distribution”
(GED) model. Each time-step, total wealth $W(t)$ is increased by a factor of
$1+\mu$, and the wealth influx $\mu W(t)$ is distributed among agents such
that agent $i$ receives $w_{i}^{\lambda}/(\sum_{j}w_{j}^{\lambda})$. For
subunitary values of $\lambda$, poorer agents disproportionately benefit from
the growth in the money supply and a quasi-stationary distribution exists;
otherwise, the system exhibits wealth condensation as in the unaltered YS
model. The system dynamics at play here are quite similar to the model of
Vallejos et al. (2018), in which growth surplus is apportioned according to a
more indirect ”wealth power” parameter.

###  III.2 Networks and preferential attachment

It is a notable and well-established result that the significance of the
introduction of adjacency networks into random asset exchange models depends
heavily on the specific model formulation. While, for instance, the specific
nature of the network has a decisive effect on the steady state wealth
distribution in BM-style models, the opposite tends to be true for KWE-style
models. Networks of exchange are an important aspect of real economic systems,
and as such there has been a significant effort to study the effect they have
on various types of RAE models.

Interestingly, models characterized by unidirectional exchange exhibit greater
sensitivity to network structure than bidirectional exchange models.
Chatterjee (2009), for example, introduces a toy model in which agents
exchange fixed fractions of their wealth on a directed network characterized
by a disorder parameter $p$. Higher values of $p$ produced networks where more
agents had similar incoming and outgoing connections; the distributions
obtained therefrom were more Gamma-like, as opposed to the Boltzmann-like
distributions obtained from lower values of $p$. Martínez-Martínez and López-
Ruiz (2013) study a unidirectional model with random exchange fractions, meant
to represent payments on a non-complete graph. This ”directed random exchange”
(DRE) model thus has the exchange rule:

$\begin{split}M=\begin{bmatrix}\varepsilon&0\\\
1-\varepsilon&1\end{bmatrix}\end{split}$ (31)

As with Chatterjee (2009), the choice of adjacency network affects the
equilibrium distribution of the DRE model. For the fully-connected case, the
equilibrium distribution $p(x)$ is again exactly Gamma, with shape parameter
$\frac{1}{2}$ Katriel (2015). Notably, this implies that $p$ possesses a
singularity at 0, explaining why Martínez-Martínez and López-Ruiz observed a
condensation-like phenomenon even on fully-connected networks.

Sánchez et al. (2007) investigate a model in which agents populate a one-
dimensional lattice. Each agent’s wealth grows in a deterministic fashion as a
product of a linear ”natural growth” term and an exponential ”control” term,
which retards growth as the difference between an agent’s wealth and the
average wealth of its neighbors increases. While this system produces a pure
power law distribution, it was later demonstrated that, for different values
of the system’s endogenous parameters or a rearrangement of agents’
neighborhoods, either a Boltzmann-Gibbs or a Pareto distribution could be
obtained González-Estévez  _et al._ (2008, 2009).

A handful of models have included the additional possibility of agents
exchanging connections or positions on a lattice as well as units of wealth.
Gusman et al. (2005) define an IGAV model on a random network in which the
winner of an exchange is rewarded with additional connections on the network,
producing a power law regime. Aydiner et al. (2019) examine a CCM-style
bidirectional exchange model on a one-dimensional lattice, with the twist that
some fraction of agents exchange lattice position each iteration of the
simulation. Fernandes and Tempere (2020) likewise consider a variation of the
CC model in which agents on a two-dimensional lattice randomly switch
positions on the lattice such that the average wealth difference between
neighboring nodes is reduced. This ultimately results in perfect wealth
segregation and uniformly higher inequality.

