Collaborative Insurance Sustainabilityand Network Structure

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

a Universite du Quebec a Montreal (UQAM), Montreal (Quebec), Canada

b EURo Institut d’Actuariat (EURIA), Universite de Brest, France

c University of Munster, Germany

∗Corresponding author charpentier.arthur@uqam.ca

July 7, 2021

arX

iv:2

107.

0276

4v1

[q-

fin.

RM

] 5

Jul

202

1

Abstract

The peer-to-peer (P2P) economy has been growing with the advent of the Internet, withwell known brands such as Uber or Airbnb being examples thereof. In the insurance sectorthe approach is still in its infancy, but some companies have started to explore P2P-basedcollaborative insurance products (eg. Lemonade in the U.S. or Inspeer in France). The actu-arial literature only recently started to consider those risk sharing mechanisms, as in Denuitand Robert [2020b] or Feng et al. [2021]. In this paper, we describe and analyse such a P2Pproduct, with some reciprocal risk sharing contracts. We consider the case where policyhold-ers 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 (throughthe distribution of degrees) on the risk reduction. We consider also some optimal settingof the reciprocal commitments, and discuss the introduction of contracts with “friends offriends” to mitigate some possible drawbacks of having people without enough connectionsto exchange risks.

Keywords: collaborative insurance; convex order; column-stochastic matrix; degrees; doubly-stochastic matrix; deductible; majorization; network; peer-to-peer insurance; reciprocal con-tracts; risk sharing.

Acknowledgement: The authors wish to thank Romuald Elie, Mathieu Lauriere, TranViet Chi, Miguel Campista, Luis Costa, Matteo Sammarco and Harpreet Kang as wellas participants of the ASTIN 2021 Colloquium for discussions and feedbacks, as well asThomas Richard. Arthur Charpentier received financial support from the Natural Sciencesand Engineering Research Council of Canada (NSERC-2019-07077) and the AXA ResearchFund. Matthias Lowe’s research was funded by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587,Mathematics Munster: Dynamics - Geometry - Structure.

1 Introduction

1.1 Collaborative insurance

In recent years, the global financial crisis of 2008 and the explosion of digital technologies haveinvolved the rise of a collaborative (or peer-to-peer) economy. Indeed, consumers are morelikely to sell, buy, rent or make available goods or services to other individuals through theInternet, thanks to the emergence of platforms such as Airbnb, Uber or Lyft (see Fargere andSouki [2019]). The insurance sector is not left out with several initiatives called collaborativeinsurance around the world, for example Friendsurance in Germany since 2010, Inspeer inFrance 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 peopletogether and pooling their risks so that they are mutually insured. Furthermore, this insur-ance model is more efficient when policyholders make up a community with similar behaviorsor needs according to the homophily principle which structures network ties of every type,including marriage, friendship, work, and other types of relationship (see McPherson et al.[2001]). Indeed, Biener et al. [2018] showed that group policies with pro-social preferencescan alleviate moral hazard of agents.

Several technology-based brands have emerged about collaborative insurance with theaim of challenging the historical players in the insurance sector by significantly improvingthe customer experience. In addition, they all operate with a bulk purchasing method anda relative participation of traditional insurance players. Rego and Carvalho [2020] breaksdown those last ones into three different classes: the broker model (eg. Friendsurance), thecarrier model (eg. Lemonade) and the self-governing model (eg. Teambrella). The brokermodel (see also Clemente and Marano [2020]) and the carrier model (see also Fargere andSouki [2019]) rely on traditional insurance players but allow customers to take on part ofthe risks insured by the group they happen to fall into or choose to adhere to and take backa portion of their profits, or at least make customers feel like they are taking on those risksand taking back such profits. The leading characters in such models appear to play, to alarge extent, the same roles traditionally ascribed to insurers and insurance intermediaries.The self-governing model is the final stage of collaborative insurance where all insuranceintermediaries are removed to self-insure. Vahdati et al. [2018] proposes a self-organizingframework for insurance based on IoT (Internet of Things) and blockchain, which eliminatesthe main problems of traditional insurers such as the transparency of their operations. Thislast 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”,from Denuit and Charpentier [2004]. More specifically, in developed countries, a centralizedinsurance company offers contracts to policyholders, and risk is then shared among the in-surer, 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 guaranteesthat the relative variability of individual contracts decreases with the number of insured inthe pool. Conventional structures are either stock companies (owned by shareholders), and

1

mutual companies, which are owned by policyholders, the later might create some sort ofsolidarity among policyholders, as discussed for instance in Abdikerimova and Feng [2019].But in emerging economies, where the insurance sector is neither properly regulated, nordeveloped, alternatives can be considered to provide risk sharing among agents, that couldbe either based on small groups (as in tontines) or based on multiple one-to-one reciprocalengagements. In this article, we will focus on those reciprocal mechanisms, where policy-holders will cross-insure each other (see Norgaard [1964], and the comment by Reinmuth[1964], on actuarial and legal aspects of reciprocal contracts).

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

1.3 Motivations for deductibles in insurance

Deductibles are a technique insurance companies use to share costs with policyholders whenthey claim a loss, in order to reduce moral hazards. Insurers expect that deductibles helpmitigate the behavioral risk of moral hazards, meaning that either policyholder may notact in good faith, or that they may engage in risky behavior without having to suffer thefinancial consequences. From an actuarial and statistical perspective, given a loss yi for apolicyholder, and a deductible s,

yi = min{yi, s}︸ ︷︷ ︸policyholder

+ (yi − s)+︸ ︷︷ ︸insurer

=

{yi + 0, if yi ≤ s

s+ (yi − s), if yi ≥ s

(we use s – for self insurance – instead of d to denote the deductible, since d will be intensivelyused afterwards to denote the number of reciprocal commitments with friends – related to thedegrees of nodes on the network, as discussed in the next section). One might assume thatif yi ≤ s, some policyholders may not report the loss, since the insurance company will notrepay anything (and in a no-claim bonus systems, there are strong incentives not to declareany small claim, if they don’t involve third party, as discussed in Charpentier et al. [2017]).In this case, the true process of accident occurrences is not the same as the process of claims.

2

claim amount

(a)

(b)

self insurance traditional insurance reinsurance

self insurance traditional insurance reinsurance

deductibleexcess

coverage

p2p

self insurance traditional insurance reinsurancep2p

self insurance traditional insurance reinsurancep2p

(c)

(d)

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

For instance, if the accident process is an homogeneous Poisson process with intensity λ, theclaiming process is an homogeneous Poisson process with intensity λP[Y > s]. Therefore,in the context of risk sharing with a centralizing insurance company, the number of claimsreported might increase since a loss below s might now be claimed since friends might repaya share of it, as discussed in Lee [2017].

From an economic perspective, it is usually assumed that without deductibles, someinsured could be tempted to damage their own property, or act recklessly, leading to moralhazard, as discussed intensively in Eeckhoudt et al. [1991], Meyer and Ormiston [1999], Halekand Eisenhauer [2001] and in the survey by Winter [2000]. In the context of asymmetricinformation, Cohen and Einav [2007] mention that if the policyholder can chose the level ofthe deductible, it might be a valid measure of the underlying risk, even if it is not possibleto distinguish risk aversion and the true level of the risk (known by the policyholder, andunobservable for the insurance company, without additional assumption). Dionne and Gagne[2001] also mention fraud as the most important motivation for insurance companies tointroduce deductibles. Following the seminal work paper of Duarte et al. [2012], Larrimore

3

et al. [2011] or Xu et al. [2015], trust is a key issue in peer-to-peer risk sharing, that mightyield two contradictory behaviors. On the one hand, Artıs et al. [1999] shows that knownrelatives and friends might fraud together; on the other hand, Albrecher et al. [2019] claimsthat fraudulent claims should decrease, and Paperno et al. [2015] claims that bad-faithpractices can be significantly mitigated by implementing a peer-to-peer (anti-fraud measuresbeing costly and hostile).

But even if there are many theoretical justifications for introducing deductibles, they canbe an important financial burden for policyholders. Since the function s 7→ E

[(Y − s)+

]is decreasing, the higher the deductible s, the smaller the premium. The mechanism wewill study in this paper is based on the following idea: policyholders purchase insurancecontracts with a deductible s (that will be less expensive), and they consider a first layerof collaborative insurance, with peers (or friends), as in Figure 1.1, with some possible selfretention. If this layer is a risk sharing among homogeneous peers, this will have no additionalcost for the insured.

1.4 Agenda

This article will be divided in four main parts. In the first one, sections 2 and 3, we willprovide some definitions and mathematical properties, on networks and risk comparisons. Inthe second one, section 4, we will start with a simple presentation of those reciprocal contractdesigns. More specifically, we will assume that we have a network of friends, homogeneousin risks and contracts, and they all share risks, through reciprocal contracts, up to a givenmagnitude γ (if my friend i experiences a loss, all his friend will contribute with the sameamount of money, up to the upper limit γ). In this second part, we will briefly describesuch a risk sharing scheme and run some simulations to understand sustainability1. Thosedesigns can be visualized on Figure 1.2 (introducing possible extensions that will be discussedlater on). Then, in the third part, we will discuss optimization of the design. A naturalapproach would have been to study network specification, that is removing unnecessaryreciprocal contracts (as we will see, having too many contracts can actually add more riskto some insured), but the results have not been computationally conclusive. Thus, we willconsider some optimal computation of individual magnitudes γ, that can depend on bothfriends, in section 5. That approach is promising, its main drawback is that we optimize thescheme from a ‘social planer’ perspective: we optimize the overall coverage, and it mightnot be optimal for each individual. Constrains in the linear program will make the schemesustainable, but possibly not optimal for agents. And finally, in the last part, in section 6, weconsider 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.

2 Mathematical Formalisation of Networks

A network (or graph) G is a pair (E ,V), where V = {1, 2, · · · , n} is a set of vertices (alsocalled nodes) that correspond to insured and E is a set of edges (also called links), which arepairs of vertices, e.g. (i, j). Here we will consider undirected networks, where an edge (i, j)

1Codes used in this paper can be found on https://github.com/phi-ra/collaborative insurance.

