\documentclass[11pt]{article}
\usepackage[top=2.5cm, bottom=2.5cm, left=2.5cm, right=2.5cm]{geometry}
\usepackage[english]{babel}
\usepackage{graphicx}

\usepackage{sectsty}
\chaptertitlefont{\huge}

\usepackage[combine]{ucs} % For German characters in references (solidarität)
\usepackage[super]{nth} % Creates superscripts for numbers (e.g. 1st 2nd)
%\usepackage[utf8x]{inputenc}
\usepackage{pdflscape} % Allow landscape pages
\usepackage{afterpage} % For correlation table
\usepackage{enumitem}
\usepackage[singlelinecheck=false, % Overwrite latex default of centering single line table titles
labelfont = sl, 
labelsep = period]{caption} 
\usepackage[bottom]{footmisc} % allow footnotes on bottom of page
\usepackage{endnotes}
%	\let\footnote=\endnote

% Blind footnotes (without number) for chapter footnotes
\newcommand\blfootnote[1]{%
	\begingroup
	\renewcommand\thefootnote{}\footnote{#1}%
	\addtocounter{footnote}{-1}%
	\endgroup
}

\usepackage{authblk} % Author notes and affiliations
\usepackage{setspace}
\doublespacing % or singlespacing/doublespacing
\usepackage{eurosym}
\usepackage{gensymb} % degree symbol

\usepackage[fleqn]{amsmath}
\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu} % in math mode
\newcommand{\textoverline}[1]{$\overline{\mbox{#1}}$} % in text


% Tables and figures
\usepackage[section]{placeins} % tables and figures within section
%\usepackage{siunitx}
\usepackage{dcolumn} % align on decimal marker
\newcolumntype{Z}{D{.}{.}{3}} % round at 3 decimals
\newcolumntype{K}{D{,}{,}{5}} % comma as decimal marker
\newcolumntype{Y}{D{.}{.}{-2}} % round at 2 decimals
\newcolumntype{b}{X}
\newcolumntype{s}{>{\hsize=.5\hsize}X}
\newcolumntype{t}{>{\hsize=.5\hsize}s}
\newcolumntype{L}{>{\raggedright\arraybackslash}X}
\newcolumntype{Q}{>{\hsize=.5\hsize}L}
\newcolumntype{R}{>{\hsize=.75\hsize}L}	
%\newcolumntype{n}{>{\hsize=.5\hsize}m}
%\usetikzlibrary{arrows.meta, decorations.pathreplacing, shadows}
\usepackage{threeparttable}
\usepackage{longtable}
\usepackage{tabularx}
\usepackage{multirow} % Make cell span multiple rows
\usepackage{makecell} % Multiple lines in cell
\usepackage{stackengine} % change row height
\newcommand\xrowht[2][0]{\addstackgap[.5\dimexpr#2\relax]{\vphantom{#1}}}
\usepackage{colortbl} % Color cells in table
\usepackage{xcolor} % allows text in different colors
\definecolor{rpurple}{RGB}{68, 1, 84}
\definecolor{rgreen}{RGB}{39, 126, 142}
\definecolor{ryellow}{RGB}{154, 217, 60}
\definecolor{expred}{RGB}{222, 22, 22}
\definecolor{expgreen}{RGB}{76, 175, 80}
\definecolor{hglgrey}{RGB}{89,89,89}
\definecolor{hghgrey}{RGB}{140,140,140}
\definecolor{gray70}{RGB}{179,179,179}
\definecolor{gray45}{RGB}{115,115,115}
\definecolor{gray15}{RGB}{38,38,38}
\usepackage{soul} %highlight
\usepackage{pgfplots}
\usepackage{subfig} % subfigures


\usepackage{geometry}
\raggedbottom % No extra space between paragraphs
\usepackage[thmmarks, amsmath, thref]{ntheorem}
\makeatletter %

% References
%\usepackage[pagebackref=true]{hyperref}
\usepackage[hidelinks]{hyperref}
\usepackage{cleveref}
\urlstyle{rm}


\usepackage[nottoc,numbib]{tocbibind} % Bibliography as chapter
\usepackage[nodoi,mask,nosectionbib]{apacite}
\renewcommand\refname{\bf References}
%\let\cite\shortcite % >3 authors immediately abbreviated to et al
\let\citeA\shortciteA
\let\citeauthor\shortciteauthor
%\makeatletter
%\renewcommand{\APACbVolEdTR}[2]{%
%	\ifx\@empty#1\@empty
%	\ifx\@empty#2\@empty
%	\else%
%	\unskip\hbox{}%            (Technical Report No.\ <no>)
%	\fi%
%	\else%
%	({#1}%               (2nd ed., Vol.~1
%	\ifx\@empty#2\@empty%
%	\else%
%	\unskip; {#2}%     ; Technical Report No.\ <no>
%	\fi%
%	)%                   Final parenthesis.
%	\fi%
%}
%\makeatother
\bibliographystyle{apacite}

% Hypotheses
\newtheoremstyle{hypotheses}%
{\item[\hskip\labelsep \theorem@headerfont ##1##2\theorem@separator]}%
{\item[\hskip\labelsep \theorem@headerfont ##1##2\ (##3)\theorem@separator]}
\makeatother

\theoremstyle{hypotheses}
\theoremseparator{:}
\theoremindent1\parindent
\newtheorem{hyp}{Hypothesis } 
%\numberwithin{hyp}{chapter}
\newtheorem{ass}{Assumption }
%\numberwithin{ass}{chapter}

\usepackage{etoolbox}
%\patchcmd{\chapter}{plain}{empty}{}{} % Don't number chapter title pages	

%\usepackage{minted}
%	\usemintedstyle[netlogo]{Nlogo}
\usepackage{listings}

\lstset{ 
	basicstyle=\footnotesize,
	%comment={;}
	morecomment=[l][\color{gray45}];
	%commentstyle=\color{gray}
}


\begin{document}

\title{Sharing Risk under Heterogeneity: Exploring Participation Patterns in Situations of Incomplete Information
}

%\titlerunning{Short form of title}        % if too long for running head

\author{ }

%\authorrunning{Short form of author list} % if too long for running head


\maketitle

\begin{abstract}
Motivated by the emergence of new Peer-to-Peer insurance organizations in many countries globally (see, e.g., Friendsurance in Germany and Broodfonds in the Netherlands), we propose a model of individual decision-making in risk-sharing arrangements with risk heterogeneity and incomplete information about the risk distribution as core features. The model puts forward participation as a utility maximizing alternative for agents with higher risk levels, who are more risk averse, are driven more by solidarity motives, and less susceptible to day-to-day fluctuations. We use this basic micro-level model to simulate decision-making by agent populations in an interdependent setting. The result is a dynamic behavioral model where one agent's decision to participate depends on the characteristics and decisions of other agents in the risk-sharing group. By varying parameter settings with respect to need heterogeneity and group size on the population level, and risk aversion, solidarity, and susceptibility to fluctuations on the agent level, we predict the resilience or decay of participation. Simulation results show that successful risk-sharing arrangements are less likely in more heterogeneous populations, as alternative factors (e.g., risk aversion) can less often make up for the larger cost deficiencies. At the same time, more heterogeneous groups deal better with uncertainty and temporary cost fluctuations than more homogeneous populations do.

{\noindent \bf Keywords:} Risk-sharing; Mutualism; Support; Uncertainty; Heterogeneity; Adverse selection

%{\noindent \bf JEL classification:} C63; D81; G22; O12
%{\noindent \bf JEL classification:} D81 Criteria for Decision-Making under Risk and Uncertainty; \and C63 Computational Techniques  \and Simulation Modeling; \and 	G22	Insurance, Insurance Companies, Actuarial Studies; \and O12 Microeconomic Analyses of Economic Development; 
% \subclass{MSC code1 \and MSC code2 \and more}
% D81 Criteria for Decision-Making under Risk and Uncertainty 
% C63 Computational Techniques • Simulation Modeling 
\end{abstract}







\section{Introduction} \label{intro}
%%%% REF BLINDED --> CHANGE TO \citep{Vriens2020}
In recent years many new insurance initiatives such as Friendsurance (Germany) and Broodfonds (the Netherlands) were launched in countries that traditionally have strong insurance systems. These initiatives, which often refer to themselves as Peer-to-Peer (P2P) insurance, aim to rethink how insurance is organized \cite{Abdikerimova2019}. Inspired by mutual insurance associations (mutuals) from the 19\textsuperscript{th} century, they go back to insurance in its most basic form: fair, transparent, and social \cite{Vriens2020}. Rather than placing responsibility and trust with corporate insurers, they return responsibility and trust to the policyholders by having them arrange their own safety net within small risk-sharing groups.\footnote{The new insurance initiatives cover a variety of things, such as de deductable excess of, e.g., liability insurance (Friendsurance) or an income replacement for self-employed people in case of sickness (Broodfonds). They act as umbrella organizations, with their members (or policyholders) subdivided into smaller risk-sharing groups (of 10, 50, or up to a few hundred members).} 

Other than regular insurance, which is paid in fixed premiums or through salary taxation, the participation costs in mutuals relying on risk-sharing groups fluctuate depending on how many or how often group members request support. When more is saved through contributions than needed for payouts to members in need of support, (a share of) the fund's reserves get redistributed among all members \cite{Breer2015}. Thus, while participants pay a contribution fee to build the insurance fund, the actual participation costs are determined by the number of claims. The lower the number of claims, the lower the costs for all participants. This introduces an interdependency between group members in the sense that one person's actions (support requests) directly affect the costs of participation for others. Under the basic assumption that costs mainly drive willingness to participate, it follows that willingness to participate likewise varies over time, thereby making participation in this type of risk-sharing insurance organizations more fragile. Members could at any time decide to revoke their membership if they no longer want to pay to help others (e.g., because more support is requested than expected; \citeNP{Platteau1997}), and if they do, they take from the common fund the share of their contributions not spent on payouts. 

With this in mind, it is important to deepen our understanding of why people participate in these new risk-sharing arrangements and when participation can be resilient to cost fluctuations. While many scholars have developed and refined risk-sharing models \cite{Coate1993,Genicot2003,Kimball1988,Ligon2002}, these models generally assume homogenous risk probabilities or---if the homogeneity assumption is relaxed---complete information on the distribution of risk in the population. In these models, a rational agent bases membership on the comparison between one's own risk probability and that of other participants, which is constant over time as long as risks do not change. However, a heterogeneous model assuming complete information is not only very constraining in terms of agents' cognitive abilities, it is also unrealistic that agents would have such detailed information about all other agents in the population beforehand. Finally, and perhaps most importantly, it comes with the hidden assumption that agents are not affected by realized outcomes (people requesting support), which is unlikely to hold in practice.  