The dynamics of wealth exchange coupled with extremal dynamics was thoroughly
studied in the ”conservative exchange market” (CEM) model Pianegonda  _et al._
(2003); Iglesias  _et al._ (2003); Pianegonda and Iglesias (2004). Said model
populates a lattice with agents who possess wealth levels in the range
$[0,1]$, and each time step sees the poorest agent’s wealth randomly re-
randomized at the expense or benefit of its two closest neighbors. The
selection rule in this model induces self-organizing behavior such that almost
all agents end up with wealth levels above a ”poverty line,” which proved to
be higher in the restricted lattice case than in the fully-connected case.
This model has been extended by a number of follow-up papers over the years.
Iglesias et al. (2010) used this model to compare two different redistribution
schemes, and Ghosh et al. (2011) considered its mean-field approximation.
Chakraborty et al. (2012) and Braunstein et al. (2013) studied the same
dynamics on various other networks.

Figure 4: Stationary distribution produced by the CEM model, with the best
log-linear fit of the right tail of the distribution shown. Simulation was
performed with $N=5000$ over $10^{7}$ iterations.

A concept closely related to adjacency is that of preferential attachment,
which defines the likelihood of two agents interacting as a function of
endogenous variables. The variable chosen is usually wealth, representing the
fact that, in real economies, both the rich and the poor tend to interact more
often with people of similar socioeconomic status to themselves. Because of
its non-discrete nature, preferential attachment can allow for somewhat more
dynamic interactions than adjacency networks can, permitting agents who become
wealthy to access the networks of the rich and not totally disallowing chance
rich-poor interactions. In fact, adjacency networks can be viewed as a special
case of preferential attachment.

Laguna et al. (2005) study the effect of this phenomenon on the IGAV model by
imposing the restriction that a given agent is only permitted to interact with
another agent if the difference between their two wealth levels is less than a
given threshold value $u$. Large values of $u$ more or less replicate the IGAV
model and small values freeze the system entirely, as one would expect.
Intermediate values, however, produced a self-organizing separation within the
distribution of wealth, with a gap separating rich agents from poor ones
spontaneously arising. This bimodal distribution persisted even for high
values of the poor-bias parameter $f$.

Chakraborty and Manna (2010) introduce a model with simple preferential
attachment behavior, such that richer agents engage in exchange more
frequently. That is, the probability that agent $i$ is selected as the first
trader is proportional $w_{i}^{\alpha}$, and the probability that $j$ is
selected as the second trader is proportional $w_{j}^{\beta}$. The limit as
either exponent goes to infinity yields purely extremal mechanics, while
$\alpha=\beta=0$ is the CCM model. Goswami and Sen (2014) defines a more
complicated attachment function, wherein the probability of a given pair of
agents $(i,j)$ interacting depends on $i$’s total wealth, the difference in
wealth between $i$ and $j$, and the number of past interactions between agents
$i$ and $j$. The strength of each factor is modulated by a corresponding
exponent, and, when applied to the classical BDY model, the choice of
modulation has a significant effect on the Pareto index of the steady-state
distribution.

###  III.3 Goods and rationality

Despite their reductive nature, many of the simplifying assumptions discussed
above are not uncommon to find in the economics literature as well. Many
neoclassical models of simple ”exchange economies” study the distribution of
endowed and conserved assets absent wealth creation, and comparatively few
consider the effect of exchange networks or other kinds of barriers to freely-
associating exchange between agents (an important source of imperfect
competition and thus market inefficiency). Rather, the main distinction
between RAE-style models and those found in the mainstream economics
literature lies in the fact that the models preferred in the former typically
study ensembles of agents exchanging money directly in a stochastic fashion,
while the latter typically study ensembles of rational (i.e. utility-
maximizing) agents exchanging goods, with money exchange being an implicit
consequence of goods exchange. A number of attempts have been made to
partially bridge this difference between these two literatures by introducing
goods and rationality into RAE models.

