Cooperation between individuals is emergent in all parts of society; yet, mechanistic reasons for this emergence are ill understood in the literature. A specific example of this is insurance. Recent work has, though, shown that assuming the risk individuals face is proportional to their wealth and optimizing the time average growth rate rather than the ensemble average results in a non-zero-sum game, where both parties benefit from cooperation through insurance contracts. In a recent paper, Peters and Skjold present a simple agent-based model and show how, over time, agents that enter into such cooperatives outperform agents that do not. Here, we extend this work by restricting the possible connections between agents via a lattice network. Under these restrictions, we still find that all agents profit from cooperating through insurance. We, though, further find that clusters of poor and rich agents emerge endogenously on the two-dimensional map and that wealth inequalities persist for a long duration, consistent with the phenomenon known as the poverty trap. By tuning the parameters that control the risk levels, we simulate both highly advantageous and extremely risky gambles and show that despite the qualitative shift in the type of risk, the findings are consistent.

In this work, we present a model of agents who face a multiplicative stochastic gamble with reward $r$ and cost $c$ on their wealth. By allowing cooperation between nearest neighbors in a lattice network, we observe the emergence of rich and poor spatial clusters. Varying $r$ and $c$ shows that the clustering into poor neighborhoods only occurs for specific parameter values. Also, we show that the model has a tendency to keep rich agents rich and poor agents poor, similar to the well-known poverty trap observed in real societies.

## I. INTRODUCTION

Self-organization of individuals is observed in many systems and is a cornerstone of complexity sciences.^{1–4} Mutual cooperation between agents exists in many forms, one such being insurance. The insurance industry is one of the largest industries in the modern economy and has existed for centuries. Yet, we find that mainstream economics, which is rooted in expected value theory, does not provide a satisfactory reason for its emergence. As an alternative approach, Peters and Adamou argue that the system in which insurance takes place is more reminiscent of a multiplicative system (i.e., that the risk is proportional to the wealth), and note that the increments in such a system are not ergodic.^{5} Using time averages rather than ensemble averages, they show that there exist regimes in which insurance contracts are favorable for both buyer and seller, which calls for such contracts to be made. In a recent paper,^{6,7} Peters and Skjold explore this idea using a simple agent-based model, which shows that agents who cooperate via insurance contracts over time systematically outperform agents who do not.

Inspired by the 2D Ising model on a lattice and its many applications to socio-economic phenomena,^{8–18} we expand the model proposed in Refs. 6 and 7 to large ensembles on a lattice network. Now, each agent is placed in a neighborhood of other agents and can only cooperate with nearby agents, giving rise to the formation of spatial patterns in our model similar to Refs. 10 and 19.

In this paper, we first present a brief background of non-ergodicity in economics and illustrate the implication of broken ergodicity with a simple coin toss example before detailing the insurance paradox and its treatment in the context of ergodicity economics in Sec. II. Next, we describe the setup of our implementation of the agent-based insurance model on a network and the mathematical techniques used to analyze its results in Sec. III. Results for a specific parametric setting are presented in Sec. IV before we perform a parameter scan by varying the parameters $c$ and $r$ representing the relative costs or rewards of the risk in Sec. V, which reveals different clustering regimes in our model. Finally, we conclude with a summary and discussion of our work in Sec. VI.

## II. INTRODUCTION TO ERGODICITY ECONOMICS

In an ergodic system, an individual trajectory at different time steps $ x i( t 1),\u2026, x i( t n)$ has the same statistical properties as an ensemble observed at a single point in time $ x 1(t),\u2026, x n(t)$. This property makes many calculations easier, and thus, the ergodic hypothesis is often explicitly or implicitly assumed for many statistical methods. The core of ergodicity economics is a careful analysis of whether the ergodic hypothesis is true and, thus, makes an explicit distinction between ensemble averages $\u27e8x\u27e9= 1 N \u2211 i x i(t)$ and time averages $ x \xaf= 1 T \u2211 t x i(t)$ for, e.g., financial time series.^{20} Assuming ergodicity implies that $\u27e8x\u27e9= x \xaf$ in the limit of large $T$ and $N$. However, it is the exception rather than the norm that this is justified in real-world systems. A simple example of a non-ergodic system is the reinvesting coin toss gamble taken from Ref. 21.