4

A

B

C

D

50

50

50

50

50

(1)

A

B

C

D

33

33

50

33

50

(2)

A

B

C

D

50

50

50

50

(3)

A

B

C

D

20

20

20

20

(4)

A

B

C

D

50

50

50

50

50

(5)

30

30

30

30

10

10

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

means that vertices i and j have a reciprocal engagement. A convention is that (i, i) is notan edge. Note further that n is the network size, i.e. the number of nodes, that correspondsto the classical notation of being the portfolio size, in insurance. In this article, the networkis static: neither nodes nor edges can added.

Classical examples of networks are email exchange, where nodes are email accounts, andwe have (directed) edges (denoted i → j) when account i sent an email to j, genealogicaltrees, where nodes are individuals, and we consider a fatherhood directed edge (i → jmeaning that i is the father of j) or a sibling undirected edge (i ↔ j or (i, j) if i andj are siblings), as in Cabrignac et al. [2020]. (Note that risk sharing within families hasalways been important, as discussed in Hayashi et al. [1996] or more recently S. Nuray Akınand Leukhina [2015]). On social media, such as Twitter, nodes are Twitter accounts, andi → j could mean that i follows j; on Facebook or Linkedin, we have undirected edges,i ↔ 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, wherereciprocal contracts are undirected edges. Some visual representations of networks are givenin Figure 2.1.

Formally, such a network can be represented by an n × n matrix A, called adjacency

5

matrix, where Ai,j ∈ {0, 1} with Ai,j = 1 iff (i, j) ∈ E . Let Ai· (and A·i) denote the vector in{0, 1}n that corresponds to row i (and column i) of matrix A. Given a network G = (E ,V)and the associated adjacency matrix A, the degree of a node i is

di = A>i·1 =n∑j=1

Ai,j.

In this article, we do not consider isolated nodes, with no connection (di = 0). Actually, wewill assume that policyholders should have a minimum number of friends to start sharing.Note that when someone is connected to everyone we have di = n− 1. For convenience, letVi denote connections of node i, i.e.

Vi = {j ∈ V : (i, j) ∈ E or Ai,j = 1}

such that card(Vi) = di. A regular network is a graph in which every vertex has the samedegree. On the left of Figure 2.1, the two graphs, with n = 20 vertices, are regular since allvertices have degree 4. Observe here that the degree function does not uniquely describe thenetwork.

A random network is obtained when assuming that edges are randomly chosen, accordingto some process, that will yield different families of networks (as discussed afterwards).Thus, the degree of a node taken at random in V is now a random variable, with somedistribution on {0, 1, 2, · · · , n−1} (or {1, 2, · · · , n−1} if vertices cannot be isolated). Randomnetworks are usually described by a random process which generates them, such as theclassical Erdos–Renyi (see Erdos and Renyi [1959]), where each edge (i, j) in V×V (excluding(i, i)) is included in the network with the same probability p, independently from every otheredge. Then D has a binomial distribution B(n − 1, p) which can be approximated by aPoisson distribution P(p/(n− 1)), when n is large, and p is not too large. In that case, notethat Var[D] ∼ E[D]. This can be visualized in the middle of Figure 2.1. The preferentialattachment method was proposed by Barabasi and Albert [1999] (and might be related tothe small world model by Watts and Strogatz [1998]). In this model, starting from somesmall network, a new node is created at each time step and connected to existing nodesaccording to the preferential attachment principle: at a given time step, the probability p ofcreating an edge between an existing node i and the new node is [(di + 1)/(|E|+ |V|)], whereE is the set of edges between nodes V (or in a nutshell, it is proportional to di, i.e. it ismore likely to connect to popular nodes). This technique produces some scale-free networkswhere the degree distribution follows a power law (see also Caldarelli [2007]). example ofsuch graphs are given in the column right of Figure 2.1. As discussed in Barabasi and Albert[1999] Caldarelli [2007], Aparicio et al. [2015], or more recently Charpentier and Flachaire[2019], most social networks are scale-free networks (with a power law type of distributionfor degrees). Even if being “friend ” on Facebook is not as strong as a financial reciprocalcommitment (that we want to model here), intuitively, insurance networks we consider couldsatisfy that feature.

As we will see in this paper, the shape of the network, described through the distributionof the degrees will have a major influence on the sustainability of the system. And interest-ingly, given a distribution for D, or more specifically a collection of {d1, d2, · · · , dn} that be

6

(a) (b) (c)

Figure 2.1: Three type of networks, with identical expected value, here E[D] = 4 (on average,a node chosen at random is connected to four others). (a) are regular networks, where D = 4(and Var[D] = 0), (b) are Erdos–Renyi networks, with Var[D] ∼ E[D] and (c) are scale-freenetworks with some nodes highly connected, Var[D] > E[D] (and others with possibly onlyone connection)

obtained by drawing independently D1, D2, · · · , Dn, with a minor additional constraint (thesum of the degrees should be even) one can generate a network with such degrees. Thesenetworks have become known under the name configuration model (cf. [Newman, 2010,Chapter13.2]). However, in such networks one has to allow for self-loops and multi-edges(see Chatterjee et al. [2011] for a probabilistic perspective). Chung and Lu [2002] suggesteda simple and easy technique to generate some random graph of given (node) size, with someexpected given degree distribution. Heuristically, given two nodes i and j with respectivedegrees di and dj, the probability that (i, j) is an edge is proportional to the product didj,

Ai,j = Aj,i =

{1, with probability proportional to didj

0 otherwise

More recently, Bayati et al. [2009] suggests some algorithms to generate networks with a givensize n and a given vector of degrees, d (possibly drawn randomly from a given probabilisticdistribution), inspired by Bender and Canfield [1978], Molloy and Reed [1995] and Newman[2002]. More recently These algorithms are implemented in the igraph librarie, in R andPython, see Csardi and Nepusz [2006]. More specifically, degree.sequence.game generates

7

Figure 2.2: Distribution of D, when d = 20, with three different standard-deviations: 5, 10and 15. Such distributions will be used in the numerical applications. Note that the supportof D is here {5, 6, · · · , 1000}, where rounded and shifted Gamma distributed were used.

a network of size n from a vector d = (d1, · · · , dn) where the degrees of the generated networkis exactly d. Note that there is also the randomNodeGraph function of R package graph, seeGentleman et al. [2019]. Therefore, if we generate random graphs and compute (for instance)the variance of the degree, it will be the variance of d, so, the only source of uncertainty herewhen using generations of random graphs is simply coming from the random generation ofd 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 shapeof the network, it is necessary to make some assumptions. Consider n nodes, indexed withi = 1, · · · , n. Each node corresponds to a policyholder, facing a random yearly loss, thatmight occur with probability p, and cost Yi, that has distribution F . For convenience, let Zidenote the indicator for the event that i claims a loss, as in the individual model in actuarialscience. Zi is a Bernoulli variable with probability p. Thus, here we consider the simple casewhere 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 ofcontributions is bounded (by the magnitude denoted γ). This will not change much theanalysis, and it will only make notations more complex to read.

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

Xi = Zi ·min{s, Yi} =

{0 if no claim occurred (Zi = 0)

min{s, Yi} if a claim occurred (Zi = 1)

8

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

Hypothesis 3.2. All risks Xi’s are assumed to be independent and identically distributedrandom variables. Furthermore, all policyholders have similar insurance contracts with thesame deductible s.

Hypothesis 3.2 is important to have a simple reciprocal and fair mechanism. If twopolicyholders 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 become more difficult toderive a fair contribution for some risk exchange (this was discussed recently in Denuit andRobert [2020a]).

3.1 Risk preferences and ordering of risk

Ohlin [1969], inspired by Karlin and Novikoff [1963], suggested to use the convex order tosolve the optimal insurance decision problem, as discussed in Denuit et al. [2006] or Cheunget al. [2015]. Given a random loss X, and an insurance scheme offering indemnity x 7→ I(x)against a premium p, an agent will purchase the insurance if ξ �CX X for the convex order,where ξ = X + π − I(X) (where π is the associated premium to transfer that risk). Thisformalism is a natural translation of the expected utility model in insurance, where agenthaving wealth w and (concave) utility u agree to purchase a contract offering utility I againsta premium π if E[u(w −X)] ≤ E[u(w + I(X) − π −X)] = E[u(w − ξ)]. As in Shaked andShanthikumar [2007] or Denuit et al. [2006], define the convex order as follows,

Definition 3.3 (Convex order). Consider two variables X and Y such that

E[g(X)] ≤ E[g(Y )] for all convex functions g : R→ R,

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

As we will see below, the inequality X �CX Y intuitively means that X and Y have thesame magnitude, as E[X] = E[Y ], but that Y is more variable than X. More specifically:

Proposition 3.4. If X �CX Y , then E[X] = E[Y ] and Var[X] ≤ Var[Y ].

Proof. For the expected value, consider g(x) = ±x (Equation (3.A.2) in Shaked and Shan-thikumar [2007]) and for the variance g(x) = x2 (Equation (3.A.4) in Shaked and Shanthiku-mar [2007])

If X �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 YL= X + Z where Z is such that

E[Z|X] = 0, as discussed in Carlier et al. [2012]. An example of variables ordered via theconvex order is given in

Lemma 3.5. Let X = (X1, · · · , Xn) denote a collection of i.i.d. variables, and p somen-dimensional probability vector. Then p>X �CX Xi for any i.