Chakraborti et al. (2001) study a model with both a fixed commodity supply $Q$
and money supply $M$ distributed among a population of agents. These agents
first seek to ensure their level of goods $q_{i}$ exceeds some subsistence
level $q_{0}$, and afterwards seek to maximize their money holdings $m_{i}$;
thus, agents with $q_{i}>q_{0}$ find agents with $q_{j}<q_{0}$ to sell their
excess goods to at a fixed price of 1. Not surprisingly, the steady-state
distribution of this system is found to be sensitive to the global quantities
$Q$ and $N$; if the commodity supply is limited ($Q/N<q_{0}$), some fraction
of agents will necessarily fall below subsistence level, while if the money
supply is limited, agents lack the ability to redistribute the commodity
supply in an efficient manner. A similar model with stochastic price
fluctuations is considered by Chatterjee and Chakrabarti (2006), in which
wealth is taken to be the sum of money and commodity holdings. In both models,
the money distribution exhibits a Pareto tail with index 1 while the commodity
distribution is exponential so long as neither $Q$ nor $M$ is restricted.

Silver et al. (2002) considers a model with a more sophisticated utility
function, in which agents possess stochastically time-varying Cobb-Douglas
utility functions of the form:

$u_{i,t}(a_{i,t},w_{i,t})=(a_{i,t})^{f_{i,t}}(w_{i,t}-a_{i,t})^{1-f_{i,t}}$
(32)

where $a_{i,t}$ represents agent $i$’s holdings of the money commodity at time
$t$, $w_{i,t}-a_{i,t}$ represents agent $i$’s holdings of non-money
commodities at time $t$, and $f_{i,t}$ is a random variable independently and
identically distributed across both indices. In an approach highly reminiscent
of the derivation of the equilibrium of the canonical Arrow-Debreu model in
economics, each agent chooses to re-allocate their wealth between money and
non-money commodities in such a way that maximizes $u_{i,t}$ subject to supply
constraints. Simulations of this system produce a wealth distribution well-fit
by a Gamma distribution with a shape parameter of 1 and a rate parameter of
$1/\alpha$, where $\alpha$ represents the global supply of the money
commodity.

However, not every exchange model with goods is paired with rational agents.
Ausloos and Pȩkalski (2007) consider a model with money, goods, and completely
stochastic agent behavior. Each time step, one agent decides via coin toss
whether to purchase a nonzero number of goods. If so, he randomly selects a
fraction of his money to spend and taps another agent to sell to him. If this
second agent has enough goods to sell and has a desire to sell (again decided
via coin toss), the exchange takes place. This model produces a distribution
of wealth which interpolates between two power laws as time progresses, while
the distribution of goods follows a static power-law. In general, those agents
that are rich in terms of money are poor in terms of goods, and vice versa.

Another interesting line of research has concerned itself with defining
traditional macroeconomic ensembles which produce equivalent results to RAE
models. For example, Chakrabarti and Chakrabarti (2009) demonstrates that the
dynamics of the CCM model can be replicated in a neoclassical framework with
rational agents producing differentiated goods and trading in order to
maximize time-varying Cobb-Douglas utility functions for goods and money. In
this case, the stochastic nature of exchange in the CCM model is represented
by random variations in agents’ utility functions in the analog model. Tao
(2015) derives the entropy-maximizing exponential distribution as the
statistical equilibrium of an Arrow-Debreu market system populated by agents
with such time-varying utility functions. More recently, Quevedo and Quimbay
(2020) have extended this formulation to permit agents to save a portion $s$
of goods possessed, naturally leading to an equivalent non-conservative RAE
model.

###  III.4 Strategic behavior

Another approach to modeling ”smarter” agent behavior attempts to integrate
game-theoretic or machine learning dynamics into RAE models. The exact nature
of this integration can take various forms, including bilateral agreement,
strategic heterogeneity, and behavioral evolution, just to name a few.

