Cooperation dynamics in networked geometric Brownian motion

Interplay of network structure and talent configuration on wealth dynamics

The economic success of individuals is often determined by a combination of talent, luck, and assistance from others. We introduce an agent-based model that simultaneously considers talent, luck, and social interaction. This model allows us to explore how network structure (how agents interact) and talent distribution among agents affect the dynamics of capital accumulation through analytical and numerical methods. We identify a phenomenon that we call the "talent configuration effect," which refers to the influence of how talent is allocated to individuals (nodes) in the network. We analyze this effect through two key properties: talent assortativity (TA) and talent-degree correlation (TD). In particular, we focus ons three economic indicators: growth rate (n_{rate}), Gini coefficient (inequality: n_{Gini}), and meritocratic fairness (n_{LT}). This investigation helps us understand the interplay between talent configuration and network structure on capital dynamics. We find that, in the short term, positive correlations exist between TA and TD for all three economic indicators. Furthermore, the dominant factor influencing capital dynamics depends on the network topology. In scale-free networks, TD has a stronger influence on the economic indices than TA. Conversely, in lattice-like networks, TA plays a more significant role. Our findings address that high socioeconomic homophily can create a dilemma between growth and equality and that hub monopolization by a few highly talented agents makes economic growth strongly dependent on their performances.

Stable cooperation emerges in stochastic multiplicative growth

Understanding the evolutionary stability of cooperation is a central problem in biology, sociology, and economics. There exist only a few known mechanisms that guarantee the existence of cooperation and its robustness to cheating. Here, we introduce a mechanism for the emergence of cooperation in the presence of fluctuations. We consider agents whose wealth changes stochastically in a multiplicative fashion. Each agent can share part of her wealth as a public good, which is equally distributed among all the agents. We show that, when agents operate with long-time horizons, cooperation produces an advantage at the individual level, as it effectively screens agents from the deleterious effect of environmental fluctuations.

The ergodicity solution of the cooperation puzzle

When two entities cooperate by sharing resources, one relinquishes something of value to the other. This apparent altruism is frequently observed in nature. Why? Classical treatments assume circumstances where combining resources creates an immediate benefit, e.g. through complementarity or thresholds. Here we ask whether cooperation is predictable without such circumstances. We study a model in which resources self-multiply with fluctuations, a null model of a range of phenomena from viral spread to financial investment. Two fundamental growth rates exist: the ensemble-average growth rate, achieved by the average resources of a large population; and the time-average growth rate, achieved by individual resources over a long time. As a consequence of non-ergodicity, the latter is lower than the former by a term which depends on fluctuation size. Repeated pooling and sharing of resources reduces the effective size of fluctuations and increases the time-average growth rate, which approaches the ensemble-average growth rate in the many-cooperator limit. Therefore, cooperation is advantageous in our model for the simple reason that those who do it grow faster than those who do not. We offer this as a candidate explanation for observed cooperation in rudimentary environments, and as a behavioural baseline for cooperation more generally. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

The ergodicity solution of the cooperation puzzle

When two entities cooperate by sharing resources, one relinquishes something of value to the other. This apparent altruism is frequently observed in nature. Why? Classical treatments assume circumstances where combining resources creates an immediate benefit, e.g. through complementarity or thresholds. Here we ask whether cooperation is predictable without such circumstances. We study a model in which resources self-multiply with fluctuations, a null model of a range of phenomena from viral spread to financial investment. Two fundamental growth rates exist: the ensemble-average growth rate, achieved by the average resources of a large population; and the time-average growth rate, achieved by individual resources over a long time. As a consequence of non-ergodicity, the latter is lower than the former by a term which depends on fluctuation size. Repeated pooling and sharing of resources reduces the effective size of fluctuations and increases the time-average growth rate, which approaches the ensemble-average growth rate in the many-cooperator limit. Therefore, cooperation is advantageous in our model for the simple reason that those who do it grow faster than those who do not. We offer this as a candidate explanation for observed cooperation in rudimentary environments, and as a behavioural baseline for cooperation more generally. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity

We explore the role of non-ergodicity in the relationship between income inequality, the extent of concentration in the income distribution, and income mobility, the feasibility of an individual to change their position in the income rankings. For this purpose, we use the properties of an established model for income growth that includes 'resetting' as a stabilizing force to ensure stationary dynamics. We find that the dynamics of inequality is regime-dependent: it may range from a strictly non-ergodic state where this phenomenon has an increasing trend, up to a stable regime where inequality is steady and the system efficiently mimics ergodicity. Mobility measures, conversely, are always stable over time, but suggest that economies become less mobile in non-ergodic regimes. By fitting the model to empirical data for the income share of the top earners in the USA, we provide evidence that the income dynamics in this country is consistently in a regime in which non-ergodicity characterizes inequality and immobility. Our results can serve as a simple rationale for the observed real-world income dynamics and as such aid in addressing non-ergodicity in various empirical settings across the globe. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Ergodicity breaking in wealth dynamics: The case of reallocating geometric Brownian motion

A growing body of empirical evidence suggests that the dynamics of wealth within a population tends to be nonergodic, even after rescaling the individual wealth with the population average. Despite these discoveries, the way in which nonergodicity manifests itself in models of economic interactions remains an open issue. Here we shed valuable insight on these properties by studying the nonergodicity of the population average wealth in a simple model for wealth dynamics in a growing and reallocating economy called reallocating geometric Brownian motion (RGBM). When the effective wealth reallocation in the economy is from the poor to the rich, the model allows for the existence of negative wealth within the population. In this work, we show that in the negative reallocation regime of RGBM, ergodicity breaks as the difference between the time-average and the ensemble growth rate of the average wealth in the population. In particular, the ensemble average wealth grows exponentially, whereas the time-average growth rate is nonexistent. Moreover, we find that the system is characterized with a critical self-averaging time period. Before this time period, the ensemble average is a fair approximation for the population average wealth. Afterwards, the nonergodicity forces the population average to oscillate between positive and negative values since the magnitude of this observable is determined by the most extreme wealth values in the population. This implies that the dynamics of the population average is an unstable phenomenon in a nonergodic economy. We use this result to argue that one should be cautious when interpreting economic well-being measures that are based on the population average wealth in nonergodic economies.

Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process

We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for nonstationary and nonergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains nonergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the nonergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable state, and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand-alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.

Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.

Power accretion in social systems

We consider a model of power distribution in a social system where a set of agents plays a simple game on a graph: The probability of winning each round is proportional to the agent's current power, and the winner gets more power as a result. We show that when the agents are distributed on simple one-dimensional and two-dimensional networks, inequality grows naturally up to a certain stationary value characterized by a clear division between a higher and a lower class of agents. High class agents are separated by one or several lower class agents which serve as a geometrical barrier preventing further flow of power between them. Moreover, we consider the effect of redistributive mechanisms, such as proportional (nonprogressive) taxation. Sufficient taxation will induce a sharp transition towards a more equal society, and we argue that the critical taxation level is uniquely determined by the system geometry. Interestingly, we find that the roughness and Shannon entropy of the power distributions are a very useful complement to the standard measures of inequality, such as the Gini index and the Lorenz curve.