Relaxing the assumption of homogeneity is of utmost importance if we want to understand real-world risk-sharing dynamics. From data collected among 230 Dutch Broodfonds groups \cite{Vriens2017a}, for instance, we know that there is substantial risk heterogeneity within these groups. And these groups are no exception. Risk-sharing has been the most common method of creating security among (rural) populations worldwide---see, e.g., \citeauthor{Lemay-Boucher2009} (2009) for examples in sub-Saharan Africa; \citeauthor{Ligon2002} (2002) on India; and \citeauthor{Fafchamps2003} (2003) on Southeast Asia. It is highly unlikely that in so many natural settings the created groups are (or have been) homogeneous in terms of risk. It is therefore of utmost importance that our theoretical ideas of when and why people engage in risk-sharing relations account for risk heterogeneity as well.

Luckily, complete information is not a necessary requirement for the realization of heterogeneous risk-sharing arrangements. As long as people assume risk levels to be approximately similar, or that the existing risk heterogeneity is somehow compensated for (e.g., because the low-risk participant is risk averse), risk-sharing arrangements can still arise \cite{Skogh2005}. We propose a model where agents know their own risk but not the true risk distribution. They base their decision-making on a personal belief about the group-level average. Based on new experiences (i.e., group members requesting support), they update this belief over time. This alternative approach combines forward-looking decision-making with backward-looking learning as a tool to deal with decisions under uncertainty. The result is that as more support is requested in some rounds than in others, the estimate of the average risk, and therefore the expected utility of participating, also fluctuates over time. Hence, where a complete information model may soon land on an equilibrium, the alternative model continues to update, thus slowing down the stabilization process. On the plus side, it may lead to larger number of people joining in the first place (and thus prevent failure from the start). On the downside, it may increase the possibility of withdrawal cascades, where agents follow each other in opting out of the risk-sharing arrangement while they may not have, had they known the true distribution.

What are the implications of this alternative approach for the effect of risk heterogeneity? The general consensus for any voluntary risk-sharing arrangement is that heterogeneity introduces adverse selection \cite{Arrow1984}: Those with the lowest risk profiles will opt out of the arrangement, followed by the second-lowest risk profiles, and so on---until either the entire risk-sharing arrangement fails or only high risks remain. %While dropout of low-risk participants can to some extent be prevented if they score higher on factors like risk aversion and solidarity \cite{Vogt2004}, under the same average risk probability a more heterogeneous group has more low-risk agents with a large(r) difference between their own risk and the group's average, and therefore a larger proportion of agents at risk of dropping out. That is, the larger the heterogeneity, the larger these alternative factors have to be to compensate.
This is, however, based on the assumption that the risk distribution is common knowledge. For our incomplete information model, the basic intuition is that participation patterns are more likely to be resilient when the agents that are part of the risk-sharing arrangement can cope with cost fluctuations. That is, when the difference in the utilities of participating and not participating is big enough that it holds even under temporary increases in the number of support requests. After all, an increase in support requests increases the estimate of the group's average risk, which decreases the utility of participation. Hence, on the one hand incomplete information may decrease the negative effect of heterogeneity (as long as people assume that others' risk levels will be similar; \citeNP{Skogh2005}). On the other hand, the negative effect may also be strengthened when, if fluctuations in support requests are more extreme in more heterogeneous group, the estimate about the group's average risk exceeds the true probability level. 

%The aims of this article are thus to investigate
The aims of this chapter are thus to investigate theoretically the extent to which (1) risk-sharing groups are indeed less likely to succeed when agents do not have complete information; (2) the reliance on realized support requests increases the chances of generating withdrawal cascades particularly in more heterogeneous groups; and (3) what levels of alternative individual factors (risk aversion, solidarity, and reinforcement learning) are needed to obtain a safe bandwidth that realizes participation patterns even under larger fluctuations in group members' claims. We introduce these additional individual factors, for otherwise participation cannot be explained for everyone whose personal risk is smaller than the (estimated) group average.

Risk aversion, firstly, reflects a preference for certain outcomes over risky situations in which outcomes are uncertain. It is deemed crucial for understanding why people take out insurances \cite{Arrow1984}, but also for participation in micro-insurance and helping arrangements \cite{Platteau2017,Vogt2004}. Solidarity, secondly, implies prosocial behavior towards members of the same group \cite{Baldassarri2015} from which personal utility (a good feeling) is derived as well \cite{Gintis2005}. The risk-sharing groups introduce many-to-many relationships that are argued to invoke solidarity, which makes people willing to (unconditionally) pay for the insurance of other group members.\footnote{Another role of solidarity, albeit not the focus of the current chapter, 
	%Another role of solidarity, albeit not the focus of the current article,
	is that it prevents excessive (fraudulent) insurance claims (i.e., it reduces moral hazard; \citeNP{VanLeeuwen2016b}), because the small risk-sharing groups make apparent that these payouts would be paid directly out of the pockets of the other group members.} As a final factor, we consider how people let current experiences influence (or override) their previous belief about the group's average risk. Based on reinforcement learning \cite{Macy2002a}, those who do so to a larger extent are more susceptible to fluctuations in support requests and therefore more likely to withdraw when costs unexpectedly (temporarily) increase. 

The assumption of incomplete information and model feature of agents updating their beliefs over time introduce new, dynamic interdependencies between agents. It means we cannot produce generalized, formally derived predictions of how agents behave under all circumstances \cite{Bianchi2015a}. After all, if (some) agents drop out because their estimate of the average risk increases, others---initially interested in participating---might follow because the total number of members decreased. Hence, the group-level outcome is more than the aggregation of individual attributes \cite{Macy2002}. We therefore rely on agent-based modelling techniques to derive predictions about agent behavior and group-level outcomes under irreducible heterogeneity for a variety of parameter combinations.

In the following sections, we take several steps to study the success of risk-sharing arrangements theoretically. \Cref{sec.lit} provides an overview of relevant literature on this topic as developed in different fields. In \Cref{sec.model}, we combine these insights in the construction of a static \textit{N}-person Risk-Sharing Model (RSM). In \Cref{sec.sim}, we build, through simulations, a dynamic, interdependent RSM to study how willingness to participate is affected by stochastic fluctuations in support requests and changes in other members' behavior. \Cref{sec.res} presents the simulation results and \Cref{sec.impl} discusses the implications from this model in light of the new mutuals and derives testable hypotheses for future research. \Cref{sec.conc} summarizes and concludes.

\section{Literature Review} \label{sec.lit}

We focus on studies that approach participation in risk-sharing arrangements as a special type of social dilemma. Other than with regular public good provision, for sharing risk there is (potentially) a long delay between contributing and actually obtaining the benefit of cooperation (support in times of need). In fact, when talking about sharing risk, ideally one never actually needs the support from the group \cite{Platteau1997}. This introduces the social dilemma, because the short-term benefit of withdrawing to avoid paying the costs to help others may easily outweigh the uncertainty surrounding the long-term benefit of perhaps needing and receiving the security oneself someday \cite{Fafchamps1992}. Moreover, one-time individually rational decisions to withdraw (e.g., because of a sudden increase in support requests by fellow group members) may translate into a collectively worse outcome where no one is insured \cite{Coate1993}.

This classifies risk-sharing arrangements as a cooperation problem under uncertainty. Participation is uncertain not only because it is unknown whether one, as an individual, ever needs a payout from the collective fund. It also derives from not knowing how many others (simultaneously) need support and whether they remain a member to pay the costs to support others \cite{Vriens2019}. We will review the main modelling approaches that take this uncertainty into account, be it from the perspective of mutual insurance, risk-sharing networks, or support games, with a special interest in whether and how these models approach heterogeneity in risk. Our review includes both formal analytical (game-theoretic) models and agent-based simulations and uses findings from empirical studies (both lab and field-based) whenever they guided theoretical refinement. 

Subsequently, we briefly address the status quo on modelling our key compensation factors of interest. While risk aversion has a central role in all models related to mutual insurance, solidarity (or prosocial motives generally) and reinforcement learning do not. For these, we base ourselves on other decision-making models.

\subsection{Risk-Sharing Models}

Theory development on risk sharing started after the seminal work of \citeA{Posner1980}, who combined Scott's \citeyear{Scott1977} descriptions of mutual insurance based on solidarity networks and reciprocity with Popkin's \citeyear{Popkin1979} counterexamples of opportunistic behavior. His argument, that an informal risk-sharing arrangement can be sustained over time when lasting relationships develop between self-interested members, was formalized by \citeA{Kimball1988} and \citeA{Coate1993}. Their models form the starting point for subsequent model building of mutuals and risk-sharing agreements.

The central question in these first models was under what conditions self-interested agents will enter an informal risk-sharing arrangement voluntarily \textit{ex ante} without defecting \textit{ex post}. The set-up is one where agents do not pay an initial contribution, but only divide the available resources after someone indeed needs support. \textit{Ex post} defection thus means that agents refuse to share their payoff if they end up with the higher one. The models start from a basic assumption that players are homogeneous in their risk probability: Each player is equally likely to end up with a high or low income. For an infinitely repeated game, the theoretical optimum entails full income pooling and equally sharing the aggregate available resources each period. This can, as long as the gain from defection is small, be achieved regardless of group size. 

Empirical tests refute this prediction, observing small groups, partial sharing of risk, and less than full insurance instead \cite{Fafchamps2003,Murgai2002,Townsend1994,Udry1994}. To accommodate these findings, alternative models predict cooperation rates when participation only requires limited commitment \cite{Ligon2002}, when cooperation should be robust not only to individual deviations but also to deviations by subgroups \cite{Genicot2003}, when arrangements are made in networked, rather than group settings \cite{Bloch2008}, or when participation can take place through threshold models (i.e., not all agents have to join in round 1; \citeNP{Breer2015}). 