9

Proof. Let X+i = Xi1, which is a commonotonic version of vector X, in the sense that all

marginals have the same distribution (since components of X are identically distributed),and Xi = p>X+

i . A simple extension of Theorem 6 in Kaas et al. [2002] (and Proposition3.4.29 in Denuit et al. [2005]) implies that p>X �CX p>X+

i for any i (since Xi’s are i.i.d.),i.e. p>X �CX Xi. Note that it can also be seen as a corollary of Property 3.4.48 inDenuit et al. [2005], using the fact that p ≺ ei for the majorization order (where classicallyei = (0, 0, · · · , 0, 1, 0, · · · , 0, 1)>)

If such a concept can be used to compare two risks, for a single individual, we needa multivariate extension to study the risk over the entire portfolio. Following Denuit andDhaene [2012], define the componentwise convex order as follows

Definition 3.6 (Componentwise convex order). Given two vectors X and Y , X is said tobe smaller than Y in the componentwise convex order if Xi �CX Yi for all i = 1, 2, · · · , n,and is denoted X �CCX Y . And if at least one of the convex order inequalities is strict, wewill say that Y dominated X, and is denoted X ≺CCX Y .

This inequality, X ≺CCX Y , means that, for this pool of n insured, each and everyoneof them prefers Xi to Yi. But as we will see later one, when optimizing the mechanism, wewill consider the perspective of some central planer, maximizing the total coverage. Thus itcould be possible to define some sort of global interest for a mechanism. This means thatwe might be slightly less restrictive in order to study some possible collective gain of sucha 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, buthere, since there might be heterogeneity of agents, we can consider some randomly chosenagent. A weaker condition than the one in Proposition 3.4 would be : let I denote a randomvariable, uniformly distributed over {1, 2, · · · , n}, and define ξ′ = ξI , then we wish to have

E[ξ′] = E[E[ξI |I]

]= E[Xi] and Var[ξ′] = E

[Var[ξI |I]

]=

1

ntrace

[Var[ξ]

]≤ Var[Xi] = σ2,

and not necessarily

E[ξi] = E[Xi] and Var[ξi] ≤ Var[Xi], ∀i ∈ {1, 2, · · · , n}.

Note that Chatfield and Collins [1981] and Johnson and Wichern [2008] call trace[Var[ξ]

]the

total variation of vector ξ. If we will not use that name here, such a quantity is a standardvariability measure for random vectors.

3.2 Risk comparison for linear risk sharing schemes

Risk sharing schemes were properly defined in Denuit and Dhaene [2012]:

Definition 3.7 (Risk sharing scheme). Consider two random vectors ξ = (ξ1, · · · , ξn) andX = (X1, · · · , Xn) on Rn

+. Then ξ is a risk-sharing scheme ofX if X1+· · ·+Xn = ξ1+· · ·+ξnalmost surely.

10

For example, the average is a risk sharing principle: ξi = X =1

n

n∑j=1

Xj, for any i. A

weighted average, based on credibility principles, too: ξi = αXi + [1− α]X, where α ∈ [0, 1](we will get back on those averages in the next section). Observe also that the order statisticsdefines a risk sharing mechanism: ξi = X(i) where X(1) ≤ X(2) ≤ · · · ≤ X(n). Given apermutation σ of {1, 2, · · · , n}, ξi = Xσ(i) defines a risk sharing (that we we write, usingvector notations, ξ = PσX, where Pσ is the permutation matrix associated with σ). Thisexample 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 ξ = (ξ1, · · · , ξn)and X = (X1, · · · , Xn) on Rn

+, such that ξ is a risk-sharing scheme of X. It is said to be alinear risk sharing scheme if there exists a matrix M , n× n, with positive entries, such thatξ = MX, almost surely.

For example, if ξi = X for i = 1, 2, · · · , n, characterized by matrix M = [Mi,j] withMi,j = 1/n. For the credibility inspired principle, Mα = αI + (1 − α)M . Another examplecan be obtained with any doubly stochastic matrices: a n×n matrix D is said to be doublystochastic2 if

D = [Di,j] where Di,j ≥ 0, D>·,j1 =n∑i=1

Di,j = 1 ∀j, and D>i,·1 =n∑j=1

Di,j = 1 ∀i.

Then ξ = DX is a linear risk sharing of X. Row and column conditions on matrix D arecalled zero-balance conservation in Feng et al. [2021]. A particular case can be when D is apermutation matrix, associated with a permutation of {1, · · · , n}. This example with doublystochastic matrices is important because of connections with the majorization concept (seeSchur [1923] and Hardy et al. [1934]). Furthermore Chang [1992], Chang and Yao [1993]extended the deterministic version to some stochastic majorization concept, with resultsthat we will use here.

Inspired by the componentwise convex ordering, the following definition can be consid-ered, as in Denuit and Dhaene [2012] (a similar concept can be found in Carlier et al. [2012])

Definition 3.9 (Ordering of risk sharing schemes (1)). Consider two risk sharing schemesξ1 and ξ2 of X. ξ1 dominates ξ2, for the convex order, if ξ2 �CCX ξ1.

Definition 3.10 (Desirable risk sharing schemes (1)). A risk sharing scheme ξ of X isdesirable if ξ �CCX X.

The order statistics of some i.i.d. random vector (ξi = X(i) where X(1) ≤ X(2) ≤ · · · ≤X(n)) is not a desirable risk sharing principle (if Xi are non-deterministic ordered values),and since E[ξn] > E[Xn] so we cannot have ξn �CX Xn. For a (deterministic) permutation,observe that agents are indifferent, ξ ∼CCX X (in the sense that (ξ �CCX X andX �CCX ξ)

2In this section, D will denote some doubly stochastic matrix n×n, not to be confused with the randomdegree vector D.

11

Proposition 3.11 (Desirable linear risk sharing schemes (1)). If ξ is a linear risk sharingof X, ξ = DX for some doubly stochastic matrix D, then ξ is desirable.

Proof. Assume that ξ = DX for some double stochastic matrix D, with rows Di·’s. Since Dis doubly stochastic, Di· is a probability measure over {1, · · · , n} (with positive componentsthat sum to one), and from Lemma 3.5, D>i·X �CX Xi, for all i.

Proposition 3.12 (Ordering of Linear risk sharing schemes (1)). Consider two linear risksharing schemes ξ1 and ξ2 of X, such that there is a doubly stochastic matrix D, n×n suchthat ξ2 = Dξ1. Then ξ2 �CCX ξ1.

Proof. This can be seen as an extention of Proposition 3.11. Here ξ1 = M1X and ξ2 = M2X,since we consider linear risk sharings, and we can write the later ξ2 = M2X = DM1X, thusM2 = DM1, since equality is valid for all X. As defined in Dahl [1999] and Beasley andLee [2000], this corresponds to the matrix majorization concept, in the sense that M2 ≺M1,that implies standard majorization per row, thus for all i, M2:i· ≺M1:i·, from Lemma 2.8 inBeasley and Lee [2000]. But from Property 3.4.48 in Denuit et al. [2005], we know that ifa ≺ b for the majorization order, a>X �CX b>X. Thus, if a = M2:i· and b = M1:i·, we havethat for any i, ξ2:i �CX ξ1:i, and therefore we have the componentwise order ξ2 �CCX ξ1.

Proposition 3.13. Consider two linear risk sharing schemes ξ1 and ξ2 of X, such thatξ2 = Dξ1, for some doubly-stochastic matrix. Then Var[ξ2:i] ≤ Var[ξ1:i] for all i.

Proof. This is simply the fact that from Proposition 3.2, ξ2 �CCX ξ1, i.e. from Definition3.1, ξ2:i �CCX ξ1:i for all i and Var[ξ2:i] ≤ Var[ξ1:i] follows from the property on the convexorder mentioned earlier.

A simple corollary is the following

Proposition 3.14. Consider two linear risk sharing schemes ξ1 and ξ2 of X, such thatξ2 = Dξ1, for some doubly-stochastic matrix. Then trace

[Var[ξ2]

]≤ trace

[Var[ξ1]

].

Proof. Observe first that Var[ξ2] = DVar[ξ1]D>. Since the variance matrix Var[ξ1] is posi-

tive, the application h : M 7→ trace[M Var[ξ1]M

>] is convex. By the Birkhoff-von Neumanntheorem (see Birkhoff [1946]), every doubly stochastic matrix is a convex combination of per-mutation matrices, i.e. D = ω1P1 + · · · + ωkPk for some permutation matrices P1, · · · , Pk(and some positive weights ωi that sum to 1). By the convexity of h,

trace[

Var[ξ2]]

= h(D) ≤k∑i=1

ωih(Pi) =k∑i=1

ωi trace[Pi Var[ξ1]P

>i

],

and therefore

trace[

Var[ξ2]]≤

k∑i=1

ωi trace[

Var[ξ1]]

= trace[

Var[ξ1]].

A discussion of these results is provided in Appendix 8.1.

12

3.3 Risk sharing on cliques

Such a design was briefly mentioned in Feng et al. [2021], and called Hierarchical P2PRisk Sharing. In a network, a clique is a subset of nodes such that every two distinctvertices in the clique are adjacent. Assume that a network with n nodes has to (dis-tinct) cliques, with respectively k and m nodes. One can consider some ex-post contri-butions, to cover for losses claimed by connected policyholders. If risks are homogeneous,contributions are equal, withing a clique. Assume that policyholders face random lossesX = (X1, · · · , Xk, Xk+1, · · · , Xk+m) and consider the following risk sharing

ξi =

1

k

k∑j=1

Xj if i ∈ {1, 2, · · · , k}

1

m

k+m∑j=k+1

Xj if i ∈ {k + 1, k + 2, · · · , k +m}

This is a linear risk sharing, with sharing matrix Dk,m, so that ξ = Dk,mX, defined as

Dk,m =

1 · · · k k + 1 k + 2 · · · k +m

1 k−1 · · · k−1 0 0 · · · 0...

......

......

...k k−1 · · · k−1 0 0 · · · 0

k + 1 0 · · · 0 m−1 m−1 · · · m−1

k + 2 0 · · · 0 m−1 m−1 · · · m−1...

......

......

