# Collaborative Insurance Sustainability
and Network Structure

Arthur Charpentiera∗, Lariosse Kouakoub, Matthias Löwec,
Philipp Ratza & Franck Vermetb

a Université du Québec à Montréal (UQAM), Montréal (Québec), Canada
b EURo Institut d’Actuariat (EURIA), Université de Brest, France
c University of Münster, Germany

∗Corresponding author<EMAIL_ADDRESS>
###### Abstract

The peer-to-peer (P2P) economy has been growing with the advent of the
Internet, with well known brands such as Uber or Airbnb being examples
thereof. In the insurance sector the approach is still in its infancy, but
some companies have started to explore P2P-based collaborative insurance
products (eg. Lemonade in the U.S. or Inspeer in France). The actuarial
literature only recently started to consider those risk sharing mechanisms, as
in Robert2 or Feng. In this paper, we describe and analyse such a P2P product,
with some reciprocal risk sharing contracts. We consider the case where
policyholders still have an insurance contract, but the first self-insurance
layer, below the deductible, can be shared with “friends”. We study the impact
of the shape of the network (through the distribution of degrees) on the risk
reduction. We consider also some optimal settings of the reciprocal
commitments, and discuss the introduction of contracts with “friends of
friends” to mitigate some possible drawbacks of having people without enough
connections to exchange risks.
Keywords: collaborative insurance; convex order; column-stochastic matrix;
degrees; doubly-stochastic matrix; deductible; majorization; network; peer-to-
peer insurance; reciprocal contracts; risk sharing.
Acknowledgement: The authors wish to thank Romuald Élie, Mathieu Laurière,
Tran Viet Chi, Miguel Campista, Luis Costa, Matteo Sammarco and Harpreet Kang
as well as participants of the IME 2021 Conference,  the Quantact “summer
seminar” and the ASTIN 2021 Colloquium for discussions and feedbacks, as well
as Thomas Richard,  two anonynous reviewers and the editor. Arthur Charpentier
received financial support from the Natural Sciences and Engineering Research
Council of Canada (NSERC-2019-07077) and the AXA Research Fund. Matthias
Löwe’s research was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587,
Mathematics Münster: Dynamics - Geometry - Structure.

##  1 Introduction

###  1.1 Collaborative insurance

In recent years, the global financial crisis of 2008 and the explosion of
digital technologies have involved the rise of a collaborative (or peer-to-
peer) economy. Indeed, consumers are more likely to sell, buy, rent or make
available goods or services to other individuals through the Internet, thanks
to the emergence of platforms such as Airbnb, Uber or Lyft (see
Fargere_Souki_2019). The insurance sector is not left out, with several
initiatives called collaborative insurance around the world, for example
Friendsurance in Germany since 2010, Inspeer in France in 2014, and Lemonade
in the U.S. since 2015.

The business model of peer-to-peer insurance is based on the principle of
bringing people together and pooling their risks so that they are mutually
insured. Furthermore, this insurance model is more efficient when
policyholders make up a community with similar behaviors or needs according to
the homophily principle ( birds of a feather flock together) which structures
network ties of every type, including marriage, friendship, work, and other
types of relationship (see McPherson_Smith-Lovin_Cook_2001). Indeed,
biener2018can showed that group policies with pro-social preferences can
alleviate moral hazard of agents.

Several technology-based brands have emerged about collaborative insurance
with the aim of challenging the historical players in the insurance sector by
significantly improving the customer experience. In addition, they all operate
with a bulk purchasing method and a relative participation of traditional
insurance players. rego2020insurance breaks down those last ones into three
different classes: the broker model (eg. Friendsurance), the carrier model
(eg. Lemonade) and the self-governing model (eg. Teambrella). The broker model
(see also clemente2020broker) and the carrier model (see also
Fargere_Souki_2019) rely on traditional insurance players but allow customers
to take on part of the risks insured by the group they happen to fall into or
choose to adhere to and take back a portion of their profits, or at least make
customers feel like they are taking on those risks and taking back such
profits. The leading characters in such models appear to play, to a large
extent, the same roles traditionally ascribed to insurers and insurance
intermediaries. The self-governing model is the final stage of collaborative
insurance where all insurance intermediaries are removed to self-insure.
vahdati2018self proposes a self-organizing framework for insurance based on
IoT (Internet of Things) and blockchain, which eliminates the main problems of
traditional insurers such as the transparency of their operations. This last
model of peer-to-peer insurance is also know as decentralized insurance.

###  1.2 Centralized and decentralized insurance

Insurance is usually defined as “the contribution of the many to the
misfortune of the few”, as said in charpentierdenuit. More specifically, in
developed countries, a centralized insurance company offers contracts to
policyholders, and risk is then shared with the insurer, or more specifically,
risks are transferred from individuals to the centralized insurance. Such a
collective transfer is possible because of the central limit theorem that
guarantees that the relative variability of individual contracts decreases
with the number of insured in the pool. Conventional structures are either
stock companies (owned by shareholders), and mutual companies, which are owned
by policyholders, the later might create some sort of solidarity among
policyholders, as discussed for instance in Abdikerimova. But in emerging
economies, where the insurance sector is neither properly regulated, nor fully
developed, alternatives can be considered to provide risk sharing among
agents, that could be either based on small groups (as in tontines) or based
on multiple one-to-one reciprocal engagements. In this article, we will focus
on those reciprocal mechanisms, where policyholders will cross-insure each
other (see Norgaard, and the comment by Reinmuth, on actuarial and legal
aspects of those reciprocal contracts).

It should be mentioned that decentralized insurance platforms now reappear in
developed economies, mainly under the scope of technological solutions,
usually through some blockchain approaches, so that even the control is
performed collectively. In this paper, we will assume that a standard
insurance company has authority to accept or deny a claim, and then set its
value. So we will not discuss possible bargaining among policyholders to
settle a claim. All those peer-to-peer contracts are typically administered by
an independently owned managing agent called an attorney-in-fact, and for
simplicity, it will be the insurance company. Furthermore, such reciprocal
engagement have very limited liability, so there is still a need for an
insurance company to settle larger claims (possibly also more complicated from
a legal perspective, such as bodily injuries claims in motor insurance). The
peer-to-peer insurance mechanism is here to provide an additional layer of
risk sharing in the standard insurance layer scheme, as considered by Inspeer,
see Figure 1.1. We will prove here that such additional layer will lower the
variability of the losses for the insured, but its efficiency depends on the
network of reciprocal commitments considered to share the risk among peers.
Since the original first layer in insurance is based on the deductible of
insurance contracts, a natural application will be on to share part of that
deductible layer with friends through reciprocal engagements.

Figure 1.1: $(a)$ is the standard 3 layer scheme of insurance coverage, with
self insurance up to some deductible, then traditional insurance up to some
upper limit, and some excess coverage provided by the reinsurance market.
$(b)$ is the scheme we will study in this paper, with some possible first
layer of self-insurance, then a peer-to-peer layer is introduced, between self
insurance and traditional insurance. With our design, on a regular network, a
full coverage of that second layer is possible  for all participants. Above
the deductible, claims are paid using traditional insurance. $(c)$ corresponds
to the case where the number of reciprocal contracts signed are heterogeneous:
in that case, it is possible that the peer-to-peer coverage is smaller than
the second layer, and a remaining self insurance layer appears, on top.
Finally $(d)$ is the case we describe at the end of the paper, where contracts
with friends of friends can be considered, as an additional layer.

###  1.3 Motivations for deductibles in insurance

Deductibles are a popular technique insurance companies use to share costs
with policyholders when they claim a loss, in order to reduce moral hazards.
Insurers expect that deductibles help mitigate the behavioral risk of moral
hazards, meaning that either policyholder may not act in good faith, or that
they may engage in risky behavior without having to suffer the financial
consequences. From an actuarial and statistical perspective, given a loss
$y_{i}$ for a policyholder, and a deductible $s$, we can write

$y_{i}=\underbrace{\min\\{y_{i},s\\}}_{\text{policyholder}}+\underbrace{(y_{i}-s)_{+}}_{\text{insurer}}=\begin{cases}y_{i}+0\text{,
if }y_{i}\leq s\\\ s+(y_{i}-s)\text{, if }y_{i}\geq s\\\ \end{cases}$

(we use $s$ – for self insurance – instead of $d$ to denote the deductible,
since $d$ will be intensively used afterwards to denote the number of
reciprocal commitments with friends – related to the degrees of nodes on the
network, as discussed in the next section). One might assume that if
$y_{i}\leq s$, some policyholders may not report the loss, since the insurance
company will not repay anything (and in a no-claim bonus systems, there are
strong incentives not to declare any small claim, if they don’t involve third
party, as discussed in arthurRisks). In this case, the true process of
accident occurrences is not the same as the process of claims. For instance,
if the accident process is an homogeneous Poisson process with intensity
$\lambda$, the claiming process is an homogeneous Poisson process with
intensity $\lambda\mathbb{P}[Y>s]$. Therefore, in the context of risk sharing
with a centralizing insurance company, the number of claims reported might
increase when lowering the deductible since a loss below $s$ might now be
claimed since friends might repay a share of it, as discussed in
lee2017general.

From an economic perspective, it is usually assumed that without deductibles,
some insured could be tempted to damage their own property, or act recklessly,
leading to moral hazard, as discussed intensively in eeckhoudt1991increases,
meyer1999analyzing, Halek and in the survey by winter2000optimal. In the
context of asymmetric information, cohen2007estimating mention that if the
policyholder can chose the level of the deductible, it might be a valid
measure of the underlying risk, even if it is not possible to distinguish risk
aversion and the true level of the risk (known by the policyholder, and un-
observable for the insurance company, without additional assumption).
dionne2001deductible also mention fraud as the most important motivation for
insurance companies to introduce deductibles. Following the seminal work
papers of Duarte, Larrimore or XuFraud2015, trust is a key issue in peer-to-
peer risk sharing, that might yield two contradictory behaviors. On the one
hand, Artis shows that known relatives and friends might fraud together; on
the other hand, Albrecher claims that fraudulent claims should decrease, and
Paperno claims that bad-faith practices can be significantly mitigated by
implementing a peer-to-peer (classical anti-fraud measures being costly and
hostile).

But even if there are many theoretical justifications for introducing
deductibles, they can be an important financial burden for policyholders.
Since the function $s\mapsto\mathbb{E}\big{[}(Y-s)_{+}\big{]}$ is decreasing,
the higher the deductible $s$, the smaller the premium. The mechanism we will
study in this paper is based on the following idea: policyholders purchase
insurance contracts with a deductible $s$ (that will be less expensive than
having no deductible), and they consider a first layer of collaborative
insurance, with peers (or friends), as in Figure 1.1, with some possible self
retention. If this layer is a risk sharing among homogeneous peers, this will
have no additional cost for the insured. But it can lower individual
uncertainty.

###  1.4 Contributions

In this article, we will instead consider a risk-sharing mechanism which is
entirely based on the network structure itself. In that regard, our
contribution can be understood as a generalization of the mechanisms proposed
in Feng and abdikerimova2021peer. Whereas these contributions consider the
network as a basis on which a standard sharing mechanism is implemented (i.e.,
a pooled risk sharing with heterogeneous risks) some rather restrictive
hypotheses are imposed. For example, the graph needs to be complete or it
needs to be partitioned into a set of complete subgraphs. We depart from that
assumption and propose a mechanism that takes into account the underlying
network structure when setting up the sharing mechanism. This enables us to
depart from the traditional pooling scheme and instead rely entirely on a
decentralized system.

Further, and as described above, we believe that taking account of the network
structure might come with additional benefits, for example to counter moral
hazard, as described in biener2018can. In that sense, this paper serves as an
introduction into the proper mathematical formulation of a risk sharing
mechanism given a specific network structure. We provide a mathematical
framework and simulate the proposed mechanism in a wide variety of settings.
Throughout the paper we assume i.i.d. risks for each member of our network. In
a sense this might seem like an overly restrictive assumption to make, but
given the novel nature of the approach, we believe the relaxing of this
hypothesis would go beyond the scope of this paper. Further, as will be shown,
i.i.d. risks does not imply homogeneous losses, which weakens the hypothesis
somewhat.

###  1.5 Agenda

This article will be divided in four main parts. In the first one,
corresponding to sections 2 and 3, we will provide some definitions and
mathematical properties, about networks and risk comparisons. In the second
one, section 4, we will start with a simple presentation of those reciprocal
contract designs. More specifically, we will assume that we have a network of
friends, homogeneous in risks and contracts, and they all share risks, through
reciprocal contracts, up to a given magnitude $\gamma$ (if a policyholder $i$
experiences a loss, all his friends will contribute with the same amount of
money, each of them up to the upper limit $\gamma$). In this second part, we
will briefly describe such a risk sharing scheme and run some simulations to
understand sustainability111Codes used in this paper can be found on
https://github.com/phi-ra/collaborative_insurance.. Those designs can be
visualized on Figure 1.2 (introducing possible extensions that will be
discussed later on). Then, in the third part, we will discuss optimization of
the design. A natural approach would have been to study network specification,
that is removing unnecessary reciprocal contracts (as we will see, having too
many contracts can actually add more risk to some insured), but the results
have not been computationally conclusive. Thus, we will consider some optimal
computation of individual magnitudes $\gamma$, now denoted $\gamma_{(i,j)}$,
that can depend on both friends $i$ and $j$, in section 5. That approach is
promising, its main drawback is that we optimize the scheme from a ‘social
planer’ perspective: we optimize the overall coverage, and it might not be
optimal for each individual. Constrains in the linear program will make the
scheme sustainable, but possibly not optimal for agents. And finally, in the
last part, in section 6, we consider the introduction of contracts with
friends of friends (with a smaller commitment), for policyholders who might
not have enough connections to get an optimal risk coverage.

Figure 1.2: The five risk sharing mechanisms, with (1) fixed engagements, (2)
personalized engagements, (3) edges removal, (4) self contributions and (5)
friends of friends.

##  2 Mathematical Formalisation of Networks

A network (or graph) $\mathcal{G}$ is a pair $(\mathcal{E},\mathcal{V})$,
where $\mathcal{V}=\\{1,2,\cdots,n\\}$ is a set of vertices (also called
nodes) that correspond to insured and $\mathcal{E}$ is a set of edges (also
called links), which are pairs of vertices, e.g. $(i,j)$. Here we will
consider undirected networks, where an edge $(i,j)$ means that vertices $i$
and $j$ have a reciprocal engagement. A convention is that $(i,i)$ is not an
edge. Note further that $n$ is the network size, i.e. the number of nodes,
that corresponds to the classical notation of being the portfolio size, in
insurance. In this article, the network is static: neither nodes nor edges can
added.

Classical examples of networks are email exchange, where nodes are email
accounts, and we have (directed) edges (denoted $i\to j$) when account $i$
sent an email to $j$, genealogical trees, where nodes are individuals, and we
consider a fatherhood directed edge ($i\to j$ meaning that $i$ is the father
of $j$) or a sibling undirected edge ($i\leftrightarrow j$ or $(i,j)$ if $i$
and $j$ are siblings), as in cabrignac2020modeling. (Note that risk sharing
within families has always been important, as discussed in Hayashi or more
recently AKIN2015270). On social media, such as Twitter, nodes are Twitter
accounts, and $i\to j$ could mean that $i$ follows $j$; on Facebook or
Linkedin, we have undirected edges, $i\leftrightarrow j$ meaning either that
nodes $i$ and $j$ are “friends”, “followers”, or simply “connected ”
(depending on the platform). In this article, we will consider a friendship
network, where reciprocal contracts are undirected edges. Some visual
representations of networks are given in Figure 2.1.

