We propose a simple agent-based model on a network to conceptualize the allocation of limited wealth among more abundant expectations at the interplay of power, frustration, and initiative. Concepts imported from the statistical physics of frustrated systems in and out of equilibrium allow us to compare subjective measures of frustration and satisfaction to collective measures of fairness in wealth distribution, such as the Lorenz curve and the Gini index. We find that a completely libertarian, law-of-the-jungle setting, where every agent can acquire wealth from or lose wealth to anybody else invariably leads to a complete polarization of the distribution of wealth vs. opportunity. This picture is however dramatically ameliorated when hard constraints are imposed over agents in the form of a limiting network of transactions. There, an out of equilibrium dynamics of the networks, based on a competition between power and frustration in the decision-making of agents, leads to network coevolution. The ratio of power and frustration controls different dynamical regimes separated by kinetic transitions and characterized by drastically different values of equality. It also leads, for proper values of social initiative, to the emergence of three self-organized social classes, lower, middle, and upper class. Their dynamics, which appears mostly controlled by the middle class, drives a cyclical regime of dramatic social changes.

Funding: This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

Copyright: © 2017 Mahault et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Simple examples abound. Most of the lower class’ wealth is stored in real estate, the acquisition of which involves financial instruments that effectively deflect towards other agents more than half of the painfully saved wealth, or indeed all of it, when epidemic foreclosures followed a bubble inflated by financial deregulation [ 22 , 23 ]. Also, in the aftermath of a market crisis the least wealthy are at loss and cannot capitalize on the new low prices of assets for the ensuing market rebound as the wealthy can, providing a ratchet mechanism toward wealth inequality at each significant market oscillation. And indeed wealth inequality in the USA worsened after the 2007 financial crisis even though income inequality mildly ameliorated [ 24 ]. Finally, as deregulated financial instruments offset wage stagnation and the widening gap between productivity and living standards [ 12 ], a steady increase in household debt to finance consumption further contributes to wealth polarization [ 12 , 25 ]. Indeed, the USA, at the forefront of financial deregulation, also tops most countries in wealth inequality, with a Gini index of ∼0.8 (worse than any African country except Namibia, as of year 2000) [ 17 ]. Furthermore, the American trend inversion concerning the share of the top 1% [ 13 ] began in the late 70s, closely tracking the deregulation of the financial market [ 26 ].

It has been generally understood that wealth inequality follows income inequality, the delay corresponding to accumulation of savings among the upper-income strata [ 1 , 19 ]. This process can be long (ref. [ 15 ] and references therein) impinging on issues of inherited wealth and social stratification [ 20 ]. However, faster, more direct pathways to wealth polarization might be possible in the fluid contemporary world where increasingly deregulated and sophisticated financial tools can provide new means for wealth transfer [ 21 ].

While convincing econometric studies have reported increasing polarization of income in western societies [ 11 , 12 ], more recent data document a worldwide increase in wealth inequality [ 12 – 16 ]. In general “wealth is unequally distributed, more so than wages or incomes” [ 12 ] and “Gini coefficients for wealth typically lie in the range of about 0.6–0.8. In contrast, most Gini coefficients for disposable income fall in the range 0.3–0.5” [ 17 ]. The Gini index [ 18 ] for the wealth of the entire world (and indeed of the USA) is estimated at a dramatic 0.8 with the bottom 50% of the world population owning 3.7% of wealth (ref. [ 17 ] and references therein).

Ever since Vilfredo Pareto provided the earliest quantitative analyses of wealth distribution [ 1 , 2 ], the inequality of societies has been a matter of debate and concern for economists, social scientists, and politicians alike. Its correlation, anti-correlation, or lack thereof, with growth is historically among the most debated issues in political economics [ 3 – 6 ]. Wealth inequality has recently caused growing concerns for the functioning of democratic institutions [ 7 – 9 ] and occupied the political debate [ 10 ].