### A. Reinvesting coin tosses

^{20}While almost all agents have negligible wealth, few have exponentially increasing wealth, which dominates the ensemble average.

### B. Insurance paradox

Consider a very fundamental and simple type of insurance, namely, an agent, $A$, faces a risk and another agent, $B$, offers to take over that risk in exchange for receiving a fee, $F$, typically known as the insurance premium. The classical economic analysis of the insurance problem is based on the expected value model, which posits that agent $A$ should accept to pay a fee lower than the expected value of the risk, whereas $B$ should demand a fee higher than the expected value of the risk. This anti-symmetry results in no fee where both agents simultaneously are satisfied, and thus, no contracts should be agreed upon. The orthodox solution is usually either that the agents must have an information asymmetry or that “risk aversion” makes the agents deviate from the supposedly optimal decision.^{22,23} We find this unsatisfactory as the former is unjustified ad hoc reasoning and the latter simply raises a new question as to why such risk aversion exits.

Assuming that the risk an agent faces is proportional to their wealth and optimizing time averages; however, Peters and Adamou show in Ref. 5 that there exist many situations in which agent $A$ is willing to pay a higher fee than agent $B$ demands. This framework provides a mechanistic reasoning for the existence of insurance contracts and shows that insurance is theoretically advantageous in the long run. These arguments are discussed in more detail in Refs. 5 and 6, and a broader scope of cooperation in non-ergodic systems can be found in Ref. 24. We highlight that behavioral experiments show evidence that humans are capable of heuristically optimizing the time average growth rate when faced with risky decisions in laboratory settings,^{25–27} which gives credibility to the time solution of the insurance puzzle.

## III. METHODS AND MODELS

### A. Insurance among agents

As we expand on the model from Refs. 6, 7, and 28, it is prudent to review its results alongside its model specifications. This model consists of $n$ agents $i=1,\u2026,n$, which are observed at time $t\u2208(0,T)$. Each agent’s wealth is initialized at $1$, $ w i(0)=1$ $\u2200i$, and in each time step, an agent $A$ with wealth $ w A(t)$ is randomly chosen to face a risky gamble: with probability $p$, $A$ either loses a relative amount of its wealth $c\u22c5 w A(t)$ or with probability $1\u2212p$ wins a relative amount $r\u22c5 w A(t)$. Note that the simulations in Refs. 6 and 7 use $c=0.95$, $r=0$, and $p=0.05$ as parameters, but that the qualitative results do not depend on the exact parameter choice.^{28}

When faced with a risky gamble, agent $A$ approaches another agent $B$ for an insurance contract. Agent $B$ will have a minimum fee, $ F min$, where they will accept to take over $A$’s risk and agent $A$ will have a maximum fee, $ F max$, they will accept to pay. If $ F min< F max$, there exist a range of fees in which signing the contract is mutually beneficial for the agents such that the risk and the fee is transferred from agent $A$ to agent $B$.^{29} To ensure the results are not conflated by the number of agents used, we distinguish between a time step, $t$, and a time unit, $ t u$, with a time unit being $ N 2$ time steps; i.e., each agent is on average chosen to face a risk ones per time unit. Note that the original model in Refs. 6 and 7 is well-mixed; i.e., all other agents are equally likely to be approached as agent $B$ without any spatial restriction. The network model presented in this paper lifts this assumption and restricts the agents’ interaction to a lattice network.

^{30}it follows that additive wealth increments are not ergodic, but logarithmic wealth increments are and, therefore, better indicate what happens over time and allow us to assume an underlying growth rate model $g(\u22c5)$.