...k +m 0 · · · 0 m−1 m−1 · · · m−1

Since Dk,m is a doubly-stochastic matrix, ξ ≺CCX X. In order to illustrate Proposition 3.14,let I denote a uniform variable over {1, 2, · · · , n}, and ξ′ = ξI ,

E[ξ′] = E[E[ξI |I]] =1

n

n∑i=1

E[ξi] =1

n

n∑i=1

E[Xi] = E[X]

as expected since it is a risk sharing, while

Var[ξ′] = E[Var[ξI |I]] =1

n

n∑i=1

Var[ξi] =1

n

(k

Var[X]

k2+m

Var[X]

m2

)=

Var[X]

n

(1

k+

1

m

)=

Var[X]

k(n− k)

since n = k+m, so that Var[ξ′] < 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 risksharing. Observe further that k 7→ k(n − k) is maximal when k = bn/2c, which meansthat risk sharing benefit is maximal (socially maximal, for a randomly chosen representativepolicyholder) when the two cliques have the same size.

13

3.4 Some weaker orderings for risk sharing scheme

Following Martınez Perıa et al. [2005], a weaker ordering can be considered with column-stochastic matrices, instead of doubly stochastic matrices. A column-stochastic matrix (orleft-stochastic matrix) C satisfies

C = [Ci,j] where Ci,j ≥ 0, andn∑i=1

Ci,j = 1 ∀j.

Proposition 3.15. Let C be some n× n column-stochastic matrix, and given X, a positivevector in R+, define ξ = CX. Then ξ is a linear risk sharing of X.

Proof. First, observe that since C is a matrix with positive entries, ξ ∈ Rn+. Then, simply

observe thatn∑i=1

ξi =n∑i=1

n∑j=1

Ci,jXj =n∑j=1

(n∑i=1

Ci,j

)Xj =

n∑j=1

Xj.

In order to illustrate why such a representation – Y = CX instead of Y = DX, asconsidered earlier – can be seen as a possible ordering, consider the two following matrices(where D1 corresponds to C2 when α = 0),

D1 =

1 0 00 1/2 1/20 1/2 1/2

and C2 =

1 α α0 (1− α)/2 (1− α)/20 (1− α)/2 (1− α)/2

, with α ∈ (0, 1).

Consider some non-negative random losses X = (X1, X2, X3) where the three componentsare independent and identically distributed. Consider the following risk sharing scheme:ξ = (ξ1, ξ2, ξ3) such that ξ = D1X, that is ξ1 = X1 and ξ2 = ξ3 = X23, where X23 =(X2 + X3)/2. Assume now that we randomly select one of the components. Let X ′ denotethat one, or more specifically, let I denote a uniform variable over {1, 2, 3}, and let X ′ = XI .And similarly, let ξ′ = ξI . Note that if µ = E[Xi] = E[X ′i], and σ2 = Var[Xi] = Var[X ′i],

E[ξ′] =1

3

3∑i=1

E[ξi] =1

3[µ+ 2 · µ] = µ = E[X ′],

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

Var[ξ′] =1

3

3∑i=1

Var[ξi] =1

3

[σ2 + 2 · 2σ2

4

]=

2

3σ2 < Var[X ′].

So it seems that the vector ξ of individual losses after sharing risks has a lower variancethan vector X. Furthermore, we can easily prove that ξ1 �CCX X since ξ1 ∼CX X1 whileξ2 �CX X2 and ξ3 �CX X3.

For the second one, consider ξ = (ξ1, ξ2, ξ3) such that ξ = C2X, that is ξ1 = X1 + 2αX23

and ξ2 = ξ3 = (1 − α)X23, where X23 = (X2 + X3)/2. Then X and ξ cannot be compared

14

using the componentwise convex order, even if ξ1+ξ2+ξ3 = X1+X2+X3. Assume now thatwe randomly select of the of the components. Let X ′ denote that one, or more specifically,let I denote a uniform variable over {1, 2, 3}, and let X ′ = XI . And similarly, let ξ′ = ξI .Note that if µ = E[Xi] = E[X ′i], and σ2 = Var[Xi] = Var[X ′i],

E[ξ′] =1

3

3∑i=1

E[ξi] =1

3[(1 + 2α)µ+ 2 · (1− α)µ] = µ = E[X ′],

Var[ξ′] =1

3

3∑i=1

Var[ξi] =1

3

[(1 + 2α2)σ2 +

4 · (1− α)2σ2

4

]=

1 + 2α2 + (1− α)2

3σ2 < Var[X ′].

Note that this case can be seen as extreme, since there is probably no interest for the firstagent 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, wewill discuss additional assumptions later on. Following Martınez Perıa et al. [2005], we candefine an ordering based those column-stochastic matrices

Definition 3.16 (Weak Ordering of Linear risk sharing schemes (2)). Consider two linearrisk sharing schemes ξ1 and ξ2 of X. ξ1 weakly dominates ξ2, denoted ξ2 �wCX ξ1 if andonly if there is a column-stochastic matrix C, n× n such that ξ2 = Cξ1.

In the examples above, we did prove that ξ1 �wCX X and ξ2 �wCX X, but can wecompare ξ1 and ξ2? One can write{

ξ2,1 = X1 + 2αX23 = ξ1,1 + 2αξ1,2 = ξ1,1 + αξ1,2 + αξ1,3

ξ2,2 = ξ2,3 = (1− α)X23 = (1− α)ξ1,2 = (1− α)/2ξ1,2 + (1− α)/2ξ1,3

so we can actually write ξ2 = C2ξ1, since C2D1 = C2.Note that Proposition 3.14 cannot be extended when matrices are only column-stochastic,

in the sense that ξ2 = Cξ1 for some column-stochastic matrix C is not sufficient to guaranteetrace[Var[ξ2]] = trace[CVar[ξ1]C

>] ≤ trace[Var[ξ1]]. Some heuristic interpretation of thatresult are given in Appendix 8.2. It does hold when ξ1 are independent risks.

Proposition 3.17. If ξ is a linear risk sharing of X, where Xi’s are independent risks withvariance σ2, associated with some column-stochastic matrix C, then

trace[Var[ξ]

]= σ2 trace

[CC>

]≤ σ2n

and therefore Var[ξ′] ≤ Var[X] = σ2.

Proof. In dimension n,

trace[CC>

]=

n∑i,j=1

C2i,j ≤

n∑i,j=1

Ci,j =n∑i=1

(n∑j=1

Ci,j

)=

n∑i=1

1 = n.

15

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

Definition 3.18 (Weak Ordering of Linear risk sharing schemes (3)). Consider two linearrisk sharing schemes ξ1 and ξ2 of X. ξ1 weakly dominates ξ2, denoted ξ2 ≺WBCX ξ1 ifand only if there is a column-stochastic matrix C, n× n such that ξ2 = Cξ1 and such thattrace[Var[ξ2]] ≤ trace[Var[ξ1]].

This weaker order, trace[Var[ξ2]] ≤ trace[Var[ξ1]], means that individually, all agents donot prefer scheme 2 over 1, but globally, scheme 2 is preferred. This issue will be discussedafterwards.

3.5 On nonlinear risk sharing mechanisms

Risk sharing mechanisms that we will consider with reciprocal contracts are not linear inX since contributions between two agents will be capped. This will be discussed in thenext section. A tractable way to compare risks will simply be based on the variances of risksharing mechanisms, that we will discuss on simulations on networks.

Definition 3.19 (Weak Ordering of Nonlinear risk sharing schemes (4)). Consider two risksharing schemes ξ1 and ξ2 of X. ξ1 weakly dominates ξ2 if trace[Var[ξ2]] ≤ trace[Var[ξ1]].

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 X is a collection of i.i.d.variables (as in hypothesis 3.2) Xi, based on i.i.d. Zi’s that are Bernoulli variables withsuccess probability 10%, and where Yi’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 onlyon policyholder A.

Indemnity and contributions are piecewise linear, as we can see on Figure 3.1. Firstof all, A has a first layer with a self contribution of 40: instead of having a deductible of100 (that is the contract A signed with the insurance company), A now has a deductibleof 40. Then reciprocal contracts are based on an upper limit γ (this will be formalized inDefinition 4.2). For instance, A and B share risk with a reciprocal contract with magnitudeγ of 20, and magnitude 30 with C. Thus, for claims x between 40 and 90, (x− 40) is sharedproportionally between B and C, with proportions 40%-60% respectively. Here we highlightthat there is a remaining part for A, for claims between 90 and 100 (some sort of residualself insurance). Of course, since A has reciprocal contracts with B and C, A is committedto 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. Inthe next section, we will consider globally a simple case with 4 policyholders.

3.6 A brief example

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’

16

BA

10

4020

C

30

A

A

C

B

reciprocals

100co

ntr

ibu

tion

s

amount of the claim

0

ded

uct

ibles

40 90

40%

60%

firs

tla

yer

(sel

fin

sura

nce

)re

sid

ual

self

insu

ran

ce

Figure 3.1: Subdivision of payments when A claims a loss, while sharing risks with B andC, based on the description on the left (with two non-identical reciprochal contracts). The xaxis is the value of the loss X. On the y axis we can visualize the amount paid respectivelyby A, B, C, and A back again. There is a first layer of self-contribution of 40, then tworeciprocal 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 ≤ 40, it is fully paidby A. Between 40 and 90, 40 is paid by A, then X − 40 is shared in two, B paying 40% andC 60%. Above 90, B and C pay respectively 20 and 30, while A pays min{X − 50, 50}.

variance - that is the trace of the variance matrix over n) for these eight mechanisms thatcan described as follows:

– the benchmark corresponds to risks Xi with mean 10 (that is 10% of E[X]), andstandard deviation 30 if the loss is deterministic; for a uniform random loss Y (uniformbetween 0 and 200), the mean of Xi is 7.5 (that is 10% of E[min{100, Y }]) and thestandard deviation is now 24.71.

(a) corresponds to a regular network, with d = (2, 2, 2, 2). Here all insured decided toshare risks with two friends, through reciprocal contracts. If A claims a loss of 80, Awill 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 E[X] (thatis 10 is the loss is 100, or 7.5 is the loss is uniform over [0,200] with a deductible at100). Since the risk is always shared with two friends, the standard deviation is herestdev[X]/

√2.