In Heinsalu and Patriarca’s original paper introducing the immediate exchange
model, the authors consider the effect of introducing an acceptance
criterion—a probabilistic factor defining the odds a given agent will agree to
engage in a given transaction as a function of the difference in wealths
between the agent and his partner, with both agents needing to agree to a
transaction for it to take place Heinsalu and Patriarca (2014). In both the
BDY and IE models, the choice of any symmetrical acceptance criterion (whether
linear, exponential, etc.) only impacts the time of relaxation to equilibrium,
but not the shape equilibrium itself. Asymmetrical decision criteria cause the
equilibrium distribution to lose its universal form and to depend instead on
the rule chosen. For the CC model, however, introducing even a symmetric
criterion causes the equilibrium to lose its Gamma-like shape.

Sun et al. (2008) investigate a KWE model in which each agent can follow one
of four strategies, chosen at random before the simulation begins. The
exchange rule between two agents depends on their strategy and the strategy of
their partner: two of the strategies are passive and tend towards equalizing
the wealth of the two agents, while the other two are aggressive and tend
towards classical theft-and-fraud exchange. As with Heinsalu and Patriarca
(2014), the introduction of heterogeneous trading strategies leads to a
steady-state distribution which depends heavily on the model parameters,
specifically those defining the rate of success of the aggressive strategies
against the passive strategies.

Heterogeneity in strategies is often studied alongside dynamics for updating
agents’ strategies, representing a rudimentary form of learning. Hu et al.
(2006, 2007, 2008), for example, consider a model in which agents begin as
either cooperators or defectors and play a series of prisoner’s dilemma or
snowdrift-style games with their neighbors. After each game, an agent
identifies the strategy of its richest neighbor and adopts it with some
probability defined by his most recent payout, leading on average to more
successful strategies propagating throughout the network. In a similar vein,
da Silva and de Figueirêdo (2014) investigate an adaptive variation of the CCM
model in which each agent $i$ has a fixed probability $\gamma_{i}$ of being
able to update their savings parameter according to a pre-defined rule each
time step. Neñer and Laguna (2021b) study a variation on a poor-biased YS
model with non-zero saving propensity, in which a fraction of agents are
subjected to a genetic evolutionary algorithm after each Monte Carlo
simulation step to update their exchange parameters, which approach the
optimal values determined by Neñer and Laguna (2021a). A BM-style model
coupled with game-theoretic dynamics is extensively analytically studied by
Degond et al. (2014).

###  III.5 Class division

As mentioned above, one of the key features of empirical income distributions
which RAE models attempt to capture is the bifurcation of the overall
distribution into distinct exponential and Pareto (”thermal” and
”superthermal,” following the terminology of Silva and Yakovenko (2004))
components. While some models attempt to replicate this two-regime behavior
while preserving homogeneity of system dynamics (e.g. by distributing a
behavioral parameter throughout the population or imposing a specific network
structure), a number of authors have instead sought an explanation by means of
a bifurcation in system dynamics for agents with large wealth. It is very
natural to identify the exponential bulk of the income distribution with labor
income and the power law tail with capital gains, seeing as Pareto’s original
observations came from data for property incomes. In this way, asymmetric
system dynamics represent the fact that, in real economies, the rich do indeed
have access to economic mechanisms not available to the majority of the
population Montroll and Badger (1974).

Simple models which have this class division ”baked in” are easily able to
replicate two-regime structures of income. Yarlagadda and Das (2005) and Das
and Yarlagadda (2005), for instance, introduce a model in which trading
dynamics differ for agents with wealths on either side of a fixed wealth
threshold. Poorer agents engage in bilateral exchange exactly as in the model
of Chakraborti and Chakrabarti (2000), while richer agents engage in
exchange—with a different saving parameter—against the system-totality,
representing forms of leverage only available to the wealthy. Quevedo and
Quimbay (2020) also study a trading model in which a fixed fraction of the
population acts as ”producers,” who employ the remainder of the population as
”workers.” Producers trade wealth and pay their associated workers a portion
of the exchanged quantity, creating two differently-shaped Gamma distributions
for producer and worker income which, when combined, create a clear two-regime
distribution.