None of these models, however, start from an assumption of risk heterogeneity.\footnote{While \citeA{Breer2015} ultimately do relax the assumption of risk homogeneity, they only derive from it that participants whose risk is higher than the group average are more likely to join, not under which conditions those whose risk is lower than the group average still might.} For predictions resulting explicitly from risk heterogeneity, we can borrow from the insights obtained from so-called support games and solidarity games. These games distil conditions under which players \textit{ex ante} agree to help others if by some probability they might be the one who needs the other's help \cite{Hegselmann1994,Vogt2004}. They predict an optimal support relation under homogeneity, but show that cooperation is likewise possible under heterogeneity in needing support, for instance when (low-risk) players are risk averse. 

These dynamics have been corroborated in lab experiments. While \citeA{Tausch2014} find support for adverse selection in experiments with heterogeneity in risk levels, \citeA{Vogt2006} find that this can be compensated for by risk aversion. Simulation studies that extend this model to N-person settings predict that when players endogenously choose with whom to engage in (dyadic) support relations, stable support relations arise under heterogeneity in needing support as long as heterogeneity is modest and risk probabilities are average \cite{Hegselmann1998}. While these results emerge in N-person settings of multiple one-on-one support relations, it suggests that cooperation in N-person risk-sharing groups can, to a certain degree and under certain conditions, be successful in heterogeneous groups as well.

%To conclude, while we can build on a large tradition of risk-sharing models, our current aims do deviate on several aspects. The main deviation from the classic models \cite{Coate1993,Kimball1988} is that we do not assume homogeneous risks. Compared to the models that do include heterogeneity \cite{Hegselmann1998,Vogt2006}, we relax the assumption of complete information about the risk distribution, but include uncertainty about the risk distribution instead. This means that where the classic models aim to solve what the most efficient version of sharing risk is (i.e., what share of the income agents would be willing to pool after someone indeed needs support), we are interested in predicting when participants are willing to participate at all. We therefore refrain from making comparisons in differences in income, potential loss, or the share to be insured and use instead a fixed contribution parameter that is used to build an insurance fund regardless of realized support \cite<cf.>{Breer2015}. If by some independent and identically distributed (i.i.d.) probability an agent loses their income (which could, in the broader context of risk-sharing applications, also reflect failed harvests, a broken product, stolen goods, poor health, etc.), they receive a fixed benefit from this fund to cover (some share of) the loss. To capture the collective good aspect, we model a redistribution parameter such that the part of the common fund that is not used for benefit payouts is shared over all participants. 

\subsection{Risk Aversion}

Central to sharing risk is the assumption that agents are risk averse, operationalized using a concave utility function. Applied to an insurance context, it means agents are willing to incur larger insurance costs today to ascertain an income in the future \cite{Arrow1984}. For risk-sharing settings, however, some scholars have questioned the validity of the assumption, primarily due to the low participation rates that are observed in field studies (see \citeNP{Platteau2017}, for a review). They argue that risk-sharing arrangements might not solve the uncertainty problem, because they introduce additional uncertainty with respect to how many other participants will need support, whether the fund is sufficient to meet these requests, and whether others remain a member after receiving support to reciprocate the favor in the future.

Several alternative assumptions have been suggested, such as loss aversion \cite{Kahneman1979}, hyperbolic discounting of future utility \cite{Platteau2017}, or ambiguity aversion \cite{Elabed2015}. Following this line of reasoning, some models (e.g. \citeNP{Dercon2014}) have posed additional assumptions of prudence and temperance, which help selection of the most preferred alternative in situations when there is more than one source of uncertainty. 

Despite these advances in the use of more sophisticated risk aversion measures in recent studies, we start from a plain concave utility function in which we assume players to be non-satiated and risk averse. Extra differentiation, such as loss aversion \cite{Kahneman1979} or ambiguity aversion \cite{Elabed2015}, could be introduced at a later stage, but given the substantial number of parameters to be modelled, we keep the model simple in this regard.

\subsection{Solidarity}

A plethora of evidence, both lab and field based \cite{Andreoni2002,Camerer2003a,Chaudhuri2011,Fehr2002,Ledyard1995,Ostrom1990} signals that cooperation in social dilemmas extends far beyond basic self-interested, reciprocal, and reputation-based cooperation. The number of alternative explanations that have been proposed is likewise towering, with other-regarding preferences interpreted, among others, as encompassing motives of altruism \cite{Levine1998}, solidarity \cite{Baldassarri2015}, strong reciprocity \cite{Fehr2003}, guilt \cite{Snijders1996}, inequity aversion \cite{Bolton2000,Fehr1999}, social welfare \cite{Charness2002}, or fairness \cite{Cox2007,Rabin1993}. Nonetheless, despite some notable exceptions \cite{Charness2009,Lin2019}, most theoretical models on risk-sharing arrangements revolve exclusively around the self-interest assumption. 

For other social dilemma settings, the most applied models are models of inequity aversion \cite{Bolton2000,Fehr1999}, which assume that an agent is altruistic towards others if their material payoff falls below an equitable benchmark, but feels envy when their payoffs exceed this benchmark. Subsequently, some studies make distinctions between altruism (directed to any other agent) or solidarity (directed to members of the same group; \citeNP{Baldassarri2015}). While inequity aversion models do not accurately account for behavior in all social dilemma settings \cite{Andreoni2002}, nor adequately explain individual-level behavior \cite{Blanco2011}, the common conception is that due to its simplicity, it is generally the preferable model in terms of broad applicability.

In our model, we label our social preferences `solidarity', reflecting willingness to help others within the same mutual group. We assume that solidarity is triggered once agents are in need of support. In that sense, it resembles a simplified (one-sided) version of inequity aversion, disregarding envy and as such relating also to models of guilt aversion (\citeNP{Snijders1996}; applied to risk sharing in \citeNP{Lin2019}) and social welfare \cite{Charness2002}. It is implemented such that players who perceive solidarity motives are willing to suffer some costs (i.e., have smaller shares of money redistributed from the common fund) to cover the loss of others. 

\subsection{Reinforcement Learning}

Standard analytical game theory uses forward-looking rationality, which assumes high cognitive capabilities of all actors involved. Its alternative, backward-looking rationality, not only makes it easier to derive predictions, but also poses fewer constraints on the agents' cognitive capabilities \cite{Macy2002a}. In situations of incomplete information, it entails learning about the profitability of certain strategies over time. Many learning models have been developed (see \citeNP{Camerer2003a} for a review), but most (e.g., belief learning; \citeNP{Fudenberg1995}) focus on agents adapting their strategies towards best-replies against beliefs about the strategies other agents have.  

In our model, the main information deficiency of agents concerns the risk distribution within the risk-sharing group. This requires a learning model that focuses on outcomes, rather than strategy beliefs. For this purpose, reinforcement learning \cite{Bush1955} and Bayesian learning \cite{Jordan1991} can be considered: two models that differ in the cognitive abilities they impose on the agents. Reinforcement learning has agents following a simple updating rule \cite{Macy2002a}. Every round, they update by some weight their estimate of the profitability of a certain strategy (here: of participating) based on realized payoffs in the previous rounds (here: driven by the number of support requests). Hence, when in a particular round more agents request support, the estimate of average risk is revised upwards. 

With Bayesian learning, instead, agents are assumed to consider every possible collection of agent characteristics (here: of risk distributions). Each collection is given a certain weight, and over time these weights are updated based on realized outcomes \cite{Jordan1991}. Hence, the number of support requests are used to add more or less credibility to each potential risk distribution: the more support requests, the more weight is given to distributions that have larger group averages or more extreme distributions. 

The general consensus is that while models of reinforcement learning are often too slow to match the pace of human learning, Bayesian learning models are too complex \cite{Camerer2003a}. Especially in our case, since we focus not on a discrete set of strategy profiles, but on an (in theory) infinite number of possible risk distributions, Bayesian learning is too demanding (cf. a complete information model). We therefore start from a simple model of reinforcement learning (cf. \citeNP{Macy2002a}) and assume that by some weight agents let current experiences (i.e., support requests) override their previous estimate. Essentially, by introducing learning we assume agents to be boundedly rational \cite{Camerer1998} such that they maximize their utility based on an estimate of the average risk of other agents, but do not know or incorporate all individual risk profiles. 

\section{Model Construction} \label{sec.model}

We define an \textit{N}-person Risk-Sharing Model (RSM) in which $N \geq 2$ agents indexed by $i \in \{1,\ldots,N \}$ choose simultaneously to become a member $(m = 1)$ or not $(m = 0)$ at time point $\tau = 1$. In each next time period $1 < \tau \leq T$, agents that opted to join decide whether or not to remain. If at any time point $\tau$ an agent decides to opt out, this decision is irrevocable. After each decision, a random draw by Nature determines realized events (i.e., which agents, if anyone, need support) and resulting payoffs. We use $n$ to denote the number of agents that are part of the Risk-Sharing Group (RSG), with $0 \leq n \leq N$. A boundedly rational, utility maximizing agent joins the RSG as long as the short-term expected payoff under $m = 1$ exceeds that of $m = 0$. 

Each agent receives an income $Y_i$ with probability $(1-p_i)$ and $y_i$ with probability $p_i$, where $Y_i - y_i$ represents the loss that can be insured\footnote{In the broader context of risk-sharing applications, this loss could also reflect failed harvests, a broken product, stolen goods, poor health, etc.} and $p_i$ is an independent and identically distributed (i.i.d.) risk probability. If agents join the RSG, they pay a contribution $c_i$ for membership and receive a benefit $b_i$ under $p_i$, where $c_i < b_i$ and $y_i + b_i \leq Y_i$. For simplicity, we assume homogeneous incomes, losses, contributions, and benefits ($Y_i = Y$, $y_i = y$, $c_i = c$, and $b_i = b$) and leave heterogeneity only with respect to $p_i$. 