Formally, such a network can be represented by an $n\times n$ matrix $A$,
called adjacency matrix, where $A_{i,j}\in\\{0,1\\}$ with $A_{i,j}=1$ iff
$(i,j)\in\mathcal{E}$. Let $A_{i\cdot}$ (and $A_{\cdot i}$) denote the vector
in $\\{0,1\\}^{n}$ that corresponds to row $i$ (and column $i$) of matrix $A$.
Given a network $\mathcal{G}=(\mathcal{E},\mathcal{V})$ and the associated
adjacency matrix $A$, the degree of a node $i$ is

$d_{i}=\displaystyle{A_{i\cdot}^{\top}\boldsymbol{1}=\sum_{j=1}^{n}A_{i,j}}.$

In this article, we do not consider isolated nodes, with no connection
($d_{i}=0$). Actually, we will assume that policyholders should have a minimum
number of friends to start sharing. Note that when someone is connected to
everyone we have $d_{i}=n-1$. For convenience, let $\mathcal{V}_{i}$ denote
connections of node $i$, i.e.

$\mathcal{V}_{i}=\left\\{j\in\mathcal{V}:(i,j)\in\mathcal{E}\text{ or
}A_{i,j}=1\right\\}$

such that $\text{card}(\mathcal{V}_{i})=d_{i}$. A regular network is a graph
in which every vertex has the same degree. On the left of Figure 2.1, the two
graphs, with $n=20$ vertices, are regular since all vertices have degree $4$.
Observe here that the degree function does not uniquely describe the network.

A random network is obtained when assuming that edges are randomly chosen,
according to some process, that will yield different families of networks (as
discussed afterwards). Thus, the degree of a node taken at random in
$\mathcal{V}$ is now a random variable, with some distribution on
$\\{0,1,2,\cdots,n-1\\}$ (or $\\{1,2,\cdots,n-1\\}$ if vertices cannot be
isolated).

Random networks are usually described by a random process which generates
them, such as the classical Erdős–Rényi (see ER59), where each edge $(i,j)$ in
$\mathcal{V}\times\mathcal{V}$ (excluding $(i,i)$) is included in the network
with the same probability $p$, independently from every other edge. Then $D$
has a binomial distribution $\mathcal{B}(n-1,p)$ which can be approximated by
a Poisson distribution $\mathcal{P}(p/(n-1))$, when $n$ is large, and $p$ is
not too large. In that case, note that $\text{Var}[D]\sim\mathbb{E}[D]$. This
can be visualized in the middle of Figure 2.1. The preferential attachment
method was proposed by Barabasi_albert_1999 (and might be related to the small
world model by Watts_strogatz_1998). In this model, starting from some small
network, a new node is created at each time step and connected to existing
nodes according to the preferential attachment principle: at a given time
step, the probability $p$ of creating an edge between an existing node $i$ and
the new node is $[(d_{i}+1)/(|\mathcal{E}|+|\mathcal{V}|)]$, where
$\mathcal{E}$ is the set of edges between nodes $\mathcal{V}$ (or in a
nutshell, it is proportional to $d_{i}$, i.e. it is more likely to connect to
popular nodes). This technique produces some scale-free networks where the
degree distribution follows a power law (see also Caldarelli2007). Examples of
such graphs are given in the column right of Figure 2.1. As discussed in
Barabasi_albert_1999 Caldarelli2007, Aparicio, or more recently
charpentier2019extended, most social networks are scale-free networks (with a
power law type of distribution for degrees). Even if being “friend ” on
Facebook is not as strong as a financial reciprocal commitment (that we want
to model here), intuitively, insurance networks we consider could satisfy that
feature.

$(a)$ | $(b)$ | $(c)$
---|---|---
|  |
|  |
Figure 2.1: Three type of networks, with identical expected value, here
$\mathbb{E}[D]=4$ (on average, a node chosen at random is connected to four
others). $(a)$ are regular networks, where $D=4$ (and $\text{Var}[D]=0$),
$(b)$ are Erdös–Rényi networks, with $\text{Var}[D]\sim\mathbb{E}[D]$ and
$(c)$ are scale-free networks with some nodes highly connected,
$\text{Var}[D]>\mathbb{E}[D]$ (and others with possibly only one connection)
Figure 2.2: Distribution of $D$, when $\overline{d}=20$, with three different
standard-deviations: 5, 10 and 15. Such distributions will be used in the
numerical applications. Note that the support of $D$ is here
$\\{5,6,\cdots,1000\\}$, where rounded and shifted Gamma distributed were
used.

As we will see in this paper, the shape of the network, described through the
distribution of the degrees will have a major influence on the sustainability
of the system. To simulate a variety of different scenarios, we will not
restrict the network generating process to a single method described above.
Instead, by introducing the constraint that the sum of degrees is even and
using the approach provided by viger_latapy, in igraph, it is possible to
create a random graph given a specific degree vector
$\boldsymbol{d}=(d_{1},\cdots,d_{n})$ that is obtained by drawing
independently from some distribution. In essence this reverses the simulation
process above. Instead of generating a network and deriving the degree
distribution from there we generate the degree distribution and derive (a)
graph from it.

This has the advantage that the only source of uncertainty for mean and
variance of the degrees is coming from the random generation of
$\boldsymbol{d}$ from a distribution with _given_ mean and variance. For our
numerical applications we will use a rounded and shifted gamma distribution
such that every node has at least five connections to simulate our degree
vector:

$D\overset{\mathcal{L}}{=}\min\\{5+[\Delta],(n-1)\\},$ (2.1)

where $\Delta$ follows a gamma distribution with mean $\mu-5$ and variance
$\sigma^{2}$. The shift and minimum condition ensures that every node has at
least five connections and no multiple edges exist. The advantage of this
approach is that by changing the $\sigma$ parameter, degree distributions that
resemble a Poisson distribution can be generated just as well as ones that
resemble an exponential distribution (ie. models generated by Erdős–Rényi or
the preferential attachment method respectively). This allows to test the
proposed framework in a variety of settings by changing one parameter only. In
the extreme case this will result in graph building a regular mesh on the left
hand side ($\sigma=0$) or a power-law degree distribution graph ($\sigma>>0$).
Figure 2.2 visualises the degree distribution from three networks generated
from equation 2.1 with a constant mean but changing standard deviation.

As we will see in this paper, the shape of the network, described through the
distribution of the degrees will have a major influence on the sustainability
of the system. And interestingly, given a distribution for $D$, or more
specifically a collection of $\\{d_{1},d_{2},\cdots,d_{n}\\}$ that be obtained
by drawing independently $D_{1},D_{2},\cdots,D_{n}$, with a minor additional
constraint (the sum of the degrees should be even) one can generate a network
with such degrees. These networks have become known under the name
configuration model (cf. [newman2010networks, Chapter13.2]). However, in such
networks one has to allow for self-loops and multi-edges (see Chatterjee for a
probabilistic perspective). chung2002connected suggested a simple and easy
technique to generate some random graph of given (node) size, with some
expected given degree distribution. Heuristically, given two nodes $i$ and $j$
with respective degrees $d_{i}$ and $d_{j}$, the probability that $(i,j)$ is
an edge is proportional to the product $d_{i}d_{j}$,

$A_{i,j}=A_{j,i}=\begin{cases}1,\text{ with probability proportional to
}d_{i}d_{j}\\\ 0~{}\text{ otherwise}\\\ \end{cases}$

More recently, Bayati_2009 suggests some algorithms to generate networks with
a given size $n$ and a given vector of degrees, $\boldsymbol{d}$ (possibly
drawn randomly from a given probabilistic distribution), inspired by BENDER,
MolloyReed and Newman2002. These algorithms are implemented in the igraph
library, in R and Python, see igraph. More specifically, degree.sequence.game
generates a network of size $n$ from a vector
$\boldsymbol{d}=(d_{1},\cdots,d_{n})$ where the degrees of the generated
network is exactly $\boldsymbol{d}$. Note that there is also the
randomNodeGraph function of R package graph, see Gentleman. Therefore, if we
generate random graphs and compute (for instance) the variance of the degree,
it will be the variance of $\boldsymbol{d}$, so, the only source of
uncertainty here when using generations of random graphs is simply coming from
the random generation of $\boldsymbol{d}$ from a distribution with a given
mean and a given variance.

##  3 Linear Risk Sharing Principles

In order to obtain some general results on the sharing schemes and the impact
of the shape of the network, it is necessary to make some assumptions.
Consider $n$ nodes, indexed with $i=1,\cdots,n$. Each node corresponds to a
policyholder, facing a random yearly loss, that might occur with probability
$p$, and cost $Y_{i}$, that has distribution $F$. For convenience, let $Z_{i}$
denote the indicator for the event that $i$ claims a loss, as in the
individual model in actuarial science. $Z_{i}$ is a Bernoulli variable with
probability $p$. Thus, here we consider the simple case where only one loss
might occur.

######  Hypothesis 3.1.

Policyholders can face only one loss, or none, over a year.

We mainly assume Hypothesis 3.1 to facilitate the model and to describe the
mechanism. It is also possible to assume that many claims might occur, and
that the total sum of contributions is bounded (by the magnitude denoted
$\gamma$). This will not change much the analysis, and it will only make
notations more complex to read.

We assume also that any insured can purchase an insurance contract with a
fixed deductible $s$, so that the random wealth of insured $i$ at the end of
the year is

$X_{i}=Z_{i}\cdot\min\\{s,Y_{i}\\}=\begin{cases}0\text{ if no claim occurred
}(Z_{i}=0)\\\ \min\\{s,Y_{i}\\}\text{ if a claim occurred }(Z_{i}=1)\\\
\end{cases}$

Note that those notations will be used when describing the algorithms used to
simulate such insurance schemes.

######  Hypothesis 3.2.

All risks $X_{i}$’s are assumed to be independent and identically distributed
random variables. Furthermore, all policyholders have similar insurance
contracts with the same deductible $s$.

Hypothesis 3.2 is important to have a simple reciprocal and fair mechanism. If
two policyholders $i$ and $j$ do not have the same probability to claim a loss
(distribution of $Z$), or the same distribution of individual claims
(distribution of $Y$), it becomes more difficult to derive a fair contribution
for some risk exchange (this was discussed recently in Robert).

We note at this point, that capping the loss (with a deductible) will result
in a non-linear risk sharing. Nevertheless, in this Section we will discuss
linear risk sharing, this is done as it helps us to derive some heuristics for
our proposed risk sharing mechanism. Though we note that in the limiting case
where the deductible $s\rightarrow\infty$ (corresponding to the case where the
risk is _only_ shared within the graph community) the findings would apply to
our case.

###  3.1 Risk preferences and ordering of risk

Ohlin1969, inspired by Karlin1963, suggested to use the convex order to solve
the optimal insurance decision problem, as discussed in denuit2006actuarial or
CHEUNG2015409. Given a random loss $X$, and an insurance scheme offering
indemnity $x\mapsto I(x)$ against a premium $p$, an agent will purchase the
insurance if $\xi\preceq_{CX}X$ for the convex order, where $\xi=X+\pi-I(X)$
(where $\pi$ is the associated premium to transfer that risk). This formalism
is a natural translation of the expected utility model in insurance, where
agent having wealth $w$ and (concave) utility $u$ agree to purchase a contract
offering utility $I$ against a premium $\pi$ if
$\mathbb{E}[u(w-X)]\leq\mathbb{E}[u(w+I(X)-\pi-X)]=\mathbb{E}[u(w-\xi)]$. As
in shaked2007stochastic or denuit2006actuarial, define the convex order as
follows,

######  Definition 3.3 (Convex order).

Consider two variables $X$ and $Y$ such that

$\mathbb{E}[g(X)]\leq\mathbb{E}[g(Y)]\text{ for all convex functions
}g:\mathbb{R}\rightarrow\mathbb{R},$

(provided the expectations exist). Then $X$ is said to be smaller than $Y$ in
the convex order, and is denoted as $X\preceq_{CX}Y$.

As we will see below, the inequality $X\preceq_{CX}Y$ intuitively means that
$X$ and $Y$ have the same magnitude, as $\mathbb{E}[X]=\mathbb{E}[Y]$, but
that $Y$ is more variable than $X$. More specifically:

######  Proposition 3.4.

If $X\preceq_{CX}Y$, then $\mathbb{E}[X]=\mathbb{E}[Y]$ and
$\text{Var}[X]\leq\text{Var}[Y]$.

###### Proof.

For the expected value, consider $g(x)=\pm x$ (Equation (3.A.2) in
shaked2007stochastic) and for the variance $g(x)=x^{2}$ (Equation (3.A.4) in
shaked2007stochastic). ∎

If $X\preceq_{CX}Y$, we can also say that $Y$ is a mean preserving spread of
$X$, in the sense that it satisfies a martingale property, that can be written
$Y\overset{\mathcal{L}}{=}X+Z$ where $Z$ is such that $\mathbb{E}[Z|X]=0$, as
discussed in CARLIER2012207. An example of variables ordered via the convex
order is given in

######  Lemma 3.5.

Let $\boldsymbol{X}=(X_{1},\cdots,X_{n})$ denote a collection of i.i.d.
variables, and $\boldsymbol{p}$ some $n$-dimensional probability vector. Then
$\boldsymbol{p}^{\top}\boldsymbol{X}\preceq_{CX}X_{i}$ for any $i$.

###### Proof.

Let $\boldsymbol{X}^{+}_{i}=X_{i}\boldsymbol{1}$, which is a commonotonic
version of vector $\boldsymbol{X}$, in the sense that all marginals have the
same distribution (since components of $\boldsymbol{X}$ are identically
distributed), and $X_{i}=\boldsymbol{p}^{\top}\boldsymbol{X}^{+}_{i}$. A
simple extension of Theorem 6 in kaas2002 (and Proposition 3.4.29 in order)
implies that
$\boldsymbol{p}^{\top}\boldsymbol{X}\preceq_{CX}\boldsymbol{p}^{\top}\boldsymbol{X}^{+}_{i}$
for any $i$ (since $X_{i}$’s are i.i.d.), i.e.
$\boldsymbol{p}^{\top}\boldsymbol{X}\preceq_{CX}X_{i}$. Note that it can also
be seen as a corollary of Property 3.4.48 in order, using the fact that
$\boldsymbol{p}\prec\boldsymbol{e}_{i}$ for the majorization order (where
classically $\boldsymbol{e}_{i}=(0,0,\cdots,0,1,0,\cdots,0,1)^{\top}$). ∎

If such a concept can be used to compare two risks, for a single individual,
we need a multivariate extension to study the risk over the entire portfolio.
Following DENUIT2012265, define the componentwise convex order as follows

######  Definition 3.6 (Componentwise convex order).

Given two vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, $\boldsymbol{X}$ is
said to be smaller than $\boldsymbol{Y}$ in the componentwise convex order if
$X_{i}\preceq_{CX}Y_{i}$ for all $i=1,2,\cdots,n$, and is denoted
$\boldsymbol{X}\preceq_{CCX}\boldsymbol{Y}$. And if at least one of the convex
order inequalities is strict, we will say that $\boldsymbol{Y}$ dominated
$\boldsymbol{X}$, and is denoted $\boldsymbol{X}\prec_{CCX}\boldsymbol{Y}$.