(b) this network, with d = (3, 2, 2, 1), is obtained from (a) by changing an edge: edge(C,D) becomes (A,C). Reciprocal commitments of γ = 50 are maintained. As we will

17

A

B

C

D

50

50

50

50

50

A

B

C

D

50

50

50

50

(a) (b)

A

B

C

D

33

33

50

33

(c)

A

B

C

D50

50

(f)

50

50

50

50

A

B

C

D

50

50

50

(d) 50

17

17

100

67

A

B

C

D

50

50

(e)

50

50

50

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

formalize with Hypothesis 4.1, whatever the commitments, an insured cannot collectmore than the actual sum of the loss. Since s = 100, if A claims a loss, the otherthree nodes will only contribute by paying up to 33.333. But if any of those nodesclaim 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, theremaining part, between 50 and 100 is now a self contribution of D. As for (a), the sumof γ’s is 200. Observe here that with reciprocal contracts of 50, A might collect morethat the actual loss: we will add an assumption so that the sum of collected moneycannot exceed s. If A claims a loss, B will pay 33.333 to A, but conversely, if B claimsa loss, A will pay 50. Therefore, those contracts do not yield a desirable risk sharingfor 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 exceedsE[X], with also a larger variable.

(c) corresponds to the case where A lowers his reciprocal engagements, with min{s/di, γ},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

18

and C can still have a risk sharing with magnitude 50 (this will not be possible in thenext section, and will be considered only in section 5). The remaining part is a selfcontribution that exists on top of the risk sharing scheme.

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

(d) corresponds to some optimal risk sharing, result of a linear program, where we maximizethe sum of γ(i,j)’s – this will be discussed in Section 5. Note that here, the sum of γ(i,j)’sis equal to 150 (as mechanism (c)). Here one node is on the side, and the other threefor a regular network, d′ = ({2, 2, 2}, {0}). As discussed earlier, we will try to avoidsuch 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 theother hand, with variant (e), the variance of D (as well as the other three) is now thesame as the one in (a) if the loss is not random. If the loss is random, the variance isslightly 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 issufficient (50 + γ = s). The sum of reciprocals is here 100.

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

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-insuranceand the traditional insurance company. In Section 4.1 we will consider the case where thereis no first layer of self-insurance, and the peer-to-peer layer is the very first one. We will seethat this scheme is not optimal in the sense that it usually yields to ex-post unfairness.

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

Hypothesis 4.1. Given a network (E ,V), policyholder i will agree to share risks with allpolicyholders j such that (i, j) ∈ E (called i’s friends). For each policyholder, the total sumcollected from the friends cannot exceed the value of the actual loss.

This assumption means that the risk sharing scheme is backed up on the original network.In the next section, reciprocal contracts can be personalized, and it will be possible to areciprocal contact with contribution equal to 0, and we will consider some sub-network.

19

Deterministic loss, y = 100

nodes A B C D overall

E[Xi] 10.00 10.00 10.00 10.00 10.00stdev[Xi] 30.00 30.00 30.00 30.00 30.00

E[ξ(a)i ] 10.00 10.00 10.00 10.00 10.00

stdev[ξ(a)i ] 21.21 21.21 21.21 21.21 21.21

E[ξ(b)i ] 15.00 8.33 8.33 8.33 10.00

stdev[ξ(b)i ] 26.00 18.00 18.00 18.00 20.50

E[ξ(c)i ] 10.00 10.00 10.00 10.00 10.00

stdev[ξ(c)i ] 17.33 18.70 18.70 22.33 19.33

E[ξ(d)i ] 10.00 10.00 10.00 10.00 10.00

stdev[ξ(d)i ] 21.21 21.21 21.21 30.00 23.71

E[ξ(e)i ] 10.00 10.00 10.00 10.00 10.00

stdev[ξ(e)i ] 21.21 21.21 21.21 21.21 21.21

E[ξ(f)i ] 10.00 10.00 10.00 10.00 10.00

stdev[ξ(f)i ] 21.21 21.21 21.21 21.21 21.21

Random loss, Y ∼ U[0,200]nodes A B C D overall

E[Xi] 7.50 7.50 7.50 7.50 7.50stdev[Xi] 24.71 24.71 24.71 24.71 24.71

E[ξ(a)i ] 7.50 7.50 7.50 7.50 7.50

stdev[ξ(a)i ] 17.47 17.47 17.47 17.47 17.47

E[ξ(b)i ] 11.88 6.25 6.25 5.62 7.50

stdev[ξ(b)i ] 22.24 14.85 14.85 14.28 17.07

E[ξ(c)i ] 8.33 7.36 7.36 6.94 7.50

stdev[ξ(c)i ] 15.36 15.70 15.70 17.88 16.20

E[ξ(d)i ] 7.50 7.50 7.50 7.50 7.50

stdev[ξ(d)i ] 17.47 17.47 17.47 24.71 19.53

E[ξ(e)i ] 7.50 7.50 7.50 7.50 7.50

stdev[ξ(e)i ] 17.47 17.47 18.03 18.03 17.75

E[ξ(f)i ] 7.50 7.50 7.50 7.50 7.50

stdev[ξ(f)i ] 18.03 18.03 18.03 18.03 18.03

Table 3.1: First two moments of Xi’s and ξi’s, and XI and ξI (denoted ‘overall’), withYi = 100 on the left, Yi uniform on [0, 200] on the right, with deductible s = 100, andp = 10%, on the six sharing schemes described in Figure 3.2.

4.1 Risk sharing with friends and identical reciprocal contracts

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

Definition 4.2. A reciprocal contract between two policyholders, i and j, with magnitudeγ implies that i will pay an amount Ci→j to j if j claims a loss, with Ci→j ∈ [0, γ], and whereall friends who signed a contract with j should pay the same amount, and conversely fromj to i. Thus, Ci→j can be denoted Cj, and for a loss yj,

Cj = Ci→j = min

{γ,

min{s, yj}dj

}= min

{γ,xjdj

}, ∀i ∈ Vj,

where xi = min{yi, s} is the value of the loss that will be shared among friends (up to limitγ).

Insured i is connected with di friends (where d stands for the degree function), and let Videnote the set of friends. Since there is a commitment with the friends, insured i will pay upto a fix amount γ = s/d to all friends that claim a loss, and if i claims a loss, all connectionswill pay a share of it. Thus

ξi = Zi ·min{s, Yi}+∑j∈Vi

Zj min

{γ,

min{s, Yj}dj

}− Zi ·min{diγ,min{s, Yi}}

20

The first term is the part of the loss below the deductible s, because of the claim experiencedby insured i, Xi = Zi · min{s, Yi}. The third term is a gain, in case Zi = 1 because allconnections will give money, where all friends will contribute by paying Ci: insured i cannotreceive more than the loss Xi, that cannot be smaller than sum of all contributions diγ.Thus, the individual contribution per connection is Ci = min{γ,Xi/di}. Finally, the secondterm corresponds to additional payments insured i will make, to all connections (in Vi) whoclaimed a loss and the payment to j is, at most γ (because of the commitment), but it canbe less if the loss of j was not too large. With the previous notations, insured i should payCj to insured j if j ∈ V i. So, this risk-sharing mechanism leads to loss ξi (instead of Xi)

ξi = Xi +∑j∈Vi

ZjCj − ZidiCi (4.1)

A

B

C

D

50

50

50

50

50

(a)

A

B

C

D

50

50

50

50

50

(b)

A

B

C

D

50

50

50

50

50

(c)

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

Consider the networks of Figure 4.1. Consider insurance contracts with a deductible s of100. Here, degrees are d = (2, 2, 2, 2) on the left, and d = (3, 2, 2, 1) for the other two, so thatd = 2, so γ = 50. In Table 4.1, we have individual losses (denoted Y ) for two policyholders.Xi would be the self contribution for policyholder if there were no risk sharing with friends.ξi is the loss of policyholder i in each scenario. Since the sum of ξi’s equal the sum of Xi’s,we consider here some risk sharing, and observe that in all cases, Var[ξI ] ≤ Var[XI ].

Proposition 4.3. The process described by Equation (4.1) and Algorithm 1 is a risk sharingprinciple.

Proof. With the notations of Algorithm 1, ξi = Xi + Si − ZidiCi, where Si is the totalamount of money collected by policyholder i, while ZidiCi is the money given to policyholderi. Observe thatn∑i=1

Si =n∑i=1

∑j∈Vi

ZjCj =n∑i=1

∑j=1n

Ai,jZjCj =n∑j=1

∑i=1n

Ai,jZjCj =n∑j=1

ZjCj∑i=1n

Ai,j =n∑j=1

ZjCjdj

son∑i=1

ξi =n∑i=1

Xi + Si − ZidiCi =n∑i=1

Xi

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

21

(a)i A B C D

di 2 2 2 2Zi 0 1 0 1Yi 200 60min{Yi, s} 100 60Xi 0 100 0 60Ci 0 50 0 30ξi 80 0 80 0

(b)A B C D3 2 2 10 1 0 1

200 60100 60

0 100 0 600 50 0 50

100 0 50 10

(c)A B C D3 2 2 10 1 0 160 20060 10060 100 0 020 50 0 050 20 70 20

Table 4.1: Scenarios of Figure 4.1, with deductible s = 100 and (maximal) contributionγ = 50. The average of x’s is x = ξ = 40, empirical standard deviation of x’s is hereproportional to

√7200, while for ξ(a) it is

√6400,

√6200 for ξ(b) and finally

√1800 for ξ(c).

