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Sharing Risk under Heterogeneity:
Participation in Settings of Incomplete
Information
Eva Vriens1,2 and Vincent Buskens3
1Institute of Cognitive Sciences and Technologies, National Research Council, Rome Via
Palestro, 32 (205)Rome 00185 Italy
2Institute for Future Studies, Stockholm, Sweden
3Department of Sociology, Utrecht University, Padualaan 14, Utrecht 3581CH, Netherlands
Correspondence should be addressed to eva.vriens@istc.cnr.it
Journal of Artificial Societies and Social Simulation 25(2) 5, 2022
Doi: 10.18564/jasss.4789 Url: http://jasss.soc.surrey.ac.uk/25/2/5.html
Received: 10-12-2020 Accepted: 24-02-2022 Published: 31-03-2022
Abstract: Motivated by the emergence of new Peer-to-Peer insurance organizations that rethink how insurance
is organized, we have proposed a theoretical model of decision-making in risk-sharing arrangements with risk
heterogeneity and incomplete information about the risk distribution as core features. For these new, informal
organizations, the available institutional solutions to heterogeneity (e.g., mandatory participation or price dif-
ferentiation) are either impossible or undesirable. Hence, we need to understand the scope conditions under
which individuals are motivated to participate in a bottom-up risk-sharing setting. The model considers partic-
ipation as a utility-maximizing alternative for agents with higher risk levels, agents who are more risk averse,
are driven more by solidarity motives, and less susceptible to cost fluctuations. This basic micro-level model
is used to simulate decision-making for agent populations in a dynamic, interdependent setting. Simulation
results show that successful risk-sharing arrangements may work if participants are driven by motivations of
solidarity or risk aversion, but this is less likely in populations more heterogeneous in risk, as individual moti-
vations can less frequently make up for larger cost deficiencies. At the same time, more heterogeneous groups
deal better with uncertainty and temporary cost fluctuations than more homogeneous populations do. In the
latter, cascades following temporary peaks in support requests more oen result in complete failure, while un-
der full information about the risk distribution this would not have happened.
Keywords: Risk-sharing, Risk aversion, Solidarity, Uncertainty, Heterogeneity, Adverse selection
Introduction
1.1 Over the last decade, new (Peer-to-Peer) insurance organizationslike Friendsurance (Germany) and Broodfonds
(the Netherlands) have emerged in countries characterized by strong insurance systems. They respond to in-
creasing privatization of social insurances, which resulted in social exclusion. In many countries welfare states
do not provide social benefits to all demographic groups (e.g., excluding self-employed workers or labor mi-
grants Baldini et al. 2016), and when insurances are oered by private insurance companies, these not rarely
refuse to insure the highest risks (or ask very high insurance premiums; Natalier & Willis 2008; Taylor-Gooby
2006). To illustrate, aer the Dutch welfare state abolished social benefits for self-employed workers in 2006,
the latter started to self-organize in voluntary risk-sharing organizations (called Broodfonds groups) in which at
most 50 people per group organize their own sickness and disability insurance (Van Leeuwen 2016). In Febru-
ary 2022 28,215 self-employed workers in 622 groups help each other by providing monetary support in case of
sickness on the basis of solidarity and trust.
1.2 While born out of need, the organizations also seek to organize insurance dierently. Inspired by mutual insur-
ance associations (mutuals) from the 19th century, the new organizations return responsibility and trust to the
policyholders by having them arrange their own safety net within small risk-sharing groups (Abdikerimova &
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
Feng 2021).1Risks are thus not insured top-down through a third party but are shared locally (people who par-
ticipate act simultaneously in the role of the insured and the insurer), which changes the dynamics underlying
participation (Jiang & Faure 2020; Liu & Faure 2018). Other than state-regulated welfare benefits, participation
in these organizations is voluntary. Apart from private insurance companies, many organizations do not dif-
ferentiate premium levels based on expected risk on the premise that risks cannot be assessed perfectly, and
benefits should be available for everyone (Vriens & De Moor 2020).
1.3 This means that the default solutions to prevent adverse selection (i.e., the tendency to attract mainly high-risk
people Arrow 1984) do not apply and begs the question why and when people are willing to share risk. If manda-
tory solidarity is impossible and price dierentiation undesirable, how can risk-sharing groups be created that
remain stable in the long term? Long-term stability is not guaranteed, because the participation costs in risk-
sharing groups fluctuate depending on how many or how oen group members request support. When more
is saved through contributions than is needed to support members in need, (a share of) the fund’s reserves are
redistributed (Breer & Novikov 2015). Thus, the lower the number of support requests, the lower the costs for
everyone. Yet, participation is costly if many request support simultaneously. This introduces an interdepen-
dency where one person’s actions (support requests) directly aect the costs for others. Assuming that costs
drive willingness to participate, the latter should also vary over time, thus making risk-sharing groups more
fragile. Members initially interested in participating could at any time revoke their membership (e.g., because
more support is requested than expected Platteau 1997), and if they do so, they take from the common fund
the share of their contributions not spent on insurance benefits.
1.4 With this in mind, we need a better understanding of why people participate in voluntary risk-sharing groups
without risk dierentiation and when participation can be resilient to cost fluctuations. While many scholars
have developed and refined (informal) risk-sharing models (Coate & Ravallion 1993; Genicot & Ray 2003; Kimball
1988; Ligon et al. 2002), these models generally assume homogeneous risk probabilities. If they do introduce
heterogeneity, formal models assume complete information about the risk distribution: Rational agents base
their own membership on a comparison between one’s own risk and that of other participants. Heterogeneity
then introduces adverse selection (Arrow 1984). Those with the lowest risk profiles will opt out of the arrange-
ment, followed by the second-lowest risk profiles, and so on—until either the entire risk-sharing arrangement
fails or only high risks remain. However, these risk comparisons are very constraining in terms of agents’ cog-
nitive abilities. Furthermore, it is unrealistic that agents have such detailed information about all other agents
in advance. Finally, this situation comes with the hidden assumption that agents are not aected by realized
outcomes (people requesting support), which is unlikely to hold in practice.
1.5 Luckily, complete information is not a requirement. As long as people assume risks to be broadly similar, risk-
sharing arrangements can still arise (Skogh & Wu 2005). We propose a dynamic agent-based model where
agents know their own risk 2but not the risk distribution. They base their decisions on an estimate of the group-
level average. This estimate is updated every time group members do or do not request support. This approach
combines forward-looking decision-making with backward-looking learning as a tool to deal with decisions un-
der uncertainty. It implies that as more support is requested in some rounds than in others, the average risk
estimate, and therefore the expected utility of participating, also fluctuates. Hence, where a complete informa-
tion model has a stable equilibrium, the alternative model continues to update. On the plus side, it may lead
to more people joining initially (and thus preventing failure from the start). On the downside, it increases the
possibility of withdrawal cascades, where agents follow each other in withdrawing from the risk-sharing group
while they may not have, had they known the true distribution.
1.6 What are the implications of this alternative approach for the eect of risk heterogeneity? The aims of this arti-
cle are to investigate theoretically (1) the extent to which heterogeneous risk-sharing groups may succeed when
agents do not have complete information; (2) whether the reliance on realized support requests increases the
chances of generating withdrawal cascades; and (3) what alternative individual factors (risk aversion, solidar-
ity, and reinforcement learning) are needed to obtain a safe bandwidth that enables stable participation rates
despite cost fluctuations. We introduced these additional individual factors, for otherwise participation cannot
be explained for everyone whose personal risk is smaller than the (estimated) group average (Coate & Ravallion
1993; Vogt & Weesie 2004).