In the third case, however, agents are in fact left free to “act”, in a kinetic model (subsection: “Emergent Social Classes and Kinetic Transitions in Market Coevolution”), thus establishing new connections with new neighbors, and severing old ones. A new global parameter is introduced, initiative, which measures the agents’ likelihood of acting on their frustration. We find that initiative controls the interplay of power and frustration. Above a critical value of initiative, society becomes dynamical, rather than converging to a more or less ameliorated inequality, with formation (and competition) of two social classes, and an alternation of times of equality and times of inequality. Equality can emerge, at least periodically, from the individual behavior of sufficiently motivated, frustrated agents.

The previous two cases pertain to static societies that equilibrate at maximum power. We have called “agents” the members of such society, but that was in a sense a misnomer in a framework where we were seeking the equilibrated state that optimizes power. Individual frustration/satisfaction played no role in determining the allocation of wealth.

In the second case (subsection: “Constrained, Static Societies”) we study how the imposition of constraints on the transfer of wealth can greatly ameliorate the inequality profile. We assume that each agent can only “play” within a network where it can exchange wealth with neighbors. We consider two classical cases (random network and scale free network) and show that they lead to rather different equality profiles, both in terms of wealth distribution and of personal satisfaction. Counterintuitively, the case least fair from the point of view of wealth distribution is also the fairest from the point of view of individual satisfaction.

To explore these questions we consider three cases, which will be studied in detail in the section “Results”. We call the first the “Law of the Jungle” (of the homonymous subsection) as every agent can gain or lose wealth to every other, without constraints: there are no limits to the exchange. In such setting we look for the distribution of wealth which maximizes power in the society. This turns out to correspond, as one would expect, to the most savage inequality, with a lower class—made up of half or more of the population—comprised of utterly dispossessed individuals, with no wealth at all, and an upper class that sees all of its opportunities satisfied. This is true regardless of the particular distribution of opportunities in the society. It also follows that a seemingly reasonable recipe often touted by many, the one which typically reads “let’s increase opportunities for all”, does not really change anything in a “Law of the Jungle” setting: more than half of the society would still remain dispossessed.

In particular, our framework allows us to compare subjective (frustration, satisfaction) as well as global (Lorenz curve, Gini index) measures of fairness, and study their interplay in the evolution of society.

In this work we thus explore how wealth inequality appears naturally in a minimal model of agents endowed with opportunities to grab or lose fractions of limited available wealth. The approach is inspired by our research in physics, chemistry and statistical mechanics [ 27 – 32 ] which provides rigorous means to predict statistical equilibria, in contrast to general equilibria in economics, and to also follow frustrated, out of equilibrium dynamics, which is relevant for inequality.

While the issue is certainly very much complex, it motivates a conceptualization of direct pathways to wealth accumulation/polarization, in a framework reflecting the essential features: power, inequality, frustration, and initiative. We seek here not (or not yet!) to provide precise quantitative predictions but rather to conceptualize these very complex issues within a simple model which nonetheless provides a rich phenomenology, one that lends itself to reasonable interpretations, analogous to those of social realities. Our aim here is thus to explore a new framework which can be later developed into a tool with predictive power.

We can recover the information lost in the Lorenz curve and the Gini index for wealth and by introducing average individual measures of fairness to describe the return on opportunity, such as average frustration and average satisfaction . From that we can also draw the Lorenz curve for personal satisfaction or frustration in addition to those for wealth, as well as the corresponding Gini index G s . A little thought shows that a Gini index of zero for personal satisfaction G s = 0 implies that all agents have the same satisfaction (and thus frustration) equal to 1/2. All the agents have half of their opportunities satisfied, a generalization of what physicists call the “ice rule” [ 27 , 29 – 32 , 36 – 38 ].

We employ such characterization tools as they are rather intuitive and ubiquitous in social sciences. However, we also realize that they are global indicators, somehow “coarser” than our framework, insofar as they disregard all the underlying local information pertaining to the individual opportunities/expectations of agents. This other set of information is however twofold important. Firstly, it provides very useful information on the agent’s “return on opportunity”: the global distribution of wealth might be irrelevant to the satisfaction of the individual, insofar most of its expectations are satisfied. Secondly, as we will see, it is reasonable to consider the individual satisfaction/frustration as a motor of society, which leads to interesting and complex dynamics.