Algorithm 1: Risk Sharing, from Equation (4.1).

initialisation: generate a network, with adjacency matrix A ;

γ ← s/d ;for i← 0 to n do

di ← A>i·1 # computing the degree ;Vi ← {j : Ai,j = 1} # set of friends of polycyholder i ;Zi ← B(p) and Yi ← F # random generation of losses ;Xi ← Zi ·min{s, Yi} # loss for policyholder i (capped at s) ;Ci ← min{γ,Xi/di} # contribution paid to policyholder i by friends ;

endfor i← 0 to n do

Si ← 0 ;for j ∈ Vi do

Si ← Si + ZjCj # sum of contributions paid by i to friends ;endξi ← Xi + Si − ZidiCi # net loss of policyholder i ;

end

4.2 Illustration with a simulated example

Consider a toy example, with which we generate a network with n nodes, so that the distri-

bution of the degrees is such that DL= min{5 + [∆], n}, where [∆] has a rounded Gamma

distribution with mean d − 5 and variance σ2. For the application d = 20, while the stan-dard deviation parameter σ will be a parameters that will take values from 0 to 4d. Sincewe want to generate some non-directed networks, the sum of degrees should be odd (weuse the degree.sequence.game function of R package igraph). The occurrence of claims(here Z) are drawn from some Bernoulli variables with probability p = 1/10, and claim

size is such that YL= 100 + Y where Y has a Gamma distribution, with mean µ = 900

and variance 20002, so that Y has mean 1000 and variance 20002. We will consider here a

22

deductible s = µ = 1000 (corresponding to the average price of a claim). Note that withsuch a distribution, 78% of the claims fall into the interval [100; 1000], and an insured facinga 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, exceed10000 (but those large claims are covered by the insurance company). Finally, note that hereE[X] = 45.2 while 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{di}: 5Maximum contribution per node, γ: 50Deductible, s: 1000

Number of connections to fully recover the deductible, d: 20Number of nodes, n 5000Probability to claim a loss: 10%

Table 4.2: Brief summary of the parameters from the toy-example of Section 4.2.

Consider generations of networks as described above, with n = 5000 nodes. The expectedloss per policy, E[X], is 4.5% of the deductible (i.e. 45.2 here). Without any risk sharing,the standard deviation of individual losses, stdev[X], is 17.3% of s, that is a limiting case forstdev[ξ], 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 networkis a regular mesh, where everyone has 20 connections to share the risk (i.e. 17.3/

√20 %

since the risk is shared with exactly 20 friends). Then it increases to 5% of s when thestandard deviation of network is close to

√20 (the network is close to some Erdos-Renyi

graph). When the variance is extremely large (standard deviation close to 80), the standarddeviation can actually exceed 17% which is the standard deviation without any risk sharing.So, with some power law type of network, it is possible that the risk sharing scheme makesit more risky (here, it is extremely risky if someone is connected to 500 nodes because thatpolicyholder is very likely to pay a lot to connected nodes).

In this example, we have seen two important drawbacks of that mechanism. The first oneis that people with a lot of connections (more than d, i.e. 20 in the examples) can potentiallysuffer from big losses, so it might make sense to engage into some peer-to-peer contract onlywith d of them: they can get the maximum payback, and minimise potential losses. Thesecond one is that, with such a mechanism, people claiming a loss have a null net loss, whileconnections, claiming no loss, will have a net loss, which might be seen as unfair. In Section4.3 we will discuss more that issue.

4.3 On fairness and risk-sharing

As discussed in Section 3, such risk-sharing techniques can be seen as interesting because ofconnections with the convex ordering. Unfortunately, from a fairness perspective, this mightnot provide proper incentives to participants. For example, we have seen that a permutationof risks should leave policyholders indifferent. Consider some regular network (E ,V) sothat everyone is connected to exactly d friends, where s = dγ. Consider the mechanism

23

Figure 4.2: Evolution of stdev[ξ] when the standard deviation of D increases from 0 to80 (and d = 20, using the design of the toy example of Section 4.2), using simulations ofrandom networks of size n = 5000, for the line with a positive slope. Points are obtainedon the various scenarios, and the plain line is a smooth (possibly nonlinear) regression. Thehorizontal 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 is17% of the deductible.

described in Equation (4.1) and Algorithm 1, and, for simplicity, suppose that value of lossesis deterministic (and equal to s, for convenience) so that all randomness is coming from Zi’s,i.e. who will get a claim. If policyholder i had an accident, Zi = 1 and the contribution Ciis exactly γ. Without any risk sharing, X = (X1, · · · , Xn) is a collection of i.i.d. randomvariables taking values in {0, s}, with probability 1 − p and p, respectively. For each nodei let Ni denote the number of connected nodes claiming a loss. Hence, Ni ∼ B(d, p). Then,for any node i,

ξi = Xi +∑j∈Vi

ZjCj − ZidCi = Zis+Niγ − Zis = Niγ (≤ s)

Observe that if i claimed no loss Xi = 0 while ξ = Niγ ≥ 0, and more precisely,

P[ξ = 0] = P[Ni = 0] = (1− p)d

(with numerical values of the previous section, this probability is (1 − 1/10)20 ∼ 12%) andotherwise, ξ > Xi. On the other hand, if i did claim a loss, Xi = s while

P[ξ = s] = P[Ni = d] = pd

(which is non-significant here) and and otherwise, ξ < Xi. Thus,{if i claims a loss, ξ ≤ Xi, and ξ < Xi with probability 1− (1− p)d

if i claims no loss, ξ ≥ Xi, and ξ > Xi with probability 1− pd

To go further, consider now a node j so that (i, j) ∈ E , and assume that i claims a loss.Recall that ξj = Njγ. If n is large enough, we can assume that Ni and Nj are ”almost”

24

independent, but we know that Nj ≥ 1, more precisely, Nj = 1 +Mj with Mi ∼ B(d− 1, p),independent of Ni (as a first order approximation, when n is assumed to be large enough).And ξj ≥ ξi if and only if Nj ≥ Ni,

P[Nj ≥ Ni] =d∑s=0

P[Nj ≥ Ni|Ni = s] · P[Ni = s] =d∑s=0

P[Nj ≥ s] · P[Ni = s].

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

P[ξi < ξj] ∼ 59% and P[ξi ≤ ξj] ∼ 78%, when (i, j) ∈ E .

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

P[Xi > Xj] ∼ 90% and P[ξi < ξj] ∼ 59%, when (i, j) ∈ E .

This potential, and very likely, reordering of actual losses, that is fair ex-ante (prior torandom dices,) can be seen as unfair, ex-post. Thus it will be natural to ask for someself-contribution, just to make it more likely that ξi > ξj if Xi > Xj. Note that anotherassumption 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 assumethat they can be different for all policyholder: the main constraint is that both i and j, thatare connected, agree on the same amount of money (since contracts are reciprocals) that wewill denote γ(i,j). But it is impossible to assume that all agents will individually optimisesome criteria, independently of of the other. If i is connected to j, the commitment γ(i,j)should be optimal for i (thus, function of other γ(i,·)’s) but also optimal for j (thus, functionof other γ(·,j)’s). A natural idea will be to consider some global planner (the insurancecompany), maximizing the overall coverage through reciprocal contracts.

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 magnitudeγ(i,j) implies that i will pay an amount Ci→j to j if j claims a loss, with Ci→j ∈ [0, γ(i,j)],and where Ck→j’s (for various k’s) should be proportional to the engagement γ(k,j), andconversely from j to i. Thus, for a loss yj

Ci→j = min

{γ(i,j),

γ(i,j)γj·min{s, yj}

}, where γj =

∑k∈Vj

γ(k,j)

γ = {γ(i,j), (i, j) ∈ E} is the collection of magnitudes.

25

A

B

C

D

30

50

30

30

A

B

C

D

20

50

30

30

20

30

30

30

20

(a) (b)

10

70

20

20

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 γ = 50.

If we assume that all γ(i,j) are equal, we are back in the situation of definition 4.2. InFigure 5.1 we visualize a situation with heterogeneous commitments. An alternative to thoseidentical contracts, is to consider an optimisation problem, where the goal is to maximizethe total coverage of those reciprocal contracts, under simple contracts, such as γ(i,j) ≤ γ(a given upper bound) for all edges. Thus, formally, we consider the following the followinglinear programming problem,

max

∑(i,j)∈E

γ(i,j)

s.t. γ(i,j) ∈ [0, γ], ∀(i, j) ∈ E∑

j∈Vi

γ(i,j) ≤ s, ∀i ∈ V

(5.1)

where s is the deductible of the insurance contract, so that the second constraint is simplyrelated to Hypothesis 4.1. With classical linear programming notations, we want to findz? ∈ Rm

+ where m is the number of edges in the network (we consider edges (i, j), withi < j), so that z

? = argmaxz∈Rm+

{1>z

}s.t. A>z ≤ a

when A is a (n+m)×m matrix, and a is a (n+m) vector,

A =

[TIm

]and a =

[γ1ns1m

]where T is the incidence matrix, T = [Tu,w], with w ∈ V i.e. w = (i, j), the Tu,w = 0 unlesseither w = i or w = j (satisfying d = T1n).

26

5.2 Optimal risk sharing

Definition 5.2. Let γ? denote a solution of the linear program (5.1). Then γ? is an optimalrisk sharing coverage. The magnitude of the coverage for policyholder i is

γ?i =∑j∈Vi

γ?(i,j)

For given lossesX = (X1, · · · , Xn), define contributions C?i→j = min

{ γ?(i,j)∑i∈Vj

γ?(i,j)·Xj, γ

?(i,j)

},

andξ?i = Xi +

∑j∈Vi

[ZjC?i→j − ZiC?

j→i].

Then ξ?i is a risk sharing, called the optimal risk sharing.Note that on Figure 5.1, risk sharing is not optimal, if γ = 50: for instance, A can

increase the commitment with any friend by +10, as on (b) in Figure 5.2. The solutionobtained from a linear programming solver (namely function lp from the lpSolve packagein R) is the one on the right, on (c). Overall, the total coverage for (b) and (c) is the same(with a sum of γ’s of 150).

A

B

C

D

30

50

30

30

(a)

10

70

20

20

A

B

C

D

30

50

40

30

(b)

60

20

20

A

B

C

D

50

50

50

(c)

50

50

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

5.3 Sparsification of the network

In the case of optimal reciprocal engagements, some contributions can be close to zero but notexactly zero. In this case it might be beneficial to “shrink” these contributions to effectivelyzero, especially if those reciprocal contracts have some management fixed costs, withoutdecreasing γ∗ too much. This can be achieved by reformulating the linear programmingproblem into a mixed-integer optimization.