Lim and Min (2020) consider the case in which the CCM model is partitioned
into two classes by a wealth percentile threshold and a ”solidarity effect”
among agents below said threshold is introduced. If two agents belong to the
same class, then exchange proceeds according to the familiar CCM system
dynamics. But if the agents belong to different classes then the lower-class
agent gathers ”partners” equal to some fraction of the size of the class, and
wins a fraction of the upper-class agent’s wealth with a probability equal to
the percent wealth his coalition possesses in the exchange. This solidarity
factor turns out to be crucial for the generation of a realistic wealth
distribution, as without it the middle income stratum collapses and one
obtains a bimodal distribution, as with Laguna et al. (2005).

Imposing a fixed boundary differentiating the upper class from the lower is
not necessarily the best approach here, however, as analysis has shown that
the ”superthermal” component of the income distribution is highly volatile,
fluctuating in size with the stochastic movements of financial markets Silva
and Yakovenko (2004). A number of models consequently attempt to capture this
out-of-equilibrium aspect of the distribution’s right tail by setting class
boundaries dynamically. Russo (2014) investigates a model without exchange in
which a new wealth percentile threshold defining the size of the upper class
is chosen from the uniform distribution at each time step. Agents above that
threshold then see their wealth augmented by a multiplicative stochastic
process, while agents below it have their wealth augmented by an additive
stochastic process. A different approach is forwarded by Smerlak (2016), who
constructs a Markov process defining transition probabilities between a finite
number of stratified classes. Agents in higher classes derive proportionally
greater amounts of income from a multiplicative process subject to shocks, and
consequently exhibit much greater fluctuations in wealth compared to the
majority of agents, who persist at low levels of wealth indefinitely.

Finally, we wish to highlight here the unique and striking ”social
architecture” (SA) model of Wright (2005), which sees agents spontaneously
self-organize into three distinct classes. Wright defines an ensemble with
three types of agents—employers, employees, and the unemployed—and in each
iteration, an agent $i$ is randomly chosen to be ”active.” The activities
agent $i$ engages in depends on its status: if $i$ is an employer, it pays as
many of its employees as it can afford; if $i$ is an employee, it receives a
wage and spends it on consumption goods produced by an employer; and if agent
$i$ is unemployed, a random (wealthy) agent is chosen to hire $i$, assuming
their level of wealth is sufficient to pay $i$’s wages. Although the initial
conditions of the simulation posited complete equality of agents (all agents
began with equal wealth and no employer or employees), the population quickly
restructured itself into a three-class regime with a distribution of wealth
characterized by an exponential bulk and a Pareto tail. The exact nature of
this distribution becomes clear when disaggregated for class: the wealth of
”employee” agents was completely governed by an exponential distribution,
while that of ”employer” agents was well-fit by a power law. This result is
consonant the argument forwarded by Montroll and Shlesinger (1982) and in
contradistinction to explanations of the two-regime distribution which rely on
endogenous differences between agents. Unfortunately, Wright’s model has seen
few direct extensions, though a similar self-organizing model was studied in
Lavička et al. (2010).

Figure 5: Stationary distributions of wealth, income, and class size produced
by the SA model of Wright (2005). Simulation was performed with the
parameterization $N=5000$, $\langle w\rangle=100$, $[w_{a},w_{b}]=[10,90]$,
over $10^{2}$ ”year rule” iterations ($6\cdot 10^{6}$ time-steps).

###  III.6 Taxation and redistribution

A good deal of attention has also been dedicated to the potential usefulness
of RAE models in the analysis of the efficiency of redistributive mechanisms.
Early studies such as Guala (2008) and Toscani (2009) considered the effect of
a simple ”income tax,” in which a fixed fraction is withdrawn from each
exchange by an external body and uniformly redistributed, on mean-conservative
KWE models, which was found to not alter the exponential nature of the steady
state distribution. Diniz and Mendes (2012) extend this result to multiple
different taxation rules on a CC model, representing both income taxes (taxes
on transaction amounts) and wealth taxes (taxes on wealth level). Bouleau and
Chorro (2017) contrast the effect of income and wealth taxes on YS-like
models, demonstrating analytically that the income taxes alone are not
sufficient to prevent condensation. Similarly, Burda et al. (2019) investigate
the dynamics of a BM-style model with the parameterization $J<0$, which would
normally cause the system to condense, paired with a redistributive mechanism.
A sufficiently strong mechanism succeeded in preventing condensation and
recovering a heavy-tailed wealth distribution, with a multimodal critical
phase also being observed. A number of non-standard redistribution rules in a
YS model were examined by Lima et al. (2022).