This inequality, $\boldsymbol{X}\prec_{CCX}\boldsymbol{Y}$, means that, for
this pool of $n$ insured, each and everyone of them prefers $X_{i}$ to
$Y_{i}$. But as we will see later one, when optimizing the mechanism, we will
consider the perspective of some central planer, maximizing the total
coverage. Thus it could be possible to define some sort of global interest for
a mechanism. This means that we might be slightly less restrictive in order to
study some possible collective gain of such a coverage, for the group of $n$
insured. Some weaker conditions will be considered here. The first one will be
related to the classical idea in economics of a representative agent, but
here, since there might be heterogeneity of agents, we can consider some
randomly chosen agent. A weaker condition than the one in Proposition 3.4
would be : let $I$ denote a random variable, uniformly distributed over
$\\{1,2,\cdots,n\\}$, and define $\xi^{\prime}=\xi_{I}$, then we wish to have

$\mathbb{E}[\xi^{\prime}]=\mathbb{E}\big{[}\mathbb{E}[\xi_{I}|I]\big{]}=\mathbb{E}[X_{i}]\text{
and
}\text{Var}[\xi^{\prime}]=\mathbb{E}\big{[}\text{Var}[\xi_{I}|I]\big{]}=\frac{1}{n}\text{trace}\big{[}\text{Var}[\boldsymbol{\xi}]\big{]}\leq\text{Var}[X_{i}]=\sigma^{2},$

and not necessarily

$\mathbb{E}[\xi_{i}]=\mathbb{E}[X_{i}]\text{ and
}\text{Var}[\xi_{i}]\leq\text{Var}[X_{i}],~{}\forall i\in\\{1,2,\cdots,n\\}.$

Note that Chatfield1981 and Johnson2008 call
$\text{trace}\big{[}\text{Var}[\boldsymbol{\xi}]\big{]}$ the total variation
of vector $\boldsymbol{\xi}$. If we will not use that name here, such a
quantity is a standard variability measure for random vectors.

###  3.2 Risk comparison for linear risk sharing schemes

Risk sharing schemes were properly defined in DENUIT2012265:

######  Definition 3.7 (Risk sharing scheme).

Consider two random vectors $\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{n})$ and
$\boldsymbol{X}=(X_{1},\cdots,X_{n})$ on $\mathbb{R}_{+}^{n}$. Then
$\boldsymbol{\xi}$ is a risk-sharing scheme of $\boldsymbol{X}$ if
$X_{1}+\cdots+X_{n}=\xi_{1}+\cdots+\xi_{n}$ almost surely.

For example, the average is a risk sharing principle:
$\xi_{i}=\overline{X}=\displaystyle{\frac{1}{n}\sum_{j=1}^{n}}X_{j}$, for any
$i$. A weighted average, based on credibility principles, too: $\xi_{i}=\alpha
X_{i}+[1-\alpha]\overline{X}$, where $\alpha\in[0,1]$ (we will get back on
those averages in the next section). Observe also that the order statistics
defines a risk sharing mechanism: $\xi_{i}=X_{(i)}$ where $X_{(1)}\leq
X_{(2)}\leq\cdots\leq X_{(n)}$. Given a permutation $\sigma$ of
$\\{1,2,\cdots,n\\}$, $\xi_{i}=X_{\sigma(i)}$ defines a risk sharing (that we
we write, using vector notations, $\boldsymbol{\xi}=P_{\sigma}\boldsymbol{X}$,
where $P_{\sigma}$ is the permutation matrix associated with $\sigma$). This
example of permutations is actually very important in the economic literature.

Here, we will consider more specifically linear risk sharing schemes.

######  Definition 3.8 (Linear Risk sharing scheme).

Consider two random vectors $\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{n})$ and
$\boldsymbol{X}=(X_{1},\cdots,X_{n})$ on $\mathbb{R}_{+}^{n}$, such that
$\boldsymbol{\xi}$ is a risk-sharing scheme of $\boldsymbol{X}$. It is said to
be a linear risk sharing scheme if there exists a matrix $M$, $n\times n$,
with positive entries, such that $\boldsymbol{\xi}=M\boldsymbol{X}$, almost
surely.

For example, if $\xi_{i}=\overline{X}$ for $i=1,2,\cdots,n$, characterized by
matrix $\bar{M}=[\bar{M}_{i,j}]$ with $\bar{M}_{i,j}=1/n$. For the credibility
inspired principle, $M_{\alpha}=\alpha\mathbb{I}+(1-\alpha)\bar{M}$. Another
example can be obtained with any doubly stochastic matrices: a $n\times n$
matrix $D$ is said to be doubly stochastic222In this section, $D$ will denote
some doubly stochastic matrix $n\times n$, not to be confused with the random
degree vector $D$. if

$D=[D_{i,j}]\text{ where }D_{i,j}\geq
0,~{}D_{\cdot,j}^{\top}\boldsymbol{1}=\sum_{i=1}^{n}D_{i,j}=1~{}\forall
j,~{}\text{and}~{}D_{i,\cdot}^{\top}\boldsymbol{1}=\sum_{j=1}^{n}D_{i,j}=1~{}\forall
i.$

Then $\boldsymbol{\xi}=D\boldsymbol{X}$ is a linear risk sharing of
$\boldsymbol{X}$. Row and column conditions on matrix $D$ are called zero-
balance conservation in Feng. A particular case can be when $D$ is a
permutation matrix, associated with a permutation of $\\{1,\cdots,n\\}$. This
example with doubly stochastic matrices is important because of connections
with the majorization concept (see schur and HLP). Furthermore Chang1992,
Chang93 extended the deterministic version to some stochastic majorization
concept, with results that we will use here.

Inspired by the componentwise convex ordering, the following definition can be
considered, as in DENUIT2012265 (a similar concept can be found in
CARLIER2012207)

######  Definition 3.9 (Ordering of risk sharing schemes (1)).

Consider two risk sharing schemes $\boldsymbol{\xi}_{1}$ and
$\boldsymbol{\xi}_{2}$ of $\boldsymbol{X}$. $\boldsymbol{\xi}_{1}$ dominates
$\boldsymbol{\xi}_{2}$, for the convex order, if
$\boldsymbol{\xi}_{2}\preceq_{CCX}\boldsymbol{\xi}_{1}$.

######  Definition 3.10 (Desirable risk sharing schemes (1)).

A risk sharing scheme $\boldsymbol{\xi}$ of $\boldsymbol{X}$ is desirable if
$\boldsymbol{\xi}\preceq_{CCX}\boldsymbol{X}$.

The order statistics of some i.i.d. random vector ($\xi_{i}=X_{(i)}$ where
$X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(n)}$) is not a desirable risk sharing
principle (if $X_{i}$ are non-deterministic ordered values), and since
$\mathbb{E}[\xi_{n}]>\mathbb{E}[X_{n}]$ so we cannot have
$\xi_{n}\preceq_{CX}X_{n}$. For a (deterministic) permutation, observe that
agents are indifferent, $\boldsymbol{\xi}\sim_{CCX}\boldsymbol{X}$ (in the
sense that ($\boldsymbol{\xi}\preceq_{CCX}\boldsymbol{X}$ and
$\boldsymbol{X}\preceq_{CCX}\boldsymbol{\xi}$)

######  Proposition 3.11 (Desirable linear risk sharing schemes (1)).

If $\boldsymbol{\xi}$ is a linear risk sharing of $\boldsymbol{X}$,
$\boldsymbol{\xi}=D\boldsymbol{X}$ for some doubly stochastic matrix $D$, then
$\boldsymbol{\xi}$ is desirable.

###### Proof.

Assume that $\boldsymbol{\xi}=D\boldsymbol{X}$ for some double stochastic
matrix $D$, with rows $D_{i\cdot}$’s. Since $D$ is doubly stochastic,
$D_{i\cdot}$ is a probability measure over $\\{1,\cdots,n\\}$ (with positive
components that sum to one), and from Lemma 3.5,
$D_{i\cdot}^{\top}\boldsymbol{X}\preceq_{CX}X_{i}$, for all $i$. ∎

######  Proposition 3.12.

[Ordering of Linear risk sharing schemes (1)] Consider two linear risk sharing
schemes $\boldsymbol{\xi}_{1}$ and $\boldsymbol{\xi}_{2}$ of $\boldsymbol{X}$,
such that there is a doubly stochastic matrix $D$, $n\times n$ such that
$\boldsymbol{\xi}_{2}=D\boldsymbol{\xi}_{1}$. Then
$\boldsymbol{\xi}_{2}\preceq_{CCX}\boldsymbol{\xi}_{1}$.

###### Proof.

This can be seen as an extention of Proposition 3.11. Here
$\boldsymbol{\xi}_{1}=M_{1}\boldsymbol{X}$ and
$\boldsymbol{\xi}_{2}=M_{2}\boldsymbol{X}$, since we consider linear risk
sharings, and we can write the later
$\boldsymbol{\xi}_{2}=M_{2}\boldsymbol{X}=DM_{1}\boldsymbol{X}$, thus
$M_{2}=DM_{1}$, since equality is valid for all $\boldsymbol{X}$. As defined
in DAHL199953 and BEASLEY2000141, this corresponds to the matrix majorization
concept, in the sense that $M_{2}\prec M_{1}$, that implies standard
majorization per row, thus for all $i$, $M_{2:i\cdot}\prec M_{1:i\cdot}$, from
Lemma 2.8 in BEASLEY2000141. But from Property 3.4.48 in order, we know that
if $\boldsymbol{a}\prec\boldsymbol{b}$ for the majorization order,
$\boldsymbol{a}^{\top}\boldsymbol{X}\preceq_{CX}\boldsymbol{b}^{\top}\boldsymbol{X}$.
Thus, if $\boldsymbol{a}=M_{2:i\cdot}$ and $\boldsymbol{b}=M_{1:i\cdot}$, we
have that for any $i$, $\xi_{2:i}\preceq_{CX}\xi_{1:i}$, and therefore we have
the componentwise order
$\boldsymbol{\xi}_{2}\preceq_{CCX}\boldsymbol{\xi}_{1}$. ∎

######  Proposition 3.13.

Consider two linear risk sharing schemes $\boldsymbol{\xi}_{1}$ and
$\boldsymbol{\xi}_{2}$ of $\boldsymbol{X}$, such that
$\boldsymbol{\xi}_{2}=D\boldsymbol{\xi}_{1}$, for some doubly-stochastic
matrix. Then
$\operatorname{Var}[\boldsymbol{\xi}_{2:i}]\leq\operatorname{Var}[\boldsymbol{\xi}_{1:i}]$
for all $i$.

###### Proof.

This is simply the fact that from Proposition 3.12,
$\boldsymbol{\xi}_{2}\preceq_{CCX}\boldsymbol{\xi}_{1}$, i.e. from Definition
3.6, $\boldsymbol{\xi}_{2:i}\preceq_{CCX}\boldsymbol{\xi}_{1:i}$ for all $i$
and
$\operatorname{Var}[\boldsymbol{\xi}_{2:i}]\leq\operatorname{Var}[\boldsymbol{\xi}_{1:i}]$
follows from the property on the convex order mentioned earlier. ∎

A simple corollary is the following

######  Proposition 3.14.

Consider two linear risk sharing schemes $\boldsymbol{\xi}_{1}$ and
$\boldsymbol{\xi}_{2}$ of $\boldsymbol{X}$, such that
$\boldsymbol{\xi}_{2}=D\boldsymbol{\xi}_{1}$, for some doubly-stochastic
matrix. Then
$\operatorname{trace}\big{[}\operatorname{Var}[\boldsymbol{\xi}_{2}]\big{]}\leq\operatorname{trace}\big{[}\operatorname{Var}[\boldsymbol{\xi}_{1}]\big{]}$.

###### Proof.

Observe first that
$\operatorname{Var}[\boldsymbol{\xi}_{2}]=D\operatorname{Var}[\boldsymbol{\xi}_{1}]D^{\top}$.
Since the variance matrix $\operatorname{Var}[\boldsymbol{\xi}_{1}]$ is
positive, the application
$h:M\mapsto\operatorname{trace}\big{[}M\operatorname{Var}[\boldsymbol{\xi}_{1}]M^{\top}\big{]}$
is convex. By the Birkhoff-von Neumann theorem (see birkhoff1946three), every
doubly stochastic matrix is a convex combination of permutation matrices, i.e.
$D=\omega_{1}P_{1}+\cdots+\omega_{k}P_{k}$ for some permutation matrices
$P_{1},\cdots,P_{k}$ (and some positive weights $\omega_{i}$ that sum to 1).
By the convexity of $h$,

$\operatorname{trace}\big{[}\operatorname{Var}[\boldsymbol{\xi}_{2}]\big{]}=h(D)\leq\sum_{i=1}^{k}\omega_{i}h(P_{i})=\sum_{i=1}^{k}\omega_{i}\operatorname{trace}\big{[}P_{i}\operatorname{Var}[\boldsymbol{\xi}_{1}]P_{i}^{\top}\big{]},$

and therefore

$\operatorname{trace}\big{[}\operatorname{Var}[\boldsymbol{\xi}_{2}]\big{]}\leq\sum_{i=1}^{k}\omega_{i}\operatorname{trace}\big{[}\operatorname{Var}[\boldsymbol{\xi}_{1}]\big{]}=\operatorname{trace}\big{[}\operatorname{Var}[\boldsymbol{\xi}_{1}]\big{]}.$

∎

A discussion of these results is provided in Appendix 8.1.

###  3.3 Risk sharing on cliques

Such a design was briefly mentioned in Feng, and called Hierarchical P2P Risk
Sharing although their model results by partitioning an already complete graph
into several complete subgraphs.

######  Definition 3.15 (Clique).

A clique $\mathcal{C}_{i}$ within an undirected graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ is a subset of vertices
$\mathcal{C}_{i}\subseteq\mathcal{V}$ such that every two distinct vertices
are adjacent. That is, the induced subgraph of $\mathcal{G}$ by
$\mathcal{C}_{i}$ is complete. A _clique cover_ is a partition of the graph
$\mathcal{G}$ into a set of cliques.