^{5,21}Comparing agents’ growth rates with $ g A , B w i t h$ and without $ g A , B w i t h o u t$ signing an insurance contract at some fee $F$, we can calculate the maximum fee $ F max$ the agent facing the risk (agent $A$) is willing to pay, as well as the minimum fee $ F min$ another agent (agent $B$) is willing to accept. We, therefore, get that

^{31}

For suitable $ w A$ and $ w B$, $ F min< F max$ is fulfilled and therefore, an interval of fees $F$ in $[ F min, F max]$ exists for which insurance contracts are mutually beneficial for both agents. For simplicity, choose then the midpoint fee $F= 1 2( F min+ F max)$ to sign the contract. As shown in Fig. 3 of Ref. 6, the agents who sign contracts based on this criterion outperform agents who do not (for example, expected value optimizers). Interestingly, the simulations show how “large” agents, who act as the insurance company for all other agents, emerge. However, if such an agent becomes “too large,” they lose the ability for other agents to insure its large risk, and thus, its growth rate declines until it is no longer larger than the other agents.

### B. Simulation on a network

The model in Ref. 6 is deliberately simple but showcases two very interesting findings: First, all agents participating in the insurance contracts profit compared to the uninsured agents. Second, even though the cooperation reduces the overall wealth inequality compared to a model without insurance, it still endogenously creates large wealth differences, leading to the emergence of a large insurance-company-like agent.

However, the well-mixed setup in which all agents can approach each other is only realistic for small systems. Therefore, we propose to perform the simulation on a lattice and allow each agent to approach only its nearest neighbors to negotiate an insurance contract. The lattice simulation allows us to investigate the spatial clustering of agents based on relative wealth levels. We use a lattice coordination number of four and periodic boundary conditions such that the first and last agents of each row and column are treated as neighbors. Throughout the simulations, we use a square lattice with length $N=64$; i.e., $N\xd7N= 2 12$ agents.^{32}

### C. Clustering evaluation methods

#### 1. Temporal clustering

To evaluate the temporal clustering of agents, i.e., whether the rich stay rich and the poor stay poor, we use Spearman’s rank correlation. All agents are ordered in an enumeration $i=1,\u2026, N 2$, and we record their ranks $ \rho i( t u)$ as the $k$th richest agent for each time unit $ t u$. Spearman’s rank correlation is now the regular Pearson’s correlation between $\rho ( t u)$ and $\rho ( t u \u2032)$. For each time lag $\Delta t u$, multiple pairs $ t u\u2212 t u \u2032=\Delta t u$ exist, and therefore, we calculate Spearman’s correlation for all pairs with a given time lag and compute the mean and standard deviations for all of them. Interestingly, Spearman’s correlation and Pearson’s correlation of the logarithmic wealth values show almost the same results.

#### 2. Spatial clustering

^{33}the expected clustering coefficient for a purely random ensemble can be calculated as

An alternative way to evaluate the spatial clustering closer to the framework of Ising-like models is to compute the spatial autocorrelation. To this end, the ensemble matrix is transformed into a binary matrix **B** with $ B i j=1$ if a member of the group under consideration (i.e., the richest or poorest decile) is in the $i$th row of the $j$th column and $0$ otherwise. The autocorrelation matrix is then computed via the function *scipy.signal.correlate2d*^{34} by correlating **B** with itself and using the settings *mode* = *“full”,* *boundary* = *“wrap”,* *fillvalue* = *0*. This method computes the full discrete linear cross-correlation of **B** with itself as a two-dimensional array. For each offset $(k,l)$, it overlays the matrix **B** over itself with an offset of $(k,l)$. The value of the $(k,l)$th entry of the two-dimensional correlation matrix is then the sum of the element-wise products, which can be positive or negative and can be normalized to lie between $\u2212$1 and 1. Note that the boundary condition *boundary* = *“wrap”* means that if the matrix shifted by $(k,l)$ does not fully lie on the original matrix, the original matrix is periodically extended with its own entries.