27

z? = argmaxz∈Rm+

{1>z

}s.t. T>z ≤ s1m

zi ≤Myi, ∀i∑i

yi ≤ m

where yi ∈ {0, 1} and m maximum desired non-zero engagements. M is an upper boundon each zi ∈ z which comes naturally as we look for reciprocal engagements smaller thanthe specified γ. This mechanism will effectively shrink small contributions to zero. Figure5.3 depicts the trade-off. This mechanism also becomes important in section 6, where net-works with a high degree standard deviation result in second degree adjacency matrices withextremely large |E|.

Figure 5.3: 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 lessedge results in −γ on the total sum, the function plateaus on the right hand side, resultingin 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 toshare their entire deductible and will hence have a need to remove unnecessary edges. Butthe converse is also true as many nodes do not have enough connections to share all or atleast most of their risk.

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

Hypothesis 6.1. Consider a policyholder i such that his coverage γ?i is strictly smaller thanthe overall loss limit s then policyholder i will try get connections with friends of friends.

28

Figure 5.4: Evolution of stdev[ξ?] when the standard deviation of D increases from 0 to 80(and d = 20, design of Example 4.2), using simulations of random networks of size n = 5000.Here the deductible is s = 1000, and various scenarios are considered.

Definition 6.2. Given a network (E ,V), with degrees d and some first-level optimizationvector γ1, define the subset of vertices who could not share their whole deductible as,

Vγ1={i ∈ V : γi =

∑j∈V

γ(i,j) < s}.

The γ1-sub-network of friends of friends is (E (2)γ1,Vγ1

),

E (2)γ1={

(i, j) ∈ Vγ1× Vγ1

: ∃k ∈ V such that (i, k) ∈ E , (k, j) ∈ E and i 6= j}

Again, from Hypothesis 6.1, policyholders with enough connections (to recover fully froma 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 thedeductible.

In Figure 6.2, we visualize the evolution of stdev[ξ] as a function of the standard deviationof D, when risks can be shared with friends of friends, when all policyholders have thesame self contribution. In view of Figure 6.2, sharing with friends of friends always lowerthe standard deviation of individual remaining losses stdev[ξ]. Furthermore, Figure 6.3illustrates the relative importance of each layer. This graph corresponds to the case whereζ = 200. On a regular network, a bit less than 40% of the claim amount is self insured, whichcorresponds to E[min{Y, 200}]/E[min{Y, 1000}] ∼ 38%. Then, the contribution of friendstends to decrease with the variance of the degrees, simply because an increasing fraction donot have enough friends to fully recover the remaining part of the deductible. On the otherhand, 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 isactually increasing with the variance of the degree.

Note that it is possible to use a linear programming algorithm similar to the one seenin section 5.2 for the friends of friends problem. From a numerical perspective, it is com-putationally intensive, since we need γ

(1)(i,j) for all first-order edges (i, j) ∈ E (friends) and

γ(2)(i,k) ∈ E

(2)γ1 for all second-order edges (i, k) (friends of friends).

29

A

B

C

D

20

50

30

30

20

40

30

30

1010

Figure 6.1: Network with five nodes {A,B,C,D,E}, with some self-contribution, and somepossible friends-of-friends risk sharing (with a lower maximal contribution), from the originalnetwork of Figure 4.1.

Figure 6.2: Evolution of stdev[ξ] when the variance of D increases from 0 to 80 (and d = 20,with the design of Example 4.2), 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 selfcontribution of 1000 (on top) mean that no risk is shared.

6.2 Optimal friends of friends

The optimization of the problem can be obtained via a two stage algorithm: The first stageis the same as previously in section 5.2, where the initial contribution among friends, γ(i,j)is bounded by γ1

γ?1 = argmax

∑(i,j)∈E(1)

γ(i,j)

s.t. γ(i,j) ∈ [0, γ1], ∀(i, j) ∈ E (1)∑

j∈V(1)i

γ(i,j) ≤ s, ∀i ∈ V

30

Figure 6.3: Evolution of amount paid in each layer, as a proportion of the deductible whena self contribution of 200 is considered, as a function of the variance of D.

In the second stage, γ(i,j) is bounded by γ2. E (2)γ?1 can be obtained by squaring the adjacencymatrix and setting nonzero entries to 1 as defined in definition 6.2.

γ?2 = argmax

∑(i,j)∈E(2)

γ(i,j)

s.t. γ(i,j) ∈ [0, γ2], ∀(i, j) ∈ E (2)γ?1∑

j∈V(1)i

γ?1:(i,j) +∑j∈V(2)

i

γ(i,j) ≤ s, ∀i ∈ V

Figure 6.4: Distribution of the risk for γ1 = 50 and γ2 = 5, without any initial self-contribution. Note how the total shareable risk first decreases but then again increasesagain as the share of friends first drops faster than the share of friends-of-friends can in-crease. As networks with a high degree variance generate “hubs” (similar to a star shapednetwork), these hub-nodes can help to connect most of the nodes in the network by passingthrough these central nodes.

As can be seen in Figure 6.4, highly connected nodes which come to exist in networks

31

with high degree distribution lead to less risk being shared among friends. For the part beingshared 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 A2 increases for networks with a higherdegree variance. This can be explained by considering the highly central nodes (with adegree). In the extreme case of a star shaped network, every node can reach any other nodeby “passing” through the central node. 3

Networks with a high degree variance typically posses multiple of these central nodesthat act as hubs. As seen for example in figure 6.4, due to these central nodes, less risk canbe shared between friends, as more edges will have a zero weight. On the other hand, thesehubs 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∑(i,j)∈E

γ?1:(i,j) +∑

(i,j)∈Eγ?1

γ?2:(i,j)). Here we considered numerical values of Section 4.2, with γ1 ≤

50 and γ2 ≤ 5. As the standard deviation of the degrees increases, more risk is shared viafriends-of-friends and more contracts are needed to cover the shared risk. In the extremecase of zero degree variance, only 50000 contracts are needed to cover all the risk.

7 Conclusion

Recently, Denuit [2020], Denuit and Robert [2020a], Denuit and Robert [2020b] and Fenget al. [2021] 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 riskswith “friends”, more specifically the amount of loss below the deductible of some classicalinsurance 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

3For our application that would mean that for the first optimization stage, at most d edges would benonzero, 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.

32

Figure 6.6: Evolution of the number of edges in A2 for different standard deviations in thedegree distribution in the original network. Duplicated edges in A2 are removed, such thatpair of nodes can be connected with at most one edge.

by the original design of Inspeer. Assuming that all policyholders have similar contracts withthe same deductible, we investigated the shape of the network, or more specifically, giventhe total number of connections (or equivalently the average degree of a node), the impact ofthe variability of the degrees. We did observe that the more regular the network, the moreefficient the risk sharing, in the sense that it will decrease the risks for individuals. Theoriginal mechanism we described, with identical contracts among policyholder is interestingsince it can evaluate with time easily: policyholder might decide to leave, or join by signingnew 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 manyfriends, who might select those they wish to sign reciprocal contracts with, and policyholderswith too few friends.

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

33

8 Appendix

In this appendix, we briefly discuss what trace[CΣC>] > trace[Σ] could mean, in dimension2, when C is column-stochastic matrix.

8.1 trace[DΣD>]gainst trace[Σ]n dimension 2

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

D =

(x 1− x

1− x x

)and Var[ξ1] =

(a2 rabrab b2

)

trace[

Var[ξ2]]

= trace[DVar[ξ1]D

>] = trace

[(x 1− x

1− x x

)(a2 rabrab b2

)(x 1− x

1− x x

)>]

= trace

[(x2a2 + 2rab(1− x)x+ b2(1− x)2 ∗∗ (1− x)2a2 + 2rab(1− x)x+ b2x2

)]≤ (a2 + b2)

(x2 + (1− x)2

)+ 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 theinterval [0, 1] is obtained either when x = 0 or x = 1, and it takes value a2 + b2, so, the traceis lower than

a2 + b2 = trace

[(a2 rabrab b2

)]

8.2 trace[CΣC>]gainst trace[Σ]n dimension 2

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

ξ2 = Cξ1with C =

(x 1− y

1− x y

), while Var[ξ1] =

(a2 rabrab b2

)

Then, since Var[ξ′2] =1

2trace[CVar[ξ1]C

>], write trace[CVar[ξ1]C>] as

trace

[(x 1− y

1− x y

)(a2 rabrab b2

)(x 1− x

1− y y

)]

= trace

[(x2a2 + 2rab(1− y)x+ b2(1− y)2 ∗∗ (1− x)2a2 + 2rab(1− x)y + b2y2

)]= a2

(x2 + (1− x)2

)+ b2

(y2 + (1− y)2

)+ 2rab

[y(1− x) + x(1− y)

]that we can write

a2

2

[(2x− 1)2 + 1] +

b2

2

[(2y − 1)2 + 1] + rab

[(2x− 1)(2y − 1)− 1

]34

that is a quadratic form in Z = (X, Y ) = (2x− 1, 2y − 1)

a2

2

[X2 + 1] +

b2

2

[Y 2 + 1] + rab

[XY − 1

]=

1

2Z>

(a2 rabrab b2

)Z

which is an elliptic paraboloid function, minimal in (X, Y ) = (0, 0) – or (x, y) = (1/2, 1/2).Since (x, y) ∈ [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 a2 + b2, and in thesecond case, it can exceed a2 + b2. More specifically, on Figure 8.1, we can see that values of(x, y) such that trace[CVar[ξ]C>] > trace[Var[ξ]] (in red, on the right), that is either whenone is close to 1, and the other close to 0.

Figure 8.1: Level curves of (x, y) 7→ trace[CVar[ξ]C>] where C is the column-stochasticmatrix 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,trace[CVar[ξ]C>] ≤ trace[Var[ξ]]. For a positive correlation, the red are corresponds to caseswhere trace[CVar[ξ]C>] > trace[Var[ξ]].