From P z and 〈w z 〉 we can then compute the classical—although coarser—economic measure of wealth distribution, the Lorenz curve for wealth [ 35 ], which graphs the cumulative share of wealth vs. cumulative population. A point (x, y) on the curve tells us the fraction y% of wealth owned by the bottom x% of the population. From it we can compute another classical indicator of equality, the Gini index G w [ 18 ] to assess the global fairness of a society. We remind the reader that the Gini index is the normalized difference of the area between the Lorenz curve and the curve of perfect equality y = x, such that G w = 0 denotes perfect equality in wealth distribution, while G w = 1 denotes perfect inequality.

In general we will study 〈w z 〉, the distribution of average wealth for agents with z opportuinities, as well as its fluctuations within agents of the same level of opportunities/expectations, or . From it we can compute the social motivation m z = (〈w z+1 〉 − 〈w z 〉)/〈w z 〉, a collective, averaged quantity describing the marginal increase in wealth for an increase in opportunities.

An obvious ingredient is still missing: the agent’s power to aquire wealth. We seek to model the cumulative advantage idea that power to acquire wealth itself derives from accumulated wealth (the so called “Matthew effect” [ 33 , 34 ]). It follows that power must scale higher than linearly with wealth. We define the power of an agent of wealth w as (3) The physically inclined reader will recognize in the previous formula a total “energy” from the pairwise attraction of w units of wealth, rather than a power, although we call it “power” in reference to the layman understanding of “economic power”.

To frustrate the system we declare that there are twice as many expectations as there is wealth ( where and are the average wealth and opportunities). Thus half of all the expectations go frustrated. Frustration is therefore built into the system and cannot be eliminated. Using this concept we can thus quantify the individual satisfaction s of an agent, defined as its gathered wealth w relative to its expectations z, and similarly its individual frustration f as the fraction of expectations that go unfulfilled, or (1) (2) Thus f = 1, s = 0 means complete frustration (no satisfied expectations), f = 0, s = 1 complete satisfaction (all expectations are satisfied). (Strictly speaking, it is redundant to use both these terms, as they are trivially related to each other. But, depending on the situation, one or the other seems much more natural.) While the system will be overall frustrated, a class of agents might very well be non-frutrated and thus quite satisfied, whereas another class might be extremely frustrated.

The “Law of the jungle” where every agent (in blue) can grab wealth (in red) from any other to fill its available opportunities (circles). The market power is optimized by a polarized society where agents with most opportunities get all their opportunities satisfied.

We consider a system of agents endowed with opportunities. Each agent has z opportunities to obtain units of wealth, where z changes among agents, and is distributed via P z , the fraction of agents endowed with z opportunities (see Fig 1 ). Each opportunity can be fulfilled by a unit of wealth, and the agent with z opportunities will possess w ≤ z units of wealth. Of course, what are objectively “opportunities” can also be seen, subjectively by the agent, as its “expectations”.

In social sciences, where unlike in Physics, frustration cannot be easily quantified, nor indeed defined, it is typically expelled from modeling. Yet, few would doubt its role on social dynamics. In our model, inspired by Physics, we attempt to bring back frustration to the social realm from which the concept initially originated.

Indeed in life too, frustration begets compromise, i.e. acceptance of partiality of success in the achievement of a goal. In life as in Physics, the set of multiple compromises among the numerous degrees of freedom of a problem typically composes into a lively manifold of configurations which exhibit a non-trivial collective dynamics. There, each change or movement has repercussions on the choice of compromise of every other agent. Instead, in absence of frustration, by definition everything converges to the optimal, and there is no dynamics, but only crystallized order.

Precisely for that reason, in Mathematics, where frustration can be more precisely characterized, it is understood as the result of a set of constraints which cannot be satisfied simultaneously. In Physics these constraints are generally related to the optimization of some energy, usually a pairwise interaction [ 28 ]. Thus, for instance, in the antiferromagnetic Ising model on a triangular lattice, it is impossible to find a configuration in which all the interactions are optimized, and thus all the neighboring spins are antiparallel to each other: an extensive number of couples of neighbors will have to be parallel (compromise) and they can be chosen with extensive freedom, resulting in a quite complex manifold.