1.7 Risk aversion reflects a preference for certain outcomes over risky situations in which outcomes are uncertain.
It is deemed crucial for understanding why (low-risk) people take out insurances (Arrow 1984), but also for par-
ticipation in (heterogeneous) risk-sharing groups and helping arrangements (Platteau et al. 2017; Vogt & Weesie
2004). Risk averse people participate even under low risk. Solidarity implies prosocial behavior towards mem-
bers of the same group (Baldassarri 2015) from which personal utility is derived (Gintis et al. 2005). The risk-
sharing groups introduce many-to-many relationships that are argued to invoke solidarity, which makes peo-
ple willing to (unconditionally) pay for the insurance of other group members.3Reinforcement learning (Macy
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
& Flache 2002), finally, is used to explain how people let current experiences influence (or override) their pre-
vious estimate about the group’s average risk. Those who do so to a larger extent are more susceptible to cost
fluctuations and therefore more likely to withdraw when costs unexpectedly (temporarily) increase.
Model Construction
2.1 We defined risk-sharing as a social dilemma under uncertainty. The creation of a common insurance fund is
a collective eort, so rather than principal-agent insurance models that predict how principals oer the insur-
ance to heterogeneous policyholders (see, e.g., Sellgren 2001 for a learning model in a heterogeneous insur-
ance market), we took a bottom-up approach with agents acting both as the insurer and the insured (Jiang &
Faure 2020) and ask when they are willing to contribute to the common fund to receive the benefit (support)
in times of need. There may be a long delay between contributing to the common fund and actually obtaining
the benefit of cooperation. In fact, since losses are undesirable even if they are (partially) covered, ideally one
never actually needs the support from the group (Platteau 1997). Hence, the short-term benefit of withdrawing
to avoid the costs of helping others may easily outweigh the uncertainty surrounding the long-term benefit of
perhaps needing and receiving the security oneself someday (Fafchamps 1992). One-time individually rational
decisions to withdraw (e.g., because of a sudden increase in support requests) may then translate into a collec-
tively worse outcome where no one is insured (Coate & Ravallion 1993). Hence, 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 under what circumstances they
remain a member to pay the costs to support others (Vriens et al. 2021).
2.2 The first models on risk-sharing as a social dilemma were formalized by Kimball (1988) and Coate & Ravallion
(1993). They formulated conditions under which self-interested agents enter informal risk-sharing arrange-
ments voluntarily ex ante without defecting ex post, where ex post defection means that agents refuse to share
their payo if they end up with the higher one. The models start from a homogeneous risk assumption: Each
agent is equally likely to end up with a high or low income. For an infinitely repeated game, the theoretical op-
timum entails full income pooling and equally sharing the aggregate available resources each period. As long
as the gain from defection is small, this can be achieved regardless of group size. Empirical tests refuted this
prediction, observing small groups, partial risk-sharing, and less than full insurance instead (Fafchamps & Lund
2003; Murgai et al. 2002; Townsend 1994). Hence, alternative models predict cooperation when participation
only requires limited commitment (Ligon et al. 2002), when cooperation should be robust also to deviations by
subgroups (Genicot & Ray 2003), when arrangements are made in networks (Bloch et al. 2008), or when par-
ticipation can take place through threshold models (i.e., not all agents have to join in round 1 Breer & Novikov
2015).
2.3 Another explanation for the lack of full insurance in large groups—and the one central in this model—is the
heterogeneity of the risk distribution. Formal models that start from risk heterogeneity (Attanasio et al. 2012;
Hegselmann 1994; Lin et al. 2019; Skogh & Wu 2005; Vogt & Weesie 2004) predicted an optimal support relation
under homogeneity, but showed that cooperation is likewise possible under heterogeneity in needing support
as long as other factors may compensate for this dierence. Apart from institutional features (like mandatory
participation or lower contribution levels for low-risk agents), risk aversion is most oen included as an indi-
vidual motivation to explain participation despite low risk levels (Vogt & Weesie 2004). Operationalized using
a concave utility function, risk averse agents are willing to incur larger insurance costs today to ascertain an
income in the future (Arrow 1984). These dynamics have been corroborated in lab experiments. While Tausch
et al. (2014) found adverse selection in risk-sharing experiments with heterogeneity in risk levels, Vogt & Weesie
(2006) found that this can be compensated for by risk aversion. Simulation studies that extend this model to
N-person settings predict thatwhen players endogenously choose with whom to engage in dyadic support rela-
tions, stable support relations arise under heterogeneity in needing support as long as heterogeneity is modest
and risk probabilities are average (Hegselmann & Flache 1998).
2.4 For risk-sharing settings, however, some scholars have questioned the validity of the risk aversion assumption,
primarily due to the low participation rates observed in field studies (see Platteau et al. 2017 for a review). They
argue that risk-sharing arrangements might not suiciently solve the uncertainty problem. While for a formal
insurance, a risk averse person chooses to pay more to be certain of not losing everything, in a risk-sharing
group participation does not guarantee support. Others may withdraw or there may not be enough money
available in the common insurance fund to cover everyone’s needs. Hence, participating and not participating
are both uncertain strategies. Following this line of reasoning, some models (e.g., Dercon et al. 2014) have posed
additional assumptions of prudence and temperance, which help selection of the most preferred alternative in
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
situations when there is more than one source of uncertainty. While such extensions could be included at a later
stage, given the large number of parameters our basic model uses plain risk aversion following a concave utility
function (Kimball 1993).
2.5 Secondly, individuals oen act out of some sort of other-regarding preferences (Chaudhuri 2011; Fehr & Gächter
2002; Ostrom 1990). A popular operationalization of other-regarding preferences is through models of inequity
aversion (Bolton & Ockenfels 2000; Fehr & Schmidt 1999), which assume that an agent is altruistic towards
others if their material payo falls below an equitable benchmark, but feels envy when their payos exceed
this benchmark. We label our social preferences ‘solidarity’ and assume that solidarity is triggered when other
participants in the risk-sharing group are in need of support. In that sense, it resembles a simplified (one-sided)
version of inequity aversion, disregarding envy and resembling models of guilt aversion (Snijders 1996; applied
to risk-sharing in Lin et al. 2019). Like in Fehr & Schmidt (1999), it is implemented additively, such that solidarity
makes agents willing to suer certain costs (i.e., have smaller shares redistributed from the common fund) to
cover the loss of others.
2.6 This implementation distinguishes within-group solidarity from general altruism (see also Baldassarri 2015)
and enables model applications where solidarity dynamically increases or decreases depending on in-group
developments (e.g., changes in the number or distribution of support requests) and relations to other group
members. It is oen argued that when solidarity is costly, there is a decay in the salience of solidarity motiva-
tions over time. Especially for repeated solidarity, the feedback of the recipient (in terms of approval) tends to
be more intermittent than the costs to execute it (Lindenberg 1998). On the other hand, if relationships across
members tighten over time and groups become more cohesive, solidarity may also increase (Vriens et al. 2021).
We start with a basic (static) definition of solidarity, but will—to illustrate how the model parameters may be
extended to particular contexts—perform sensitivity checks to show what happens to membership rates and
the success of risk-sharing arrangements when solidarity depends on the number of support requests. That is,
in an alternative model solidarity motivations decrease when the estimated dierence between one’s own risk
and that of the group increases. Simulation models of dyadic support relations have shown, for instance, that
support relations may arise even when people dier in their need for support, but that support relations are
stronger when dierences in need levels are smaller (Bianchi et al. 2020; Flache & Hegselmann 1999).