In a network, a clique is a subset of nodes such that every two distinct
vertices in the clique are adjacent. Assume that a network with $n$ nodes has
two (distinct) cliques, with respectively $k$ and $m$ nodes. One can consider
some ex-post contributions, to cover for losses claimed by connected
policyholders. If risks are homogeneous, contributions are equal, within a
clique. Assume that policyholders face random losses
$\boldsymbol{X}=(X_{1},\cdots,X_{k},X_{k+1},\cdots,X_{k+m})$ and consider the
following risk sharing

$\xi_{i}=\begin{cases}\displaystyle{\frac{1}{k}\sum_{j=1}^{k}X_{j}}\text{ if
}i\in\\{1,2,\cdots,k\\}\vspace{.2cm}\\\
\displaystyle{\frac{1}{m}\sum_{j=k+1}^{k+m}X_{j}}\text{ if
}i\in\\{k+1,k+2,\cdots,k+m\\}\end{cases}$

This is a linear risk sharing, with sharing matrix $D_{k,m}$, so that
$\boldsymbol{\xi}=D_{k,m}\boldsymbol{X}$, defined as

$D_{k,m}=\qquad\bordermatrix{~{}&~{}1&~{}\cdots&~{}k&k+1&k+2&~{}\cdots&k+m\cr~{}~{}~{}1&k^{-1}&\cdots&k^{-1}&0&0&\cdots&0\cr~{}~{}~{}\vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\cr~{}~{}~{}k&k^{-1}&\cdots&k^{-1}&0&0&\cdots&0\cr
k+1&0&\cdots&0&m^{-1}&m^{-1}&\cdots&m^{-1}\cr
k+2&0&\cdots&0&m^{-1}&m^{-1}&\cdots&m^{-1}\cr~{}~{}~{}\vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\cr
k+m&0&\cdots&0&m^{-1}&m^{-1}&\cdots&m^{-1}\cr}$

Since $D_{k,m}$ is a doubly-stochastic matrix,
$\boldsymbol{\xi}\prec_{CCX}\boldsymbol{X}$. In order to illustrate
Proposition 3.14, let $I$ denote a uniform variable over $\\{1,2,\cdots,n\\}$,
and $\xi^{\prime}=\xi_{I}$,

$\mathbb{E}[\xi^{\prime}]=\mathbb{E}[\mathbb{E}[\xi_{I}|I]]=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[\xi_{i}]=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[X_{i}]=\mathbb{E}[X]$

as expected since it is a risk sharing, while

$\text{Var}[\xi^{\prime}]=\mathbb{E}[\text{Var}[\xi_{I}|I]]=\frac{1}{n}\sum_{i=1}^{n}\text{Var}[\xi_{i}]=\frac{1}{n}\left(k\frac{\text{Var}[X]}{k^{2}}+m\frac{\text{Var}[X]}{m^{2}}\right)\\\
=\frac{\text{Var}[X]}{n}\left(\frac{1}{k}+\frac{1}{m}\right)=\frac{\text{Var}[X]}{k(n-k)}$

since $n=k+m$, so that $\text{Var}[\xi^{\prime}]<\text{Var}[X]$, meaning that
if we randomly pick a policyholder, the variance of the loss while risk
sharing is lower than the variance of the loss without risk sharing. Observe
further that $k\mapsto k(n-k)$ is maximal when $k=\lfloor n/2\rfloor$, which
means that risk sharing benefit is maximal (socially maximal, for a randomly
chosen representative policyholder) when the two cliques have the same size.

####  3.3.1 Cliques in practical problems

Cliques have attractive properties, in that they allow to use a linear risk
sharing mechanism on the resulting subgraphs. Further, they can be used to
represent a (more) homogeneous subgroup within a larger network, as for
example Feng propose (in their case, they partition the network based on a
hierarchical structure). As we depart from the idea of a complete graph to
begin with, partitioning the graph into cliques becomes a _clique problem_.
Given a number of cliques $j$, the number of nodes $n$ and the number of
members of clique $i$ $n_{i}$ (where $n_{j}=n-\sum_{i=1}^{j-1}n_{i}$), the
variance is always minimized as:

$\frac{\partial\text{Var}[\xi^{\prime}]}{\partial
n_{k}}=\frac{1}{n_{j}^{2}}-\frac{1}{n_{k}^{2}}=0$

implying $n_{i}=\tilde{n}\forall i$ and $\tilde{n}=\frac{n}{j}$. This in turn
means that:

$\text{Var}[\xi^{\prime}]=\text{Var}[X]\left(\frac{1}{n}\frac{j}{\frac{n}{j}}\right)=\text{Var}[X]\left(\frac{j}{n}\right)^{2}$

So the variance decrease is increasing in $n$ as in standard partitioning
problems, but decreasing in $j$ up to the extreme case where $j=n$ (ie. every
node is its own clique). Even when abstracting from the fact that the set of
all cliques $\mathcal{C}$ in graph $\mathcal{G}$ might not contain ideally
sized cliques, the problem of finding a clique cover that minimizes $j$
becomes a _minimal clique problem_ and this was shown to be NP-complete by
karp1972reducibility. Optimizing both clique size and clique number becomes
computationally infeasible even with small graphs.

So unless a very specific network structure (such as a complete graph) is
given or a separation into cliques can be translated into a hierarchical
structure as in Feng, working on cliques might present attractive mathematical
properties but does not seem practical. Instead, one could consider working
with other graph partitioning methods. In the i.i.d. case this might not lead
to desirable results, but in the case where there is heterogeneity present in
the graph but less so in a densely connected subgroup, there might be some
value. For example, if the network considered is of the social kind, densely
connected subgroups can represent family or neighbourly ties, which would be
interesting (bridge2002neighbourhood or hanneman2005introduction) for life-
insurance. Subgroup finding on graphs generally works well and a range of fast
algorithms exist (e.g. clauset2004fastgreedy) and would allow to tune
optimizations such as those in section 5 to depart from hypothesis 3.2.

###  3.4 Some weaker orderings for risk sharing scheme

Following MARTINEZPERIA2005343, a weaker ordering can be considered with
column-stochastic matrices, instead of doubly stochastic matrices. A column-
stochastic matrix (or left-stochastic matrix) $C$ satisfies

$C=[C_{i,j}]\text{ where }C_{i,j}\geq 0,~{}\text{and
}\sum_{i=1}^{n}C_{i,j}=1~{}\forall j.$

######  Proposition 3.16.

Let $C$ be some $n\times n$ column-stochastic matrix, and given
$\boldsymbol{X}$, a positive vector in $\mathbb{R}^{+}$, define
$\boldsymbol{\xi}=C\boldsymbol{X}$. Then $\boldsymbol{\xi}$ is a linear risk
sharing of $\boldsymbol{X}$.

###### Proof.

First, observe that since $C$ is a matrix with positive entries,
$\boldsymbol{\xi}\in\mathbb{R}^{n}_{+}$. Then, simply observe that

$\sum_{i=1}^{n}\xi_{i}=\sum_{i=1}^{n}\sum_{j=1}^{n}C_{i,j}X_{j}=\sum_{j=1}^{n}\left(\sum_{i=1}^{n}C_{i,j}\right)X_{j}=\sum_{j=1}^{n}X_{j}.$

∎

In order to illustrate why such a representation –
$\boldsymbol{Y}=C\boldsymbol{X}$ instead of $\boldsymbol{Y}=D\boldsymbol{X}$,
as considered earlier – can be seen as a possible ordering, consider the two
following matrices (where $D_{1}$ corresponds to $C_{2}$ when $\alpha=0$),

$D_{1}=\begin{pmatrix}1&0&0\\\ 0&1/2&1/2\\\ 0&1/2&1/2\\\ \end{pmatrix}\text{
and }C_{2}=\begin{pmatrix}1&\alpha&\alpha\\\ 0&(1-\alpha)/2&(1-\alpha)/2\\\
0&(1-\alpha)/2&(1-\alpha)/2\\\ \end{pmatrix},\text{ with }\alpha\in(0,1).$

Consider some non-negative random losses $\boldsymbol{X}=(X_{1},X_{2},X_{3})$
where the three components are independent and identically distributed.
Consider the following risk sharing scheme:
$\boldsymbol{\xi}=(\xi_{1},\xi_{2},\xi_{3})$ such that
$\boldsymbol{\xi}=D_{1}\boldsymbol{X}$, that is $\xi_{1}=X_{1}$ and
$\xi_{2}=\xi_{3}=\overline{X}_{23}$, where
$\overline{X}_{23}=(X_{2}+X_{3})/2$. Assume now that we randomly select one of
the components. Let $X^{\prime}$ denote that one, or more specifically, let
$I$ denote a uniform variable over $\\{1,2,3\\}$, and let $X^{\prime}=X_{I}$.
And similarly, let $\xi^{\prime}=\xi_{I}$. Note that if
$\mu=\mathbb{E}[X_{i}]=\mathbb{E}[X^{\prime}_{i}]$, and
$\sigma^{2}=\text{Var}[X_{i}]=\text{Var}[X_{i}^{\prime}]$,

$\mathbb{E}[\xi^{\prime}]=\frac{1}{3}\sum_{i=1}^{3}\mathbb{E}[\xi_{i}]=\frac{1}{3}\left[\mu+2\cdot\mu\right]=\mu=\mathbb{E}[X^{\prime}],$

which is consistent with the fact that it is a risk sharing principle, and

$\text{Var}[\xi^{\prime}]=\frac{1}{3}\sum_{i=1}^{3}\text{Var}[\xi_{i}]=\frac{1}{3}\left[\sigma^{2}+2\cdot\frac{2\sigma^{2}}{4}\right]=\frac{2}{3}\sigma^{2}<\text{Var}[X^{\prime}].$

So it seems that the vector $\boldsymbol{\xi}$ of individual losses after
sharing risks has a lower variance than vector $\boldsymbol{X}$. Furthermore,
we can easily prove that $\boldsymbol{\xi}_{1}\preceq_{CCX}\boldsymbol{X}$
since $\xi_{1}\sim_{CX}X_{1}$ while $\xi_{2}\preceq_{CX}X_{2}$ and
$\xi_{3}\preceq_{CX}X_{3}$.

For the second one, consider $\boldsymbol{\xi}=(\xi_{1},\xi_{2},\xi_{3})$ such
that $\boldsymbol{\xi}=C_{2}\boldsymbol{X}$, that is
$\xi_{1}=X_{1}+2\alpha\overline{X}_{23}$ and
$\xi_{2}=\xi_{3}=(1-\alpha)\overline{X}_{23}$, where
$\overline{X}_{23}=(X_{2}+X_{3})/2$. Then $\boldsymbol{X}$ and
$\boldsymbol{\xi}$ cannot be compared using the componentwise convex order,
even if $\xi_{1}+\xi_{2}+\xi_{3}=X_{1}+X_{2}+X_{3}$. Assume now that we
randomly select of the of the components. Let $X^{\prime}$ denote that one, or
more specifically, let $I$ denote a uniform variable over $\\{1,2,3\\}$, and
let $X^{\prime}=X_{I}$. And similarly, let $\xi^{\prime}=\xi_{I}$. Note that
if $\mu=\mathbb{E}[X_{i}]=\mathbb{E}[X^{\prime}_{i}]$, and
$\sigma^{2}=\text{Var}[X_{i}]=\text{Var}[X_{i}^{\prime}]$,

$\mathbb{E}[\xi^{\prime}]=\frac{1}{3}\sum_{i=1}^{3}\mathbb{E}[\xi_{i}]=\frac{1}{3}\left[(1+2\alpha)\mu+2\cdot(1-\alpha)\mu\right]=\mu=\mathbb{E}[X^{\prime}],$
$\text{Var}[\xi^{\prime}]=\frac{1}{3}\sum_{i=1}^{3}\text{Var}[\xi_{i}]=\frac{1}{3}\left[(1+2\alpha^{2})\sigma^{2}+\frac{4\cdot(1-\alpha)^{2}\sigma^{2}}{4}\right]=\frac{1+2\alpha^{2}+(1-\alpha)^{2}}{3}\sigma^{2}<\text{Var}[X^{\prime}].$

Note that this case can be seen as extreme, since there is probably no
interest for the first agent to agree to share the risks with such a
mechanism. First of all, we can think of parents, agreeing to share the risks
of their children to motive that case; but more realistically, we will discuss
additional assumptions later on. Following MARTINEZPERIA2005343, we can define
an ordering based those column-stochastic matrices

######  Definition 3.17 (Weak Ordering of Linear risk sharing schemes (2)).

Consider two linear risk sharing schemes $\boldsymbol{\xi}_{1}$ and
$\boldsymbol{\xi}_{2}$ of $\boldsymbol{X}$. $\boldsymbol{\xi}_{1}$ weakly
dominates $\boldsymbol{\xi}_{2}$, denoted
$\boldsymbol{\xi}_{2}\preceq_{wCX}\boldsymbol{\xi}_{1}$ if and only if there
is a column-stochastic matrix $C$, $n\times n$ such that
$\boldsymbol{\xi}_{2}=C\boldsymbol{\xi}_{1}$.

In the examples above, we did prove that
$\boldsymbol{\xi}_{1}\preceq_{wCX}\boldsymbol{X}$ and
$\boldsymbol{\xi}_{2}\preceq_{wCX}\boldsymbol{X}$, but can we compare
$\boldsymbol{\xi}_{1}$ and $\boldsymbol{\xi}_{2}$? One can write

$\begin{cases}\xi_{2,1}=X_{1}+2\alpha\overline{X}_{23}=\xi_{1,1}+2\alpha\xi_{1,2}=\xi_{1,1}+\alpha\xi_{1,2}+\alpha\xi_{1,3}\\\
\xi_{2,2}=\xi_{2,3}=(1-\alpha)\overline{X}_{23}=(1-\alpha)\xi_{1,2}=(1-\alpha)/2\xi_{1,2}+(1-\alpha)/2\xi_{1,3}\end{cases}$

so we can actually write $\boldsymbol{\xi}_{2}=C_{2}\boldsymbol{\xi}_{1}$,
since $C_{2}D_{1}=C_{2}$.

Note that Proposition 3.14 cannot be extended when matrices are only column-
stochastic, in the sense that $\boldsymbol{\xi}_{2}=C\boldsymbol{\xi}_{1}$ for
some column-stochastic matrix $C$ is not sufficient to guarantee
$\text{trace}[\text{Var}[\boldsymbol{\xi}_{2}]]=\text{trace}[C\text{Var}[\boldsymbol{\xi}_{1}]C^{\top}]\leq\text{trace}[\text{Var}[\boldsymbol{\xi}_{1}]]$.
Some heuristic interpretation of that result are given in Appendix 8.2. It
does hold when $\boldsymbol{\xi}_{1}$ are independent risks.

######  Proposition 3.18.

If $\boldsymbol{\xi}$ is a linear risk sharing of $\boldsymbol{X}$, where
$X_{i}$’s are independent risks with variance $\sigma^{2}$, associated with
some column-stochastic matrix $C$, then

$\operatorname{trace}\big{[}\text{Var}[\boldsymbol{\xi}]\big{]}=\sigma^{2}\operatorname{trace}\big{[}CC^{\top}\big{]}\leq\sigma^{2}n$

and therefore
$\operatorname{Var}[\xi^{\prime}]\leq\operatorname{Var}[X]=\sigma^{2}$.

###### Proof.

In dimension $n$,

$\text{trace}\big{[}CC^{\top}\big{]}=\sum_{i,j=1}^{n}C_{i,j}^{2}\leq\sum_{i,j=1}^{n}C_{i,j}=\sum_{i=1}^{n}\left(\sum_{j=1}^{n}C_{i,j}\right)=\sum_{i=1}^{n}1=n.$

∎

Thus, it seems that the ordering we introduced might not be interesting if we
do not add that ordering on the variance, that is not always satisfied.

######  Definition 3.19 (Weak Ordering of Linear risk sharing schemes (3)).

Consider two linear risk sharing schemes $\boldsymbol{\xi}_{1}$ and
$\boldsymbol{\xi}_{2}$ of $\boldsymbol{X}$. $\boldsymbol{\xi}_{1}$ weakly
dominates $\boldsymbol{\xi}_{2}$, denoted
$\boldsymbol{\xi}_{2}\prec_{WBCX}\boldsymbol{\xi}_{1}$ if and only if there is
a column-stochastic matrix $C$, $n\times n$ such that
$\boldsymbol{\xi}_{2}=C\boldsymbol{\xi}_{1}$ and such that
$\text{trace}[\text{Var}[\boldsymbol{\xi}_{2}]]\leq\text{trace}[\text{Var}[\boldsymbol{\xi}_{1}]]$.