Frustration is understood in everyday life as the emotional outcome of failure to achieve predetermined goals, despite proper investment in time and effort—not to mention one’s best intentions. Outside of the realm of emotions, it has come to signify situations in which full success is impossible. The failure—or indeed more commonly the only partial success—that originates frustration is often the result of limitations and constraints which are intrinsic and cannot be lifted—hence the bitter feeling.

Given the multidisciplinary aim of this work, it might be useful to start by introducing some fundamental concepts to which we will return later in the manuscript. We begin with frustration.

Results and discussion

The Law of the Jungle Consider a population of agents all endowed with varying number of “opportunities” z. These are defined as available slots, which might or might not be filled by w units of wealth, as in Fig 1. P z is the normalized distribution of expectations and represents the market. As stated above, what are, objectively, “opportunities” can also be seen, subjectively by the agent, as its “expectations” and there are twice as many opportunities as there is wealth ( where and are the average wealth and opportunities). The distribution of wealth which maximizes the total power of the market can be found easily by filling with wealth the agents of largest opportunities and continuing in decreasing order of opportunity, until all wealth is stored. The result is a completely polarized society in which agents with opportunity above a critical value z c have zero frustration and all their opportunities satisfied, whereas the others have nothing as in Fig 1. In such setting there is clearly no discernible “middle class”. The critical opportunity z c is implicitly defined by (4) which follows from . Now, is the distribution of expectations, that is the total number of opportunities for all the agents of equal opportunities. Thus, Eq (4) implies that all the agents with opportunities below the median fall in the class of the haves-not. Clearly the fraction of the utterly dispossessed depends on the structure of the market, or P z . Yet, a little thought shows that the dispossessed are always more than 50% of the population (they are exactly 50% when all the agents have the same number of opportunities [29]: even then, half of the population gets nothing). In particular, one must always make the assumption that the overall wealth is large, as it means that the unit of transaction must be small compared with the overall wealth of the society. Then, for large , we can safely take the continuum limit on z, and find that a scaling P z → kP kz does not change the overall fraction of haves-not, as the critical opportunity also scales in Eq (4). In other words, in a Law of the Jungle setting, where power dominates and each agent can gain or lose wealth to anybody else, the fraction of dispossessed is scale invariant in the opportunities, and recipes often touted by politicians and experts, such as “increasing opportunities for everybody” are not viable policies in the absence of structural changes in the societal topology. One way to grow a small middle class in this setting could be to make society less efficient in the optimization of power. We can introduce “thermal” disorder into the system via standard statistical physics techniques [29] already applied to social settings [39, 40], where a parameter T, an effective temperature, describes the deviation of the market from optimal behavior. Fig 2 shows that as expected, thermal disorder smears the curve, providing for a small middle class with opportunities centering around the critical value, situated among the haves and haves-not. Not surprisingly, the middle class is the only class endowed with motivation (Fig 2, right inset), as it is the only class which can experience a sensible marginal wealth increase following an increase in opportunity. PPT PowerPoint slide

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larger image TIFF original image Download: Fig 2. The distribution of average wealth 〈w z 〉 vs. opportunities z at different temperatures in the Law of the Jungle setting of The distribution of average wealth 〈w〉 vs. opportunities z at different temperatures in the Law of the Jungle setting of Fig 1 for a binomial distribution of opportunities of mean The left and right insets plot respectively the average frustration and average motivation vs. opportunities z. https://doi.org/10.1371/journal.pone.0171832.g002 As society becomes more optimized for power (T → 0), the middle class shrinks to nothing, while its motivation skyrockets. (At very large thermal disorder power play no role and the ice-rule wins with 〈f z 〉 = 1/2 leading to an egalitarian society, but that is a trivial and not very realistic case.) We see that in this jungle setting a limited improvement in equality can only come from disorder that prevents complete optimization of power. However, we will see below, that even non-perfectly optimized jungles still lead to very strong polarization both in cumulative wealth and in individual frustration, at least compared to constrained societies.