2.7 Finally, we implemented learning to the model so that agents agents do not know the risk levels of all group
members, but update their estimate of the group’s average risk over time following new support requests by
group members (or the lack thereof). In doing so, we assume bounded, backward-looking rationality (Camerer
1998). This makes deriving predictions easier and poses fewer constraints on the agents’ cognitive capabilities
(Macy & Flache 2002). With reinforcement learning (Bush & Mosteller 1955), agents follow a simple updating
rule. Each round, they update by some weight their estimate of a strategy’s profitability (here: of participating)
based on realized payos in the previous round (here: driven by support requests). Hence, when more agents
request support, the average risk estimate is revised upwards. The alternative, Bayesian learning, has agents
consider every possible collection of agent characteristics (here: of risk distributions). Each collection receives
a weight, and over time these weights are updated based on realized outcomes (Jordan 1991). Since we have an
infinite number of possible risk distributions, Bayesian learning is too demanding. We therefore started from
a simple model of reinforcement learning (cf. Macy & Flache 2002), where by some weight agents let current
experiences (i.e., support requests) override previous estimates.
Formal model
2.8 We defined an N-person Risk-Sharing Model (RSM) in which N≥2agents indexed by i∈ {1, . . . , N}choose
simultaneously to become a member (m= 1) or not (m= 0) at time τ= 0. At each time point τ≥1,
agents that joined decide whether or not to stay. If at any time point τan agent withdraws this decision is
irrevocable. In each round τ≥1, a random draw by Nature determines realized events (i.e., which agents,
if any, need support) and resulting payos. ndenotes the number of agents that are part of the Risk-Sharing
Group (RSG), with 0≤n≤N. A boundedly rational, utility maximizing agent joins the RSG as long as the
short-term expected payo under (m= 1) exceeds that of (m= 0).
2.9 Each agent receives income Yiwith probability (1 −pi)and yiwith probability pi, where Yi−yirepresents
the loss that can be insured (reflecting, e.g., failed harvests, a broken product, stolen goods, or poor health)
and piis an independent and identically distributed (i.i.d.) risk probability. If agents join the RSG, they pay
contribution cifor membership and receive benefit biunder pi, where ci< biand yi+bi≤Yi. For simplicity,
we assume homogeneous incomes, losses, contributions, and benefits (Yi=Y,yi=y,ci=c, and bi=b) and
leave heterogeneity only with respect to pi.
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
2.10 To represent a collective non-profit fund, the pooled contributions that were not needed to pay benefits get
redistributed among all agents with an individual share δ. That is, the expected profit on the level of the mutual
corresponds to4
P=
n
X
i=1
(c−pib−δ) = 0.(1)
2.11 Bounded rationality implies that players cannot foresee the risk probabilities of other agents j. Instead, we
assumed that agents know their own risk pi5but make an estimate ˆpiof the average risk probability of all
other agents. ˆpiis based on an intuition about the population’s risk probability and is updated over time aer
observing the number of support requests. We denoted for each agent ithe number of other group members
j6=ithat needed support as kτ
i, where kτ
iinfluences the agent’s average risk estimate ˆpiwith some weight as
determined by the learning parameter 0< ω ≤1, i.e.
ˆpτ
i= (1 −ω) ˆpτ−1
i+ωkτ
i
nτ.(2)
2.12 Thus, the average risk estimate at time τis a function of the previous estimate and the proportion of members
that requested support ki/n. Combining Equation 1 and Equation 2, the expected redistribution ˆ
δiboils down
to
ˆ
δi=nc −pib−ˆpi(n−1)b
n=c−ˆpib−(ˆpi−pi)b
n.(3)
2.13 In this equation, nc reflects the maximum available amount for redistribution. This means, on an individual
level, that if no one needs support everyone gets their contribution cto the common fund returned. ˆpibreflects
the estimated average loss on payments to other players, and the dierence (ˆpi−pi)either decreases or in-
creases this share spent on payments depending on whether player i’s risk is lower or higher than the estimated
other players’ average risk ˆpi. If we include that each agent receives benefit bwith probability pi, we can rewrite
this such that pib−b
nrepresents the expected net benefit from participation, while −ˆpib−b
nrepresents
the expected net costs (see Appendix A for detailed derivations). Thus, without additional assumptions of soli-
darity and risk aversion, agents would participate only when the expected net benefits are equal or larger than
the expected net costs, which is the case when pi≥ˆpi. We rewrite β=b−b
nand introduce solidarity αas the
utility obtained by agent ifrom supporting any of agents ki>0. Hence, solidarity interacts with the net costs
ˆpiβagents expect to pay to support group members, lowering by some factor 0≤α≤1these subjective costs
to (1 −α)ˆpiβ. Solidarity can then explain participation even if pi<ˆpias long as α≥ˆpi−pi
ˆpi(see Appendix A).
2.14 Finally, we included risk aversion, which to obtain a concave utility function is operationalized by adding an
exponent (1 −r)to the utility function for both strategies (m= 1) and (m= 0). In this function, 0< r < 1,
where a higher score for r means agents are more risk averse. That is, the utility of each strategy is discounted
more. This yields the final expected utility functions of
EU =((1 −pi)Y(1−r)+piy(1−r),if (m= 0)
(1 −pi)(Y−(1 −α)ˆpiβ)(1−r)+pi(y+β−(1 −α)ˆpiβ)(1−r)if (m= 1).(4)
2.15 Put dierently, when player idoes not need support, i.e. under (1 −pi), we always have Y(1−r)≥(Y−
(1 −α)ˆpiβ)(1−r). No benefits are obtained, so participation is costly only. Under pi, contrarily, we always have
(y+β−(1 −α)ˆpiβ)(1−r)> y(1−r). Hence, the crucial evaluation lies in the dierence between the subjective
values attached to the net benefits and losses. In essence, the model states that players will participate when
the individual risk pi, solidarity α, risk aversion r, and the size of the loss Y−yare suiciently large, while
estimated others’ risk ˆpiand the learning parameter ωare suiciently small. What ‘suiciently large’ and ‘sui-
ciently small’ mean, is analyzed using simulations to accommodate for the stochasticity in group-level support
requests.
Simulations
3.1 The aggregate outcomes of the RSM are not evident because of the stochasticity in the number of support re-
quests and the interdependencies between agents. The utility of participation might lie above the threshold
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for some agent iat time τ, yet sudden peaks in kior changes in nand ˆpiwhen other agents jwithdraw might
move the utility below the threshold at time point τ+ 1. Thus, the global patterns of interest are more than the
aggregation of individual attributes (Bianchi & Squazzoni 2015; Macy & Willer 2002).
3.2 We used ABM simulations to understand group-level outcomes. Under which conditions are risk-sharing groups
stable and when are they subject to withdrawal cascades? On the micro-level, the agents at risk for withdrawal
are those with a negative pi−ˆpidierence (i.e., low-risk agents). On the group-level, the severity of this risk is
captured by risk heterogeneity. The more heterogeneous agent groups are, the more likely it is to find agents
with substantial negative pi−ˆpidierences, and thus the higher the likelihood of agents withdrawing when
other (low-risk) agents do or fluctuations in kiare more extreme.