To make the RSG represent a collective non-profit fund, the pooled money that is not needed to pay benefits gets redistributed among all agents with an individual share $\delta$. That is, the expected profit on the level of the mutual corresponds to\footnote{The basic set-up assumes there are no operation costs involved in the organization of the mutual insurance fund. For an institutionalized setting, the profit parameter could easily be extended to include some administration fee that is paid by all members and reflects a fixed cost (i.e. that is not redistributed at the end of the term).}

\begin{align} \label{eq.1}
P = \sum\limits_{i = 1}^{n} (c - p_i b - \delta) = 0.
\end{align}

%Let us define $k_i$ as the number of members other than $i$ that is expected to need support each round, with $k_i = \sum\limits_{j \neq i}^{n - 1} p_j$ and $0 \leq k_i \leq n - 1$. 
The bounded rationality assumption implies that players cannot foresee the risk levels of other agents $j$.  Instead, we assume that agents have an estimate $\hat{p}_i$ of the average risk probability of all other agents.  $\hat{p}_i$ is based on an intuition about the risk probability of the population and is updated over time $\tau$ after observing the number of support requests by other group members in the previous round. We denote for each agent $i$ the number of other group members $j \neq i$ that needed support in the previous round as $k_i^{\tau-1}$, where $k_i^{\tau-1}$ influences the agent's average risk estimate $\hat{p}_i$ with some weight as determined by the learning parameter $0 < \omega \leq 1$, i.e.
\begin{align} \label{eq.2}
\hat{p}_i^{\tau} = (1 - \omega) \hat{p}_i^{\tau-1} + \omega \frac{k_i^{\tau-1}}{n^{\tau-1}}.
\end{align}

Thus, the estimate of others' risk at time $\tau$ is a function of the previous estimate as well as the proportion of members that requested support $k_i/n$ in the previous time point $\tau - 1$. Combining (\ref{eq.1}) and (\ref{eq.2}), the expected redistribution boils down to 
\begin{align} \label{eq.3}
\hat{\delta}_i = \frac{n c - p_i b - \hat{p}_i (n - 1) b}{n} = c - \hat{p}_i b - \frac{(\hat{p}_i - p_i)b}{n}.
\end{align}

In this equation, $c$ reflects the maximum available amount for redistribution (if no one needs support everyone simply gets their contribution to the common fund returned), $\hat{p}_i b$ reflects the estimated average loss on payments to other players, and the difference $(\hat{p}_i-p_i)$ either decreases or increases this share spent on payments depending on whether player $i$'s risk is lower or higher than the estimated other players' average risk $\hat{p}_i$, respectively. If we include that each agent receives benefit $b$ with probability $p_i$, we can rewrite this such that $p_i \left(b - \frac{b}{n}\right)$ represents the expected net benefit from participation, while $-\hat{p}_i \left( b - \frac{b}{n}\right)$ represents the expected net costs (see Supplementary Materials for detailed derivations).%Section \ref{s.equations} in the Appendix for detailed derivations).

Thus, without additional assumptions of solidarity and risk aversion, agents would participate only when $p_i > \hat{p}_i$. We rewrite $\beta = b - \frac{b}{n}$ and introduce solidarity $\alpha$ as the utility obtained by agent $i$ from supporting any of agents $k_i > 0$. Hence, solidarity interacts with the net costs $\hat{p}_i \beta$ agents expect to pay to support group members, lowering by some factor $0 \leq \alpha \leq 1$ these subjective costs to $(1-\alpha)\hat{p}_i \beta$. Solidarity can then explain participation even if $p_i < \hat{p}_i$ as long as $\alpha \geq \frac{\hat{p}_i - p_i}{\hat{p}_i}$. 

Finally, that players do participate even if $p_i < \hat{p}_i$ and $\alpha < \frac{\hat{p}_i - p_i}{\hat{p}_i}$ is explained by assuming that players are risk averse. We capture this by adding an exponent $(1 - r)$ to the utility function: $EU = EV^{(1-r)}$ with $0 < r < 1$ for both strategies $m = 1$ and $m = 0$ to obtain
\begin{equation} \label{eq.6}
EU = 
\begin{cases}
(1 - p_i)Y^{(1-r)} + p_i y^{(1-r)}, & \text{if } m = 0\\
(1-p_i) (Y - (1 - \alpha)\hat{p}_i \beta)^{(1-r)} + p_i (y + \beta - (1-\alpha)\hat{p}_i \beta)^{(1-r)} & \text{if } m = 1.
\end{cases}
\end{equation}

Put differently, it follows that when player $i$ would not need support, i.e. under $(1-p_i)$, we always have $Y^{(1-r)} \geq (Y- (1-\alpha)\hat{p}_i \beta)^{(1-r)}$. No benefits are obtained, so participation is costly only. Under $p_i$, contrarily, we always have $(y + \beta - (1-\alpha)\hat{p}_i \beta)^{(1-r)} > y^{(1-r)}$. Hence, the crucial evaluation lies in the difference between the subjective values attached to the net benefits and net losses, which depend on the player's risk preferences. In essence, the model states that players will participate when the individual risk $p_i$, solidarity $\alpha$, risk aversion $r$, and the size of the loss $Y - y$ are sufficiently large, while estimated others' risk $\hat{p}_i$ and the learning parameter $\omega$ are sufficiently small. What `sufficiently large' and `sufficiently small' mean, will be analyzed using simulations to accommodate for the stochasticity in group-level support requests.

\section{Simulations} \label{sec.sim}

The aggregate outcomes of the RSM are far from evident due to the interdependencies between agents. While the utility of participation might lie above the threshold for some agent $i$ at time point $\tau$, sudden peaks in $k_i$ or changes in $n$ and $\hat{p}_i$ due to the withdrawal of other agents $j$ might move the utility below the threshold at time point $\tau + 1$. Thus, the global patterns of interest are more than the aggregation of individual attributes. Agent-based models (ABM) provide a solution here \cite{Bianchi2015a,Macy2002}.

We use ABM simulations not for individual behavior, but with the purpose of understanding group-level outcomes. That is, under which conditions are risk-sharing arrangements stable and when are they subject to withdrawal cascades? Through simulations, we can dynamically model the micro-level foundations of the RSM to compare the outcomes on the group-level. On the micro-level, the agents at risk are those with a negative $p_i - \hat{p}_i$ difference. While the RSM proposes several parameters that can compensate for this difference, in principle these are the agents whose utility may fall below the participation threshold. 

On the group-level, the severity of this risk is captured by risk heterogeneity. The more heterogeneous agent groups are in terms of risk, the more likely it is that these groups have larger sets of agents with substantial negative $p_i - \hat{p}_i$ differences. In other words, the larger the number of agents that are at risk of falling below the participation threshold when other (low-risk) agents withdraw or fluctuations in $k_i$ are more extreme. The simulation thus serves to compare different levels of risk heterogeneity, whether and how they lower group-level participation rates, and which (combinations of) individual factors (learning, solidarity, and risk aversion) may minimize or diminish this effect. 

Below we discuss the set-up of our dynamic RSM roughly following the ODD (Overview, Design concepts, Details) protocol of \citeA{Grimm2010}. The simulation was programmed in NetLogo and analyzed in R. The NetLogo code is documented in the Supplementary Materials.%\Cref{s.netlogo}.

\subsection{Parameter Settings}

The dynamic RSM consists of an agent population deciding whether or not to participate in the RSG. Agent variables modelled are individual risk $p_i$, risk aversion $r$, solidarity $\alpha$, and reinforcement learning $\omega$. Environmental variables are the size of the benefit $b$ and the size of the population $N$. The model is run over a number of discrete consecutive time steps $\tau$, which can be seen as representing months: Every month, agents evaluate whether or not to proceed in the RSG. 

\Cref{t.parcom} presents an overview of all possible and tested model parameters. The risk probabilities are randomly drawn to increase group-level variation in fluctuations in $k$, the number of agents that need support. For all other parameters, we chose an expressive selection of fixed parameter values. Note that this implies that for each parameter combination the fixed parameter values apply to all agents in the same risk-sharing group. While earlier models (e.g., \citeNP{Skogh2005}; \citeNP{Vogt2004}) suggest that heterogeneity in a second factor, such as risk aversion, is necessary to compensate, what really matters is that low-risk individuals score higher on this factor. High-risk agents will participate regardless of whether they score high or low on this factor, making this factor irrelevant. Hence, there is no need to test this under balanced heterogeneity; we can test the effects of these parameters for a homogeneous population without loss of information. This setup has the advantage that we can separate the model dynamics with regards to parameter settings from the stochastic processes that result from the differences between expected risk and realized support. 

\begin{table}[t!]
	\centering
	\footnotesize
	\caption{Overview of model and simulation parameters}
	\label{t.parcom}
	\begin{tabularx}{\textwidth}{Lll}%{@{\extracolsep{12pt}}lll}
		\\[-1.8ex]\hline 
		\hline \\[-1.8ex] 
		\textbf{Parameter} & \textbf{Possible values} & \textbf{Values simulation} \\
		\hline \\[-1.8ex]
		\textbf{Constants} 	& & \\
		Number of rounds	  & $T > 0$ 			& $T = 180$\textsuperscript{1} \\
		Number of simulations & $\sigma > 0$ 		& $\sigma = 50$ \\
		Maximum income 		  & $Y > 0$ 			& $Y = 100$ \\
		Income after loss	  & $0 \leq y < Y$ 		& $y = 0$ \\
		\\[-1.8ex]\hline
		\textbf{Varying parameters, random} & & \\
		Risk probability   & $0 \leq p_i \leq 1$ 	& $p_i \sim \begin{cases} 
		U([0.1,0.2]) \\ %& \text{if cond } = 1\\ 
		U([0.05,0.25]) \\ %& \text{if cond } = 2\\ 
		U([0,0.3]) %& \text{if cond } = 3
		\end{cases}$ \\
		Estimated average risk probability at $\tau = 1$ & $0 \leq \hat{p}_i \leq 1$ & $\hat{p}_i = p_i$ \\
		\\[-1.8ex]\hline
		\textbf{Varying parameters, fixed} & & \\
		Size subject population 	& $N \geq 2$ 				& $N \in \{10,50,90\}$ \\
		Benefit 			 		& $0 < b \leq (Y-y)$ 		& $b \in \{40, 80\}$  \\ 
		Reinforcement learning		& $0 \leq \omega \leq 1$ 	& $\omega \in \{0.2, 0.4\}$ \\
		Risk aversion 		 		& $0 \leq r \leq 1$ 		& $r \in \{0, 0.2, 0.4\}$ \\ 
		Solidarity 			 		& $0 \leq a \leq 1$ 		& $\alpha \in \{0, 0.2, 0.4\}$ \\
		\hline \\[-1.8ex]
		\textbf{Parameter combinations:} & & 324 \\
		\textbf{Total observations:} & & 2,916,000\textsuperscript{2} \\
		\\[-1.8ex]\hline 
		\hline \\[-1.8ex]
	\end{tabularx}	
	\begin{flushleft} 
		\textit{Notes:} \textsuperscript{1} Simulations could take more rounds, but per-round information was stored for up to $T = 180$ rounds; \textsuperscript{2} Total observations = parameter combinations $\times$ rounds $\times$ simulations.
	\end{flushleft}
	%(combinations $\times$ rounds $\times$ simulation runs)
\end{table}

We used two stopping rules for our simulation runs. First, the simulation ended as soon as the risk-sharing group was empty (i.e., all members dropped out). Second, the simulation ended when (after at least 120 rounds) membership rates were stable for 60 rounds. While this meant that in our results the longest simulation runs took 559 rounds (before reaching stability) and 674 rounds (before ending in failure), we only stored detailed per-round data up until round $T = 180$. This practical limit was chosen because, if rounds are months, this generates a dataset with simulated dynamics for what represents a 15-year period.\footnote{For any real-life situation it is unlikely that long-term cooperation patterns would be characterized by fixed characteristics (e.g., no changes in the risk probability over time), so we should not use this basic model to interpret cooperation dynamics on very long time scales.}

In total, we performed $\sigma = 50$ simulation runs of all parameter combinations to test within parameter combinations what share of the outcome is driven by the stochasticity resulting from the discrepancy between $p_i$ and $k_i$. Since $Y$ and $y$ merely represent the bandwidth within which the dynamics of benefits and losses take place and are not of substantial interest otherwise, we fixed these values to $Y = 100$ and $y = 0$.