This weaker order,
$\text{trace}[\text{Var}[\boldsymbol{\xi}_{2}]]\leq\text{trace}[\text{Var}[\boldsymbol{\xi}_{1}]]$,
means that individually, all agents do not prefer scheme $2$ over $1$, but
globally, scheme $2$ is preferred. This issue will be discussed afterwards.

###  3.5 On nonlinear risk sharing mechanisms

Risk sharing mechanisms that we will consider with reciprocal contracts are
not linear in $\boldsymbol{X}$ since contributions between two agents will be
capped. This will be discussed in the next section. A tractable way to compare
risks will simply be based on the variances of risk sharing mechanisms, that
we will discuss on simulations on networks.

######  Definition 3.20 (Weak Ordering of Nonlinear risk sharing schemes (4)).

Consider two risk sharing schemes $\boldsymbol{\xi}_{1}$ and
$\boldsymbol{\xi}_{2}$ of $\boldsymbol{X}$. $\boldsymbol{\xi}_{1}$ weakly
dominates $\boldsymbol{\xi}_{2}$ if
$\text{trace}[\text{Var}[\boldsymbol{\xi}_{2}]]\leq\text{trace}[\text{Var}[\boldsymbol{\xi}_{1}]]$.

Before proposing the core model of this paper, it is worth considering the
different nuts and bolts of risk sharing on a network. Contrary to earlier
contributions to the field of P2P insurance we do not assume that agents are
willing to enter contracts with everyone in the system but only with nodes
that are connected to them, which will lead to substantially different
results. Further, we only impose an _upper_ limit to the reciprocal engagement
between nodes. There are three types of contributions, which in general do not
need to be the same for each node, note that these layers are treated in a
hierarchical manner (Layer 1 is activated before layer 2 and so on - this is
relevant if the loss is smaller than the deductible).

  1. 1.

Self Insurance corresponds to the new post-P2P insurance deductible. This part
is what each node will pay by itself, regardless of the risk it shares with
other nodes. This part can be any amount from $0$ to $s$. We will later show
that having a self-insurance layer of $0$ may lead to having _ex-post_ unfair
outcomes.

  2. 2.

Reciprocal insurance corresponds to the sum of the reciprocal engagement
contracts signed by each node. These contracts can be of varying amount but
their sum cannot exceed the deductible $s$ minus the self insurance layer.

  3. 3.

Residual Self Insurance corresponds to the amount of the deductible less the
previous two layers. As the name indicates, this is also a layer that each
node will pay by itself. It differs in nature to the first layer as the agent
would like to insure this part but can not (usually because the connected
nodes do not want to enter a reciprocal contract, this will be explained in
section 3.6). If for example another node would connect to the agent, this
part could be eliminated, whereas the first layer would remain.

Let us consider a simple toy example to illustrate the different layers.
Suppose that the network under consideration has the form as depicted on the
left-hand side in Figure 3.1 and we consider the division of payments from the
view of node A. The total value of the deductible $s$ is 100 and we assume
node A makes a claim, the claim amount can take any positive value. The right-
hand side of Figure 3.1 illustrates the sharing mechanism across the three
layers. The first layer, consisting of A’s self-insurance layer, covers losses
up to the amount of 40. The second layer is only activated once that threshold
is reached. From there on, nodes B and C will share proportionally the amount
between 40 and 90 (the area corresponding to the second layer) with
proportions 40%-60% respectively. If the claim amount passes 90, A will also
need to cover the last layer (the residual self-insurance) up to the
deductible $s$.

This illustrates that the contributions are piecewise linear but added
together constitute a non-linear risk sharing mechanism. Note that in this
example we have assumed the values of the self-contribution and reciprocal
contracts, but there is a plethora of combinations possible. The next
subsection introduces different risk sharing mechanism for an extremely simple
network with only four nodes, sections 4 and 5 then generalize the findings to
larger networks introduce an approach to select the optimal sharing mechanism
for a specific network.

Let us consider a simple toy example to illustrate how to define risk sharing
on a networks, with (only) $n=4$ nodes. On networks of Figure 3.2, suppose
that $\boldsymbol{X}$ is a collection of i.i.d. variables (as in hypothesis
3.2) $X_{i}$, based on i.i.d. $Z_{i}$’s that are Bernoulli variables with
success probability $10\%$, and where $Y_{i}$’s are either constant (taking
value $100$) or uniform (over $[0,200]$). The split among participants is
described in Figure 3.1, where we focus only on policyholder A.

Figure 3.1: Subdivision of payments when A claims a loss, while sharing risks
with B and C, based on the description on the left (with two non-identical
reciprochal contracts). The $x$ axis is the value of the loss $X$. On the $y$
axis we can visualize the amount paid respectively by A, B, C, and A back
again. There is a first layer of self-contribution of 40, then two reciprocal
of 20 (with B) and 30 (with C), that yield a split prorata capita with a 2-3
ratio, and a third layer that is a remaining self-contribution, up to 10. If
$X\leq 40$, it is fully paid by A. Between 40 and 90, $40$ is paid by A, then
$X-40$ is shared in two, B paying 40% and C 60%. Above $90$, B and C pay
respectively 20 and 30, while A pays $\min\\{X-50,50\\}$.

Indemnity and contributions are piecewise linear, as we can see on Figure 3.1.
First of all, A has a first layer with a self contribution of 40: instead of
having a deductible of 100 (that is the contract A signed with the insurance
company), A now has a deductible of 40. Then reciprocal contracts are based on
an upper limit $\gamma$ (this will be formalized in Definition 4.2). For
instance, A and B share risk with a reciprocal contract with magnitude
$\gamma$ of 20, and magnitude 30 with C. Thus, for claims $x$ between 40 and
90, $(x-40)$ is shared proportionally between B and C, with proportions
40%-60% respectively. Here we highlight that there is a remaining part for A,
for claims between 90 and 100 (some sort of residual self insurance). Of
course, since A has reciprocal contracts with B and C, A is committed to pay B
or C if they experience losses (the amount depends on the contract they did
sign).

This case with non-identical reciprocal contracts will only be discussed in
Section 4. In the next section, we will consider globally a simple case with 4
policyholders.

###  3.6 A brief example

Figure 3.2: Six networks, on $n=4$ nodes. The first layer is the self
contribution (if any)  which is first paid; the second layer  is then paid by
connected nodes, prorate capita, above the first layer; finally, there can be
a third layer of self contribution

Let us consider some risk sharing mechanisms, for 4 policyholders, on various
networks, described in Figure 3.2. In Table 3.1 we can observe summary
statistics for each node (as well as ‘representative’ policyholder, with the
‘average’ average loss, and the ‘average’ variance - that is the trace of the
variance matrix over $n$) for these eight mechanisms that can described as
follows:

  * –

the benchmark corresponds to risks $X_{i}$ with mean $10$ (that is 10% of
$\mathbb{E}[X]$), and standard deviation $30$ if the loss is deterministic;
for a uniform random loss $Y$ (uniform between 0 and 200), the mean of $X_{i}$
is $7.5$ (that is 10% of $\mathbb{E}[\min\\{100,Y\\}]$) and the standard
deviation is now $24.71$.

  * (a)

corresponds to a regular network, with $\boldsymbol{d}=(2,2,2,2)$. Here all
insured decided to share risks with two friends, through reciprocal contracts.
If A claims a loss of $80$, A will collect an equal amount of money from B and
D (here $40$).

  * Note that this is a risk-sharing mechanism, all the nodes pay on average $\mathbb{E}[X]$ (that is 10 is the loss is 100, or 7.5 is the loss is uniform over [0,200] with a deductible at 100). Since the risk is always shared with two friends, the standard deviation is here $\text{stdev}[X]/\sqrt{2}$.

  * (b)

this network, with $\boldsymbol{d}=(3,2,2,1)$, is obtained from (a) by
changing an edge: edge (C,D) becomes (A,C). Reciprocal commitments of
$\gamma=50$ are maintained. As we will formalize with Hypothesis 4.1, whatever
the commitments, an insured cannot collect more than the actual sum of the
loss. Since $s=100$, if A claims a loss, the other three nodes will only
contribute by paying up to 33.333. But if any of those nodes claim a loss, A
is committed to pay 50 to each of them. This is clearly sub-optimal, for A.
Furthermore, D has only a single connection, so if the claim size exceeds 50,
the remaining part, between 50 and 100 is now a self contribution of D. As for
(a), the sum of $\gamma$’s is 200. Observe here that with reciprocal contracts
of $50$, A might collect more that the actual loss: we will add an assumption
so that the sum of collected money cannot exceed $s$. If A claims a loss, B
will pay 33.333 to A, but conversely, if B claims a loss, A will pay 50.
Therefore, those contracts do not yield a desirable risk sharing for
policyholder A, even if collectively it could be seen as interesting.

  * This asymmetry can be visualized in Table 3.1 where the average loss of A exceeds $\mathbb{E}[X]$, with also a larger variable.

  * (c)

corresponds to the case where A lowers his reciprocal engagements, with
$\min\\{s/d_{i},\gamma\\}$, with all the connections. Unfortunately, in that
case, since contracts are reciprocals, all nodes connected with node A will
have a lower coverage. Here, we assume that B and C can still have a risk
sharing with magnitude $50$ (this will not be possible in the next section,
and will be considered only in section 5). The remaining part is a self
contribution that exists on top of the risk sharing scheme.

  * This design is optimal since it maximizes the overall coverage, but it is also the one that lowers the standard deviation, globally.

  * (d)

corresponds to some optimal risk sharing, result of a linear program, where we
maximize the sum of $\gamma_{(i,j)}$’s – this will be discussed in Section 5.
Note that here, the sum of $\gamma_{(i,j)}$’s is equal to 150 (as mechanism
(c)). Here one node is on the side, and the other three for a regular network,
$\boldsymbol{d}^{\prime}=(\\{2,2,2\\},\\{0\\})$. As discussed earlier, we will
try to avoid such a mechanism that excludes a policyholder.

  * (e)

this mechanism is obtained from (d) by changing an edge : edge (A,C) becomes
(A,D). It is also an optimum of the linear program.

  * If the two seem to be equivalent, it is not the case in Table 3.1: for A, B and C, (d) is equivalent to (a) with a regular graph (variances are equal) but not for D. On the other hand, with variant (e), the variance of D (as well as the other three) is now the same as the one in (a) if the loss is not random. If the loss is random, the variance is slightly larger with (e) for C and D.

  * (f)

corresponds to the case where a first layer of self contributions of $50$ are
considered, for all policyholders. (f) is the optimal risk sharing, obtained
from the linear program. In that case, two disconnected networks are
considered, since one single contract is sufficient ($50+\gamma=s$). The sum
of reciprocals is here $100$.

All those items, discussed in a simple toy example with 4 nodes, will be
discussed more intensively in the next sections, on much larger networks.

Deterministic loss, $y=100$
---
nodes | A | B | C | D | overall
$\mathbb{E}[X_{i}]$ | 10.00 | 10.00 | 10.00 | 10.00 | 10.00
$\text{stdev}[X_{i}]$ | 30.00 | 30.00 | 30.00 | 30.00 | 30.00
$\mathbb{E}[\xi^{(a)}_{i}]$ | 10.00 | 10.00 | 10.00 | 10.00 | 10.00
$\text{stdev}[\xi^{(a)}_{i}]$ | 21.21 | 21.21 | 21.21 | 21.21 | 21.21
$\mathbb{E}[\xi^{(b)}_{i}]$ | 15.00 | 8.33 | 8.33 | 8.33 | 10.00
$\text{stdev}[\xi^{(b)}_{i}]$ | 26.00 | 18.00 | 18.00 | 18.00 | 20.50
$\mathbb{E}[\xi^{(c)}_{i}]$ | 10.00 | 10.00 | 10.00 | 10.00 | 10.00
$\text{stdev}[\xi^{(c)}_{i}]$ | 17.33 | 18.70 | 18.70 | 22.33 | 19.33
$\mathbb{E}[\xi^{(d)}_{i}]$ | 10.00 | 10.00 | 10.00 | 10.00 | 10.00
$\text{stdev}[\xi^{(d)}_{i}]$ | 21.21 | 21.21 | 21.21 | 30.00 | 23.71
$\mathbb{E}[\xi^{(e)}_{i}]$ | 10.00 | 10.00 | 10.00 | 10.00 | 10.00
$\text{stdev}[\xi^{(e)}_{i}]$ | 21.21 | 21.21 | 21.21 | 21.21 | 21.21
$\mathbb{E}[\xi^{(f)}_{i}]$ | 10.00 | 10.00 | 10.00 | 10.00 | 10.00
$\text{stdev}[\xi^{(f)}_{i}]$ | 21.21 | 21.21 | 21.21 | 21.21 | 21.21

Random loss, $Y\sim\mathcal{U}_{[0,200]}$
---
nodes | A | B | C | D | overall
$\mathbb{E}[X_{i}]$ | 7.50 | 7.50 | 7.50 | 7.50 | 7.50
$\text{stdev}[X_{i}]$ | 24.71 | 24.71 | 24.71 | 24.71 | 24.71
$\mathbb{E}[\xi^{(a)}_{i}]$ | 7.50 | 7.50 | 7.50 | 7.50 | 7.50
$\text{stdev}[\xi^{(a)}_{i}]$ | 17.47 | 17.47 | 17.47 | 17.47 | 17.47
$\mathbb{E}[\xi^{(b)}_{i}]$ | 11.88 | 6.25 | 6.25 | 5.62 | 7.50
$\text{stdev}[\xi^{(b)}_{i}]$ | 22.24 | 14.85 | 14.85 | 14.28 | 17.07
$\mathbb{E}[\xi^{(c)}_{i}]$ | 8.33 | 7.36 | 7.36 | 6.94 | 7.50
$\text{stdev}[\xi^{(c)}_{i}]$ | 15.36 | 15.70 | 15.70 | 17.88 | 16.20
$\mathbb{E}[\xi^{(d)}_{i}]$ | 7.50 | 7.50 | 7.50 | 7.50 | 7.50
$\text{stdev}[\xi^{(d)}_{i}]$ | 17.47 | 17.47 | 17.47 | 24.71 | 19.53
$\mathbb{E}[\xi^{(e)}_{i}]$ | 7.50 | 7.50 | 7.50 | 7.50 | 7.50
$\text{stdev}[\xi^{(e)}_{i}]$ | 17.47 | 17.47 | 18.03 | 18.03 | 17.75
$\mathbb{E}[\xi^{(f)}_{i}]$ | 7.50 | 7.50 | 7.50 | 7.50 | 7.50
$\text{stdev}[\xi^{(f)}_{i}]$ | 18.03 | 18.03 | 18.03 | 18.03 | 18.03

Table 3.1: First two moments of $X_{i}$’s and $\xi_{i}$’s, and $X_{I}$ and
$\xi_{I}$ (denoted ‘overall’), with $Y_{i}=100$ on the left, $Y_{i}$ uniform
on $[0,200]$ on the right, with deductible $s=100$, and $p=10\%$, on the six
sharing schemes described in Figure 3.2.