3.3 Hence, the simulation serves to compare degrees 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 eect. Below we discuss the dynamic RSM roughly following the ODD protocol
(Grimm et al. 2010). The simulation was programmed in NetLogo and analyzed in R. The NetLogo code and
model documentation can be retrieved from https://www.comses.net/codebases/3ffb006b- e50f- 4
77d-a1ba-2dce78b9b5e9/releases/1.1.0/ and the model data and R scripts are stored under https:
//osf.io/xsyb8/.
Parameter settings
3.4 The dynamic RSM consists of an agent population deciding to participate in the RSG. Agent variables are indi-
vidual risk pi, risk aversion r, solidarity α, and reinforcement learning ω. Environmental variables are benefit
size band population size N. The model is run over a number of discrete consecutive time steps τthat represent
months: Every month, agents reevaluate whether or not to proceed in the RSG.
3.5 Table 1 presents all possible and tested model parameters. Risk probabilities are randomly drawn to increase
group-level variation in support requests. For all other parameters, we chose an expressive selection of fixed
parameter values. That is, the same parameter values apply to all agents in the population. While earlier models
(e.g., Skogh & Wu 2005; Vogt & Weesie 2004 suggest that heterogeneity in a second factor is necessary to com-
pensate, what really matters is that low-risk individuals score high on this factor. High-risk agents participate
regardless, so we do not need balanced heterogeneity. Using a homogeneous population has the advantage
that we can separate the model dynamics with regards to parameter settings from the stochastic processes
resulting from dierences between expected risk and realized support.
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
Table 1: Overview of model and simulation parameters
Parameter Possible values Values simulation
Constants
Number of rounds T > 0T= 1801
Number of simulations σ > 0σ= 50
Maximum income Y > 0Y= 100
Income aer loss 0≤y < Y y = 0
Varying parameters, random
Risk probability 0≤pi≤1pi∼
U([0.1,0.2])
U([0.05,0.25])
U([0,0.3])
Estimated average risk probability at τ= 1 0 ≤ˆpi≤1 ˆpi=pi
Varying parameters, fixed
Size subject population N≥2N∈ {10,50,90}
Benefit 0< b ≤(Y−y)b∈ {40,80}
Reinforcement learning 0≤ω≤1ω∈ {0.2,0.4}
Risk aversion 0≤r≤1r∈ {0,0.2,0.4}
Solidarity 0≤a≤1α∈ {0,0.2,0.4}
Parameter combinations 324
Total observations 2,916,0002
Notes: 1Simulations could take more rounds, but per-round information was stored for up to T= 180 rounds;
2Total observations = parameter combinations ×rounds ×simulations.
3.6 We used two stopping rules: the simulation ended (1) once the RSG was empty (all members dropped out),
or (2) when, aer at least 120 rounds, membership rates were stable for 60 rounds. The longest simulation
runs took 559 rounds (before reaching stability) and 674 rounds (before ending in failure). We performed σ=
50 simulation runs of all parameter combinations to test within parameter combinations to what extent the
outcome is driven by the stochasticity resulting from the discrepancy between pand k. Since Yand ymerely
represent the bandwidth within which the dynamics take place and are not of substantial interest otherwise,
we fixed these values to Y= 100 and y= 0.
3.7 To model risk heterogeneity, we drew pifrom three uniform distributions with ranges [0, 0.3], [0.05, 0.25], and
[0.1, 0.2]. Hence, at the starting point (τ= 0), the average risk is p= 0.15 for each heterogeneity condition.
This average risk is low enough to enable fund building and payouts in small groups, yet high enough to gen-
erate suicient support requests (and thereby fluctuations). For the estimated average risk, we take ˆpi=pias
starting value, assuming that agents initially believe that others face a similar risk (cf. Skogh & Wu 2005).
3.8 For the subject population, we implemeneted N∈ {10,50,90}to compare substantially dierent group sizes,
as that might aect the severity by which fluctuations impact the estimated risk.6For benefit size, we chose b∈
{40,80}to compare a situation in which almost the entire loss is covered to the situation in which participation
is cheaper yet with a lower coverage. For risk aversion and solidarity we used parameter values {0,0.2,0.4}.
Since values > .5guarantee participation, this allowed us to compare the interesting in-between cases. Finally,
we used {0.2,0.4}as reinforcement learning weights (excluding 0 as that inhibits learning and values >.5 are
unrealistic in the sense that agents would rapidly forget about the past).
3.9 Population-level parameters n(the number of members) and p(the average risk) are the main outcome vari-
ables of interest. Groups are successful when most of the population remains a member (i.e., high n) and risk
levels remain close the population average (i.e., stable p).
Process overview and scheduling
3.10 Figure 1 presents a flowchart describing the stages of the simulation. We systematically compare all 3×3×
2×2×3×3 = 324 parameter combinations 50 times. In the first phase, the initialization phase (τ= 0),
individual risk probabilities piare randomly drawn from the range [pmin, pmax]for each agent in the population.
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
The simulation starts at round τ= 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 risk probabilities to support events. The update phase is skipped in the first round
and agents move immediately to the decision phase, where they decide, by calculating and comparing the
expected utilities of participating (m= 1) and not participating (m= 0), whether or not to join the RSG.
Figure 1: Simulation process flowchart
3.11 At this point, all agents move back to the realization phase. Only agents that chose to stay move to the update
phase, which involves updating population-level parameters n(the number of members) and p(the average
risk) as well as agent-level parameters ki(the number of members other than ithat need support) and ˆpi(the
estimated average risk). Subsequently, agents that joined the RSG are brought to another decision phase, while
agents that dropped out wait to move to another realization phase. Agents loop over these phases until one of
the two stopping conditions is met.
3.12 Note that agents only choose to join in the first round; aerwards those who joined choose whether or not
to stay. Likewise, since only RSG members move through the update phase, realizations kiare informed to
members only. Hence, agents cannot wait a few rounds to see how the RSG develops before joining, nor can
they, if they chose to leave, revoke this decision later. Without new information about ki, they do not update
ˆpi, and therefore have no incentive to participate in later rounds if they did not before. This is a strict modelling
choice, but one justified by the substantive argument that other group members will not reward leaving (a way
of defecting) by allowing the defected members to return whenever it suits better. However, it implies that as
long as no new members are introduced to the population in later rounds, the RSG can only remain stable or
decrease.
3.13 At each round, we also calculate whether agents would have participated in the RSG had they known the true
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risk distribution. In this benchmark Complete Information Model (CIM), agents know the average risk p, which
means there is no learning involved based on ωand ki. The CIM thus shows the baseline withdrawal pattern
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 fluc-
tuations and thus reflect ‘erroneous’ dropouts.
Data and analysis
3.14 Data was stored on the population level for each time point ×parameter combination for 50 simulations and
up until round T= 180. This practical limit was chosen because, if rounds are months, this generates a dataset
for what represents a 15-year period.7This resulted in a dataset with 2,916,000 observations.
3.15 Our dependent variable is membership rate: the percentage of agents that remains a member. The main pre-
dictor is risk heterogeneity, a categorical comparison of the three conditions that set the boundaries for the
populations’ risk distributions: low heterogeneity (LH), intermediate heterogeneity (IH), and high heterogene-
ity (HH). First, we explored the average stability of the dierent heterogeneity conditions visually by comparing
dierences in membership rates between the RSM and CIM. Visualizationsof the average risk pare used to assess
whether dropout follows adverse selection mechanisms (i.e., low-risk agents drop out, increasing the average
risk within the RSGs).