Recently, however, interest within the literature has grown around the problem
of identifying optimal tax rates in models, often borrowing techniques from
control theory to do so. Bouchaud (2015) extends the BM model to permit a
wealth tax capable of reallocating wealth between the private and public
sectors, with different growth rate parameters. By maximizing expected
economic growth, an optimal tax rate in the interval $(0,1)$ is obtained for
growth rate differences within an intermediate range. Düring et al. (2018)
develop a finite-horizon model predictive control mechanism for the CPT model
to derive the feasible tax regime which minimizes a cost function representing
some metric of inequality, and consider various objective functions and
redistribution schemes. Zhou and Lai (2022) investigate a novel model of
individual wealth growth and formulate both an additive and a multiplicative
control mechanism to modulate the excessive growth of the right tail of the
wealth distribution of the ensemble. Lastly, Wang et al. (2022) pairs a CPT
model with an evolutionary description of agents’ decision-making
competence—which feeds back into their saving propensities—and a model
predictive control mechanism to reduce inequality.

Notable Papers
---
Model
Feature |  Theft & Fraud |  Yard Sale |  Bouchaud-Mézard |  Other
Canonical |  Angle (1986)
Bennati (1988)
Ispolatov et al. (1998)
Drăgulescu &
Yakovenko (2000)
Chakraborti &
Chakrabarti (2000)
Chatterjee et al. (2003) |  Hayes (2002)
Iglesias et al. (2004)
Caon et al. (2007)
Moukarzel et al. (2007) |  Bouchaud & Mézard
(2000) |  Biham et al. (1998)
Solomon & Richmond
(2001, 2002)
Non-cons. |  Slanina (2004)
Cordier et al. (2005)
Coelho et al. (2008)
Bisi et al. (2009)
Bassetti & Toscani
(2010)
Chen et al. (2013)
Schmitt et al. (2014) |  Liu et al. (2021)
Klein et al. (2021) |  |  Heinsalu & Patriarca
(2014)
Networks |  Chatterjee (2009)
Martínez-Martínez &
López-Ruíz (2013)
Aydiner et al. (2019)
Fern&es & Tempere
(2020) |  Gusman et al. (2005)
Laguna et al. (2005)
Guajardo &
Moukarzel (2012) |  Souma et al. (2001)
Di Matteo et al. (2003)
Scafetta et al. (2004)
Garlaschelli &
Loffredo (2004)
Ma et al. (2013) |  Pianegonda et al.
(2003)
Sánchez et al. (2007)
Goods |  Chakraborti et al.
(2001)
Chatterjee &
Chakrabarti (2006) |  |  |  Ausloos & Pȩkalski
(2007)
Rationality |  Chakrabarti &
Chakrabarti (2009)
Tao (2015)
Quevedo & Quimbay
(2020) |  |  |  Silver et al. (2002)
Strategies |  Sun et al. (2008)
da Silva &
de Figueirêdo (2014) |  Neñer & Laguna
(2021b) |  Degond et al. (2014) |  Hu et al. (2006)
Class div. |  Yarlagadda & Das
(2005)
Lim & Min (2020) |  |  |  Wright (2005)
Lavička et al. (2010)
Russo (2014)
Smerlak (2016)
Redist. |  Guala (2008)
Toscani (2009)
Diniz & Mendes (2012) |  Boghosian (2014a)
Boghosian et al. (2017)
Bouleau & Chorro
(2017)
Düring et al. (2018)
Lima et al. (2022)
Wang et al. (2022)
Li et al. (2019) |  Bouchaud (2015)
Burda et al. (2019) |  Zhou & Lai (2022)
Table 2: Notable papers in the random asset exchange literature, disaggregated
by formulation type and prominent features.