##  4 Risk Sharing with Friends and Identical Reciprocal Contracts

In Figure 1.1 $(b)-(d)$, we said that the peer-to-peer layer should be between
self-insurance and the traditional insurance company. To justify this, we will
first build a model in section 4.1 where we consider the case with no self-
insurance layer to illustrate why this scheme is not optimal in the sense that
it usually yields to ex-post unfairness. In the present section we will study
the case where every edge in the network represents a reciprocal contract (see
4.2 for a definition thereof) and assume identical magnitude $\gamma$. In the
following section 5 we will further relax the framework and allow for
individual magnitudes $\gamma_{i}\in[0,c]$ for each edge in $\mathcal{E}$, for
some positive constant $c$333Note that setting $\gamma_{i}=0$ is equivalent to
removing the edge from the graph, hence the mechanism allows us to analyze the
framework on sub-networks.

Before explaining the basic mechanism of risk sharing, we need to agree on a
simple principle: it is not possible to gain money with a fair risk sharing
scheme. This mathematical assumption is common in most reciprocal schemes.

######  Hypothesis 4.1.

Given a network $(\mathcal{E},\mathcal{V})$, policyholder $i$ will agree to
share risks with all policyholders $j$ such that $(i,j)\in\mathcal{E}$ (called
$i$’s friends). For each policyholder, the total sum collected from the
friends cannot exceed the value of the actual loss.

###  4.1 Risk sharing with friends and identical reciprocal contracts

Let us formalize here the simple risk sharing principle described previously,
based on identical reciprocal contracts (without first layer self-
contribution)

######  Definition 4.2.

A reciprocal contract between two policyholders, $i$ and $j$, with magnitude
$\gamma$ implies that $i$ will pay an amount $C_{i\rightarrow j}$ to $j$ if
$j$ claims a loss, with $C_{i\rightarrow j}\in[0,\gamma]$, and where all
friends who signed a contract with $j$ should pay the same amount, and
conversely from $j$ to $i$. Thus, $C_{i\rightarrow j}$ can be denoted $C_{j}$,
and for a loss $y_{j}$,

$C_{j}=C_{i\rightarrow
j}=\min\left\\{\gamma,\frac{\min\\{s,y_{j}\\}}{d_{j}}\right\\}=\min\left\\{\gamma,\frac{x_{j}}{d_{j}}\right\\},~{}\forall
i\in\mathcal{V}_{j},$

where $x_{i}=\min\\{y_{i},s\\}$ is the value of the loss that will be shared
among friends (up to limit $\gamma$).

Insured $i$ is connected with $d_{i}$ friends (where $d$ stands for the degree
function), and let $\mathcal{V}_{i}$ denote the set of friends. Since there is
a commitment with the friends, insured $i$ will pay up to a fix amount, which
we set as the total deductible divided by the average number of connections
$\overline{d}$ (ie. $\gamma=s/\overline{d}$) to all friends that claim a
loss444 This seems to be a good starting value, as it would correspond to the
optimal $\gamma$ in the case where we have regular network - in this case the
optimal response of each node would be to share its risk with all possible
nodes, which is what we propose here. This is a well known result when working
with i.i.d. losses. We note that this amount might not correspond to the
optimal value in the case where the graph is _not_ regular, this will be
illustrated in this section and an alternative proposed in the subsequent
section.. Conversely and if $i$ claims a loss, all connections will pay a
share of it. Thus

$\xi_{i}=Z_{i}\cdot\min\\{s,Y_{i}\\}+\sum_{j\in\mathcal{V}_{i}}Z_{j}\min\left\\{\gamma,\frac{\min\\{s,Y_{j}\\}}{d_{j}}\right\\}-Z_{i}\cdot\min\\{d_{i}\gamma,\min\\{s,Y_{i}\\}\\}$

The first term is the part of the loss below the deductible $s$, because of
the claim experienced by insured $i$, $X_{i}=Z_{i}\cdot\min\\{s,Y_{i}\\}$. The
third term is a gain, in case $Z_{i}=1$ because all connections will give
money, where all friends will contribute by paying $C_{i}$: insured $i$ cannot
receive more than the loss $X_{i}$, that cannot be smaller than sum of all
contributions $d_{i}\gamma$. Thus, the individual contribution per connection
is $C_{i}=\min\\{\gamma,X_{i}/d_{i}\\}$. It is important to point out that the
third term can only cancel out the first term if
$d_{i}\gamma\geq\min\\{s,Y_{i}\\}$, that is, node $i$ has at least
$\overline{d}$ friends. Finally, the second term corresponds to additional
payments insured $i$ will make, to all connections (in $\mathcal{V}_{i}$) who
claimed a loss and the payment to $j$ is, at most $\gamma$ (because of the
commitment), but it can be less if the loss of $j$ was not too large. With the
previous notations, insured $i$ should pay $C_{j}$ to insured $j$ if
$j\in{\mathcal{V}}_{i}$. So, this risk-sharing mechanism leads to loss
$\xi_{i}$ (instead of $X_{i}$)

$\xi_{i}=X_{i}+\sum_{j\in\mathcal{V}_{i}}Z_{j}C_{j}-Z_{i}d_{i}C_{i}$ (4.1)

Figure 4.1: Network with four nodes $\\{$A,B,C,D$\\}$ and three configurations
for nodes claiming a loss: nodes  will claim no loss, but will contribute to
loss of connections (if any)  that will claim a loss.

###  4.2 Illustration with a simple example

Consider the networks of Figure 4.1. Consider insurance contracts with a
deductible $s$ of 100. Here, degrees are $\boldsymbol{d}=(2,2,2,2)$ on the
left, and $\boldsymbol{d}=(3,2,2,1)$ for the other two, so that
$\overline{d}=2$, so $\gamma=50$. In Table 4.1, we have individual losses
(denoted $Y$) for two policyholders. $X_{i}$ would be the self contribution
for policyholder if there were no risk sharing with friends. $\xi_{i}$ is the
loss of policyholder $i$ in each scenario. Since the sum of $\xi_{i}$’s equal
the sum of $X_{i}$’s, we consider here some risk sharing, and observe that in
all cases, $\text{Var}[\xi_{I}]\leq\text{Var}[X_{I}]$.

| (a)
---|---
$i$ | A | B | C | D
$d_{i}$ | 2 | 2 | 2 | 2
$Z_{i}$ | 0 | 1 | 0 | 1
$Y_{i}$ |  | 200 |  | 60
$\min\\{Y_{i},s\\}$ |  | 100 |  | 60
$X_{i}$ | 0 | 100 | 0 | 60
$C_{i}$ | 0 | 50 | 0 | 30
$\xi_{i}$ | 80 | 0 | 80 | 0

(b)
---
A | B | C | D
3 | 2 | 2 | 1
0 | 1 | 0 | 1
| 200 |  | 60
| 100 |  | 60
0 | 100 | 0 | 60
0 | 50 | 0 | 50
100 | 0 | 50 | 10

(c)
---
A | B | C | D
3 | 2 | 2 | 1
0 | 1 | 0 | 1
60 | 200 |  |
60 | 100 |  |
60 | 100 | 0 | 0
20 | 50 | 0 | 0
50 | 20 | 70 | 20

Table 4.1: Scenarios of Figure 4.1, with deductible $s=100$ and (maximal)
contribution $\gamma=50$. The average of $x$’s is
$\overline{x}=\overline{\xi}=40$, empirical standard deviation of $x$’s is
here proportional to $\sqrt{7200}$, while for $\xi_{(\text{a})}$ it is
$\sqrt{6400}$, $\sqrt{6200}$ for $\xi_{(\text{b})}$ and finally $\sqrt{1800}$
for $\xi_{(\text{c})}$.

initialisation: generate a network, with adjacency matrix $A$ ;

$\gamma\leftarrow s/\overline{d}$ ;

for  _$i\leftarrow 0$ to $n$_ do

$d_{i}\leftarrow A_{i\cdot}^{\top}\boldsymbol{1}$ # computing the degree ;

$V_{i}\leftarrow\\{j:A_{i,j}=1\\}$ # set of friends of polycyholder $i$ ;

$Z_{i}\leftarrow\mathcal{B}(p)$ and $Y_{i}\leftarrow F$ # random generation of
losses ;

$X_{i}\leftarrow Z_{i}\cdot\min\\{s,Y_{i}\\}$ # loss for policyholder $i$
(capped at $s$) ;

$C_{i}\leftarrow\min\\{\gamma,X_{i}/d_{i}\\}$ # contribution paid to
policyholder $i$ by friends ;

end for

for  _$i\leftarrow 0$ to $n$_ do

$S_{i}\leftarrow 0$ ;

for  _$j\in V_{i}$_ do

$S_{i}\leftarrow S_{i}+Z_{j}C_{j}$ # sum of contributions paid by $i$ to
friends ;

end for

$\xi_{i}\leftarrow X_{i}+S_{i}-Z_{i}d_{i}C_{i}$ # net loss of policyholder $i$
;

end for

Algorithm 1 Risk Sharing, from Equation (4.1).

######  Proposition 4.3.

The process described by Equation (4.1) and Algorithm 1 is a risk sharing
principle.

###### Proof.

With the notations of Algorithm 1, $\xi_{i}=X_{i}+S_{i}-Z_{i}d_{i}C_{i}$,
where $S_{i}$ is the total amount of money collected by policyholder $i$,
while $Z_{i}d_{i}C_{i}$ is the money given to policyholder $i$. Observe that

$\sum_{i=1}^{n}S_{i}=\sum_{i=1}^{n}\sum_{j\in\mathcal{V}_{i}}Z_{j}C_{j}=\sum_{i=1}^{n}\sum_{j=1}^{n}A_{i,j}Z_{j}C_{j}=\sum_{j=1}^{n}\sum_{i=1}^{n}A_{i,j}Z_{j}C_{j}=\sum_{j=1}^{n}Z_{j}C_{j}\sum_{i=1}^{n}A_{i,j}=\sum_{j=1}^{n}Z_{j}C_{j}d_{j}$

so

$\sum_{i=1}^{n}\xi_{i}=\sum_{i=1}^{n}X_{i}+S_{i}-Z_{i}d_{i}C_{i}=\sum_{i=1}^{n}X_{i}$

so the process described by Equation (4.1) and Algorithm 1 is a risk sharing
principle. ∎

###  4.3 Illustration with a simulated example

Consider a toy example,  where we generate data according to 2.1. For the
example we will assume $\overline{d}=20$ and vary the standard deviation
parameter $\sigma$ between 0 and $4\overline{d}$ (which covers a wide range of
degree distributions from a regular grid to scale free networks - cf. Figure
2.2). The occurrence of claims (here $Z$) are drawn from some Bernoulli
variables with probability $p=1/10$, and claim size is such that
$Y\overset{\mathcal{L}}{=}100+\tilde{Y}$ where $\tilde{Y}$ has a Gamma
distribution, with mean $\mu=900$ and variance $2000^{2}$, so that $Y$ has
mean $1000$ and variance $2000^{2}$. We will consider here a deductible
$s=\mu=1000$ (corresponding to the average price of a claim). Note that with
such a distribution, $78\%$ of the claims fall into the interval $[100;1000]$,
and an insured facing a loss will have a claim below the deductible in 78% of
the cases (if the variance was $1000$, it would be $66\%$ of the claims). Note
that a bit more than $1\%$ of the claims, here, exceed $10000$ (but those
large claims are covered by the insurance company). Finally, note that here
$\mathbb{E}[X]=45.2$ while $\text{stdev}[X]=170$ (which is respectively $45\%$
and $17\%$ of the deductible $s$). A brief summary of that example if given in
Table 4.2.

Minimum number of connections to participate, $\min\\{d_{i}\\}$: | 5
---|---
Maximum contribution per node, $\gamma$: | 50
Deductible, $s$: | 1000
Number of connections to fully recover the deductible, $\overline{d}$: | 20
Number of nodes, $n$ | 5000
Probability to claim a loss: | 10%
Table 4.2: Brief summary of the parameters from the toy-example of Section
4.3.

Consider generations of networks as described above, with $n=5000$ nodes. The
expected loss per policy, $\mathbb{E}[X]$, is $4.5\%$ of the deductible (i.e.
$45.2$ here). Without any risk sharing, the standard deviation of individual
losses, $\text{stdev}[X]$, is $17.3\%$ of $s$, that is a limiting case for
$\text{stdev}[\xi]$, without any contribution, in or out. When introducing
risk sharing on a network, as we can see on Figure 4.2, the standard deviation
is close to $3.9\%$ of $s$ when the network is a regular mesh, where everyone
has 20 connections to share the risk (i.e. $17.3/\sqrt{20}~{}\%$ since the
risk is shared with exactly 20 friends). Then it increases to $5\%$ of $s$
when the standard deviation of network is close to $\sqrt{20}$ (the network is
close to some Erdös-Rényi graph). When the variance is extremely large
(standard deviation close to 80, ie. a scale free network), the loss standard
deviation can actually exceed 17% which is the loss standard deviation without
any risk sharing. So, with some power law type of network, it is possible that
the risk sharing scheme makes it more risky (here, it is extremely risky if
someone is connected to 500 nodes because that policyholder is very likely to
pay a lot to connected nodes).

Figure 4.2: Evolution of $\text{stdev}[\xi]$ when the standard deviation of
$D$ increases from $0$ to $80$ (and $\overline{d}=20$, using the design of the
toy example of Section 4.3), using simulations of random networks of size
$n=5000$, for the line with a positive slope. Points are obtained on the
various scenarios, and the plain line is a smooth (possibly nonlinear)
regression. The horizontal line is the standard deviation of individual
losses, if no risk sharing is considered. We obtain (on average since we use
simulations here) about a standard deviation that is 17% of the deductible.

In this example, we have seen two important drawbacks of that mechanism. The
first one is that people with a lot of connections (more than $\overline{d}$,
i.e. 20 in the examples) can potentially suffer from big losses, so it might
make sense to  not engage with every connection they have. The second one
being that, with such a mechanism, people claiming a loss have a null net
loss, while connections, claiming no loss, will have a net loss, which might
be seen as unfair. In the next section we will discuss the latter issue and
introduce possible solutions for the former in Section 5.