## IV. RESULTS: TYPICAL OBSERVATIONS

### A. Temporal clustering

In Fig. 1, we show Spearman’s rank correlation, which reveals that even for large time differences, there is a great ranking correlation showcasing that there is a large memory in the agents’ wealth ranking. This means that the general wealth ranking is preserved for extended periods of time, i.e., that, for example, “large” agents stay as the “large” agents for long. This is similar to what is observed in Ref. 6 and can be related to the well-documented poverty trap observed in real societies.^{35,36}

### B. Spatial clustering

After $ T u=200$ time units, we observe a notable pattern when plotting the spatial distribution of the richest or poorest 10% of the agents (Fig. 2). Both deciles form small clusters of neighborhoods where they have mostly neighbors from within their own group. Using the clustering coefficient defined in Section III, we find that in the given parameter setting, both quantiles converge to the same level of clustering and far exceed the expectation for random matrices, indicating that a significant level of clustering emerges from the system’s dynamics (Fig. 3 left panel). Note that even within a neighborhood, the wealth levels can still be heterogeneous and vary wildly, even though they all fall in the same decile of the wealth distribution.

We further find that (after normalizing the autocorrelation matrix with its maximum) there are no noteworthy differences between the row-wise and column-wise autocorrelation (corresponding to the x and y directions). Both directions for both the top and bottom decile show a notably higher autocorrelation for the first nontrivial lag than the random matrix, thereby confirming the analysis of the clustering coefficients that rich and poor neighborhoods emerge in the system (Fig. 3 right panel).

## V. PARAMETER SCAN

To check if the clustering observed in Sec. IV is a general feature of the model or an artifact of the specific parameter configuration, we perform a parameter scan across the space of $(r,c)$. For constant $N=64$ and $ T u=200$, we vary $c$ across $0,0.05,\u2026,0.95$ and $r$ across $0,0.05,\u2026,2$ and calculate the clustering coefficients for each pairing. We show the results in the $(r,c)$ phase space in Fig. 4. By repeating the analysis with different values for $ T u$, we show that the qualitative behavior is robust, though with the clustering in both regimes slowly increasing with larger $ T u$.

For both the top and bottom decile, we find a gradual increase in clustering with $c$ and $r$ (Fig. 4). However, for the bottom decile, we find a much more pronounced behavior with a well-defined region of high clustering in the center of the phase space. We explore this further in Sec. V A, where we show that its borders (black lines on the right panel of Fig. 4) can be derived analytically and reflect the onset of degenerate values for the fees $ F min / max$.

To visualize how sudden the clustering emerges, we scan through a slice of the phase space at $r=0.5$ by varying $c$ and plot the autocorrelation function for the top and bottom decile. Normalizing the first nontrivial value of the ACF by its maximum $A C 1/A C 0$ reveals a well-defined maximum for the clustering of the bottom decile as shown in Fig. 5. On the contrary, the top decile (cf. Fig. 7 in Appendix A) only has a linear increase of $A C 1/A C 0$ with $c$ without any indications of a discontinuity. An exploration of the highly volatile regime in the top right corner of the phase space is also given in the appendix.

### A. Degenerate regimes

Insurance can be thought of as a special form of cooperation in the face of uncertainty. The model described in Refs. 5–7 includes a stochastic risk, which is entirely negative ( $c>0$, $r=0$), but the parameter scan in this paper explores a more general setup, where both $c\u22650$ and $r\u22650$ are varied. Hence, we also simulate regimes where the “risk” is favorable to agent $A$ ( $r\u226bc\u22730$). Though even in such a positive regime, growth rate optimization might incentivize an agent $A$ to rather accept a deterministic fixed payment instead of a fluctuating reward. Because this regime is reached by a simple parameter variation, it is not a qualitatively new model and can still be simulated and analyzed with the same methods as above. Because these regimes describe a different kind of cooperation in an uncertain environment than what we traditionally think of as “insurance,” we decided to call this a degenerate regime to stress that the model has adopted a new behavior. The question arises: how does this situation with beneficial risk parameters $r\u226bc\u22730$ affect the fees?