3.16 Subsequently, we predicted membership rates for each population using multilevel OLS regressions with sim-
ulation runs nested in parameter combinations. Hence, level 1 variance represents stochasticity resulting from
the translation of probabilities to events, whereas level 2 variance represents variation in outcomes depending
on specific parameter combinations. The analyses are used to give a qualitative description of the simulation
results; i.e., to numerically infer when more agents remain part of the RSG. To understand whether population
size, risk aversion, solidarity, and learning compensate or strengthen the eect of fluctuations for dierent risk
heterogeneity conditions, we tested for interactions between these variables and risk heterogeneity. Benefit
size, previous round dropout (relative to the population size), average estimated risk, and the time period were
included as controls. We centered and standardized all continuous variables (i.e., all but the heterogeneity
conditions) to compare eect sizes. The analyses were conducted on observations of τ > 1(because interde-
pendencies in decision-making start from round 2 onwards) and τ≤60 (because most groups were stable by
that time point).
3.17 As a sensitivity check, we ran 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
dierence between risk and realized support as well as the misconception between true risk and estimated
risk. On the aggregate level, however, these eects are not clearly visible. By running multiple simulations
over many parameter combinations, we have smoothed this process. Moreover, since shocks are based on the
average risk, they are rarely truly out of bound. Hence, we introduce correlated risks—i.e., the simultaneous
occurrence of many losses from a single event—to see whether extreme shocks set in motion new withdrawal
cascades. We compared three variations where we introduce an external shock of ps= 0.5(making 50% of the
RSG members reliant on support) in one, two, or three consecutive time periods starting from round τ= 50.
Since we were primarily interested in what happens in the rounds following the shock(s) a new stopping rule
ended all simulations aer 80 runs. We ran 30 simulations for each parameter combination, resulting in a new
dataset with 3×324 ×80 ×30 = 2.332.800 observations.
3.18 A second sensitivity check involved an alternative specification of solidarity. Solidarity in the basic model is
a fixed trait. To see how sensitive the results are we also studied what happens if solidarity depends on the
perceived average risk, decreasing when the negative dierence between pi−ˆpiincreases by multiplying α
with min(1,pi
ˆpi). We ran new simulations using the same criteria to obtain a dataset with another 2,916,000
observations (324 ×180 ×50) to compare to the original.
Results
4.1 Figure 2 plots the decay in membership rates (le panel) and increase in average risk p(right panel) for the three
heterogeneity conditions. The general pattern signals adverse selection: as the membership rate decreases, p
increases, so the decrease in membership is the result of low-risk agents dropping out. Adverse selection is
more severe for the RSM than for the CIM. Under the static CIM, stable participation patterns are still predicted
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for a substantial share of the population (i.e., 84% for LH, 69% for IH, and 56% for HH under our chosen param-
eter settings).
Figure 2: Membership rates and risk probability over time for the three risk conditions
4.2 For the RSM, conversely, we can see that despite the agents’starting assumption that the group’s average risk re-
sembles their own, support requests rapidly increase the average estimated risk ˆp(dotted line in right panel) for
all but the LH condition, causing membership rates to drop below the CIM pattern within 10 rounds already (le
panel). Hence, while roughly speaking the decay observed for the RSM in the first 10 rounds can be attributed
to the fact that the negative pi−pdierence is too big to compensate for by any of the other parameters, the
continued drop in membership rate aerwards results from the uncertainty surrounding the true risk p, the
temporary peaks in support requests k, and the resulting cost fluctuations. Simultaneously, agents do under-
estimate the average risk. The estimate only starts to catch up to the true average risk when participation decay
starts to slow down (aer approximately τ= 60). The slower decrease aer τ= 60 suggests that a temporary
increase in kmight drive an occasional agent to drop out, but that this no longer invokes cascading eects—a
pattern we do observe in earlier rounds.
4.3 Figure 3 plots the average and distribution of membership rates at time τ= 60 (see Appendix B for the distri-
bution at dierent times). It signals the importance of considering not only aggregate membership rates, but
also the variation between groups. Lower aggregate averages largely result from failed RSGs. For each hetero-
geneity condition, the RSM distribution has roughly two peaks (right panel). For LH, the highest peak remains
around full membership, but if members withdraw this nearly always ends in complete failure. The other two
conditions both moved their main peak from full membership to group failure, but did stabilize more oen on
alternative, in-between membership rates. Hence, high heterogeneity is less attractive for low-risk agents, but
does not automatically lead to cascades. Or rather, while LH groups are more successful in terms of aggregate
membership rates, some degree of heterogeneity might be beneficial for the group’s resilience, for dropout of
a few members does not necessarily result in complete failure.
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Figure 3: Average percentage and distribution of expected and realized members (in round τ= 60)
4.4 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,
regardless of heterogeneity condition. In the RSM, contrarily, 3625 groups (22%) failed—1297, 1228, and 1100,
in the LH, IH, and HH conditions. Hence, about half of the failed groups did so due to cost fluctuations and
incomplete information, and complete failure occurs more oen in LH groups than in HH groups (a dierence
of 197 groups or a 18% higher failure rate).
Predicting membership rates from individual motivations
4.5 Table 2 outputs the multilevel OLS regressions. Model 1 estimates the main eects and Model 2 includes the
interaction terms for r,α,ω, and Nwith risk heterogeneity (taking the LH condition as reference category).
Most variance lies on the second level: Model 0 has an intraclass correlation of ρ= 0.73. Hence, most variance
is explained from the model parameters. Nonetheless, 27% of the results are driven by stochasticity in piand
ki. This is an important finding: despite favorable starting conditions, RSGs may still risk failure depending on
the realization of support requests.
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Table 2: Multilevel OLS regressions on membership rate for 813,845 simulation runs ×round combinations.
Model 0 Model 1 Model 2
Intercept 64.902∗∗∗ (1.388) 75.625∗∗∗ (0.924) 75.624∗∗∗ (0.909)
Range [0.05, 0.25] −13.887∗∗∗ (1.307) −13.887∗∗∗ (1.286)
Range [0, 0.3] −23.038∗∗∗ (1.307) −23.035∗∗∗ (1.286)
Risk aversion r17.718∗∗∗ (0.534) 18.630∗∗∗ (0.909)
r×Range [0.05, 0.25] 0.537 (1.286)
r×Range [0, 0.3] −3.271∗∗ (1.286)
Solidarity α13.431∗∗∗ (0.534) 14.546∗∗∗ (0.909)
α×Range [0.05, 0.25] −0.380 (1.286)
α×Range [0, 0.3] −2.965∗∗ (1.286)
Reinforcement learning ω−3.892∗∗∗ (0.534) −3.874∗∗∗ (0.909)
ω×Range [0.05, 0.25] −0.226 (1.286)
ω×Range [0, 0.3] 0.171 (1.286)
Population size N1.567∗∗∗ (0.534) 2.483∗∗∗ (0.909)
N×Range [0.05, 0.25] −0.970 (1.286)
N×Range [0, 0.3] −1.777 (1.286)
Benefit b−2.846∗∗∗ (0.534) −2.846∗∗∗ (0.525)
% of members that le −0.030∗(0.016) −0.030∗(0.016)
Estimated average risk ˆpi−2.239∗∗∗ (0.018) −2.239∗∗∗ (0.018)
Time point τ−7.463∗∗∗ (0.015) −7.463∗∗∗ (0.015)
Residual variance 232.24 172.68 172.68
Random intercept 624.19 100.14 97.37
Intraclass correlation 0.73 0.35 0.34
Log Likelihood −3,373,026 −3,252,181 −3,252,163
Akaike Inf. Crit. 6,746,059 6,504,389 6,504,369
Bayesian Inf. Crit. 6,746,094 6,504,540 6,504,612
Notes: ∗p < 0.05;∗∗p < 0.01;∗∗∗ p < 0.001; Range [0.1, 0.2] used as reference category.