[b]

###  III.7 Miscellanea

Though the above list enumerates the most widely-studied modifications to RAE
models, it should by no means be considered exhaustive. The flexibility of the
random asset exchange framework makes it easy to introduce new system dynamics
and isolate the effects of a given modification. Just to name a few examples,
Pareschi and Toscani (2014) investigate the effect of variable agent knowledge
on the CPT model, obtaining the intriguing result that the most knowledgeable
agents tend not to be the richest ones; Trigaux (2005) examines the effect of
introducing altruistic behavior to a subpopulation and finds a very strong
equalizing effect when combined with redistribution; Coelho et al. (2005) and
Patrício and Araújo (2021) model the propagation of wealth on a generational
network to study the stratifying effect of inheritance; and Dimarco et al.
(2020) use a class-based framework to characterize the effect of pandemics on
wealth inequality..

The RAE literature has also given rise to a number of wholly new analytical
techniques. Ballante et al. (2020) demonstrate that fitting the distribution
of saving propensities to real-time economic data in a generalized CCM model
via statistical sampling may be useful as a leading indicator of economic
stressors which have the potential to increase inequality. Luquini et al.
(2020) establish a formal equivalence between KWE models and population-based
random search algorithms in computer science, and speculate that said
formulation could ultimately be used as a benchmark model in cybernetics.
Finally, dos Santos et al. (2022) propose a computational technique by which
the crossover point between the exponential and Pareto regimes can be
identified within data sets of real income distributions, aiding in the
empirical study of economic inequality.

##  IV Discussion

From the ambition and breadth of recent publications such as those mentioned
above, it is clear that random asset exchange modeling is being increasingly
recognized as a highly versatile tool which has the potential to find wide
application even beyond its original use as a descriptive econophysical model.
It is also clear that, in seeking to explain the characteristic features of
wealth and income distributions, such models have highlighted the existence of
a number of more fundamental economic phenomena underlying those features,
such as the inherently diffusive nature of exchange economies and the
emergence of apparent power laws from overlapping exponential functions.

Furthermore, it has also become apparent in the course of this investigation
that the random asset exchange modeling literature has a number of bigger-
picture implications. Namely, all of the models discussed indicate that a
large proportion of observed economic inequality is the result of luck and the
inherently diffusive (entropy-increasing) nature of exchange itself. While
some authors have taken this to mean that the ”natural,” entropy-maximizing
level of inequality is by definition fair, such a conclusion is far too strong
and veers into the territory of naturalistic fallacies. Instead, the
conclusion one ought to draw from this cardinal result of the random asset
exchange literature depends on one’s own (subjective) beliefs concerning the
”ideal” level of inequality—however that is determined—as compared to the
level currently prevailing. For proponents of relatively unrestrained
capitalism, who have argued that inequality plays an important role in the
economy by encouraging people to work harder in the hopes of achieving better
economic outcomes, the implications of said result are quite positive:
statistics seem to naturally guarantee such inequality without the help of
market-distorting conditions such as the formation of monopolies or the
institutionalization of economic thievery! On the other hand, for those policy
makers who aim to reduce the degree of inequality in modern, developed
economies, the corresponding implication may be somewhat more dismal. For
them, the main implication of these models is that altering government
policies to make market economies operate more ”fairly” by, for example,
introducing progressive taxation can only do so much. At the end of the day,
large scale regimes of wealth redistribution, such as wealth taxes, may be
necessary in order to reduce inequality below the level that is endogenous to
exchange-based systems.