^{37}The condition $c=r$ is shown as a dashed line in the right panel of Fig. 4.

As seen in Fig. 4 (left panel), there is no immediate connection between the degeneracy regimes and the clustering in the top decile. However, for the poorest decile (Fig. 4, right panel), the degeneracy lines approximate the contours of the regime of high clustering. Clustering into poor neighborhoods, thus, happens if $c\u22651\u2212 1 1 + r$ (9) and if $r>c$ (10) under the condition that $ w B\u226b w A$.

## VI. DISCUSSION

### A. Summary

The model presented in this paper is an extension of the agent-based model presented in Refs. 6 and 7. We find that despite restricting the agents’ ability to make contracts to their nearest neighbors on a spatially restricted grid, the overall conclusion from Ref. 6, i.e., that insurance is advantageous in the long run for all agents, still holds. However, we find that the general wealth ranking is preserved for shorter periods of time, indicating that not having spatial constraints leads to what can be thought of as monopoly-like structures, consistent with tendencies in society with increasingly global access through the Internet. We, though, highlight that the memory in the system with spatial constraints is still high, which is consistent with the well-documented poverty trap observed in real societies.^{35,36} We further find that this restriction leads to a rich phenomenology of spatial cluster formations, again similar to what we see in the real world. While the clustering of the wealthiest agents increases continuously for larger parameters $r$ and $c$, we find that the poorest agents only exhibit high clustering for particular combinations of $(r,c)$ and that this regime can approximated by the degeneracy lines.

### B. Model relevance

While including spatial constraints to the insurance model presented in Ref. 6 is an attempt to explore a more realistic system, we recognize that the system is still a very simplified representation of the world. A question we have been faced with is why we do not include specific insurance companies rather than having the insurer simply be another agent in the system. This admittedly is closer to the real world. However, this is an unnecessary restriction to the model: Even if a large insurance company exist, one still has to rely either on the non-satisfactory solutions discussed earlier or on the time solution provided here to solve the insurance paradox, and as the argument often is that such an insurance company can offer contracts as expected value (either through having such high wealth that the time average of the risk approaches the expected value or through having access to the actual ensemble), cooperation through insurance would only be stronger than what we present. We, therefore, see our model as more general.

### C. Relation to classical cooperation

This form of insurance can be considered a restricted form of cooperation. Although the setup on the lattice network restricts the agents in our model, the cooperation is still beneficial for them, which is in line with previous findings. Studies on evolutionary game theory have shown that cooperative strategies can become dominant over the defectors’ strategy purely due to the effect of fluctuations on the reward.^{38,39} This has also been studied within the time average framework, where cooperation is similarly found to be stable, i.e., defecting is not advantageous,^{40} which has been backed by empirical studies.^{41–43} An important note, however, on this pure form of cooperation is that trust does play a role. Despite the fact that full cooperation is mathematically stable and defecting can be shown not to be advantageous, this is only in the long run; if you give a large part of your wealth to someone in an attempt to cooperate and they decide to defect, it is, of course, not advantageous for you. This is not a consideration in our model, as both agents improve their growth rate in every time step when they sign a contract and, therefore, need not worry about what the other agent does in the future: taking the deal is good regardless. Whether there is a path from non-cooperation through insurance-like cooperation to full cooperation is the subject of ongoing research. Expanding the neighborhood of potential insurance partners in an exploratory analysis indicates only minor effects on the median wealth (Fig. 6).

### D. Spatial clustering

The main feature of this model compared to Ref. 6 is the study of spatial cluster formations. Whenever the volatility (i.e., the variation in the gamble’s payoff) is high enough, the top decile of agents tends to cluster into neighborhoods, while the bottom decile displays a similar behavior in a specific parameter regime. The emergence of rich and poor neighborhoods might superficially resemble kin selection.^{44} However, such a mechanism is not present in our model’s algorithm. Instead, the clustering emerges endogenously: Rich agents do not choose to connect with other rich agents but rather become and stay rich because there are other rich agents in their neighborhoods. Indeed, the original model already shows that a rich agent eventually collapses if no other agent is rich enough to insure their risk.^{6,7,45} Hence, with our spatial constraint and whenever the volatility is high enough, only agents with other rich neighbors have a chance to stay rich in the long run. This phenomenon might be interpreted as a special case of network reciprocity described by Nowak^{44} and is not restricted to a narrow regime in the phase space, but a general feature that emerges whenever the volatility (quantified by $c$ and $r$) is high.