4.6 The main eects follow the pattern expected from the model input: More risk heterogeneity generates lower
membership rates; risk aversion and solidarity increase it; and membership rates decrease the larger the rein-
forcement learning weight. Population size has a positive influence, indicating that (the eects of) fluctuations
are more modest in larger groups. Benefit size has a negative eect, implying that on average costs are higher
than the benefits (reflecting low-risk agents that remain part of the RSG anyway). There is a small negative ef-
fect of the percentage of members that le at round τ−1, but the main driver of the decay observed in Figure 2
is the average estimated risk. The main eects sharply reduce the variance on level 2 (the random intercept
drops from 624 to 100), leaving most unexplained variance the result of stochasticity on level 1.
4.7 The interactions with the heterogeneity conditions in Model 2 do little to improve the explained variance on
level 2 (the random intercept goes from 100 to 97). The only significant interaction eects are related to risk
aversion and solidarity. For both, the eects are about 3% points smaller in the HH condition compared to the
LH condition, which means that higher levels of risk aversion and solidarity strengthen the negative eect of
heterogeneity. In other words, risk aversion and solidarity are more eective in maintaining participation of
low-risk agents when risk heterogeneity is small. An example will help to illustrate this finding: let us say an
agent has a risk of pi= 0.13 and solidarity α= 0.4(assuming no risk aversion). This agent is willing to accept
ˆpi> piup until ˆpi= 0.13 + 0.4×0.13 = 0.182 (Figure 4). Since average risk increases more under high
heterogeneity, solidarity is less likely to compensate for the increasing costs (they may more easily exceed the
acceptable range). For more homogeneous groups, contrarily, minimal solidarity (or risk aversion) is already
suicient to participate even if ˆpi> pi.
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Figure 4: Illustration of extent to which solidarity can compensate for larger costs
Sensitivity checks
4.8 Figure 5 plots the membership decay (top row) and average risk increase (bottom row) for the three correlated
shock conditions. The correlated shock has a strong eect in round τ+ 1, but does not have a long aermath
in terms of starting new cascades. The learning mechanism, which updates based on the number of support
requests, causes the estimated risk to increase steeply, but to decrease equally fast once the correlated shock(s)
passed. The eects are detrimental nonetheless, for they greatly reduce membership rates. The longer the
shock lasts (and thus the more it impacts average estimated risk), the steeper the total dropout. This holds
particularly for the LH condition, where average membership rates aer three shocks even end up below the
other heterogeneity conditions (Table 3).
Figure 5: Membership rates and average risk over time aer one, two, or three rounds of high correlated shocks
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Table 3: Percentage of agents that remains member at τ= 60 aer high correlated shocks
No shock 1 at τ= 50 2 at τ= [50,51) 3 at τ= [50,52)
Range [0.1, 0.2] 63% 44% 28% 20%
Range [0.05, 0.25] 49% 37% 27% 22%
Range [0, 0.3] 40% 35% 27% 24%
4.9 That LH groups are aected most can be explained from dierences both in risk average and distribution. Since
mostly low-risk agents drop out initially, the average risk of the HH groups has increased more (approaching
p= 0.22 by round τ= 50). Hence, the dierence between the average risk and the temporary shock is smaller
in the HH condition, which increases 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, in LH groups most
agents have approximately the same risk, so once the utility drops below the threshold for one, it also does for
most others, because costs and benefits are largely the same for all members.
4.10 The second sensitivity check changes the operationalization of solidarity. Recall that we see solidarity not as
general altruistic preferences, but as prosocial motives directed towards other members in the risk-sharing
group. Solidarity may therefore also increase or decrease depending on group events. For the purpose of ad-
dressing limits of risk-sharing arrangements, we considered what happens if in-group solidarity decreases if
more people request support. The larger the dierence between the estimate of the group’s average risk and
one’s own, the smaller the eect of solidarity. Naturally, this has a negative eect on the number of members
and on the success of the risk-sharing groups. Figure 6 shows how the membership rates under dynamic soli-
darity (solid lines) decrease faster than the membership rates under static solidarity (dashed lines). While the
drop is substantial, a comparison of membership rates and % of failed groups in Table 4 suggests that the size
of the decrease does not depend on the degree of heterogeneity.
4.11 The stricter measure of solidarity does not undermine its eect entirely though. Regression analyses (see Table
5 in Appendix C for detailed results) suggest a drop of b= 14.546 (Table 2) to b= 10.258 (Table 5) for the LH
condition reference category—i.e., a decrease of about one third. The interaction eect for the HH condition
remains more or less the same (b=−3.874 to b= 3.385). The positive eects of risk aversion rand population
size Nand the negative eect of reinforcement learning wall slightly increase in strength. Yet in qualitative
terms, the main conclusions do not change.
Figure 6: Membership rates compared for RSM with dynamic and static solidarity
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Table 4: Percentage of agents that remained a member and percentage of groups that failed under static and
dynamic solidarity.
% member % failed
Static Dynamic Static Dynamic
Range [0.1, 0.2] 63% 54% 24% 30%
Range [0.05, 0.25] 49% 39% 23% 28%
Range [0, 0.3] 40% 33% 20% 25%
Implications and Hypotheses
5.1 From the simulation results we can derive a set of testable hypotheses. The basic implication of risk-sharing un-
der incomplete information corresponds to classic risk-sharing models (Coate & Ravallion 1993; Kimball 1988).
Members whose risk is lower than the estimated group’s risk are at risk of dropping out, showing the general
tendency of adverse selection. Moreover, we established that if agents update their ideas about the risk of other
group members based on support requests (rather than the true risk distribution), participation rates are signif-
icantly lower and 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
also by Skogh & Wu (2005).
5.2 Beyond these general results, several dynamics deserve further attention. First, continued membership de-
pends not only on (changes in) support requests, but also on (changes in) the number of members. 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 because of new
support requests, but because of the decreased number of members. If the costs of supporting group mem-
bers have to be carried by fewer people, the costs of participation likewise increase. Hence, we derive (H1)
that larger numbers of support requests increase the probability of withdrawal cascades, where both other
members withdrawing and new support requests cause members to follow each other in deciding to leave the
risk-sharing group.
5.3 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 thatif something happens that make participation less attrac-
tive for one agent, this is probably true for most of them. Hence, (H2) the more homogeneous the risk-sharing
group, the more agents ‘erroneously’ drop out aer sudden, exceptional increases in support requests.
5.4 Finally, similar to earlier models that study risk heterogeneity (Attanasio et al. 2012; Skogh & Wu 2005; Vogt &
Weesie 2004), we included individual factors (risk aversion and solidarity) that can compensate for risk het-
erogeneity. While these factors could indeed compensate for heterogeneity, they only do so to some extent.
The more heterogeneous the risk distribution of a risk-sharing group, the higher risk aversion and/or solidarity
have to be to compensate. Thus, (H3) the lower the risk heterogeneity, the stronger the positive eect of (a)
risk aversion and (b) solidarity on the likelihood that members stay in the risk-sharing group.