%Moreover, we chose an arbitrary, yet fixed number of $T = 180$ iterations to explore whether and when participation rates stabilize. That is, we implemented a stop-strategy that ended the simulation run if the number of agents had not changed for at least 60 iterations after iteration $\tau = 120$. Hence, for all groups that failed or stabilized before that iteration we have 180 observations; for the other groups simulations continued until stabilization was reached. If time points were to represent months, 180 iterations yield patterns over a 15-year period. Hence, while we stored the datapoints for groups that needed more iterations to either fail or stabilize, we consider any failure beyond this threshold less relevant. 
%\footnote{If time points were to represent months, this yields patterns over a 15-year period.}

To model risk heterogeneity, we draw $p_i$ from three uniform distributions with ranges [0,0.3], [0.05,0.25], and [0.1,0.2]. Hence, at the starting point ($\tau = 0$), the average risk probability per agent population is $\overbar{p} = 0.15$ for each heterogeneity condition. We chose this average probability level, because it is low enough to enable fund building and payouts in small groups, whilst simultaneously high enough to ascertain sufficient translations into actual support events (and thereby fluctuations). For the estimated average probability, we take $\hat{p}_i = p_i$ as starting value, assuming that agents initially believe that other agents face a similar risk as they do themselves (cf. \citeNP{Skogh2005}). 

For the remaining parameters we implement two or three fixed values. For the subject population, we chose values of 10, 50, and 90 to compare substantially different group sizes, as that might affect the severity by which fluctuations impact the estimated risk and how costly participation is. With respect to the benefit parameter, we chose 40 and 80 to compare a situation in which almost the entire loss is covered versus the situation in which participation is cheaper yet with a smaller loss coverage. For risk aversion and solidarity we use parameter values 0, 0.2, and 0.4. Since values $> .5$ guarantee participation, this allows us to compare the interesting in-between cases. Finally, we use 0.2 and 0.4 as weights for reinforcement learning (excluding 0 as that inhibits learning).

\subsection{Process Overview and Scheduling}

\Cref{f.sim-flow} presents a flowchart describing the different stages of the simulation. We systematically compare all $3\times3\times2\times2\times3\times3 = 324$ parameter combinations from matrix $P$, that contains the unique parameter combinations on the rows, with input parameters $Y$, $y$, $T$, $p_{min}$, $p_{max}$, $N$, $b$, $\omega$, $r$, and $\alpha$ on the columns, i.e.

\begin{align*}
P = 
\begin{bmatrix}
100    & 0 		& 180 	 & 0 	  & 0.3    & 10     & 40     & 0.2    & 0      & 0      \\
100    & 0 		& 180 	 & 0      & 0.3    & 10     & 40     & 0.2    & 0      & 0.2    \\	
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
100    & 0      & 180    & 0.1    & 0.2    & 90     & 80     & 0.4    & 0.4    & 0.4    
\end{bmatrix} 
\end{align*}

\begin{figure}[t!]
	\centering
	\includegraphics[width = 0.7\textwidth]{sim_flowchart.png}
	\caption{Simulation process flowchart}
	\label{f.sim-flow}
\end{figure}

All stages of the simulation are run 50 times for each row of matrix P. 
In the first phase, the initialization phase ($\tau = 0$), individual risk probabilities $p_i$ are randomly drawn from the range $[p_{min}, p_{max}]$ for each agent in the population. After this phase, the simulation starts at round $\tau = 1$, using synchronous updating. All agents move through the flowchart simultaneously and wait for other agents to arrive before proceeding to the next phase. In the realization phase, a random draw translates the individual risk probabilities to realized support events. The update phase is skipped in the first round and agents move immediately to the decision phase. In the decision phase all agents decide, by calculating and comparing the expected utilities of participating ($m = 1$) and not participating ($m = 0$), whether or not to join the RSG. 

Regardless of the decision, all agents move back to the realization phase. However, only agents that chose to stay move to the update phase, which involves updating population-level parameters $n$ (the total number of agents that chose to stay) and $\overbar{p}$ (the average risk probability of the RSG) as well as agent-level parameters $k_i$ (the number of group members other than $i$ that need support) and $\hat{p}_i$ (the estimated average risk probability). Subsequently, the agents that are part of the RSG are brought to the decision phase, while the agents that dropped out are moved to another realization phase. Agents loop over these phases until one of the two stopping conditions ($ n = 0$ or \# stable rounds $= 60$) is met. 

Note that agents only choose to join in the first round; afterwards they choose (if they joined) whether to stay or not. Likewise, since only RSG members move through the update phase, any realization $k_i$ is common knowledge for members only. Given this model set-up, agents cannot wait a few rounds to see how the RSG develops and decide whether or not to join, nor can they, if they chose to leave, revoke this decision at a later time point. Without new information about $k_i$, they do not update $\hat{p}_i$, and therefore have no incentive to participate in later rounds if they did not in round $\tau = 1$ or in the round they decided to leave. This is a strict modelling choice, but one justified by the substantive argument that leaving (which is essentially a way of defecting) will not be rewarded by other group members by allowing the defected members to return whenever it suits them better. At the same time, it does imply that as long as we do not introduce new members to the population in later rounds, the RSG can only remain stable or decrease in size. In the discussion we discuss implications of this design feature as well as several (simple and more complex) solutions.

%Even if agents would wait a few rounds before making the decision to join, they have no information about the RSG besides the number of members. Hence, if the threshold for participating does not exceed that of not participating in the first round, it will not in later rounds either. 

%The same holds for the decision to leave, which we set to be irrevocable, for leaving is a way of defecting. Defection will not be rewarded by other group members by allowing the defected members to leave and return whenever it suits them better. Apart from this substantive argument, agents would have no incentive to return. They no longer update their belief of the group's average risk probability (as they are not informed about new realizations), meaning that the expected utility of participating remains lower than that of not participating regardless of new developments in the RSG. This means that, 

At each round, we also calculate whether agents would have decided to participate in the risk-sharing group if they had known the true risk distribution (i.e., under complete information). In this benchmark Complete Information Model (CIM), agents know the distribution of $p$ of all other members in their RSG, which means they update based on $\overbar{p}$ rather than their estimate $\hat{p}_i$. Hence, while they update by recalculating expected benefits after other agents drop out, there is no learning involved based on $\omega$ and $k_i$. The CIM thus shows the baseline withdrawal pattern that can be expected from the initial distribution and input parameters, recognizing that for some agents the RSG is not interesting to begin with. Deviations between the RSM and the CIM are the result of agents' responses to fluctuations and thus reflect `erroneous' dropouts. This way, we can compare the results of the simulation model based on reinforcement learning (the RSM) to that of earlier, static modelling approaches to understand how this alternative interpretation affects predictions about risk-sharing behavior.

\subsection{Data and Analysis}

We logged data on the population level for each time point $\times$ parameter combination for 50 simulations and, while storing the final round number, only kept observations up until round $T = 180$. This brings the total number of observations to 2,916,000. 

Our variable of interest is the membership rate: the percentage of agents that decides to remain a member. The main predictor is risk heterogeneity, a categorical comparison of the three conditions under which the risk distribution of an agent population was drawn. First, we explored the average stability of the agent populations of different heterogeneity conditions visually by comparing differences in membership rates between the RSM and CIM. Here, we also include visualizations of the average risk $\overbar{p}$ for the three heterogeneity conditions over time, to assess whether dropout follows adverse selection mechanisms (i.e., low-risk agents are more likely to drop out, increasing the average risk within the risk-sharing groups).

Subsequently, we used multilevel OLS regression models with simulation runs nested in parameter combinations to predict membership rates for each agent population. The multilevel specification was used to separate the effects of the different model parameters from the effects of which particular distribution is drawn and how probabilities were translated to support requests. Hence, the variance on level 1 represents stochasticity resulting from the translation of probabilities to events within parameter combinations\footnote{After all, for each population a risk distribution is randomly drawn and risks are randomly translated into support requests, meaning that for the same parameter combination the risk distribution looks different every simulation run and two agents with the same risk may not have needed support equally often.}, whereas the variance on level 2 represents variation in outcomes depending on specific parameter combinations. 

The goal of this analysis was not to statistically draw conclusions about a sample, but to give a qualitative description of the results of the simulation runs; i.e., to numerically infer under which conditions more agents remain part of the RSG. Primarily, we were interested in the extent to which population size $N$ and individual factors risk aversion $r$, solidarity $\alpha$, and learning $\omega$ compensate or strengthen the effect of fluctuations for different risk heterogeneity conditions. In other words, we tested for interactions between these variables and risk heterogeneity. The size of the benefit $b$, the dropout in the previous round (relative to the population size $N$), the average estimated risk $\hat{p}$, and the time period $\tau$ were included as control variables. We centered and standardized all continuous variables (i.e., all but the heterogeneity conditions) to be able to compare effect sizes. The analyses were conducted on all observations of $\tau > 1$ (as interdependencies in decision-making only start from round 2 onwards) and $\tau \leq 60$ (as we found most groups to be stable by that time point, with occasional dropouts not influencing other agents to follow). 