### E. Persistent inequality

Real economies have a strong memory effect; i.e., rich people tend to stay rich and poor people tend to stay poor and remain in the so-called poverty trap. The well-mixed and network model both have such memory effects as shown in Fig. 1. We find that the inequality memory quantified via the Spearman correlation is higher in the well-mixed model from Ref. 6 than in the spatially constrained network model. This is an interesting finding when we compare it to the two theories for the emergence of poverty traps discussed in Ref. 35: scarcity-driven and friction-driven. The scarcity-driven poverty traps describe a different behavior of agents under the pressure of extreme scarcity, whereas the friction-driven poverty traps describe situations in which market inefficiencies or different initial conditions lead agents with identical decision criteria to different wealth trajectories.

Interestingly, our findings seem to contradict the friction-driven theory of poverty traps: Including spatial constraints clearly a market inefficiency, but we find that the inequality persistence of the system is greater without it. We suspect that this can be explained by monopoly-like structures, which emerge in the well-mixed system but are not possible with the spatial constraint. We, though, highlight that in both our and the well-mixed system, there are no inefficiencies in forming contracts (such as administration costs), nor any agents acting in bad faith (such as agents demanding higher prices than based on their own assessment), which might change these results.

### F. Generalization of the original model gives rise to negative fees

Another extension of Ref. 6 is to include a positive outcome of the gamble with reward $r>0$ in addition to the harmful cost $c$. We show that this can lead to regimes where it might be mutually beneficial to have $F\u22640$; i.e., the “insurer” pays an amount to take over the risk. The phase space spanned by the two parameters $(r,c)$ in Fig. 4 shows that the bottom quantile’s clustering corresponds to the two degeneracy lines [Eqs. (9) and (10)] derived in Section V A. Equation (9) marks the regime at which $ F max<0$ and the time average growth rate of the gamble becomes neutral, while Eq. (10) indicates the regime where the ensemble average becomes neutral and, under the condition $ w A\u226a w B$, $ F min$ can become negative. It might be tempting to dismiss the regimes with negative fees as beyond the scope of an insurance model. However, insurance is just a special case of cooperation in the face of uncertainty and, as will be discussed at the end of this section, this degenerate regime can be interpreted as a relationship between employee $A$ and employer $B$. Our analysis shows that there are no qualitative differences or irregularities in the model behavior in the different regimes:

First, the condition $ F min< F max$ is still a meaningful model of reality. Consider $ F min<0< F max$ as discussed above. The negative $ F min<0$ of agent $B$ means that they want to acquire the risk, while the positive $ F max>0$ of agent $A$ means that they want to get rid of it. In this scenario, a transfer at, e.g., $F=0$ (fulfilling $ F min<F=0< F max$) is beneficial for both of them. Alternatively, if both $ F min< F max<0$, then one can more easily understand the setup by reversing the signs and considering the price $ P B=\u2212 F min$ that agent $B$ is willing to pay and $ P A=\u2212 F max$ that agent $A$ demands. Now, $0< P A< P B$ and agent $B$ is willing to match agent $A$’s demanded price. The interested reader can play around with the parameters for $ F min<0< F max$ and $ F min< F max<0$ to verify that the condition $ F min< F max$ still yields sensible results.