In that case, for instance with x = ε and y = 1− ε, it means that(ξ2,1ξ2,2

)=

(x 1− y

1− x y

)(ξ1,1ξ1,2

)=

(ε ε

1− ε 1− ε

)(ξ1,1ξ1,2

)=

(ε(ξ1,1 + ξ1,2)

(1− ε)(ξ1,1 + ξ1,2)

)Even if it is a risk-sharing, it can be seen as an unfair, or unbalanced, one, since policyholder2 is now taking all most of the risks.

35

References

S. Abdikerimova and R. Feng. Peer-to-peer multi-risk insurance and mutual aid. SSRN,(3505646), 2019.

H. Albrecher, A. Bommier, D. Filipovic, P. Koch-Medina, S. Loisel, and H. Schmeiser.Insurance: models, digitalization, and data science. European Actuarial Journal, 9:349–360, 2019.

S. Aparicio, J. Villazon-Terrazas, and G. Alvarez. A model for scale-free networks: Appli-cation to twitter. Entropy, 17(8):5848–5867, 2015. doi: 10.3390/e17085848.

M. Artıs, M. Ayuso, and M. Guillen. Modelling different types of automobile insurance fraudbehaviour in the spanish market. Insurance: Mathematics and Economics, 24(1):67–81,1999.

A. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286(5439):509–512, 1999.

M. Bayati, J. H. Kim, and A. Saberi. A sequential algorithm for generating random graphs.Algorithmica, 58(4):860–910, 2009.

L. B. Beasley and S.-G. Lee. Linear operators preserving multivariate majorization. LinearAlgebra and its Applications, 304(1):141–159, 2000.

E. A. Bender and E. Canfield. The asymptotic number of labeled graphs with given degreesequences. Journal of Combinatorial Theory, Series A, 24(3):296–307, 1978.

C. Biener, M. Eling, A. Landmann, and S. Pradhan. Can group incentives alleviate moralhazard? the role of pro-social preferences. European Economic Review, 101:230–249, 2018.

G. Birkhoff. Three observations on linear algebra, univ. Revista (Universidad Nacional deTucuman) Series A, 5:147–151, 1946.

O. Cabrignac, A. Charpentier, and E. Gallic. Modeling joint lives within families. arXiv,2006.08446, 2020.

G. Caldarelli. Scale-Free Networks: Complex Webs in Nature and Technology. CambridgeUniversity Press, 2007.

G. Carlier, R.-A. Dana, and A. Galichon. Pareto efficiency for the concave order and multi-variate comonotonicity. Journal of Economic Theory, 147(1):207–229, 2012.

C.-S. Chang. A new ordering for stochastic majorization: Theory and applications. Advancesin Applied Probability, 24(3):604–634, 1992.

C.-S. Chang and D. D. Yao. Rearrangement, majorization and stochastic scheduling. Math-ematics of Operations Research, 18(3):658–684, 1993.

A. Charpentier and E. Flachaire. Extended scale-free networks. arXiv, (1905.10267), 2019.

36

A. Charpentier, A. David, and R. Elie. Optimal claiming strategies in bonus malus systemsand implied markov chains. Risks, 5(4), 2017.

C. Chatfield and A. Collins. Introduction to multivariate analysis. Chapman & Hall, 1981.

S. Chatterjee, P. Diaconis, and A. Sly. Random graphs with a given degree sequence. TheAnnals of Applied Probability, 21(4):1400 – 1435, 2011.

K. Cheung, W. Chong, and S. Yam. Convex ordering for insurance preferences. Insurance:Mathematics and Economics, 64:409–416, 2015.

F. Chung and L. Lu. Connected components in random graphs with given expected degreesequences. Annals of combinatorics, 6(2):125–145, 2002.

G. P. Clemente and P. Marano. The broker model for peer-to-peer insurance: an analysisof its value. The Geneva Papers on Risk and Insurance-Issues and Practice, 45:457–481,2020.

A. Cohen and L. Einav. Estimating risk preferences from deductible choice. Americaneconomic review, 97(3):745–788, 2007.

S. Nuray Akın and O. Leukhina. Risk-sharing within families: Evidence from the health andretirement study. Journal of Economic Dynamics and Control, 52:270–284, 2015.

G. Csardi and T. Nepusz. The igraph software package for complex network research. In-terJournal, Complex Systems:1695, 2006. URL https://igraph.org.

G. Dahl. Matrix majorization. Linear Algebra and its Applications, 288:53–73, 1999.

M. Denuit. Investing in your own and peers’ risks: the simple analytics of p2p insurance.European Actuarial Journal, 10:335–359, 2020.

M. Denuit and A. Charpentier. Mathematiques de l’Assurance Non-Vie, tome 1. Economica,2004.

M. Denuit and J. Dhaene. Convex order and comonotonic conditional mean risk sharing.Insurance: Mathematics and Economics, 51(2):265–270, 2012.

M. Denuit and C. Robert. Conditional mean risk sharing for dependent risks using graphicalmodels. LIDAM Discussion Papers ISBA, (2020029), 2020a.

M. Denuit and C. Robert. Stop-loss protection for a large p2p insurance pool. Insurance:Mathematics & Economics, 100:210–233, 2020b.

M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas. Actuarial Theory for Dependent Risks:Measures, Orders and Models. Wiley, 2005.

M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas. Actuarial theory for dependent risks:measures, orders and models. John Wiley & Sons, 2006.

37

G. Dionne and R. Gagne. Deductible contracts against fraudulent claims: evidence fromautomobile insurance. Review of Economics and Statistics, 83(2):290–301, 2001.

J. Duarte, S. Siegel, and L. Young. Trust and Credit: The Role of Appearance in Peer-to-peerLending. The Review of Financial Studies, 25(8):2455–2484, 08 2012.

L. Eeckhoudt, C. Gollier, and H. Schlesinger. Increases in risk and deductible insurance.Journal of economic Theory, 55(2):435–440, 1991.

P. Erdos and A. Renyi. On random graphs. I. Publ. Math. Debrecen, 6:290–297, 1959. ISSN0033-3883.

A. Fargere and A. Souki. L’assurance collaborative, de la promesse a l’execution. Master’sthesis, CEA, 2019.

R. Feng, C. Liu, and S. Taylor. Peer-to-peer risk sharing with an application to flood riskpooling. mimeo, 2021.

R. Gentleman, E. Whalen, W. Huber, and S. Falcon. graph: graph: A package to handlegraph data structures, 2019. R package version 1.64.0.

M. Halek and J. G. Eisenhauer. Demography of risk aversion. The Journal of Risk andInsurance, 68(1):1–24, 2001.

G. H. Hardy, J. E. Littlewood, and G. Polya. Inequalities. Cambridge University Press,1934.

F. Hayashi, J. Altonji, and L. Kotlikoff. Risk-sharing between and within families. Econo-metrica, 64(2):261–294, 1996.

R. A. Johnson and D. W. Wichern. Applied Multivariate Statistical Analysis. Pearson, 2008.

R. Kaas, J. Dhaene, D. Vyncke, M. Goovaerts, and M. Denuit. A simple geometric proofthat comonotonic risks have the convex-largest sum. ASTIN Bulletin, 32(1):71–80, 2002.

S. Karlin and A. Novikoff. Generalized convex inequalities. Pacific Journal of Mathematics,13(4):1251 – 1279, 1963.

L. Larrimore, L. Jiang, J. Larrimore, D. Markowitz, and S. Gorski. Peer to peer lending: Therelationship between language features, trustworthiness, and persuasion success. Journalof Applied Communication Research, 39(1):19–37, 2011.

G. Y. Lee. General insurance deductible ratemaking. North American Actuarial Journal, 21(4):620–638, 2017.

F. D. Martınez Perıa, P. G. Massey, and L. E. Silvestre. Weak matrix majorization. LinearAlgebra and its Applications, 403:343–368, 2005.

M. McPherson, L. Smith-Lovin, and J. M. Cook. Birds of a feather: Homophily in socialnetworks. Annual Review of Sociology, 27(1):415–444, 2001.

38

J. Meyer and M. B. Ormiston. Analyzing the demand for deductible insurance. Journal ofRisk and Uncertainty, 18(3):223–230, 1999.

M. Molloy and B. Reed. A critical point for random graphs with a given degree sequence.Random Structures & Algorithms, 6(2-3):161–180, 1995.

M. Newman. Networks: an introduction. Oxford University Press, Inc., 2010.

M. E. J. Newman. Random graphs as models of networks, chapter 2, pages 35–68. JohnWiley & Sons, Ltd, 2002.

R. L. Norgaard. What is a reciprocal? The Journal of Risk and Insurance, 31(1):51–61,1964.

J. Ohlin. On a class of measures of dispersion with application to optimal reinsurance. AstinBuletin, 5(2):249–266, 1969.

A. Paperno, V. Kravchuk, and E. Porubaev. A peer-to-peer coverage system. Teambrella:White Paper, 2015.

M. L. Rego and J. C. Carvalho. Insurance in today’s sharing economy: New challenges aheador a return to the origins of insurance? In InsurTech: A Legal and Regulatory View, pages27–47. Springer, 2020.

D. F. Reinmuth. What is a reciprocal?: Comment. The Journal of Risk and Insurance, 31(4):641–646, 1964.

I. Schur. Uber eine klasse von mittelbildungen mit anwendungen auf die determinantenthe-orie. Sitzungsberichte der Berliner Mathematischen Gesellschaft, 1923.

M. Shaked and J. G. Shanthikumar. Stochastic orders. Springer Science & Business Media,2007.

M. Vahdati, K. G. HamlAbadi, A. M. Saghiri, and H. Rashidi. A self-organized frameworkfor insurance based on internet of things and blockchain. In 2018 IEEE 6th internationalconference on future internet of things and cloud (FiCloud), pages 169–175. IEEE, 2018.

D. Watts and S. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 398:440–442, 1998.

R. A. Winter. Optimal insurance under moral hazard. Handbook of Insurance, pages 155–183,2000.

J. J. Xu, Y. Lu, and M. Chau. P2p lending fraud detection: A big data approach. InM. Chau, G. A. Wang, and H. Chen, editors, Intelligence and Security Informatics, pages71–81, Cham, 2015. Springer International Publishing.

39