Finally, as a sensitivity check, we analyzed the stability of the results by running additional simulations in which we relaxed the core assumption of i.i.d. risks and introduced external correlated shocks. Shocks are inherently present in the simulation model due to the difference between expected risk and realized support as well as the misconception between true risk and estimated risk. In our analyses on the aggregate level, however, the effects of these shocks are not clearly visible. By running multiple simulations over many parameter combinations we have smoothed this process. Moreover, as the shocks remain based on the average risk probability, they will rarely be truly out of bound. To see whether extreme shocks can set in motion new withdrawal cascades, we tested a variation of the model where we include correlated risks. Correlated risk refers to the simultaneous occurrence of many losses from a single event. We compared three variations where we introduce an external shock of $p_s = 0.5$ (i.e., making 50\% of the RSG members reliant on support) in one, two, or three consecutive time periods starting from round $\tau = 50$. Since we were primarily interested in what happens in the rounds following the shock(s) we implemented a new stopping rule ending all simulations after 80 runs. We repeated each parameter combination for 30 simulations, resulting in a new dataset with another $3 \times 324 \times 80 \times 30 = 2.332.800$ observations.

\section{Results}
\label{sec.res}

\begin{figure}[t]
	\centering
	\includegraphics[width = 0.8\textwidth]{time.pdf}
	\caption[Membership rates and risk probability over time]{Membership rates and risk probability over time for the three risk conditions}
	\label{f.time}
\end{figure}

\Cref{f.time} plots the decay in membership rates (left panel) and the increase in the average risk probability $\overbar{p}$ (right panel) for the three risk heterogeneity conditions. The two panels show patterns that are generally in line with adverse selection arguments: while the membership rate decreases, $\overbar{p}$ increases, so the decrease in membership is the result of low-risk agents dropping out. Strikingly, adverse selection is much more severe for the RSM than for the CIM. Under the static complete information model, the decision whether or not to participate is made in the first round. While membership rates are indeed lower when heterogeneity increases---indicating that cooperation is more difficult to establish---stable participation patterns are still predicted for a substantial share of the population (i.e., 84\% for the low heterogeneity condition, 69\% for the intermediate heterogeneity condition, and 56\% for the highest heterogeneity condition under our chosen parameter settings).   

When comparing these participation patterns to the results we obtain with the RSM, we see that despite agents' starting assumption that the group's average risk resembles their own, support requests rapidly increase the average estimated risk probability $\hat{p}$ (dotted line in right panel) for all but the low heterogeneity condition, which causes membership rates to drop below that of the CIM within 10 rounds already (left panel). Hence, while roughly speaking the decay patterns observed for the RSM in the first 10 rounds can be attributed to the fact that the negative $p_i - \overbar{p}$ difference is too big to compensate for by any of the other parameters, the fact that the relative number of members continues to drop afterwards in the RSM is explained by the uncertainty surrounding the true risk distribution $\overbar{p}$ and the temporary peaks in the number of support requests $k$ and the resulting cost fluctuations.

At the same time, the right panel of \Cref{f.time} also shows that, because agents start from a similarity assumption, agents tend to underestimate the average risk. The estimate only starts to catch up to the true average risk when participation decay starts to slow down (after approximately $\tau = 60$). The slower decrease after $\tau = 60$ suggests that a temporary increase in $k$ might drive an occasional agent to drop out every now and then, but that this no longer takes place through cascading effects where agents follow each other in opting out---a pattern we do observe for the first rounds. In other words, the steep decrease up until round $\tau = 60$ represents overall success versus failure; for any changes after these rounds, it should be relatively easy to draft countermeasures. 


\begin{figure}[t]
	\centering
	\includegraphics[width = \textwidth]{avg_dist.pdf}
	\caption[Average percentage and distribution of expected and realized members]{Average percentage and distribution of expected and realized members (in round $\tau = 60$) for three risk heterogeneity conditions}
	\label{f.avg-dist}
\end{figure}

\Cref{f.avg-dist} plots the average and distribution of membership rates for time point $\tau = 60$, chosen because it represents a period where most major changes have occurred (see \Cref{f.dist-memb} in the Supplementary Materials
%\Cref{s.distributions} 
for an overview of the distribution in different time points). It signals the importance of not only considering the average membership rates, but also the variation between groups. Lower average membership rates are largely driven by RSGs that failed.  

The right panel of \Cref{f.avg-dist} shows that for each heterogeneity condition, there are roughly two peaks. For the low heterogeneity condition, the highest peak remains around full membership, but when members start to withdraw this nearly always sets in motion a cascade that ends in complete failure. The other two heterogeneity conditions both have moved their main peak from full membership to group failure, but also seem to have stabilized more often on alternative, in-between membership rates. %Particularly for the high heterogeneity condition, very few observations remain on the far right side of the distribution. %Instead, an alternative, flatter peak emerged around 70\%. 
This signals not only that high heterogeneity is less attractive for low-risk agents, but also that withdrawal does not automatically lead to cascades as long as some degree of heterogeneity exists. A substantial number of groups continues to exist and seems to have found more or less stable participation rates. %The intermediate heterogeneity condition, finally, is most spread out, but finds its new, secondary peak at around 85\%.
Hence, while low-heterogeneity RSGs are more successful in terms of average membership rates, some degree of heterogeneity might be beneficial for the group as a whole, for dropout of a few members is less likely to result in complete failure.

Apart from showing that RSGs either fail or retain quite large membership rates, zooming in on the distribution also shows that failure is much higher than it would have been under complete information. With complete information, total failure would only be expected for 1805 (11\%) of 16200 groups (approximately 600 per heterogeneity condition). In the RSM, contrarily, 3625 groups (22\%) failed---1297, 1228, and 1100, in the low, intermediate, and high heterogeneity condition, respectively. This means about half the groups that failed did so due to cost fluctuations and incomplete information, and more in low-heterogeneity RSGs than in high-heterogeneity RSGs.



\subsection{Predicting Membership Rates from Individual Motivations}

\begin{table}[t] \centering \footnotesize
	\caption{Multilevel OLS regressions on membership rate for 813,845 simulation run $\times$ round combinations.} 
	\label{t.reg} 
	\begin{tabular}{@{\extracolsep{-5pt}}lD{.}{.}{-3} D{.}{.}{-3} D{.}{.}{-3} D{.}{.}{-3} D{.}{.}{-3} D{.}{.}{-3}} 
		\\[-1.8ex]\hline 
		\hline \\[-1.8ex] 
		\\[-1.8ex] & \multicolumn{2}{c}{M0} & \multicolumn{2}{c}{M1} & \multicolumn{2}{c}{M2}\\ 
		\hline \\[-1.8ex] 
		Intercept & 64.902^{***} & (1.388) & 75.625^{***} & (0.924) & 75.624^{***} & (0.909) \\ 
		\hspace{3mm}Range [0.05,0.25] & &  & -13.887^{***} & (1.307) & -13.887^{***} & (1.286) \\ 
		\hspace{3mm}Range [0.05,0.25] & &  & -23.038^{***} & (1.307) & -23.035^{***} & (1.286) \\ 
		Risk aversion $r$ & & & 17.718^{***} & (0.534) & 18.630^{***} & (0.909) \\ 
		\hspace{3mm}$r$ * Range [0.05,0.25]& & &  &  & 0.537 & (1.286) \\ 
		\hspace{3mm}$r$ * Range [0.05,0.25] && &  &  & -3.271^{**} & (1.286) \\ 
		Solidarity $\alpha$ & & & 13.431^{***} & (0.534) & 14.546^{***} & (0.909) \\ 
		\hspace{3mm}$\alpha$ * Range [0.05,0.25] & &&  &  & -0.380 & (1.286) \\ 
		\hspace{3mm}$\alpha$ * Range [0.05,0.25] & &  &&  & -2.965^{**} & (1.286) \\ 
		Reinforcement learning $\omega$ & & & -3.892^{***} & (0.534) & -3.874^{***} & (0.909) \\ 
		\hspace{3mm}$\omega$ * Range [0.05,0.25] &&&  &  & -0.226 & (1.286) \\ 
		\hspace{3mm}$\omega$ * Range [0.05,0.25] && &  &  & 0.171 & (1.286) \\ 
		Population size $N$ &  && 1.567^{***} & (0.534) & 2.483^{***} & (0.909) \\ 
		\hspace{3mm}$N$ * Range [0.05,0.25] && &  &  & -0.970 & (1.286) \\ 
		\hspace{3mm}$N$ * Range [0.05,0.25] && &  &  & -1.777 & (1.286) \\ 
		Benefit $b$ &  && -2.846^{***} & (0.534) & -2.846^{***} & (0.525) \\ 
		\% of members that left& &  & -0.030^{*} & (0.016) & -0.030^{*} & (0.016) \\ 
		Estimated average risk $\hat{p}_i$ &  && -2.239^{***} & (0.018) & -2.239^{***} & (0.018) \\ 
		Time point $\tau$ & & & -7.463^{***} & (0.015) & -7.463^{***} & (0.015) \\ 
		\hline \\[-1.8ex] 
		Residual variance & 232.24 & & 172.68 & & 172.68 & \\ 
		Random intercept & 624.19 & & 100.14 & & 97.37 & \\ 
		Intraclass correlation & 0.73 & &0.35 & &0.34 &\\ 
		\hline \\[-1.8ex] 
		Log Likelihood & \multicolumn{2}{Y}{-3,373,026} & \multicolumn{2}{Y}{-3,252,181} & \multicolumn{2}{Y}{-3,252,163} \\ 
		Akaike Inf. Crit. & \multicolumn{2}{Y}{6,746,059} & \multicolumn{2}{Y}{6,504,389} & \multicolumn{2}{Y}{6,504,369} \\ 
		Bayesian Inf. Crit. & \multicolumn{2}{Y}{6,746,094} & \multicolumn{2}{Y}{6,504,540} & \multicolumn{2}{Y}{6,504,612} \\ 
		\hline 
		\hline \\[-1.8ex] 
	\end{tabular} 
	\raggedright \textit{Notes:}  $^{*}p<0.05$; $^{**}p<0.01$; $^{***}p<0.001$; Range [0.1,0.2] used as reference category. \\ 
\end{table} 


\Cref{t.reg} shows the results of the multilevel OLS regression analyses. Model 1 estimates the main effects and Model 2 includes the interaction terms for $r$, $\alpha$, $\omega$, and $N$ with risk heterogeneity (taking the lowest heterogeneity condition as reference category). First of all, the results indicate that most variance lies on the second level: The null model has an intraclass correlation of $\rho = 0.73$. This means that most variance is explained from the model parameters, but at the same time that 27\% of the results are driven by stochasticity in $p_i$ and $k_i$. This is an important finding: despite favorable starting conditions, RSGs can still be at risk for failure. Depending on the realization of support requests RSGs may lose a lot of members regardless of the parameter combination. 