However, econophysical models are not without their own problems. Most are
still incapable of replicating all of the characteristic features of wealth
and income distributions. For wealth distributions, as has been discussed,
these include non-negligible segments of the population with non-positive
wealth and possibly a power law right tail; for income distributions, these
include an exponential or log-normal bulk, and an at least apparent power law
tail with exponent between -2 and -3.

More importantly, while all of the models discussed above serve as excellent
demonstrations of the role random chance plays in generating the inequalities
observed in market economies, the literature has not yet been able to provide
an adequate explanation for the emergence of these distributional features
which it posits to be universal. That is to say, it has not yet been able to
identify and describe the concrete system dynamics, common to all market
economies, which generate the characteristic features of inequality.

Models which impose specific distributions on an endogenous parameter
throughout the population (thriftiness, size of social network, etc.) clearly
have the capability of producing nearly any desired distribution, but such
results have far less explanatory power seeing as they merely defer the
question. If one observes a given distribution of wealth because there exists
an underlying distribution of a certain behavioral parameter, why is this
parameter distributed the way it is throughout the population? One ultimately
returns to Pareto’s unsatisfying explanation for his own law—that economic
inequality is purely the result of intrinsic differences between
individuals—and finds oneself no closer to actually understanding the crux of
the issue.

More promise is shown by models which offer, as Reddy (2020) calls it, a
”processual” account of inequality Reddy (2020). Such models introduce the
elements of production and class relationships into the mix as fundamental
processes of economic systems. This approach reflects concrete asymmetries in
the economy, reduces the degree of reductionism present within the models, and
permits the identification of different sections of wealth and income
distributions with different social positions. Unfortunately, only a handful
of studies in this direction have been performed, with Wright (2005) and
Lavička et al. (2010) remaining the two most notable examples.

Another significant problem pertains to the relationship between the
distribution of wealth and the distribution of income, the nature of which the
literature has not consistently grasped. Wealth and income are two linked but
quite distinct quantities. Wealth can take a wide variety of forms—money,
consumption goods, real estate, debts, and even information or skills can all
be considered forms of wealth. Income, on the other hand, typically refers to
the amount of ”wealth,” however it is enumerated, received by an individual in
a given time period, prior to expenses. This quantity is of course added to
one’s existing wealth, but the straightforward relation of income as the time-
derivative of wealth only holds under the simplifying assumption that wealth
is not subject to any endogenous changes: that is, no articles of wealth are
consumed, fluctuate in value, are traded for articles of differing value, etc.

For the most part, random asset exchange models are concerned with the
distribution of an undifferentiated, non-consumable, exchangeable
asset—usually a stand-in for money—throughout an ensemble of agents. Thus, the
distributions of said asset throughout the population are best interpreted as
wealth distributions, and it is inapt to compare them to empirical
distributions of income. In fact, Xu et al. (2010) note that reconstructing
time-series of agents’ income within canonical KWE models actually produces
income distributions which are Gaussian, as opposed to exponential, directly
contrary to the available data. Once again, a greater focus on developing
models in a more processual vein, which explicitly link income to salaries and
wages paid out by firms to employees, could be immensely useful in clearing up
this confusion.

Further progress towards an econophysical explanation for inequality is sorely
needed. Per Horowitz et al. once again, 61% of American adults believe that
there exists too much inequality in the United States today. Horowitz  _et
al._ (2020) Of that number, 81% believe that this problem will require either
major policy interventions or a complete restructuring of the economy to
address. There exists a clear political will, at least in the U.S., to reduce
the degree of inequality that has been allowed to develop over the past few
decades. But despite that fact, the same Pew survey demonstrated that there
exists no consensus on what the major contributors to economic inequality in
the U.S. even are. While there are a number of potential explicators commonly
cited—such as industrial outsourcing, the country’s tax structure, and
intrinsic differences between individuals—no single one is viewed by a
majority of the population as a decisive factor. Needless to say, determining
what policies would be required to create a fairer society and economy
necessitates a clearer understanding of the principal processes responsible
for generating inequality in the first place, and much work remains to be done
before such a satisfactory understanding is reached.

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