5.5 Altogether, these hypotheses make for an interesting theory on the role of risk heterogeneity. More homoge-
neous groups are, on many accounts, more likely to be established, to succeed, and to maintain high mem-
bership rates. Heterogeneous groups, on the other hand, suer 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.
5.6 Since this dierence is the result of incomplete information about the risk distribution, the real threat is not
high risk, but a perception of high risk. Each time a sudden increase in support requests results in a spike in
estimated risk, agents are at risk of dropping out. This has important implications for the new Peer-to-Peer
insurances. To prevent members reacting to sudden and temporary increases in support requests, the organi-
zations must provide clear information about the long-term perspectives of the risk-sharing group. Only then
can the impact of exceptionally high support requests 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.
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Conclusion and Discussion
6.1 Engaging in risk-sharing arrangements can be uncertain, fragile, and unstable—but so is the uninsured alterna-
tive. So therefore, what are the conditions that underlie participation? What allows stable participationpatterns
to emerge even under more heterogeneous distributions of risk? A recent revival of mutualism, where so-called
Peer-to-Peer insurances set up insurance systems in small risk-sharing groups (RSGs), revived the importance of
gaining better theoretical understanding of these questions. Aer all, other than with regular insurance, RSGs
lack the standard institutional arrangements to govern behavior (e.g., mandatory participation or risk dieren-
tiation in premium levels) and introduce uncertainty with respect to how oen other group members will need
support and whether they will continue to participate. That is, two sources of uncertainty that influence the
costs of participation.
6.2 We constructed a dynamic Risk-Sharing Model (RSM) where members have incomplete information about the
risk distribution. Through agent-based simulations we compared how dierent degrees of risk heterogeneity
aects agents’ willingness to be part of the risk-sharing group. Our model showed that membership rates in
heterogeneous population are lower on average. One reason is that in more homogeneous groups individual
motivations such as solidarity and risk aversion can better compensate for cost fluctuations, for these fluctua-
tions 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 (large) increases in support requests, mak-
ing them fragile to internal or external shocks. These results provide potentially important clarifications on the
role of heterogeneity in risk-sharing arrangements. While homogeneous RSGs have larger membership rates
overall, if members do withdraw this more oen results in complete failure. Heterogeneous groups that do sur-
vive the start-up phase, on the other hand, are more resilient to sudden external or internal shocks in the long
term.
6.3 At the same time, it should be noted that with an eye on tractability the model introduced several severe sim-
plifications. 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 theories on the dynamics of participation.
Empirical calibration
6.4 The simulations have resulted in several hypotheses that could be tested experimentally, using a risk-sharing
setting that follows the set-up of this risk-sharing model. Participants to the experiment would by some prob-
ability risk losing their income (and thus earn only the show-up fee for their participation in the experiment).
They have the possibility to share this risk with the other (anonymous) participants to the experiment. By as-
signing them an individual risk but not informing them about the risk of the other group members, it can be
estimated whether participation patterns vary across treatments that apply dierent degrees of risk hetero-
geneity. The experimental data can be used to test the hypotheses and empirically calibrate the ABM to explore
risk-sharing dynamics under realistic values of risk aversion and solidarity.
Stabilizing participation oatterns
6.5 The current model set-up is such that member rates can never increase. One model extension to avoid failure
would be to introduce a random new batch of agents at several time points. These new agents get the choice
whether or not to join the existing RSG, knowing only their own risk and the number of agents that are already
participating. While this allows for membership rates to increase, it probably would not solve the heterogeneity
problem. Low-risk agents might join initially, but would eventually drop out again, just like the other low-risk
agents did before them. High-risk agents do remain, meaning that ultimately we would end up with homoge-
neous groups of high-risk agents.
6.6 The alternative is to implement a more advanced learning parameter. Plain reinforcement learning assumes
naive agents (Camerer 2003). More realistic would be to decrease the weight agents attach to support request
over time. As they get a better idea of what the true group-level risk is, they should be less aected by sudden
peaks in support requests, thus stabilizing participation rates.
Dynamic risk perception
6.7 In the current model, people update their estimate of the group’s average risk, but their individual risk remains
stable. It is assumed that people know their own risk, or at least have an estimate that is more accurate than
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their estimate of other people’s risk. Agents participate as long as their estimate of the group risk does not
exceed their (estimate of the) individual risk (much). In practice, people’s individual risk perception may vary
because despite an objectively low risk probability, they worry about the consequences (Wol et al. 2019). Such
worries may also be driven by changing external consequences, such as increasing risks in other domains (Ab-
dulkareem et al. 2020) or increasing risks of other group members. Such an implementation would, however,
ultimately mean that as the estimate of the group’s average risk increases, so does the estimate of an agent’s
personal risk. Hence, it would be easier to generate stable participation levels in risk-sharing groups.
Endogenous group formation
6.8 Several studies have shown theoretically and empirically that homogeneity preferences play an important role
in the formation of RSGs. The simulation results of Hegselmann & Flache (1998), for instance, signal that agents
are more likely to engage in risk-sharing when their risk probabilities lie closer together, especially for agents
with more extreme risks (either high or low). Attanasio et al. (2012) find experimentally that people are more
likely to join a risk-sharing group with close friends and relatives or with people with similar risk attitudes.
6.9 In the current setup, agents who drop out are treated as uninterested in risk-sharing arrangements. How-
ever, they might have participated had the group’s composition been dierent. Rather than providing a single
risk-sharing setting as an all-or-nothing decision, another possibility would therefore be to let agents choose
whether to join one of many RSGs. This does introduce the possibility that agents who now agreed to participate
in relatively costly RSGs decide to leave these groups for another, more homogeneous alternative. 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 aect participation.
Dynamic solidarity
6.10 In the current model, solidarity is a personal characteristic that is independent of earlier experiences. In a sen-
sitivity check, we studied what happens if solidarity not only compensates for cost fluctuations, but is also
aected by them. It might feel good to help one or two others, but what if one has to support multiple group
members without needing help in return? Or what if the same group member repeatedly needs support? Does
this only aect participation costs or also one’s solidarity?
6.11 While we illustrated the eects of a decay in solidarity over time (Lindenberg 1998), earlier experiences may
also increase solidarity. Solidarity could be implemented as following a Markov chain process where current
solidarity depends in part on earlier experiences. Solidarity could increase the longer members cooperate, for
instance because of increased social embeddedness. Alternatively, when the same agent repeatedly needs
support, other agents’ willingness to provide it may have an expiration date. Is the level of solidarity towards
the same agent equal aer one or ten support requests? An interesting next step would be to model how such
dynamics aect solidarity motives over time.
Moral hazard
6.12 In the current model, to participate in the RSG is the cooperative strategy, while not participating is considered
defection. Another means to defect, however, is not to (no longer) participate in the RSG, but to make fraud-
ulent use of the common fund. A model extension that includes this possibility requires parameters reflecting
the probability that such misuse is caught, e.g. through random institutional checks or informal social con-
trol. Moreover, the social preferences parameter would have to be extended with a guilt parameter that makes
opportunistic strategies less attractive for agents with high solidarity motives (Fehr & Schmidt 1999; Snijders
1996).
Model Documentation
The simulation was programmed in NetLogo (version 6.1.1) and analyzed in R (version 4.0.2). The NetLogo code
and model documentation can be retrieved from https://www.comses.net/codebases/3ffb006b-e50f
-477d-a1ba-2dce78b9b5e9/releases/1.1.0/ and the model data and R scripts for the visualizations and
analyses are stored under hhttps://osf.io/xsyb8/.