The main effects follow the expected pattern from the model input: More risk heterogeneity generates lower membership rates; risk aversion and solidarity increase membership rates; and membership rates are smaller the larger the reinforcement learning weight. Population size has a positive influence, which means that (the effects of) fluctuations are more modest in larger groups. The effect of the benefit parameter is negative, implying that in general monetary participation costs are higher than the benefits (due to low-risk agents that remain part of the RSG anyway). There is a small negative effect of the percentage of members that left at round $t - 1$, but the main driver of the cascades observed in \Cref{f.time} is the average estimated risk. Finally, member rates are smaller the more time has passed. The main effects drastically reduce the variance on the parameter combination level, sharply decreasing the random intercept and leaving most unexplained variance the result of stochasticity on level 1. 


\begin{figure}
	\centering
	\begin{tikzpicture}
	[scale=100]
	%\draw[step=1cm, gray] (9.5,-1) grid (20.5,1);
	\draw [draw = lightgray,fill=lightgray] (0.14,0) rectangle (0.175,0.005);
	\draw[very thick,{latex}-{latex}] (0.085,0) -- (0.215,0);
	\foreach \x in {0.10,0.20}
	\draw (\x,0.03pt) -- (\x,-0.03pt) node[below]{\x};
	\foreach \x in {0.14,0.175}
	\draw [<-] (\x,0.2pt) -- (\x,-0.03pt) node[below]{\x};
	\node at (0.14,0.01) {$p_i$};
	\node at (0.175,0.01) {max. $\hat{p}_i$};
	\draw [->] (0.15,0.03pt) -- (0.15,-0.25pt) node[below, color = gray70]{\textbf{\textit{\textoverline{p}} = 0.15}};
	\draw [->] (0.17,0.03pt) -- (0.17,-0.25pt) node[below, color = gray45]{\textbf{\textit{\textoverline{p}} = 0.17}};
	\draw [->] (0.19,0.03pt) -- (0.19,-0.25pt) node[below, color = gray15]{\textbf{\textit{\textoverline{p}} = 0.19}};
	%	\draw [->] (0.15,0.03pt) -- (0.15,-0.25pt) node[below, color = ryellow]{\textbf{\textit{\textoverline{p}} = 0.15}};
	%	\draw [->] (0.17,0.03pt) -- (0.17,-0.25pt) node[below, color = rgreen]{\textbf{\textit{\textoverline{p}} = 0.17}};
	%	\draw [->] (0.19,0.03pt) -- (0.19,-0.25pt) node[below, color = rpurple]{\textbf{\textit{\textoverline{p}} = 0.19}};
	\draw [decorate,decoration={brace,amplitude=5pt,raise=0.04ex}]
	(0.14,0.015) -- (0.175,0.015) node[midway,yshift=1.5em]{Acceptable range};
	\draw [decorate,decoration={brace,amplitude=5pt,raise=0.04ex}]
	(0.087,0.015) -- (0.139,0.015) node[midway,yshift=1.5em]{Guaranteed participation};
	\end{tikzpicture}
	\caption{Illustration of extent to which solidarity can compensate for larger costs}
	\label{f.eff-sol}
\end{figure}

Model 2 adds the interactions with the heterogeneity conditions. These effects do not improve the explained variance much (residual variance drops from 100 to 97), for the only dynamics we find are those related to risk aversion and solidarity. For both, the effects are about 3\% points smaller in the highest heterogeneity condition compared to the lowest heterogeneity condition, thus strengthening the negative effect of high heterogeneity. This can be understood if we realize how these factors compensate for increases in cost. Say that an agent has a risk probability of $p_i = 0.14$ and solidarity $\alpha = 0.2$ (assuming no risk aversion). In that case, this agent is willing to accept every $\hat{p}_i > 0.14$ up until $\hat{p}_i = 0.175$ (Figure \ref{f.eff-sol}). Since higher risk heterogeneity conditions have higher maximum risk probabilities, solidarity is less likely to compensate for the increasing costs in more heterogeneous groups---the costs may more easily exceed the acceptable range. For more homogeneous groups, contrarily, limited solidarity (or risk aversion) is already sufficient to be willing to participate even if $\hat{p}_i > p_i$.
%\begin{figure}
%	\centering
%	\includegraphics[width = 0.9\textwidth]{int_effs.pdf}
%	\caption{Predicted effects on proportion of members and average risk probability for different risk heterogeneity conditions}
%	\label{f.int-effs}
%\end{figure}


\subsection{Sensitivity Check}

\begin{figure}[t]
	\centering
	\includegraphics[width = 0.9\textwidth]{shock_effs.pdf}
	\caption[Membership rates and average risk over time after high correlated shocks]{Membership rates and average risk over time after one, two, or three rounds of high correlated shocks}
	\label{f.shock-effs}
\end{figure}

\Cref{f.shock-effs} plots the decay in membership (top row) and increase in average risk (bottom row) for the three correlated shock conditions, tested with the same parameter combination matrix $P$ over thirty simulations. It shows that such a shock has a strong effect in next round, but does not have a long aftermath in terms of starting new cascades. The learning mechanism, which updates based on the number of support requests, causes the estimated risk to increase rapidly, but to decrease just as fast once the correlated shock(s) passed. However, the effect of such large shocks are still detrimental, for they lead to large decreases in membership rates. The more consecutive rounds the shock lasts (and thus the more it impacts the average estimated risk), the steeper the total dropout. This holds particularly for the low-risk heterogeneity condition, whose average membership rates after three shock rounds even end up below those of the other heterogeneity conditions (\Cref{t.prop.shock}).

That particularly RSGs low in heterogeneity are affected by such sudden and temporary shocks can be explained both from differences in the average risk and differences in the variation. Since mostly low-risk agents drop out initially, the average risk of the most heterogenous groups has increased more (approaching $\overbar{p} = 0.22$). Hence, the difference between the average risk and the temporary shock is smaller for the high heterogeneity groups, thereby increasing the probability that other factors can compensate for this gap (i.e., the scores on risk aversion and solidarity now also matter for the high-risk agents). Second, since in low-heterogeneity groups most agents have approximately the same risk, once the utility drops below the threshold for one, it mostly likely also does for most others, because the costs and benefits are largely the same for all members.

\begin{table}[t]
	\centering \footnotesize
	\caption[Percentage of agents that remains member after high correlated shocks]{Percentage of agents that remains member at $\tau = 60$ after high correlated shocks}
	\label{t.prop.shock}
	\begin{tabular}{lZZZ}
		\\[-1.8ex]\hline 
		\hline \\[-1.8ex] 
		& \multicolumn{1}{c}{Range [0.1,0.2]} & \multicolumn{1}{c}{Range [0.05,0.25]} & \multicolumn{1}{c}{Range [0,0.3]} \\
		\hline \\[-1.8ex]
		No shock 					  & 63\% & 49\% & 40\% \\
		1 shock at $\tau = 50$  	  & 44\% & 37\% & 35\% \\
		2 shocks at $\tau = [50,51)$  & 28\% & 27\% & 27\% \\
		3 shocks at $\tau = [50,52)$  & 20\% & 22\% & 24\% \\
		\\[-1.8ex]\hline 
		\hline \\[-1.8ex]	
	\end{tabular}	
\end{table}


\section{Implications and Hypotheses} \label{sec.impl}

From the simulation results we can derive a set of testable hypotheses. We start by comparing our model to the classic risk-sharing models that assume complete information about the risk distribution. \citeA{Skogh2005} proved theoretically that under incomplete information participation can still arise, based on the assumption that agents believe the risk probabilities of other agents will be more or less similar to their own. If we instead assume that agents update their ideas about the risk probabilities of other group members based on realized support requests, we found participation rates to be significantly lower and much less stable. As long as the weight attached to these new realizations is small enough, however, decisions remain largely based on the similarity assumption, resulting in the high participation levels predicted by \citeauthor{Skogh2005}. Hence, the following basic hypotheses can be derived:

\begin{hyp}
	Incomplete information about the risk distribution in the population decreases the likelihood that members stay in the risk-sharing group. 
\end{hyp}
\begin{hyp}
	The more susceptible members are to support realizations, the smaller the likelihood that members stay in the risk-sharing group. 
\end{hyp}

Incomplete information was primarily introduced to explore what this meant for the effect of heterogeneity. Classic risk-sharing models \cite{Coate1993,Kimball1988} assume homogeneous risk distributions. Risk heterogeneity hinders participation, for members who know their risk to be lower than the group's average will opt out. We focus on perceived rather than true risk and compare variations in heterogeneity rather than homogeneity versus heterogeneity, but the same mechanism applies. Members whose risk is lower than the estimated group's risk are at risk of dropping out. Hence, these hypotheses are in line with earlier models (albeit stronger because estimated risk increases rapidly) and show the general adverse selection tendency. % ADD BIT ABOUT CASCADES

\begin{hyp}
	The lower the risk heterogeneity, the larger the likelihood that members stay in the risk-sharing group. 
\end{hyp}

It is important to realize that continued membership is a decision that depends not only on (changes in) the number of support requests, but on (changes in) the total number of members as well. Hence, if a sudden increase in support requests means that for some members the utility of participating no longer outweighs that of not participating, in the next round other members---who did not have a problem with the increase in the group's average risk---might also prefer not participating over participating. Not only because of new support requests, but also because of the decreased number of members. After all, the risk estimate is updated based on the number of support requests relative to the number of members. If the costs of supporting group members have to be carried by fewer people, the costs of participation likewise increase. This sets in motion withdrawal cascades, where both earlier dropouts and continued support requests cause members to drop out round after round.

\begin{hyp}
	The larger the number of support requests, the higher the probability that withdrawal cascades arise where members follow each other in deciding to leave the risk-sharing group.
\end{hyp}