###  4.4 On fairness and risk-sharing

As discussed in Section 3, such risk-sharing techniques can be seen as
interesting because of connections with the convex ordering. Unfortunately,
from a fairness perspective, this might not provide proper incentives to
participants. For example, we have seen that a permutation of risks should
leave policyholders indifferent. Consider some regular network
$(\mathcal{E},\mathcal{V})$ so that everyone is connected to exactly
$\overline{d}$ friends, where $s=\overline{d}\gamma$. Consider the mechanism
described in Equation (4.1) and Algorithm 1, and, for simplicity, suppose that
value of losses is deterministic (and equal to $s$, for convenience) so that
all randomness is coming from $Z_{i}$’s, i.e. who will get a claim. If
policyholder $i$ had an accident, $Z_{i}=1$ and the contribution $C_{i}$ is
exactly $\gamma$. Without any risk sharing,
$\boldsymbol{X}=(X_{1},\cdots,X_{n})$ is a collection of i.i.d. random
variables taking values in $\\{0,s\\}$, with probability $1-p$ and $p$,
respectively. For each node $i$ let $N_{i}$ denote the number of connected
nodes claiming a loss. Hence, $N_{i}\sim\mathcal{B}(\overline{d},p)$. Then,
for any node $i$,

$\xi_{i}=X_{i}+\sum_{j\in\mathcal{V}_{i}}Z_{j}C_{j}-Z_{i}\overline{d}C_{i}=Z_{i}s+N_{i}\gamma-
Z_{i}s=N_{i}\gamma~{}(\leq s)$

Observe that if $i$ claimed no loss $X_{i}=0$ while $\xi_{i}=N_{i}\gamma\geq
0$, and more precisely,

$\mathbb{P}[\xi_{i}=0]=\mathbb{P}[N_{i}=0]=(1-p)^{\overline{d}}$

(with numerical values of the previous section, this probability is
$(1-1/10)^{20}\sim 12\%$) and otherwise, $\xi_{i}>X_{i}$. On the other hand,
if $i$ did claim a loss, $X_{i}=s$ while

$\mathbb{P}[\xi_{i}=s]=\mathbb{P}[N_{i}=\overline{d}]=p^{\overline{d}}$

(which is non-significant here) and otherwise, $\xi_{i}<X_{i}$. Thus,

$\begin{cases}\text{if }i\text{ claims a loss},~{}\xi_{i}\leq X_{i},~{}\text{
and }\xi_{i}<X_{i}\text{ with probability }1-(1-p)^{\overline{d}}\\\ \text{if
}i\text{ claims no loss},~{}\xi_{i}\geq X_{i},~{}\text{ and
}\xi_{i}>X_{i}\text{ with probability }1-p^{\overline{d}}\end{cases}$

To go further, consider now a node $j$ so that $(i,j)\in\mathcal{E}$, and
assume that $i$ claims a loss. Recall that $\xi_{j}=N_{j}\gamma$. If $n$ is
large enough, we can assume that $N_{i}$ and $N_{j}$ are ”almost” independent,
but we know that $N_{j}\geq 1$, more precisely, $N_{j}=1+M_{j}$ with
$M_{i}\sim\mathcal{B}(\overline{d}-1,p)$, independent of $N_{i}$ (as a first
order approximation, when $n$ is assumed to be large enough). And
$\xi_{j}\geq\xi_{i}$ if and only if $N_{j}\geq N_{i}$,

$\mathbb{P}[N_{j}\geq N_{i}]=\sum_{s=0}^{\overline{d}}\mathbb{P}[N_{j}\geq
N_{i}|N_{i}=s]\cdot\mathbb{P}[N_{i}=s]=\sum_{s=0}^{\overline{d}}\mathbb{P}[N_{j}\geq
s]\cdot\mathbb{P}[N_{i}=s].$

Numerically, using numerical values from the previous section, if $i$ did
claim a loss,

$\mathbb{P}[\xi_{i}<\xi_{j}]\sim 59\%\text{ and
}\mathbb{P}[\xi_{i}\leq\xi_{j}]\sim 78\%,\text{ when }(i,j)\in\mathcal{E}.$

Thus, if someone claims a loss, there are here between 60% and 80% chances
that this person will finally pay less than any of his or her connections
(depending if less is strict, or not). Or we might also write it as

$\mathbb{P}[X_{i}>X_{j}]\sim 90\%\text{ and }\mathbb{P}[\xi_{i}<\xi_{j}]\sim
59\%,\text{ when }(i,j)\in\mathcal{E}.$

This potential, and very likely, reordering of actual losses, that is fair ex-
ante (prior to random dices,) can be seen as unfair, ex-post. Thus it will be
natural to ask for some self-contribution, just to make it more likely that
$\xi_{i}>\xi_{j}$ if $X_{i}>X_{j}$. Note that another assumption is that the
first layer of self contribution should be the same, for all policyholders.

##  5 Optimal Personalized Reciprocal Engagements

So far, we did assume that all reciprocal contracts were identical. But it is
possible to assume that they can be different for all policyholder: the main
constraint is that both $i$ and $j$, that are connected, agree on the same
amount of money (since contracts are reciprocals) that we will denote
$\gamma_{(i,j)}$. But it is impossible to assume that all agents will
individually optimise some criteria, independently of the other. If $i$ is
connected to $j$, the commitment $\gamma_{(i,j)}$ should be optimal for $i$
(thus, function of other $\gamma_{(i,\cdot)}$’s) but also optimal for $j$
(thus, function of other $\gamma_{(\cdot,j)}$’s). A natural idea will be to
consider some global planner (the insurance company), maximizing the overall
coverage through reciprocal contracts.

It is important to keep in mind that we have the strong assumption of
homogeneous risks across all agents in the network, which means that a global
planner can optimize the total coverage of the network, irrespective of the
underlying risk. This can somewhat be justified by the homophily property, if
birds of a feather flock together the feather usually also contains many
characteristics that would typically be used in risk assessment. In this case,
one could consider the application presented here as an approach for a
homophile sub-network within a larger graph. Nevertheless, many possible
extensions which relax the homogeneity assumption can be thought of, but a
thorough discussion would go beyond the scope of this paper.

###  5.1 Some notations

First let us generalize the definition of reciprocal contracts from definition
4.2:

######  Definition 5.1.

A reciprocal contract between two policyholders, $i$ and $j$, with magnitude
$\gamma_{(i,j)}$ implies that $i$ will pay an amount $C_{i\rightarrow j}$ to
$j$ if $j$ claims a loss, with $C_{i\rightarrow j}\in[0,\gamma_{(i,j)}]$, and
where $C_{k\rightarrow j}$’s (for various $k$’s) should be proportional to the
engagement $\gamma_{(k,j)}$, and conversely from $j$ to $i$. Thus, for a loss
$y_{j}$

$C_{i\rightarrow
j}=\min\left\\{\gamma_{(i,j)},\frac{\gamma_{(i,j)}}{\overline{\gamma}_{j}}\cdot\min\\{s,y_{j}\\}\right\\},\text{
where }\overline{\gamma}_{j}=\sum_{k\in\mathcal{V}_{j}}\gamma_{(k,j)}$

$\boldsymbol{\gamma}=\\{\gamma_{(i,j)},~{}(i,j)\in\mathcal{E}\\}$ is the
collection of magnitudes.

Figure 5.1: Two risk sharing schemes, with an initial layer of self
contributions on the right (of level 30), for four reciprocal contracts,
$s=100$ and $\gamma=50$.

If we assume that all $\gamma_{(i,j)}$ are equal, we are back in the situation
of definition 4.2. In Figure 5.1 we visualize a situation with heterogeneous
commitments. An alternative to those identical contracts, is to consider an
optimisation problem, where the goal is to maximize the total coverage of
those reciprocal contracts subject to some simple constraints:

  1. 1.

Every reciprocal engagement $\gamma_{i,j}$ is lower than some threshold
$\gamma$

  2. 2.

Every node will only enter reciprocal engagements such that the sum is lower
than or equal the deductible

Thus, formally, we consider the following linear programming problem,

$\begin{cases}\max\left\\{\displaystyle{\sum_{(i,j)\in\mathcal{E}}\gamma_{(i,j)}}\right\\}\\\
\text{s.t. }\gamma_{(i,j)}\in[0,\gamma],~{}\forall(i,j)\in\mathcal{E}\\\
\hskip 18.49411pt\displaystyle{\sum_{j\in\mathcal{V}_{i}}}\gamma_{(i,j)}\leq
s,~{}\forall i\in\mathcal{V}\\\ \end{cases}$ (5.1)

where $s$ is the deductible of the insurance contract and assumed to be the
same across all nodes, so that the second constraint is simply related to
Hypothesis 4.1. With classical linear programming notations, we want to find
$\boldsymbol{z}^{\star}\in\mathbb{R}_{+}^{m}$ where $m$ is the number of edges
in the network (we consider edges $(i,j)$, with $i<j$), so that

$\begin{cases}\boldsymbol{z}^{\star}=\displaystyle{\underset{\boldsymbol{z}\in\mathbb{R}_{+}^{m}}{\text{argmax}}\left\\{\boldsymbol{1}^{\top}\boldsymbol{z}\right\\}}\\\
\text{s.t. }\boldsymbol{A}^{\top}\boldsymbol{z}\leq\boldsymbol{a}\end{cases}$

when $\boldsymbol{A}$ is a $(n+m)\times m$ matrix, and $\boldsymbol{a}$ is a
$(n+m)$ vector,

$\boldsymbol{A}=\left[\begin{matrix}\boldsymbol{T}\\\
\mathbb{I}_{m}\end{matrix}\right]\text{ and
}\boldsymbol{a}=\left[\begin{matrix}\gamma\boldsymbol{1}_{n}\\\
s\boldsymbol{1}_{m}\end{matrix}\right]$

where $\boldsymbol{T}$ is the incidence matrix, $\boldsymbol{T}=[T_{u,w}]$,
with $w\in\mathcal{V}$ i.e. $w=(i,j)$, the $T_{u,w}=0$ unless either $w=i$ or
$w=j$ (satisfying $\boldsymbol{d}=\boldsymbol{T}\boldsymbol{1}_{n}$).

###  5.2 Optimal risk sharing

######  Definition 5.2.

Let $\boldsymbol{\gamma}^{\star}$ denote a solution of the linear program
(5.1). Then $\boldsymbol{\gamma}^{\star}$ is an optimal risk sharing coverage.
The magnitude of the coverage for policyholder $i$ is

$\gamma_{i}^{\star}=\sum_{j\in\mathcal{V}_{i}}\gamma_{(i,j)}^{\star}$

For given losses $\boldsymbol{X}=(X_{1},\cdots,X_{n})$, define contributions
$\displaystyle{C^{\star}_{i\rightarrow
j}=\displaystyle{\min\Big{\\{}\frac{\gamma^{\star}_{(i,j)}}{\displaystyle{\sum_{i\in\mathcal{V}_{j}}\gamma^{\star}_{(i,j)}}}\cdot
X_{j},\gamma^{\star}_{(i,j)}\Big{\\}}}}$, and

$\xi_{i}^{\star}=X_{i}+\sum_{j\in\mathcal{V}_{i}}[Z_{j}C^{\star}_{i\rightarrow
j}-Z_{i}C_{j\rightarrow i}^{\star}].$

Then $\xi_{i}^{\star}$ is a risk sharing, called the optimal risk sharing.

Note that on Figure 5.1, risk sharing is not optimal, if $\gamma=50$: for
instance, A can increase the commitment with any friend by $+10$, as on (b) in
Figure 5.2. The solution obtained from a linear programming solver, is the one
on the right, on (c). Overall, the total coverage for (b) and (c) is the same
(with a sum of $\gamma$’s of 150).

Figure 5.2: Some optimal risk sharing mechanisms - (b) and (c) - where (a) was
the network from Figure 5.1.

The results for a larger simulation are visualized in Figure 5.3. Here we
consider a network of with $n=5000$ nodes, $\overline{d}=20$, deductible
$s=1000$ and a degree distribution that follows model 2.1. We simulate
different graphs for $\sigma$ values between 0 and 80 and calculate the
optimal reciprocal engagements according to 5.1 and calculate the standard
deviation of $\xi^{*}$ under different self-contribution regimes. A risk
averse node would always prefer a risk sharing regime where $\xi^{*}$ is lower
than the no-insurance case (ie. a self-contribution equal to the deductible).
This is the case for all analysed graphs here (note that here, contrary to the
simple case analysed in Figure 4.2 the proposed mechanism is preferable to the
no-insurance case even in the high-degree-variance case).

###  5.3 Sparsification of the network

In the case of optimal reciprocal engagements, some contributions can be close
to zero but not exactly zero, as even a minimal engagement will increase the
global optimization problem. In this case it might be beneficial to “shrink”
these contributions to effectively zero, especially if those reciprocal
contracts have some management fixed costs, without decreasing
$\boldsymbol{\gamma}^{*}$ too much. This can be achieved by reformulating the
linear programming problem into a mixed-integer optimization.

$\begin{cases}\boldsymbol{z}^{\star}=\displaystyle{\underset{\boldsymbol{z}\in\mathbb{R}_{+}^{m}}{\text{argmax}}\left\\{\boldsymbol{1}^{\top}\boldsymbol{z}\right\\}}\\\
\text{s.t. }\boldsymbol{T}^{\top}\boldsymbol{z}\leq s\boldsymbol{1}_{m}\\\
~{}~{}~{}~{}z_{i}\leq My_{i},~{}\forall i\\\
~{}~{}~{}~{}\displaystyle{\sum_{i}y_{i}\leq m}\end{cases}$

where $y_{i}\in\\{0,1\\}$ and $m$ maximum desired non-zero engagements. $M$ is
an upper bound on each $z_{i}\in\boldsymbol{z}$ which comes naturally as we
look for reciprocal engagements smaller than the specified $\gamma$. This
mechanism will effectively shrink small contributions to zero. Figure 5.4
depicts the trade-off. This mechanism also becomes important in section 6,
where networks with a high degree standard deviation result in second degree
adjacency matrices with extremely large $|\mathcal{E}|$. Intuitively, this
allows the global planner to introduce a trade-off between having a large
total coverage but a restricted number of contracts.

Figure 5.3: Evolution of $\text{stdev}[\xi^{\star}]$ when the standard
deviation of $D$ increases from $0$ to $80$ (and $\overline{d}=20$, design of
Example 4.3), using simulations of random networks of size $n=5000$. Here the
deductible is $s=1000$, and various scenarios are considered. Figure 5.4:
Percentage of the sum of deductibles being shared with different number of
non-zero edges. Whereas there is a close to linear trade-off in the middle
ground, where a one less edge results in $-\gamma$ on the total sum, the
function plateaus on the right hand side, resulting in a sparser solution with
almost the same percentage shared.

##  6 Risk Sharing with Friends of Friends

###  6.1 Some notations

When the variance of the degrees is high, some nodes will have too many
connections to share their entire deductible and will hence have a need to
remove unnecessary edges. But the converse is also true as many nodes do not
have enough connections to share all or at least most of their risk.

In order to compensate for this, we will suggest to add an additional layer
based on connections with other nodes via a pathway of length two, as
described on Figure 1.1. This new scheme is based on an additional hypothesis

######  Hypothesis 6.1.

Consider a policyholder $i$ such that his coverage $\gamma^{\star}_{i}$ is
strictly smaller than the overall loss limit $s$ then policyholder $i$ will
try get connections with friends of friends.

######  Definition 6.2.

Given a network $(\mathcal{E},\mathcal{V})$, with degrees $\boldsymbol{d}$ and
some first-level optimization vector $\boldsymbol{\gamma}_{1}$, define the
subset of vertices who could not share their whole deductible as,

$\mathcal{V}_{\boldsymbol{\gamma}_{1}}=\big{\\{}i\in\mathcal{V}:\gamma_{i}=\sum_{j\in\mathcal{V}}\gamma_{(i,j)}<s\big{\\}}.$

The $\boldsymbol{\gamma}_{1}$-sub-network of friends of friends is
$(\mathcal{E}_{\boldsymbol{\gamma}_{1}}^{(2)},\mathcal{V}_{\boldsymbol{\gamma}_{1}})$,

$\mathcal{E}_{\boldsymbol{\gamma}_{1}}^{(2)}=\big{\\{}(i,j)\in\mathcal{V}_{\boldsymbol{\gamma}_{1}}\times\mathcal{V}_{\boldsymbol{\gamma}_{1}}:\exists
k\in\mathcal{V}\text{ such that
}(i,k)\in\mathcal{E},~{}(k,j)\in\mathcal{E}\text{ and }i\neq j\big{\\}}$

Again, from Hypothesis 6.1, policyholders with enough connections (to recover
fully from a loss) will not need to share their risk with additional people.
So only people in that sub-network will try find additional resources though
friends of friends in order to fully cover the deductible.