Second, the regime of high clustering is characterized by $ F max>0$ and the possibility that $ F min<0$ if $ w A\u226a w B$, i.e., if $B$ is much wealthier than $A$, it will consider the gamble (whose reward and cost are relative to $A$’s wealth) so advantageous that $B$ offers $A$ money ( $ F min<0$) to reap the potential rewards. At the same time, the risk $c$ is so large relative to $A$’s wealth that $A$ is willing to pay money in order to get rid of the risk. This is just a more extreme version of the risk assessment $0< F min< F max$ in the non-degenerate regime and illustrates that purely by considering time averages, the agents can come to mutually beneficial deals without assuming the existence of any subjective utility functions or subjective risk assessment.^{5} Both agents operate under the same rules, but the time average considerations lead them to different risk assessments based on their individual wealth $ w A / B$.

Third, if $ F min< F max<0$, one can rethink this setup as modeling employment rather than insurance contracts. While insurances are used to replace $A$’s overall detrimental risk with a fixed payment from $A$ to $B$, one can think about employment in similar terms. Now, $A$ can work as a self-sustained freelancer and, while being mostly profitable, has varying success. Alternatively, $A$ can be hired by another agent $B$ and give $B$ the (varying) rewards of its work while, in return, getting a fixed income (a negative fee $F<0$ corresponds to $A$ getting $\u2212F$ from $B$). This regime is considered in more detail in Ref. 46 but arises naturally from our model’s parameter scan as another regime of cooperation in the face of uncertainty.

### G. Further work

We consider this a first attempt to generalize the model presented in Ref. 6. First, we find qualitatively similar results on the benefits in the long run of evaluating the risk using time average growth rates, which leaves the question of whether any non-trivial network structures lead to qualitatively different results open. Second, our spatial constraint reveals a decreased memory in wealth ranking. The question now is: How much does the temporal autocorrelation change if the neighborhood of the agents is expanded or if the network structure is changed? And are there any configurations that lead to qualitatively different results? Last, we recognize that consumption is an important feature in real systems.^{47} This leads to the possibility of bankruptcy, which in this model is equal to being erased from the system. We are aware of no solutions for dealing with such a problem on a non-*ad hoc* basis, but we would be interested in discussing possible solutions. In general, we hope this paper will inspire researchers to explore the insurance problem further through the lens of time average considerations.

## ACKNOWLEDGMENTS

The authors thank the organizers and participants of the Ergodicity Economics 2024 conference as well as the anonymous referees for their valuable feedback and discussions. Tobias Wand is supported by the Studienstiftung des deutschen Volkes (German Academic Scholarship Foundation). Benjamin Skjold is supported by a Novo Nordisk Exploratory Interdisciplinary Synergy grant (Ref. No. NNF20OC0064869).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Tobias Wand:** Conceptualization (lead); Formal analysis (lead); Investigation (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (equal). **Oliver Kamps:** Supervision (equal); Writing – review & editing (equal). **Benjamin Skjold:** Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study. The simulation code is available via Ref. 32.

### APPENDIX A: ACF FOR THE RICHEST DECILE

### APPENDIX B: HIGHLY VOLATILE REGIME

The phase space in Fig. 4 reveals that particularly high clustering for both the top and bottom quantile can be found in a region of high $c$ and high $r$, i.e., where the volatility of the gamble is very large in both directions. While the low-volatile regime transitions into the deterministic case, it is not obvious how the model behaves in the high-volatile regime. To investigate this, we explore a highly volatile system with $c=0.95$ and $r=2$.

In Fig. 8, left panel, we show the richest (white), middle (gray), and poorest (black) third of the agents and the ACF (right panel) of the richest and poorest thirds. We see that the ACF for both have fat tails compared to the ACF of a random ensemble, reflecting the high degree of clustering seen in the left panel. The formation of large, macroscopic neighborhood areas for both the richest and poorest third is clearly visible in this plot, whereas the middle third mostly forms a thin boundary layer between the two extremes instead of manifesting into large-scale neighborhoods. Hence, the neighborhood map shows high polarization, reminiscent of the magnetic domains in ferromagnets. This superficial similarity should not be taken too seriously, though. The magnetic domains form because of energetic considerations of the macroscopic magnetic field, whereas our model and the Ising model are microscopic descriptions.

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