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Notes
1The new insurance initiatives cover, e.g., the deductible excess of liability insurance (Friendsurance) or an
income replacement for self-employed people in case of sickness (Broodfonds).
2While in reality people do not know their precise individual risk, they at least know more about their own
risk than about the risk of other group members. What matters is that they perceive it to be lower or higher
than the group’s average, which is captured in this model. The model could be elaborated by adding an error
term to the risk such that agents over- or underestimate their own risk, but as long as they use this estimate to
compare their own risk to the group’s average that would not change the results.
3Another role of solidarity is that it prevents excessive (fraudulent) insurance claims (i.e., it reduces moral
hazard (Van Leeuwen 2016)). This behavior is not captured in our model.
4The basic set-up assumes there are no operation costs involved. For an institutionalized setting, the profit
parameter could be extended to include some administration fee that is paid by all members and reflects a fixed
cost.
5While agents may not know their precise risk level, they at least more accurately predict their own risk than
that of others.
6In practice, most Peer-to-Peer insurance organizations have similar-sized groups. Friendsurance, for in-
stance, groups members in groups of 10. Broodfonds uses 50 members as a maximum.
7In real-life situations parameters would not remain constant (e.g., risk probability can change over time),
we should therefore not use this basic model to interpret cooperation dynamics on very long time scales.
Appendix A: Formal models and derivation of participation conditions
The Risk-Sharing Model is built in 3 steps: The baseline model RSM0compares the utility of participating m= 1
and not participating m= 0 based on rational costs and benefits given individual risk piand the estimated av-
erage group risk ˆpi. Model RSM1includes the individual parameter solidarity αthat osets the costs of support-
ing group members. The complete model RSM2includes the individual parameter risk aversion rthat discounts
uncertain outcomes over certain ones.
In RSM0, the expected utilities of participating (m= 1) and not participating (m= 0) are:
EU =((1 −pi)Y+piyif m= 0
(1 −pi)(Y−c+ˆ
δi) + pi(y+b−c+ˆ
δi)if m= 1.
If we rewrite ˆ
δi=c−ˆpib−( ˆpi−pi)b
nwe obtain for m= 1
EU(m=1) = (1 −pi)(Y−c+c−ˆpib−( ˆpi−pi)b
n) + pi(y+b−c+c−ˆpib−(ˆpi−pi)b
n)
= (1 −pi)(Y−ˆpi(b−b
n)−pi
b
n) + pi(y+b−ˆpi(b−b
n)−pi
b
n)
= (1 −pi)Y+pi(y+b)−ˆpi(b−b
n)−pi
b
n.
For RSM0we may therefore assume that agents participate if:
EU(m=1) ≥EU(m=0)
(1 −pi)Y+pi(y+b)−ˆpi(b−b
n)−pi
b
n≥(1 −pi)Y+piy
pib−ˆpi(b−ˆpi
n)−pi
b
n≥0
pi(b−b
n)≥ˆpi(b−b
n)
pi≥ˆpi.
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
In other words, pi(b−b
n)represent the net benefits of participation and ˆpi(b−b
n)the net costs. In RSM0, the
net benefits outweigh the net costs only if pi>= ˆpi.
In RSM1solidarity is included as parameter αthat defines the extent to which agents are willing to pay the costs
of support for their group members. Implemented as (1 −α)ˆpi(b−b
n)it compensates for the net costs of
participation. This means that for m= 1
EU(m=1) = (1 −pi)Y+pi(y+b)−(1 −α)ˆpi(b−b
n)−pi
b
n.
For RSM1we can therefore assume that agents participate if:
EU(m=1) ≥EU(m=0)
pi(b−b
n)≥(1 −α)ˆpi(b−b
n).
If we take β=b−b
n, we obtain
piβ≥(1 −α)ˆpiβ
αˆpiβ≥ˆpiβ−piβ
α≥ˆpi−pi
ˆpi
.
Hence, in RSM1solidarity αexplains that agents may participate even if pi<ˆpias long as α≥ˆpi−pi
ˆpi.
Finally, RSM2includes risk aversion r. This is operationalized using a concave utility function by adding an
exponent (1 −r)to the utility function for both strategies (m= 1) and (m= 0). That is, the utility of each
strategy is discounted more for agents who are more risk averse. This yields the final expected utility functions
of
EU =((1 −pi)Y(1−r)+piy(1−r),if (m= 0)
(1 −pi)(Y−(1 −α)ˆpiβ)(1−r)+pi(y+β−(1 −α)ˆpiβ)(1−r)if (m= 1).
Since the participation conditions of RSM2cannot be derived analytically, they are studied through agent-based
simulations.
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Appendix B: Distributions
Figure 7: Distribution of proportion of members n/N at time points τ∈ {2,6,12,24,60,180}
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
Figure 8: Distribution of average need probability pat time points τ∈ {2,6,12,24,60,180}
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
Appendix C: Multilevel OLS regressions with dynamic solidarity
Table 5: Multilevel OLS regressions on membership rate for 787,650 simulation run ×round combinations with
dynamic solidarity.
Model 0 Model 1 Model 2
Intercept 59.175∗∗∗ (1.308) 69.658∗∗∗ (0.847) 69.654∗∗∗ (0.800)
Range [0.05, 0.25] −15.233∗∗∗ (1.197) −15.229∗∗∗ (1.131)
Range [0, 0.3] −22.994∗∗∗ (1.197) −22.987∗∗∗ (1.131)
Risk aversion r19.133∗∗∗ (0.489) 22.143∗∗∗ (0.800)
r×Range [0.05, 0.25] −2.636∗∗ (1.131)
r×Range [0, 0.3] −6.394∗∗∗ (1.131)
Solidarity α8.524∗∗∗ (0.489) 10.258∗∗∗ (0.800)
α×Range [0.05, 0.25] −1.815 (1.131)
α×Range [0, 0.3] −3.385∗∗∗ (1.131)
Reinforcement learning ω−4.726∗∗∗ (0.489) −4.962∗∗∗ (0.800)
ω×Range [0.05, 0.25] 0.083 (1.131)
ω×Range [0, 0.3] 0.625 (1.131)
Population size N1.770∗∗∗ (0.489) 3.067∗∗∗ (0.800)
N×Range [0.05, 0.25] −1.512 (1.131)
N×Range [0, 0.3] −2.377∗∗ (1.131)
Benefit b−3.376∗∗∗ (0.489) −3.376∗∗∗ (0.462)
% of members that le −0.313∗∗∗ (0.018) −0.312∗∗∗ (0.018)
Estimated average risk ˆpi−2.504∗∗∗ (0.020) −2.504∗∗∗ (0.020)
Time point τ−9.239∗∗∗ (0.017) −9.239∗∗∗ (0.017)
Residual variance 289.91 201.38 201.38
Random intercept 554.56 77.29 68.97
Intraclass correlation 0.66 0.28 0.26
Log Likelihood −3,351,798 −3,208,028 −3,207,998
Akaike Inf. Crit. 6,703,603 6,416,081 6,416,038
Bayesian Inf. Crit. 6,703,638 6,416,232 6,416,281
Notes: ∗p < 0.05;∗∗p < 0.01;∗∗∗ p < 0.001; Range [0.1, 0.2] used as reference category.
JASSS, 25(2) 5, 2022 http://jasss.soc.surrey.ac.uk/25/2/5.html Doi: 10.18564/jasss.4789
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