In Figure 6.2, we visualize the evolution of $\text{stdev}[\xi]$ as a function
of the standard deviation of $D$, when risks can be shared with friends of
friends, when all policyholders have the same self contribution. In view of
Figure 6.2, sharing with friends of friends always lower the standard
deviation of individual remaining losses $\text{stdev}[\xi]$. Furthermore,
Figure 6.3 illustrates the relative importance of each layer. This graph
corresponds to the case where $\zeta=200$. On a regular network, a bit less
than 40% of the claim amount is self insured, which corresponds to
$\mathbb{E}[\min\\{Y,200\\}]/\mathbb{E}[\min\\{Y,1000\\}]\sim 38\%$. Then, the
contribution of friends tends to decrease with the variance of the degrees,
simply because an increasing fraction do not have enough friends to fully
recover the remaining part of the deductible. On the other hand, the
contribution of friends of friends tends to increase with the variance of
degrees, probably up to a given plateau. Observe finally that the proportion
of the first layer is actually increasing with the variance of the degree.

Figure 6.1: Network with five nodes $\\{$A,B,C,D,E$\\}$, with some self-
contribution, and some possible friends-of-friends risk sharing (with a lower
maximal contribution), from the original network of Figure 4.1. Figure 6.2:
Evolution of $\text{stdev}[\xi]$ when the variance of $D$ increases from $0$
to $80$ (and $\overline{d}=20$, with the design of Example 4.3), using
simulations of random networks of size $n=5000$. Eight levels of self
contributions are considered, at $0$ (no self-contribution) $200$, $300$,
$400$, 500 and $700$, when friends, as well as (possibly) friends of friends,
can share risks. A self contribution of $1000$ (on top) mean that no risk is
shared. Figure 6.3: Evolution of amount paid in each layer, as a proportion of
the deductible when a self contribution of $200$ is considered, as a function
of the variance of $D$.

Note that it is possible to use a linear programming algorithm similar to the
one seen in section 5.2 for the friends of friends problem. From a numerical
perspective, it is computationally intensive, since we need
$\gamma^{(1)}_{(i,j)}$ for all first-order edges $(i,j)\in\mathcal{E}$
(friends) and $\gamma^{(2)}_{(i,k)}\in\mathcal{E}_{\gamma_{1}}^{(2)}$ for all
second-order edges $(i,k)$ (friends of friends).

###  6.2 Optimal friends of friends

The optimization of the problem can be obtained via a two stage algorithm: The
first stage is the same as previously in section 5.2, where the initial
contribution among friends, $\gamma_{(i,j)}$ is bounded by $\gamma_{1}$

$\begin{cases}\gamma_{1}^{\star}=\text{argmax}\left\\{\displaystyle{\sum_{(i,j)\in\mathcal{E}^{(1)}}\gamma_{(i,j)}}\right\\}\\\
\text{s.t.
}\gamma_{(i,j)}\in[0,\gamma_{1}],~{}\forall(i,j)\in\mathcal{E}^{(1)}\\\ \hskip
18.49411pt\displaystyle{\sum_{j\in\mathcal{V}_{i}^{(1)}}}\gamma_{(i,j)}\leq
s,~{}\forall i\in\mathcal{V}\\\ \end{cases}$

In the second stage, $\gamma_{(i,j)}$ is bounded by $\gamma_{2}$.
$\mathcal{E}_{\boldsymbol{\gamma}_{1}^{\star}}^{(2)}$ can be obtained by
squaring the adjacency matrix and setting nonzero entries to 1 as defined in
definition 6.2.

$\begin{cases}\gamma_{2}^{\star}=\text{argmax}\left\\{\displaystyle{\sum_{(i,j)\in\mathcal{E}^{(2)}}\gamma_{(i,j)}}\right\\}\\\
\text{s.t.
}\gamma_{(i,j)}\in[0,\gamma_{2}],~{}\forall(i,j)\in\mathcal{E}_{\boldsymbol{\gamma}_{1}^{\star}}^{(2)}\\\
\hskip
18.49411pt\displaystyle{\sum_{j\in\mathcal{V}_{i}^{(1)}}\gamma^{\star}_{1:(i,j)}+\sum_{j\in\mathcal{V}^{(2)}_{i}}}\gamma_{(i,j)}\leq
s,~{}\forall i\in\mathcal{V}\\\ \end{cases}$

Figure 6.4: Distribution of the risk for $\gamma_{1}=50$ and $\gamma_{2}=5$,
without any initial self-contribution. Note how the total shareable risk first
decreases but then again increases again as the share of friends first drops
faster than the share of friends-of-friends can increase. As networks with a
high degree variance generate “hubs” (similar to a star shaped network), these
hub-nodes can help to connect most of the nodes in the network by passing
through these central nodes.

As can be seen in Figure 6.4, highly connected nodes which come to exist in
networks with high degree distribution lead to less risk being shared among
friends. For the part being shared with friends-of-friends, highly central
nodes play the opposite role.

###  6.3 The role of central nodes

As can be seen in figure 6.6, the number of edges in $A^{2}$ increases for
networks with a higher degree variance. This can be explained by considering
the highly central nodes (with a degree). In the extreme case of a star shaped
network, every node can reach any other node by “passing” through the central
node. 555For our application that would mean that for the first optimization
stage, at most $\overline{d}$ edges would be nonzero, as the central node
would not engage with more nodes as it can already share its entire
deductible. On the other hand, every other node would share most of its risk
via the friends of friends mechanism.

Networks with a high degree variance typically posses multiple of these
central nodes that act as hubs. As seen for example in figure 6.4, due to
these central nodes, less risk can be shared between friends, as more edges
will have a zero weight. On the other hand, these hubs enable more nodes to
profit from the friends of friends mechanism.

Figure 6.5: Number of contracts needed to cover the maximum possible amount
(that is
$\displaystyle{\sum_{(i,j)\in\mathcal{E}}\gamma^{\star}_{1:(i,j)}+\sum_{(i,j)\in\mathcal{E}_{\gamma^{\star}_{1}}}\gamma^{\star}_{2:(i,j)}}$).
Here we considered numerical values of Section 4.3, with $\gamma_{1}\leq 50$
and $\gamma_{2}\leq 5$. As the standard deviation of the degrees increases,
more risk is shared via friends-of-friends and more contracts are needed to
cover the shared risk. In the extreme case of zero degree variance, only
$50000$ contracts are needed to cover all the risk.  Figure 6.6: Evolution of
the number of edges in $A^{2}$ for different standard deviations in the degree
distribution in the original network. Duplicated edges in $A^{2}$ are removed,
such that pair of nodes can be connected with at most one edge.

##  7 Conclusion

Recently, denuit2020, Robert, Robert2 and Feng studied actuarial properties of
peer-to-peer insurance mechanisms on networks. In this paper, we consider the
use of reciprocal contracts, as a technique to share risks with “friends”,
more specifically the amount of loss below the deductible of some classical
insurance contract (as visualized on Figure 1.1). Under strong assumptions of
homogeneity, with identical risks over the network, we described a simple
peer-to-peer mechanism, inspired by the original design of Inspeer. Assuming
that all policyholders have similar contracts with the same deductible, we
investigated the shape of the network, or more specifically, given the total
number of connections (or equivalently the average degree of a node), the
impact of the variability of the degrees. We did observe that the more regular
the network, the more efficient the risk sharing, in the sense that it will
decrease the risks for individuals. The original mechanism we described, with
identical contracts among policyholder is interesting since it can evaluate
with time easily: policyholder might decide to leave, or join by signing new
contracts with friends. This flexibility is interesting, but it is clearly
sub-optimal, especially when the network is not regular: in that case, there
are policyholder with too many friends, who might select those they wish to
sign reciprocal contracts with, and policyholders with too few friends.

To overcome that problem, we considered an optimal computation of commitments,
contract per contract, from a global planer’s perspective (in the sense that
we simply want to maximize the overall coverage through those contracts). With
this approach, the dynamics of the approach becomes more complex, since it is
necessary to compute the optimal values when someone joins or leaves, which
yields practical issues. But those contracts clearly lower the risks for
policyholder, who can now share risks with their friends. And finally, we
considered a possible extension, were additional edges could be considered:
policyholders can also agree to share risks with friends of friends. Even if
we assume that policyholders might have less interest to share risks with
people they don’t know, and therefore assume that the commitment will be
financially much smaller, we see that it is possible to increase the total
coverage, and decrease more the variability of the losses for policyholders.

##  8 Appendix

In this appendix, we briefly discuss what $\text{trace}[C\Sigma
C^{\top}]>\text{trace}[\Sigma]$ could mean, in dimension 2, when $C$ is
column-stochastic matrix.

###  8.1 $\text{trace}[D\Sigma D^{\top}]$gainst $\text{trace}[\Sigma]$n
dimension 2

(i) In dimension 2, we can prove that the result holds. There is $x\in[0,1]$
such that

$D=\begin{pmatrix}x&1-x\\\ 1-x&x\end{pmatrix}\text{ and
}\operatorname{Var}[\boldsymbol{\xi}_{1}]=\begin{pmatrix}a^{2}&rab\\\
rab&b^{2}\end{pmatrix}$
$\operatorname{trace}\big{[}\operatorname{Var}[\boldsymbol{\xi}_{2}]\big{]}=\operatorname{trace}\big{[}D\operatorname{Var}[\boldsymbol{\xi}_{1}]D^{\top}\big{]}=\text{trace}\left[\begin{pmatrix}x&1-x\\\
1-x&x\end{pmatrix}\begin{pmatrix}a^{2}&rab\\\
rab&b^{2}\end{pmatrix}\begin{pmatrix}x&1-x\\\
1-x&x\end{pmatrix}^{\top}\right]$
$=\text{trace}\left[\begin{pmatrix}x^{2}a^{2}+2rab(1-x)x+b^{2}(1-x)^{2}~{}~{}~{}*\\\
*~{}~{}~{}(1-x)^{2}a^{2}+2rab(1-x)x+b^{2}x^{2}\end{pmatrix}\right]$
$\leq(a^{2}+b^{2})\big{(}x^{2}+(1-x)^{2}\big{)}+4abx(1-x)$

This parabolic function (in $x$) is symmetric in $x$ and $(1-x)$, so it is
symmetric in $x=1/2$, which is the minimum of that function. The maximum of
that parabolic function on the interval $[0,1]$ is obtained either when $x=0$
or $x=1$, and it takes value $a^{2}+b^{2}$, so, the trace is lower than

$a^{2}+b^{2}=\text{trace}\left[\begin{pmatrix}a^{2}&rab\\\
rab&b^{2}\end{pmatrix}\right]$

###  8.2 $\text{trace}[C\Sigma C^{\top}]$gainst $\text{trace}[\Sigma]$n
dimension 2

In order to understand, consider the case in dimension 2. Consider
$x,y\in[0,1]$ so that

$\boldsymbol{\xi}_{2}=C\boldsymbol{\xi}_{1}\text{with
}C=\begin{pmatrix}x&1-y\\\ 1-x&y\end{pmatrix},\text{ while
}\text{Var}[\boldsymbol{\xi}_{1}]=\begin{pmatrix}a^{2}&rab\\\
rab&b^{2}\end{pmatrix}$

Then, since
$\text{Var}[\xi_{2}^{\prime}]=\displaystyle{\frac{1}{2}}\text{trace}[C\text{Var}[\boldsymbol{\xi}_{1}]C^{\top}]$,
write $\text{trace}[C\text{Var}[\boldsymbol{\xi}_{1}]C^{\top}]$ as

$\text{trace}\left[\begin{pmatrix}x&1-y\\\
1-x&y\end{pmatrix}\begin{pmatrix}a^{2}&rab\\\
rab&b^{2}\end{pmatrix}\begin{pmatrix}x&1-x\\\ 1-y&y\end{pmatrix}\right]$
$=\text{trace}\left[\begin{pmatrix}x^{2}a^{2}+2rab(1-y)x+b^{2}(1-y)^{2}~{}~{}~{}*\\\
*~{}~{}~{}(1-x)^{2}a^{2}+2rab(1-x)y+b^{2}y^{2}\end{pmatrix}\right]$
$=a^{2}\big{(}x^{2}+(1-x)^{2}\big{)}+b^{2}\big{(}y^{2}+(1-y)^{2}\big{)}+2rab\big{[}y(1-x)+x(1-y)\big{]}$

that we can write

$\frac{a^{2}}{2}\big{[}(2x-1)^{2}+1]+\frac{b^{2}}{2}\big{[}(2y-1)^{2}+1]+rab\big{[}(2x-1)(2y-1)-1\big{]}$

that is a quadratic form in $\boldsymbol{Z}=(X,Y)=(2x-1,2y-1)$

$\frac{a^{2}}{2}\big{[}X^{2}+1]+\frac{b^{2}}{2}\big{[}Y^{2}+1]+rab\big{[}XY-1\big{]}=\frac{1}{2}\boldsymbol{Z}^{\top}\begin{pmatrix}a^{2}&rab\\\
rab&b^{2}\end{pmatrix}\boldsymbol{Z}$

which is an elliptic paraboloid function, minimal in $(X,Y)=(0,0)$ – or
$(x,y)=(1/2,1/2)$. Since $(x,y)\in[0,1]^{2}$, the maximum is either attained
in $(0,0)$ or $(1,1)$ when $r<0$ or $(0,1)$ or $(1,0)$ when $r>0$. In the
first case, we obtain that the trace is lower than $a^{2}+b^{2}$, and in the
second case, it can exceed $a^{2}+b^{2}$. More specifically, on Figure 8.1, we
can see that values of $(x,y)$ such that
$\text{trace}[C\text{Var}[\boldsymbol{\xi}]C^{\top}]>\text{trace}[\text{Var}[\boldsymbol{\xi}]]$
(in red, on the right), that is either when one is close to 1, and the other
close to 0.

Figure 8.1: Level curves of
$(x,y)\mapsto\text{trace}[C\text{Var}[\boldsymbol{\xi}]C^{\top}]$ where $C$ is
the column-stochastic matrix with diagonal $(x,y)$, for some variance matrix
with negative correlation on the left, no correlation in the middle, and
positive correlation on the right. For a negative correlation,
$\text{trace}[C\text{Var}[\boldsymbol{\xi}]C^{\top}]\leq\text{trace}[\text{Var}[\boldsymbol{\xi}]]$.
For a positive correlation, the red are corresponds to cases where
$\text{trace}[C\text{Var}[\boldsymbol{\xi}]C^{\top}]>\text{trace}[\text{Var}[\boldsymbol{\xi}]]$.

In that case, for instance with $x=\epsilon$ and $y=1-\epsilon$, it means that

$\begin{pmatrix}\xi_{2,1}\\\ \xi_{2,2}\end{pmatrix}=\begin{pmatrix}x&1-y\\\
1-x&y\\\ \end{pmatrix}\begin{pmatrix}\xi_{1,1}\\\
\xi_{1,2}\end{pmatrix}=\begin{pmatrix}\epsilon&\epsilon\\\
1-\epsilon&1-\epsilon\\\ \end{pmatrix}\begin{pmatrix}\xi_{1,1}\\\
\xi_{1,2}\end{pmatrix}=\begin{pmatrix}\epsilon(\xi_{1,1}+\xi_{1,2})\\\
(1-\epsilon)(\xi_{1,1}+\xi_{1,2})\\\ \end{pmatrix}$

Even if it is a risk-sharing, it can be seen as an unfair, or unbalanced, one,
since policyholder 2 is now taking all most of the risks.