With regards to these fluctuations in support requests, subsequently, our sensitivity checks indicate that while more heterogeneous groups are less successful overall, they are better equipped to deal with sudden, exceptional increases in support requests. In more homogeneous groups, the high similarity between agents means that if something happens that make participation less attractive for one agent, this is probably true for most others as well. Hence, under extreme fluctuations, more agents `erroneously' drop out in more homogeneous groups.

\begin{hyp}
	The more homogeneous the risk-sharing group, the more membership rates are effected by sudden, exceptional increases in support requests.
\end{hyp}

Finally, as in other models that study the effect of risk heterogeneity \cite{Attanasio2012,Skogh2005,Vogt2004}, we included individual factors that can compensate for risk heterogeneity. While risk aversion is often used as compensatory factor, we also proposed to include solidarity in the risk-sharing utility function. The simulations pointed out that while these factors could indeed compensate for heterogeneity, they can only do so to some extent. Hence, in comparing homogeneous versus heterogeneous groups, it is true that a heterogeneous group in general is more likely to succeed if its members are driven by solidarity and risk aversion motives. However, once we start to compare degrees of heterogeneity, it becomes clear that less heterogeneous groups are still more successful. Or rather, the more heterogeneous the risk distribution of a risk-sharing group, the higher the demands posed on the extent of risk aversion and solidarity of it members. This results in the following hypotheses:

\begin{hyp}
	The lower the risk heterogeneity, the stronger the positive effect of \textbf{(a)} risk aversion and \textbf{(b)} solidarity on the likelihood that members stay in the risk-sharing group.
\end{hyp}

Altogether, these hypotheses make for an interesting theory on the role of risk heterogeneity. More homogeneous groups are, on many accounts, more likely to be established, to succeed, and to maintain high membership rates. Heterogeneous groups, on the other hand, are less likely to succeed overall and suffer a bigger loss in terms of adverse selection. Yet when they do manage to succeed, they are more resilient to extreme fluctuations in support requests. 

This difference is the result of assuming incomplete (as opposed to complete) information about the distribution of risk in the group. It signals that the real threat for the success of risk-sharing groups is not high risk, but a perception of high risks. Every time a sudden increase in support requests results in a spike in estimated risk, more agents are at risk of dropping out. Therefore, many more agents dropped out than they would have under complete information. The low-heterogeneity condition had fewer of these peaks, but if they do occur, the results are more detrimental.

This has important implications for the new mutuals initiatives. In general, communication and information are key. To prevent members reacting to sudden and temporary increases in support requests, the organizations need to be clear about the long-term perspectives of the risk-sharing group. Only then can the weight of one-time realizations be decreased. Another solution is to boost solidarity by stimulating dense and cohesive RSGs. However, these measures may be easier to implement if the group size is smaller, while larger groups were better able to smooth support requests and deal with the occasional drop-out. 

Hence, in the end it might be more important to try to keep heterogeneity within reasonable bounds, precisely because this type of RSG is generally quite small. To avoid the risk of withdrawal cascades, heterogeneity should be controlled as much as possible, even if that means separate RSGs will be formed for high- and low-risk subgroups. 
%%%% REF BLINDED --> CHANGE TO \cite{Vriens2020}
Alternatively, despite many new mutuals initiatives starting from the premise that the insurance should be available to everyone, and therefore refrain from applying risk differentiation in contribution levels based on (in their opinion) subjective risk profiles \cite{Vriens2020}, it may help to follow regular insurance companies in asking different premium levels to different people. Such differentiation  might help to counter the tendency of low-risk members to drop out.


\section{Conclusion and Discussion} \label{sec.conc}
%%%% REF BLINDED --> CHANGE TO \cite{Vriens2020}
Engaging in risk-sharing (or mutual insurance) arrangements can be uncertain, fragile, and unstable---but so is its uninsured alternative. So what, then, explains long-term successful participation? What allows stable participation patterns to emerge even under more heterogeneous distributions of risk? We modelled participation in risk-sharing arrangements such as informal mutual insurance organizations. The interest for this endeavor is derived from the recent revival of mutualism, that takes shape through initiatives that experiment with alternative ways of organizing insurance \cite{Vriens2020}. Under the label of Peer-to-Peer insurance, these organizations set up insurance in risk-sharing groups (RSGs). Given the large uncertainty in such initiatives, as well as the emphasis these initiatives themselves place on motives of solidarity, it seems vital to better understand the mechanisms underlying cooperation in such settings. 

We constructed a dynamic Risk-Sharing Model (RSM) that is based on the idea that members have incomplete information about the risk distribution. Through agent-based simulations we compared several degrees of risk heterogeneity to see how that affects agents' willingness to be part of the mutual. We included several individual motivations (risk aversion, solidarity, and lower reinforcement learning weights) that may compensate for the cost increases that come with higher heterogeneity. Our model showed that more heterogeneous populations are less likely to realize successful, stable RSGs, which may be explained from the fact that in more homogeneous groups individual motivations such as solidarity, risk aversion, and lower susceptibility can better compensate for cost fluctuations, for these fluctuations will be less extreme. At the same time, the model predicts that more homogeneous populations---precisely because of their similarity---are less capable of dealing with sudden, exceptionally large increases in support requests, making them fragile to sudden internal or external shocks. % At the same time, the model predicts that more homogeneous populations lose more members than necessary due to uncertainty. That is, a larger number of members that dropped out would have stayed had they known the true probability distribution. This makes more homogeneous groups more fragile with respect to (severe) temporary shocks, as it increases the probability that these introduce a second withdrawal cascade. 

These results provide potentially important clarifications on the role of heterogeneity in risk-sharing arrangements. %While homogeneous RSGs are overall more likely to succeed, heterogeneous groups that do survive are more resilient to sudden external or internal shocks. 
A logical next step would be to test these mechanisms in an empirical setting. Lab experiments are arguably most relevant, because they enable isolated tests of the core mechanisms. Moreover, lab experiments make it possible to not only analyze the basic behavior (e.g., by manipulating risk heterogeneity and comparing participation levels), but also to study the role of endogenous motivations, and how they change over the course of the experiment. 

At the same time, it should be noted that with an eye on tractability the model introduced several severe simplifications, particularly with respect to individual motivations and preferences. We therefore end with a discussion of possible extensions, in increasing order of how much they change the basic RSM, that would further fine-tune insights in the dynamics of participation. 

\subsection{Stabilizing Participation Patterns}
The current model set-up is such that member rates can never increase. One solution to this problem would be to extend the model by introducing a random new batch of agents at several time points. These new agents get the choice whether or not to join the RSG, knowing only their own risk and the number of agents that are currently part of the RSG. While this allows for membership rates to increase as well, it likely would not solve the heterogeneity problem. As long as agents do not know the risk distribution beforehand, low-risk agents might join initially, but would eventually drop out again, just like the other low-risk agents did before them. The high-risk agents do remain, meaning that if the simulation is continued infinitely, ultimately we would end up with completely homogeneous groups of high-risk agents.

The alternative would be to change the learning parameter in a way that moves reinforcement learning more in the direction of Bayesian learning. Plain reinforcement learning assumes naive agents \cite{Camerer2003a}, and it would make sense to model learning in such a way that over time agents are less susceptible to support realizations, because they get a better idea of what the true group-level probability is. At that point, they should no longer be much affected by sudden peaks in support realizations, thus stabilizing participation rates.

\subsection{Endogenous Group Formation}
Several studies, both theoretically and empirically, have shown that endogenous preferences play an important role in the formation of RSGs. The simulation results of \citeA{Hegselmann1998}, for instance, signal that agents are more likely to engage in risk-sharing relations when their risk probabilities lie closer together, and that this effect is stronger for agents with more extreme risk probabilities (both high and low). \citeA{Attanasio2012} find experimentally that close friends and relatives group assortatively on risk attitudes and are more likely to join the same risk-sharing group, while unfamiliar participants group less. This might be due to a lack of trust in a group of strangers, which relates to the social dilemma aspect of risk-sharing arrangements. Players do not trust that they can reap the benefits of cooperation if the other group members (and their risk attitudes) are not known. 

In the current setup, agents who drop out are treated as uninterested in such risk-sharing arrangements. However, they might have participated had the group composition been different (e.g., when homogeneity is larger). Rather than providing a single risk-sharing setting as an all-or-nothing decision, another possibility would therefore be to let agents choose between different RSGs. However, this does introduce the possibility that agents who currently reside in RSGs even though it is relatively costly would decide to leave these groups for another, more homogeneous RSG. Introducing endogeneity in group formation increases the number of alternative strategies and could therefore drastically alter the results with respect to how parameters like risk aversion and solidarity affect participation in RSGs. 


\subsection{Dynamic Solidarity}
In the current model, solidarity is treated as a personal characteristic that is independent of earlier experiences. There is, however, no reason to assume beforehand that solidarity is constant. Solidarity motivations can compensate for cost fluctuations, as in the current RSM, but may in turn also be affected by them. It might feel good to help one or two others, but what if one has to support multiple group members without needing the help in return? Or what if one group member repeatedly needs support? Does this only affect one's participation costs or do such different degrees of helping intensity also affect one's degree of solidarity? 

The model could be extended with solidarity implemented as following a Markov chain process in the sense that current solidarity depends in part on earlier experiences. Solidarity motives could grow as members cooperate for longer time periods, for instance because of increased social embeddedness: The better agent $i$ gets to know their fellow group members, the more willing they might be to help them, as social norms regarding helping behavior are established among them. Alternatively, when the same agent repeatedly needs support, it is also conceivable that other agents' willingness to provide it has an expiration date. Is the level of solidarity towards the same agent equal after 1, 5, or 20 support requests? An interesting next step would be to model how such dynamics affect solidarity motives over time. 

\subsection{Moral Hazard}
In the current model set-up, to participate in the RSG is the cooperative strategy, while not participating is the strategy of defection. Another means to defect, however, is not to (no longer) participate in the RSG, but to make fraudulent use of the money saved in the common fund. As in most risk-sharing models, this strategy is not accounted for in the basic model. A model extension that includes this possibility involves the implementation of parameters reflecting the probability that such misuse is caught, e.g. through random institutional checks or because of informal social control mechanisms. Moreover, the social preferences parameter would have to be extended. For agents with high solidarity motives, individual opportunistic strategies might be less attractive because of an additional guilt parameter \cite{Fehr1999,Snijders1996}.

\bibliography{Z:/research-vriens/2-Papers/1-Proefschrift}
%\bibliography{/Users/Eva/Documents/PhD/Dissertation.bib}

\end{document}